LIBRARY OF
ALLEN KNIGHT
CERTIFIED PUBLIC ACCOUNTANT
502 California Street
SAN FRANCISCO. CALIFORNIA
m MEMQR.IANI
Digitized by the Internet Archive
in 2007 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/accountanctyofinOOsprarich
THE ACCOUNTANCY OF
INVESTMENT
BY
Charles Ezra Sprague, a.m., Ph.d., Litt.d., c.p.a.
Late Professor of Accountancy in New York University ;
Late President of Union Dime Savings Bank, New York Ctty
With which are incorporated "Logarithms to 12 Places
and Their Use in Interest Calculations" and
"Amortization" by the same author
REVISED BY
LEROY L. PeRRINE, Ph.B., B.C.S., C.P.A.
Lecturer on Accounting at the Nexv York University School
of Commerce, Accounts and Finance; Meynber of the
staff of Haskins &> Sells, Certified Public AccountaMis
RONALD ACCOUNTING SERIES
NEW YORK
THE RONALD PRESS COMPANY
1914
Copyright, 1904, by Charles E. Sprague
Copyright, 1905, by Charles E. Sprague
Copyright, 1907, by Charles E. Sprague
Copyright, 1909, by Charles E. Sprague
Copyright, 1910, by Charles E. Sprague
Copyright, 1914,
by
THE RONALD PRESS COMPANY
(Sjenjeral SMtmmt oi ^laxiaxml §0arir
Applicable to all books of the
Ronald Accounting Series
THE manuscripts of the books forming the
Ronald Accounting Series have been sub-
mitted to us and have been approved by us for
publication.
In some cases the authors express views that
are not fully in accord with those entertained by
us, but in no instance are such differences of
sufficient importance, in our judgment, to warrant
the withholding from publication of a meritorious
work.
J. E. Sterrett
Robert H. Montgomery
PREFACE
Among the published works of the late Colonel Sprague,
there are four which deal particularly with certain mathe-
matical phases of accounting, viz. : *Text Book of the Ac-
countancy of Investment" ; "Amortization" ; "Logarithms to
12 Places and Their Use in Interest Calculations"; and
"Extended Bond Tables." Since the author's death in
March, 1912, it has become desirable to combine the first
three of these publications into one volume, in order to
serve more effectively the needs of business men and students
of accounting by presenting the material in compact and
convenient form.
The present volume is the result of this consolidation. In
it has been incorporated everything of practical value con-
tained in the three works mentioned, while at the same time
the special features of those books have been amplified by
additional text matter and problems, wherever such addi-
tions have seemed desirable for the sake of more adequate
treatment.
In conformity with the usual and commendable practice
of Colonel Sprague, the reviser has avoided as far as pos-
sible the use of the more difficult mathematical demonstra-
tions and formulas, believing that thereby the book will
prove of greater utility to practicing accountants, bankers,
and other business men. On this point we quote from the
author's original preface : "Treatises on this subject (Mathe-
matics of Investment), written for actuarial students, are
invariably too difficult, except for those who have not only
been highly trained in algebra, but are fresh in its use, and
this makes the subject forbidding to many minds. I have
vi PREFACE
made all my demonstrations arithmetical and illustrative,
but, I think, none the less convincing and intelligible."
It is believed that for a considerable number of readers,
the tables of logarithms given in Part III will prove of great
utility in those cases where more than ordinary accuracy is
required, and where special tables are, at times, imperative.
To quote again from the preface of Colonel Sprague:
"Rough results will answer for approximative purposes ; but
where it is desirable, for instance, to construct a table of
amortization, sinking fund, or valuation of a lease at an
unusual rate, for a large amount and for a great many years,
exactness is desirable and becomes self-proving at the end."
A whole book is required for the ordinary tables of loga-
rithms of six or seven places, while the tables here presented
are contained in a few pages and give accurate results to
twelve places of decimals. The processes with these tables
are necessarily somewhat slower than with those of six or
seven places, but their use is fully justified where greater
accuracy in results is desirable.
The tables of compound interest, present worth, an-
nuities, and sinking funds, carried to eight places of decimals,
have been retained in this edition. Such tables are prac-
tically indispensable in securing accurate computations. The
index of subjects at the end of the book is a new feature and
will facilitate quick reference to any information desired.
Leroy L. Perrine.
New York City, January, 1914.
CONTENTS
PART I— THE MATHEMATICS OF INVESTMENT
Chapter I. Capital and Revenue
• 1. Definition of Capital
2. The Use of Capital
3. Sources of Capital
4. Investment ^
5. Revenue
6. Interest and Rent
7. Dividends
8. Laws of Interest
Chapter II. Interest
I 9. Interest
10. Essentials of Interest Contract
11. Interest Rate
12. Principal
13. Simple and Compound Interest
14. Punctual Interest
15. Computation of Interest
16. Simple Interest
17. Compound Interest
18. Computation of Compound Interest
19. Comparison of Simple and Compound Interest
20. The Day as a Time Unit
21. The Month as a, Time Unit
22. Half and Quarter Years
23. Partial Interest Periods
24. Changing the Day Basis
25. The Amount — First Period
26. The Amount — Subsequent Periods
27. Exponents and Powers
28. Finding the Amount — Compound Interest
29. Present Worth
30. Present Worth and Amount Series
31. Relation Between Present Worth and Amount
iii CONTENTS
32. Formation of Series
33. Discount
34. Computing Compound Discount
35. Formulas for Interest Calculations
36. Use of Logarithms
Chapter III. The Use of Logarithms
§ 37. Purpose of Logarithms
38. Exponents, Powers, and Roots
39. Logarithms as Exponents
40. Rules and Symbols of Logarithms ^
41. The Two Parts of a Logarithm
42. Mantissa Not Affected by Position of the Decimal Point
43. Four-Place Table of Logarithms
44. Multiplication by Logarithms
45. Division by Logarithms
46. Powers by Logarithms
47. Roots by Logarithms
48. Fractional Exponents
49. Use of Logarithms in Computing Compound Interest
50. Accuracy of Logarithmic Results
51. Logarithms to Fifteen Places
52. Use of Logarithms in Present Worth Calculations
Chapter IV. Amount of an Annuity
§ 53. Evaluation of a Series of Payments
54. Annuities
55. Amount of Annuity
56. Calculation of Annuity Amounts
57. Formation of Tables
58. Use of Tables
59. Compound Interest as a Base for Annuity Calculations
60. Rule and Formula for Finding Amount
61. Operation of Rule
Chapter V. Present Worth of an Annuity
§ 62. Method of Calculation
63. Tables of Present Worth
64. Short Method for Finding Present Worth of an Annuity
65. Present Worth Obtained
66. Rule for Present Worth
67. Formulas for Present Worth
68. Analysis of Annuity Payments
69. Components of Annuity Instalments
CONTENTS ix
70. Amortization
71. Amortization and Present Worth
72. Development of a Series of Amortizations
7Z. Evaluation by Logarithms
Chapter VI. Special Forms of Annuities
74. Ordinary or Immediate Annuities
75. Annuities Due
76. Present Worth of Annuities Due
77. Present Worth of Deferred Annuities
78. Rule for Finding Present Worth of Deferred Annuity
79. Present Worth of Perpetuities
80. Perpetuity in Stock Purchased for Investment
81. When Annuity Periods and Interest Periods Differ
82. Varying Annuities
Chapter VII. Rent of Annuity and Sinking Fund
83. Rent of Annuity
84. Rule for Finding Rent of Annuity
85. Alternative Method of Finding Rent
86. Rent of Deferred Payments
87. Annuities as Sinking Funds
88. Rule for Finding Sinking Fund Contributions
89. Verification Schedule
90. Amortization and Sinking Fund
Chapter VIII. Nominal and Effective Rates
91. Explanation of Terms
92. Semi-Annual and Quarterly Conversions
93. Limit of EflFective Annual Rate
94. Rule for EflFective Rate
95. Logarithmic Process
Chapter IX. Bonds and the Proper Basis of Bond Accounts
I 96. Provisions of Bonds
97. Interest on Bonds
98. Hov^r Bonds Are Designated
99. Relation of Cost to Net Income
100. Coupon and EflFective Rate of Interest on Bonds
101. Present Worth of Bonds
102. Considerations in the Purchase of Bonds
103. Present Worth and Earning Capacity of Bonds
104. Cost and Par of Bonds
CONTENTS
105. Intermediate Value of Bonds
106. True Investment Basis for Bonds
107. Various Bond Values
108. Investment Value the True Accounting Basis
Chapter X. Valuation of Bonds
109. Cash Rate and Income Rate of Bonds
110. Elements of a Bond
111. Valuation of Bonds — First Method
112. (a) Finding Present Worth of Principal
113. (b) Present Worth of Coupons
114. Schedule of Evaluation
115. Valuation of Bonds — Second Method
116. Evaluation When Cash Rate Is Less Than Income Rate
117. Second Method by Schedule
118. Rule for Second Method of Evaluation
119. Principles of Investment
120. Solution by Logarithms
121. Amortization Schedule
122. Use of Schedules in Accountancy
123. Book Values in Schedules
124. Checks on Accuracy of Schedules
125. Tables Derivable from Bond Values
126. Methods of Handling Interest
127. Schedule of Bond Values
Chapter XI. Valuation of Bonds (Concluded)
128. Bond Purchases at Intermediate Dates
129. Errors in Adjusting Bond Prices
130. Schedule of Periodic Evaluation
131. Objection to Valuation on Interest Dates
132. Interpolation Method of Periodic Valuation
133. Multiplication Method of Valuation
134. Computation of Net Income for Partial Period
135. Purchase Agreements
136. Approximation Method of Finding Income Rate
137. Application of Method
138. First Method of Eliminating Residues
139. Second Method of Eliminating Residues
140. Third Method of Eliminating Residues
141. Short Terminals
142. Rule for Short Terminals
143. Discounting
144. Last Half- Year of Bond
CONTENTS , xi
145. Serial Bonds
146. Irredeemable Bonds
147. Optional Redemption
148. Bonds as Trust Fund Investments
149. Payments to Life Tenant
150. Effect of Varying Rates on Investments
151. Example of Payments to Life Tenant
152. Cullen Decision
153. Cullen Decision Scheduled
154. Unjust Feature of Cullen Decision
155. Bond Tables
156. Features of the Bond Table
Chapter XII. Summary of Compound Interest Processes
§ 157. Rules and Formulas
158. Rules
159. Formulas
Chapter XIII. Accounts — General Principles
§ 160. Relation of General Ledger to Subordinate Ledgers
161. The Interest Account
162. Mortgage and Loan Accounts
Chapter XIV. Real Estate Mortgages
§ 163. Nature of Loans on Bond and Mortgage
164. Separate Accounts for Principal and Interest
165. Interest Debits and Credits
166. Characteristics of Modern Ledger
167. The Mortgage Ledger
168. Identification of Mortgages by Number
169. The "Principal" Account
170. Special Columns for Mortgagee's Disbursements
171. The Interest Account
172. Interest Due
173. Books Auxiliary to Ledger
174. The "Due" Column
175. Interest Account Must Be Analyzed
176. Form of "Interest Due" Account
177. Forms for Mortgage Account
178. Loose-Leaf and Card Records
179. Forms of Mortgage Loan Accounts
180. Reverse Posting of Interest Register
181. Handling Receipts and Notices
182. Mortgages Account in General Ledger
ii CONTENTS
183. Tabular Register
184. Equal Instalment Method
Chapter XV. Loans on Collateral
185. Short Time Loans on Personal Property
186. Forms for Loan Accounts
187. Requirements for Interest Account
188. Forms for Collateral Loan Accounts
Chapter XVI. Interest Accounts
189. Functions of the Three Interest Accounts
190. A Double Record for Interest Earned
191. Example of Interest Income
192. Daily Register of Interest Accruing
193. Monthly Summary
194. Method and Importance of Interest Earned Account
195. Interest Accounts in General Ledger
196. Payment of Accrued Interest
Chapter XVII. Bonds and Similar Securities
197. Investments with Fluctuating Values
198. Amortization Account
199. Effect on Schedule of Additional Purchases
200. The Bond Sales Account
201. Requirements as to Bond Records
202. Form of Bond Ledger
203. Interest Due Account
204. Interest Account — Bond Ledger
205. Amortization Entries
206. Bond Entries in General Ledger
207. Accounts Where Original Cost Is Disregarded
208. Amortization Reserve
209. Premiums and Amortization
210. Writing Off Premiums
211. Disposal of Amortization
212. Amortization Accounting — Comparison of Methods
213. Irredeemable Bonds a Perpetual Annuity
214. Bond Accounts for General Ledger
(Plans I to V)
Chapter XVIII. Discounted Values
215. Securities Payable at Fixed Dates Without Interest
216. Rates of Interest and Discount
217. Rate of Discount Named in Notes
CONTENTS xiii
218. Form as AfiFecting Legality
219. Entry of Notes Discounted
220. Discount and Interest Entries
221. Total Earnings from Discounts
PART II— PROBLEMS AND STUDIES
Chapter XIX. Interest and Discount
§ 222. Problems in Simple Interest
223. Notes on the One Per Cent Method
224. Answers to Problems in Simple Interest
225. Problems in Compound Interest
226. Answers to Problems in Compound Interest
227. Proof of Amount and Present Worth
228. Contracted Multiplication
229. Problems in the Use of Logarithms
230. Problems Requiring Use of More Extended Tables of
Logarithms
231. Answers to Problems in Logarithms
Chapter XX. Problems in Annuities and in Nominal and Effective
Rates
§ 232. Problems in Annuities
233. Answers to Problems in Annuities
234. Problems in Rent of Annuity and Sinking Fund
235. Answers to Problems in Rent of Annuity and Sinking Fund
236. Problems in Nominal and Effective Rates
237. Answers to Problems in Nominal and Effective Rates
238. Constant Compounding
239. Finding Nominal Rate
240. Approximate Rules
Chapter XXI. Equivalent Rates of Interest— Bond Valuations
§ 241. Annual and Semi-Annual Interest
242. Semi-Annual and Quarterly Interest
243. Problems in Valuation of Bonds
244. Successive Method of Bond Valuation — Problems
245. Answers to Bond Valuation Problems
246. Bond Valuations by the Use of Logarithms
247. Finding Initial Book Values
248. Tabular Multiplication and Contracted Division
249. Formation of Successive Amortizations
xiv CONTENTS
250. Test by Differencing
251. Successive Columns
252. Intentional Errors
253. Rejected Decimals
254. Limit of Tolerance
Chapter XXII. Broken Initial and Short Terminal Bonds
§ 255. Problems in Valuation
256. Answers to Valuation Problems
Chapter XXIII. The Use of Tables in Determining the Accurate
Income Rate
§ 257. Bond Tables as Annuity Tables
258. Premium and Discount as a Present Worth
259. Present Worth by Differences
260. Present Worth by Division
261. Compound Discount and Present Value of a Single Sum
262. Use of Bond Tables in Compound Interest Problems
263. Determination of Accurate Income Rate
264. Assumed Trial Rate
265. Application of Assumed Trial Rate — Bond Above Par
266. Variations in Assumed Rates
267. Application of Assumed Trial Rate — Bond Below Par
268. Trial Rates from Bond Tables
269. Use of Bond Tables
Chapter XXIV. Discounting
§ 270. Table of Multiples
271. Present Worths of Interest-Difference
272. Discounts from Tables
273. Reussner's Tables
Chapter XXV. Serial Bonds
§ 274. Problem in Valuation of Serial Bonds
275. Inter-rates
276. Table of Differences
277. Successive Method
278. Balancing Period
279. First Payment in Series
280. Elimination of Residue
281. Schedule
282. Uneven Loans
283. Tabular Methods
CONTENTS XV
284. Formula for Serials
285. Problems in Valuation of Serial Bonds
286. Answers to Problems in Valuation of Serial Bonds
Chapter XXVI. Option of Redemption
287. Method of Calculating Income Rate
288. Advantageous Redemption Ignored
289. Disadvantageous Redemption Expected
290. Change in Principal
291. Approximate Location
292. Problems Involving Optional Redemption Dates
293. Rule for Determining Net Income
294. Answers to Problems Involving Optional Redemption Dates
Chapter XXVII. Bonds at Annual and Other Rates
295. Standard of Interest
296. Semi-Annual and Quarterly Coupons
297. Shifting of Income Basis
298. Problems — Bonds at Varying Rates
299. Answers to Problems — Bonds at Varying Rates
300. Bonds with Annual Interest — Semi-Annual Basis
301. Annualization
302. , Semi-Annual Income Annualized
303. Comparison of Annual and Semi-Annual Bonds
304. Finding Present Worth of an Annuity
305. Rule for Bond Valuation
306. Multipliers for Annualizing
307. Formula for Annualizer
308. Conventional Process
309. Scientific Process
310. Values Derived from Tables
311. Successive Process
312. Problems and Answers — Varying Time Basis
313. Bonds at Two Successive Rates
314. Calculation of Immediate Premium
315. Calculation of Deferred Premium
316. Symbols and Rule
317. Analysis of Premiums
318. Problems and Answers — Successive Rates
Chapter XXVIII. Repayment and Reinvestment
319. Aspects of Periodic Payment
320. Integration of Original Debt
321. Use of the Reinvestment Point of View
322. Replacement
xvi CONTENTS
323. Diminishing Interest Rates
324. Proof of Accuracy
325. Varying Rates of Interest
326. Dual Rate for Income and Accumulation
327. Instalment at Two Rates
328. Amortization of Premiums at Dual Rate
329. Modified Method for Valuing Premiums
330. Rule for Valuation of a Premium
331. Computation at Dual Rate
332. Dual Rate in Bookkeeping
333. Utilization of Dual Principle
334. Installation of Amortization Accounts
335. Scope of Calculations
336. Method of Procedure When Same Basis Is Retained
PART III— LOGARITHMS
Chapter XXIX. Finding a Number When Its Logarithm Is Given
§ 2Z7. Logarithmic Tables
338. Discussion of Logarithms
339. Standard Tables of Logarithms
340. United States Coast Survey Tables
34L Gray and Steinhauser Tables
342. A Twelve-Place Table
343. The "Factoring" Method
344. Finding a Number from Its Logarithm
345. Procedure in an Unusual Case
346. Rule for Finding Number When Logarithm Is Given
347. Method by Multiples
Chapter XXX. Forming Logarithms; Tables
§ 348. Explanation of Process
349. Rule for Finding a Logarithm
350. Examples of Logarithmic Computations
351. Logarithms to Less Than Twelve Places
352. Tables with More Than Twelve Places
353. Multiplying Up
354. Process of Multiplying Up
355. Supplementary Multiplication
356. Multiplying Up by Little's Table
357. Different Bases
358. Table of Factors
359. Table of Interest Ratios
CONTENTS xvii
360. Table of Sub-Reciprocals
361. Table of Multiples
PART IV— TABLES
Chapter XXXI. Explanation of Tables Used
§ 362. Object of the Tables
363. Degree of Accuracy
364. Rates and Periods
365. Tables Shown
366. Annuities — When Payable
367. Table I— Amount
368. Compound Interest
369. Table II— Present Worth
370. Table III— Amount of Annuity
371. Table IV— Present Worth of Annuity
372. Table V— Sinking Fund
373. Rent of Annuity
374. Extension of Time
375. Subdivision of Rates
376. Interpolation
377. Table VI — Reciprocals and Square Roots
Chapter XXXII. Tables of Compound Interest, Present Worth,
Annuities, Sinking Funds, and Other Computations
§ 378. Table I — Amount of $1 at Compound Interest
(a) Part 1
(b) Part 2
379. Table II— Present Worth of $1 at Compound Interest
(a) Part 1
(b) Part 2
380. Table III— Amount of Annuity of $1 at End of Each Period
(a) Part 1
(b) Part 2
381. Table IV— Present Worth of Annuity of $1 at End of Each
Period
(a) Part 1
(b) Part 2
382. Table V — Sinking Fund or Annuity Which, Invested at the
End of Each Period, Will Amount to $1
(a) Part 1
(b) Part 2
383. Table VI — Reciprocals and Square Roots
EXPLANATION OF SYMBOLS
For the sake of brevity and clearness, certain constantly-
recurring expressions have been represented in the text by
symbols. The following list comprises all of those which are
not self-explanatory.
1 = $1, £l, or any other unit of value.
a = the amount of $1 for a given time at a given rate.
A = the amount of an annuity of $1 for a given time at a
given rate.
c = the cash, or coupon, rate of interest (or the cash pay-
ment) for a single period.
d = the rate of discount for a single period.
D = the discount on $1 for a given time at a given rate.
i = the rate of interest (or the income) for a single period.
I =the compound interest on $1 for a given time at a
given rate..
/ = the effective rate of interest for one year.
M = an indefinite number of units.
p = the present worth of $1 for a given time at a given rate.
P = the present worth of an annuity of $1 for a given time
at a given rate.
r = (1+ 1), the periodic ratio of increase.
xvm
THE ACCOUNTANCY OF INVESTMENT
Part I — The Mathematics of Investment
CHAPTER I
CAPITAL AND REVENUE
§ I. Definition of Capital
That portion of wealth which is set aside for the pro-
duction of additional wealth is capital. The capital of a
business, therefore, is the whole or a part of the assets of
the business, and of course appears on the active or debit
side of its balance sheet. This is the sense in which the
word "capital" is used in economics ; but in bookkeeping the
term "Capital account" is often used in quite another sense
to mean accounts on the credit or passive side, which denote
proprietorship. To prevent confusion, the use of the ex-
pression "Capital account" will be avoided.
§ 2. The Use of Capital
In active business, capital must be employed, and, in
order to produce more wealth, it must be combined with
skill and industry. Businesses, and consequently their ac-
counting methods, vary as to the manner in which capital
is used. Cash is convertible into potential capital of any
kind desired. In a manufacturing business it is exchanged
^9
"20' ^'' ' ^'^^riE^MAXriEMATICS OF INVESTMENT
for machinery, appliances, raw materials, and labor which
transforms these raw materials into finished products. In
a mercantile business cash is expended for goods, bought at
one price to sell at another, and for collecting, displaying,
caring for, advertising, and delivering goods. To bridge
over the time between selling and collecting, additional
capital is required, usually known as "working capital," but
which might more appropriately be styled "waiting capital."
Thus we may analyze each kind of business, and show that
the nature of its capital assets depends on the character of
the business.
§ 3. Sources of Capital
On the credit side of the balance sheet the capital must
be accounted for in such a manner as to show its sources.
Here there are two sharply divided classes : loan-csipitsl, or
liability, and ow^-capital, or proprietorship. The great
distinction is that the latter participates in the profits and
bears the losses, while the former takes its share irrespective
of the success of the concern. It is the own-capital which
is referred to in the phrase "Capital account."
§ 4. Investment
While we often speak of a man's capital as being in-
vested in a business, we use the word "investment" more
strictly when we confine it to the non-participating sense.
Thus we say, "He not only owns a business, but he has
some investments besides." In the strictest sense, then, in-
vestment implies divesting one's self of the possession and
control of one's assets, and granting such possession and
control to another. The advantage of the use of capital
must be great enough to enable the user to earn more than
the sum which he pays to the investor, or capitalist. There
are many cases where the surrender is not absolute, and
CAPITAL AND REVENUE 21
more or less risk is assumed by the investor. This is not
absolute investment, but to some extent partnership. The
essence of strict investment is the vicarious earning of a
share in gains which do not depend on the business skill of
the investor.
§ 5. Revenue
All investments are made with a view to obtaining
revenue, which is the share of the earnings given for the
use of capital. Revenue takes three forms: interest, rent,
and dividends — the first two corresponding to strict invest-
ment, and the latter to participation.
§6. Interest and Rent
These do not essentially differ. Both are stipulated pay-
ments for the use of capital ; but in case of rent the identical
physical asset received by the lessee must be returned by
him on the completion of the contract. If you borrow a
dollar, you may repay any dollar you please ; if you hire a
house or a horse, you may not return any house or any
horse, but must produce the identical one you had. Interest
and rent are both proportionate to time.
§ 7. Dividends
These are profits paid over to the owners of the own-
capital, whether partners or shareholders. The amount is
subtracted from the collective assets and paid over to the
separate owners. Theoretically there is neither profit nor
loss in this distribution. I have more cash, but my share
in the collective assets is exactly that much less. The cash
is distributed partly because it is needed by the participants
for consumption ; and partly because no more capital can be
profitably used in the enterprise. Some concerns, however,
such as banks, which can profitably use more capital and
22 THE MATHEMATICS OF INVESTMENT
whose shareholders do not require cash for consumption,
frequently refrain from dividing the periodical profits, or
distribute but a small portion of them.
The accumulation of the profits, however, inures just as
surely to the benefit of the shareholders, and is usually
realizable through increased value of the shares upon sale.
Thus, dividends are not strictly revenue, though the share-
holder may treat them as such; his dividend may be so
regular as practically to be fixed, or his shares may be
preferential, so that to some extent he is receiving an
ascertained amount; or, as in case of a leased railway, the
dividend may be expressly stipulated in a contract. Still,
legally speaking, the dividend is instantaneous, and does
not accrue, like interest and rent.
§ 8. Laws of Interest
As all investments are really purchases of revenue, and
as the value of an investment depends largely upon the
amount of revenue derivable therefrom, and as the typical
form of revenue is interest, it is necessary to study the laws
of interest, including those more complex forms — annuities,
sinking funds, and amortization. Although there is a
special branch of accountancy — ^the actuarial — which deals
not only with these subjects, but with life and other con-
tingencies, it is yet very necessary for the general accountant
to understand at least their fundamental principles.
CHAPTER II
INTEREST
§ g. Interest
As ordinarily defined, interest is "money paid for the
use of money." A better definition from a mathematical
standpoint would be, "the increase of indebtedness through
lapse of time." Since the production of additional wealth is
dependent on the processes of nature, and since these
processes require time, it is equitable that compensation for
an increase in time should be made by an increase in in-
debtedness. The "money paid" of the first definition is a
payment on account of the general debt (including in-
terest) ; the direct effect of interest is to increase the debt,
while the direct effect of a payment is to reduce it.
§ 10. Essentials of Interest Contract
The contract, express or implied, regarding an interest
transaction, must take into consideration the following
items:
(1) Principal. The number of units of value (dollars,
pounds, francs, marks, etc.) originally loaned or invested.
(2) Rate. The part of the unit of value (usually a
small number of hundredths) which is added to each such
unit by the lapse of one unit of time.
(3) Frequency. The length of the unit of time, measured
in years, months, or days. Weeks are not used as time
units, nor are parts of a day.
(4) Time. The number of units of time during which
the indebtedness is to continue.
23
24
THE MATHEMATICS OF INVESTMENT
§11. Interest Rate
The rate is usually spoken of as so much per cent per
period, or term. Thus, if the contract provides for the
payment of three cents each year for the use of each dollar
of principal, the rate may be expressed, .03 per annum, 3
per centum per annum, 3 per cent, or simply 3%. Where
the period is not a year, but a smaller unit of time, it is
nevertheless customary to speak of the annual rate. For
instance, instead of saying, "3% per half-year,'' we say,
"6%, payable semi-annually." In the same way, 1% per
quarter would be 4%, payable quarterly. In our discus-
sions of interest, however, we shall treat of periods, and of
the rate per period, in order to avoid confusion. The in-
terest rate will be designated by the small letter i\ as,
z = .06. At the end of the first period the increased in-
debtedness, corresponding to the original unit of indebted-
ness at the beginning of the term, is 1 + i (1.06), a very-
important quantity in computation. The subject of rates
of interest will be discussed in greater detail in Chapter
VIII, "Nominal and Effective Rates."*
§ 12. Principal
Since each dollar increases just as much as every other
dollar, the general practice is to consider the principal as
one dollar and, when the proper interest thereon has been
found, to multiply it by the number of dollars.
§ 13. Simple and Compound Interest
Interest is assumed to be paid when due. If it is not
so paid, it ought to be added to the principal, and interest
should be computed on the increased principal. But the
law does not directly sanction this compounding of interest,
*For discussion of the causes of higher or lower interest rates, see "The Rate
of Interest," by Prof. Irving Fisher.
INTEREST
25
and simple interest is spoken of as if it were a distinct
species in which the original principal remains unchanged,
even though interest is in default. There is really no such
thing as simple interest, since the interest money which is
wrongfully withheld by the borrower, may be by him em-
ployed, and thus compound interest be earned. But the
wrong party gets the benefit of the compounding. All the
calculations of finance depend upon compounding interest,
which is the only rational and consistent method. When
there is occasion hereafter to speak of the interest for one
period, it will be called "single interest."
§ 14. Punctual Interest
The usual interest contract provides that the increase
shall be paid off in cash at the end of each period, restoring
the principal to its original amount. Let c denote the cash
payment ; then l + i — c = l; and the second term would
repeat the same process. The payment of cash for interest
must not be regarded as the interest ; it is a cancellation of
part of the increased principal. Many persons, and even
courts, have been misled by the old definition of interest —
"money paid for the use of money" — into treating uncol-
lected or unmatured interest as a nullity, though secured in
precisely the same way as the principal.
§ 15. Computation of Interest
But the interest money may not be paid exactly at the
end of each term, either in violation of the contract or by
a special clause permitting it to run on, or by the debt being
assigned to a third party at a price which modifies the true
interest rate. In this case the question arises : How shall
the interest be computed for the following periods? This
gives rise to a distinction between simple and compound
interest.
26 THE MATHEMATICS OF INVESTMENT
§ 1 6. Simple Interest
During the second period, although the borrower has in
his hands an increased principal, 1 + i, he is at simple
interest charged with interest only on 1, and has the free
use of i, which, though small, has an earning power pro-
portionate to that of 1. His indebtedness at the end of the
second term is 1 + 2i, and thereafter 1 + 3f, 1 + 4z', etc.
After the first period he is not charged with the agreed per-
centage of the sum actually employed by him, and this to
the detriment of the creditor. For any scientific calculation,
simple interest is impossible of application.
§17. Compound Interest
The indebtedness at the end of the first period is 1 + 1,
and up to this point punctual, simple, and compound interest
coincide. But in compound interest the fact is recognized
that the increased principal, 1 + i, is all subject to interest
during the next period, and that the debt increases by geo-
metrical progression, not arithmetical. The increase from
1 to 1 + i is regarded, not as an addition of i to 1, but as a
multiplication of 1 by the ratio of increase (1 + i). We
shall designate the ratio of increase by r when convenient,
although this is merely an abbreviation of 1 + i, and the
two expressions are at all times interchangeable.
§ 18. Computation of Compound Interest
At the end of the first period (which is equivalent to the
beginning of the second period), the actual indebtedness is
1 + i. This amount is the equitable principal for the second
period, and it should be again increased in the ratio 1 + i.
The total indebtedness at the end of the second period
(which is equivalent to the beginning of the third period)
is therefore 1 X (1 + 1) X (1 + ?-)• For the sake of brevity,
this may be written 1 X (1 + O^j the figure 2 (called an ex-
INTEREST
27
ponent) indicating that the expression (I'+t) is to ,be
taken twice as a factor. Since the expression (1 + i) equals
the rate, a still simpler way of indicating the indebtedness at
the end of the first period is r; at the end of the second
period, r^. At the end of the third period the indebtedness
will have become t^ ; and at the end of period t, it will have
become r\
§ 19. Comparison of Simple and Compound Interest
The following schedule shows the accumulations of in-
terest for several periods, giving a comparison between the
simple interest computations and the compound interest
computations :
■
Indebtedness
Indebtedness
Time
Based on
Based on
Simple Interest
Compound Interest*
Beginning of 1st period..
1
1
Beginning of 2nd period.
1-bi
l + t
Beginning of 3rd period.
l + 2i
(i + »T
Beginning of 4th period. .
l + 3i
(i+»r
Beginning of 5th period. .
l + 4i
(1 + 0*
etc.
•For the benefit of students familiar with algebra, it may be pointed out that
(1 + O* = 1 + 2i + t*. This differs from the simple interest computation by the
small quantity t*. Similarly, (1 + t)^ = 1 + 3t + 3i^ + »^, which differs from the
simple interest result, 1 + Si, by the quantity Si'^ + *"'• Tests may be readily made
of the computations by substituting a numerical rate, say .06, in place of i. If this
be done, the simple interest result at the beginning of the 4th period is found to be
14- (3 times .06), or 1.18. The compound interest result would be:
1 =1.
plus 3 times .06 = .18
plus 3 timco .06', or
3 times .0036 = .0108
plus .06» = .000216
That is, (1.06)« = 1.191016
28 THE MATHEMATICS OF INVESTMENT
§ 20. The Day as a Time Unit
Coming now to a discussion of frequency and time, in
connection with the subject of interest, we find that the
smallest unit of time is one day, since the law does not
recognize interest for fractions of a day. The legal day
begins at midnight and ends on the following midnight. In
reckoning from one day to another, the day from which
should be excluded. Thus, if a loan is made at any hour
on the third day of the month and is paid at any hour on
the fourth, there is one day's interest due, the interest being
for the fourth day and not for the third. Practically it is
the nights that count. If five midnights have passed since
the loan was made, then the accrued interest is for a period
of five days.
§ 21. The Month as a Time Unit
As has been previously stated, weeks are not used as
time units. The next longer interest period after a day is
a month. Calendar months are computed as follows : Com-
mence at the day from which the reckoning is made, and
exclude that day; then the day in the next month having
the same number will at its close complete the first month ;
the second month will end with the same numbered day,
and so on to the same day of the final month. A difficulty
arises in the case where the initial date is the 31st, while the
last month has only thirty days or less. In this case the
interest month ends with the last day of the calendar month.
For example, one month from January 31st, 1912, was
February 29th; one month from January 30th or 29th, in
the same year, also terminated on February 29th ; in a com-
mon year, not a leap year, the last day of a period one
month from January 28th, 29th, 30th or 31st, would be
February 28th.
INTEREST
29
§ 22. Half and Quarter Years
Since there are no fractions of a day in interest compu-
tations, it becomes necessary to inquire what is meant by
a half-year or by a quarter. In the State of New York the
Statutory Construction Law (Laws of 1892, Chapter 677,
§ 25) solves this difficulty by prescribing that a half-year
is not 182% days, but six calendar months; and that a
quarter is not 91^/4 days, but three calendar months.
§ 23. Partial Interest Periods
In practice any fraction of an interest period is com-
puted at the corresponding fraction of the rate, although
theoretically this is not quite just. For example, if the
interest rate is 6% per annum, payable annually, making
the ratio of increase 1.06, then it is customary to consider
the ratio of increase for a half-year as 1.03; whereas
theoretically it should be the square root of 1.06, or slightly
over 1.029563.
If the regular period is one year, any odd days should
be reckoned as 365ths of a year. Also, if the contract is for
days only and there is no mention of months, quarters, or
half-years, then also a day should be regarded as 1/365 of
a year. But when the contract is for months, quarters, or
half-years, any fractional time should be divided into months,
and there is usually an odd number of days left over. In
New York, doubt exists as to how these odd days should
be treated, whether on a 365-day basis or on a 360-day basis.
Before 1892 there was no doubt. The statute distinctly
stated that a number of days less than a month should be
estimated for the purpose of interest computations as 30ths
of a month, or, consequently, 360ths of a year. This was
a most excellent provision, and merely enacted what had
been the custom long before. The so-called "360-day" in-
terest tables are based upon this rule. In 1892, however,
30
THE MATHEMATICS OF INVESTMENT
the revisers of the statutes of the State of New York
dropped this sensible provision and left the question open.
No judicial decision has since been rendered on the subject,
but many good lawyers think that the odd days should be
computed as 365ths of a year. In business nearly every
one calls the odd days 360ths, and it is only in legal account-
ings that there can be any question. It would be well if
the old provision could be re-enacted by law or re-established
by the courts.
§ 24. Changing the Day Basis
If the interest for a certain number of odd days has been
computed on a 360-day basis, a change may be readily made
to a 365-day basis by subtracting from such interest 1/73
of itself. On the other hand, if the interest for an odd
number of days has been ascertained on a 365-day basis,
the addition of 1/72 of itself to this amount will give the
interest on a 360-day basis.
§ 25. The Amount — First Period
The principal and interest taken together constitute the
amount. At the end of the first half-year period, the
amount of $1.00 at 6% interest, payable semi-annually, is
$1.03. Instead of considering the $1.00 and the 3 cents as
two separate items to be added together, it is best to con-
sider the operation as the single one of multiplying $1.00
by the ratio of increase, 1.03. Sometimes the error is made
of considering that the original principal of $1.00 is multi-
plied by $1.03, or, in other words, that a certain number
of dollars is multiplied by another number of dollars. It
is well to emphasize, in this connection, the old principle
given in arithmetic, that one concrete number cannot be
multiplied by another concrete number. We cannot multi-
ply dollars by dollars, or feet by feet, or horses by dollars.
The multiplicand may be either a concrete or an abstract
number, but the multiplier must always be abstract.
INTEREST
31
§ 26. The Amount — Subsequent Periods
The principal which is employed during the second
period is $1.03. It is evident that this, like the original
$1.00, should be multiplied by the ratio 1.03. The new
amount will be the square of 1.03, which we may write:
1.03 X 1.03
or, 1.03=^
or, 1.0609
This is the new amount on interest during the third period.
At the end of the third period the amount will be :
1.03 X 1.03 X 1.03
or, 1.03'
or, 1.092727
At the end of the fourth period the amount becomes :
1.03*
or, 1.12550881
Possibly at this point the number of decimal places may be
unwieldy. If we desire to have only seven decimal places,
we reject the final 1, rounding the result oif to 1.1255088;
if we prefer to use only six places, we round the result up
to 1.125509, which is more nearly correct than 1.125508.
§ 27. Exponents and Powers
In some of the following paragraphs, it will be neces-
sary to speak occasionally of exponents and powers. In
the expression 1.03^, the figure 2 is called an exponent, and
it means (as indicated in the preceding paragraph) that
1.03 is to be taken twice as a factor. In other words, the
number is to be multiplied by itself. The result, 1.0609
(which equals 1.03^), is said to be the second power of 1.03 ;
1.092727 is the third power of 1.03, and so on. (See also
§38.)
32 THE MATHEMATICS OF INVESTMENT
§ 28. Finding the Amount — Compound Interest
The amount of $1.00 at the end of any number of
periods is obtained by taking such a power of the ratio of
increase as is indicated by the number of periods; or, in
other words, by muhiplying $1.00 by the ratio as many
times as there are periods. If the original principal be sub-
tracted from the amount, the remainder is the compound
interest. For example, in § 26, the amount of $1.00 at the
ratio 1.03, for four periods, is $1.12550881; and the com-
pound interest is $.12550881.
§ 29. Present Worth
The present worth of a future sum is a smaller sum
which, put at interest, will amount to the future sum. The
present worth of $1.00 is such a sum as, at the given rate
and for the given period, will amount to $1.00. In order
to illustrate the method of ascertaining the present worth,
let us suppose that it is desired to find the present worth
of $1.00, due in four years, the ratio of increase being 1.03
per annum. The required figure must evidently be such
that, when multiplied four times in succession by 1.03, the
result will be $1.00. Therefore, by using the reverse
process, division, the required figure may be obtained. The
first operation, by ordinary long division, results as follows :
1.03 ) 1.00000000 ( .970873
927
730
721
900
824
760
721
390
309
81
INTEREST 33
The result, rounded up at the 6th place, is .970874, this
being the present worth of $1.00 due in one period at 3%
interest. The present worth for two periods may be ob-
tained either by again dividing .970874 by 1.03, or by mul-
tiplying .970874 by itself, or by dividing 1 by 1.0609 (the
square of 1.03), each of which operations gives the same
result, .942596. The present worth for three periods may
also be obtained in several ways, the result being the same
in all cases, .915142, or ^^. The present worth for four
periods is ^, or .888487.
§ 30. Present Worth and Amount Series
If we arrange these results in reverse order, followed
by $1.00 and by the amounts computed in § 26, we have a
continuous series :
1-
- 1.03*
-= .888487
1-
- 1.03'
-= .915142
1-
-1.03'
-= .942596
1-
-1.03
=- .970874
1.
= 1.03
1 X 1.03
1 X 1.03'
= 1.0609
1 X 1.03'
= 1.092727
1>
< 1.03*
= 1.125509
§ 31. Relation between Present Worth and Amount
In the foregoing series, which might be extended in-
definitely upward and downward, every term is a present
worth of the one which immediately follows it, and an
amount of the one which immediately precedes it. When
one number is the amount of another, the latter number is
the present worth of the former. For example, .888487 is
the present worth of 1.125509 for 8 interest periods; and,
on the other hand, 1.125509 is the amount of .888487, for
the same number of interest periods and at the same ratio.
34
THE MATHEMATICS OF INVESTMENT
In some instances in this series, a present worth and its
corresponding amount are reciprocals (that is, their product
is 1), but this is true only when the two figures are distant
an equal number of periods from 1, the present-worth
figure being upward from 1 and the amount figure being
downward from 1. Thus, .915142, or 1 -^- 1.03^, is the
reciprocal of 1.092727, or 1.03^
§ 32. Formation of Series
K any term of the series be multiplied by 1.03, the
product will be the next following term ; if it be divided by
1.03 or (which amounts to the same thing) be multiplied by
.970874, the result will be the next preceding term. Since
multiplying by 1.03 is easier than dividing by it, and also
easier than multiplying by .970874, the easiest way of
obtaining the different numbers in the series is to compute
first the smallest number (in this case, .888487), and then
perform successive multiplications by 1.03. A brief process
for finding this initial number will be explained in the next
chapter.
§ 33. Discount
In considering the present worth of $1.00 for a single
period (.970874), it is evident that the original $1.00 has
been diminished by .029126, which is a little less than .03;
in fact it is .03 -^ 1.03. This difference, .029126, is called
the discount. In the present worth for two periods, the dis-
count is 1 — .942596, or .057404. This discount for two
periods, and likewise the discounts for three or more
periods, are called compound discounts.
§ 34. Computing Compound Discount
The compound discount for any number of periods may
be found either by subtracting the present worth from 1, or
INTEREST
35
by finding the present worth of the compound interest for
the same time and at the same rate. As an illustration,
suppose that it is desired to find the compound discount of
$1.00 for three periods at 3%. First, we may subtract the
present worth, .915142, from. 1, which gives the compound
discount as .084858. Second, we may divide the compound
interest (.092727) by the amount of $1.00 for the three
periods (1.092727), which gives the compound discount the
same as before, .084858.
§ 35. Formulas for Interest Calculations
We may reduce the rules to more compact form by the
use of symbols. Let a represent the amount of $1.00 for
any number of periods (it periods) ; p the present worth;
i the rate of interest per period ; d the rate of discount per
period, and n the number of periods. Let the compound
interest be represented by I, and the compound discount by
D. Then, by §17, the ratio of increase is (1 + i). By
§28, a=(l-}-iy; and I==a — 1. By §29, /> = 1-^
(1 + 0"; and by § 34, D = 1 - /), or I -^ a.
§ 36. Use of Logarithms
The method of ascertaining the values of a and p
through successive multiplications and divisions for a large
number of periods, is intolerably slow. A much briefer way
is by the use of certain auxiliary numbers called logarithms
as explained in the next chapter.
CHAPTER III
THE USE OF LOGARITHMS
§ 37. Purpose of Logarithms
For multiplying or dividing a great many times by the
same number, or for finding powers and roots, there is no
device superior to a table of logarithms. Although the
computation of logarithms — as in the formation of a table of
logarithms — requires a knowledge of algebra, the practical
use of logarithmic tables does not require such knowledge.
The aid derived from such tables is purely arithmetical, and
the occasional prejudice against logarithms as something
mysterious or occult is without reasonable foundation.
§ 38. Exponents, Powers, and Roots
We have seen in § 27 that an exponent is a number writ-
ten at the right and slightly above another number to indi-
cate how many times the latter is to be taken as a factor;
and also that a power is the result obtained by taking any
given number a certain number of times as a factor. We
now add that a root is the number repeated as a factor to
form a power. The following table exemplifies roots,
exponents, and powers :
Roots and ^^
T, , Powers
Exponents
22 = 2X2 =4
32 = 3x3 =9
4' = 4 X 4 =16
etc.
2^ = 2X2X2= 8
3' = 3X3X3 = 27
43 = 4X4X4 = 64
etc.
36
THE USE OF LOGARITHMS
37
The root of a number is called its first power. When the
root is taken twice as a factor, the result is called the second
power, or square ; when taken three times, the result is the
third power, or cube; we may in like manner obtain the
fourth, fifth, or any power of a root by repeating it as a
factor the required number of times.
§ 39. Logarithms as Exponents
Now, logarithms are merely exponents of certain roots
which are called bases. The common system of logarithms
is based upon the number 10, this number being the basis of
our decimal system of numeration.
Taking a specific illustration, let us multiply six lO's
together, 10 X 10 X 10 X 10 X 10 X 10 ; we may write the
result as :
1,000,000
or, 10"
or, the sixth power of ten.
The small figure "6'* is the exponent of the power. A
series of some of the powers of 10 might be represented as
follows :
1,000,000 or 10"
100,000
10°
10,000
10*
1,000
10'
100
10^
10
10'
1
tt
10"
From the above series, the following observations may
be made :
(1) The number of zeroes in any number in the first
column is the same as the exponent in the second column.
(2) Each term in the first column is one-tenth of the
2fi THE MATHEMATICS OF INVESTMENT
one above it, while in the second column each exponent is one
less than the exponent above it. This leads to the result
that 10*^ = 1, which at first seems impossible. It is difficult
to understand how 10, taken zero times as a factor, equals
1, but such nevertheless is a fact, as can be easily demon-
strated by algebra.*
(3) By adding any two exponents in the second column,
w^e may find the result of multiplying together the two
corresponding numbers in the first column. For example,
10' (or 100) times 10' (or 1,000) equals 10' + ^ i.e., 10'
(or 100,000) ; in other words, by adding the logarithms of
two numbers we obtain the logarithm of their product.
Again, by finding the difference between any two logarithms
in the second column, we may find the quotients of the
corresponding numbers in the first column. For example,
10' - ' = 10' = 1,000, which = 100,000 -f- 100; i.e., by sub-
tracting logarithms the logarithms of quotients are found.
Suppose we should wish to obtain the second power of
10'; the exponent (or index) of the second power is 2; and
10^ X 2 == ;^Q6 ^ -^QQQ ^ -^QQQ^ ^j. ^^000,000 ; from which it
appears that by multiplying the logarithm 3 by the index 2
we have obtained the square of 10', or 1,000. Again 10^ "^ ^
= 10' = 1,000, or the square root of 1,000,000; from
which it appears that, by dividing the logarithm 6 by the
index 2, we obtain the square root of 10^.
§ 40. Rules and Symbols of Logarithms
Summarized very briefly, the rules of logarithms, de-
duced from the foregoing illustrations, are as follows:
By the use of the equation :
Therefore, x<> = 1
or, if X =10
10»= 1
— = !• — =x*-» — x»
THE USE OF LOGARITHMS 39
(1) By adding logarithms, numbers are multiplied.
(2) By subtracting logarithms, numbers are divided.
(3) By midtiplying logarithms, numbers are raised to
powers.
(4) By dividing logarithms, the roots of numbers are
extracted.
The last two of these rules are the only ones necessary
to be employed in the calculations of compound interest.
With this preliminary explanation of logarithms, the
series in § 39 may be rewritten and "extended"; the symbol
nl meaning *'the number whose logarithm is."
The base being 10,
1,000,000 is the number whose logarithm is 6,
or, in contracted form.
10« =
1,000,000.
nl
6
10' =
100,000.
nl
5
10* =
10,000.
nl
4
10' =
1,000.
nl
3
10" =
100.
nl
2
10^ =
10.
nl
1
10« =
1.
nl
.1
nl ■
-1
.01
nl
-2
.001
nl
-3
.0001
nl
-4
.00001
nl ■
-5
Occasionally the symbol In will also be used, its meaning
being *'the logarithm of the number."
From the above, it will be seen that the logarithm of a
number is merely the exponent which indicates the power
to which some given number, called the base, would have
to be raised in order to give that number. If the number
40
THE MATHEMATICS OF INVESTMENT
were 100 and the base 10, then 10 would have to be raised
to the second power to give 100; in other words, the loga-
rithrn of 100, with 10 as a base, is 2. If the number were
100,000, the base still being 10, then 10 would have to be
raised to the fifth power to give the number; or we may
say, in different language, that the logarithm of 100,000,
with 10 as a base, is 5.
§ 41. The Two Parts of a Logarithm
The logarithms of a few numbers have already been
given, but for practical use in calculating we need the loga-
rithms of a great many others. From the series in § 39, it
may be readily inferred that the numbers between 1 and 10
must have their logarithms between and 1 ; that is, the
logarithms of these numbers must be fractions. In tables
of logarithms, these fractions are expressed as decimals,
the usual number of decimal places being seven. Similarly,
the numbers between 10 and 100 have their logarithms be-
tween 1 and 2 ; that is, these logarithms are 1 plus a decimal
fraction.
To give a few illustrations:
The logarithm of 10 = 1.
12 = 1.0792
20 = 1.3010
50 = 1.6990
90 = 1.9542
99 = 1.9956
100 = 2.
It will be observed that there are usually two parts to a
logarithm — ^the decimal part and the whole number preced-
ing the decimal. The decimal part is known as the mantissa,
and the whole number as the characteristic. From the man-
tissa of a logarithm, we are able (through the aid of loga-
THE USE OF LOGARITHMS
41
rithmic tables) to determine the corresponding number,
except as to the position of its decimal point. This latter
is determined by the characteristic, which, for numbers
greater than 1, is always one less than the number of places
to the left of the decimal point. For numbers less than 1,
the characteristic is said to be negative, and it is equal to
the number of places to the right from the decimal point to
the place occupied by the first significant figure of the
decimal. A negative characteristic is indicated by a short
dash placed above it. A characteristic may thus be either
positive or negative, but a mantissa is always positive.
§42. Mantissa Not Affected by Position of the Decimal
Point
In the logarithms of 20, 200, 2,000, 20,000, 200,000,
2,000,000, etc., we shall find the same decimal part, .301 030
(which is the logarithm of 2), preceded by the figures 1, 2,
3, 4, 5, 6, etc. This same thing is true of any combination
of figures; that is to say, whatever may be the position of
the decimal point in a number, the logarithm of that num-
ber always has the same decimal fraction, or mantissa.
Thus, if the logarithm of 2.378 is .376 212, then,
.0002378 nl 4.376 212
.002378 nl 3.376 212
.02378
nl 2.376 212
.2378
nl 1.376 212
2.378
nl .376 212
.23.78
nl 1.376 212
237.8
nl 2.376 212
2,378.
nl 3.376 212
23,780.
nl 4.376 212
237,800.
nl 5.376 212
etc.
etc.
42
THE MATHEMATICS OF INVESTMENT
§ 43. Four-Place Table of Logarithms
Illustrations will now be given of the properties of loga-
rithms, and for this purpose a table of the logarithms of
numbers from 10 to 99, inclusive, to four places of decimals,
is given on the following pages. This is a very simple table
of logarithms, and is known as a four-place table.
-The ordinary tables of logarithms are calculated to seven
places of decimals. If it is desired to multiply the number
82 by 1.03 fifty times in succession, ordinary logarithm
tables would give only the first seven figures of the answer.
If this operation were performed accurately by the simple
processes of multiplication, the answer vrould contain 103
figures, 3 in front of the decimal point and 100 after it.
Since the figures after the first seven are for most purposes
insignificant, the result obtained by logarithms will be near
enough even if rounded off at the sixth figure.
N
1
2
3
4
5
6
7
8
9
10
0000
043
086
128
170
212
253
294
334
374
11
0414
453
492
531
569
607
645
682
719
755
12
0792
828
864
899
934
969
*004
*038
*072
♦106
13
1139
173
206
239
271
303
335
367
399
430
14
1461
492
523
553
584
614
644
67Z
703
732
15
1761
790
818
847
875
903
931
959
987
*014
16
2041
068
095
122
148
175
201
227
253
279
17
2304
330
355
380
405
430
455
480
504
529
18
2553
577
601
625
648
672
695
718
742
765
19
2788
810
833
856
878
900
923
945
967
989
20
3010
032
054
075
096
118
139
160
181
201
21
3222
243
263
284
304
324
345
365
385
404
22
3424
444
464
483
502
522
541
560
579
598
23
3617
636
655
674
692
711
729
747
766
784
24
3802
820
838
856
874
892
909
927
945
962
25
3979
997
*014
*031
*048
*065
*082
*099
*116
*133
26
4150
166
183
200
216
232
249
265
281
298
27
4314
330
346
362
378
393
409
425
440
456
28
4472
487
502
518
533
548
564
579
594
609
29
4624
639
654
669
683
698
713
728
742
757
See explanation at the end of this section.
THE USE OF LOGARITHMS
43
N
1
2
3
4
5
6
7
8
9
30
4771
786
800
814
829
843
857
871
886
900
31
4914
928
942
955
969
983
997
*011
*024
*038
32
5051
065
079
092
105
119
132
145
159
172
33
5185
198
211
224
237
250
263
276
289
302
34
5315
328
340
353
366
378
391
403
416
428
35
5441
453
465
478
490
502
514
527
539
551
36
5563
575
587
599
611
623
635
647
658
670
37
5682
694
705
717
729
740
752
763
775
786
38
5798
809
821
832
843
855
866
877
888
899
39
5911
922
933
944
955
966
977
988
999
*010
40
6021
031
042
053
064
075
085
096
107
117
41
6128
138
149
160
170
180
191
201
212
222
42
6232
243
253
263
274
284
294
304
314
325
43
6335
345
355
365
375
385
395
405
415
425
44
6435
444
454
464
474
484
493
503
513
522
45
6532
542
551
561
571
580
590
599
609
618
46
6628
637
646
656
665
675
684
693
702
712
47
6721
730
739
749
758
767
776
785
794
803
48
6812
821
830
839
848
857
866
875
884
893
49
6902
911
920
928
937
946
955
964
972
981
50
6990
998
^007
*016
*024
*033
*042
*050
*059
*067
51
7076
084
093
101
110
118
126
135
143
152
52
7160
168
177
185
193
202
210
218
226
235
53
7243
251
259
267
275
282
292
300
308
316
54
7324
332
340
348
356
364
372
380
388
396
55
7404
412
419
427
435
443
451
459
466
474
56
7482
490
497
505
513
520
528
536
543
551
57
7559
566
574
582
589
597
604
612
619
627
58
7634
642
649
657
664
672
679
686
694
701
59
7709
716
723
731
738
745
752
760
767
774
60
7782
789
796
803
810
818
825
832
839
846
61
7853
860
868
875
882
889
896
903
910
917
62
7924
931
938
945
952
959
966
973
980
987
63
7993
*000
*007
*014
*021
*028
*035
*041
*048
*055
64
8062
069
075
082
089
096
102
109
116
122
* See explanation at the end of this section.
44
THE MATHEMATICS OF INVESTMENT
N
1
2
3
4
5
6
7
8
9
65
8129
136
142
149
156
162
169
176
182
189
66
8195
202
209
215
222
228
235
241
248
254
67
8261
267
274
280
287
293
299
306
312
319
68
8325
331
338
344
351
357
363
370
376
382
69
8388
395
401
407
414
420
426
432
439
445
70
8451
457
463
470
476
482
488
494
500
506
71
8513
519
525
531
537
543
549
555
561
567
72
8573
579
585
591
597
603
609
615
621
627
73
8633
639
645
651
657
663
669
675
681
686
74
8692
698
704
710
716
722
727
7ZZ
739
745
75
8751
756
762
768
774
779
785
791
797
802
76
8808
814
820
825
831
837
842
848
854
859
77
8865
871
876
882
887
893
899
904
910
915
78
8921
927
932
938
943
949
954
960
965
971
79
8976
982
987
993
998
*004
*009
*015
*020
*025
80
9031
036
042
047
053
058
063
069
074
079
81
9085
090
096
101
106
112
117
122
128
133
82
9138
143
149
154
159
165
170
175
180
186
83
9191
196
201
206
212
217
222
227
232
238
84
9243
248
253
258
263
269
274
279
284
289
85
9294
299
304
309
315
320
325
330
335
340
86
9345
350
355
360
365
370
375
380
385
390
87
9395
400
405
410
415
420
425
430
435
440
88
9445
450
455
460
465
469
474
479
484
489
89
9494
499
504
509
513
518
523
528
533
538
90
9542
547
552
557
562
566
571
576
581
586
91
9590
595
600
605
609
614
619
624
628
633
92
9638
643
647
652
657
661
666
671
675
680
93
9685
689
694
699
703
708
713
717
722
727
94
9731
736
741
745
750
754
. 759
763
768
77Z
95
9777
782
786
791
795
800
805
809
814
818
96
9823
827
832
836
841
845
850
854
859
863
97
9868
872
877
881
886
890
894
899
903
908
98
9912
917
921
926
930
934
939
943
948
952
99
9956
961
965
969
974
978
983
987
991
996
See explanation at the end of this section.
THE USE OF LOGARITHMS .45
In the table preceding, the figures in the column headed
"N" denote the numbers whose logarithms are given. These
numbers must be considered in conjunction with the num-
bers at the top of the remaining ten columns ; in other words,
we can find the logarithm not only of the number (say) 34,
but also of 34.1, 34.2, 34.3, etc., and similarly of .34, .341,
.342, .343, etc., and of 3,400, 3,410, 3,420, 3,430, etc.
The figures in the columns headed "0," "1,'' "2," etc.,
represent simply the decimal parts (or mantissas) of the
logarithms; the whole (or integral) parts of the logarithms
must always be determined by inspection (§41).
The column headed "0" has four places of figures, while
the following columns have only three places. This is done
to save space, since a fourth figure is assumed to be pre-
fixed, this figure being the same as the first figure in the
four-place column. There is an exception to this rule in the
case of figures prefixed by an asterisk, and three or four
examples will serve to make this clear:
Corresponding
Number Logarithm
2.50 .3979
2.51 .3997
2.52 .4014 (not .3014)
2.53 .4031 (not .3031)
etc. etc.
§ 44. Multiplication by Logarithms
As stated in § 40, there are four general rules regarding
logarithms, and these will now be illustrated in order.
Rule 1: The sum of the logarithms of two or more
numbers is the logarithm of their product.
46 THE MATHEMATICS OF INVESTMENT
2 nl .3010
3 nl .4771
2X3
6 nl
.778t
4 nl
.6021
14 nl
1.1461
4X14
56 nl
1.7482
5 w/
.6990
20 nl
1.3010
6X20
100 w/
2.0000
In these and other illustrations, there may be apparent
errors in the final decimal figure, due to throwing away or
adding on parts of decimals, as the case might be. For
example, in the first illustration given above, the logarithm
of 2 to six places is .301030, and the logarithm of 3 is
.477121; the logarithm of 6, the product, is .778151.
§ 45. Division by Logarithms
The logarithm of a product is obtained by finding the
sum of the logarithms of the factors, and, as division is the
converse of multiplication, the logarithm of a quotient is
obtained by finding the difference between the logarithms
of the dividend and divisor.
Rule 2 : The difference of the logarithms of two num-
bers is the logarithm of their quotient.
Required the quotient of 6 -^ 2.
6 nl .7782
2 nl .3010
6-^2 nl .4771 In 3
Required the quotient of 42 -^ 14.
42 nl 1.6232
14 nl 1.1461
42-^14 nl .4771 In 3
THE USE OF LOGARITHMS 47
Required the quotient of 100 -^- 4.
100 nl 2.0000
4 nl .6021
100-^4 fil 1.3979 In 25
§ 46. Powers by Logarithms
Rule 3 : The logarithm of the power of a number is equal
to the logarithm of the number multiplied by the exponent
of the power.
Let it be required to find the third power of 2, that is,
2', or the product of 2 X 2 X 2.
2 nl .3010
2' w/ 3 X .3010
or, .9030 In 8
Required the fourth power of 5, that is, 5*.
5 nl .6990
5' nl 4 X. 6990
or, 2.7960 In 625
It has been observed in this connection (§ 38), that the
second power is usually called the square, and the third
power the cube.
§ 47. Roots by Logarithms
The fourth general rule regarding logarithms refers to
the extraction of roots. We have seen in §38 that, if a
certain number is a power of another, we call the latter
number a root of the former. For example, since 2 X 2 X
2 X 2 X 2 = 32, it is said that 32 is the 5th power of 2, and
that the 5th root of 32 is 2. The usual way of expressing
this latter fact is :
V32 = 2
or, 32^ = 2
48 THE MATHEMATICS OF INVESTMENT
With the above explanation, the fourth rule is now
stated, and it will be observed that it is the converse of the
third rule.
Rule 4 : The logarithm of the root of a number is equal
to the logarithm of the number divided by the index of the
root.
As an illustration, let it be required to find the square
root of 49.
49 • nl 1.6902
V49, or 49'^ nl 1/2 of 1.6902
or, .8451 In 7
Required the cube root of 512.
512 nl 2.Y093
V512, or 512^ nl Ys of 2.7093
or, .9031 In 8
§ 48. Fractional Exponents
Such an exponent as % requires explanation. It signi-
fies the third power of the fourth root, or the fourth root
of the third power. Thus, the value of 10^* may be ascer-
tained by finding the fourth root of 10, and then getting the
cube of this root; or by finding the cube of 10, which is
1,000, and then taking the fourth root (or the square root
of the square root) of 1,000. By the methods of arithmetic,
the value of lO''^ is thus found to be 5.62+; or, in other
words, the logarithm of 5.62 is approximately .75. It is
interesting to compare this result with the table in § 43,
where it is indicated that the logarithm of 5.62 is .7497,
which is very close to .7500. Fractional exponents may be
expressed as decimal, instead of common, fractions ; and, in
fact, that is what most logarithms are : simply fractional
exponents of 10, expressed decimally.
THE USE OF LOGARITHMS
49
§ 49. Use of Logarithms in Computing Compound Interest
To demonstrate the use of logarithms in compound in-
terest, let us take an example and work it out, illustrating
each step. We will take 3% as the rate, the same as already
used (§§25-30), but endeavor to find the amount for 50
periods, instead of for 4 periods.
The ratio of increase is 1.03. Looking for the logarithm
(to eight decimal places) of this ratio (Chambers' or Bab-
bage's tables, page 192) we find this line :
No. 0123456789
10300 0128 3722 4144 4566 4987 5409 5831 6252 6674 7096 7517
The meaning of this line is that the logarithms are as
follows :
1.03
nl
.01283722
1.03001
nl
.01284144
1.03002
nl
.01284566
1.03003
nl
.01284987,
1.03009 nl .01287517
The first figures of both the numbers and the logarithms
are given only once in the table, which saves space in print-
ing and time in searching.
Since 1.03 is to be taken as a factor 50 times, we must
multiply its logarithm by 50, as stated in Rule 3 (§46).
This gives :
50 X.01283722 = .6418610
The result is the logarithm of the answer; for, when we
have found the corresponding number, we shall know the
value of 1.03^^
We must now look in the right-hand columns for the
logarithm figures .6418610. We first look for the 641,
which stands out by itself, overhanging a blank space
so
THE MATHEMATICS OF INVESTMENT
(Chambers' or Babbage's tables, page 73), and we find that
the nearest approach to .6418610 is .6418606, which latter
is indicated as the logarithm of the number 4.3839. The
next nearest logarithm is .6418705, which corresponds to
the number 4.3840. The following tabulation shows the
details more clearly :
Corresponding
Logarithm Number
.6418606 4.3839
.6418610 To be determined
.6418705 4.3840
It is evident that the number to be determined lies be-
tween 4.3839 and 4.3840, which differ by .0001. The dif-
ference between the first and third logarithms is .0000099,
and between the first and second logarithms is .0000004.
For practical purposes, we take 4/99 of the difference
between the numbers (.0001), and add this amount to
the smaller number, thus obtaining the required number
4.383904. In order to assist in determining the decimal
value of 4/99 and similar fractions, little difference-tables
are usually given in the margins of the pages of logarithm
tables, the table for 99 reading as follows:
99
1
10
2
20
3
30
4
40
5
50
6
59
7
69
8
79
9
89
The meaning of this table is that 40/99 = .40; 4/99 =
THE USE OF LOGARITHMS 5I
.04; 8/99 = .079 ; etc. By the use of these small tables, the
labor of dividing is thus avoided.
§ 50. Accuracy of Logarithmic Results
The amount of $1.00 compounded for 50 periods at 3%
is seen to be $4.383904. The result is slightly inaccurate
in the last figure, for the reason that two decimal places were
lost by multiplying. Had we taken the ten-figure logarithm
on page XVIII of Chambers' tables 0128372247
this multiplied by 50 would give 641861235
or, rounded off at the 7th place 6418612
which gives the more accurate result 4.383906
§ 51. Logarithms to Fifteen Places
Since it is necessary, for problems involving many
periods, to use a very extended logarithm, there is given in
Part III of the present volume, tables of fifteen-place loga-
rithms for a number of different ratios of increase (1 +i).
These are at much closer intervals than any table previously
published, and, with a ten-figure book of logarithms, will
give exact results to the nearest cent on $1,000,000.00.
§ 52. Use of Logarithms in Present Worth Calculations
We will further exemplify the advantage of the loga-
rithmic method by solving a present worth problem. Let
it be required to find the present worth of $1.00 due in 50
periods, compounded at 3% per period. Multiplying the
logarithm of 1.03 by 50, just as in § 50, we obtain
.641861235. But it is the reciprocal of 1.03'^ or 1 -^
1.03^^ which we wish to obtain; hence we must subtract
.641861235 from the logarithm of 1, which is 0.
0.000000000
0.641861235
Remainder, 1.358138765
52 THE MATHEMATICS OF INVESTMENT
In subtracting a greater from a less logarithm, we get a
negative whole number (as shown by the minus above), the
decimal part being positive and obtained by ordinary
subtraction.
Neglecting the 1, for the moment, we search in the right-
hand column for .358138765, and find that .3581253 is the
logarithm of 2.2810; and proceeding as in §50, we find
that .3581388 is the logarithm of 2.281071. The decimal
point, however, must be moved one place to the left, as
directed by the characteristic 1; thus giving as the final
result, .2281071.
By means of multiplication, we may check the results
shown in this and the foregoing sections.
By §50, 1.03'Ms 4.383906
As above, 1 -^ 1.03'" is. . . .2281071
Since these two results are reciprocals, their product should
equal unity, or 1. The result of the multiplication is
1.0000000843326, which verifies the accuracy of the
previous computations.
CHAPTER IV
AMOUNT OF AN ANNUITY
§ 53. Evaluation of a Series of Payments
We have now investigated the two fundamental prob-
lems in compound interest, viz. : to find the amount of a
present worth, and to find the present worth of an amount.
The next question is a more complex one: to find the
amount and the present worth of a series of payments. If
these payments are irregular as to period, value, and rate
of interest, the only way of finding the amount or the present
worth of the series is to make as many separate computa-
tions as there are payments, and then find the sum of the
results obtained. But, if the payments, periods, and rates
of interest are uniform, we can devise a method for finding
by one operation the amount or present worth of the
whole series.
§ 54. Annuities
A series of payments of like amounts, made at regular
periods, is called an annuity ; the period does not necessarily
need to be a year, but may be a half-year, a quarter, or any
other length of time. Thus, if an agreement is made pro-
viding for the following payments :
September 9, 1914 $100.00
March 9, 1915 100.00
September 9, 1915 100.00
March 9, 1916 100.00
53
54 THE MATHEMATICS OF INVESTMENT
there would be an annuity of $200.00 per annum, payable
semi-annually; or, in other words, an annuity of $100.00 for
each half-year period, terminating after four periods. As-
-suming the rate of interest to be 6% per annum, payable
semi-annually (3% per period), let us suppose that it is
required to find the total amount to which the annuity will
have accumulated on March 9, 1916, and the present worth,
on March 9, 1914, of this series of future payments. It is
evident that the answer to the first question will be greater
than $400.00, and that the answer to the second question, as
shown in the next chapter, will be less than $400.00.
§ 55. Amount of Annuity
It is easy, in this case, to find the separate amounts of
the payments, since the number of terms is very small and
since we may avail ourselves of the computations in § 30.
A schedule could be made as follows :
Date of Payment Amount at March 9, 1916
March 9, 1916 $100.00
September 9, 1915 103.00
March 9, 1915 106.09
September 9, 1914 109.2727
Total, $418.3627
§ 56. Calculation of Annuity Amounts
If, however, there were 50 terms instead of 4, the work
of computing these 50 separate amounts, by the use of loga-
rithms, or by the shorter process (in this case) of simple
multiplication, would be very tedious. To shorten the process
let us make up three columns of amounts for four periods,
the first being amounts of $1.00, the second being amounts
of $1.03, and the third being amounts of $.03. The figures
in the second column will accordingly be 1.03 times the
AMOUNT OF AN ANNUITY
55
corresponding figures in the first column, while the figures
in the third column will be the difference between the corre-
sponding figures in the first two columns.
(1>
Amounts
of $1.00
(2)
Amounts
of $1.03
(3)
Amounts
of $.03
$1.00
1.03
1.0609
1.092727
Total, $4.183627
$1.03
1.0609
1.092727
1.12550881
$4.30913581
$.03
.0309
.031827
.03278181
$.12550881
§ 57. Formation of Tables
We may take the diflFerence between the totals of
columns (1) and (2) without actually finding these totals.
It will be observed that the first three items in column (2)
are the same as the last three items of column (1). The
difference between the totals of the two columns, therefore,
is the same as the difference between the last item of (2) and
the first item of (1); that is, $1.12550881 less $1.00, or
$.12550881. This latter figure equals the total of column
(3).
§ 58. Use of Tables
It is evident that an annuity of three cents will amount,
under the conditions assumed, to twelve cents and the
decimal .550881. Accordingly, an annuity of one cent will
amount to one-third of $.12550881, or $.04183627. An
annuity of $1.00 will amount to 100 times as much, or
$4.183627, while an annuity of $100.00 will amount to
$418.3627, which agrees exactly with the result obtained by
addition, in § 55.
56 THE MATHEMATICS OF INVESTMENT
§ 59. Compound Interest as a Base for Annuity Calculations
The amount $.12550881 (obtained by subtracting $1.00
from $1.12550881) is the compound interest on $1.00 for
the given rate and time, and the amount $.03 is the single
interest. The compound interest on $1.00, compounded
semi-annually at 6%, up to any time corresponds with the
amount of an annuity of three cents, payable on exactly
the same plan. The amount of the annuity of $1.00 is
$.12550881-^.03, or $4.183627; and from this we formu-
late the rule given in the following section.
§ 60. Rule and Formula for Finding Amount
To find the amount of an annuity of $1.00 for a given
time and at a given rate, divide the compound interest for
the total number of periods, by the single interest for one
period, both expressed decimally.
To express the rule in a formula, let A represent the
amount, not of a single $1.00, but of an annuity of $1.00;
then A = I -T- i.
§ 61. Operation of Rule
To illustrate, let us take the case worked out in § 50,
where we found the amount of a single dollar at 3%, for 50
periods, to be. $4.383906
Subtracting one dollar 1.000000
The compound interest is , $3.383906
Divide this by .03 and we have $112.79687
which is the amount to which 50 payments of $1.00 each,
at 3% per period, would accumulate.
CHAPTER V
PRESENT WORTH OF AN ANNUITY
§ 62. Method of Calculation
To find the present worth of an annuity, we can, of
course, find the present worth of each payment, and then,
by addition, find the total present worth of all the payments ;
but it will save much labor if we compute the total in one
operation, as we computed the amount, and a similar course
of reasoning will lead to the desired result.
§ 63. Tables of Present Worth
In the second column of the following table is shown
the present worth of $1.00 for 4, 3, 2 and 1 period, respec-
tively, at 3% per period ; and in the third and fourth columns
are shown similar values of $1.03 and $.03, respectively.
(1)
Number
of
Periods
(2)
Present
Worths
of $1.00
(3)
Present
Worths
of $1.03
(4)
Present
Worths
of $.03
4
3
2
1
$.888487
.915142
.942596
.970874
$3.717099
$.915142
.942596
.970874
1.000000
$.026655
.027454
.028278
.029126
Total,
$3.828612
$.111513
57
58 THE MATHEMATICS OF INVESTMENT
§ 64. Short Method for Finding Present Worth of an
Annuity
Since the last three items in column (2) are the same
as the first three items in column (3), it is evident that, in
order to obtain the difference between the totals of columns
(2) and (3), it is not necessary to make the actual additions
of these columns, but merely to find the difference between
the items not found in both columns. These items are only
two, viz., $.888487 in the second column, and $1.000000 in
the third column. Their difference is $.111513, which agrees
with the total found by the addition of column (4).
§ 65. Present Worth Obtained
The difference between the $.888487 of the second
column and $1.000000 of the third column, amounting to
$.111513, is the compound discount of $1.00 for four
periods at 3%. When this difference is divided by the single
interest (.03), we obtain $3.71710, which is the same result
(rounded up) as that obtained by adding column (2). From
this observation, we construct the rule given in the following
section :
§ 66. Rule for Present Worth
To find the present worth of an annuity of $1.00 for a
given time at a given rate, divide the compound discount for
that time and rate by the single interest for one period, both
expressed decimally.
§ 67. Formulas for Present Worth
In symbols, the rule may be expressed, P = D -^ /. Since,
by § 35, D = I -^ a, we obtain F = l-^a-^i, or P = I-^
i -f- a. And since, by § 60, A = I -^- i, there comes the re-
sulting symbolic rule, P = A -^- a, the latter part of this
equation signifying the present worth of the amount of the
PRESENT WORTH OF AN ANNUITY 59
annuity. Summarizing, therefore, we have the two symbolic
rules :
P = A-f-a
§ 68. Analysis of Annuity Payments
It may assist in acquiring a clear idea of the working of
an annuity, if an analysis is given of a series of annuity pay-
ments from the point of view of the purchaser. For this
purpose we will suppose that a person investing $3.7171
at 3%, in an annuity of $1.00 per period, payable at the
end of each period, expects to receive at each payment, be-
sides 3% on his principal to date, a portion of that principal,
and thus to have his entire principal gradually repaid.
His original principal is , $3.7171
At the end of the first period, he receives :
3% on $3.7171 $.1115
Payment on principal 8886 .8885
Total $1.0000
Leaving new principal (which is equiva-
lent to the present worth at three periods) $2.8286
At the end of the second period, he receives :
3% on $2.8286 $.0849
Payment on principal ,..,.. .9151 .9151
Total $1.0000
Leaving new principal $1.9135
At the end of the third period, he receives :
3% on $1.9135 $.0574
Payment on principal 9426 .9426
Total $1.0000
Leaving new principal $.9709
6o THE MATHEMATICS OF INVESTMENT
At the end of the last period, he receives :
3% on $.9709 $.0291
Payment on principal in full 9709 .9709
Total $1.0000
In the above manner we find that the annuitant has re-
ceived interest in full on the principal outstanding, and has
also received the entire original principal. The correctness
of the basis on which we have been working is thus
corroborated.
§ 69. Components of Annuity Instalments
It is usual to form a schedule showing the components
of each instalment in tabular form :
Date
Total
Payment
Payments
of
Interest
Payments
on
Principal
Principal
Out-
standing
March 9, 1914.
September 9, 1914.
March 9, 1915.
September 9, 1915.
March 9, 1916.
$1.00
1.00
1.00
1.00
$4.00
$.1115
.0849
.0574
.0291
$.2829
$.8885
.9151
.9426
.9709
$3.7171
$3.7171
2.8286
1.9135
0.9709
0.0000
Had the purchaser reinvested each instalment at 3%, he
would have, at the end, $4.1836 (§55), which is equivalent
to his original investment compounded ($3.7171 X 1.1255 =
$4.1836).
§ 70. Amortization
The payments on principal are known as amortization,
which may be defined as the gradual repayment of a principal
sum through the resultant operation of two opposing forces
— ^periodical payments and compound interest. The effect
PRESENT WORTH OF AN ANNUITY 6l
of the periodical payments is to reduce the principal sum,
while the effect of the compound interest is to increase it.
In ordinary compound interest, each new principal is greater
than the preceding principal ; while in the case of amortiza-
tion, each principal is less than the preceding one.
§ 71. Amortization and Present Worth
It will be noticed, from § 69, that each payment on
principal, or amortization, for one period, is the present
worth of the instalment at the beginning of its period. For
example, at the end of the first period, September 9, 1914, a
payment on principal is made amounting to $.8885, which
is the present worth of the instalment paid on that date
($1.00) for four periods at 3%. From this fact, it follows
that, if we know the amount of the instalment, the rate, and
the number of remaining periods, we can calculate the
amortization included in the instalment.
§ 72. Development of a Series of Amortizations
It will also be noticed that each amortization multiplied
by 1.03 becomes the next following, these being a series of
present worths ; and that thus they may be derived from one
another, upwards or downwards.
§ 73. Evaluation by Logarithms
In § 52, by the use of logarithms, we found the present
worth of $1.00 for 50 periods, at 3%, to be. . $.2281071
Subtracting this from 1.0000000
we have the compound discount ,. ., , $.7718929
Dividing this by .03, we have $25.72976 +
which is the present worth of an annuity of $1.00 for 50
periods, at 3%. Thus we see that the process of finding the
present worth of an annuity, or, as it is termed, evaluation, is
rendered easy — no matter how long the time — by using
logarithms.
CHAPTER VI
SPECIAL FORMS OF ANNUITIES
§ 74. Ordinary or Immediate Annuities
The annuities heretofore spoken of are payable at the
end of each period, and are the kind most frequently occur-
ring. To distinguish them from other varieties, they are
spoken of as ordinary or immediate annuities.
§ 75. Annuities Due
When the instalments of an annuity are payable at the
beginning of their respective periods, the annuity is called
an annuity due, although prepaid would seem more natural.
It is evident that this is merely a question of dating. The
instalments compared with those in § 56 are as follows :
Immediate
Annuity
Immediate
Annuity
4 Periods
Due
4 Periods
Annuity
5 Periods
r
$1.00
$1.03
$1.00
Amounts of
1.03
1.0609
1.0609
1.0927
1.03
1.0609
$1.00
1.0927
1.1255
1.0927
1.1255
$5.3091
— 1.0000
$4.1836
$4.3091
$4.3091
62
SPECIAL FORMS OF ANNUITIES
63
Hence, to find the amount of an annuity due, for any
number of periods, say t periods, find the amount of an
immediate annuity for ^ + 1 periods, and subtract one
instalment.
§ 76. Present Worth of Annuities Due
In regard to present worths, the instalments compared
with those in § 63 would be as follows :
Immediate
Annuity
4 Periods
Annuity-
Due
4 Periods
Immediate
Annuity
3 Periods
Present
Worths of -
$1.00
$.888487
.915142
.942596
.970874
$.915142
.942596
.970874
1.000000
$.915142
.942596
.970874
$2.828612
+ 1.000000
$3.717099
$3.828612
$3.828612
Therefore, to find the present worth of an annuity dtw
for / periods, find the present worth of an immediate annuity
for ; — 1 periods, and add one instalment.
§ 77. Present Worth of Deferred Annuities
A deferred annuity is one which does not commence to
run immediately, but only after a certain number of periods
have elapsed. Thus, an annuity of 5 terms, 4 terms deferred,
would commence at the beginning of the fifth period, and
continue to the end of the ninth period.
If there were nine terms in the annuity, none being de-
ferred, and if the ratio of increase were assumed to be r and
64 THE MATHEMATICS OF INVESTMENT
the present worth of the first term were assumed to be
unity, the present worth of the annuity for nine terms
would be :
l+7 + ^ + ;;:i + ;:5 + J + ^ + J + ^(§§ 18, 66)
The present worth of the annuity for four terms would be :
The present worth of the annuity for the five deferred terms
would, of course, be the difference between the above two
sums, or:
i+l+i.i.i
^4 ~ ^5 ~ ^6 I ^7 I ^8
§ 78. Rule for Finding Present Worth of Deferred Annuity
From the foregoing, we derive the rule: To find the
present worth of an annuity for m terms, deferred for n
terms, subtract the present worth of an annuity for n terms
from the present worth of an annuity for m-\- n terms.
§ 79. Present Worth of Perpetuities
A perpetual annuity, or a perpetuity, is one which never
terminates. Its amount is infinity, but its present worth can
be calculated at any given rate of interest. If each instalment
of an annuity is $1.00* and the rate 5%, the value of the
annuity is such a sum as will produce $1.00 at that rate.
This sum is $20.00, being $1.00 -f- 6%. The compound dis-
count is the entire $1.00, being for an infinite number of
terms. Therefore, the rule of § 66 still holds true : divide
the compound discount by the single rate of interest, in order
to find the present worth of the annuity.
SPECIAL FORMS OF ANNUITIES 65
§ 80. Perpetuity in Stock Purchased for Investment
A share of stock may be treated in the same manner as a
perpetuity, provided its dividend is assumed to continue at
a fixed rate. If the dividend is $4.00 per share, and if it is
desired to purchase at such a basis as to yield 6% on the
investment, the price per share should be $4.00-^-6%, which
equals $66.67. This price is irrespective of the nominal or
par value of the stock. Both in perpetuities and in shares of
stock, the price = c-^i.
§81. When Annuity Periods and Interest Periods Differ
In all of these examples of annuities, it has been assumed
that the term or interval between payments is the same
length of time as the interest period. It frequently happens,
however, that the rate of interest is stated to be so much per
year, while the payments are half-yearly or quarterly; or
there may be yearly payments, while the desired interest
rate is to be on a half-yearly basis. We shall defer the treat-
ment of these latter cases until the subject of nominal and
effective rates of interest has been discussed.
§ 82. Varying Annuities
There may also be varying annuities, where the instal-
ment changes by some uniform law. These seldom occur
in practice. Where the change is simple, as in arithmetical
progression, the total annuity may be regarded as the sum
of several partial annuities; otherwise the values must be
separately calculated for each term. An annuity running
for five terms, as follows : 13, 18, 23, 28, 33, may be re-
garded as the sum of the following :
(1) an annuity of 13 for 6 terms;
(2) an annuity of 5 for 4 terms;
66 THE MATHEMATICS OF INVESTMENT
(3) an annuity of 5 for 3 terms;
(4) an annuity of 5 for 2 terms; and
;(5) an annuity of 5 for 1 term.
In actual practice, in a case of this kind, in order to find
the amount or the present worth of the annuity, it would
probably be easiest to find the amount or the present worth
of each term, and then find the total of these separate items.
CHAPTER VII
RENT OF ANNUITY AND SINKING FUND
§ 83. Rent of Annuity
The number of dollars in each separate payment of an
annuity is called the rent of the annuity.
In § 63, we saw that $3.Y171 is the present worth, at
3%, of an annuity composed of 4 payments of $1.00 each.
We may reverse this and say that $1.00 is the rent of
$3.7171 invested in an annuity of 4 payments at 3%. What,
then, is the rent to be obtained by investing $1.00 in the
same way? Since the present worth has been reduced in
the ratio of 3.7171 to 1, evidently the rent must be reduced
in the same ratio, that is, 1 -^ 3.7171. By ordinary division
or by logarithms, this quotient is .26903. Therefore,
$.26903 is the rent of an annuity of 4 terms at 3%, for
every $1.00 invested; or $1.00 is the present worth at 3%
of an annuity for 4 years of $.26903. This may be illus-
trated by making up a schedule :
Rent
Interest
Reduction
or Amorti-
zation
Value
Beginning of first period.
End of first period
End of second period
End of third period
End of fourth period
$ .26903
.26903
.26903
.26903
$.03
.02283
.01544
.00785
$ .23903
.24620
.25359
.26118
$1.00000
.76097
.51477
.26118
0.
$1.07612
$.07612
$1.00000
67
(^ THE MATHEMATICS OF INVESTMENT
§ 84. Rule for Finding Rent of Annuity
To find the rent of an annuity valued at $1.00, divide
$1.00 by the present worth of an annuity of $1.00 for the
given rate and time. Rent = 1 -> P ; and since, by § 67,
P = D ^ i^ and F = A-7- a, we obtain two other symbolic
rules :
Rent = i-^D
Rent = a -^- A
§ 85. Alternative Method of Finding Rent
An alternative method of determining the value of the
rent of an annuity is to form a proportion, as in arithmetic,
and then solve the proportion. For example :
Rent of Annuity Present Worth of Annuity
$1.00 : X :: $3.7171 : $1.00
In other words, if a rent of $1.00 produces a present worth
of $3.7171, then what quantity of rent will produce a present
worth of $1.00? Multiplying the two extremes together,
and dividing the product by the mean, we find the other
mean to be $.26903, which is the rent required.
§ 86. Rent of Deferred Payments
The problem of finding the rent of an annuity may be
regarded as equivalent to another problem — that of finding
how much per period for n periods, at the rate i, can be
bought for $1.00. A borrower may agree to pay back a loan
in instalments, each of which comprises both principal and
interest. Suppose that a loan of $1,000 were made under
the agreement that such a uniform sum should be paid
annually as would pay off (amortize) the entire debt with
3% interest in 4 years. The present worth is, of course,
$1,000, and by the above process each instalment or con-
tribution would be $269.03. In countries imposing an in-
RENT OF ANNUITY AND SINKING FUND
69
come tax, it is usual to incorporate in agreements of this
nature a schedule showing what part of the instalment is
interest — since that alone is taxable — somewhat as follows :
Annual
Instalment
[nterest on
Balance
Payment on
Principal
Principal
Outstanding
January 1, 1914
December 31, 1914
December 31, 1915
December 31, 1916
December 31, 1917
$269.03
269.03
269.03
269.03
$30.00
22.83
15.44
7.85
$239.03
246.20
253.59
261.18
$1,000.00
760.97
514.77
261.18
0.
$1,076.12
$76.12
$1,000.00
§ 87. Annuities as Sinking Funds
One other question arises with regard to annuities, and
that is in the cnse of an annuity so constructed as to accumu-
late to a certain amount at a certain time. The amount to
be accumulated is called a sinking fund. Frequently the
uniform periodical contribution is itself called the sinking
fund, but, more strictly speaking, it should be called the
sinking fund contribution.
In the case exhibited in the schedule of § 86, the debt
was amortized, with the assent of the creditor, by gradual
payments. Let us suppose, however, that the creditor pre-
fers to wait until the day of maturity, and receive his $1,000
all at one time, instead of by partial payments. The debtor
must pay interest amounting to $30.00 each year, but, in
addition to this, in order to provide for the principal on a
sinking fund plan, he must transfer from his general assets
to a special account (or into the hands of a trustee) such
an annual sum as will accumulate, in 4 years at 3%, to
$1,000. Since $1.00, set aside annually, amounts, after 4
70
THE MATHEMATICS OF INVESTMENT
years on a 3% basis, to $4.183627 (§56), to find what
sum will similarly amount to $1,000, we must divide 1,000
by 4.183627. In this manner the sinking fund contribution
is found to be $239.03.
§ 88. Rule for Finding Sinking Fund Contributions
To find what annuity will amount to $1.00, or what
should be each sinking fund contribution to provide for
$1.00: divide $1.00 by the amount of an annuity of $1.00
for the given rate and time. In symbols, sinking fund con-
tribution, or S. F. C, = 1 -^ A ; or (since A = I ^i, per
§ 60) it also equals i -h I.
Put in the form of a proportion, the question of § 87
would appear as follows :
Sinking Fund Contribution
$1.00 : X ::
Sinking Fund
$4.183627 : $1,000.00
The unknown quantity, x, would be the same as before,
$239.03.
§ 89. Verification Schedule
The correctness of the result found in § 87 may be proved
by a schedule constructed in the following manner :
Annual Sink-
ing Fund
Contribution
Interest Dur-
ing Preced-
ing Year
Total
Addition
to Sinking
Fund
TotalAmount
Accumulated
in Sinking
Fund
January 1, 1914
December 31, 1914
December 31, 1915
December 31, 1910
December 31, 1917
$239.03
239.03
239.03
239.03
$ 7.17
14.56
22.15
$239.03
246.20
253.59
261.18
$ 0.
239.03
485.23
738.82
1,000.00
$056.12
$43.88
$1,000.00
RENT OF ANNUITY AND SINKING FUND
71
§ 90. Amortization and Sinking Fund
On comparing the schedules in §§ 86 and 89, we find
that the annual instalments or contributions are respectively
$269.03 and $239.03, the difference of which is $30.00, or
exactly the yearly interest on the original loan of $1,000.00.
Hence, the amount paid in the second case, if interest be
included, is just the same as in the first case. Gradual pay-
ments on account of a debt, or gradual accumulations hav-
ing in view one single final payment in full, therefore amount
to the same thing.
As a provision for liquidating indebtedness, or for re-
placing vanishing assets, sinking fund and amortization are
two different applications of the same principle. Formerly,
the terms were used interchangeably, but more recently they
are distinguished as follows :
(1) The sinking fund method permits the debt to stand
until maturity, but in the meantime accumulates a fund
which at maturity pays off the entire debt, the interest on
the original sum being paid separately.
(2) The amortization method accumulates nothing, but
gradually reduces the debt, the amount of the reduction
being the excess of the periodical payment over the
periodical interest.
CHAPTER VIII
NOMINAL AND EFFECTIVE RATES
§ 91. Explanation of Terms
In the previous chapters, all of our computations re-
garding interest have been based upon a certain number of
periods and upon a certain rate per period. In the business
world, it is usual to speak of interest rates as so much per
annum. In the vast majority of instances, however, the in-
terest, although it is either designated or understood to be
per annum, is, nevertheless, not paid by the year (that is,
once a year), but in semi-annual or quarterly instalments.
Where the interest is payable otherwise than annually, the
rate per annum is only nominally correct. For example, if
on May 1, 1914, we lend $1,000.00 at 6%, interest to be paid
semi-annually, the interest account for the year would be
as follows :
November 1, 1914, Interest earned. . ., ,. . $30.00
May 1, 1915, Interest earned:
On original loan , 30.00
On the $30.00 received on November 1, 1914,
for 6 months at an assumed rate of 6% ... . .90
Total $60.90
The total interest earnings during the year, therefore,
would be $60.90, which is at the effective rate of 6.09%
on the original investment, as compared with a nominal
rate of 6%.
72
fjj
NOMINAL AND EFFECTIVE RATES
73
§ 92. Semi-Annual and Quarterly Conversions
In the example given in the preceding section, the in-
terest is payable (or, as it is frequently called, convertible)
semi-annually. The true or effective rate for each half-
yearly period is therefore 3%, and the ratio of increase is
1.03. The amount at the end of the year would be the
square of 1.03, or 1.0609, thus giving 6.09% as the effective
rate per annum. In the case of quarterly conversion, the
amount at the end of the year v^ould be the fourth power of
1.015, or 1.061364, giving 6.1364=% as the effective annual
rate. The following table shows the effective annual rates
for various periods of conversion, the nominal annual rate
being 6% :
Period of ^^ . . , r> .
^ . Effective Annual Rate
Conversion
Yearly 1.06 — 1 or 6.0000%
Semi-annually 1.03' — 1 or 6.0900%
Quarterly 1.015* — 1 or 6.1364%
Monthly , 1.005'^ — 1 or 6.1678%?
Y 365/
Daily f 1+-^ I —1 or 6.1826%
§ 93. Limit of Effective Annual Rate
It will be seen that the effective rate increases as the
conversions become more frequent. There is a limit, how-
ever, beyond which this acceleration will not go. If an in-
vestment on a 6% nominal annual rate is compounded every
minute, or every second, or every millionth of a second, or
constantly, the effective annual rate could never be so great
as 6.184%.*
'See § 238.
74
THE MATHEMATICS OF INVESTMENT
§ 94. Rule for Effective Rate
From observation of the table shown in § 92, we may
deduce the following symbolic rule for finding the effective
rate, m representing the number of payments per annum,
and y the effective rate :
=(i+i)'"-
§ 95. Logarithmic Process
In order to exemplify logarithmic processes in working
out the foregoing rule, let it be required to find the effective
rate of interest when the nominal rate is 6% per annum,
compounded daily. The rule in § 94 then becomes :
By the use of logarithms, we obtain :
log. .06 =2.7Y81513
log. 365 =2.5622929
Hence, log. (.06 -^ 365) =4.2158584
4.2158584 is, we find, the logarithm of .0001643835; and,
therefore, the value found thus far is :
y=(l + .0001643835) ^^^ — 1
or, ; = 1.0001643835'«' — 1
The logarithm of 1.0001643835 is .00007138 ; and 365 times
this latter figure is .02605370, which we find to be the loga-
rithm of 1.061826. The value for the effective rate then
becomes :
y = 1.061826 — 1
or, y= .061826, or 6.1826%
CHAPTER IX
BONDS AND THE PROPER BASIS OF BOND
ACCOUNTS
§ 96. Provisions of Bonds
The most common forms of interest-bearing securities
are bonds. Every bond contains a complex promise to pay :
(1) A certain sum of money at a stipulated future time,
this sum being known as the principal, or par.
(2) Certain smaller sums, proportionate to the principal,
and payable at various earlier times than the
principal.
These smaller sums are usually known as the interest
payments, but, as they do not necessarily correspond to the
true rate of interest, it will be better to speak of them as the
coupons.
Bonds also contain provisions as to the time, place, and
manner of these payments, and usually refer, also, to the
mortgage, if any, made to insure their fulfillment, and to the
law, if any, authorizing the issue.
§97. Interest on Bonds
The rate of interest named in a bond is usually an integer
per cent, or midway between two integers: as, 2%, 2%%,
3%, 31/2%, 4%, 41/2%, 5%, 6%, 7%, etc. Occasionally
such odd rates occur as 3%%, 3.60%, 3.65%, 33/4%, but
these are unusual and inconvenient. Most bonds provide for
semi-annual payments of interest ; a considerable number of
75
^6 THE MATHEMATICS OF INVESTMENT
issues, however, pay interest quarterly, and a very few
annually. With- most bonds, the interest is payable on the
first day of the month. In the case of a very few bonds
the interest falls due on- the 15th or on the last day of the
month. In some respects it would be better if bond interest
were payable on the last day of a calendar month, instead
of on the first day of the succeeding month, since the entire
transaction (including* the payment of cash for the accrued
interest) would thus be brought inside of a calendar period.
The item of "Interest Accrued" on monthly balance sheets
would in this manner frequently be eliminated, or at least
substantially reduced.
§ 98. How Bonds Are Designated
Bonds are usually designated according to the obligor,
the rate of interest, the date of maturity, and sometimes the
initials of the months when interest is payable. Thus, "Man-
hattan 4's of 1990, J J" indicates the bonds of the Manhat-
tan Railway Company, bearing 4% interest per annum, the
principal being due in 1990, and the interest coupons being
payable semi-annually in January and July.
§ gg. Relation of Cost to Net Income
Bonds are seldom bought or sold at their exact par
value, and this fact has an effect on the rate of net income
derived from the original investment. If the amount in-
vested is greater than the par value, the difference is known
as the premium. This premium is not repaid at maturity,
as is the par value or principal of the bond, and hence must
be provided for out of the various interest payments. Thus,
a. bond purchased above par produces a lower rate of in-
come than the rate of interest represented by the coupons.
Conversely, if the purchase is below par, the investor will,
at maturity, receive not only the amount of his original in-
BONDS AND BOND ACCOUNTS 77
vestment, but also the difference between this amount and
the par value of the bond. This difference, technically
known as the discount, has the effect of making the rate of
income higher than the rate of interest shown by the
coupons.
§ 100. Coupon and Effective Rate of Interest on Bonds
The following are some of the expressions used to de-
note an investment made above par : "6% bond to net 5%";
"6% bond on 5% basis"; "6% bond yielding 5%"; "6%
bond paying 5%"; etc. In the cases of bonds bought below
par, the income rate would be larger than the coupon rate,
as, for example, "3% bond to net 4%," etc. In all of the
above instances, the percentage immediately preceding the
word "bond" signifies the coupon rate of interest, while the
other percentage signifies the true or effective rate of
interest.
§ loi. Present Worth of Bonds
It will be seen, therefore, that the sale of a bond involves
the transfer of the right to receive, at the stipulated times,
both the principal and the periodical amounts of interest.
None of these various sums is ever worth its face value, or
par, until the arrival of its stipulated date of payment. The
principal is never worth its face value until its maturity,
and the coupons are never worth their face values until
their respective maturities. Yet, while both principal and
coupons are always at a discount, except at their respective
dates of maturity, the aggregate value or present worth of
the principal and coupons at any one time prior to maturity
is frequently more than the par value of the principal alone
(as in the case of a bond bought at a premium) ; and it is
this aggregate present worth of both principal and coupons
which is always the question at issue in connection with the
purchases and sales of bonds.
y8 THE MATHEMATICS OF INVESTMENT
§ 102. Considerations in the Purchase of Bonds
In fixing the price which he is wiUing to pay, the pur-
chaser is guided by several considerations, among them the
following :
(1) The amount of the principal.
(2) The date of maturity of the principal.
(3) The amount of each coupon.
(4) The number of coupons.
(5) The dates of maturity of the various coupons.
(6) The rate of interest which can be earned upon
securities of a similar grade.
This last point also involves a determination of the de-
gree of probability that the principal and the various coupons
will be promptly paid at their dates of maturity; or, in
other words, consideration must be given to the financial
reputation and integrity of the obligor.
§ 103. Present Worth and Earning Capacity of Bonds
In effect, the purchaser of a bond discounts, at a certain
fixed rate, the principal and each coupon at compound inter-
est, for the periods which they respectively have to run, and
the sum of these partial present worths is the value o'f the
bond. If he can buy at a price below this value, he will re-
ceive a higher rate of Interest than he anticipated. If he has
to pay more than this value, his rate of Interest will be lower.
As he cashes each coupon, he receives what he paid for it,
plus compound interest at the uniform rate; thenceforward
he earns interest on a diminished Investment as far as cou-
pons are concerned, but on an increased investment as to
principal. If the par value of his coupons Is less than the
total interest earned during the period, there Is an Increase
in the total Investment; if such par value is greater, then
there Is a surplus which operates to reduce the investment or
to amortize the premium.
BONDS AND BOND ACCOUNTS 79
§ 104. Cost and Par of Bonds
There are, therefore, two fixed points in the history of a
bond : the original cost, or money invested, and the principal,
or par — the money to be received at maturity. Between
these two points there is a gradual change : if bought below
par, the bond must rise to par; if bought above par, it must
sink to par. This gradual change is the resultant effect of
two opposing forces, the interest earned tending to increase
the investment value, while the payment of coupons reduces
the investment value. At any intermediate moment between
these two points there is an investment value which can be
calculated, and which is just as true as the original cost and
the par. In fact, these latter are merely special cases of in-
vestment value ; the investment value at the date of purchase
is cost, and at the date of maturity it is par.
§ 105. Intermediate Value of Bonds
The gradual change in investment value of bonds be-
tween purchase and maturity is ignored by some investors,
who, during the whole period, use either the original cost or
the par value. In the former case they suppose that the in-
vestment value remains at its original figure until the very
day of maturity, and is then instantly changed to par, either
by a loss of all of the premium or by a sudden gain of all of
the discount. Those who use par as the investment value
also assume that there is this sudden change of value, the
difference being that the change occurred at the instant of
purchase instead of at maturity. These methods of treat-
ment are manifestly fictitious and unreal, and are only re-
sorted to on account of the labor involved in computing in-
termediate values. Experience would tell us, if theory did
not, that there is no such violent change. The cost and the
par value, while entirely correct at the beginning and at the
end, respectively, of the period of ownership, are entirely
incorrect during the interim.
8o THE MATHEMATICS OF INVESTMENT
§ 1 06. True Investment Basis for Bonds
The true standard of investment value for bonds is the
present worth, at compound interest, of all recipiends, or
sums of cash to be received, whether such sums be called
coupons or principal. Neither the original cost of a bond
nor its ultimate par is a proper permanent investment basis.
The bond should enter into the accounts at cost, which is a
fact, and should go out of the accounts at par, which is an-
other fact. During the interim, the change from cost to par
should take place gradually by the processes of amortization
or accumulation, at the rate of the true interest on the
original investment.
§ 107. Various Bond Values
There are thus three values in the life of a bond which
resemble three tenses in grammar : The past tense represents
the cost, that is, the amount originally paid ; the future tense
represents the par, which is the amount ultimately to be re-
ceived; while the present tense represents the investment
value, intermediate between the values of the past and future,
except in the special case of a bond bought at par.
There is also a fourth value of a bond, that is, the
amount which might be obtained on sale at the present time.
This is the market value, and is a matter of judgment,
opinion, and inference. Although the market value of a
bond has great utility in some respects, it has no place,
strictly speaking, in accounts kept with regard to invest-
ments. It is not an act or a fact of the business; it is a
statement of what might be done. The market value con-
templates a possibility, or a probability — ^but never an
actuality, in so far as the accounts are concerned, unless a
sale is actually consummated. If an investor has had an
opportunity to make a sale of a bond, but has allowed it to
pass by, the mere fact that he has been offered such an op-
BONDS AND BOND ACCOUNTS 8l
portunity to sell has not the slightest effect on his financial
status.
§ 1 08. Investment Value the True Accounting Basis
Unless accounts with respect to bonds and similar securi-
ties are kept on the investment-value basis, an investor is
unable to tell whether a contemplated selling price will result
in a loss or a gain. If the books are kept on* the basis of
par, every sale above par will appear as a gain, even though
it may be a losing bargain; while a comparison with the
Driginal cost will be equally delusive and unsatisfactory.
CHAPTER X
VALUATION OF BONDS
§ 109. Cash Rate and Income Rate of Bonds
With respect to all bonds bought above or below par,
there are always two rates of interest involved: first, a
nominal or cash rate, which is a certain percentage of par,
and which is indicated by the coupons ; and second, an effec-
tive or income rate, which is a certain percentage of the
amount originally invested and remaining invested. For the
sake of greater clearness, we shall use the terms cash rate
and income rate, since they are more readily understood
than the terms nominal and effective. The symbols c and
i will respectively designate the cash rate and the income
rate. 1 + i is the ratio of increase as heretofore. The
symbol 1 + c will not be required, since c is not an accumula-
tive rate, but merely an annuity purchased with the bond,
the number of periods of the annuity being the same as the
number of coupons attached to the bond. The difference of
rates is c — i, or i — c.
§110. Elements of a Bond
In a bond purchased above or below par, we have, there-
fore, the following elements : the par, or principal, payable
after n periods ; an annuity of c per cent of par for n periods ;
and a ratio of increase, 1 + i. With these elements given,
there are two distinct methods for finding the value of the
entire security, and these must give the same result.
82
VALUATION OF BONDS 83
§111. Valuation of Bonds — First Method
As an illustration of this method, let us take the case
of a 7% bond, having 25 years (50 periods) to run, interest
payable semi-annually, the par being $1,000. Suppose that
it is required to compute the value of the bond at the be-
ginning of its first interest period. This present value is
composed of two parts : (a) the present worth of $1,000
due 50 periods hence; and (b) the present worth of an an-
nuity of $35 for 50 terms. We cannot ascertain the value
of these two parts until we know the income rate current
upon securities of a similar grade. Let us assume that this
income rate is 3% per period, or what is usually called a
6% basis. The ratio of increase is thus 1.03 per period.
§ 112. (a) Finding Present Worth of Principal
The first part of the solution is to find the present worth
of $1,000 due in 50 periods, at 3% per period. In § 52,
we have found the present worth of $1.00, under the same
conditions, to be $.2281071 ; hence the similar present worth
of $1,000 is $228.1071. This result, it will be noticed, has
not the slightest reference to the 7% rate of the bond. For
the purposes of the first part of the solution, the cash or
coupon rate is absolutely immaterial; the bond might be
equally well a 10% bond or a 0% bond, in the latter case
bearing no coupons at all.
§113. (b) Present Worth of Coupons
We next have to find the present value of an annuity of
$35 for 50 terms at 3%. In § 73, we found the present value
of a similar annuity of $1.00 to be $25.72976 +. An
annuity of $35, therefore, has a present value of $900.5417.
Hence, we have the following :
84
THE MATHEMATICS OF INVESTMENT
Present worth of the par $228.1071
Present worth of the coupons 900.5417
Present worth of the entire bond $1,128.6488
The ordinary tables, which give the values of a $100 bond
only, read $112.86, which is the same as the above, rounded
off. The above computation gives a result which is correct
to the nearest cent on $100,000, viz. : $112,864.88.
§114. Schedule of Evaluation
In order to present the subject still more clearly, in a
schedule form, let it be required to find the value, as at Jan-
uary 1, 1913, of a 7% bond for $1,000, interest payable
semi-annually, due at January 1, 1915, the income rate being
3% per period. In § 30, we have found that the present
worth of $1.00. for 1, 2, 3, and 4 periods is $.970874,
$.942596, $.915142, and $.888487, respectively. The re-
spective present worths of $35.00 are, therefore, $33.980590,
$32.990860, $32.029970, and $31.097045. The following
schedule may then be formed :
Periods from
Present
Items to be
Dates of
Jan. 1, 1013,
Worth
Evaluated
Maturity-
to Dates of
at Jan. 1,
Maturity
1913
Coupon, $35
35
35
35
July
January
July
January
1, 1913
1, 1914
1, 1914
1, 1915
1
2
3
4
$33.980590
32.990860
32.029970
31.097045
Total .
. . $130.098465'
Par, $1,000
January
rotal
1, 1915
4
888.487
Grand 1
.$1,018.585465
VALUATION OF BONDS 85
The total present value of the four coupons
($130.098465) could have been found by one operation, as
was done in the preceding section, and this is the usual
method of finding the present worth of an annuity. The
foregoing schedule, however, sets forth the details clearly,
although it is not a practicable method of evaluation when
the number of coupons is large.
§ 115. Valuation of Bonds — Second Method
In illustration of this method, we shall assume the same
facts as presented in § 111. Each semi-annual payment of
$35 may be considered as made up of two parts : $30 and
$5. The $30 is the income on the $1,000 par value at the
assumed semi-annual income rate of 3%. We may disre-
gard this, and consider only the $5, which is a surplus over
and above the income rate, and, in fact, is an annuity which
must be paid for and which is represented by the premium
paid on the bond. Having devoted $30 to the payment of
our expected income-rate on par, we have a remainder of
$5, the difference in rates per period ; this annuity of $5, in
excess of the income rate, is a semi-annual benefit the value
of which is to be ascertained.
We have already found the present value of an annuity
of $1.00 for 50 terms at 3% to be $25.72976. The present
value of a similar annuity of $5.00 would therefore be
$128.6488, which is the premium and which agrees with
the result found in § 113. The method is not only quicker
than the first method presented, but also often gives one
more place of decimals.
§116. Evaluation when Cash Rate Is Less than Income
Rate
In the case of a bond sold below par, and where, ac-
cordingfly, the cash rate is less than the income rate, the
86
THE MATHEMATICS OF INVESTMENT
same procedure is followed for finding the present worth
of an annuity of the difference in rates. In the above illus-
tration, if the bond had a cash rate of 5% instead oi 7%,
the annuity to be evaluated would still be $5 (that is, $30
less $25). In this case, however, the value of the annuity
($128.6488) would have to be subtracted from the par,
giving $871.3512 as the value of a 5% bond, due in 25
years and having an income rate of 3% per period. This
would be commonly known as a 6% basis, although the
effective annual income is 6.09%, as pointed out in § 91.
§117. Second Method by Schedule
As a further illustration of the second method of
evaluation, let us take the case of the bond described in
§ 114. Under the second method the schedule would be :
Differences
Between Cash and
Income Rates
(c-i)
Dates of
Maturity
Periods from
Jan. 1, 1913,
to Dates of
Maturity
Present
Worth
at Jan. 1,
1913 •
$5
5
5
5
July 1, 1913
January 1, 1914
July 1, 1914
January 1, 1915
al
1
2
3
4
$4.854370
4.712980
4.575710
4.442435
Tot
. .$18.585495
The premium above found disagrees slightly with that
shown in § 114, since in the latter case there is a loss of
three decimal places in finding the present worth of the
$1,000 par value. In examining the above schedule, it must
be borne in mind that the total can be ascertained by a
single operation, and that the details are here presented only
for the sake of additional clearness.
VALUATION OF BONDS 87
§ 118. Rule for Second Method of Evaluation
. Since the second method is superior to the first, it will
hereafter be considered as the standard; and we give ac-
cordingly the following rule: The premium (or discount)
on a bond bought above (or below) par is the present worth,
at the income rate, of an annuity equal to the difference be-
tween the cash and income rates for the life of the bond.
§ 119. Principles of Investment
We have found the value of a 7% bond for $1,000,
paying 6% (semi-annually), due in 25 years, to be
$1,128.65 to the nearest cent. This is the amount which
must be invested if the 6% income is to be secured. At the
end of the first half-year, the holder of the bond receives, as
income, 3% interest on the $1,128.65 originally invested,
which is $33.86. But he actually collects $35.00, and after
deducting $33.86 as revenue, there remains $1.14, which
must be applied in amortizing the premium. This will
leave the value of the bond at the end of the first half-year,
at the same income rate, $1,127.51. If our operations have
been correct, the value of a 7% bond to net 6% (payable
semi-annually), having 24% years or 49 periods to run,
will be $1,127.51. To test this, and to exemplify the method
through the use of logarithms, the entire operation is pre-
sented in the following section.
§ 120. Solution by Logarithms
The logarithm of 1 is zero
The logarithm of 1.03 is .01283722
The logarithm of 1.03*' is therefore _.6290238
The logarithm of (1 ^ 1.03"") is therefore. . 1.3709762
We find that the logarithm of .23495 is 1.3709754
Remainder 8
This gives the additional decimal figures 02.
88
THE MATHEMATICS OF INVESTMENT
Hence, $.2349502 is the present value of $1.00 at 3%
per period for 49 periods. The compound discount is there-
fore $.7650498, and this divided by the single rate of in-
terest, 3%, gives the result $25.50166, which is the present
value of an annuity of $1.00 per period. The difference be-
tween the cash and income rates is $5, i.e., $35 — $30.
Therefore, the present value of a $5 annuity for 49 periods
at 3% would be $127,508, or, rounded off, $127.51, which
is the premium desired. Adding this to the par, we have
$1,127.51, which agrees with the result obtained in § 119.
§ 121. Amortization Schedule
When bonds are purchased for investment purposes, a
Schedule of Amortization should be constructed, showing
the gradual extinction of the premium by the application of
the surplus interest. The form shown below is recom-
mended for this purpose, although it is merely suggestive
and not complete. The calculations should be continued to
the date of maturity, and at intervals corrected in the last
figure by a fresh logarithmic computation.
Schedule of Amortization
7% Bond of the. , , payable January 1,
1939. Net 6%. J J.
Date
Total
Interest
7%
Net
Income
Amortiza-
tion
Book
Value
Par
1914, Jan. 1
Cost
..$1 128 65
$1,000.00
Julyl
1915, Jan. 1
Julyl
$35.00
35.00
35.00
$33.86
33.83
33.79
$1.14
1.17
1.21
1,127.51
1,126.34
1,125.13
Strictly speaking, the net income rate is not 6% per
annum, but 3% for each semi-annual period, or an effective
VALUATION OF BONDS
89
annual rate of 6.09%. The column headed "Total Interest"
could be changed to "Cash Receipts," and the term "Book
Value" might also be called "Investment Value."
§ 122. Use of Schedules in Accountancy
The foregoing schedule is the source of the entry which
should be made each half-year for "writing off" the premium
or "writing up" the discount, in order that at maturity the
bond may stand exactly at par. Two other schedules are
set forth below, in which the semi-annual steps in the chang-
ing value of the bond are shown in detail from the date of
purchase until maturity, one schedule being for a bond
bought above par, and the other for a bond bought below
par. Since the formation of schedules is the basis of the
accountancy of amortized securities, we shall present the
same material in various forms, and shall attach to the
schedules the letters (A), (B), etc., for the purposes of
ready reference.
Schedule (A) — ^Amortization
5% Bond of the ,. ., payable May 1,
1919. M N.
Date
Total
Interest
5<
Net
Income
Amortiza-
tion
Book
Value
Par
1914, May 1
Nov. 1
1915, May 1
Nov. 1
1916, May 1
Nov. 1
1917, May 1
Nov. 1
1918, May 1
Nov. 1
1919, May 1
Cost
$104,491.29
104,081.12
103,662.74
103,236.00
102,800.72
102,356.73
101,903.86
101,441.94
100,970.78
100,490.20
100,000.00
$100,000.00
$ 2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
$ 2,089.83
2,081.62
2,073.26
2,064.72
2,056.01
2,047.13
2,038.08
2,028.84
2,019.42
2,009.80
$ 410.17
418.38
426.74
435.28
443.99
452.87
461.92
471.16
480.58
490.20
$25,000.00
$20,508.71
$4,491.29
90
THE MATHEMATICS OF INVESTMENT
Schedule (B) — Accumulation
3% Bond of the , payable May 1,
1919. M N.
Date
Total
Interest
3^
Net
Income
4^
Accumula-
tion
Book
Value
Par
1914, May 1
$95,508.71
$100,000.00
Nov. 1
$ 1,500.00
$ 1,910.17
$ 410.17
95,918.88
1915, May 1
1,500.00
1,918.38
418.38
96,337.26
Nov. 1
1,500.00
1,926.74
426.74
96,764.00
1916, May 1
1,500.00
1,935.28
435.28
97,199.28
Nov. 1
1,500.00
1,943.99
443.99
97,643.27
1917, May 1
1,500.00
1,952.87
452.87
98,096.14
Nov. 1
1,500.00
1,961.92
461.92
98,558.06
1918, May 1
1,500.00'
1,971.16
471.16
99,029.22
Nov. 1
1,500.00
1,980.58
480.58
99,509.80
1919, May 1
1,500.00
1,990.20
490.20
100,000.00
$15,000.00
$19,491.29
$4,491.29
§ 123. Book Values in Schedules
In the foregoing two schedules, (A) and (B), it will
be observed that at any given date the book value in
Schedule (A) is always exactly as much above par as the
book value in Schedule (B) is below par. During any
given period, the "amortization" and the "accumulation"
are exactly the same in both, being deducted in Schedule
(A) and added in Schedule (B).
§ 124. Checks on Accuracy of Schedules
There are three internal checks which are of value in
verifying the accuracy of the schedules. For example, in
Schedule (B), the following facts may be observed:
(1) The total interest plus the total accumulation equals
the total net income.
(2) The total accumulation equals the par less the book
value; or, in other words, it equals the inaugural discount.
VALUATION OF BONDS
91
(3) Each item of accumulation equals the preceding one
multiplied by the semi-annual ratio of increase 1.02, the
semi-annual net income being 2%. That is :
$461.92X1.02 = $471.16
$471.16 X 1.02 = $480.68
etc.
In some instances in these computations, there will be an
apparent error of one cent, which is accounted for by the
fact that the number of decimal places is not carried out
sufficiently far.
§ 125. Tables Derivable from Bond Values
The figures in the column headed "Book Value" might
be taken from tables of bond values published in book form.
If Sprague's Eight-Place Bond Tables were used, and if
the column "Book Value" were copied directly from the
tables, the other columns could be derived by the processes
of addition or subtraction. The result arrived at by this
method would be exactly the same as the results shown in
Schedules (A) and (B). The successive amounts of
amortization or accumulation would be found by finding
the differences between successive book values; while the
net income for any period would be found by either adding
the accumulation to the total interest, or by deducting the
amortization from the total interest.
§ 126. Methods of Handling Interest
It will be observed that in Schedules (A) and (B), the
entire interest is accounted for, both in the case of the
interest on par plus premium, and also in the case of the in-
terest on par minus discount. We may easily construct the
schedules so as to eliminate the par and the interest thereon
92
THE MATHEMATICS OF INVESTMENT
at the rate i. In this manner we would have to deal only
with the surplus interest or the deficient interest, according
to the theory explained in § 115. Since this method may be
preferable for some forms of accounts, a new schedule is
presented below, based on the same facts as those shown in
Schedule (A) :
Schedule (C) — Amortization; Premium Only
Date
Surplus
Interest
on Par
1%
Interest
on
Premium
4^
Amortiza-
tion
Premium
1914, May
$4,491.29
Nov.
$ 500.00
$ 89.83
$ 410.17
4,081.12
1915, May
500.00
81.62
418.38
3,662.74
Nov.
500.00
73.26
426.74
3,236.00
1916, May
500.00
64.72
435.28
2,800.72
Nov.
500.00
56.01
443.99
2,356.73
1917, May
500.00
47.13
452.87
1,903.86
Nov.
500.00
38.08
461.92
1,441.94
1918, May
500.00
28.84
471.16
970.78
Nov.
500.00
19.42
480.58
490.20
1919, May
500.00
9.80
490.20
0.
$5,000.00
$508.71
$4,491.29
§ 127. Schedule of Bond Values
Another way of setting forth the value of bonds at the
successive interest dates is shown in the following table,
which indicates clearly the steps taken in computing the
value. This table, however, is not nearly so compact as the
preceding ones, and for this reason is not recommended, for
most purposes. We will take as an illustration Schedule
(A), shown in §122.
VALUATION OF BONDS 93
Value of bond at May 1, 1914 (cost) $104,491.29
Amortization for ensuing 6 months :
Nominal interest at 2%% on
$100,000.00 $2,500.00
Effective interest at 2% on
$104,491.29 '. 2,089 .83
Difference, being the amor-
tization to be subtracted
from the investment value 410.17
Value of bond at November 1, 1914 $104,081.12
Amortization for ensuing 6 months :
Nominal interest at 2^/2% on
$100,000.00 $2,500.00
Effective interest at 2% on
$104,081.12 2,081.62
Difference, being the amor-
tization to be subtracted
from the investment value , 418.38
Value of bond at May 1,1915 $103,662.Y4
etc., etc.
A slight variation of the above form is to put all of the
figures of the schedule in one column, as follows :
Value, May 1, 1914 $104,491.29
Plus effective interest 2,089.83
$106,581.12
Minus amortization 2,500.00
Value, November 1, 1914 $104,081.12
Plus effective interest 2,081.62
$106,162.74
Minus amortization 2,500.00
Value, May 1, 1915 $103,662.74
etc., etc.
94
THE MATHEMATICS OF INVESTMENT
By using red ink for the subtrahends (which are indi-
cated by italic figures), the addition and subtraction can be
performed at one operation, viz. :
$104,491.29
2,089.83
2,500.00
$104,081.12
2,081.62
2,500.00
$103,662.74
2,073.26
2,500.00
$103,236.00
etc., etc.
It will be noticed that the computation of the interest
may be done without using any other paper. Even with a
fractional rate, such as 2.7% per annum, or 1.35% per
period, the 1%, the .3%, and the .05% may be successively
written down direct without further computation. For
example :
Assumed inaugural value $120,039.00
1,200.39
360.117
60.019
2,500.00
Value at end of 6 months ,. . .$119,159,526
1,191.595
357.479
59.580
2,500.00
Value at end of 1 year $118,268,180
etc., etc.
CHAPTER XI
VALUATION OF BONDS (Concluded)
§ 128. Bond Purchases at Intermediate Dates
It has hitherto been assumed that the purchase of the
bond took place exactly upon an interest date. In the vast
majority of purchases, however, the purchase date differs
from the interest date, and we will now consider cases of
this character. Let us suppose that the interest dates are
May 1 and November 1, whereas the purchase took place on
July 1, after one-third of the interest period had elapsed.
The business custom is to adjust the matter as follows : The
buyer pays to the seller the (simple) interest accrued for the
two months, acquiring thereby the full interest rights, which
will fall due on November 1, and the premium (or the dis-
count, as the case may be) is also considered as vanishing
by an equal portion each month, so that one-third of the half-
yearly amortization takes place by July 1. Taking as an
illustration the bond considered in Schedule (A) (§122),
the amortization from May 1, 1914, to November 1, 1914,
is $410.17; the amortization up to July 1 would therefore
be one-third of this amount, or $186.72. The book value at
July 1 is $104,491.29 minus $136.72, plus $833.33 (the
accrued interest for two months), giving a net figure of
$105,187.90. This last amount is called the nat price, that
is, it is the price including interest ; if the interest is not in-
cluded, the price is said to be at so many per cent and
interest. These are the two methods in most common use
for indicating the prices of bonds. The flat price as above
95
C|6 THE MATHEMATICS OF INVESTMENT
computed might also have been obtained in the following
manner :
To the value on May 1, 1914 $104,491.29
add simple interest thereon for 2 months at 4%,
which is the effective income rate 696.61
giving the flat price at July 1, 1914 $105,187.90
§ 129. Errors in Adjusting Bond Prices
This practice of adjusting the price of bonds at inter-
mediate dates by simple interest is conventionally correct,
but is scientifically inaccurate, and always works a slight
injustice to the buyer. The seller is having his interest
compounded at the end of two months instead of six months,
and receives a benefit therefrom at the expense of the buyer.
It will be readily seen that the buyer does not net the effec-
tive rate of 4% semi-annually on his investment of $105,-
187.90. In order to give both buyer and seller a return at
the effective rate of 2% semi-annually (or 4.04% annually),
with a bimonthly conversion for the seller and a four-month-
ly conversion for the buyer, the true price would be $105,-
183.31.* In practice, however, for any time under six
months, simple interest is generally used, to the slight dis-
advantage of the buyer, who may claim that the value at
November 1, ($104,081.12) + interest due ($2,500.00),
should have been discounted at 4%. This would give $106,-
581.12 -^- 1.01 >^, or $105,178.74. This latter figure is al-
most exactly as much too low ($4.57) as the $105,187.90
is too high ($4.59).
•This price is found by finding the cube root of 1.02, which is 1.00662271.
This last figure is the rate for a two-months period at the effective rate of 2%
semi-annually, or 4.04% annually. When the value at May 1 ($104,491.29) is
multiplied by this figure, the result is $105,183.31, which is the true price on an
effective income basis of 4.04% annually.
VALUATION OF BONDS
97
§ 130* Schedule of Periodic Evaluation
The schedule would therefore, in practice, read as
follows :
Schedule (D) — Periodic Valuation; Simple Interest
Date
Total
Interest
5^
Net
Income
A%
Amortiza-
tion
Book
Value
Par
1914, July 1
Nov. 1
Cost
$104,354.57
104,081.12
$100,000.00
$ 1,666.67
$ 1,393.22
$ 273.45
1915, May 1
2,500.00
2,081.62
418.38
103,662.74
Nov. 1
2,500.00
2,073.26
426.74
103,236.00
1916, May 1
2,500.00
2,064.72
435.28
102,800.72
Nov. 1
2,500.00
2,056.01
443.99
102,356.73
1917, May 1
2,500.00
2,047.13
452.87
101,903.86
Nov. 1
2,500.00
2,038.08
461.92
101,441.94
1918, May 1
2,500.00
2,028.84
471.16
100,970.78
Nov. 1
2,500.00
2,019.42
480.58
100,490.20
1919, May 1
2,500.00
2,009.80
490.20
100,000.00
$24,166.67
$19,812.10
$4,354.57
§ 131. Objection to Valuation on Interest Dates
The interest dates may not always be the most con-
venient dates for periodical valuation. In the case of an
investment consisting of several kinds of bonds, there would
generally be some interest coupons falling due in every
month of the year, and yet on a certain annual or semi-
annual date the entire holdings must be simultaneously
valued, irrespective of the varying interest dates. In cases
of this kind, it will therefore be convenient if the schedules
can be arranged in such a manner that, without recalculation,
every book value will be ready to place in the balance sheet.
Fortunately, this is easier than would be supposed.
98 THE MATHEMATICS OF INVESTMENT
§ 132. Interpolation Method of Periodic Valuation
As an illustration, we will again take the bond described
in § 130 ; but we will now assume that the investor closes
his books on the last days of June and December. We will
suppose that the purchase is made on August 1, 1914. Since
August 1 is midway between May 1 and November 1, the
price must be adjusted as shown in § 128. The price at
August 1 would therefore be midway between $104,491.29
and $104,081.12 — namely, $104,286.20 — plus, of course,
the accrued interest ($1,250.00), this being the customary,
not the theoretical, method. The value at November 1 need
not enter into the schedule, but we must compute the De-
cember 31 value in the same manner as we found the July 1
value in § 128. One-third of the difference between $104,-
081.12 and $103,662.74, or $418.38, is $139.46; $104,-
081.12 — $139.46 = $103,941.66. Our schedule so far,
the headings being the same as in § 130, reads :
1914, Aug. 1 Cost $104,286.20 $100,000.00
Dec. 31 $2,083.33 $1,738.79 $344.54 103.941.66
Proceeding in the same way to find the value on June 30,
1915, from those on May 1 and November 1, we get $103,-
520.49. To these values at dates when interest does not
fall due, there must be added the accrued interest to find the
total values. This method of finding the value of bonds be-
tween interest dates is called interpolation.
§ 133. Multiplication Method of Valuation
There is another method of finding the intermediate
values, however, which might be called the multiplication
method. Having found the value at December 31 to be
$103,941.66, the interest for six months thereon at 4% is
$2,078.83, which, subtracted from the coupon interest
VALUATION OF BONDS 99
($2,500.00), gives as the amortization $421.17. This latter
amount, written off from $103,941.66, gives $103,520.49 as
the value at June 30, which is precisely the same result as
was obtained by interpolation between May 1 and November
1 in § 132. In practice, the method of multiplication will
be found more convenient than the method of interpolation.
Having once adjusted the value at one of the balancing
periods, we can derive all of the values at the remaining
balancing periods by finding the net income, subtracting it
from the cash interest and reducing the premium by the
difference, completely ignoring the values on interest days
(M N).
§ 134. Computation of Net Income for Partial Period
No difficulty arises until we reach the broken period,
January 1 to May 1, 1919. Here the computation of the
net income is peculiar; the par and the premium must be
treated separately. The net income on $100,000.00 is taken
at Yz of 2% for the ^ time, giving $1,333.33. The
premium. $326.80, however, must always be multiplied by
the full 2%, giving $6.54. Adding $1,333.33 and $6.54,
we have $1,339.87, which, used as heretofore, reduces the
principal to par. The reason for this peculiarity is that
$490.20 (the premium at November 1, 1918), and not
$326.80 (the premium at December 31, 1918), is the con-
ventional premium on which 4% is to be computed. Hence,
instead of taking $490.20 for ^ of a period, we take
$326.80 itself for a whole period; these two methods reach
the same result, since $490.20 is 3/2 of $326.80, and tv^^o-
thirds of three-halves is unity. In other words, Yz of
a dollar for a whole period is equivalent in value to the whole
of the dollar for ^ of a period. On the basis outlined, the
completed schedule would therefore be as follows :
lOO THE MATHEMATICS OF INVESTMENT
Schedule (E) — Periodic Valuation by
Multiplication
Date
Total
Interest
Net
Income
4%
Amortiza-
tion
Book
Value
Par
1914, Aug. 1
Cost
$104,286.20
103,941.66
103,520.49
103,090.90
102,652.72
102,205.77
101,749.89
101,284.89
100,810.59
100,326.80
100,000.00
$100,000.00
Dec. 31
1915, Jun. 30
Dec. 31
1916, Jun. 30
Dec. 31
1917, Jun. 30
Dec. 31
1918, Jun. 30
Dec. 31
1919, May 1
$ 2,083.33
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
2,500.00
1,666.67
$23,750.00
$ 1,738.79
2,078.83
2,070.41
2,061.82
2,053.05
2,044.12
2,035.00
2,025.70
2,016.21
1,339.87
$ 344.54
421.17
429.59
438.18
446.95
455.88
465.00
474.30
483.79
326.80
$19,463.80
$4,286.20
§ 135. Purchase Agreements
In all the foregoing examples it has been assumed that
the bond has been bought "on a basis," which means that
the buyer and seller have agreed upon the income rate which
the bonds shall pay, and that from this the price has been
adjusted. But in probably the majority of cases the bargain
is made "at a price," and then the income rate must be found.
This is a more difficult problem.
§ 136. Approximation Method of Finding Income Rate*
The best method of ascertaining the basis, when the
price is given, is by trial and approximation — in fact, all
methods more or less depend upon that. The ordinary
tables will locate several figures of the rate, and one more
figure can safely be added by simple proportion. But it is
an important question to what degree of fineness we should
try to attain. It seems to be the consensus of opinion and
*For a new method of approximation, see Chapter XXIII.
VALUATION OF BONDS lOl
practice that to carry the decimals to hundredths of one per
cent is far enough, although in some cases, by introducing
eighths and sixteenths, two-hundredths and four-hundredths
may be required. Sprague's Tables, by the use of auxiliary
figures, give values for each one-hundredth of one per cent.
§ 137. Application of Method
Let us suppose that $100,000 of 5% bonds, 5 years to
run, M N, are offered at the round price of 104% on May
1, 1914. It is evident that this is nearly, but not quite, a
4% basis. Trying a 3.99% basis we find that the premium
is $4,537.39, which is further from the price than is $4,-
491.29, the 4% basis. Hence, 4% is the nearest basis within
1/100 of one per cent. In fact, by repeated trials, we find
that the rate is about .0399812 per annum. It is manifest
that such a ratio of increase as 1.0199906 would be very un-
wieldy and impracticable, and that such laborious exactness
would be intolerable. Yet here we have paid $104,500, and
the nearest admissible basis gives $104,491.29; what shall
be done with the odd $8.71 ? It must disappear before ma-
turity, and on a 4% basis it will be even larger at maturity
than now. Three ways of ridding ourselves of it may be
suggested.
§ 138. First Method of Eliminating Residues
Add the residue $8.71 to the first amortization, thereby
reducing the value to an exact 4% basis at once. In Schedule
(A) — shown in § 122 — the first amortization would be
$418.88, instead of $410.17. This is at the income rate of
about 3.983% for the first half-year and thereafter at 4%.
For short bonds the result is fairly satisfactory.
§ 139. Second Method of Eliminating Residues
Divide $8.71 into as many parts as there are periods.
I02
THE MATHEMATICS OF INVESTMENT
This would give $.87 for each period, except the first, which
would be e$.88 on account of the odd cent. Set down the 4%
amortization in one column, the $.87 in the next, and the
adjusted figures in the third :
$410.17
$.88
$411.05
418.38
.87
419.25
426.74
.87
427.61
435.28
.87
436.15
443.99
.87
444.86
452.87
.87
453.74
461.92
.87
462.79
471.16
.87
472.03
480.58
.87
481.45
490.20
.87
491.07
The following will then be the schedule :
Schedule (F) — Elimination of Residues;
Second Method
Date
Total
Interest
Net
Income
4%i-)
Amortiza-
tion
Book
Value
Par
1914, May 1
$104,500.00
$100,000.00
Nov. 1
$ 2,500.00
$ 2,088.95
$ 411.05
104,088.95
1915, May 1
2,500.00
2,080.75
419.25
103,669.70
Nov. 1
2,500.00
2,072.39
427.61
103,242.09
1916, May 1
2,500.00
2,063.85
436.15
102,805.94
Nov. 1
2,500.00
2,055.14
444.86
102,361.08
1917, May 1
2,500.00
2,046.26
453.74
101,907.34
Nov. 1
2,500.00
2,037.21
462.79
101,444.55
1918, May 1
2,500.00
2,027.97
472.03
100,972.52
Nov. 1
2,500.00
2,018.55
481.45
100,491.07
1919, May 1
2,500.00
2,008.93
491.07
100,000.00
$25,000.00
$20,500.00
$4,500.00
VALUATION OF BONDS 103
In this schedule the income rate varies from 3.997995^
to 3.99822%; hence the approximation is sufficiently close
for any holdings, except large ones for long maturities.
§ 140. Third Method of Eliminating Residues
For still greater accuracy, we may divide the $8.71 in
parts proportionate to the amortization. The amortization
on the 4% basis amounts to $4,491.29, and v^e have an
extra amount of $8.71 to exhaust. Dividing the latter by
the former, we have as the quotient .00194, which is the
portion to be added to each dollar of amortization. With
this we form a table for the 9 digits :
100194
200388
300582
400776
500970
601164
701358
801552
901746
From this table it is easy to adjust each item of amortiza-
tion, writing down, for example, to the nearest mill :
410.17 418.38 426.74 435.28
400.776
400.776
400.776
400.776
10.019
10.019
20.039
30.058
.100
8.016
6.012
5.010
.070
.301
.080
.701
.040
.200
410.97
.080
419.19 427.57 436.12
I04
THE MATHEMATICS OF INVESTMENT
The respective amounts of amortization, in Schedule (G),
vary (at the most) but 8 cents from those shown in Schedule
(F).
Schedule (G) — Elimination of Residues
Third Method
Date
Total
Interest
5^
Net
Income
4%(-)
Amortiza-
tion
Book
Value
Par
1914, May 1
$104,500.00
$100,000.00
Nov. 1
$ 2,500.00
$ 2,089.03
$ 410.97
104,089.03
1915, May 1
2,500.00
2,080.81
419.19
103,669.84
Nov. 1
2,500.00
2,072.43
427.57
103,242.27
1916, May 1
2,500.00
2,063.88
436.12
102,806.15
Nov. 1
2,500.00
2,055.15
444.85
102,361.30
1917, May 1
2,500.00
2,046.25
453.75
101,907.55
Nov. 1
2,500.00
2,037.18
462.82
101,444.73
1918, May 1
2,500.00
2,027.93
472.07
100,972.66
Nov. 1
2,500.00
2,018.49
481.51
100,491.15
1919, May 1
2,500.00
2,008.85
491.15
100,000.00
$25,000.00
$20,500.00
$4,500.00
§ 141. Short Terminals
It sometimes happens (though infrequently) that the
principal of a bond is payable, not at an interest date, but
from one to five months later, making a short terminal
period. The following is a very simple method of obtain-
ing the present value in this case. It will not be necessary
to demonstrate it, but an example will test it.
Suppose the 5% bond, M N, yielding 4%, bought May
1, 1914, were payable October 1, instead of May 1, 1919,
that is, in 10 5/6 periods. The short period is 5/6. The
VALUATION OF BONDS 105
short ratio (at 4%) will be 1.0166 J^. The short interest
(at 5%) will be .02083 J^.
We first ascertain the value for the ten full
periods, viz., for $1 1.0449129*
Add to this the short interest 0208333
1.0657462
and divide by the short ratio 1.0166667
To perform this division it will be easier to divide 3 times
the dividend by 3 times the divisor.
3.05 ) 3.1972386 ( Quotient 1.0482750
3.05
1472
1220
2523
2440
838
610
2286
2135
151
152
Multiplying down by the usual procedure, we have the
following schedule:
•See Schedule (A), § 122.
k
I06 THE MATHEMATICS OF INVESTMENT
Schedule (H) — Short Terminals
Date
Total
Interest
5!<
Net
Income
4^
Amortiza-
tion
Book
Value
Par
1914, May 1
$104,827.50
$100,000.00
Nov. 1
$ 2,500.00
$ 2,096.55
$ 403.45
104,424.05
1915, May 1
2,500.00
2,088.48
411.52
104,012.53
Nov. 1
2,500.00
2,080.25
419.75
103,592.78
1916, May 1
2,500.00
2,071.86
428.14
103,164.64
Nov. 1
2,500.00
2,063.29
436.71
102,727.93
1917, May 1
2,500.00
2,054.56
445.44
102,282.49
Nov. 1
2,500.00
2,045.65
454.35
101,828.14
1918, May 1
2,500.00
2,036.56
463.44
101,364.70
Nov. 1
2,500.00
2,027.29
472.71
100,891.99
1919, May 1
2,500.00
2,017.84
482.16
100,409.83
Oct. 1
2,083.33
1,673.50
$22,255.83
409.83
100,000.00
$27,083.33
$4,827.50
§ 142. Rule for Short Terminals
Ascertain the value of the bond for the full number of
periods, disregarding the terminal. To this value add the
short interest, and divide by the short ratio.
It may be remarked that this same process applies to
short initial periods. It even applies to bonds originally
issued between interest dates, and also maturing between
interest dates ; in the case of bonds of this description, the
process would be applied twice.
§ 143. Discounting
Hitherto we have calculated the value of the bond at its
earliest date, and then obtained the successive values at
later dates by multiplication and subtraction. We can also
work backwards, however, obtaining each value from the
VALUATION OF BONDS
107
next later one by addition and division. Let us take, for
illustration, the bond shown in Schedule (A), § 122:
Principal to be received at maturity, May 1,
1919 $100,000.00
Coupon to be received at May 1, 1919 2,500.00
Total amount receivable at May 1, 1919 $102,500.00
Discounted value at November 1, 1918, exclud-
ing the coupon receivable at that date, found
by dividing $102,500.00 by 1.02. $100,490.20
Coupon to be received at November 1, 1918 .... 2,500.00
Total value at November 1, 1918, including the
coupon receivable at that date $102,990.20
Discounted value at May 1, 1918, excluding the
coupon receivable at that date, found by divid-
ing $102,990.20 by 1.02 $100,9Y0.Y8
etc., etc.
In this manner successive terms may be obtained as far as
desired.
§ 144. Last Half- Year of Bond
In the last half-year of a bond, its value should be dis-
counted, and not found as in § 128. Thus, if the bond men-
tioned in § 128 were sold three months prior to maturity, its
value would be found by dividing $102,500.00 by 1.01,
which would give $101,485.15 "flat," equivalent to $100,-
235.15 and interest; whereas by the ordinary rule it would
be $100,245.10 (that is, midway between $100,490.20 and
$100,000.00). The theoretically exact value (recognizing
effective rates, which is never done in business) would be
$100,240.13. This is found by multiplying the value six
months prior to maturity ($100,490.20) by the square root
of 1.02, this root to ten decimal places being 1.0099504938.
Io8 THE MATHEMATICS OF INVESTMENT
The product is $101,490.13, which, less the accrued interest
amounting to $1,250.00, gives $100,240.13. To "split the
difference" would be an easy way of adjusting the matter,
and would be almost exact.
§ 145. Serial Bonds
Bonds are often issued in series so that they mature at
various dates. For example, there may be an issue of
$30,000.00, of which $1,000.00 is payable after one year,
another $1,000.00 after two years, and so on, the final
$1,000.00 being payable after thirty years. Other series are
more complex, as, for example, $2,000.00 payable each
year for five years, and $4,000.00 each year thereafter for
ten years. The initial value of a series on any given basis
cannot be found by one operation ; the initial value of each
instalment must first be found, and the sum of these separate
initial values gives the initial value of the entire series.
After the aggregate initial value has been ascertained, it
may, for the purposes of deriving values at succeeding in-
terest periods, be treated as a unit, as if the bonds were
not in series. At the end of each of the yearly periods, the
ordinary amortization or accumulation would have to be
computed, and it would also be necessary to deduct from the
total value the par value- of the bonds cancelled or retired.
In offering serial bonds for sale, they are often listed as
of "average maturity — 15^/2 years." This is entirely de-
lusive, and frequently causes the buyer to believe that he is
getting a more favorable basis than will be realized. The
only correct valuation of a series is the sum of all its separate
values. If we assume that the $30,000.00 above referred to
was a series of 5% bonds bought on an assumed 3.50%
basis, the true value would be $35,005.00
whereas the value for the "average time," i.e.,
151/2 years, would be 35,348.22
VALUATION OF BONDS
109
In computing the rate of income yielded by a series, the
income rate corresponding to the average time may be taken
as a point of departure, but it will be found that it is in-
variably too high. For example, let us assume that the
thirty 5% serial bonds mentioned above were purchased
when first issued at 116.68, and that the income rate thereon
is desired.
Looking in Sprague's "Extended Bond Tables," under
the 5% bonds in the 15% year column (average date), we
find that the value nearest to 116.68 is $1,165,200.00
which is at a yield of approximately 3.60%.
If, however, we take off on an adding machine,
the value of a 5% bond due in 30 years at
3.60% $1,255,549.38
the value of a 5% bond due in 29 years at
3.60% 1,250,705.95
the value of a 5% bond due in 28 years at
3.60% 1,245,686.60
and so on, to and including the bond due in
1 year, we shall find that the total value of
the 30 bonds is 34,531,390.28
and that the average price is therefore 115.10.
An income yield of 3.55% for the above bonds will re-
sult in a value of 116.06, while a yield of 3.50% will dis-
close a value of 116.68. The true yield on these bonds, if
bought at 116.68, is therefore not 3.60%, as seems at first
apparent, but is 3.50%.
§ 146. Irredeemable Bonds
Sometimes, as in the case of British Consols, there is
no right nor obligation of redemption. If the government
wishes to pay off any of its bonds, it has to buy them at the
market price. With this class of bonds, there is no question
of amortization; the investment is simply a perpetual an-
no THE MATHEMATICS OF INVESTMENT
nuity. The cash interest is all revenue, and the original cost
is the constant book value. If £100 of 4% Consols be
bought at 96, the income is £4 per annum, and the book
value is £96. Since the investment of £96 produces £4 an-
nually, the rate of income is 4 -j- 96, or 4 1/6%.
§ 147. Optional Redemption
Sometimes the issuer of a bond has the right to redeem
at a certain date earlier than the date at which he must
redeem. It must always be expected that this right will be
exercised if profitable to the issuer; hence, to be conserva-
tive, a purchaser, when buying this class of bonds at a
premium, must always consider them as maturing, or reach-
ing par, at the earlier date. On the other hand, bonds of
this character bought at a discount must be considered as
running to the longer date. If the bonds bought at a
premium run to the very latest date, or if the bonds bought
at a discount are called for redemption at an earlier date
than was anticipated by the investor, he will, in either case,
receive a higher yield in his income rate than he would have
received on the more conservative basis. The element of
chance enters in here, but, to be safe, the purchaser should
always consider that the chances will go against him ; he will
then have all to gain and nothing to lose.
The option of redemption is sometimes attended by a
premium. For example, the issuer of a thirty-year bond
reserves the right to redeem after twenty years at 105.
Where bonds are bought at such an income yield that after
twenty years the book value will be more than 105, the right
of redemption at 105 is a detriment to the purchaser. In
such a case as this, the safe and conservative purchaser
should buy at such an income basis as will bring the book
value at 105 or below at the end of twenty years.
There is also a form of bond issue, not uncommon in
VALUATION OF BONDS HI
Europe, where a certain or indefinite number of bonds is
drawn by lot each year for the purposes of retirement. As
these bonds are usually issued at a discount, those which are
drawn at the earlier dates are the more profitable. The in-
vestor, however, in estimating his income, must assume that
his particular bonds will be among the last ones drawn. If
drawn at earlier dates, there is a profit exactly the same as
that arising from a sale above book value.
§ 148. Bonds as Trust Fund Investments
A bond which has been purchased by a trustee at a
premium is subject to amortization in the absence of testa-
mentary instructions to the contrary. The trustee has no
right to pay over the full cash interest to the life tenant,
because he must keep the principal intact for the remainder
man. If, for example, the trustee were to invest $104,-
491.29 in a 5% bond having five years to run, and if he
were to pay over the full amount of the coupons to the life
tenant for the period of five years, the fund at the end of
the period would simply be the par value of the bonds,
$100,000.00, and would therefore be depleted to the extent
of $4,491.29, to the manifest injury of the remainder man.
Since the investment is on a 4% basis, the trustee should pay
over at the end of the first half-year only 2% of $104,491.29
(or $2,089.83),. and not 21/2% of $100,000.00 (or $2,-
500.00). He then has $410.17 cash to reinvest, and the fund,
including this, is still $104,491.29. It may be dif^cult to
invest the $410.17 at as favorable a rate as the bonds, very
small and very large amounts being most diflficult to invest.
The trustee can deposit it in a trust company, at least, and
receive interest at some rate, however small.
§ 149. Payments to Life Tenant
At the end of the second half-year, the net income on
112 THE MATHEMATICS OF INVESTMENT
the bond is only $2,081.62; but, in addition to this amount,
the life tenant is also entitled to the interest on the $410.17.
If this has been reinvested at exactly 4% (interest payable
semi-annually), the interest thereon is $8.20, and the total
amount payable to the life tenant is $2,081.62 + $8.20 =
$2,089.82. This is practically the same amount of income
as in the first half-year. The difference between the coupons
received ($2,500.00) and the net income ($2,089.82) is the
amortization ($418.38), which is deposited or invested as
before. The trustee now has in the fund :
Book value of the bonds ,. . .. .$103,662.74
Invested in Trust Company or otherwise at end
of first half-year , ,. . 410.17
Invested in Trust Company or otherwise at end
of second half-year 418.38
Total $104,491.29
He has paid over all of the new interest earned, and he
has kept the corpus or principal intact.
§ 150. Effect of Varying Rates on Investments
Suppose, however, that the trustee was not able to get
4% on the $410.17, but only 3%, so that from this source
would come only $6.15, making the total income $2,081.62
plus $6.15, or $2,087.77. This shows a slight falling off in
income, but that is to be expected when part of an investment
is returned and reinvested at a lower rate. If the reinvest-
ment had been at 41/2%, the income would have been $2,-
090.85, slightly more than the first half-year, owing to the
improved demand for capital. It might be urged that the
life tenant ought to receive $2,089.83 semi-annually — no
more, no less — being at a 4% rate on $104,491.29. This
would leave $410.17 each half-year to be invested in a
VALUATION OF BONDS
"3
Sinking fund, from which no interest should be drawn, but
which should be left to accumulate to maturity, when it
would exactly replace the premium, if compounded at 4%,
But this hope might not be realized. Very likely the average
rate would be less or more than 4% and not exactly 4%.
If less, the original fund would be to some extent depleted,
and the remainder man wronged; if more, there would be
too much in the fund, and the life tenant would receive too
little. It seems, therefore, that the sinking fund principle
is not correct in a case like this, and that, at all events, the
original fund should be kept constant, neither increased nor
diminished. So much of the semi-annual receipts as are not
necessary to maintain the constancy of the fund due to the
remainder man should be paid over as income to the life
tenant.
§ 151. Example of Payments to Life Tenant
A two-year, 4% bond, par value $10,000, bought at
$10,192.72, would be scheduled thus :
Coupon
Income
Cash
Bond
$10,192.72
$200.00
$162.89
$47.11
10,146.61
200.00
152.18
47.82
10,097.79
200.00
151.47
48.63
10,049.26
200.00
150.74
49.26
10,000.00
$800.00
$607.28
$192.72
The life tenant would receive, at the end of the first half-
year, $152.89; at the end of the second, $152.18 + what-
ever the $47.11 cash had earned; at the end of the third,
$151.47 + whatever $94.93 had earned; at the close,
$150.74 -f whatever $143.46 had earned. If the cash bal-
114
THE MATHEMATICS OF INVESTMENT
ance were periodically deposited in a trust company at 3%
(payable semi-annually), the life tenant would receive a uni-
form income of $152.89.
§ 152. Cullen Decision
In a New York case (38 App. Div. 419), Justice Cullen
very clearly lays down the law as to the duty of the trustee
to reserve a part of the interest to provide for the premium,
and says that "any other view would lead to the impair-
ment of the principal of the trust, to protect the integrity
of which has always been the cardinal rule of courts of
equity." He says further : *'If one buys a ten-year five per
cent bond at one hundred and twenty, the true income or
interest the bond pays is not 4 1/6% on the amount in-
vested, nor 5% on the face of the bond, but 2 7/10% on
the investment, or 3 24/100% on the face of the bond. The
matter is simply one of arithmetical calculation, and
tables are readily accessible, showing the result of the
computation."
§ 153. Cullen Decision Scheduled
Consulting one of the tables referred to by Justice Cullen
(Sprague's "Extended Bond Tables"), and looking in the
5% tables under the column headed "10 Years," we find
that the value nearest to $120,000.00 is $120,038,997,
which is opposite the net income rate of 2.70%. As stated
by Justice Cullen, the bonds, while bearing a coupon rate of
5%, actually net, therefore, only 2.7% on account of the
high premium. With a slight correction in the initial figures
in order to make the income rate exactly 2.7%, and assum-
ing a par value of $100,000.00, the illustration as given in
the above case by Justice Cullen, when tabulated to show
the present value, the income, and the amount reinvested,
would work out as follows :
VALUATION OF BONDS
115
Total
Interest
Income
Paid Over
Reinvested
Present
Value
$120,039.00
$2,500.00
$1,620.53
$879.47
119,159.53
2,500.00
1,608.65
891.35
118,268.18
2,500.00
1,596.62
903.38
117,364.80
2,500.00
1,584.42
915.58
116,449.22
2,500.00
1,572.07
927.93
115,521.29
2,500.00
1,559.53
940.47
114,580.82
2,500.00
1,546.85
953.15
113,627.67
2,500.00
1,533.97
966.03
112,661.64
2,500.00
1,520.93
979.07
111,682.57
2,500.00
1,507.72
992.28
110,690.29
2,500.00
1,494.31
1,005.69
109,684.60
2,500.00
1,480.75
1,019.25
108,665.35
2,500.00
1,466.93
1,033.07
107,632.28
2,500.00
1,453.08
1,046.92
106,585.36
2,500.00
1,438.91
1,061.09
105,524.27
2,500.00
1,424.57
1,075.43
104,448.84
2,500.00
1,410.06
1,089.94
103,358.90
2,500.00
1,395.35
1,104.65
102,254.25
2,500.00
1,380.43
1,119.57
101,134.68
2,500.00
1,365.32
1,134.68
100,000.00
$50,000.00
$29,961.00
$20,039.00
§ 154. Unjust Feature of Cullen Decision
The foregoing schedule is perfectly correct, but we can
scarcely agree with the method described further on in the
same opinion, as follows: "There is, however, a simpler
way of preserving the principal intact — the method adopted
by the learned referee. He divided the premium paid for
the bonds by the number of interest payments, which would
ii6
THE MATHEMATICS OF INVESTMENT
be made up to the maturity of the bonds, and held that the
quotient should be deducted from each interest payment
and held as principal. These deductions being principal,
the life tenant would get the benefit of any interest that
they might earn. We do not see why this plan does not
work equal justice between the parties." The reason "why
it does not work equal justice" is that the life tenant in the
earlier years receives much less than his due share of the
income, but from year to year he gradually receives more
and more, until he receives more than his share; but not
until the very last payment does he overtake his true share.
Thus, if he dies before the maturity of the bonds, it is cer-
tain that "equal justice" will not have been done, and that
the remainder man will have had altogether the best of it.
The schedule under the referee's plan would work out as
follows :
Total
Interest
Income
Paid Over
Reinvested
Present
Value
$120,039.00
$2,500.00
$1,498.05
$1,001.95
119,037.05
2,600.00
1,498.05
1,001.95
118,035.10
2,500.00
1,498.05
1,001.95
117,033.15
2,500.00
1,498.05
1,001.95
116,031.20
2,500.00
1,498.05
1,001.95
115,029.25
2,500.00
1,498.05
1,001.95
114,027.30
2,500.00
1,498.05
1,001.95
113,025.35
2,500.00
1,498.05
1,001.95
112,023.40
2,500.00
1,498.05
1,001.95
111,021.45
3,500.00
1,498.05
1,001.95
110,019.50
etc.
etc.
etc.
etc.
Assuming that, under each plan, the reinvested funds
would earn the same rate of income as the oriynal invest-
VALUATION OF BONDS II7
ment (i.e., 2.7%), the total semi-annual income of the re-
mainder man would be as follows :
Under Plan Under Plan
in § 153 in § 154
$1,620.53 $1,498.05
1,620.53 1,511.58
1,620.53 1,525.10
1,620.53 . 1,538.63
1,620.53 1,552.16
1,620.53 1,565.68
1,620.53 1,579.21
1,620.53 1,592.73
1,620.53 1,606.26
1,620.53 1,619.79
etc. etc.
A comparison of the two columns will show the injus-
tice of the referee's plan toward the life tenant, and sub-
stantiate the equity of the plan of scientific amortization set
forth in the schedule in § 153.
§ 155. Bond Tables
Mention has been made heretofore of bond tables. These
tables show the values of bonds at various coupon rates,
yielding various rates of net income, and due in different
periods from one-half year to one hundred years. The usual
tables refer to bonds whose coupons are payable semi-
annually, but there are generally supplementary tables by
the use of which values of those bonds may be ascertained
whose coupons are payable annually or quarterly. The fol-
lowing table is taken from page 80 of Sprague's "Extended
Bond Tables," and sets forth (in part) the values of the
bond mentioned in Schedule (A), §122:
ii8
THE MATHEMATICS OF INVESTMENT
Bond Table
Values, to the Nearest Cent, of a Bond for $1,000,000 at
6% Interest, Payable Semi-Annually
Net
In-
come
3 Years
3J^ Years
4 Years
4^ Years
5 Years
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.05
3.10
3.15
3.20
3.25
3.30
3.35
3.40
3.45
3.50
3.55
3.60
3.65
3.70
3.75
3.80
3.85
3.90
3.95
4.00
4.05
4.10
4.15
4.20
4.2S
,071,825.12
,070,328.46
,068,834.33
,067,342.73
,065,853.65
,064,367.09
,062,883.04
,061,401.50
,059,922.46
,058,445.92
,056,971.87
,055,500.31
,054,031.24
,052,564.64
,051,100.52
,049,638.87
,048,179.68
,046,722.96
,045,268.68
,043,816.86
,042,367.48
,040,920.54
,039,476.04
,038,033.97
,036,594.33
,035,157.11
,033,722.30
,032,289.91
,030,859.92
,029,432.34
,028,007.15
,026,584.36
,025,163.96
,023,745.94
,022,330.31
,020,917.04
1,083,284.07
1,081,538.84
1,079,796.97
1,078,058.45
1,076,323.28
1,074,591.45
1,072,862.96
1,071,137.78
1,069,415.93
1,067,697.38
1,065,982.14
1,064,270.19
1,062,561.54
1,060,856.16
1,059,154.06
1,057,455.22
1,055,759.65
1,054,067.33
1,052,378.25
1,050,692.42
1,049,009.81
1,047,330.43
1,045,654.27
1,043,981.31
1,042,311.57
1,040,645.01
1,038,981.65
1,037,321.47
1,035,664.46
1,034,010.63
1,032,359.96
1,030,712.44
1,029,068.07
1,027,426.84
1,025,788.74
1,024,153.77
1,094,601.55
1,092,608.09
1,090,618.92
1,088,634.05
1,086,653.46
1,084,677.14
1,082,705.08
1,080,737.28
1,078,773.71
1,076,814.37
1,074,859.25
1,072,908.34
1,070,961.63
1,069,019.11
1,067,080.77
1,065,146.59
1,063,216.58
1,061,290.71
1,059,368.98
1,057,451.38
1,055,537.90
1,053,628.52
1,051,723.25
1,049,822.06
1,047,924.95
1,046,031.91
1,044,142.93
1,042,258.00
1,040,377.11
1,038,500.25
1,036,627.41
1,034,758.58
1,032,893.74
1,031,032.90
1,029,176.04
1,027,323.16
1,105,779.31
1,103,537.98
1,101,302.00
1,099,071.36
1,096,846.04
1,094,626.03
1,092,411.33
1,090,201.90
1,087,997.74
1,085,798.84
1,083,605.17
1,081,416.74
1,079,233.51
1,077,055.49
1,074,882.64
1,072,714.97
1,070,552.46
1,068,395.09
1,066,242.85
1,064,095.73
1,061,953.71
1,059,816.78
1,057,684.92
1,055,558.13
1,053,436.38
1,051,319.67
1,049,207.98
1,047,101.30
1,044,999.62
1,042,902.92
1,040,811.18
1,038,724.41
1,036,642.57
1,034,565.67
1,032,493.68
1,030,426.59
1,116,819.07
1,114,330.27
1,111,847.97
1,109,372.18
1,106,902.85
1,104,439.98
1,101,983.56
1,099,533.55
1,097,089.94
1,094,652.71
1,092,221.85
1,089,797.33
1,087,379.13
1,084,967.25
1,082,561.66
1,080,162.34
1,077,769.27
1,075,382.44
1,073,001.82
1,070,627.41
1,068,259.17
1,065,897.10
1,063,541.18
1,061,191.38
1,058,847.70
1,056,510.11
1,054,178.59
1,051,853.13
1,049,533.71
1,047,220.32
1,044,912.93
1,042,611.52
1,040,316.09
1,038,026.61
1,035,743.07
1,033,465.45
VALUATION OF BONDS
119
Net
In-
come
3 Years
354 Years
4 Years
454 Years
5 Years
4.30
1,019,506.15
1,022,521.93
1,025,474.23
1,028,364.40
1,031,193.73
4.35
1,018,097.62
1,020,893.20
1,023,629.26
1,026,307.08
1,028,927.90
4.40
1,016,691.46
1,019,267.57
1,021,788.23
1,024,254.63
1,026,667.93
4.45
1,015,287.65
1,017,645.05
1,019,951.13
1,022,207.03
1,024,413.82
4.50
1.013,886.19
1,016,025.62
1,018,117.96
1,020,164.27
1,022,165.54
4.55
1,012,487.08
1,014,409.27
1,016,288.70
1,018,126.33
1,019,923.08
4.60
1,011,090.32
1,012,796.01
1,014,463.35
1,016,093.21
1,017,686.42
4.65
1,009,695.89
1,011,185.82
1,012,641.90
1,014,064.89
1,015,455.55
4.70
1,008,303.80
1,009,578.70
1,010,824.33
1,012,041.35
1,013,230.44
4.75
1,006,914.03
1,007,974.64
1,009,010.63
1,010,022.60
1,011,011.08
4.80
1,005,526.59
1,006,373.63
1,007,200.81
1,008,008.60
1,008,797.46
4.85
1,004,141.48
1,004,775.67
1,005,394.84
1,005,999.36
1,006,589.56
4.90
1,002,758.67
1,003,180.75
1,003,592.72.
1,003,994.85
1,004,387.36
4.95
1,001,378.18
1,001,588.86
1,001,794.45
1,001,995.07
1,002,190.85
5.00
1,000,000.00
1,000,000.00
1,000,000.00
1,000,000.00
1,000,000.00
§ 156. Features of the Bond Table
The relations existing between succeeding values on the
same horizontal line of the table are readily seen, these
values being computed at the same income rate but for
different semi-annual periods. For example, on a 2.50%
basis, the value of this 5% bond 5 years prior to maturity is
$1,116,819.07. At 4% years prior to maturity, its value is
1.0125 (which is the semi-annual income rate) times $1,-
116,819.07, producing $1,130,779.31, from which must be
deducted the semi-annual coupons ($25,000.00), giving as
a final result $1,105,779.31.- The net income yields shown
in the preceding table vary to the extent of .05%. Usually
there are supplementary tables by the use of which values
of bonds may be calculated at different income yields differ-
ing by only .01%. In this manner, the values of the bond
shown in the preceding table may be computed on an income
yield of 2.51%, 2.52%, 2.53%, etc.
CHAPTER XII
SUMMARY OF COMPOUND INTEREST
PROCESSES
§ 157. Rules and Formulas
In the present chapter are given in condensed and sym-
bolic form the rules and formulas which have been explained
in the preceding chapters.
§ 158. Rules
(1) To find the ratio of increase :
Add 1 to the rate of interest.
(2) To find the amount of $1 :
Multiply 1 by the ratio as many times as there
are periods.
(3) To find the present worth of $1, or to discount $1 :
Divide 1 by the ratio as many times as there
are periods.
(4) To find the total interest on $1 :
Subtract 1 from the amount.
(5) To find the total discount on $1 :
Subtract the present worth from 1.
(6) To find the amount of an annuity of $1 :
Divide the total interest by the rate of interest.
(7) To find the present worth of an annuity of $1 :
Divide the total discount by the rate of interest.
120
COMPOUND INTEREST PROCESSES 12 1
(8) To find the rent of an annuity worth $1, or what
annuity can be bought for $1 :
Divide 1 by the present worth of the annuity.
(9) To find what annuity (sinking fund) will produce
$1:
Divide 1 by the amount of the annuity,
(10) To find the premium on a bond, or the discount on
a bond :
Consider the difference between the cash and
income rates as an annuity to be valued, and
find its present worth at the income rate.
( 11 ) To find the value of a bond :
In case the cash rate is greater than the income
rate, the bond is at a premium; therefore,
add par to the premium.
In case the income rate is greater than the cash
rate, the bond is at a discount; therefore,
subtract the discount from par.
§ 159. Formulas
i is the rate of interest or the interest on unity for 1
period, n is the number of periods, c is the cash rate on a
bond.
(1) Ratio of Increase =
1 + i
(2) Amount =
(1+ir
(3) Present Worth &i ^C-"^**^
1
(i+ir
(4) Total Interest =
(i+ir-i
(5) Total Discount =
1- 1
122 THE MATHEMATICS OF INVESTMENT
(6) Amount of Annuity =
(7) Present Worth of An-
nuity
(8) Rent of Annuity
(9) Sinking Fund
i
1
X —
(1+ir
i
i
1-
1
i
(l+i)**-!
(10) Premium on Bond = — : — ( 1 — .^ , .. „ )
(11) Discount on Bond = — ; — ( 1 — .^ , ..„ )
(12) Value of Bond (at a ^^ , c-j / _ 1 \
Premium) i \ (l + i)**/
(13) Value of Bond (at a ^ i- c / _ 1 \
Discount)* ^ * \ (l + iy)
• This formula is equivalent to Formula (12).
CHAPTER XIII
ACCOUNTS— GENERAL PRINCIPLES
§ 1 60. Relation of General Ledger to Subordinate Ledgers
In any extensive system of accountancy, in order to
fulfill the opposite requirements of minuteness and compre-
hensiveness, it is necessary to keep, in some form, a general
ledger and various subordinate ledgers. Each account in
the general ledger, as a rule, comprises or summarizes the
entire contents of one subordinate ledger. Each account of
the general ledger comprises groups of similar accounts,
which are handled in the subordinate ledgers as individual
assets or as groups which may be treated as individuals. It
is the province of the general ledger to give information
in grand totals as an indicator of tendencies; while the
function of the subordinate ledger is to give all desired in-
formation as to details, even beyond the figures required
for balancing — facts not only of numerical accountancy, but
descriptive, cautionary, or auxiliary. Thus the general
ledger may contain a ''Mortgages" account, which will show
the increase or decrease of the amount invested on mortgage,
and the resultant or present amount; the mortgage ledger
will contain an account for each separate mortgage, with
additional information as to interest, taxes, insurance, title,
ownership, security, valuation, or any other thing useful
or necessary to be known.
§ 161. The Interest Account
We shall assume that a general ledger exists with subor-
dinate or class ledgers. We shall also assume that the
123
124 THE MATHEMATICS OF INVESTMENT
accounts are to be so arranged as to give currently the
amount of interest earned up to any time, and the amount
outstanding and overdue at any time. It would hardly
seem necessary to argue this point, were it not that many
large investors pay no attention to interest until it matures,
and some do not carry it into their accounts until it is paid.
They are compelled to make an adjustment on their
periodical balancing dates "in the air," compiling it from
various sources without check, which seems as crude as it
would be to take no account of cash, except by counting it
occasionally. The Profit and Loss account depends for its
accuracy upon the interest earned, not upon the interest
falling due, nor upon the interest collected ; and the accruing
of interest is a fact which should be recognized and recorded.
§ 162. Mortgage and Loan Accounts
In considering the forms of account for investments, we
will first take up, as being simpler, those in which there is
never any value to be considered other than par, such as
mortgages and loans upon collateral security. Both of these
classes of investments are for comparatively short terms,
and are usually the result of direct negotiation between
borrower and lender, and not the subject of purchase and
sale; hence, changes in rate of interest are readily effected
by agreement, and do not result in a premium or discount.
CHAPTER XIV
REAL ESTATE MORTGAGES
§163. Nature of Loans on Bond and Mortgage
The instruments which we have spoken of as "bonds"
are very often secured by a mortgage of property. But one
mortgage will secure a great number of bonds, the mort-
gagee being a trustee for all the bondholders. In contrast
with these instruments, those of which we now speak are
the ordinary *'bond and mortgage," by which the investor
receives from the borrower two instruments : the one an
agreement to pay, and the other conferring the right, in
case default is made, to have certain real estate sold, and
the proceeds used to pay the debt. As only a portion of the
value of the real estate is loaned, the reliance is primarily
on the mortgage rather than on the bond. Therefore, the
mortgagee must be vigilant in seeing that his margin is
not reduced to a hazardous point. This may happen by the
depreciation of the land for various economic reasons; by
the deterioration of the structures thereon through time or
neglect; by destruction through fire; or by the non-payment
of taxes, which are a lien superior to all mortgages. By
reason of these risks a mortgage loan is seldom made for
more than a few years ; but after the date of maturity, ex-
tensions are made from time to time; or, even more fre-
quently, without formal extension, the loan is allowed to
remain "on demand," either party having the right to ter-
minate the relation at will. A large proportion of outstand-
ing mortgages are thus "on sufferance," or payable on
125
126 THE MATHEMATICS OF INVESTMENT
demand. The market rate of interest seldom causes the
obHgation to change hands at either a premium or a dis-
count; hence we may ignore that feature, referring the
exceptional cases, where it occurs, to the analogy of bonds.
The two instruments, bond and mortgage, relate to the
same transaction, are held by the same owner, and for most
purposes are treated as a unit. In bookkeeping, the invest-
ment must likewise be treated as a unit, both as to principal
and income.
§ 164. Separate Accounts for Principal and Interest
It is desirable to know at any time how much is due on
principal, allowing for any partial payments. It is also de-
sirable to know what interest, if any, is due and payable, and
to be able to look after its collection. An account with
principal and an account with interest are therefore requisite.
It is better, however, if these two accounts for the same
mortgage be adjacent.
§ 165. Interest Debits and Credits
Accrued interest need not be considered as to each
mortgage. It should be treated in bulk, in the same manner
as the revenue of the aggregate mortgages, as will be ex-
plained hereafter. The Interest account (on the investor's
books) here referred to is debited on the day when the inter-
est becomes a matured obligation, and credited when that
obligation is discharged.
§ 166. Characteristics of Modern Ledger
Those who adhere to the original form of the Italian
ledger will probably be averse to combining with the ledger
account any general business information ; in fact, that form
is not suited for such purposes, and is not adapted to con-
taining anything but the bare figures that will make the
J J
REAL ESTATE MORTGAGES
127
trial balance prove. But the modern conception of a ledger
is broader and more practical : it should be an encyclopedia
of information bearing on the subject of the account; it
should be specialized for every class ledger ; it should be of
any form which will best serve its purposes, regardless of
custom or tradition.
§ 167. The Mortgage Ledger
The form of mortgage ledger which seems best to the
author contains four parts :
(1) Descriptive.
(2) Account with principal.
(3) Account with interest.
(4) Auxiliary information.
These may occupy four successive pages, or two pages, if
preferred. In the latter case, if kept in a bound volume, the
arrangement whereby two of these parts are on the left-
hand page and two on the right, confronting each other, is
a convenient one, giving all the facts at one view. For a
loose-leaf ledger, the order (1), (2), (3), (4) will generally
be found the best.
§ 168. Identification of Mortgages by Number
Mortgages should be numbered in chronological order,
and every page or document should bear the number of the
mortgage loan to which it refers.
§ 169. The "Principal" Account
The account with principal may be in the ordinary
ledger form; but what is known as the balance-column, or
three-column form, will be found more convenient. It con-
tains but one date column, so that successive transactions,
whether payments on account, or additional sums loaned,
appear in their proper chronological order.
128 THE MATHEMATICS OF INVESTMENT
§ 170. Special Columns for Mortgagee's Disbursements
The mortgage usually contains clauses which permit the
mortgagee, when the mortgagor fails to make any necessary
payment for the benefit of the property, like taxes and in-
surance premiums, to step in and advance the money, which
he has the right to recover with interest. It will be use-
ful to have columns for these disbursements and the corre-
sponding reimbursements (§ 179, Form II).
§ 171. The Interest Account
The Interest account (§ 179, Form III) may be very
simple. It contains two columns, one for debits on the day
when interest falls due, the other for crediting when it is
collected. The entries in the Interest account will naturally
be much more numerous than in the Principal account;
hence, this pair of columns may be repeated several times.
The arrangement shown has been found advantageous.
§ 172. Interest Due
Experience has shown that the safest way to insure at-
tention to the punctual and accurate collection of interest is
to charge up, systematically, under the due date, every item,
and to let it stand as a debit balance until collected. Many
attempt to accomplish the same purpose by merely marking
"paid" on a list; but this is apt to lead to confusion, and
it is difficult to verify afterward the state of the accounts
on any given date.
§ 173. Books Auxiliary to Ledger
It is not proposed in. this treatise to prescribe the
forms of posting mediums (cash book, journal, etc.) from
which the postings in the ledger are made, because these
forms are so largely dependent upon the peculiarities of the
REAL ESTATE MORTGAGES 1 29
business, and have deviated so far from the traditional
Italian form, that no universal type could be presented. We
shall, however, give the debit and credit formulas underlying
the postings, and will suggest auxiliary books or lists for
making up the entries.
§ 174. The "Due" Column
The formula for the "Due" column of the Interest
account is :
Interest Due / Interest Accrued $
It is a transfer from one branch of interest receivable, viz.,
that which is a debt, but not yet enforceable, to another
branch, viz., that which is a matured claim.
§ 175. Interest Account Must Be Analyzed
In the general ledger the entry will be simply as above :
Interest Due / Interest Accrued $
and this may be a daily, weekly, or monthly entry, or for
any other space of time, according to the general practice
of the business ; the monthly period is most in use, and we
shall take that as the standard. The credit side of the entry
(/ Interest Accrued) is not regarded in the subordinate
ledger (§160), but the debit entry (Interest Due /) must
be somewhere analyzed into its component parts; in other
words, there must be somewhere a list, the total of which is
the aggregate falling due on all mortgages, and the items
of which are the interest falling due on each mortgage.
§ 176. Form of "Interest Due" Account
The following heading will suggest the requirements
for such a list, the form to be modified to conform to the
general system.
130 THE MATHEMATICS OF INVESTMENT
Register of Interest Due
Mortgages
Date
No.
Principal Rate
Time Interest Total
§ 177. Forms for Mortgage Account
Form I (§ 179) of the Mortgage account is descriptive.
Its elements may be placed in various orders of arrange-
ment. Form IV (§ 179) combines all the particulars or-
dinarily required in the State of New York.
§ 178. Loose-Leaf and Card Records
Form IV (§ 179) is not an essential feature of mortgage
loan accounts, and may be replaced by card lists, if pre-
ferred. Yet, if there is space, there are advantages in hav-
ing all the information about a certain mortgage accessible
at one time, and concentrated in one place. The changing
names and addresses of the mortgagors and owners, and the
successive policies of insurance require for their record
considerable space, which may be conveniently arranged
under the headings in Form IV.
The card form of mortgage ledger is very convenient
in many respects, and the forms here given may be re-
arranged to suit different sizes of cards. Both in cards and
loose leaves it will be helpful to use different colors for
pages of different contents. Where interest on different
mortgages fall due in different months, tags marked "J J,"
"F A,'* "M S," "A O," "M N," and "J D," may project
from the interest-sheet like an index, the tags of each
month at the same distance from the top. This will greatly
facilitate the compiling of the register of interest due.
REAL ESTATE MORTGAGES
§ 179. Forms of Mortgage Loan Accounts
131
Pi
o
^
i2
en
g
a
6 H
1
<4
a
M
>
fl
'S
«
1^
II - II i
i
i
!
i
1
(0 S
^"
ZI I
5
II - 1
1 1
m
Q
H
OS .
1— (
K 1
W
fu
«
P^
: i
pq
. ....
1
2; 6 • : -^ 6 i V
Oh
/3 ^ pq H:r ^ rt H^
So 5
« 5
H A
o
© ©
o « S
cq a -<
i
^
1^
a
»
P.
00
n
«
d
^
J£
n
iS
»
t-
on
o»
r»
0*
O)
Oi
Oi
a
S
a»
gj
(^
?S
Oi
o>
Oi
f-i
0*
G<
r»
CM
Oi
Oi
Oi
p^
i-t
-2
'a
b-
CD
Oi
e
-^
^
a>
Oi
b
1— 1
'"'
1-1
e8
(fi
c
H
IX!
ed
<«1
s
H
p.
Tf"
*o
CO
Oi
Ci
Oi
<
13
32
4
^
HE MATHEMATICS OF INVESTMENT
w^
GO
H
>5
H
a
8
H
Bt3
g
lg
H
tf
Mft
^C,
» H
D C
1
2 !5
»5
>4
<
P
>«
<
hJ
t!
J
n
2
S;
M
O
Q
M
■<
fa
Oh
g
<J
o
O
55
<U
n
J
<
Q
6
55
REAL ESTATE MORTGAGES
U3
o
i
134
THE MATHEMATICS OF INVESTMENT
O
(3h
■^'
g
a
^
H
Z
a
»
s;
o
H
M
S
X
a
z
H
-ll
Z
«
o
z
6
SS
t«
o
1
3
o
»«
|H
25
<!
(1<
a
o
o
H
«
Q
g
<
2:
O
H
o
•:]
I
Q
H
^
Z
»
»
M
O
P4
<
p
H
Ph
O
1
00
s
cc
z
»
^
M
o
Q
Q
<*!
»
H
■"U
6
Q
S5
1
REAL ESTATE MORTGAGES
135
§ 180. Reverse Posting of Interest Register
The interest register should always be made up and
proved (subjected to modifications) in advance. In doing
this, instead of making the computations in the register and
posting thence to the ledger, a surer way is by "reverse
posting" ; that is, making the computation from the data in
the ledger and entering it there at once, in pencil if pre-
ferred; then copying the items into the register, where the
total can be proved. When this has been done, we can be
sure without further check that the ledger is correct. (See
also §183.)
§ 181. Handling Receipts and Notices
It is desirable, also, to have receipts prepared in advance
ready for signature. The correctness of these receipts may
be assured by introducing them into the "reverse posting"
process, as follows: Having made the computation on the
ledger, prepare the receipts from the ledger, copying down
the figures just as they appear; from the receipts make up
the register, which prove as before. This method may be
extended to the notices, if any are sent to the mortgagors,
the notice being derived from ledger account, the receipt
from the notice, and the register from the receipt; if the
register proves correct, the correctness of its antecedents is
established. These interest notices may be made of as-
sistance in the bookkeeping, if their return is insisted upon
and made convenient. Below the formal notification of the
sum falling due on such a date, with all particulars, is a
blank form somewhat as follows :
In payment of the above interest I inclose check on the
for $ and
request you to acknowledge receipt as per the address below.
Signature
Address
136
THE MATHEMATICS OF INVESTMENT
The notice upon its being received, together with the
check, becomes a "voucher-with-cash," and the entries on
the cash book and the interest page of the mortgage ledger
are made directly from the documents. Book-to-book post-
ing, which formerly was the only method of rearranging
items, is becoming obsolete, being superseded in many busi-
nesses by voucher or document posting. By the carbon
process the notice and the receipt may be filled in simulta-
neously in fac-simile.
§ 182. Mortgages Account in General Ledger
The class account "Mortgages" in the general ledger is
simply kept to show aggregates. Its entries are, as far as
possible, monthly, the posting mediums being so arranged
as to give a monthly total of the same items which have al-
ready been posted in detail to the mortgage ledger. The
standard form of ledger account may be used, or the three
column. In the former, the debits and credits of the same
month should be kept in line, even though one line of paper
be wasted.
(Standard Form)
Mortgages
1914
1914
Jan.
Balance
$169,000
00
1-31
Total loaned
12,000
00
Jan.
1-31
Total paid in
$ 7,000
00
Feb. 1-28
<( «
10,000
00
March 1-31
« X
50,000
00
March
1-31
« « <t
32,000
00
April 1-30
..
20,000
00
April
1-30
<< << <<
40,000
00
May 1-31
<l «
5,000
00
May
1-31
« « t<
12,000
00
June 1-30
<( t<
10,000
00
June
1-30
<t << <<
3,000
00
Balance
00
30
Balance
182,000
00
$276,000
$276,000
00
July
$182,000
00
REAL ESTATE MORTGAGES
137
(Three-Column Form)
Mortgages
Dk.
Cr.
Balance
1914
Jan.
$169,000
00
"
Transactions for month
$ 12,000
00
$ 7,000
00
174,000
00
Feb.
♦. .t
'
10,000
00
184,000
00
March
«
'
50,000
00
32,000
00
202,000
00
April
« «
'
20,000
00
40,000
00
182,000
00
May
« «
'
5,000
00
12,000
00
175,000
00
June
Transactions for half-year
10,000
00
00
3,000
00
00
182,000
00
$107,000
$94,000
+ $ 13,000
00
July
$182,000
00
§ 183. Tabular Register
In order to keep the fullest control of the interest accru-
ing and falling due periodically, it is useful to keep tabular
registers, classifying the mortgages, first, by rates of in-
terest; and second, by the months in which the interest
comes due. Those investors who require all interest to be
paid at the same date can dispense with the latter. The two
presentations or developments may be on opposite pages,
both proved by the same totals.
Form I
Mortgages Classified by Rates of Interest
Date
Total
Changes
3^%
4%
4H%
5%
6%
1914
Jan.O
$169,000
7,000
262-
984 +
$11,000
$43,000
7,000
$50,000
12,000
$60,000
$5,000
$162,000
12,000
Feb.O
$174,000
$11,000
$36,000
$62,000
$60,000
$5,000
138
THE MATHEMATICS OF INVESTMENT
Form II
Mortgages Classified by Interest Dates
Date
Total
Changes
JJ
FA
MS
AG
MN
i -J.
JD
1914
Jan.O
$169,000
7,000
262-
984 +
$23,000
$30,000
12,000
$4,000
$8,000
$90,000
$14,000
7,000
$162,000
12,000
Feb.O
$174,000,
$23,000
$42,000
$4,000
$8,000
$90,000
$ 7,000
The numbers in the column headed "Changes" are the
serial numbers of the respective mortgages.
§ 184. Equal Instalment Method
Mortgages payable in equal instalments, each covering
the interest and part of the principal, present no special
difficulty. The value of the periodical instalment should
first be ascertained, as shown in § 76 ; then it should be
separated by means of a schedule into "Interest on Balances"
and "Payments on Principal," down to the final payment.
CHAPTER XV
LOANS ON COLLATERAL
§ 185. Short Time Loans on Personal Property
Short time investments are often made upon the security
or pledge of bonds, stocks, goods, or other personal property
valued at more than the amount of the loan. Frequently
these are payable on demand, and are known as "call loans."
It is evident that the rate of interest may be readjusted
every day, or as often as either party is dissatisfied, and,
if an agreement cannot be reached, the loan will be paid
off. Hence, neither premium nor discount will occur in this
kind of investment, and, as in the case of mortgages, we
need only concern ourselves with principal (at par) and
interest.
§ 186. Forms for Loan Accounts
The accountancy of loans is even simpler than that of
mortgages, and it is only necessary to give two forms, one
for Principal account and Interest account, and the other for
the ''Register of Collateral" (§ 188). The latter account, at
least, is often kept on cards or on envelopes, and there is
great danger of the history becoming confused and unin-
telligible through erasures and changes in the amounts of
collateral, when substitutions are made. When part of a
certain security is withdrawn, the' entire line should be
ruled out, and the reduced quantity rewritten on a new line.
When a card becomes at all complicated, it is better to insert
a fresh one, rewriting all collateral, but keeping the former
card with the new one until the loan is entirely liquidated.
139
I40 THE MATHEMATICS OF INVESTMENT
§ 187. Requirements for Interest Account
The Interest account may be kept concurrent with the
Principal account — that is, using up the same number of
lines in each. In the suggested form there is a column for
interest accrued as well as for interest due. The interest
accrued column is merely a preparatory calculation column,
entered up at each change of rate or principal, so that there
may be only one computation to make when the interest
becomes due. With this exception, the mechanism of the
loan ledger is practically the same as that of the mortgage
ledger, and the general ledger account of loans will be
similar to that of mortgages.
As the principal and the interest in bond accounts are
so intimately connected, it will be advisable to consider the
accounting of interest revenue more fully before taking
up the subject of bond accounts. This is done in the follow-
ing chapter.
§ 188. Forms for Collateral Loan Accounts
On the following page are shown two suggested forms
for use in connection with the records for collateral loans.
These forms are merely suggestive, and in this respect are
like other forms presented in the various chapters on book-
keeping records. Very few banks or trust companies handle
their accounts in exactly the same way, and changes and
additions will therefore be necessary or advisable in making
use of the forms suggested, in order to meet the particular
requirements of individual companies :
LOANS ON COLLATERAL
141
o
n
0)
U
S
Q
—
85
Ti
55
<
9
a}
«
•J
f3
«
ca
OQ
(H
•0
<s
iJ
<U
PK
f)
s
2
M
m
n
PL4
m
ja
u
>
.^
aJ
S
a>
»
H
©
55
es
«^
Q
a
c
1
(D
'
Form II
Collateral Security
(4
1
s
6
CHAPTER XVI
INTEREST ACCOUNTS
§ 189. Functions of the Three Interest Accounts
Interest is earned and accrues every day; then, at con-
venient periods, it matures and becomes collectible; then or
thereafter it is collected and takes the form of cash. These
three stages may be represented by the bookkeeping
formulas :
(1) Interest Accrued / Interest Revenue
(2) Interest Due / Interest Accrued
(3) Cash / Interest Due
Frequently the three accounts, Interest Revenue, Interest
Accrued, and Interest Due, are confused under the one title
"Interest," although they have three distinct functions. In-
terest Revenue (which alone may be termed simply "Inter—,
est") shows how much interest has been earned during the
current fiscal period. The balance of Interest Accrued shows
how much of those earnings and of the earnings in previous
periods has not yet fallen due. The balance of Interest Due
shows how much of that which has fallen due remains
uncollected.
The first of the three entries is the only one which imports
a modification in the wealth of the proprietor; the other
two are merely permutative, representing a shifting from
one kind of asset to another. It is not the mere collecting
of interest which increases wealth; nor is it merely the
coming-due of the interest : it is the earning of it from day
to day.
142
INTEREST ACCOUNTS 143
§ 190. A Double Record for Interest Earned
Interest accrued need not, and cannot conveniently, be
computed on each unit of investment, as we have already
stated. But it can readily be computed on all investments
of the same kind and rate of interest, and the aggregate
(say for a month) will form the basis for the entry ''Interest
Accrued / Interest Revenue." Or a daily rate for the entire
investment may be established, and this may be used without
change, day after day, until some change in the principal
or in the rate causes a variation in the daily increment. The
most complete and accurate method is to keep a double
register of interest earned : first, by daily additions ; second,
by monthly aggregates, classified under rates and time.
§ 191. Example of Interest Income
To illustrate this, we will take a period of ten days
instead of a month, and assume that the investments are in
mortgages only. On the first day of the period there is
$100,000 running at 4%, $60,000 at 41/2%, and $150,000
at 5%. On the second day, $10,000 at 4% is paid off, and
on the fifth day $5,000 at 5%. On the seventh day a loan
of $15,000 is made at 41/2%, and one of $6,000 at 5%. We
begin by computing the daily increment, as follows :
One day at 4% on $100,000 $11.1111
One day at 41/2% on $ 60,000 7.50
One day at 5% on $150,000 20.8333
Total daily increment $39.4444
§ 192. Daily Register of Interest Accruing
The decimals are carried out two places beyond the
cents, and rounded only in the total. The daily register will
then be conducted as follows :
144
THE MATHEMATICS OF INVESTMENT
Daily Register of Interest Accruing
For the month of , 1914
Date
No.
of
Loan
Decrease in
Principal
Increase in
Principal
Rate
Working
Column
Daily
Increment
1
2
3
4
5
6
7
8
9
10
647
453
981
982
$10,000
5,000
$ 15,000
6,000
4
5
4^
5
$39.4444
1.1111
$ 39.4444
39.4444
38.3333
38.3333
38.3333
37.6388
37.6388
40.3472
40.3472
40.3472
$38.3333
.6944
$37.6388
1.875
.8333
$15,000
$ 21,000
$390.21
Balances
at Close
$ 90,000
75,000
151,000
4
4H
5
Proof
of Rate
One day
$ 10.
9.375
20.9722
$316,000
$ 40.3472
>
§ 193. Monthly Summary
The monthly register or summary takes up, first, the
mortgages upon which payments are made, then those re-
maining to the end of the month, whether old or new. Its
result will corroborate that of the daily register.
The monthly register or summary of interest accruing
may be kept in the following form. As the loans are paid
off, the interest accrued is entered up in the last column.
New loans negotiated, or increases in principal, are entered
in column four, and the interest accruing to date of pay-
ment is carried to the last column in a similar manner.
INTEREST ACCOUNTS
Monthly Summary of Interest Accruing
For the month of , 1914
145
Date
No.
of Loan
PakioflE
Remaining
Rate
Days
Monthly
Increment
2
5
7
«
10
<•
««
647
453
981
982
$10,000
5,000
$ 15,000
6,000
90,000
60,000
145,000
4
5
4J^
5
4
5
2
5
3
3
10
10
10
$ 2.2222
3.4722
5.625
2.50
100.00
75.00
201.3888
$316,000
$390.21
§ 194. Method and Importance of Interest Earned Account
The daily and monthly registers of interest earned may
be in separate books or in one book — ^preferably the latter
in most cases. A convenient arrangement would be to use
two confronting pages for a month, one and one-half pages
for the daily register, and one-half page for the monthly
register. If an accurate daily statement of affairs is kept,
the daily interest accrued will form part of that system.
Again, the interest on mortgages, on bonds, on loans, or on
discounts may be separated or be all thrown together. In
all such respects the individual circumstances must govern,
and no precise forms can be prescribed. Our main conten-
tion is that in some manner interest should be accounted for
when earned rather than when collected, or when due.
§ 195. Interest Accounts in General Ledger
The general ledger accounts of Interest, Interest Ac-
crued, and Interest Due will now be illustrated in simple
form as to mortgages only. It is easier to combine the
several kinds of interest, when carrying them to the Profit
and Loss account, than to separate them if they are all
thrown in together at first.
146
THE MATHEMATICS OF INVESTMENT
Form I — Interest Revenue
Mortgages
1914
June 30
Carried to Profit
and Loss
$4270
60
60
$4270
— II
1914
Jan.
Feb.
March
April
May
June
1-31
1-28
1-31
1-30
1-31
1-30
Total Earnings
$ 654
708
723
756
719
708
$4270
Form II — Interest Accrued
Mortgages
1914
1914
Jan.
Balance
$2362
50
1-31
Earnings
654
58
Jan. 1-31
Due
$1272
50
Feb.
1-28
"
708
25
Feb. 1-28
"
125
00
March
17
Cash for Accrued
on No. 987
58
33
1-31
Earnings
723
34
March 1-31
"
875
00
April
1-30
"
756
67
April 1-30
"
625
00
May
1-31
"
719
44
May 1-31
1200
00
June
1-30
"
708
33
June 1-30
"
65
00
Balance
44
30
Balance
2528
94
$6691
$6691
44
July
$2528
94
Form III — Interest Due
Mortgages
1914
1914
Jan.
Balance
$ 125
00
1-31
Due
1272
50
Jan. 1-31
Collections
$1325
00
Feb. 1-28
"
125
00
Feb. 1-28
197
50
March 1-31
«
875
00
March 1-31
850
00
April 1-30
"
625
00
April 1-30
600
00
May 1-31
«
1200
00
May 1-31
1200
00
June 1-30
It
65
00
June 1-30
100
00
Balance
50
30
Balance
15
00
$4287
$4287
50
July
$ 15 Ool
INTEREST ACCOUNTS
147
§ 196. Payment of Accrued Interest
There is one entry in Interest Accrued account which
does not arise from earnings : the accrued interest on Mort-
gage No. 987, which is paid for in cash on March 17, the
mortgage not having been made direct with the mortgagor,
but purchased from a previous holder. This case occurs
frequently in bond accounts, but not so often in connection
with mortgages.
CHAPTER XVII
BONDS AND SIMILAR SECURITIES
§ 197. Investments with Fluctuating Values
The investments heretofore considered are interest bear-
ing, but bear no premium nor discount ; the variation from
time to time is in the rate of interest, while the principal is
invariable. When we consider investments whose price fluc-
tuates, while the cash rate of interest is constant, the problem
is more difficult, because there are several prices which it
may be desired to record, viz., the original cost, the market
value, the par, and the book value or amortized value. The
original cost and the par are the extremes : one at the be-
ginning, and one at the end of the investment. The book
values are intermediate between these, and represent the
investment value, falling or rising to par by a regular law,
which maintains the net income at a constant rate. The
market value is not an investment value, but a commercial
one ; it is the price at which the investor could withdraw his
investment, but until he has done so, he has not profited by
its rise, nor lost by its fall. So long as he retains his
investment, the market value does not affect him, nor should
it enter into his accounts. It is valuable information, how-
ever, from time to time, if he has the privilege of changing
mvestments, or the necessity of realizing.
§ 198. Amortization Account
The account with principal, showing at each half-year
the result of amortization, is very suitably kept in the three-
148
BONDS AND SIMILAR SECURITIES
149
column or balance-column form recommended in § 169 for
mortgages. Thus, the history of the bonds in Schedule (F),
§ 139, would be thus recorded in ledger form :
$100,000 Smithtown 5's of May 1, 19
19
Date
Dr.
Cr.
Balance
1914 May 1
Nov. 1
1915 May 1
Nov. 1
1916 May 1
Purchased from A. B. & Co.
Amortization (4%)
$104,500
$410.97
419.19
427.57
436.12
$104,089.03
103,669.84
103,212.27
102,806.15
§ 199. Effect on Schedule of Additional Purchases
In case of an additional purchase the account will, of
course, be debited and cash credited. It will then be neces-
sary to reconstruct the schedule from that point on. This
may be done in either of two ways: (1) make an indepen-
dent schedule of the new purchase, and then consolidate this
with the old one, adding the terms; or (2) add together the
values of the old and new bonds at the next balance date;
find what the basis of the total is, eliminate any slight resi-
due (§§137 to 140, inclusive), and proceed with the
calculation.*
§ 200. The Bond Sales Account
In case of a sale, the procedure is different. Instead of
crediting the Bond account by cash, it is best to transfer the
amount sold to a Bond Sales account at its book value com-
puted down to the day of sale ; Bond Sales account will then
show a debit, and the cash proceeds will be credited to the
same account. The resultant will show a gain or loss on the
sale, and at the balancing date the account will be closed into
* Bonds purchased flat should be separated into principal and interest.
I50
THE MATHEMATICS OF INVESTMENT
Profit and Loss. Thus, in the example in § 198, we will
suppose a sale on August 1, 1916, of half the $100,000 at
102.88, or $51,440. We find the book value of the $50,000
on August 1, which is $51,291.86; we transfer this to the
debit of the Bond Sales account in the general ledger, which
account we credit with the $51,440 cash proceeds. Bond
Sales is purely a Profit and Loss account, and at the proper
time will show the actual profit realized on the sale, $51,440
— $51,29L86 = $148.14.
Form I — Bond Ledger
$100,000 Smithtown 5's of May 1, 1919
Date
Dr.
Cr.
Balance
1914 May 1
Purchased from A. B. & Co.
$104,500
Nov. 1
Amortization
$ 410.97
$104,089.03
1915 May 1
««
419.19
103,669.84
Nov. 1
"
427.57
103,242.27
1916 May 1
"
436.12
102,806.15
Aug. 1
Sale to C. D. & Co.
$50,000 @ 102.88
51,291.86
51,514.29
«
Amortization on $50,000
111.21
51,403.08
Nov. 1
" on balance
222.43
51,180.65
Form II — General Ledger
Bond Sales
1914
Aug. 1
Smithtown 5's
$51,291.86
1914
Aug 1
Proceeds
$51,440.00
To adjust the profit in the Bond account itself would be
as unphilosophical as the old-fashioned Merchandise account
before the Purchases and Sales accounts were introduced,
and even more awkward.
BONDS AND SIMILAR SECURITIES
151
§ 201. Requirements as to Bond Records
Besides the book value of a bond, the par is also needed
because the cash interest is reckoned upon the par. For
some purposes, also, it is useful to show the original cost.
We must, therefore, provide means for exhibiting these three
values: the par, the original cost, and the book value. A
mere memorandum of par and cost at the top would be
sufficient where the group of bonds in question will all be
held to the same date; but this is not always the case, and
provision must be made for increase and decrease. The
three-column form of ledger (§169), constantly exhibiting
the balance, is the most suitable for this purpose also. But
if we endeavor to display all of these forms side by side, we
require nine columns, and this makes an unwieldy book.
The most practical way is to abandon the use of debit and
credit columns, and proceed by addition and subtraction, or
in what the Italians term the scalar (ladder-like) form,
which gives a perfectly clear result, especially if the balances
are all written in red. Headed by a description of the bonds,
and embracing, also, a place for noting the market value at
intervals (not as matter of account, but of information), the
Principal account will appear as shown in Form I (page
152).
§ 202. Form of Bond Ledger
As far as the bond ledger is concerned, the transfer of
the $50,000 sold to Sales account is final ; we have, however,
in the example indicated (§ 200), a way of incorporating a
statement of the profit or loss in the margin for historical
purposes. The amortization of November 1, 1916, is com-
posed of two parts : 3 months on the $50,000 sold, $111.21;
and the regular 6 months on the $50,000 retained, $222.43.
In the example given in § 200, these are entered separately;
either method may be pursued, but on the whole there are
greater advantages in postponing all entries of amortization
152
THE MATHEMATICS OF INVESTMENT
Jz;
8
<
<
u
o
o ^
Is
Q
#,
<U
-^7r^
0)
3
g
8
8
8
811
1
V
8 S § i §
M
Tt^ CO CO CO '-H
u
O O O O m
a
»— 1 T-H r-( »-H
§
^9-
O t^
fO o\
Tj- t^
ts. <\i
in vo
0\ Tf
m
V
o o\
O -H
00 m
CM 1-H
'I °9
CM VO
vo
^
o o
On 0\
On rvl
CM vd
vo '-<
Tt CO
g
>
00 '-H
VO CM
Tj- CO
O ON
-H CO
s ^
O TT
^. -^
^.^
00 (N
in CO
M
Tj-"
Tf
CO
CO
cm" ^
,_^
,_r
g
o
o
o
O
O m
m
«n
T-H
PQ
</3-
m-
^
m-
^.^
<«■
*«-
In
o
s
8
8
c3
8
O
lO
m
"rt
in
CM
CM
.s
S
CM*^
m
"C
o
^
4/^
V
8
S.
8
r3
3
g
g
8
>
o
O
•^i.
u
o"
o"
o
fs
o
in
m
^
€^
8 2
1^
,
»-?
o
in
u
<«-
o3
*«
'♦H O
1
M
£
<
a §
o -^
-« c
Q -2
N
r ^ ^
\J N
-6 "^
' ' ' 2 -g
bo o
iij o
§ a
-S B
pq<
c/3 <;
u
•g
1
o
>
CVl
•— >
>^ >
>» > >. bb >
■M
rt o
rt o t^ a o
CO
Q
s;^;
S ^ ^ < 55
"■t
in vo
T-H ,-<
o\
0\ ON 11
T-«
i-i
U
BONDS AND SIMILAR SECURITIES
153
till the end of the half-year. The three months' amortiza-
tion of the bonds sold is in effect implied in the price $51,-
291.86, which is reduced by the amortization ($111.21)
from $51,403.07, the half of $102,806.15, but it need not
be entered till November 1.
§ 203. Interest Due Account
The register of interest due on bonds is conducted on
precisely the same principles as that described for mortgages
in § 183 ; in fact, they are but subdivisions of the same
register. Of course, only the cash interest is considered.
§204. Interest Account — Bond Ledger
The interest pages of the bond ledger are also similar to
those of the mortgage ledger (§ 183), but the dates of in-
terest due may be printed in advance, there being but little
chance of partial payments disturbing their orderly ar-
rangement.
The paging of the bond ledger will probably be geo-
graphical, as far as possible, in respect of public issues, and
alphabetical in respect of those of private corporations. The
loose-leaf plan permits an indefinite number of classifica-
tions from which to choose. The date tags suggested in
§ 178 are especially useful for pointing out dates for interest
falling due, as "J J," "F A," etc.
§ 205. Amortization Entries
The entries of amortization are made directly from the
schedules of amortization, the preparation of which was
discussed in Chapter X. But it is necessary, also, to make
up a list of these several amortizations in order to form the
general ledger entry :
Amortization / Bonds
or. Amortization / Premiums
154
THE MATHEMATICS OF INVESTMENT
according to the form of the general ledger. This list
should be in the same order as the bond ledger. Probably
the most practical way is to combine it with the trial balance
of the bond ledger, thus giving at each fiscal period a com-
plete list of the holdings, which may give the par, cost, book,
and market values, the titles of the securities being written
but once. The total of the second column will form the basis
of the entry for amortization. The next three columns will
corroborate the general ledger balances.
Bond Statement for the Half- Year Ending
Name and
Description
Amorti-
zation
Book Value
Par Value
Original
Cost
Market
Value
In the cases of bonds bought at a discount, the analogous
general ledger entry would be :
Bonds / Accumulation
or. Discounts / Accumulation
We have provided in the preceding form for amortization
only and not for accumulation on bonds below par. Where
the latter values are few in number they may be embraced
in the same column, but distinguished as negatives by being
written in red or encircled. If the bonds below par are
numerous, there should be two columns : * 'amortization" and
"accumulation."
§ 206. Bond Entries in General Ledger
While the book value is the proper one to be introduced
BONDS AND SIMILAR SECURITIES 155
into the general ledger, the par is very necessary, and some-
times the cost, and these requirements inevitably introduce
some complexity. There are two methods effecting the
purpose :
(1) By considering the par and cost as extraneous in-
formation, and ruling side columns for them
beside the book value.
(2) By dividing the account into several accounts, by
the proper combination of which the several
values may be obtained.
The first plan will preserve the conformity of the Bonds
account with the bond ledger better than the other. The
Bonds account may, if necessary, be extended across both
pages of the ledger, to allow for three debit and three credit
columns, if all are required.
The second method will commend itself more to those
having a repugnance to introducing into the general ledger
any figures beyond those actually forming part of the trial
balance. The theory on which it is based is that the premium
is not part of the bond, but is a sum paid in advance for
excess interest, while the discount is a rebate returned to
make good deficient interest. This is a perfectly admissible
way of looking at the matter, especially from the personalis-
tic point of view; for the debtor does not owe us the
premium and has nothing to do with it. Still the other view,
which regards the investment as a whole, is also correct,
and wx may adopt whichever is most suitable to our purpose.
§ 207. Accounts Where Original Cost Is Disregarded
If original cost is disregarded, or deemed easily obtain-
able when required, the accounts may be :
156 THE MATHEMATICS OF INVESTMENT
(a) Bonds at Par
(b) Premiums
(c) Discounts
or,
(a) Bonds at Par
(b) Premiums and Discounts
If premiums and discounts are kept separate, Premiums
account must always show a debit balance, being credited
for amortization; Discounts account must show a credit
balance, being debited for accumulation. If the two are
consolidated, only the net amortization will be credited
(§ 205) ; or if the greater part of the bonds were below par,
the net accumulation only would be debited. The choice be-
tween one account and two for premiums and discounts is
largely a question of convenience.
The management of such a double or triple account is
obvious, entries of transactions being divided between par
and premiums, or par and discounts, but we give in § 214 an
example of each.
We shall hereafter confine the discussion to premiums,
leaving the cases of discount to be determined by analogy.
§ 208. Amortization Reserve
Where it is deemed necessary to keep account of cost
also, as well as of par and book value, the difficulty is some-
what greater, as we have a valueless or extinct quantity to
record, namely, so much of the original premium on bonds
still held as has not yet been absorbed in the process of
amortization. This carrying of a dead value, which is some-
what artificial, necessitates the carrying, also, of an artificial
annulling or offsetting account, the sole function of which
is to express this departed value. We may call this credit
BONDS AND SIMILAR SECURITIES
157
account "Reserve for Amortization." It is analogous to
Depreciation and Reserve for Depreciation. The part of the
premiums which has been extinguished bytcredits to Reserve
for Amortization may be designated as "Premiums Amor-
tized," or "Ineffective Premiums," while the live premiums
may be styled "Effective Premiums," being what in § 207
we called simply "Premiums." A double operation takes
place in these accounts : first, the absorption of effective
premiums by lapse of time; and second, the cancellation of
ineffective premiums upon redemption or sale.
§ 209. Premiums and Amortization
There are two ways of handling these accounts, differ-
ing as to premiums. We may keep two accounts : "Effec-
tive Premiums" and "Amortized Premiums," or we may
combine these in one, "Premiums, at Cost." The entire
scheme will be :
(a) Bonds at Par
(b) Premiums at Cost
(e) Reserve for. Amortization
or,
(a) Bonds at Par
(c) Effective Premiums
(d) Amortized Premiums
(e) Reserve for Amortization
"a" will in both schemes be the same ; "e" will also be the
same, "b" is the sum of "c" and "d." In the former, the
cost is a + b, while the book value is a + b — e. In the latter
the book value is a + c, while the cost is a + c + d. The
former gives the cost more readily than the latter, and the
book value less readily. The former might be considered the
more suitable for a trustee; the latter, for an investor.
158
THE MATHEMATICS OF INVESTMENT
Account (a), Bonds at Par, is debited for par value of
purchases and credited for par value of sales. Its only two
entries are :
Bonds at Par / Cash (or some other asset)
Cash (or some other asset) / Bonds at Par
In case of purchase at a premium, the premium is
charged to Premiums at Cost or to Effective Premiums, as
the case may be, there being no ineffective premiums at this
time.
§ 210. Writing Off Premiums
When premiums are written off, on the first plan illus-
trated in § 209 there is but one entry : crediting Reserve for
Amortization and debiting the Profit and Loss account or
its subdivision.
Amortization / Reserve for Amortization
The second plan involves not only this process, but a
transfer from Effective to Amortized Premiums. Thus, the
aggregate of premiums written off is posted four times as a
consequence of the separation of premiums at cost into two
accounts: if
Premiums Amortized / Effective Premiums
Amortization / Reserve for Amortization
§ 211. Disposal of Amortization
The word "Amortization" has been used in the illustra-
tive entries as the title of an account tributary to Profit and
Loss. At the balancing period it may be disposed of in
either of two ways : It may be closed into Profit and Loss
direct ; or it may be closed into Interest account, the balance
of which will enter into Profit and Loss at so much les-
•BONDS AND SIMILAR SECURITIES icq
sened a figure. By the former method the Profit and Loss
account will show, on the credit side, the gross cash inter-
est, and on the debit side the amount devoted to amortiza-
tion ; the second method exhibits only the net income from
interest on bonds. Whether it be preferable to show both
elements, or only the net resultant, will be determined by
expediency.
§ 212. Amortization Accounting — Comparison of Methods
In §§ 200 and 202 we discussed two methods of keeping
account of amortization: the first (in §200), where any
incidental amortization occurring in the midst of the period
is at once entered; the second (in §202), where all such
entries are deferred to the end of the period., and comprised
in one entry in the general ledger. If the latter method be
adopted, the Amortization account may be dispensed with
altogether, and the total amount amortized (which is
credited to Bonds, or to Premiums, or to Reserve for
Amortization) may be debited at once to Profit and Loss or
to Interest, without resting in a special account. A single
item, of course, needs no machinery for grouping.
§ 213. Irredeemable Bonds a Perpetual Ai^nuity
Irredeemable bonds (§ 146) merely lack the element of
amortization, and require no special arrangement of ac-
counts. The par is purely ideal, as it never can be demanded
and is merely a basis for expressing the interest paid. What
the investor buys is a perpetual annuity. If he buys an
annuity of $6 per annum, it is unimportant whether it is
called 6% on $100 principal, or 4% on $150 principal; and
this $150 may be the par value, or it may b.e $100 par at
50% premium, or $200 par at 25% discount. The par value
is really non-existent.
l6o THE MATHEMATICS OF INVESTMENT
§ 214. Bond Accounts for General Ledger
In the present section are shown the forms for the gen-
eral ledger outlined in §§ 206-212. We will suppose that
on January 1, 1915, the following lots of bonds are held :
January 1, 1915
Par Book Value
$100,000 5% Bonds, J J,
due Jan. 1, 1925, net 2.7% ; value. .$120,039.00
Original cost, $124,263.25
100,000 3% Bonds, M N,
due May 1, 1918, net 4%; value. . 96,909.10
Original cost, $93,644.28
10,000 4% Bonds, A O,
due Oct. 1, 1916, net 3%; value.. 10,169.19
Original cost, $10,250.00
$210,000 Totals $227,117.29
The premiums on the 5% and 4% bonds amount to
$20,208.19. The discount on the 3% bonds is $3,090.90.
The net premium is $17,117.29. The total original cost
was $228,157.53.
BONDS AND SIMILAR SECURITIES
i6i
o
<
W
O
«
o
P^
H
O
u
u
<
o
M>
^ o
o o th eo CO
tT" O CI O C5
d d \6 ci d
O C I- lO l-Ii
-f C -^ r}< Tf
O
O
s.a
I o
l62 THE MATHEMATICS OF INVESTMENT
Bond Accounts for General Ledger — Plan II (§ 207)
Dr. Bonds at Par Cr.
1915
Jan. 0,
Balance $210,000.00
1916
Oct. L
Redeemed. $10,000.00
Dr.
Premiums
Cr.
1915
1915
Jan. 0,
Balance $20,208.19
June 30,
Amortization.
..$926.94
Dec. 31,
«
.. 939.54
1916
June 30,
((
.. 952.28
Dec. 31,
i(
.. 940.21
1917
June 30,
u
... 927.93
Dec. 31,
it
.. 940.47
Dr.
Discounts
Cr.
1915
1915
June 30,
Accumulation.. .
.$438.18
Jan. 0,
Balance $3,090.90
Dec. 31,
«
. 446.95
1916
June 30,
«
. 455.88
Dec. 31,
it
. 465.00
1917
i
June 30.
«
. 474.30
1
Dec. 31.
«
. 483.79
1
BONDS AND SIMILAR SECURITIES
163
Bond Accounts for General Ledger — Plan III (§ 207)
(Original cost omitted)
Dr.
Bonds at Par
Cr,
1915
Jan. 0,
Balance $210,000.00
1916
Oct. 1
Redeemed $10,000.00
Dr.
Premiums and Discounts
Cr.
1915
Jan. 0,
Balance $17,117.29
1915
June 30,
Dec. 31,
Amortization.
..$488.76
.. 492.59
1916
June 30,
Dec. 31,
u
u
.. 496.40
.. 475.21
1917
June 30,
Dec. 31,
It
.. 453.63
... 456.68
164
THE MATHEMATICS OF INVESTMENT
Bond Accounts for General Ledger — Plan IV (§ 209)
("Bonds at Par" as in foregoing plans)
Dr.
Premiums at Cost
Cr.
1915
Jan. 0,
1918
Jan. 0,
Balance $18,157.53
Balance,
$18,157.53
.$17,907.53
1916
Oct. 1,
1917
Dec. 31,
Canceled at Re-
demption... .$ 250.00
Balance 17.907.53
$18,157.53
Dr.
Reserve for
Amortization Cr.
1916
1915
Oct. 1,
Canceled at Re-
Jan. 0.
Balance $1,040.24
demption.. . .$ 250.00
June 30,
Amortization. . . 488.76
Dec. 31,
... 492.59
1917
Dec. 31,
Balance 3,653.51
1916
June 30,
... 496.40
Dec. 31,
... 475.21
1917
June 30,
... 453.63
Dec. 31,
... 456.68
$3,903.51
$3,903.51
1918
Jan. 0,
Balance $3,653.51
BONDS AND SIMILAR SECURITIES
i6S
Bond Accounts for General Ledger — Plan V (§ 210)
(By the balance column method)
Bonds at Par
Dr.
Cr. Balance Dr.
1915
Jan.
1916
Oct. 1
Balance
Redemption.
$210,000.00
$10,000.00
$210,000.00
200,000.00
Effective Premiums
Dr,
Cr. Balance Dr.
1915
Jan.
June 30
Dec. 31
1916
June 30
Dec. 31
1917
June 30
Dec. 31
Balance
Amortized
$ 17,117.29
$ 488.76
492.59
496.40
475.21
453.63
456.68
$ 17,117.29
16,628.53
16,135.94
15,639.54
15,164.33
14,710.70
14,254.02
Ineffective or Amortized
Premiums
Dr.
Cr. Balance Dr.
1915
Jan.
June 30
Dec. 31
1916
June 30
Oct. 1
Dec. 31
1917
June 30
Dec. 31
Balance
Amortized
«
u
Canceled by Redemption
Amortized
$ 1,040.24
488.76
492.59
496.40
475.21
453.63
456.68
$ 250.00
1,040.24
1,529.00
2,021.59
2,517.99
2,267.99
2,743.20
3,196.83
3,653.51
i66
THE MATHEMATICS OF INVESTMENT
Reserve for Amortization
Dr.
Cr. Balance Cr.
1915
Jan.
June 30
Dec. 31
1916
June 30
Oct. 1
Dec. 31
1917
June 30
Dec. 31
Balance
Amortized
((
Canceled by Redemption
Amortized
$ 250.00
$ 1,040.24
488.76
492.59
496.40
475.21
453.63
456.68
1,040.24
1,529.00
2.021.59
2,517.99
2,267.99
2,743.20
3,196.83
3.653.51
CHAPTER XVIII
DISCOUNTED VALUES
§ 215. Securities Payable at Fixed Dates Without Interest
The securities heretofore considered have all carried a
stipulated rate of interest or annuity. There is another class
to which no periodical interest attaches, but the obligation is
simply to pay a single definite sum on a certain date. The
present vakie of that sum at the current or contractual rate
of income is, of course, obtained by discounting according
to the principles explained in Chapter XL If the maturity
were more than one year distant at the time of discount, it
would be necessary to compute the compound discount ; but
in practice this never occurs, such discounts being for a few
months.
The obligations discounted in this manner are almost
invariably promissory notes. Formerly they consisted large-
ly of bills of exchange; hence the survival in bookkeeping
of the words "Bills Receivable," "Bills Payable," and "Bills
Discounted."
These obligations belong rather to mercantile and bank-
ing accountancy than to investment accountancy. The
arrangement of accounts for recording their amounts, classi-
fication, and maturity has been so fully treated in works on
those branches that we refer to them here only for the
purpose of illustrating another phase of the process of secur-
ing income.
§ 216. Rates of Interest and Discount
The difference between the rate of interest and the rate
167
l68 THE MATHEMATICS OF INVESTMENT
of discount has been pointed out in Chapter II. It was
there shown that in a single period the rate of interest 3%
corresponds to the rate of discount .029126. Hence, if we
discount a note for $1.00 at 2.9126%, we acquire interest at
the rate of 3% on the $.970874 actually invested. The rate
of interest is always greater than the rate of discount.
§ 217. Rate of Discount Named in Notes
It is usual to name a rate of discount rather than a rate
of interest in stipulating for the acquisition of notes. For
example, a three months' note for $1,000 is taken for dis-
count at 6% (per annum). This means that $.015 is to be
retained by the payee of the note from each dollar, and the
amount actually paid over is $985. The income from this
is the $15, and by dividing 15 by 985 we readily ascertain
that the rate of interest realized is 6.09%. It is sometimes
believed that there is a kind of deception in this; that the
borrower agrees to pay 6% and actually has to pay 6.09%.
But this is not so : the bargain is not to pay 6 % interest, but
to allow 6% discount, which is a different thing.
§ 218. Form as Affecting Legality
Curiously, the lawfulness or unlawfulness of a trans-
action sometimes depends upon the mere form of words in
which it is expressed. Thus, suppose that A lends $985 to
B, who promises to repay $1,000 at the end of 3 months. If
B's promise reads : *T promise to pay $1,000," A is a law-
abiding citizen; but if B writes : "I promise to pay $985 and
interest at 6.09% per annum," the statute prohibiting usury
is violated.
§ 219. Entry of Notes Discounted
Notes discounted are usually entered among the assets
at the full face, and the discount credited to an offsetting
DISCOUNTED VALUES
169
account, "Discounts," the latter having precisely the same
effect as the Discounts account used in connection with
bonds. The difference of the two is the net amount of the
asset. Strictly speaking, the discount is at first an offset to
the note, and represents at that time nothing earned what-
ever; as time goes on, the earning is effected by diminution
of this offset, which is equivalent to a rise in the net value
of the note, from cost to par. In § 220 the process is shown
by the state of the accounts at the initial date and at the end
of each month up to maturity, for a 3 months' note for
$1,000, discounted at 6%.
§ 220. Discount and Interest Entries
Note
$1000.00
(i) When Discounted
Discount
$15.00
(2) At the End of One Month
Note Discount Interest Revenue
$1000.00
$ 5.00
$15.00
$5.00
(3) At the End of Two Months
Note Discount Interest Revenue
$1000.00
$ 5.00
5.00
$15.00
$5.00
5.0U
Note
$1000.00
(4) At Maturity
Discount Interest Revenue
$ 5.00
5.00
5.00
$15.00
$15.00
$15.00
$5.00
5.00
5.00
lyo THE MATHEMATICS OF INVESTMENT
§ 221. Total Earnings from Discounts
Since notes are issued generally for short periods, the
gradual crediting of earnings illustrated in § 220 is usually
ignored. At the date when the books are closed, an inven-
tory should be taken of the discounts unearned. The differ-
ence between the amount of this inventory and the net credit
in the Discounts account represents the earning from dis-
counts during the fiscal period, and this earning should then
be transferred to Profit and Loss. The unearned discounts
may be easily computed by finding the discount on each
note from the date of closing the books to the respective
dates of maturity. The investment value of the notes on
hand at the close of the fiscal period will be the difference
between the par and the unearned discount. Expressed in
a formula, the earnings from discounts may be found as
follows :
Unearned discounts at beginning of fiscal period,
+ discounts credited during period,
— unearned discounts at end of period,
= earnings from discounts during period.
Part II — Problems and Studies
CHAPTER XIX
INTEREST AND DISCOUNT
§ 222. Problems in Simple Interest*
(1) What is the time in months and days from January
10th to :
(a) June 12th?
(b) July 4th?
(c) September 1st?
(2) What date is:
(a) Two months after June 30th ?
(b) Four months after May 31st?
(c) Two months after December 31st?
(d) Five months and seven days after September
26th?
(3) On a loan of $54,750, interest payable semi-annually
at 4% per annum, interest was last paid to and including No-
vember 1 : compute the interest accrued on the following
February 25th :
(a) In the customary manner, legal in New York
before 1892.
(b) Assuming that the odd days are 365ths of a
year.
• In connection with the text of Chapter II. For answers see § 224.
171
172
PROBLEMS AND STUDIES
(c) Compute the same by both methods at 4%%.
(d) " " " " " " " 5%.
(e) " " " " " " " 6%.
(4) On a 365-day basis, the interest for 17 days, on a
certain sum, at a certain rate, was $83.73 ; what would have
been the interest on a 360-day basis ?
(5) The interest for 19 days on a certain sum at a
certain rate was $2,185.00 on a 360-day basis; compute
the interest on a 365-day basis.
§ 223. Notes on the One Per Cent Method
Observe that when days are considered as 360ths of a
year, it is useful to know how many days correspond to one
per cent. For example, if the rate is 3%, it takes 120 in-
terest days to earn 1% interest.
At 3 % the number of days for 1% is 120
At 4 %
(( ((
" 90
At 41/2%
({ (f
" 80
At 5 %
(t ((
" 72
At 6 %
(( (t
" 60
At 8 %
(t ((
" 45
At 9 %
(( t(
" 40
For purposes of calculation we may set down the num-
ber of days corresponding to 1% at the given rate, and in
line with it the principal, pointing off two places from the
right in the principal in order to obtain 1%. Thus, in
Problem (3) of the preceding section :
90 days $547.50
meaning that the interest for 90 days at 4% is $547.50.
Knowing the interest for 90 days, we can build up that for
114 days (3 months and 24 days on the 360-day basis).
24 days = 15 days + 9 days. 15 days is 1/6 of 90 days; 9
INTEREST AND DISCOUNT
173
days, 1/10. Dividing the interest for 90 days by 6 to secure
the interest for 15 days, and by 10 to secure the interest
for 9 days, and adding, gives the result :
90 days $547.50
15 " 91.25
9 " 54.Y5
114 " $693.50
The same result may be obtained, and just as easily, by
the combination 90 + 18 + 6. Sometimes the work may be
shortened by the use of subtraction ; in the present case, no
time would be saved by this method, the result working out
as follows :
90 days $547.50
30 " $182.50
less 6 (1/5 of 30) 36.50 146.00
114 $693.50
In the case of problem (3-d), on the 5% basis, the result
would work out as follows :
72 days $547.50
18 " (1/4 of 72) 136.875
24 " (Ys of 72) 182.50
114 " $866,875
Rates like 7% or 3%%, which are not exact divisors of
360, must be obtained from the exact rates by division and
addition. Thus, 7% is derived by adding 1/6 to 6%, which
is obtained as follows :
ly^ PROBLEMS AND STUDIES
60 days $547.50
30 " 273.75
20 " 182.50
4 " 36.50
114 " $1,040.25 (interest at Q%)
add 1/6 173.375
$1,213,625 (interest at 7%)
§ 224. Answers to Problems in Simple Interest
Problem (1)
(a) 5 months, 2 days
(b) 5 months, 24 days
(c) 7 months, 22 days
Problem (2)
(a) August 30th
(b) September 30th
(c) February 28th or 29th
(d) March 4th or 5th
Problem (3)
(a) $693.50
(b) $691.50
(c) $780.19 (360-day method)
$777.94 (365-day method)
(d) $866,875 (360-day method)
$864,375 (365-day method)
(e) $1,040.25 (360-day method)
$1,037.25 (365-day method)
Problem (4)
$84.89
Problem (5)
$2,155.07
INTEREST AND DISCOUNT
175
§ 225. Problems in Compound Interest*
(6) Find the amount of $1 at 2% per period, correct to
six decimals :
(a) For one period
(b) For two periods
(c) For three periods
(d) For four periods
(e) For five periods
(7) Find the present worth of $1 at 2%, correct to six
decimals :
(a) For one period
(b) For two periods
(c) For three periods
(d) For four periods
(e) For five periods
In § 29, several methods are mentioned for finding the
present worth; assuming that the solutions for problem
(6) above have been found, the easiest method of finding
the present worth for five periods would be to divide 1 by
the amount for five periods. The present worths for 1, 2,
3, and 4 periods can then be found by multiplying the present
worth for five periods successively by 1.03.' This is much
easier than dividing 1 successively by 1.03.
(8) Find the amount of $1 at 1^^% (.0176) per
period :
(a) For one period
(b) For two periods
(c) For three periods
(d) For four periods
(e) For five periods
(f) For six periods
* In connection with the text of Chapter II, For answers see § 226.
176
PROBLEMS AND STUDIES
(9) Find the present worth of $1 at 1.75% per period :
(a) For one period
(b) For two periods
(c) For three periods
(d) For four periods
(e) For five periods
(f) For six periods
(10) Find the amount and the present worth of $1,-
000.00 for eight periods at 1.5% per period.
(11) What is the rate of discount corresponding to 2%
interest ?
(12) What is the rate of interest corresponding to the
discount rate of .0384615?
(13) Three notes for $1,000.00 each, due (without in-
terest) at three months, six months, and one year respec-
tively, are discounted at 6% :
(a) If the proceeds of the first note are $985, find
the equivalent interest rate.
(b) If the proceeds of the second note are $970,
find the equivalent interest rate.
(c) If the proceeds of the third note are $940,
find the equivalent interest rate.
(14) What is the compound interest on $1 for five
periods at 2% ?
(15) What is the compound discount on $1 for four
periods at 2% ?
§ 226. Answers to Problems in Compound Interest
Problem (6)
(a) $1.02
(b) $1.0404
(c) $1.061208
(d) $1.082432
(e) $1.104081
INTEREST AND DISCOUNT
177
Problem (Y)
(a) $.980392
(b) $.961169
(c) $.942322
(d) $.923845
(e) $.905731
Problem (8)
(f) $1.109702
Problem (9)
(f) $.901143
Problem (10)
Amount, $1,126.493 ; present worth, $887,711
Problem (11)
.0196078 (Observe that this decimal when divided
by 2%, the rate of interest, gives the present
worth for one period, .98039. This will be a
test for all similar computations.)
Problem (12)
4%
Problem (13)
(a) 1.52284% quarterly, or (nominally)
6.09137% annually
(b) 3.09278% semi-annually, or (nominally)
6.18557% annually
(c) 6.38298% annually
Problem (14)
$.104081
Problem (15)
$.076155
§ 227. Proof of Amount and Present Worth
The amount and the present worth of the same sum for
178
PROBLEMS AND STUDIES
the same time and rate should, when multiplied together,
give the product 1,
Problems (6) and (7) give the amount of $1 for 5
periods at 2% as $1.104081, and' its present value for the
same time and rate as $.905731. These numbers multiplied
together should give as a product, unity. Such multiplica-
tions of decimal numbers are best performed by beginning
at the left of the multiplier.
1.10408
1
.905731
993672
9
5520
405
772
8567
33
12243
1
104081
1.000000 388211
The vertical line is drawn to cut off the figures beyond
the 6th decimal, which have no utility except to furnish a
carrying amount for the 6th figure. They may be dis-
pensed with by using contracted multiplication.
§ 228. Contracted Multiplication
In this process the subproducts are shortened at each
step by one figure, taking into account, however, the carry-
ing amount from the rejected figures.
1.104081
.905731
(first 6 figures X 9)...
. 993673
(first 5 figures X 0) . . .
(first 4 figures X 5)...
5520
(first 3 figures X 7)
773
(first 2 figures X 3)...
33
(first figure X 1)...
1
1.000000
INTEREST AND DISCOUNT
179
Here we commence to multiply by 9 at the sixth figure,
8 ; the product would be 72, but we know that the rejected
1, X 9, would make the product nearer 73 ; this subproduct,
therefore, becomes 993673. In each of these partial prod-
ucts the last retained figure is slightly increased, if neces-
sary, by mental allowance for the next rejected figure. The
last figure of the final product will, even then, not always be
exact, but may vary one or two units from the correct prod-
uct. In all multiplications by rounded decimals, there is
an error, small it is true, in the product; this final error
may be reduced to as small a quantity as desired, by in-
creasing the decimal places in the factors to such extent
as the accuracy of the work may require.
It sometimes happens in contracted multiplication that
you "lose your place" and forget at what figure of the
multiplicand to begin next. This may be overcome by tick-
ing off each figure as you have done with it ; or by repeating
the multiplier figures from left to right and (at the same
time) the multiplicand figures from right to left. In the
above illustration the correlated figures would be 9-8, 0-0,
5-4, 7-0, 3-1, and 1-1.
§ 229. Problems in Use of Logarithms*
The following problems are elementary and the 4-place
table given in § 43 may be used in their solution.
(16) What is the logarithm of :
(a) 3 (d) 1.8 (g) .54
(b) 30 (e) .0018 (h) 1.03
(c) 3,000 (f) 5.4
(17) Give the number whose logarithm is:
(a) .1614 (c) 1.6474 (e) 3.6474
(b) 2.3838 (d) 1.6474 (f) .0212
* In connection with the text of Chapter III. For answers see § 231.
l8o PROBLEMS AND STUDIES
(18) Find the logarithm of :
(a) 291.5 (b) 4.362 (c) .027433
(19) Find the number whose logarithm is :
(a) 2.5849 (b) 1.38425 (c) 3.6931
(20) Prove by logarithms that :
(a) 9X8 = 72
(b) 7X1.12 = 7.84
(c) .032X300 = 9.6
(d) .004X4000 = 16
(21) Show by logarithms that:
(a) 72 -f- 2.4 = 30
(b) 12.5 -^ 625 = .02
(c) 5.2^.04 = 130
(22) What is the 28th power of :
(a) 1.02 (b) 1.04
(23) What is the present worth of $1 for 45 periods at :
(a) 3% (b) 5%
(24) Find by logarithms the value of the following :
829 X 76.3 X .0484 -v- 7.28 -^ 25
§ 230. Problems Requiring Use of More Extended Tables
of Logarithms*
For further exercise in logarithmic computations, Prob-
lems (14) to (18) inclusive should again be worked out,
using logarithms to the limit of such tables as may be at
hand. The logarithms of all of the ordinary ratios of in-
crease (1 + i), with which the operation always begins, will
be found in Part III. These logarithms have been com-
puted to fifteen places of decimals.
The following examples, which are for too many periods
to be worked out arithmetically, may also be worked by
For answers see § 231.
INTEREST AND DISCOUNT igi
logarithms. If no other tables are available, the four-place
tables in § 43 may be used, although these tables cannot be
relied upon to bring correct results to as many decimal places
as are given in the solutions.
(25) Find the amount and present worth of $1 ;
(a) At 1.25% for 30 periods
(b) At 1.70% for 50 periods
(c) At 2.00% for 10 periods
(d) At 2.40% for 68 periods
(e) At 2.50% for 70 periods
§231. Answers to Problems in Logarithms
Problem (16)
(a) .4771
(b) 1.4771
(c) 3.4771
Problem (17)
(a) 1.45
(b) 242
Problem (18)
(a) 2.4647
Problem (19)
(a) 384.5
Problem (20)
(a) log. 9 = .9542 ; log. 8 = .9031. The sum of
these two logarithms is 1.8573, which is
the logarithm of 72. Similarly for (b),
(c), and (d).
Problem (21)
(a) log. 72 = 1.8573 ; log. 2.4 = .3802. The first
logarithm minus the second is 1.4771,
which is the logarithm of 30. Similarly
for (b) and (c).
(d)
(e)
(f)
.2553
3.2553
.7324
(g) 1.7324
(h) .0128
(c)
(d)
44.4
.444
(e) .00444
(f) 1.05
(b)
.6397
(c) 2.4383
(b)
.24225
(c) 4933
l82 PROBLEMS AND STUDIES
Problem (22)
(a) log. 1.02 is .0086; multiplied by 28 gives
.2408, which logarithm corresponds to
the number 1.741. The correct result to
eight decimal places is given in Part IV,
being 1.74102421.
(b) 2.99; to eight decimal places, the result is
2.99870332.
Problem (23)
(a) log. 1.03 is .012.8, which multiplied by 45
gives .5760. Then log. (1^1.03"') =
zero minus .5760, or 1.4240. The num-
ber corresponding to this last logarithm
is .265 ; to eight places the result is
.26443862. .
(b) .111, and to eight places, .11129651.
Problem (24)
log. 829 =2.9186
plus log. 76.3 -=1.8825
plus log. .0484 =2.6848
minus log. 7.28 = .8621
minus log. 25 = 1.3979
Net result =1.2259, which is the logarithm
corresponding to the number 16.8; the
result by actual multiplication and divi-
sion is 16.82105.
Problem (25)
(a) Amount, $1.45161336
(b) " $2.32299164
(c) " $1.21899442
(d) " $5.01645651
(e) " $5.63210286
present worth, $.68888867
. " " $.43047938
" $.82034830
" $.19934390
" $.17755358
CHAPTER XX
PROBLEMS IN ANNUITIES AND IN NOMINAL
AND EFFECTIVE RATES
§ 232. Problems in Annuities*
(26) Find the amounts and present worths of an an-
nuity of $1 :
(a) At 1.25% for 30 periods
(b) At 1.70% for 50 periods
(c) At 2.00% for 10 periods
(d) At 2.40% for 68 periods
(e) At 2.50% for 70 periods
In Problem (26), a to e inclusive, assume that the present
worth in each case is a loan, and construct a schedule show-
ing the gradual repayment of this loan at $1 per period, for
a few periods or for the entire time.
§ 233. Answers to Problems in Annuities
Problem (26)
(a) Amount, $36.129069
(b) " $77.823037
(c) " $10.949721
(d) " $167.352355
(e) " $185.284114
present worth, $24.888906
" $33.501213
" $ 8.982585
" $33.360671
" $32.897857
In connection with the text of Chapters IV and V.
183
l84 PROBLEMS AND STUDIES
§ 234. Problems in Rent of Annuity and Sinking Fund*
(27) What is the rent of an annuity of 30 periods
valued at $1,000 if the rate of interest is 1.25% per period?
In other words, what is each term of an annuity the present
worth of which is $1,000, the interest earned being 1.25%
per period and the number of periods 30 ?
(28) Assume the same present worth as in (27), and
find the rent of an annuity under the following conditions :
(a) 1.70%, 50 periods
(b) 2.00%, 10 periods
(c) 2.40%, 68 periods
(d) 2.50%, 70 periods
(29) What is the sinking fund to be reserved at the
end of each period and invested at 1.25%, to amount to
$1,000 at the end of 30 periods ?
(30) Compute the sinking funds for the same data as
in (a), (b), (c), and (d), in (28) above.
(31) What amount should be laid aside each half-year
to amount to $100,000 at the end of 50 years at 4% per
annum, interest payable semi-annually?
(32) What amount at 3% ?
(33) A father wishing to make a gift of $10,000 to his
son, now 15 years old, on the latter's 21st birthday, deposits
a certain sum at a trust company, on a 4% annual basis, on
the 16th and each succeeding birthday, including the 21st,
sufficient to amount to the $10,000 when the last deposit is
made. Find the required annual deposit.
(34) Assume that after the annual deposit is made on
the 18th birthday, the trust company states that the interest
rate thereafter on deposits is to be only 3% annually. Find
the annual amount which should be deposited on the 19th,
* In connection with the text of Chapter VII.
ANNUITIES
i8S
20th, and 21st birthdays in order to reach the desired
$10,000.
(35) On July 1, 1914, a company decides to accumu-
late a sinking fund of $100,000 by July 1, 1921, assuming
that interest on the fund will be at the rate of 4% per
annum. It is expected that annual contributions to the fund
of $12,000 each will be made at July 1, 1917, 1918, 1919,
1920, and 1921. Find the two equal contributions re-
quired at July 1, 1915 and 1916, in order that the seven con-
tributions, with accumulated interest, may amount to $100,-
000 at July 1, 1921.
§ 235. Answers to Problems in Rent of Annuity and Sink-
ing Fund
Problem (27)
$40.17854
Problem (28)
(a) $29.84967
(b) $111.32653
(c) $29.975416
(d) $30.39712
Problem (29)
$27.67854
Problem (30)
(a) $12.84967
(b) $91.32653
(c) $5.975416
(d) $5.39712
Problem (31)
$320.27
Problem (32)
$437.06
l86 PROBLEMS AND STUDIES
Compare the answers to Problems (27) and (29)";
(28-a) and (30-a) ; (28-b) and (30-b) ; (28-c) and (30-c) ;
and (28-d) and (30-d), respectively. Note that the differ-
ences between these five pairs of answers are in proportion
to the respective five rates of income.
Problem (33)
$1,507.62
Problem (34)
$1,571.53
Problem (35)
$14,103.35
§ 236. Problems in Nominal and Effective Rates*
(36) If the interest rate is 12% per annum, payable in
monthly instalments, what is the effective annual rate?
(37) If the interest is 12% payable semi-annually, what
is the effective annual rate?
(38) What is the nominal rate per annum which, if paid
semi-annually, is equivalent to an effective rate of .99505%
per quarter?
(39) (a) If the nominal rate is 4% per annum, payable
semi-annually, what nominal rate per annum, payable
quarterly, will produce the same income ?
(b) What is the equivalent nominal annual rate, payable
monthly ?
(40) Interest being 6% per annum, payable quarterly
(the effective rate per annum being therefore 1.015^),
which is the more valuable — an income of $4,080, payable
at the end of the year, or an income of $4,000, of which
$1,000 is payable at the end of each quarter?
(41) Interest being worth 5% per annum converted
• In connection with the text of Chapter VIII.
NOMINAL AND EFFECTIVE RATES 187
quarterly, what rate should be paid annually as an equiva-
lent? (Note that the expressions "payable annually," "pay-
able quarterly," etc., signify — through custom — that the in-
terest is payable at the end of the year, quarter, etc. When
interest is paid before the end of the interest period, an
element of discounting enters in.)
(42) (a) Given 5% as the effective annual rate; de-
scribe the process of finding the effective quarterly rate
equivalent thereto.
(b) What is the quarterly rate so found?
(c) To what nominal annual rate is this quarterly
rate equivalent ?
(43) A note for $1,000, due in one year, is discounted
at the beginning of the term, the net proceeds being $940 :
(a) What is the discount rate?
(b) What is the interest rate which is actually
being paid?
(44) If the above note were for six months and the net
proceeds were $970, what would be the nominal annual in-
terest rate?
(45) Suppose the above note were for three months and
the net proceeds $985 ; find the nominal annual interest rate.
§ 237. Answers to Problems in Nominal and Effective
Rates
Problem (36)
12.68%
Problem (37)
12.36%
Problem (38)
4%
l88 PROBLEMS AND STUDIES
Problem (39)
(a) 3.98%
(b) 3.97%
Problem (40)
The latter, by $10.90
Problem (41)
5.095%
Problem (42)
(a) Find the 4th root of 1.05.
(b) 1.2272%
(c) 4.9088%
Problem (43)
(a) 6%
(b) 6.383%
Problem (44)
6.186%
Problem (45)
6.091%
§ 238. Constant Compounding
In § 93 it was stated that if an investment on a 6%
nominal annual rate were compounded every millionth of a
second, or constantly, the effective annual rate could never
be so great as 6.184%. It may be interesting to know how
to ascertain this limit. The following rule gives the method :
M^i Rule : Multiply the constant quantity .4342944819 +, or
\jjKj . so much thereof as is necessary, by the nominal rate per
^^'^' annum expressed decimally; find the logarithm of the
p V^. product ; from this logarithm, subtract 1, and the remainder
is the effective annual rate required.
For example, take a 6%nominal annual rate. .4342944819
X .06 = .026057668914. But this latter number is the
NOMINAL AND EFFECTIVE RATES
189
logarithm of 1.061837, which, diminished by 1, gives
.061837, which is the limit required.*
§ 239. Finding Nominal Rate
The opposite rule for finding a nominal rate which, if
compounded an infinite number of times, will amount to a
given effective rate at the end of the year, is as follows :
Rule: Multiply the logarithm of the effective ratio by
the constant quantity 2.302585092994 +, or so much there-
of as is necessary, and the product will be the nominal rate
itself.t
Example: What rate compounded continuously will
amount to an effective rate of 6% ? Log. 1.06 = .02530587 ;
this multiplied by 2.302585 gives .058270, the rate required.
§ 240. Approximate Rules
An approximation to the rate may also be obtained by
* For the benefit of more advanced readers, an algebraic demonstration of
the rule is here given:
(- , .08 \ n
^ "r "^ / =e«<*, when n becomes infinite.
\o^. e.o« = .06 log. e = .06 log. 2.7182818284 = .06 (.4342944819)
= .026057668914 = log. 1.061837. Therefore, e •«• = 1.061837.
(>+f)
Therefore, ( 1 + :^ ) °^ 1.061837, when n becomes infinite.
The quantity e, used above, is the base of the Napierian system of logarithms
and is the sum of the infinite series,
'"■'^i^ti^
t An algebraic demonstration of the rule is as follows :
^^y^'^n) =106, when n becomes infinite, find the value of x, i.e., the nom-
inal rate.
(1 , x\ n X X
'^ n J —^ » when n becomes infinite; or e =1.06.
Therefore, x(log. e) = log. 1.06
Therefor., x = (log 1.06)(-^-) = (log. 1.06) {:~ii^)
= (log. 1.06) (2.302585092994).
190
PROBLEMS AND STUDIES
subtracting from the rate half its square. The square of
.06 is .0036, one-half of which is .0018 ; .06 — .0018 = .0582.
Another approximation may be obtained by taking the
mean between the effective interest rate 06
and the corresponding discount rate .0566
which, added together, give 1166
Half of this is the approximate nominal rate 0583
CHAPTER XXI
EQUIVALENT RATES OF INTEREST— BOND
VALUATIONS
§ 241. Annual and Semi-Annual Interest
The great majority of investments pay interest semi-
annually. Occasionally annual-interest securities are offered,
and it will be useful, for comparison with the ordinary semi-
annual securities, to know the equivalent rates. The fol-
lowing table shows the equivalents for the more common
annual rates, the decimals being carried to the nearest one-
thousandth of one per cent.
Table of Equivalent Rates of Interest Payable
Annually and Semi-Annually
Nominal Rate
Nominal Rate
Per Annum,
Per Annum,
Payable
Payable
Annually
Semi-annually
2.50%
equivalent to
2.485%
2.55%
2.534%
2.60%
2.583%
2.65%
2.633%
2.70%
2.682%
2.75%
2.731%
2.80%
2.781%
2.85%
2.830%
2.90%
2.879%
2.95%
2.929%
191
192
PROBLEMS AND STUDIES
Nominal Rate
Nominal Rate
Per Annum,
Per Annum,
Payable
Payable
Annually
Semi-annually
(Continued)
(Continued)
3.00%
equivalent 1
:o 2.978%
3.05%
tt t
' 3.027%
3.10%
t( (
' 3.076%
3.15%
It (
' 3.126%
3.20%
(( i
' 3.174%
3.25%
it i
' 3.224%
3.30%
it I
' 3.273%
3.35%
it i
' 3.322%
3.40%
tt i
' 3.372%
3.45%
tt <
* 3.421%
3.50%
ti i
* 3.470%
3.55%
tt t
' 3.519%
3.60%
tt t
' 3.568%
3.65%
it I
' 3.617%
3.70%
tt i
' 3.666%
3.75%
tt t
* 3.715%
3.80%
tt (
* 3.765%
3.85%
tt t
' 3.814%
3.90%
it t
' 3.863%
3.95%
tt t
' 3.912%
4.00%
it t
* 3.961%
4.05%
it t
' 4.010%
4.10%
a i
' 4.059%
4.15%
(t t
' 4.108%
4.20%
it t
' 4.157%
4.25%
a i
' 4.206%
4.30%
it t
' 4.255%
4.35%
it t
' 4.304%
EQUIVALENT RATES OF INTEREST
193
Nominal Rate
Nominal Rate
Per Annum,
Per Annum,
Payable
Payable
Annually
Semi-annually
(Continued)
(Continued)
4.40%
equivalent to
4:MS%
4.45%
(( ((
4.402%
4.50%
t( it
4.450%
4.55%
tt tt
4.500%
4.60%
li t(
4.548%
4.65%
i( it
4.597%
4.70%
it ((
4.646%
4.75%
(( ((
4.695%
4.80%
(( tt
4.744%
4.85%
tt tt
4.793%
4.90%
tt tt
4.841%
4.95%
tt tt
4.890%
5.00%
tt it
4.939%
5.25%
it tj.
5.183%
5.50%
it '«
5.426%
'5.75%
se a
5.670%
6.00%
if it
5.913%
6.25%
a tt
6.155%
6.50%
tt it
6.398%
6.75%
it it
6.640%
7.00%
it tt
•6.882%
As an illustration of the use of the^above table, 'take the
annual rate 2.50%. In this case, the square of 1.012425,
which is the semi-annual effective ratio, equals approxi-
mately 1.025, the annual ratio of increase. In the case of
the annual rate 4.45%, the square of 1.02201 equals ap-
proximately 1.0445, etc.
194
PROBLEMS AND STUDIES
§ 242. Semi-Annual and Quarterly Interest
Quarterly bonds also occur, but with less frequency than
semi-annual bonds. Some companies, in order to induce
holders of bonds to register them, pay interest quarterly
after registration, but semi-annually while in coupon form.
Sometimes, therefore, it is desirable to know approximately
how much improvement in income will result from the
quarterly payments.
Table of Equivalent Rates of Interest Payable
Semi-Annually and Quarterly
Nominal Rate
Nominal Rate
Per Annum,
Per Annum,
Payable
Payable
Quarterly
Semi-annually
2.60% equivalent to
2.508%
2.55%
2.558%
2.60%
2.608%
2.65%
2.659%
2.70%'
2.709%
2.75%
2.759%
2.80%
2.810%
2.85%'
2.860%
2.90%
2.910%
2.95%
2.961%
3.00%
3.011%
3.05%
3.062%
3.10%
3.112%
3.15%
3.162%
3.20%
3.213%
3.25%
3.263%
3.30%
3.314%
3.35%
3.364%
3.40%
3.414%
EQUIVALENT RATES OF INTEREST
195
Nominal Rate
Nominal Rate
Per Annum,
Per Annum,
Payable
Payable
Quarterly
Semi-annually
(Continued)
(Continued)
3.45% equivalent to
3.465%
3.50%
3.515%
3.55%
3.566%
3.60%
3.616%
3.65%
3.667%
3.70%
3.717%
3.75%
3.768%
3.80%
3.818%
3.85%
3.869%
3.90%
3.919%
3.95%
3.970%
4.00%
4.020%
4.05%
4.071%
4.10%
4.121%
4.15%
4.172%
4.20%
4.222%
4.25%
4.273%
4.30%
4.323%
4.35%
4.374%
4.40%
4.424%
4.45%
4.475%
4.50%
4.525%
4.55%
4.576%
4.60%
4.626%
4.65%
4.677%
4.70%
4.728%
4.75%
4.778%
4.80%
4.829%
4.85%
4.879%
196
PROBLEMS AND STUDIES
Nominal Rate
Per Annum,
Payable
Quarterly
(Continued)
Nominal Rate
Per Annum,
Payable
Semi-annually
(Continued)
4.90%
4.95%
equivalent to 4.930%
" 4.981%
5.00%
t( (
' 5.031%
5.25%
(t I
' 5.284%
5.50%
t( (
' 5.538%
5.75%
(( i
' 5.791%
6.00%
i( i
' 6.045%
6.25%
t( (
6.299%
6.50%
it i
' 6.553%
6.75%
(t (
' 6.807%
7.00%
(( i
' 7.061%
In illustration of the above table, take the rate 4.20%
given in the first column. The quarterly ratio is then 1.0105.
The square of this is 1.02111025, w^hich is the semi-annual
equivalent earning ratio ; the equivalent semi-annual rate is
2.111025%, and the nominal annual rate equivalent to the
last-named figure is approximately 4.222%.
§ 243. Problems in Valuation of Bonds*
In the following problems, all bonds are supposed to be
semi-annual, unless otherwise stated.
(46) What is the difference between the cash and income
rates of:
In connection with the text of Chapter X.
BOND VALUATIONS I97
(a) 4% bond netting 21/2%
(b) 3% bond netting 21/2^0
(c) 5% bond netting 3.40%
(d) 3% bond netting 3.40%
(e) Y% bond netting 4%
(f) 5% bond netting 4.80%
(g) 3.65% bond netting 5%
(47) Remembering that the premium or discount on a
bond is the present worth of an annuity of the difference in
rates, at the income rate, and that problems have already-
been given involving the computation of present worths at
the foregoing income rates (Problem 26), find the premium
or discount on the following bonds, and hence their value,
par being $1,000 in each case :
(a) 4% bond netting 2%%, 15 years
(b) 3% bond netting 21/2%, 15 years
(c) 6% bond netting 3.40%, 25 years
(d) 3% bond netting 3.40%, 25 years
(e) 7% bond netting 4%, 10 years
(f) 5% bond netting 4.80%, 34 years
(g) 3.65% bond netting 5%, 35 years
§ 244. Successive Method of Bond Valuation — Problems
By adding the net income for one period to each of the
computed values, and subtracting the cash interest, find the
next periodic value at 141/2, 241/2, 91/2, 331/2, and 34%
years, respectively. Continue this operation as many times
as you please, and at any point you may prove your work by
a fresh computation of the annuity.
(48) Find the value of a 4%% bond having a par of
$10,000, netting 3%%, and having three years to run.
From this initial value, work out the values successively
down to par at maturity, and construct a schedule as in
§122.
198 PROBLEMS AND STUDIES
(49) Perform the same operation with:
(a) a 470 bond
(b) a 3% bond
(c) a 2% bond
(50) By the use of logarithms, find the values of the
following bonds of $1,000 each :
(a) 4% bond, netting 4.50%, 95 years
(b) 3%% bond, netting 3%, 401/2 years
(c) 7% bond, netting 4%%, 45 years
(d) 5% bond, netting 4%, 28 years
(e) 31/^% bond, netting 3.80%, 100 years
§ 245. Answers to Bond Valuation Problems
Problem (46)
(a) .75%
(c) .80%
(e) 1.5%
(b) .25%
(d) .20%
(f) .1%
(g) .675%
Problem (47)
(a) $1,186.67
(c) $1,268.01
(e) $1,245.27
(b) $1,062.22
(d) $933.00
(f) $1,033.36
(g) $777.94
Problem (48)
$10,282.45
Problem (49)
(a) $10,141.22
(b) $9,858.78
(c) $9,576.33
Problem (50)
(a) $890.51
(c) $1,480.56
(e) $922.88
(b) $1,116.77 (d) $1,167.52
§ 246. Bond Valuations by the Use of Logarithms
The following will illustrate the method of solution by
logarithms, taking (for example) Problem (50-a). Here
the number of periods is 190, the difference between the
BOND VALUATIONS Iqq
cash and income rates per period is $2.50, and the income
rate is 2.25% per period. We must therefore find the
present worth (P) of an annuity of $2.50 for 190 periods
at 2.25%, and subtract this from the par of the bond
($1,000), since this bond is at a discount, the income rate
being larger than the cash rate. The formula for the
value of the discount on a bond, as given in § 159,
is(t-g) n~ (l-^i)A
which becomes (2.50)( "^ "" 1.0225^^M
^ .0225 /
Now, log. 1.0225 = .00966331668.
Therefore, log. 1.0225''° = 190 X .00966331668 =
1.8360301692.
Hence, ^og.\z^^^^^]-=\og. 1-log. 1.0225"«
= zero — 1.8360301692 = 2.1639698308.
The number corresponding to this logarithm is .014587128.
The value of the discount thus becomes :
2.50
/l --.01458Y128 \
\ .0225 /
which equals $109.49. This discount when deducted from
the par of $1,000 gives the value of the ix)nd, $890.51.
The solution by logarithms involves considerable "figur-
ing,'* but is nevertheless far superior to any solution by
ordinary arithmetic. The labor of finding the present worth
of an annuity for 190 periods by arithmetic would be
intolerable.
200 PROBLEMS AND STUDIES
§ 247. Finding Initial Book Values
The methods of finding the initial book values of the
bonds in Schedules (A) and (B) (§ 122) are not shown
in the text. The operation is here given without logarithms,
and with some variations in method.
Take the case of the bond in Schedule (A), a 5% bond
for $100,000 to net 4%, due in 5 years. The problem is
to find the present value of an annuity of $500 for 10
periods at the ratio 1.02 ; but in the present method we also
require the separate present worths of each instalment of
$500. These ten present worths are the ten respective
amounts of amortization for the ten periods in the life of
the bond.
The present worth of the first instalment of $500 (i.e.,
the first amortization) will be $500 -^ 1.02^^ the present
worth of the second will be $500 -^ 1.02^ ; etc. Since mul-
tiplication is easier than division, it will be best to obtain
first the value of 1-^1.02"; 500 times this will give the
present worth of the first instalment, or the first entry in
the amortization column. From the first amortization, the
second and following ones may be obtained by successive
multiplications by 1.02.
To obtain the value of 1 -^ 1.02'^ we find first the 10th
power of 1.02. After multiplying 1.02 by itself, we do not
again use it as a multiplier, but square the square, giving
the fourth power. The 4th multiplied by the 4th gives the
8th, and the 8th multiplied by the 2nd gives the 10th power
of 1.02, as shown on the following page. A check on the
accuracy of the result may also be obtained by employing the
method suggested in the footnote of § 19. In this latter
case, the process consists in finding the value of (1 + .06)^^
by the use of the algebraic formula known as the binomial
theorem.
BOND VALUATIONS
201
102
102
102
204
10404
41616
41616
(102^) (It is unnecessary to
repeat the multiplier)
108243216
865945728
(102^) = (102^)^
21648643
4329729
(contracted multiplication)
324730
21649
1082
649
11716593810 1
10404
(102)«=(102*)^
11716593810
468663752
4686638
12189944200 (102)^"= (102«) X (102')
§ 248. Tabular Multiplication and Contracted Division
Next, 1 is to be divided by 1.21899442. We shall use
contracted multiplication, and further facilitate the work by
employing the tabular plan. This consists in preparing in
advance a table of the first 9 multiples of 1.21899442 in such
a way that we are certain of their correctness. The use of
a table such as this greatly facilitates accuracy and quickness
in performing the division of several numbers by the same
divisor, especially in cases where the divisor is lengthy and
no calculating machines are available.
202
PROBLEMS AND STUDIES
On the first line of the table we set down the number, and
on the second line, its double.
121899442
243798884
Proof
The third line is formed by adding the first to the second,
and all the others in succession by adding the first. The
proof line is 10 times the original, if there is no mistake in
the work.
121899442
243798884
365698326
487697768
609497210
731396652
853296094
975195536
097094978
Proof
1218994420
The contracted division consists in merely subtracting
these multiples. The quotient may as well be placed above
the dividend to save space.
BOND VALUATIONS
Quotient 820348300
203
Dividend
(8)
1000000000
975195536
(2)
24804464
24379888||
(03)
424576
365698 1
(4)
58878
48760
(8)
10118
9752
(3)
366
366||
$.8203483 is therefore the present v^orth of $1 due in
5 years ; its product by 500 is the first amortization :
$410.17415
Subtracting this from. , 500.
gives the compound discount $ 89.82585
Dividing this by .02 gives 4491.2925 (D-^t = P)
or the premium, rounded to 4491.29 1|
§ 249. Formation of Successive Amortizations
Our amortization column v^ill begin v^ith $410.17, and
each successive term will be 1.02 times the preceding, while
the sum of the column must be $4,491.29. To insure ac-
curacy in the last figure, it will be well to retain at least the
mills. Having obtained all the ten terms, the multiplication
is performed once more, giving as a test $500. The terms
204 PROBLEMS AND STUDIES
are again tested by addition, bringing the result, $4,491.29.
Then the book values beginning with $104,491.29, and end-
ing with $100,000, are formed by subtraction, still retain-
ing the mills. In making up the schedule the values are
rounded to the nearest cent, and the amortization column is
made to correspond.
$410,174
$104,491,292
410.174
418.377
$104,081,118
418.377
426.745
$103,662,741
426.745
435.280
$103,235,996
435.280
443.986
$102,800,716
443.986
452.866
$102,356,730
452.866
461.923
$101,903,864
461.923
471.161
$101,441,941
471.161
480.584
$100,970,780
480.584
490.196*
$100,490,196
490.196
$100,000,000
Total, $4,491,292
• $490,196 X 1.02 = $500
BOND VALUATIONS 205
§ 250. Test by Differencing
In a successive computation like the one just given, a
slight error increases at every step, and there is danger that
a great many terms may have to be recalculated. The
method of differencing, applied during the progress of the
work, will form an efficient check on all except the last
figure.
§ 251. Successive Columns
To difference a series, we first set down its terms in a
first column. In the second column we set down the first
differences (Di), of which the first line is the difference be-
tween the first term and the second, the second line is the
difference between the second and the third, and so on.
D2 is composed of the differences between these first differ-
ences. D3 is formed from D2 in just the same way as Da
from Di, and all succeeding differences in the same way, to
the extent required.
The terms just obtained in amortizing $104,491,292
down to par, would be differenced as follows :
Term
D.
D,
D,
410.174
8.203
.165
.002
418.377
8.368
.167
.004
426.745
8.535
.171
.003
435.280
8.706
.174
.003
443.986
8.880
.177
.004
452.866
9.057
.181
.004
461.923
9.238
.185
.004
471.161
9.423
.189
.003
480.584
9.612
.192
490.196
9.804
500.000
2o6 PROBLEMS AND STUDIES
§ 252. Intentional Errors
To demonstrate the utility of the method, introduce an
error purposely by altering one of the figures in a term at
least three or four lines from the top. Even a mill, when
all the differences are carried out, will cause violent fluctua-
tions in the column IX and instantly call attention to the
error.
§ 253. Rejected Decimals
The reason the fourth column shows some fluctuation
even though no errors have been made, is that the last
figure of a term is never accurate, but always rounded
off or up. In a third difference-column, this residue of error
increases threefold; in a fourth column, it may reach six
times the original rounding, and, in the fifth, ten times.
§ 254. Limit of Tolerance
The extent to which the last column of differences may
be allowed to "waver" will be learned by experience. The
next-to-the-last column should be progressive; that is, it
should never change its course and go backward ; it should
either constantly increase or constantly decrease.
It will be a useful exercise to take the more extended
value, $410.17415 (instead of $410,174), multiply it up to
$500, and difference the results out to 5 differences. A very
minute error will become enormously magnified and call
attention to itself.
CHAPTER XXII
BROKEN INITIAL AND SHORT TERMINAL
BONDS
§ 255. Problems in Valuation*
(51) Suppose the value of a 4% bond for 15 years on
a 2%% basis to be, as shown in Problem (4Y-a), $1,-
186.66680; what would be its value one month later, the
time prior to maturity then being 14 years, 11 months?
(Since we are dealing with half-years, this time must be
treated as 14% years, 5 months, or 15 years less 1/6 of the
semi-annual amortization period.)
The theoretical, or mathematically correct, value (§ 129)
In the above case would be ascertained as follows :
The ratio of increase is 1.0125
Its logarithm is 005 395 031 887
This must be divided by 6, giving .000 899 171 981
which is the logarithm of the 6th
root of 1.0125, or (in other words)
the logarithm of the effective ratio
for 1/6 of a semi-annual period.
The number corresponding to the last
logarithm is 1.002 072 564 8
Multiplying the value at the begin-
ning of the 15-year period ($1,-
186.66680), by this number, gives
the flat value at 14 years, 11 months,
before maturity $1,189,126 21
* In connection with the text of Chapter XI.
207
2o8 PROBLEMS AND STUDIES
Although the above method is never used in actual buy-
ing or selling, yet it is proper for estimating results of
financial operations.
(52) A firm of brokers offers $50,000 of 3% bonds, due
July 1, 1929, J & J, on a 21/2% basis. What should be the
price on September 25, 1914, flat or "and interest"?
(53) On July 10, 1913, $25,000 of 5% bonds due
April 1, 1938, A & O, are bought at a price to yield 3.40%.
(a) What is the flat price ?
(b) What is the price "and interest"?
(54) $10,000 of 3% bonds due January 1, 1938, J & J,
are purchased to net 3.40%. Find the price exclusive of
interest and the price flat, on May 16, 1913.
(55) $6,000 of 41/2% bonds were issued in 1908, due
April 1, 1928, M & N. Find the price "and interest," on
July 1, 1914, on a 4.80% basis.
(56) An investor owns the four lots of bonds men-
tioned in Problems (52), (53), (54), and (55), and has
hitherto carried them on his books at par. He desires to
have them adjusted to investment value as of December 31,
1914. What will be the investment value:
(a) Of each lot?
(b) Of the aggregate?
(57) Find the amount of amortization for the semi-
annual period ending June 30, 1915 :
(a) On each of these lots of bonds.
(b) On the aggregate of the four lots.
(58) Taking the bonds in Problem (53), ascertain their
values at April 1 and October 1, 1937, and thence at July
1, 1937. From this last value, (a) amortize to January 1,
1938, (b) and then for the broken period to April 1, 1938,
when they should reduce to par.
(59) Taking the bonds in Schedule (H) (§141), re-
SHORT TERMINAL BONDS 209
construct the schedule so that the next date after May 1,
1914, is July 1, 1914; then January 1, 1915, and so on at
balancing periods,, giving a J & J schedule instead of an
M & N schedule.
(60) A certain issue of $100,000 of 4% bonds is dated
September 1, 1913, and interest begins at that date; but in-
terest is payable on February 1 and August 1, and the prin-
cipal (with 4 months' interest) is payable December 1, 1917.
(a) What is the value of the bonds on a 3.60%
basis at the date of issue ?
(b) What is their value on the same basis if pur-
chased at December 1, 1913 ?
(c) At August 1, 1917?
(In this question, note that the period at the beginning
is for 5 months, and not the usual 6 months. )
(61) Make a schedule running from December 1, 1913,
to maturity, of the above bonds at the F & A dates.
(62) Make a schedule as above, but with J & J dates,
for balancing purposes.
(63) A broker offers the above bonds on December 1,
1913, at 101.50 (meaning $101.50 for each $100 of par,
which is the customary phrase), which he says will pay
about 3.60%. Eliminate any residue by the methods in
§§ 136 to 139, inclusive, making a J & J schedule* running
to maturity.
As will be noted, this last example contains all of the
following peculiarities: short initial period, odd purchase
date in that period, short terminal period, interpolated
balance dates, and residue to be eliminated.
§ 256. Answers to Valuation Problems
Problem (51)
$1,189.13902 (by the customary method).
2IO PROBLEMS AND STUDIES
Problem (52)
$53,420.93 flat, or $53,070.93 and interest.
Problem (53)
(a) $31,996.64 flat.
(b) $31,652.89 and interest.
Problem (54)
$9,336.43 and interest, or $9,448.93 flat.
Problem (55)
$5,820.34 and interest.
Problem (56)
(a) $53,025.00; $31,392.26; $9,365.30; $5,825.03.
(b) $99,607.59.
Problem (57)
Amortization, $87.19 and $91.33; accumulation,
$9.21 and $4.80; net amortization, $164.51.
Problem (58)
Value at January 1, 1938, $25,098.33; for the
broken period from January 1 to April 1, ■
1938, interest on premium is $1.67, interest
on par is $212.50, and cash interest is $312.50,
thus reducing the bond to par.
Problem (59)
July 1, 1914, $104,693.02 ; January 1, 1915, $104,-
286.88 ; etc. ; July 1, 1919, $100,245.90.
Problem (60)
(a) September 1, 1913, $101,563.90.
(b) December 1, 1913, $101,477.98.
(c) August 1, 1917, $100,131.75.
Problem (61)
Value at February 1, 1914, $101,420.69; etc.
SHORT TERMINAL BONDS 2II
Problem (62)
Value at January 1, 1914, $101,449.33 ; at July 1,
1914, $101,275.34; etc.
Problem (63)
The residue is $22.02, being the difference between
$101,500.00 and $101,477.98. The co-
efficient for elimination of the residue is
1.0148987, meaning that for every dollar of
amortization on the bonds bought at the exact
3.60% basis, there should be added 1.48987c.
if the bonds are bought on the approximate
3.60% basis, i.e., $101,500.00.
CHAPTER XXIII
THE USE OF TABLES IN DETERMINING THE
ACCURATE INCOME RATE
§ 257. Bond Tables as Annuity Tables
The "Extended Bond Tables"* can be used as an annuity
table in case of need, when the latter is not at hand or when
the figures in it are not sufficiently extended or the rates not
sufficiently close.
In using the "Extended Bond Tables" for this purpose,
it must be remembered that its results are based on semi-
annual payments of interest, the periods being half-years. In
the foregoing problems on annuities where periods and rates
per period are used, in order to make use of the bond tables
these "periods and rates per period" must be transformed
into years and rates per annum, payable semi-annually. In
this manner the data given in Problem (26) will be changed
as follows :
1.25%, 30 periods, becomes 2.50%, 15 years.
1.70%, 50 periods, becomes 3.40%, 25 years.
2.00%, 10 periods, becomes 4.00%, 5 years.
2.40%, 68 periods, becomes 4.80%, 34 years.
2.50%, 70 periods, becomes 5.00%, 35 years.
§ 258. Premium and Discount as a Present Worth
As explained in Chapter X, the premium or discount on
a bond is nothing more or less than the present worth, at
Spraguc's "Extended Bond Tables."
212
DETERMINING ACCURATE INCOME RATE 213
the income rate, of an annuity for the life of the bond equal
to the difference between the cash and income rates. Tak-
ing, as an illustration, the second case mentioned above,
3.40% for 25 years, we turn to the 5% bond table, page 88,*
and find the value of such a bond to be $1,268,009.70
The value of a similar bond in the 4% table,
page 54* is 1,100,503.64
Difference $167,506.06
The first amount results from a cash rate of 5% and
an income rate of 3.40% ; in the case of the second amount,
the cash rate is 4%, with the same income rate. The differ-
ence between these two amounts arises therefore on account
of the difference in cash rates, which, for a bond of $1,-
000,000, is $10,000 annually. In other words, the differ-
ence is the present worth of an annuity of $10,000 per
annum, payable semi-annually, at 3.40% for 25 years. Ex-
pressed in periods, it is the present worth of an annuity of
$5,000 per period, for 50 periods, at 1.70% per period. The
present worth of an annuity of $1 per period, under like
conditions, would therefore be 1/5000 of $167,506.06, or
$33.501212.
§ 259. Present Worth by Differences
Instead of using the coupon rates 4% and 5%, we might
have selected 3% and 4%, 31/2% and 41/2%, 5% and 6%,
or any other two rates differing by 1%. For example :
Value of 4% bond, yielding 3.40% $1,100,503.64
Value of 3% bond, yielding 3.40% 932,997.57
Difference, being value of annuity. ......... $167,506.07
*Sprag:ue'8 "Extended Bond Tables.
214 PROBLEMS AND STUDIES
There is a discrepancy of one cent in comparison with
the previous difference, owing to the rounded decimals.
The reason for the process may be explained as follows :
On a 6% bond of $1,000,000 yielding
3.40% the cash, or coupon, interest is. .$50,000
the net income is 34,000
Difference $16,000
In the case of a 4% bond, the interest at
the cash or coupon rate is $40,000
the net income is. 34,000
Difference $6,000
The term "net income,'' as here used, has a slightly
different meaning from its use in the schedules in Chapters
X and XI ; in the latter case, the income rate was applied to
the book value, while in the present instance it is applied to
the par value.
Hence, from the bond tables we may derive the present
worths of two annuities of $16,000 and $6,000 (being re-
spectively $268,009.70 and $100,503.64), and their differ-
ence must always be the present worth of an annuity of
$10,000. From the foregoing, we may state the following :
Rule : The present worth of an annuity of $10,000, pay-
able semi-annually, at a certain income rate, is equal to the
difference between the values of a 4% and a 5% bond for
$1,000,000 at the same income rate.
If it should happen that the rent of the desired annuity
were $5,000 instead of $10,000, the present worth thereof
might be obtained at once from the difference in values be-
tween 3% and 31/2% bonds, or between 31/2% and 4%
bonds. Similarly, the difference between 31/^ % and 5%
bonds would give the present worth of an annuity of $16,-
DETERMINING ACCURATE INCOME RATE
215
000; 3% and 5%, $20,000; 31/2% and 6%, $25,000; 3%
and 6%, $30,000; 31/2% and 7%, $35,000; and 3% and
7%, $40,000. These results would be a trifle more accurate
in the last figure than those obtained by multiplying the
present worths of the $10,000 annuities, since the multiplica-
tion of figures which have been rounded increases the error.
§ 260. Present Worth by Division
The present worth of an annuity may also be obtained
by division from a single bond value, instead of taking the
difference between two. We saw that the premium on a 4%
bond to net 3.40% is the present worth of an annuity of
$6,000, payable semi-annually; therefore, if the premium be
divided by 6, it will give the present worth of an annuity
of $1,000, payable semi-annually :
$100,503.64 -^- 6 = $16,750.61
§ 261. Compound Discount and Present Value of a Single
Sum
From the present worth of an annuity of $10,000 ob-
tained as above, the compound discount and the present
value of a single sum for the same time and rate can also
be ascertained. Multiplying the present worth of the an-
nuity by the number of units in the rate per cent gives the
compound discount on a single sum of $1,000,000.
$167,506.06 times 3.4 = $569,520.60 compound discount
Subtract this from 1,000,000.00
and we have $430,479.40, which is the present
worth of $1,000,000 payable in a single sum in 25 years at
3.40% compounded semi-annually. These computations are
merely applications of the two formulas, PX** = D (§67)
and p = l — D (§35). The last figure in the above present
2l6
PROBLEMS AND STUDIES
worth is unreliable; as a matter of fact, the cents should
be 38.
If necessary, in the absence of compound interest tables
or logarithms, the amount of a single sum at compound in-
terest may be obtained through the application of the
formula a^l^p (§35). The present value of $1 (or p)
is $.4304794; therefore, divide 1 by .4304794, using con-
tracted multiplication.
4304794 ) 1.0000000 ( 2.3229917
8609588
1390412
1291438 1 (The sign | indicates con-
traction or rounding.)
98974
86096|
12878
8610
4268
3874
(For explanation of
contracted multiplica-
394
tion, see § 228.)
387 1
7
4]|
3
§ 262. Use of Bond Tables in Compound Interest Problems
The amount of $1 for 50 periods at 1.70% per period, as
above computed, is $2.3229917, and the compound interest
is $1.3229917. If the latter amount be divided by .017, the
rate of income for a single period, the result ($77.82306)
will be the amount of an annuity of $1 for 50 periods at
DETERMINING ACCURATE INCOME RATE
217
1.70% ; being an application of the rule (§ 60) A = 1 -^i.
Again, when this result ($77.82306, or A) is divided by
2.3229917 (or a) the quotient is $33.5012 + (or P), which
is the present worth of an annuity of $1 for 50 periods at
1.70%. This is an application of the formula (§67)
P = A -^ a. The quotient last obtained checks very closely
with the result previously found for the value of P,
$33.501212. Thus, all of the problems in compound in-
terest are soluble through the bond tables.
§263. Determination of the Accurate Income Rate*
As stated (§ 136), values of bonds for each one-
hundredth of one per cent of gradation in the ordinary in-
come rates may be obtained from Sprague's "Extended
Bond Tables." If, however, an even more minute degree of
accuracy is desired in the income rate, as, for example, a
rate like 4.2678%, these tables are not sufficient. In order
to develop a method to accomplish this result, we will first
state the problem in symbolic form :
Given a bond on which there is a premium or discount
Q, cash rate c, and number of periods n, what is the income
rate i?
Every premium or discount is the present worth, at the
income rate, of an annuity of n terms, each instalment of
which is the difference between the cash and income rates;
in other words, it is the present worth of an annuity of $1
multiplied by the difference in rates (§118). Writing P
for the present worth of an annuity of $1, we have the
equation : Q = P X (c — i) . The terms c and i, in the great
majority of bonds, theoretically refer to the rates for semi-
annual periods. In practice, however, a 4% rate or a 5%
rate means an annual nominal rate, irrespective of the fact
that the coupons are semi-annual. In order to conform to
Compare text of §§ 135, 136.
2i8 PROBLEMS AND STUDIES
commercial usage, we will alter the equation by halving the
P and doubling the (c — i) ; the equation then becomes :
Q = 1/2? X {2c— 2i). With this change, the value of the
right-hand member is not altered, and there is the advantage
that the quantities 2c and 2i represent, respectively, the
nominal annual cash and income rates.
§ 264. Assumed Trial Rate
In the equation given above, the premium or discount Q
is known, and the cash rate c is also known. There is there-
fore, in reality, but one unknown quantity, the income rate
i, since P can be ascertained when once the value of i is
known. It is evident that if we divide Q by %P (which
latter we will hereafter call the trial divisor), we shall find
the difference in rates. Let us assume the rate of income
to be any rate whatever, and then calculate the trial divisor
at that rate. Then, since the product of %P times (2c — 2i)
is the constant, or known, quantity Q, we have the following
chain of reasoning: If the assumed income rate is too
small, P will be too large, the difference in rates will be too
small, and the ascertained income rate will be too large ; and
vice versa if the assumed income rate is too large. Taking
now this first ascertained rate as the new assumed rate, we
may find a second ascertained rate, and so on, as many times
as we please, the proceeding being something like the swing-
ing of a clock pendulum, except that each swing is shorter
than the preceding one, since the successive ascertained rates,
one after another, more nearly approach the true income
rate. We may slightly modify any rate in order to make
the work easier; if we are fortunate in selecting our first
trial rate near the true rate, fewer successive approximations
will be necessary.
For the purpose of computing the value of the tria!
divisor (%P), a table of bond values may be used for
DETERMINING ACCURATE INCOME RATE 219
the first two or three approximations, by taking the differ-
ence between the values (at the same income rate) of a
3% bond and a 4% bond, or of some other pair of bonds
whose nominal annual cash rates differ by 1%.
§265. Application of Assumed Trial Rate — Bond Above
Par
As an example, we will take a 6% semi-annual bond for
$100, due in 50 years and sold at 133, to find the income
rate. With so large a premium as 33, the income rate is
evidently much less than 6% ; let us assume 4%. From the
bond tables we find that the value of a 5% bond, due in
50 years, and earning 4%, on a par of 100, is. . . .$121,549
The value of a similar bond earning only 4% is
par, or • 100.000
The difference is the present worth of an annuity
of 50c. (the difference between the semi-annual
cash and income rates) for 100 periods at 2%
per period $21,549
The present worth of a similar annuity of $1, or
P, is $43,098
%P, the first trial divisor, is therefore $21.55
33.00^21.55 = 1.531, the difference in rates. 6% —
1.531 = 4.469%, the new trial rate. Taking 4.45% as more
convenient, the new trial divisor is 19.98. 33.00 -^ 19.98 =
1.651. 6% — 1.651% =4.349%. For this new rate (or
4.35%), we find that 20.315 is the trial divisor. 33.00-^
20.315 = 1.6244. 6% — 1.6244% = 4.3Y56%. Next using
4.37%, the trial divisor is 20.25. 33.00-^20.25 = 4.37,
almost exactly, so that the use of 4.37% as an assumed or
trial rate leads to it again as an ascertained rate; in other
words, the rate 4.37% reproduces itself, which shows that
'/'
220 PROBLEMS AND STUDIES
we have now found the correct rate. The value of the bond
at 4.37%, as computed by logarithms, is $133.0069, an
error of less than one cent.
§ 266. Variations in Assumed Rates
The example in § 265 is an illustration of what we have
previously pointed out ; that is, that the results always swing
to the opposite side of the true rate. If the trial rate is too
large, the ascertained rate will be too small, and the true
rate will lie between them. The successive rates were 4%,
4.469%, 4.349%,4.3756%, and 4.37%. 4.37% lies between
any pair of these rates except the last two, where one rate
coincides with 4.37%. The foregoing is always the case
with bonds above par. With bonds below par it is different ;
here the true rate is always larger than the last approxima-
tion. The ascertained rate may be carried to many decimal
places, but it never quite overtakes the true rate. The case
is somewhat analogous to a circle having an inscribed
polygon. We may increase the number of sides of the
polygon indefinitely, but its area will never quite equal the
area of the circle.
§267. Application of Assumed Trial Rate — Bond Below
Par
As an example of a bond below par, take a 3% bond
payable in 25 years. If purchased at 88.25, what is the
income rate ? The following may be the steps, the dividend
being always 11.75, the discount:
Trial rates 3.70% 3.725% 3.7265%
Trial divisors 16.2190 16.175 16.17245
Ascertained rates. 3.7244% 3.7264% 3.7265%
Since 3.7265% reproduces itself, it must be correct to
the 4th decimal. Tested by logarithms, the value of a 3%
bond for 25 years yielding 3.7265% proves to be $88.25018.
DETERMINING ACCURATE INCOME RATE 221
§ 268. Trial Rates from Bond Tables
While the method of trial rates is correct in theory, it
may be greatly facilitated in practice by first locating by
means of bond tables* the required income rate between two
rates one-hundredth of one per cent apart. The results will
be so close that simple interpolation (explained in Chapter
XXXI) will suffice for at least seven decimals, and the
laborious divisions necessary in the foregoing method will
be avoided.
§ 269. Use of Bond Tables
For example, let it be required to find the income rate of
a 4% bond for $1,000,000 due in 100 years, bought for $1,-
264,806.66. From the 4% table, we find that the income
rate must lie between 3.10% and 3.15%. The values corre-
sponding to these rates are as follows :
3.10% $1,276,929.04
3.15% 1,257,990.62
1/5 of the difference being $3,787,684, we may roughly
estimate the intermediate values as follows :
3.11% $1,273,141.35
3.12% 1,269,353.66
3.13% 1,265,565.98
3.14%.. 1,261,778.30
The required rate must lie between 3.13% and 3.14%;
the difference in rates lies between .87% and .86%. Correct-
ing the above intermediate values by the colored pages in
the bond tables,* we have :
Premium at 3.13% $265,505.52
Premium at 3.14% 261,738.09
Premium at the required rate. . 264,806.66
Since any two premiums at the same income rate are
•Sprague's "Extended Bond Tables."
222 PROBLEMS AND STUDIES
directly proportional to the difference between the cash and
income rates, we have the following proportions :
at 3.13%— $265,505.52 : $264,806.66 : : .87% : x% (x = .867/09998)
at 3.14% — $261,738.09 : $264,806.66 : : .86% ; x% (x = .870082484)
At the same premium on each bond ($264,806.66), we
see from the above two proportions that the following facts
prevail with reference to the rates :
Income Rate Cash Rate
3.13% corresponds with 3.997709998%
3.14% " " 4.010082484%
Our problem is to determine the income rate correspond-
ing with a cash rate of 4%, the premium still being the
same. For this purpose, the method of interpolation will be
sufficiently exact, and we may form a proportion as follows :
x% : .01% :: .002290002% : .012372486%
The unknown term of the proportion is found to be
.0018509%, which added to 3.13% gives 3.1318509% as
the income rate corresponding to a 4% cash rate. The ac-
curate value of the bond computed to ten decimal places at
the income rate of 3.131851% is $1,264,806.6645.
CHAPTER XXIV
DISCOUNTING*
§ 270. Table of Multiples
Discounting may be performed as well by multiplication
as by division, and multiplication is preferable as being the
more direct and compact process. In Table VI (§ 383) are
the reciprocals of all usual ratios of increase. Multiplying
by .9803921568, for example, will give the same result up
to a certain number of places, as dividing by 1.02. Using
the tabular plan, we have this table :
98
196
294
392
490
588
686
784
882
039
078
117
156
196
235
274
313
352
216
431
647
863
078
294
510
726
941
We will take as an illustration a 5% bond, yielding 4%,
both the coupons and the income being on a semi-annual
basis. The amounts receivable at maturity are $100,000.00
of principal and $2,500.00 of coupons, a total of
$102,500.00. The discounting process would then be as
follows :
In connection with text of § 143.
223
224 PROBLEJMS AND STUDIES
102,500.00
98,039.22
1,960.78
490.20
100,490.20
2,600.00
102,990.20
98,039.22
1,960.78
882.35
88.24
.20
100,970.79
2,500.00
103,470.79
98,039.22
2,941.18 etc.
There is an error of 1 cent in the value 100,970.79 ; this
could easily have been prevented by carrying out into mills.
For long operations it is always advantageous to use a few
spare places beyond those retained in the final result.
§ 271. Present Worths of Interest-DifiFerence
Still greater brevity will be attained by working out first
the items of amortization, or present worths of the difference
between the cash and income rates. The present worths of
the interest-difference 500 are obtained as follows, using
fewer figures and less labor than in the preceding example :
DISCOUNTING 225
600.000
490. X9 6 % year before maturity
392.157
88.235
.098
.088
6
480.584 1 year before maturity
392.157
78.431
.490
.078
4
471.160 1% years before maturity
392.157
68.627
980
98
59
461.921 2 years before maturity
Writing these down in reverse order, the amortization
column of the schedule is filled :
461.92
471.16
480.58
490.20
1903.86
226 PROBLEMS AND STUDIES
The value at two years before maturity is therefore
$101,903.86, and the schedule may be further filled :
Book
Par
Amortization
Value
Value
$101,903.86
$100,000.00
$461.92
101,441.94
471.16
100,970.78
480.58
100,490.20
490.20
100,000.00
For practice, any of the Problems (52) to (55), inclu-
sive, may be worked over backwards.
§ 272. Discounts from Tables
If the rate is one of those embraced in Table II (§ 379),
and the difference of interest is a simple number, the process
is still easier. Here the present worths of 500 for various
numbers of periods at 2% per period are required. In Table
II we find these present worths for $1 ; pointing off 3 places
to the rig-ht gives the corresponding values for $1,000, and
halving this, all in the one operation, gives the successive
figures required :
4 periods .92384643X 1000 -^ 2 =461.9227||
3 " .94232233 " 471.1612||
2 " .96116878 " 480.6844||
1 " .98039216 " 490.1961||
1903.8644|I
§ 273. Reussner's Tables
Reussner's "True Discount Tables" give multipliers for
each day, from 1 to 180, carried to 8 places, for a great
number of usu^l rates, and will much facilitate discounting
DISCOUNTING
227
for fractional periods. In the example in the text, it gives
.99009901 opposite 90 days at 4%, with the following
result :
102,600.000
99,009.901
1,980.198
495.050
101,485.149, the same as in the text of § 144.
CHAPTER XXV
SERIAL BONDS
§ 274. Problem in Valuation of Serial Bonds
(64) A city issues ten 4% bonds for $10,000 each, A&O,
on April 1, 1914, maturing as follows : $10,000 on April 1,
1916; $10,000 on April 1, 1918, and so on— $10,000 each
alternate year, the last $10,000 on April 1, 1934. They
are sold at 108.33, the purchaser believing that he has a
3.10% investment. How near right is he?
As the average time of the bonds is 11 years, it might
be inferred that the true value of the series was the value
of a single bond of $100,000 due in 1925, which would be
$108,334.54; but this is fallacious. The true price, obtained
by adding together all the separate tabular values, is always
less.
At 3.10%, the values of the bond at varying due dates
are as follows :
Due Dates
Values
1916
$10,173.24
1918
10,336.13
1920
10,489.31
1922
10,633.35
1924
10,768.79
1926
10,896.16
1928
11,015.92
1930
11,128.53
1932
11,234.43
1934
11,334.01
Total series
$108,009.87
228
SERIAL BONDS
229
It is evident that the purchaser should have paid 108.01
instead of 108.33, and that on the latter price he will earn
less than 3.10%. How much less, is to be ascertained.
The value at 3.10% might have been carried out further
in decimals to the limit of the tables, giving $108,009.8686.
The values at 3.05% will next be copied down.
Due Dates
Values
r 2
years
$10,182.9714
4
10,355.1945
6
10,517.3006
8
10,669.8840
3.05%
10
10,813.5041
Basis 1
12
10,948.6875
14
11,075.9297
16
11,195.6973
18
11,308.4294
20
11,414.5391
$108,482.1376
§ 275. Inter-rates
The inter-rates, 3.06%, 3.07%, 3.08%, and 3.09%, can
now be obtained in bulk without determining the values for
separate years, according to the directions on page 123 of
Sprague's ^'Extended Bond Tables."
Find the difference between ,. .$108,482.1376
and ; 108,009.8686
which is ., $472.2690
ys of this is 94.4538
Subtracting from $108,482.1376 succes-
sively ys, }i, Ys, and fs , we have the
approximate values for 3.06% ... $108,387.6838
for 3.07% 108,293.2300
But it is unnecessary to go further ; it is evident that the
effective rate is a little below 3.07%.
230
PROBLEMS AND STUDIES
§ 276. Table of Differences
The value given at the basis of 3.07% is approximate,
and we can get a corrected value by applying the rule given
on page 122 of the tables,* viz. : ''To correct any terminal
2 or 7, subtract II/2 times the difference and then add 1/10
the sub-difference." The following table is derived from
pages 146 to 149, inclusive, of the bond tables,* and shows
the differences and sub-differences in the case of a 4% bond
of $1,000,000 at the income rates of 3.05% and 3.10%.
Dates of
Differences at
Differences at
Sub-
Maturity of Bond
3.05% Basis
3.10% Basis
Differences
2
years
$ .09
$ .09
4
.33
.33
6
.69
.69
8
1.17
1.16
$.01
10
1.74
1.73
.01
12
2.40
2.39
.01
14
3.14
3.12
.02
16
3.95
3.92
.03
18
4.82
4.78
.04
20
Total
5.74
5.68
.06
$24.07
$23.89
$.18
On account of the fact that each of the bonds in question
has the par of $10,000 and not $1,000,000, the tabular
difference for the rate 3.07% becomes $.2407, and the sub-
difference $.0018; 1% times the difference equals $.3611,
and 1/10 of the sub-difference is $.0002. The corrected
value at 3.07% therefore becomes :
Sprague's "Extended Bond Tables.
SERIAL BONDS
$108,293.2300 — $.3611 + $.0002 = $108,292.8691
The residue to be eliminated is 37.1309
231
making the price paid $108,330.0000
§ 277. Successive Method
The values at the basis of 3.07% must next be worked
out for each period down to the last maturity.
Value at April 1, 1914 $108,292.8691
X 1.01535 1,082.9287
541.4643
32.4879
5.4146
less 2,000.0000
Value at October 1, 1914, $107,955.1646
X 1.01535 1,079.5516
539.7758
32.3865
5.3978
less 2,000.0000
Value at April 1, 1915 $107,612.2763
X 1.01535 1,076.1228
538.0614
32.2837
5.3806
less 2,000.0000
Value at October 1, 1915 . . $107,264.1248
At April 1, 1916, 1918, etc., at intervals of two years,
the book value will be further diminished to the extent of
the principal of the bonds maturing at these respective dates.
232 PROBLEMS AND STUDIES
§ 278. Balancing Period
But it may be that balancing-period figures are wanted,
say J & J. In that case, the value on July 1, 1914, is half-
way between , $108,292.8691
and 107,955.1646
or $108,124.0168
with which we continue — 1,081.2402
540.6201
32.4372
5.4062
less 2,000.0000
Value at January 1, 1915 . . $107,783.7205
1,077.8372
538.9186
32.3351
5.3892
less 2,000.0000
Value at July 1, 1915 $107,438.2006
1,074.3820
537.1910
32.2315
5.3719
less 2,000.0000
Value at January 1, 1916. . $107,087.3770
§ 279. First Payment in Series
We have now reached a point where a broken terminal
period occurs, as to the first $10,000 due April 1, 1916,
and we must follow the directions of § 88, with tfiis modifica-
tion: that the $10,000 and the remaining $97,087.38 must
be treated separately, the reason being obvious.
SERIAL BONDS
233
$107,087.3770
Amount of principal due April 1 10,000.0000
Remainder ; $97,087.3770
The usual procedure —
970.8738
485.4369
29.1262
4.8544
$10,000 X .007675 (3 months) . . 76.7500
$98,654.4183
Income 2% on $90,000
1% on $10,000 = 1,900.0000
Value at July 1, 1916 $96,754.4183
This will exemplify the process when the principal of
one of the serial bonds is paid off.
§ 280. Elimination of Residue
There is a residue of $37.1309 to be eliminated, for
which we shall use the third method. A total premium of
$8,330 is to be amortized, while the 3.07% basis will amor-
tize only $8,292.8691. The proportion is 8,330 -^ 8,292.8691
= 1.0044784. A table formed from this will give the fol-
lowing multiples :
1004478
2008957
3013435
4017914
5022392
6026870
7031349
8035827
9040306
234 PROBLEMS AND STUDIES
The amortization at 3.07% for the fractional period and
the 4 full periods is as follows :
168.8523 (April 1 to July 1, 1914)
340.2963
345.5199
350.8236
332.9587
and as adjusted for elimination as follows, the eliminands
appearing in the top line and the eliminates in the bottom
line:
1688523 3402963 3455199 3508236 3329587
1004478 3013435 3013435 3013435 3013435
602687 401791 401791 502239 301344
80358 2009 50224 8036 20090
8036 904 5022 201 9040
502 63 199 36 502
23 87
1696084 3418202 3470671 3523947 3344498
As thus computed, the adjusted amounts of amortiza-
tion would be as follows :
April 1, 1914, to July 1, 1914 $169.61
July 1, 1914, to Jan. 1, 1915 341.82
Jan. 1, 1915, to July 1, 1915 347.07
July 1, 1915, to Jan. 1, 1916 352.39
Jan. 1, 1916, to July 1, 1916... 334.45
Total for 4% years $1,545.34
§ 281. Schedule
The schedule will then be made up as follows to this
point :
SERIAL BONDS
235
o 05
O tH
as
o
H
N
H
PS
O
<
O
W
.-1
W
u
CO
05
00
tH
U CO
tn "^
O . >%
PQ ^ ^03
-^ »— .^
C/}
o
o
o
o
o
€/9-
< Oh
\ — ; CO
:3 o
l-l
2
a
Pk
0^
0^
05
T-{
m
Oi t^ tH CD
CO 10 lO- tH CO
B,
00 rH Oi rJ^
CO <^ rH 1>- rH 00
nj
CO rH 00 ThI rH Ir-
m>
00" 00" ^-" J>^ i>^
000000
tH rH rH rH rH
€€
)■
N
• rH oq Ir- Oi
*f C
CD 00 CO tH
S.2
Ci rH* i>^ Cq t:H
S rt
CD TJH ^ iO CO
<
rH CO CO CO CO
0^
Oi 00 CO rH VO
CO rH Oi CD 10
ni
06 C^* 1>^ lO
-^
CO lO iO -^ CD
00 CD CD CD iO
aJ*^
^^ T-i -rH T-i T-i
iz;
00000
00000
if
00000
00000
Oi
rH CN C^l C?^ rH
1— 1
<(->
en
u
,__,
a
TjT 10 CD
rH rH rH
Oi Oi OS
r-
< rH iH II
236 PROBLEMS AND STUDIES
The premium is now $6,784.66, and the premium at
3.07% is $6,754.42, which we test as follows:
1.004478
X 675442
602687
70313
5022
402
40
Proof 678466
§ 282. Uneven Loans
The terms of a series of bonds need not necessarily be of
like amount. Suppose the payments in the above example
were:
$10,000 in 1916
$20,000 in 1918
$30,000 in 1920
$40,000 in 1922
$100,000
and it were desired to find the value at 3.10% ; the process
would be :
$10,173.2358 $10,173.2358
10,336.1340 X 2 20,672.2680
10,489.3124 X 3 31,467.9372
10,633.3506 X 4 42,533.4024
Value of series $104,846.8434
The formation of the schedule would be precisely an-
alogous to that already given.
SERIAL BONDS
237
§ 283. Tabular Methods
Most serial bonds run by years, an equal amount being
payable annually. Where the rate is one ending in 5 or 0,
and the values for exact interest periods are required, not
for intermediate periods, a simpler process may be used,
copying values direct from the tables. For example, a series
of five 4% bonds of the par value of $1,000 each, J & J,
issued July 1, 1914, payable on each first of July, 1915 to
1919, is sold on a 3.50% basis.
Set down in two columns the first ten values from the
tables ; then add and subtract successively, as follows :
1/2 yr. $1002.457
1 yr. $1004.872
lYz 1007.245
2 1009.577
2% 1011.870
3 1014.122
31/2 1016.337
4 1018.513
4% 1020.651
5 1022.753
$5069.837
Jan. 1, 1915, $5058.560
1020.651
Jan. 1, 1916, $4037.909
1016.337
Jan. 1, 1917, $3021.572
1011.870
Jan. 1, 1918, $2009.702
1007.245
1022.753
$4047.084 July 1, 1915
1018.513
$3028.571 July 1, 1916
1014.122
$2014.449 July 1, 1917
1009.577
$1004.872 July 1, 1918
Jan. 1, 1919, $1002.457
238
PROBLEMS AND STUDIES
§ 284. Formula for Serials
The total value of an annual series may be obtained by
the following formula :
Let m be the number of different maturities and n the
number of the periods the last bond has to run. Let r, for
brevity, represent the ratio of increase, instead of 1 + i.
The powers of r are obtainable from Table I (§ 378), or by
logarithms. The principal of each bond being $1, the
formula would read :
^^^ — /^2_f y7n K ^' X (c — i)
In the preceding example m^ 6,n = 10, r = 1.0175^ i =
.0175, c = .02, c — i = .0025.
From Table I* or from the "Extended Bond Tables"t :
1.03530625
1.18944449
1.18944449
.18944449
.03530625
.18944449
r^
=
y5m =
^10 _
r^ =
^10 =
Therefore : r'^ - 1
=
r'-l
=
^2m_i
_
{r'-l)r''
=
^2in _ 1
= n
.03530625 X 1.18944449
4.511139
4.511139 = .488861
(m— /o''^~x"^n ^^ *' == -488861 -^ .0175 = 27.93491
Value of series = 5 + (27.93491 X .0025) = 5.0698373
which is the result already obtained by addition.
This formula will seldom be of use except in the case of
a very complex rate not comprised in the tables. It will then
involve the computation of three powers of r by logarithms.
§ 378. t Sprag ue's "Extended Bond Tables.
SERIAL BONDS 239
§ 285. Problems in Valuation of Serial Bonds
The following problems may be solved in either of the
ways discussed :
(65) A corporation issued a series of ten $1,000 bonds,
5%, M & N, on May 1, 1913, payable each May 1, 1921 to
1930. What is the value on a 3.60% basis :
(a) On May 1,1918?
(b) On July 1, 1918?
(c) On August 23, 1918?
(66) Find the values as above, but on a 4% basis.
§ 286. Answers to Problems in Valuation of Serial Bonds
Problem (65)
(a) $10,897.40
(b) $10,962.79 flat.
(c) $11,019.45 flat.
Problem (66)
(a) $10,630.42
(b) $10,701.29 flat.
(c) $10,762.71 flat.
CHAPTER XXVI
OPTION OF REDEMPTION
§ 287. Method of Calculating Income Rate*
The rate of income on a bond subject to a right to redeem
at an earlier date than that of actual maturity and on pay-
ment of a premium, can be ascertained by means of tables.
Only the income which is certain must be calculated upon in
advance; hence there will always be a contingent profit
which may be realized.
For example, suppose the bond to be a 4%% one abso-
lutely due in 30 years but redeemable at 105 after 20 years;
issued 1905, redeemable 1925, payable 1935.
In order to determine where the redemption is a benefit
and where it is a disadvantage, we must suppose ourselves
to be in 1925 at the redemption date. This bond now has
10 years to run. Turning to the 4%% bond table,t under
10 years, we find that 1.05 is the price almost exactly at a
3.89% basis. Therefore, if the bond is bought now on a
3.89% basis, the investment value in 1925 will be exactly
1.05 and there will be neither profit nor loss in being re-
quired to surrender at 1.05; 3.89% may be called the
neutral rate.
§ 288. Advantageous Redemption Ignored
It is necessary to bear in mind that the higher the rate
of income the lower is the premium ; if the rate be more than
• Compare § 147.
t Sprague's "Extended Bond Tables."
240
OPTION OF REDEMPTION 241
3.89%, say 4%, the option may be disregarded, for we shall
surely have 4% for 20 years, and probably for the full time.
In case the rate of interest has fallen to 3.89%, the issuer
of the bond may think it advantageous to redeem, so as to
sell his new issue at more than .05 premium. Then, as our
bond stands at less than 1.05, we get a profit besides our 4%
income. Thus, if the bond is bought at a basis which yields
more than 3.89% for 30 years, we may safely amortize at
that basis for 20 years, or until the option is exercised.
§ 289. Disadvantageous Redemption Expected
But if the rate for thirty years, which we may call the
apparent rate, or non-redemption rate, is less than 3.89%,
the bond will be worth more than 105 at the redemption date
and the issuer may be expected to redeem. If he does not,
it is because the general rate of interest has risen so that
he must pay more than 3.89%, in which case he will allow
us to continue at 3.89% till maturity. Thus, if the bond is
bought at a price which would be on an apparent basis of
less than 3.89%, redemption must be expected as being
adverse to our interests. The redemption date then becomes
the actual date of maturity, but the principal is not 1 but
1.05.
§ 290. Change in Principal
Let the par be $100,000 and the price $114,423.38, which
is at the apparent basis of 3.70%. To get the actual basis
we must consider the par as $105,000 and the time 20 years.
But if the par is $105,000, the cost is not at 1.1442 1| but at
1.1442||-^1.05 = 1.0897||. The cash rate is also trans-
formed; the cash income is still $4,500, but this is not 4^2%
of $105,000 ; it is only 4 2/7%.
Therefore, the limitation imposed by the option of re-
demption entirely changes the problem. Instead of a 4%%
242 PROBLEMS AND STUDIES
bond for $100,000, due in 30 years, bought at 1.1442, we
have a 4 2/7% bond for $105,000, due in 20 years, bought
at 1.0897.
No tables have been published for 4 2/7% bonds, presum-
ably because this exact case of 4%% bonds redeemable at
1.05 is infrequent. However, we can easily construct them
by adding to the value of a 4% bond, 2/7 of the difference
between a 4% bond and a 5% bond.
§ 291. Approximate Location
As a rough approximation, find 1.0897 as closely as pos-
sible in the 20-year tables for 4% and 5% respectively. The
nearest to 1.0897 in the 4% table is 1.08655516, which is a
3.40% income; the nearest in the 6% table is 1.08623676, a
4.35% income. The required rate will be about 2/7 of the
distance between 3.40% and 4.35%.
4.35 — 3.40= .95
2/7 of .95 = .27
3.40+ .27 = 3.67
Therefore 3.67% is the approximate rate, and we might
begin testing with that rate. We notice, however, that the
approximations 1.08656|| and 1.08624|| are both short of
1.0897; hence, probably the rate will fall short of 3.67%,
and it will be easier to start with the tabular rate 3.65%.
In fact, had we gone a little further in decimals, using the
colored pages of differences and sub-differences in the bond
tables, we should have obtained the following values in the
4% table:
Income rate, 3.38%, 1.08960122
Income rate, 3.37%, 1.09112831
The rate nearest to 1.0897 in the 4% table is therefore
3.38%. Similarly, in the 5% table the nearest rate is
OPTION OF REDEMPTION
243
4.33% . Taking 2/7 of the difference between these two rates
and adding this difference to 3.38%, gives 3.65% as the
approximate income rate.
4% table, 20 years 3.65% 1.0493748
5% " " " 3.65% 1.1904458
Difference .1410710
1/7 .0201530
2/7 .0403060
Add to 4% value 1.0493748
Giving 4 2/7% value 1.0896808
This value is very close to 1.0897.
Value of $105,000 at the same price. .$114,416.48
Actual price 114,423.38
Residue 6.90
This is the nearest approximation we can obtain without
using more decimals; therefore, 3.65% is the actual rate of
income for a 4%% bond redeemable at 1.05, 10 years before
maturity, if purchased at 114.42, 30 years before maturity.
In the diagram (page 244) the dotted line marked 3.70 is
the apparent course of a bond at 114.42, 30 years to run ; but
the option at 105 pulls it down to a 3.65 basis; during the
last 10 years it earns 3.89%, if not redeemed. The 4% line,
as it passes below the 105 point, is unaffected by the option
of redemption. The issuer would not redeem, at 105, a bond
whose value was less than 105.
To complete a schedule running from the date of issue
to that of redemption, we have the following data :
Par, $105,000.
Cash interest, semi-annually, $2,250, being at the rate of
4 2/7% per annum.
Net income, semi-annually, $1,916.25, being 3.65% per
annum on $105,000.
244
PROBLEMS AND STUDIES
Difference of interest, $2,250 — $1,916.25 = $333.75.
Present worth of 20-year annuity of $333.75 each half-
year, $9,416.48. Present value of bond at 3.65%,
$114,416.48. Actual value, $114,423.38. Eliminand,
$6.90.
We might now proceed to amortize $114,416.48 down to
maturity. Each term would then have to be corrected to
eliminate the residue, $6.90. The multiplier for this pur-
pose would be :
9423.38 ^ 9416.48 = 1.00073276| |
But we may proceed in the other direction and discount
$333.75 at various dates ; this has the great advantage that
4.1S
■
Graphic Representation
OF THE Effect of aw
1.14
N
V
Optionai, Redemption Date
%.%»
•
N;
<;
^
'^K
t.ia
•
\
*»
&.X1
t.xo
•
^
•^
&.O0
i.oe
^
^
\
^
v
t*7
>
\
^^"\^< ^\\ ' •
i.e<
X.05
1.04
•
vet
\^.
1*1
^\
VAR 1.©©
...X.
-i-l.
JU
J.
Om
....(.. . iS«i
o
&0
as
ao vsX
•Z^XUXBMXatK
SVX JL»«X>
Jkvm*
FAVASCS
OPTION OF REDEMPTION
245
$333.75 may be first multiplied by 1.00073276, thus accom-
plishing the elimination process once for all.
$333.75 X 1.00073276 = $333.99456
This last is substituted as a base in place of $333.75, and
we proceed to discount, using the factor .982077093] |, which
is the reciprocal of the semi-annual ratio 1.01825, in the
tabular method :
$333.9946
294.6231
29.4623
2.9462
8839
884
6
$328.0084 % year before maturity
294.6231
19.6415
7.8566
79
4
$322.1295 1 year before maturity
etc.
§ 292. Problems Involving Optional Redemption Dates
(67) If a 4% bond is redeemable 25 years before
maturity at 105, what is the neutral rate of income?
(68) If a bond reads at 4%, but the amount which will
be received is 1.05 of the nominal par, what is the actual
percentage of cash income ?
(69) A 50-year 4% bond is redeemable at 105 after 25
246 PROBLEMS AND STUDIES
years. Find its actual income rate if bought at (a) 105, (b)
106, (c) 107, (d) 108, and (e) 109.
(70) A 30-year 5% bond is redeemable at 110 after 15
years. Find at what price it should be bought when issued
to pay (a) 3.90%, (b) 4.40%.
§ 293. Rule for Determining Net Income
We are now prepared to formulate a rule for determin-
ing the net income yielded at a certain price, by a bond bear-
ing a certain par interest but subject to redemption at an-
other price, on the assumption that the right will be
exercised.
(1) Divide the nominal cash rate of interest by the re-
demption price per unit ; the quotient will be the actual cash
rate, consisting of a whole number and a fraction; e.g.,
4y2%-^1.05 = 4 2/7%.
(2) Divide the purchase price by the same divisor, giv-
ing the actual purchase price per unit; e.g., 1.1442 -r-
1.05 ==1.0897.
(3) Select two different bond tables, one at a lower,
one at a higher cash rate than the actual rate obtained in
paragraph 1. These should be even rates, not fractional,
and 1% apart. Find the column for the number of years
before redemption; e.g., 4% and 5%, 20 years.
(4) In each of these columns find the nearest price to
the actual purchase price in paragraph 2 ; e.g., in 4% table,
1.08656; in 5% table, 1.08624.
(5) Set down the two rates of net income found op-
posite these values, and find their difference; e.g., inter-
rates, 4.35% and 3.40% ; difference, .95%.
(6) Take such a fraction of the difference as is shown
by the fractional part of the mixed number which represents
the actual cash rate; add the result to the smaller rate and
OPTION OF REDEMPTION
247
the sum is, approximately, the desired yield; e.g., 2/7 X
.95 = .27 ; .27 + 3.40 = 3.67.
(7) Try the nearest rates from the table until one is
found which produces the desired price; e.g., 3.65 produces
1.08968.
§ 294. Answers to Problems Involving Optional Redemp-
tion Dates
Problem (67)
3.69% +
Problem (68)
3 17/21%, or 3.80952% +
Problem (69)
(a) Between 3.77% and 3.78%; (b) 3.73%+;
(c) 3.69% ; (d) 3.63% ; (e) 3.57% +
Problem (70)
(a) 118.005676; (b) 109.042757
CHAPTER XXVII
BONDS AT ANNUAL AND OTHER RATES
§ 295. Standard of Interest
In popular usage and, in fact, in legalized usage, though
not from the mathematical standpoint, the interest on a
given principal is directly proportional to the time; that is,
iif the interest is six dollars on a hundred for a year, it
must for six months be three dollars, and for three months
one dollar and a half. These three rates are popularly re-
garded as identical, but actually they are very different. A
single standard should be preserved, and when in any prob-
lem "6 per cent" is once taken as meaning "3 per cent per
half-year," it must not be arbitrarily shifted to mean "1%
per cent per quarter," which is really "3.0225 per cent per
half-year."
If the ratio of increase or income yield be kept at the
same unvarying standard, the frequency of collection, or
cash payment, affects the value of the investment. To change
the coupon from half-yearly to quarterly, must necessarily
enhance the value of the annuity made up of the coupons.
The nearer any one of them approaches to the present, or
the less time one must wait for his money, the more nearly
is it worth its par ; while the present worth of the principal
remains the same, unless we vary the income yield.
§ 296. Semi-Annual and Quarterly Coupons
A bond for $1,000,000, due in one year, bearing semi-
annual coupons at 6 per cent per annum, at a price to net
248
BONDS AT ANNUAL AND OTHER RATES 249
21/2 per cent computed semi-annually (1%% per period),
is worth, according to all tables and computations (except
the fictitious one of "reinvestment" at an arbitrary
rate) $1,034,354.52
thus,
Present worth of first coupon, $30,000,
one period, 1%% ,. $ 29,629.63
Present worth of second coupon, $30,-
000, two periods, 114% 29,263.83
Present worth of principal, $1,000,-
000, two periods, 1%% 975,461.06
By the method in § 111 $1,034,354.52
Or, using the method in § 116, we should take the
nominal interest $30,000
subtract from it the effective interest. .: 12,500
and obtain the interest-difference $17,500
An annuity of $17,500, for two terms, at
1^A%, would be the premium. $34,354.52
The company issuing the bonds is willing in return for
certain concessions to make its interest payments quarterly.
How much would this add to the value of the bond, the in-
come yield being still 2% per cent on a semi-annual basis?
If the bond be made quarterly, the same cash is received
each half-year, but $15,000 of it is received three months
earlier than before. On this $15,000 the bondholder is en-
titled to only 3 months' interest, instead of 6 months, at
1%% per half-year; therefore, a quarter's interest on this
quarterly coupon must be deducted each half-year from the
entire interest earned.
We must be careful, however, to compute the interest cor-
rectly on this advanced coupon. It must be at .00623059,
250
PROBLEMS AND STUDIES
not at .00625. Interest at a half-period is not half of the
.0125, but the square root of the ratio 1.0125, less the 1;
Vi.0125 = 1.0062305911, interest = .00623059. Otherwise
we should be using a higher rate than 1.0125 for the half-
year, nearly 1.01254. The interest to be deducted each half-
year is $15,000 X .00623059 = $93.46. The effective in-
terest is $12,500 — $93.46 = $12,406.54, and the interest-
difference $30,000 — $12,406.54 = $17,593.46. If we should
now consider each instalment of the annuity to be $17,593.46
instead of $17,500, we should have the premium for quarter-
ly coupons. Therefore, the two annuities (or, in other
words, the premiums) at any point must be to each other
as $17,593.46 : $17,500; or the ratio of the quarterly
premium to the semi-annual is 1.005340507. Hence the
multiplier .0053405 on page VII of Sprague's "Extended
Bond Tables.''
In symbols, the income rate becomes (instead of i),
i — f (Vl + ^* — 1) and the interest-difference becomes (in-
stead of c — i)y c— [i — |(VlTT— 1)] =c — i+^
(VTTl — 1), which divided by (c — i) gives the propor-
tion 1-f ^ (Vl + ^ — 1) ^ ^.^ ^^^ ^^^^ ^^^^^ 1.0053405.
c — i
The process of finding .0053405 may be briefly expressed
thus:
Rule : Divide a quarter's interest on a quarterly coupon
by the interest-difference.
The value of the bond when trimestralized (reduced to a
quarterly basis) is, therefore :
At semi-annual payments .$1,034,354.52
Added for quarterly coupon, 34,354.52 X
.0053405 183.47
Value trimestralized $1,034,537.99
BONDS AT ANNUAL AND OTHER RATES
251
This may be tested by multiplying down to
maturity, 3 months at a time, viz. :
$1,034,537.99 X .00623059 + 6,M5.Y9
$1,040,983.78
— 15,000.00
$1,025,983.78
$1,025,983.78 X .00623059 + 6,392.48
$1,032,376.26
— 15,000.00
$1,017,376.26
$1,017,376.26 X .00623059 + 6,338.86
$1,023,715.12
— 15,000.00
$1,008,715.12
$1,008,715.12 X .00623059 + 6,284.88
$1,015,000.00
Final payment, 1,015,000.00
We will now take an example where the effective rate is
greater than the cash rate. A bond of $1000 at 4% (sem.),
due in ten years, is bought so as to give a net income of 5%
(sem.) ; what will be its value if trimestralized ?
The normal or semi-annual value is by all tables.. $922,054
The discount, -^-^ X ( '^ ,, } .,^ ^ , is 77.946
I V (1 + /
The multiplier* is 0248457
77.946 X .0248457, amount of added value, = . . .$1.93661
922.054 + 1.937 = 923.991
Spragrue's "Extended Bond Tables," page VII.
252
PROBLEMS AND STUDIES
§ 297. Shifting of Income Basis
This is the correct vakie, the income basis being un-
changed. But in some recent books we find the quarterly-
value io be stated as $921,683, which is a surprising result,
for we should not expect the value of the security to be
diminished by a more frequent interest-payment. The
trouble is, that the income basis has been suddenly shifted
from 4% semi-annual to 4% quarterly, and we are given
comparisons between the following values :
(a) At 4% semi-annual basis, coupon 5% semi-annual.
(b) At 4% quarterly basis, coupon 5% quarterly.
Whereas the value really desired is :
(c) At 4% semi-annual basis, coupon 5% quarterly.
In all the tables using the basis (b), the values below par
are all apparently diminished by frequency of payment. The
author's tables are computed on the semi-annual income
basis, though the coupons may be quarterly or annual.
§ 298. Problems — Bonds at Varying Rates
(71) A 5% quarterly bond for $100,000 has 5 years
to run on a 4% semi-annual basis; what is its value?
(72) Ascertain the value of the same bond at 4I/2 years.
(73) Derive the 4% years' value from the 5 years, and
obtain the same value as in (72).
(74) Find the value of a 2% quarterly bond, 5 years to
run, which nets 1.80% semi-annually.
(75) Two issues of 20 year, 31/^ % bonds, each
$100,000, are offered; one with interest semi-annually at
95.29, the other quarterly at 95.38 ; find the better purchase.
(76) Which is the better purchase :
$1,000,000 4%' quarterly bonds, 10 years, at
104.33, or
$1,000,000 3% semi-annual bonds, 10 years, at
95.50?
BONDS AT ANNUAL AND OTHER RATES 253
§ 299. Answers to Problems — Bonds at Varying Rates
Problem (71)
$104,608.02
Problem (Y2)
$104,182.64
Problem (73)
Value at 6 years , $104,603.02
Of this $1,250 is payable in three
months.
Present worth at 4% semi-annually. . 1,237.69
The remainder $103,365.33
Produces income at .02 2,067.31
$105,432.64
Cash interest received 1,250.00
Value at 4% years > $104,182.64
An alternative solution for this problem, and the one
usually employed, is as follows :
Value at 5 years. $104,603.02
This multiplied by the quarterly effective
rate, .00995049 (which is the square
root of 1.02, less 1) gives 1,040.85
$105,643.87
Less quarterly coupon 1,250.00
Giving value at ^% years $104,393.87
This multiplied by .00995049 gives the next
quarterly income 1,038.77
$105,432.64
Less quarterly coupon. 1,250.00
Giving value at 4% years. ,. . ... . .. .$104,182.64
254
PROBLEMS AND STUDIES
Problem (74)
$100,973.61
Problem (75)
The quarterly bonds.
Problem (76)
The semi-annual bonds.
§ 300. Bonds with Annual Interest — Semi-Annual Basis
Bonds on which the interest is paid only once a year are
somewhat rarer than those where it is paid four times a
year; but, when they do occur, means should be provided
for ascertaining their value at any given rate reduced to the
standard of semi-annual income. This is somewhat easier
than finding the value of a quarterly bond on a semi-annual
income basis.
We may begin by a simple example using the discount
method, either by division or by multiplication, taking a 4%
annual bond yielding 3% semi-annually, 2 years to run, for
$100,000.
Beginning at maturity at par. .....' $100,000.00
and adding to it the annual coupon then due. . 4,000.00
$104,000.00
We discount this by dividing by the ratio, 1.015,
or, what is the same thing, multiplying
by its reciprocal, .98522167; $104,000 -f-
1.015 or X .98522167 = $102,463.05
This is the value, flat, 6 months before maturity.
If there were a -payment of interest at this
date we should add its value. But there is
none; hence we continue the process, $102,-
463.05 -^ 1.015 or X .98522167 = $100,948.82
BONDS AT ANNUAL AND OTHER RATES
255
Here we add the coupon payable one year before
maturity 4,000.00
$104,948.82
We discount this for another half-year, $104,-
948.82 -^ 1.015. $103,397.85
and again, $103,397.85 ^ 1.015 $101,869.81
which is the value required. ' =
To test this, let us multiply down to maturity :
Value at 2 years $101,869.81
Income at 1%%, % year 1,018.70
509.35
Value at 1% years, flat , $103,397.86
No coupon.
Income at 1%%, % year 1,033.98
516.99
$104,948.83
Annual coupon paid 4,000.00
Value at 1 year .$100,948.83
Add 1/2 year's income, at 11/2% 1,009.48
504.74
$102,463.05
Add last % year's income, at 1%% • • 1,024.63
512.32
Total principal and interest $104,000.00
§ 301. Annualization
We will now annualize the above process; that is, in-
stead of multiplying twice by 1.015, we will multiply once
by 1.030225, which is 1.015 X 1.015, or (1.015)^
256 PROBLEMS AND STUDIES
As before, beginning with $101,869.81
we multiply by 1.030225. 3,056.09
20.37
2.04
.61
$104,948.82
and subtract the coupon 4,000.00
$100,948.82
again multiply by 1.030225 3,028.47
20.19
2.02
.50
giving the same result $104,000.00
Thus, income has been received on all of the investment
outstanding at 1.5% per half-year, or at 3.0225% per year.
§ 302. Semi-Annual Income Annualized
Suppose now that we take the case of an ordinary half-
yearly bond paying a cash interest of 2% twice a year, and
yielding 1.5% half-yearly, with the purpose of annualizing
in this case also. The ratio, when annualized, is the same
as before, 1.030225, but there are two semi-annual coupons
of $2,000.00 each, instead of the single annual coupon of
$4,000.00 as in the previous case. The first of these cou-
pons, if deferred to the end of the year, will increase at
the semi-annual ratio of 1.015 to $2,030.00
The second coupon remains 2,000.00
The entire cash interest, when concentrated at the
end of the year, is therefore equivalent to $4,030.00
The processes of multiplying down to maturity, using
both semi-annual and annual periods, are shown below side
by side, beginning with the value $101,927.19 found from
BONDS AT ANNUAL AND OTHER RATES
257
tables or by computation. In a third column appears the an-
nuaHzed process in the case of a 4% annual coupon. In all
three cases, the net income is 1.5% per half-year, or its
equivalent, 3.0225% annually.
Cash Interest
2% PER Half- Year
Cash Interest
4% per Year
Ordinary Process
Annualized Process
Annualized Process
$101,927.19
1,019.27
509.64
$101,927.19
3,057.82
20.38
2.04
.51
$101,869.81
3,056.09
20.37
2.04
.51
$103,456.10
2,000.00
$101,456.10
1,014.56
507.28
'
$102,977.94
2,000.00
$105,007.94
4,030.00
$104,948.82
4,000.00
$100,977.94
1,009.78
504.89
$100,977.94
3,029.34
20.20
2.02
.50
$100,948.82
3,028.47
20.19
2.02
.60
$102,492.61
2,000.00
$100,492.61
1,004.93
502.46
$102,000.00
$104,030.00
$104,000.00
258
PROBLEMS AND STUDIES
§ 303' Comparison of Annual and Semi-Annual Bonds
In each of these columns the proper principal is attained
at maturity, together with its accompanying interest, either
actual or annualized. Observing the first and second
columns, we see that a semi-annual 4% bond is effectively
a 4.03% annual bond, the net income in both cases being
3.0225% per annum. Comparing the second and third
columns, the point to be noted is that their chief difference
lies in the effective cash rates, one being 4.03% and the
other 4% ; in the semi-annual bond, annualized, the interest-
difference between the cash and income rates is
$4,030.00 — $3,022.50 = $1,007.50
In the annual bond, it is.. 4,000.00 — 3,022.50= 977.50
§ 304. Finding Present Worth of an Annuity
These interest-differences, $1,007.50 and $977.50, are
important because (according to the second rule in Chapter
X) we have only to multiply these two interest-differences
by the present worth of an annuity of $1 for 2 periods at
3.0225%, in order to obtain the respective bond premiums.
We might find this present worth approximately from Table
IV* by interpolation between the 3% and 31/2% columns,
but a much more accurate result may be obtained by the
use of Table II*, where we can find the present worth of $1
for 4 periods at 1.015, which is exactly equivalent to the
present worth of $1 for 2 periods at 1.030225.
This value $ .94218423
must, according to Chapter V, be subtracted
from 1.00000000
and the remainder $ .05781577
must be divided by the income rate 030225
The quotient is $1.9128460
* In Chapter XXXII,
BONDS AT ANNUAL AND OTHER RATES
259
which is the present value of an annuity of $1 for 2 periods
at 3.0225% per period. The foregoing is an application of
the two symbolic rules, D = 1 — p and P = D -v- 1.
In order to obtain the bond premiums, we must multiply
the above present worth by 1,007.50 in the case of the semi-
annual bond, and for the annual bond by 977.50.
Premium on semi-annual
bond $1.912846 X 1,007.50 = $1,927,192
Premium on annual bond 1.912846 X 977.50 = 1,869.807
These premiums agree perfectly with the values pre-
viously obtained otherwise,* viz. : $101,927.19 and $101,-
869.81.
As another example, take that of a 4% annual bond
yielding 5%, for two years. Evidently this will be at a
discount instead of at a premium. To annualize the ratio
1.025, multiply it by itself, giving 1.050625 ; the annualized
interest rate is therefore : . . .050625
from this subtract 04
giving as the interest-difference 010625
To find the present worth of an annuity of $1 for 2 (an-
nual) periods at 5.0625%, take from Table II,* column
2%%, the value for 4 (semi-annual) periods. .$ .90595064
subtract from 1.00000000
The compound discount is therefore $ .09404936
Divide by .050625 ; the quotient is $1.8577652
which is the required present worth of an an-
nuity of $1 for 2 periods at 5.0625%.
Multiply this by .010625; $1.8577652|| X
.010625 = $ .01973876
This is the discount, which, subtracted from par 1.00000000
gives the value of a $1 bond, $ .98026124
* In Chapter XXXII,
26o PROBLEMS AND STUDIES
This may be tested by multiplying down to maturity :
X 1.025 .02450653
X 1.025
$1.00476777
.02511919
$1.02988696
.04
X 1.025
$ .98988696
.02474718
X 1.025
$1.01463414
.02536586
$1.04000000
.04
$1.00
§ 305. Rule for Bond Valuation
We are now prepared to formulate a rule for valuing an
annual bond on a semi-annual basis without reference to
the values of a corresponding ordinary (or semi-annual)
bond.
Rule 1:
(a) Annualize the rate of interest (find the equivalent
annual income rate) ; e.g., 1.015^ = 1.030225.
(b) Subtract this rate from the annual coupon, or vice
versa, to give the interest-difference; e.g., .04 — .030225 =
.009775.
(c) Multiply the latter by the present worth of an an-
nuity of $1 for the number of annual periods at the an-
nualized rate, giving the premium or the discount; e.g.,
.009775 X 1.9128453 = .0186981.
Where the values of the ordinary semi-annual bond
BONDS AT ANNUAL AND OTHER RATES 261
have already been calculated, as in the bond tables, it will
be possible to obtain therefrom the values of the annual
bond, with a saving of time.
§ 306. Multipliers for Annualizing
For each combination of a cash rate with an income rate,
a multiplier may be found which, applied to the premium or
the discount for any number of years on a semi-annual bond,
will give the depreciation caused by the collection of the
interest once a year only; and this multiplier will be con-
stant, whatever the time. A table of these multipliers will
be found in Spragne's "Extended Bond Tables," page VIII.
In the example given in § 300 we have a 4% annual
bond yielding 3% semi-annually. On page VIII* in the
column headed "4% Bond" on the line opposite "3%" is
the multiplier .0297767. The premium on the ordinary
semi-annual bond for $100,000 at 2 years, we have seen, is
$1,927.19.
$1,927.19 X .0297767 = $ 57.385
As the value, if semi-annual, would be. . . 101,927.192
the value of the annual bond is reduced to 101,869.807
In the example in § 304, the annualizer, or multiplier, for
a 4% bond to yield 5% is found from the table
to be ,. . .0493827
The value of a semi-annual bond of $1 at 2
years is $.98119013
or its discount is ,. 01880987
$.01880987 X .0493827 = 00092888
which subtracted from 98119013
gives the annualized value 98026125
This differs from the one already given 98026124
by 1 cent on a million dollars, owing to decimals having
been rounded off.
* Sprague's "Extended Bond Tables."
262 PROBLEMS AND STUDIES
These multipliers are obtained by the following formula,
in which c and i represent the nominal rates per annum.
ci
(4 + i) (c-i)
§ 307. Formula for Annualizer
The formula may be thus expressed as a rule.
Rule 2 : To find the annualizer for any two rates :
(a) Multiply the rates together for a dividend; e.g.,
.04 X .03 = .0012.
(b) Multiply 4 + the income rate, by the difference of
rates for a divisor; e.g., 4.03 X .01 = .0403.
(c) Their quotient will be the required multiplier, or an-
nualizer; e.g., .0012 ^ .0403 == .029776675.
The product of the premium by the annualizer is always
subtracted from the semi-annual value; and sometimes the
resulting value may be shifted to a discount from a premium,
even if it was a premium which was extracted from the
table. Thus, in the case of a $1,000,000 5% annual bond,
payable in one year, netting 4.95%, the premium $482.03 X
the annualizer 1.22237313 = $589.22, and the value of the
annual bond becomes $999,892.81.
It must be observed that only values for full years can
be obtained in either of these ways. An odd half-year is a
"broken" period, and must be treated as in Chapter XL
§ 308. Conventional Process
While the foregoing is the method which would doubt-
less be followed in buying and selling, a more accurate re-
sult, from a mathematical standpoint, would be obtained by
using as the half-year value the one found by multiplying
down at the effective rate.
Thus, in a bond at 4%, payable annually, on a 3% semi-
annual basis, the values are :
BONDS AT ANNUAL AND OTHER RATES 263
2 years before maturity. . .$1,018,698.07
lyear " " ... 1,009,488.22
Maturity 1,000,000.00
The amortization for the first year is $9,209.85, and for
the second $9,488.22. Halving these severally, the values
by half-years appear as follows :
Values Di
2 years $1,018,698.07 $4,604.92
11/2 years 1,014,093.15 4,604.93
1 year 1,009,488.22 4,744.11
1/2 year 1,004,744.11 4,744.11
Maturity 1,000,000.00
§ 309. Scientific Process
The foregoing result would be in accordance with the
conventionally established rule that during any period
(which is here a year) simple interest must prevail and the
amortization accrue proportionately to the time elapsed from
the beginning of the period.
But the half-year may, with equal propriety, be con-
sidered the period, since the income is on a semi-annual
basis. Under this assumption we must multiply down :
Value, 2 years $1,018,698.07
X 1.015 10,186.98
5,093.49
Value, 11/2 years $1,033,978.54 flat
Less accrued interest 20,000.00
Value, 11/2 years $1,013,978.54 and interest
Similarly, the value at one-half year is fixed at $1,004,-
630.54, and the series with differences will appear as
follows :
264 PROBLEMS AND STUDIES
Values Di
2 years $1,018,698.07 $4,719.53
11/2 years 1,013,978.54 4,490.32
1 year 1,009,488.22 4,857.68
1/2 year 1,004,630.54 4,630.54
Maturity 1,000,000.00
In the second half of each year there is less amortization,
and consequently more earning than in the first half; but
this may be defended on the ground that by the conditions
prescribed, interest is compounded semi-annually. The
earning power at compound interest must continue to in-
crease until a cash payment; and there is no cash payment
at the mid-year.
§ 310. Values Derived from Tables
This latter form of valuation at mid-years is recom-
mended for comparative (non-commercial) purposes.
The values at ^1/2 years, $1,004,630.54, $1,013,978.54,
etc., may be deduced from the ordinary extended tables by
multiplying by the annualizer, with this proviso : that the
interest-difference must first be temporarily added to the
tabular premium or discount before multiplying. Thus, in
the case just considered, the excess of .02 over .015 is .005
each half-year ; or, on $1,000,000, $5,000.
To find the value for 1% years, take from the
table the premium. . $14,561.00
add the interest-difference 5,000.00
giving the multiplicand $19,561.00
which, multiplied by the annualizer .9702233,
equals $18,978.54
from which again subtract. 5,000.00
giving the premium as above $13,978.54
BONDS AT ANNUAL AND OTHER RATES 265
§ 311. Successive Process
In general, when a schedule is to be formed for an an-
nual or a quarterly bond, on a semi-annual basis, it will be
found easier after ascertaining the initial value to multiply
down to maturity, as that will usually require fewer figures.
§312. Problems and Answers — Varying Time Basis
(Y7) $25,000 4% bonds, interest payable annually, 8
years to run ; what is the price at a 3.70% semi-annual basis?
(78) What multiplier will annualize the premium on
the above bonds as given in the regular bond table ?
(79) An offering is made of $30,000 31/2% bonds, in-
terest payable annually, of which $10,000 mature in one
year and $10,000 each year thereafter. What should be
paid for them to produce 3.40% semi-annually?
Answers :
Problem (77)
$25,452.30
Problem (78)
.8777970411
Problem (79)
$30,040.34
§ 313. Bonds at Two Successive Rates
Occasionally bonds are issued with the agreement that
the interest paid shall be at a certain rate for some years,
and at another rate for the remainder of the time to maturity.
An example is a fifty-year bond bearing 4% for 20 years
and 5% for the following 30 years. The problem is then to
find the price at which they will pay a certain income, say
3.60%.
Each of the two successive cash rates will cause a
premium, and we may calculate these premiums separately.
266 PROBLEMS AND STUDIES
§ 314. Calculation of Immediate Premium
The premium caused by the 4% rate will last only 20
years and will then vanish ; hence, this premium is just the
same as that on a plain 4% bond for 20 years, netting
3.60%, which we find by calculation or from tables to be
$56,680.10 on $1,000,000.
§315. Calculation of Deferred Premium
The premium produced by the 5% rate does not take
effect immediately, but after 20 years. It is a deferred an-
nuity. An annuity for the entire 50 years of the excess in-
terest, 1.40%, or in other words the premium on a fifty
year 5% bond to net 3.60%, is $323,568.65
But during the first 20 years there will be no
such premium ; we have already charged that at
4%. Hence we must by subtraction eliminate the
analogous 5% premium for 20 years, which is 198,380.36
leaving a remainder $125,188.29
which is the premium, or present worth, of the enjoyment of
a 5% cash rate (as against a 3.60% income rate) commenc-
ing 20 years from date and continuing till 50 years from
date.
Adding together the two premiums, $56,680.10 and
$125,188.29, we have $181,868.39 as the premium which
should be paid for the bond.
A simpler way to apply the principle is to add together
the 4% value for 20 years $1,056,680.10
and the 5% value for 50 years 1,323,568.65
$2,380,248.75
and subtract the 5% value for 20 years 1,198,380.36
giving the value of the composite bond $1,181,868.39
BONDS AT ANNUAL AND OTHER RATES 267
This procedure has the advantage that it applies alike to
bonds which are selling at a premium and to those which are
selling at a discount and automatically allows for that dis-
tinction.
§316. Symbols and Rule
We may for convenience represent the earlier rate by Ci
and the latter rate by C2, i being the net income. We may put
m for the number of years at which the rate Ci prevails, and
n for the number of years at C2; m + n is the entire time.
The rule will then be as follows :
Rule : To find the value of a bond to yield i per cent,
when by its terms it pays cash interest at the rate Ci for m
years and thereafter at Cz for n years, maturing in w + w
years. Add together the value of a Ci bond for m years
and that of a Cz bond for m-\- n years, and from the sum
subtract the value of a Cz bond for m years.
An example of a bond of very early maturity will il-
lustrate the principle of the rule and will admit of demonstra-
tion by multiplying down. A bond for $100,000 paying
5% for 1 year (2 periods) and 6% thereafter for 1% years
(3 periods) is to be valued so as to yield the annual return
of 4%.
§ 317. Analysis of Premiums
If the rate on the bond were 5% for the entire 2% years,
its value, according to the bond tables,* would be
$102,356.73. On the other hand, if the rate were 6%, its
value would be $104,713.46. Let us analyze these premiums
into their component parts, which are the present worths of
excess interest for five periods, $500 per period in the case
of the 5% bond, and $1,000 per period in the case of the
6% bond.
Sprague's "Extended Bond Tables.
268
PROBLEMS AND STUDIES
% year
1 year
Premium one year before maturity
1% years
2 years
2% years
5%
$490,196
480.584
6%
$980,392
961.168
$970.T80 J $1,941,560
471.161
461.923
452.866
942.322
923.846
905.732
Premium 2% years before maturity $2,356,730 $4,713,460
Any premium is the sum of a certain number of present
worths of $500 or of $1,000. But in the double-rate bond,
the only present worths that have an influence on the in-
augural value are the first two in the 5% column and the
last three in the 6% column, as indicated by the braces placed
opposite them.
It is evident that the values producing premiums at the
5% rate amount to $970.78, and that those in the 6%
column amount to $2,771.90 (the easiest way to obtain this
latter amount being to subtract $1,941.56 from $4,713.46).
Hence the premium is :
$970.78 + $2,771.90 = $3,742.68
The equivalent process by the rule would be :
Value of Cr bond, m years. . .$100,970.78
plus Value of d bond, m-\- n years 104,713.46
$205,684.24
less Value of Ca bond, m years. . . 101,941.56
$103,742.68
It will be interesting to multiply down to maturity and
thus test this result :
BONDS AT ANNUAL AND OTHER RATES 269
$103,742.68
+ 2,074.85
$105,817.53
— 2,500.00
$103,317.53
+ 2,066.35
$105,383.88
— 2,500.00
$102,883.88
+ 2,057.68
$104,941.56
— 3,000.00
$101,941.56
+ 2,038.83
$103,980.39
— 3,000.00
$100,980.39
+ 2,019.61
$103,000.00
— 3,000.00
'Par $100,000.00
It will sometimes be the case that in multiplying down
the values will increase for a time and then begin to de-
crease at the change of rate ; or vice versa, the values will at
first decrease and then later increase.
270 PROBLEMS AND STUDIES
§ 318. Problems and Answers — Successive Rates
(80) An issue of bonds matures on Jan. 1, 1966. In-
terest is to be at 6% till Jan. 1, 1936, and thereafter at Q'/o,
What is the price at a 3.60% basis on July 1, 1916?
(81) $10,000 of Waterworks Bonds, 5 years to run,
first 3 years at 4%, thereafter at 5%; find the value to
yield 4.40%.
(82) Find the value of the same bonds to net 4%%;
51/4%.
Answers :
Problem (80)
$1,413,422.66
Problem (81)
$9,988.49
Problem (82)
$9,824.98 ; $9,617.04
CHAPTER XXVIII
REPAYMENT AND REINVESTMENT
§319. Aspects of Periodic Payment
When a loan is payable in equal periodic instalments,
each covering the interest and part of the principal, the
most obvious way of looking at it is that the principal is
gradually paid off; and then we have this aspect:
(1) A diminishing principal;
A diminishing interest charge, and therefore
An increasing repayment.
But precisely the same result may be obtained from a
different point of view by assuming that no payment is
made at all until the final date of maturity, at which time the
sinking fund, or sum of instalments plus interest, is just
sufficient to pay off the whole debt. In this case, we will
have the following aspect :
(2) An unchanged principal;
A uniform interest charge ;
A uniform instalment devoted to reinvestment and
allowed to accumulate.
As an illustration of the first aspect, suppose we consider
a debt of $1,000, bearing interest at 3% per period. This
debt may be extinguished in four periods by uniform instal-
ments of $269.03 at the end of each period, as we have
already pointed out in Chapter VII. For convenience, how-
ever, we again set forth the details on th^ following page :
271
2^2
PROBLEMS AND STUDIES
Instalment
Interest
on Balance
Payment on
Principal
Principal
Outstanding
$1,000.00
(1) $269.03
$30.00
$239.03
760.97
(2) 269.03
22.83
246.20
514.77
(3) 269.03
15.44
253.59
261.18
(4) 269.03
7.85
261.18
0.
Total, $1,076.12
$76.12
$1,000.00
Here we see the diminishing principal, the diminishing
interest charge and the increasing repayment or amortiza-
tion.
From the reinvestment point of view, we have :
Instalment
Interest on
Entire Principal
Carried to
Sinking Fund
Principal
$1,000.00
(1)
$269.03
$30.00
$239.03
1,000.00
(2)
269.03
30.00
239.03
1,000.00
(3)
269.03
30.00
239.03
1,000.00
(4)
269.03
30.00
239.03
1,000.00
For
Reinvestment
Interest on
Previous Total
Total
Accumulated
(1)
(2)
(3)
(4)
$239.03
239.03
239.03
239.03
$ 7.17
14.56
22.15
$ 239.03
485.23
738.82
1,000.00
The amortization of principal in its two aspects as re-
payment and reinvestment should be carefully studied and
REPAYMENT AND REINVESTMENT 273
the problems in connection with Chapter VII should be
worked over into schedule form in each aspect.
§ 320. Integration of Original Debt
This principle will be found to hold : The "principal out-
standing" by the first method + the "total accumulated" by
the second method = the original debt.
The first point of view is based entirely on facts. With-
out regard to reinvestment, it is certain that the borrower
pays and the lender receives the exact rate of interest stipu-
lated for each period on the actual balance due at the be-
ginning of such period, and this balance may be represented
either by a single account or by a cost account and an
annulling account.
§ 321. Use of the Reinvestment Point of View
There are some cases where, especially from the point of
view of the debtor, it is desirable to keep in view the entire
original sum. One of these cases is where it is impossible
or impracticable to diminish or pay off the debt before
maturity and where accumulation is the only method avail-
able. Another is that of a trust where there is an obligation
to keep the corpus of the fund intact, and consequently
reinvestment in some form is necessary.
But the calculations of reinvestment are hypothetical and
prospective. They have not the same actuality as those of
repayment, but are theoretical estimates of what is expected.
Unless a contract has been made to take the instalments ofY
one's hands at a fixed rate, the amount realized is pretty sure
to differ from the amount anticipated.
§ 322. Replacement
There is a third method of considering periodic pay-
ments, which is not mentioned in the actuarial treatises,, and
274
PROBLEMS AND STUDIES
which may be called replacement to distinguish it from re-
payment and reinvestment. The successive repayments are
transferred to new investments, which are not to accumu-
late but merely to furnish new income, helping out the
diminished income on the waning principal. We have out-
lined this procedure under "Bonds as Trust Fund Invest-
ments," in § 148 ; but for purposes of comparison we will
put the materials already used in § 319 into the replacement
form, assuming at first that replacements are so invested as
to earn exactly 3%.
1
Interest on
Principal
2
Payment on
Principal
3
Principal
Unpaid
4
Replace-
ment
5
Interest on
Replace-
ments
6
Total
Income
1+5
(1) $30.00
(2) 22.83
(3) 15.44
(4) 7.85
$ 239.03
246.20
253.59
261.18
$1,000.00
760.97
514.77
261.18
0.
$ 239.03
246.20
253.59
261.18
$ 7.17
14.56
22.15
$ 30.00
30.00
30.00
30.00
$76.12
$1,000.00
$1,000.00
$43.88
$120.00
Column 4 of replacements is not accumulative, as its in-
terest is not compounded, but is used as income, supplement-
ing that in Column 1. The balance of Column 3 plus the
total of Column 4 at any point make up $1,000. The two
corresponding amounts in Columns 1 and 6 always make
up $30 (Column 6). At the close, the original $1,000 has
been exactly replaced by the new securities.
§ 323. Diminishing Interest Rates
As already remarked, it would seldom happen that
exactly 3% would be the rate secured for the replacements,
which ought to be of the same grade of security and avail-
ability as the original sum. Let us suppose that the rate of
interest was declining so that the first replacement had to
REPAYMENT AND REINVESTMENT 275
be loaned at 2.95%, the second at 2.90%, and the third at
2.75%. Columns 5 and 6 are then the only ones changed:
5
Interest on
Replacements
6
Total
Income 1+5
$ 7.05
14.19
21.16
$ 30.00
29.88
29.63
29.01
$42.40
$118.52
Here we have the principal intact, and the falling-off is a
gradual one affecting the interest. If we had proceeded on
plan No. (2), the full predicted interest would have been
consumed, but the principal would have been impaired, which
is inadmissible. Hence, in cases of this kind, we must use
the vanishing principal with actual replacement. The re-
investment scheme is a basis of calculation only and cannot,
like the repayment plan, be reduced to practice.
§ 324. Proof of Accuracy
It is interesting to note that in the repayment method
the work may at any point be tested by a fresh calculation,
showing the whole procedure to be coherent and consistent.
For instance, in our example, the principal at three periods
from maturity is $760.97. Treating this as the principal,
to find the sinking fund we divide $760.97 by 2.82861, just
as for 4 periods we divided $1,000 by 3.7171. This gives
$269.03 — the same result as before for the value of the
equivalent annuity; $22.83 as the interest ($760.97 X .03),
and $246.20 as the first repayment or the constant reinvest-
ment, in either aspect.
2^6 PROBLEMS AND STUDIES
§ 325. Varying Rates of Interest
It must not be supposed that there is at any one moment
a single rate of interest prevaiHng. Considerations of se-
curity, convenience, and availabihty give rise to different
grades of securities and different rates of interest. The
prudent investor will probably have at the same time some
capital out at high rates and some at low. The money at
high rates is not quite so secure, not quite so readily realiz-
able, and requires more effort for the collection of its in-
come. That at low rates is nearer to absolute freedom from
risk and from the labor of supervision; it almost automat-
ically collects its own income. The investor will have so
planned his investments as to endeavor to preserve a
judicious equilibrium between different grades of security,
and consequently of income. As his investments are liqui-
dated, he will try to maintain or improve this equilibrium,
and he will choose his reinvestments from a wide range,
some of low revenue but highest safety and others of the
contrary qualities. It is therefore fallacious to assume that,
as an author has said, "on the same day and under the same
circumstances money received from any one source may be
invested at the same rate as that received from any other
source." Theoretically it may be, but practically it will
usually be invested in the same grade of security as that
which it replaces.
§ 326. Dual Rate for Income and Accumulation
When the lender assumes great risk, or when the supply
of loanable capital is temporarily deficient, he will exact very
high rates, or refuse to loan. Or he may require a high
rate and also demand that the instalments of repayment shall
be large enough to secure the higher rate on the entire
original loan until fully paid; while in ordinary reinvest-
ments a lower rate is easily obtained,
REPAYMENT AND REINVESTMENT
277
§ 327. Instalment at Two Rates
Suppose that $1,000 is loaned, repayable in 4 instalments,
on such a basis that the lender will have 5% interest per
period on the entire capital, while it will be replaced by
accumulating at 3%.
The sinking fund is exactly the same as in our previous
example, $239.03. But the instalment is :
not $30 + $239.03, or $269.03
but $50 + $239.03, or $289.03
The instalment here is as much greater as the interest is
greater. The accumulation is precisely the same as hereto-
fore. The instalment provides not only 5% on the money
remaining invested, but also 2% (unearned) on that which
had been repaid.
An instalment of only $282.01 would pay 5% on the
outstanding capital, which would gradually be replaced by
3% investments. Thus it is seen that the borrower has
to pay more than 5% ; in this instance about 6.(
§ 328. Amortization of Premiums at Dual Rate
This loaning at a dual rate is of so little practical im-
portance, at least in this country, that it would not be worth
mentioning, except that a few writers have tried to apply
the same principle to the amortization of premiums. They
assume that there is no other way of ascertaining the value
of a bond than by laying aside the excess of interest and
letting it accumulate till maturity. But this is not at all
necessary. The question is, what uniform rate is yielded
by each dollar of the investment during the time it is out-
standing. When this is ascertained, it can make no differ-
ence what is done with the capital after it is returned. We
may as well say that the rate of a series of bonds payable
$1,000 each year and issued at par, cannot be determined
2^8 » PROBLEMS AND STUDIES
until we Know at what rate the amounts were reinvested up
to the date of the last maturity. Reinvestment has nothing
to do with the yield of the original investment. Neverthe-
less, two authors have constructed tables based upon a dual
rate, one a rate of income, the other a rate of accumulation,
and they have taken the latter at the arbitrary figure of 4%,
irrespective of the grade of the bond.
§ 329. Modified Method for Valuing Premiums
It is proper to give the method by which these results
seem to be obtained, or, at least, a method which will pro-
duce those results.
As a preliminary we will consider the valuation of a
premium in a slightly different way from any yet given.
We have seen that the premium on $1 is the present
worth (at the income rate) of the difference of rates. We
may modify this by saying that it is the difference of rates
(c — i) X the present worth of an annuity of $1 (P),
which may be found in Table IV.* But to multiply by P
is the same thing as to divide by 1 ^ P, or 1/P. Therefore,
another expression for the premium is (c — i)^(l/F). But
we found in § 90 that the rent (1/P) is the sum of the
sinking fund (1/A) and the single interest (i). There-
fore, we still further modify our expression:
Premium = (c — i) -^ (i + 1/A)
§ 330. Rule for Valuation of a Premium
Rule: Subtract the income rate from the cash rate, and
use this as a dividend. Add the instalment from Table V*
to the rate of income, and this will be the divisor. The
quotient will be the premium.
Example : What is the premium on a 6% bond (semi-
annual) for $1, 50 years, yielding 6%?
In Chapter XXXII.
REPAYMENT AND REINVESTMENT
279
c = .OS; i = .025 ; c — i= .005 (dividend)
1/A at 21/2%, 100 periods = .002312 (Table V*)
.025 + .002312 = .027312 (divisor)
Premium = .005 -^ .027312 = .18307
Value of bond, $1.18307
§ 331. Computation at Dual Rate
To introduce the feature of an accumulative rate differ-
ing from the income rate, it is only necessary to change one
term in the above formula. The value of 1/A must be taken
from the column of Table V,* which represents the accumu-
lative rate, i remaining as the income rate.
Example : What is the premium on a 6% bond, as above,
yielding 5% on the entire investment to maturity, the
principal being replaced by a sinking fund at 4% ?
c — i = .03 — .025 = .005 (dividend)
1/A at 2% ='.003203 (Table V*)
i + 1/A = .025 + .003203 = .028203 (divisor)
.005 -^ .028203 = .17729
Value of bond = $1.17729, agreeing with Croad's
and Robinson's tables.
The constant income is .0294322 (i.e., 2%% of the value
of the bond), which subtracted from the cash received .03,
leaves as contribution to the sinking fund .0005678. At 4%
an annuity of .0005678 will amount in 50 years to .17729||,
as may be ascertained from Table III,* thus replacing the
premium.
§ 332. Dual Rate in Bookkeeping
This form of valuation, which introduces an arbitrary
element, cannot be satisfactorily applied in the bookkeeping
processes of Chapter XVII. It is impossible to derive one
value from another consistently. The result will not agree
* In Chapter XXXII.
228o PROBLEMS AND STUDIES
with a fresh calculation, and the profit or loss on a sale will
be distorted. Any intermediate value, as shown by the
actuaries, may have three different versions.
§ 333* Utilization of Dual Principle
While tables on a fixed replacement rate are useless for
purchasing securities, the principle may occasionally be
utilized. Thus, the trustee referred to in § 148 may find
that it is impracticable to invest favorably such small
amounts as $400 or $500, and may conclude to deposit a
sinking fund in a savings bank where he may reasonably
expect that it will accumulate for the next five years at
3%%, or he may make a contract with a trust company on
the same terms. He may then decide also that it is better
for the beneficiary to receive a uniform income, rather than
one gradually decreasing.
At 13/4% per period, a sinking fund of $.092375 will, in
ten periods, amount to $1; therefore, by multiplication it
will take a sinking fund of $414,883 to accumulate to $4,-
491.29 in 10 periods. Out of the coupon of $2,500 must
be taken the instalment of $414.88, leaving for the bene-
ficiary a constant semi-annual income of $2,085.12, instead
of the $2,089.83 with which he would have begun on the
replacement plan, and which would have gradually fallen to
$2,079.82 for the last half-year.
§ 334. Installation of Amortization Accounts
When the accounts of securities have once been estab-
lished on the plan of gradual extinction of premiums and
discounts, it is not difficult to take care of each new purchase
as it comes in, and to prepare its appropriate schedule, run-
ning if desired all the way to the date of redemption. When,
however, the accounts have been previously kept on the basis
of par or of cost, and it is desired to introduce investment
values instead- the task is much greater.
REPAYMENT AND REINVESTMENT 281
§ 335- Scope of Calculations
It might be supposed at first thought that it would be
necessary to start- the schedules back at the date of pur-
chase, but this is entirely unnecessary. For example, we find
a 5% bond for $100,000 which 20 years ago was bought for
$112,650, and which has still 10 years to run. At the date
of purchase it must have had 30 years to run. Turning to
any table of 5% bonds, 30-year column, we find that this
was (within 39 cents) a 4^4% basis. Turning to the 10-
year column, it appears that the value of a 5-year bond at
4l^% is $106,058.46. It is sufficiently accurate to begin
with this value, disregarding the 39 cents residue, although
that residue might be eliminated by the proportion,
12,649.61 : 6,058.46 :: 39 : 19
This would increase the present value to $106,058.65.
§ 336. Method of Procedure when Same Basis Is Retained
So long as the same basis is preserved, any number of
intervening years may be disregarded. The following pro-
cedure may be recommended :
(1) Make an accurate list of the issues held, giving the
following particulars : dates of maturity; dates of purchase;
par value of each lot; cost of each lot, being at the rate of
$. . . . per $1,000 of par; rate of interest paid, and the in-
come basis when ascertained. Leave a column for valuation
at a date one period earlier than the proposed date of trans-
formation,
(2) Ascertain on what income basis each lot was bought.
This is done most easily by using the tables. In these and
the subsequent calculations it will be found advantageous to
use blank books and entrust nothing to loose papers. Head
each calculation with a statement of the problem which it
solves. Paper for these blank books, ruled with vertical
lineSj everj" third one of which is darker than the other two,
282 PROBLEMS AND STUDIES
will much facilitate the work, and it is desirable to have the
pages numbered in a continuous series, for reference.
(3) Find the value of each lot at the initial date, which
is, as already stated, one period earlier than the date on
which the books are to be transformed to investment values.
(4) Where different lots of the same class have been pur-
chased at various dates and prices, their values at the various
bases on the initial date should be added together, giving a
composite value. Ascertain what is the income basis for
the time yet to run on this composite value. This basis is
the average basis for the remaining time of the bond.
(5) Having carefully verified all the initial values and
the effective rates, proceed to calculate the amortization and
accumulation of each class for one period, commencing a
schedule for each. The resulting values should be again
verified with care, these being the values with which the new
accounts will begin.
(6) Continue the calculations of successive values, carry-
ing them into decimals two places beyond the cents, ignoring
slight differences in the last figure. Copy the results,
rounded to the nearest cent, into the schedules, and complete
the latter. If time allows, it is advisable to calculate each
schedule to maturity, because no better proof of the correct-
ness of the entire chain of values can be had than the fact
that the bond reduces exactly to par at maturity. But if
time presses, only a few of the values may be calculated,
but the last one should be verified by some independent
method. It is well in this case to leave in the blank book
sufficient room to complete the calculations for each schedule.
A reference on the schedule to the page of the blank book
where the calculation is made, will be useful.
(Y) Make such entries as will place the ledger or ledgers
on the investment-value basis.
Part III — Logarithms
CHAPTER XXIX
FINDING A NUMBER WHEN ITS LOGARITHM
IS GIVEN
§ 337' Logarithmic Tables
The meaning and use of logarithms have already been
discussed in a general way,* and a simple three-figure, four-
place table of logarithms given (§43). The expression
"three-figure" refers to the number of figures in each of the
numbers of the table, and the expression "four-place" refers
to the number of decimals in each of the corresponding
logarithms. In the table given, for example, the logarithm
of 7.41 is shown to be .8698.
§ 338. Discussion of Logarithms
As previously explained, every logarithm consists of the
characteristic, or whole number (which is frequently zero),
and a decimal fraction. Occasionally the decimal fraction
is zero, as in the case of the logarithms of .01, .1, 1, 10, 100,
etc. The decimal fractions which constitute that part of the
logarithm requiring tabulation are interminate ; that is, their
values may be computed to any desired number of decimal
See Chapter III.
283
284 LOGARITHMS
places and the last place will still be inexact. Thus, the
logarithm of 2 to 20 places is .301 029 995 663 981 195 21+.
In a 4-place table, this would be rounded
off to 301
in a 7-place table, to 301 030
in a 10-place table, to 301 029 995 Y
in a 12-place table, to 301 029 995 664
The terminal decimal is never quite accurate, but is nearer
the true value than either the next greater decimal or the
next smaller one. Thus, the logarithm .8698 is nearer the
true logarithm of 7.41 than either .8699 or .8697.
§339- Standard Tables of Logarithms
The tables most in use, like those of Vega, Chambers,
and Babbage, are of five figures and seven places. A six-
figure table would have to contain ten times as many loga-
rithms as a five-figure table and, even though the number of
places were not increased, the space occupied would be ten
times greater than in the case of the five-figure table. In
the tables above mentioned, two figures in addition to the
five tabulated may be obtained by interpolation.
§ 340. United States Coast Survey Tables
The tables of the United States Coast Survey have five
figures and ten places. Nine figures may be obtained by
simple proportion, but the tenth is, for most purposes,
unreliable.
It will, of course, be understood that the more decimal
places given in the tables, the more figures we can obtain
in the corresponding numbers, but the number of figures (in
the desired number) can never be more than the number of
places (in the corresponding logarithm). All of the fore-
going tables give auxiliary tables of proportionate parts or
differences*
NUMBER FROM LOGARITHM 285
§ 341. Gray and Steinhauser Tables
Tables of 24 and 20 places have been published by Peter
Gray and Anton Steinhauser, respectively, but the plan for
extending the number of figures is quite different from the
method of simple interpolation above referred to. Both of
these authors proceed on the plan of subdividing the number
into factors, and adding together the logarithms of those
factors.
§342 A Twelve-Place Table
For the accurate computation of problems in compound
interest, specially designed tables will be found in Chapter
XXX. A limit of twelve figures has been selected as the
most useful for this purpose. In the logarithms tabulated,
thirteen decimal places are given, the thirteenth place insur-
ing the accuracy of the twelfth figure of the corresponding
number, which would otherwise sometimes be 1, 2, or even
3 units in error, through the roundings being preponderant
in one direction or the other.
§ 343. The "Factoring" Method
The method used in finding logarithms within the scope
of these tables, but not directly given in them, is that of
factoring, it being possible to construct the logarithm of any
number of twelve figures or less (999,999,999,999 in all)
by some combination of the 584 logarithms given in the
table of factors (§358).
Column A contains the logarithms of numbers of two
figures, 11 to 99, both inclusive, carried to thirteen places
of decimals.
Column B contains the logarithms of four-figure numbers
1.001 to 1.099, each beginning with 1. and one zero.
Column C contains the logarithms of six-figure numbers
J.QOOOl to 1.00099, each beginning with 1. and three zeroes.
286 LOGARITHMS
Column D, 1.0000001 to 1.0000099, beginning with 1.
and five zeroes.
Column E, 1.000000001 to 1.000000099, beginning
with 1. and seven zeroes.
Column F, 1.00000000001 to 1.00000000099, beginning
with 1. and nine zeroes.
For example, opposite 34 in the table we find :
A .531 478 917 042,3 In . ., 3.4
B .014 520 538 757,9 In 1.034
C .000 147 635 027,3 In 1.00034
D .000 001 476 598,7 In 1.0000034
E .000 000 014 766,0 In 1.000000034
F .000 000 000 147,7 In . . ., 1.00000000034
By omitting all the prefixed zeroes, the printed table is
made very compact, each complete line across the table of
factors shown in § 358 containing only 57 figures instead of
82, as would otherwise be necessary. In using the tables
this must be taken into consideration, and accordingly it will
be understood hereafter that C 34, for example, means the
number 1.00034, and F 34 means 1.00000000034.
§ 344. Finding a Number from Its Logarithm
In this process there are two stages : first, to divide the
logarithm into a number of partial logarithms taken from
those contained in the table of factors; second, to multiply
together the numbers corresponding to these logarithms. Of
course only the decimal part of the logarithm is used, and
the number has the position of its units figure determined
from the characteristic of the logarithm.
Let the logarithm .753 797 472 366,5 be one which
has been obtained as the result of an operation, and let the
corresponding number be required. Search in Column A
for the highest logarithm which does not exceed the given
NUMBER FROM LOGARITHM
287
one. This is found to be .748 188 027 006,2, which stands
opposite 56.
Subtracting from ,. 753 797 472 366,5
A 56 .748 188 027 006,2
we have the remainder
This is smaller than any logarithm
in Column A. We search for it in
Column B and find opposite 13
precisely the same figures
5 609 445 360,3
6 609 445 360,3
These two logarithms added together make the given loga-
rithm ; hence the product of their numbers gives the number
required.
To multiply 56 by 1.013 :
56
1013
56
56
168
56728
1013
5065
6078
56728
This process may be greatly simplified as follows, plac-
ing the figures of the multiplier in vertical order at the side :
56
56
168
or
56
13 X 5 065
13 X 6 078
56728
56728
Notice that the first product is moved two columns to the
right of the multiplicand.
The column G used in the following example is not
given in the table of factors, but it is found by simply taking
the first two figures from E. The "G" number in this case
may be either 55 or 56, which may make the thirteenth
figure of the result doubtful, but probably not the twelfth.
288
LOGARITHMS
Now take a larger logarithm ....
and continue the subtraction A 56
B13
C26
D29
E 58
F48
G55
753 911 659 107,4
748 188 027 006,2
5 723
5 609
632
445
101,2
360,3
114
112
186
901
740,9
888,7
1
1
284
259
852,2
452,2
25
25
400,0
189,1
210,9
208,5
2,4
2,4
(See Note 2*)
(See Note 1*) 2
6
(See Note 3*) 5
8
4
8
5
_5
(See Note 4*)
5600
56
168
567280000
113456
340368
5674274928000
*
11348550
5106847
5674291383397
• 283715
• 45394
• 2270
• 454
• 28
3
567429171526
" Qlj following page,
NUMBER FROM LOGARITHM 289
Note 1 : The second multiplication jumps its right-hand
figure (6) four places to the right, which may be marked off
by four zeroes, or four dots.
Note 2 : Having extended the product to include the
13th figure, contraction begins in this multiplicand ; its first
figure used being the 7th (marked 'A') allowing for the
carrying from the 8th. Thus the starting point for this
multiplication is moved six places hack.
Note 3 : The multiplicand need no longer be extended,
as has been done at successive stages above, but remains the
same to the end. For convenience, dots may be placed in
advance under the first figure to be used in multiplication in
each line.
Note 4 : The thirteenth figures are added, but only used
for carrying to the twelfth. In this example the total of the
last column is 31, but it does not appear, except as con-
tributing 3 to the next column.
The dot below a figure indicates where the contracted
multiplication begins, all the figures to the right being
ignored, except as to their carrying power.
§ 345. Procedure in an Unusual Case
Required the number for log. Oil 253 170 127.
In this example there is no suitable logarithm in A and
we must begin with B, as shown on page 290.
This example illustrates the procedure when B furnishes
the first logarithm. It also shows the convenience of using
paper ruled for the purpose.
In order to set down the partial products without hesita-
tion, remember the numbers 2, 4, 6.
In multiplying by B, the first figure of the product moves
two places to the right.
In multiplying by C, the first figure of the product moves
four places to the right.
290
LOGARITHMS
1 — -.1
Formation of Number from Logarithm
Logarithm
A —
B26
C24:
D36
E63
F83
G67
1
1
2
5
3
1
7
1
2
7
1
1
1
4
7
3
6
7
7
5
8
1
1
5
4
8
2
1
9
8
3
1
5
7
1
2
1
1
5
6
9
6
1
3
1
4
8
5
1
7
2
3
2
2
7
7
7
3
2
6
3
9
5
3
3
6
6
3
4
5
2
2
9
9
A
B
26
C 2
4
D 3
6
E 6
3
F 8
3
G 6
7
1
1
0^
2
6
2
4
6
1
*
2
4
—
—
—
1
2
6
2
4
•
6
•
3
2
6
4
7
1
8
5
7
7
3
4
9
8
1
•
•
2
•
6
•
2
•
4
9
9
3
6
4
1
3
4
5
8
8
7
7
2
3
7
5
9
1
1
6
1
1
2
6
2 5
NUMBER FROM LOGARITHM
291
In multiplying by D, the first figure of the mtdtipliccmd
moves six places to the left.
The following rule may now be formulated for this
process.
§ 346. Rule for Finding Number when Logarithm Is Given
(a) By successive subtractions separate the given loga-
rithm into a series of partial logarithms found in the
columns of the table of factors, setting opposite each its
letter and number.
(b) By successive multiplications find the product of all
the numbers thus found, allowing, in the placing of the
partial products, for the prefixed 1 and zeroes.
The work may be made to occupy fewer lines by setting
down the factors E, F, and G as one number at the top,
multiplying it by A, and incorporating it thereafter as one
multiplicand with the preceding figures. The result will
not be affected. Let the factors be, as in the first example :
A 56, B 13, C 26, D 29, E 58, F 48, and G 55.
E F G
584855
A 56 2924275
350913
5600000327519
B13 56000003275
16800000982
5672800331776
C26 1134560066
340368029
5674275259871
D29 11348551
5106848
5674291715270
292 LOGARITHMS
Required the number whose logarithm is .6 or ^.
A 31
.500
491
000 000 000,0
361 693 834,3
B20
8
8
638 306 165,7
600 171 761,9
COS
38 134 403,8
34 742 168,9
D78
3 392 234,9
3 387 483,7
ElO
4 751,2
4 342,9
F94
G03
408,3
408,2
0,1
The resulting factors, A 31, B 20, C 08, D 78, E 10,
F 94, and G 03, when combined produce the result
3. 1622776601 7. The multiplication illustrates how
zeroes are treated when they occur in the multipliers. The
result is the square root of 10, to 12 places, as may be
demonstrated by multiplying 3. 16227766017 by itself.
§ 347. Method by Multiples
In order to facilitate the multiplication of the factors,
A, B, C, etc., the table of multiples* (§361), giving the
product of each number from 1 to 9, by every number from
2 to 99, will be found convenient. Thus, the multiples of
89 read in one line as follows :
•Devised by Arthur S, Little.
NUMBER FROM LOGARITHM 293
123456789
089 178 267 356 445 534 623 712 801
Then, if it be desired, for example, to multiply 68792341
by 89, we would select from the above table
under 6 5 3 4
8 712
7 623
9 801
2 178
3 267
4 356
1 089
6122518349
We have thus multiplied each figure of the multiplicand
by both figures of the multiplier, setting down each partial
product unhesitatingly. Three figures must be set down
for each partial product, even if the first be a zero. The
work may be made more compact by piling the partial
products like bricks, using only three lines :
5 3 4,8 1,3 5 6,
7 1 2,1 7 8,0 8 9
6 2 3,2 6 7,
6122518349
To use this method in combining the factors of a num-
ber, the letters A, B, C, etc., are written above alternate
figure spaces, which is facilitated by the use of paper properly
ruled. Then the first partial product under each letter is
placed with its middle figure under that letter at the top.
The following is an example of a combination already
performed in another form :
294 LOGARITHMS
A B C D E F G
A 56 1 684856
2 8 0,4 4 8
4 4 8,2 8
2 2 4,2 8
56 327519
B 13 6 5 3 9,0 6
7 8 2 6,2
091
6672800331778
C26 13 0,0 5 2,0 0,
1 5 6,2 8,0 7 8
1 8 2,0 0,0 8
5674275259864
D 29 1 4 5,1 1 6,1 6
1 7 4,0 5 8,1
2 3,2 3
567429171526
A process* for verifying a numerical result, by using
a different set of factors in a second operation, is as follows :
Required the number corresponding to
.305 773 384 163,0
The factors are A 20, B 10, C 97, D 21, E 94, F 94,
and G 33 ; and the number is 2.02196383809.
In order to check the result and make sure of perfect
accuracy, we may solve the problem a second time, using a
smaller factor for A, provided the first remainder be less
than B 99, or .040997692423,5. Using A 19 instead of
A 20:
Suggested by Arthur S. Little.
NUMBER FROM LOGARITHM
295
A 19
305
278
773 384 163,0
753 600 952,8
B64
27
26
019 783 210,2
941 627 959,0
C17
78 155 251,2
73 823 787,1
D99
4 331 464,1
4 299 494,1
E73
31 970,0
31 703,5
F61
G37
266,5
264,9
1,6
The new factors are A 19, B 64, C 17, D 99, E 73, F 61,
and G 37.
By multiplication, we obtain the same result as before :
A 19
B64
C17
D 99
7 361,37
6 625,233
19
11
13 986,60
4 839,2
76 65,9
202
160 014 881,7
20 216 001,5
14 151 201,1
202
194 382 084,3
1 819 749,5
181 974,9
202 196 383 809
CHAPTER XXX
FORMING LOGARITHMS; TABLES
§ 348. Explanation of Process
To form the logarithm of a given number — the table of
factors being used — ^two processes are necessary: first, the
number is separated into a series of factors corresponding to
the six columns of the thirteen-place table ; second, the loga-
rithms of these factors are taken from the table and added
together.
The factoring is effected by a progressive division, as
illustrated by the following simple example :
To find the logarithmic factors. A, B, C, etc., of 5.6728.
First extend the number to 12 places, 567 280 000 000.
The first factor. A, is always the first two figures of the
number itself.
A 56)56 7 2 80 000 000 (1.013 B
56
72
56
168
168
It will readily be seen that one 56 might have been
omitted.
A 56)7 280 000 000 (B 13
56
168
168
296
FORMING LOGARITHMS
297
Turning then to the table, we have only to set down the
logarithms of these two factors :
A 56 nl 748 188 027 006,2
B 13 m/ 5 609 445 360,3
56728 nl 753 797 472 366 5
B 13 may be regarded as an abbreviation of 1.013.
In the next example a second divisor, at least, is required.
A 56) 7 42 9 17152 6 (B 13
56
182
168
AB56728)14
The second divisor is the product of A and B. It might
be obtained in either of three ways :
By multiplication 56 X 1.013 = 56728
By addition 56
+ 56
+ 168
56728
But the easiest way is
by subtraction 56742 (first five figures of the number)
— 14 (the remainder)
56728
This is the proper method for forming all divisors after
the first; that is, subtract the remainder from the original
number so far as used.
We resume the division, bringing down four more
figures, to the ninth inclusive :
298 LOGARITHMS
AB ) 56728 )149171626(C26
113456
357155
340368
ABC ) 5674275) *1678726(D29
1134855
543871
510685
56742914 33186 (E 58
28371
4815
4539
276 (F 48, 7
227
49
45
The third divisor A B C is also formed by subtracting
from the number 5674291715
iiC the remainder 16 7 8 7
leaving 5674274928
As only six figures are needed for the divisor and
one additional figure for carrying, this is rounded up to
5 6 7 4 2 7,5
The fourth divisor is practically the number itself so far
as needed, and this lasts to the end.
The entire process is now repeated, but to insure greater
FORMING LOGARITHMS
299
accuracy in the twelfth figure we will divide out to the
thirteenth :
A 56) 742 9 17152 6,0 (B 13
56
182
168
A B 56 728) 149 171 (C26
113456
357155
340368
A B C 56 742 749 28) 16 7 8 7 2 6,0 (D 29
(Contracted division begins here) 1 1 3 4 8 5 5,0
5 4 3 8 7 1,0
6 1 6 8 4,7
56 742 92) 3 3 18 6,3 (E 58
2 8 3 7 1,4
4 8 1 4,9
4 5 3 9,4
2 7 5,5 (F48
2 2 7,0
485
454
3 1 (G55
28
300 LOGARITHMS
It remains only to add together the logarithms
A 56 (nl) 748 188 027 006,2
B 13
5 609 445 360,3
C 26
112 901 888,7
D 29
1259 452,2
E 58
25 189,1
F 48
208,5
G 65
2,4
567 429 171526 (nl) 753 911659 107
The figures in the thirteenth column are used only for
carrying to the twelfth column.
§ 349. Rule for Finding a Logarithm
We may now formulate the following rule for finding
the logarithm :
(a) Fix the number at 13 figures, by adding ciphers or
cutting off decimals.
(b) Cut off the two left-hand figures by a curve, giving
A.
(c) Divide the next three figures by A, giving the two
figures of B, and a remainder.
(d) Form the second divisor A B, by subtracting the
remainder from the first five figures of the number.
(e) Bring down four more figures to the remainder and
divide by A B, giving the two figures of C and a remainder.
(f) Form the third (and last) divisor A B C by sub-
tracting the remainder from ten figures of the number.
(g) Divide the remaining figures by the third divisor.
As there are ten figures in the divisor and only eight in the
dividend, contraction begins immediately. Having obtained
the figures of D, the divisor for E, F, and G is simply the
number itself contracted.
FORMING LOGARITHMS
301
(h) Write down the logarithms of A, B, C, D, E, and
F, obtained from the several columns of the table of factors ;
also that of G, being the first two figures of the correspond-
ing E. The sum will be the mantissa or decimal part of the
logarithm of the number, the thirteenth decimal place being
used for carrying only.
§ 350. Examples of Logarithmic Computations
It is advisable, for the sake of both convenience and
accuracy, to make all of these logarithmic computations on
paper ruled with at least thirteen vertical lines, every third
line being darker than the other two. Space should be left
on either side of these lines for writing in the divisors and
quotients, and for such other arithmetical work as may be
necessary. As a rule, however, there would be few, if any,
additional arithmetical computations which would have to
be performed at the sides.
A few examples for practice are given below with the
factors and the solution :
56Y4=A 56 B 13 C 21 D 15 E 35 F 42 G 70
log. 5674=3.753 889 331458
38.8586468578 =A 38 B 22 C 58 D 31 E 39 F 02 G 25
log. do. =1.589 487 673 453
3.1415926535898+=A 31 B 13 C 41 D 16 E 33 F 11 G 91
log. do. =497 149 872 694
(This number is the ratio of the circumference of a circle
to its diameter. )
1.02625=B 26 C 24 D 36 E 63 F 83
log. do. = .011 253 170 127
This number begins with an expression of the form B
(1.026), hence no division by A occurs. 1026 is the first
divisor.
302
LOGARITHMS
B 1026) 2 5 C 24
2052
4480
4104
B C 102624624) 3 7 6 0,0 D 36
3 Y 8 Y 3,9
6 8 1 2 6,1
6 1 5 7 4,8
6 5 5 1,3 E 63
102625) 6 1 5 7,5
3 9 3,8
3 7,9
8 5,9 F 83
8 2,1
3,8
3,1
7 G
B26
Oil 147 360 775,8
C24
104 218 170,0
D36
1 563 457,3
E63
27 360,6
F83
360,5
G70
3,0
70
log. 1.02626 = .011253 170 127*
§ 351. Logarithms to Less Than Twelve Places
The table of factors may be cut down to any lower num-
ber of places. In the example in § 348 it may be required
* This result will be found also in the Table of Interest Ratios, but even
more extended.
FORMING LOGARITHMS
303
to give 9 places only, the tenth being used for carrying. We
cut down the original logarithm to ten figures, with a comma
after the ninth,, and it becomes :
A 56
753 911 659,1
748 188 027,0
B 13
5 723 632,1
5 609 445,4
C 26
114 186,7
112 901,9
D 29
1 284,8
1 259,5
E 58
F 24
A
B 1
3
26,3
25,2
1
56
56
168
C 2
6
567280000
113456
3 4 3 6,8
D2
9
E 5
8
F 2
56742749 2,8
1 1 3 4,9
610,7
2 8,4
4,6
1
56742917 1,4
The number is slightly in error in its tenth place, but
correct to the ninth.
304
LOGARITHMS
§ 352. Tables with More Than Twelve Places
If a table of factors for 18 or some other number of
places should hereafter be prepared, the methods which have
been explained would be applicable to the new table.
§ 353- Multiplying Up
Another method for obtaining the factors of the number
in forming its logarithm* proceeds by multipHcatioa instead
of division, the latter operation being notably the more
laborious. The number, at first taken as a decimal less than 1,
is successively multiplied up to produce 1.000,000,000,000,0
and these multipliers are the A, B, C, D, E, F, and G, whose
logarithms added together make the cologarithm, or loga-
rithm of the reciprocal, from which the logarithm is easily
obtained.
§ 354. Process of Multiplying Up
A is a number of two figures, a little less than the
reciprocal of the number, which will be called the sub-
reciprocal of its two initial figures. A table of sub-reciprocals
is given in § 360. The number multiplied by A will always
give a product beginning with 9. B is always the arithmetical
complement of the two figures following the nine, or the
remainder obtained by subtracting those two figures from
99. Multiplication by B will usually give a result beginning
with 999. C is the next complement and gives five 9's,
999,99. D similarly brings 999,999,9**,***,*. No further
multiplication is necessary, after D has been used as a factor ;
the six figures in the places of the asterisks are the comple-
ments of E, F, and G.
To illustrate, let it be required to obtain the logarithm
to the 12th place of .314 159 265 359 0. The object is to
multiply .314 159 265 359 up to 1.000 000 000 000 0. The
♦ guggested by Edward S. Thomas of Cincinnati,
FORMING LOGARITHMS
305
first step is to find the sub-reciprocal of 31, or A. Turning
to the table of sub-reciprocals, opposite 31 we find 31, by
which we multiply.
A 31
99 — 73 = 26
B 26 is therefore
the next multipli-
er; dropping the
last two figures
(99—21) C 78
(99—43) D 66
.3 141592653590
.9 424777960770
314159265359
.9 738937226129 (One nine secured)
194778744523
58433623357
.9992149594009 (Three nines secured)
6994504716
799371968
.9999943470693 (Five nines secured)
49999718
5 9 9 9 9 6 6
(99—47) E 52
(99—03) F 96
(100—77) G 23
A 31 nl
B 26
C 78
D 56
E 52
F 96
G 23
colog.
log.
.9999999470377 (Seven nines secured)
52
96
23
.4913616938343
111473607758
3386176522
24320423
225833
4169
10
0.50285012 7306
f. 4 9 7149872694
3o6 LOGARITHMS
§ 355' Supplementary Multiplication
It may happen, in the course of multiplication, that the
complement of the figures following the 9 does not suffice
to secure two nines more. In this case, another supple-
mentary multiplication must take place. I'his occurs in the
following example, which has already been solved in § 348.
Required the logarithm of the number
.567 429 171 526
In this example the C multiplication also requires an ad-
ditional figure. This seldom occurs.
.567 429 171526
A 17 .397 200 420 068 2
.964 629 591594 2
B 35
28 938 887 747 8
4 823 147 958
.998 391 627 300
B 01
998 391 627 3
.999 390 018 927 3
C 60
599 634 0114
.999 989 652 938 7
C 01
9 999 896 5
.999 999 652 835 2
D 03
299 999 9
.999 999 952 835 1
E 47, F 16, G 49
47 164 9
A 17
230 448 921 378 3
/B 35
I 01
14 940 349 792 9
434 077 479 3
C 60
260 498 547 4
01
4 342 923 1
D 03
130 288 3
E47
20 411 8
F 16
69 5
G 49
21
colog.
.246 088 340 892 7
log.
1.753 911 659 107 3
FORMING LOGARITHMS
307
As the multiplication by B 35 brings only 998 instead of
999, we multiply again by B 01, which brings it up to 999-h.
In the next example there is a large defect in the product
obtained by multiplying by B 85, which requires an addi-
tional multiplication by B 7.
110 175*
A 83
881 400 (83, sub-reciprocal of 11)
33 052 5
B 85
914 452 5
73 156 200
4 572 262 5
B 07
992 180 962 5
6 945 266 737,5
C 87
999 126 229 237,5
799 300 983,4
69 938 836,0
D45
999 995 469 056,9
3 999 981,9
499 997,7
E 30, F 96, G 35
999 999 969 036,5
30 963,5
A 83
B 85
B 07
C 87
D45
E 30
F 96
G35
919 078 092 376,1
35 429 738 184,5
3 029 470 553,6
377 671 935,8
1 954 320,8
13 028,8
416,9
1,5
colog.
.957 916 940 818
log.
1.042 083 059 182
*The number no 175 was purposely selected, very slightly in excess of the
highest number in column B, so as to produce the shortage of 7.
2o8 LOGARITHMS
§ 356. Multiplying Up by Little's Table
.137 128 857 423 9
A 71 710 715 682 846 4 (71 being the
213 142 355 142 sub-reciprocal
49 756 849 721 3 of 13.)
.973 614 887 709 7
B 26 23 415 620 818 2
1 820 262 080
078 104182 2
.998 928 874 790 1
B 01 998 928 874 8
.999 927 803 664 9
C 07 69 994 946 3
.999 997 798 6112
D 22
19819818
198 198
19 815 4
.999 999 998 606 4
01 393 6
EFG
A 71
851 258 348 719 1
B 26
11 147 360 775 8
B 01
434 077 479 3
C 07
30 399 549 8
D 22
955 446 8
E 01
434 3
F 39
169 4
G 36
16
862 871 142 576 1
.137128 857 423 9
which is the log. of 1.371 288 574 239
FORMING LOGARITHMS 309
In the preceding example, Little's table of multiples
(§ 361) is used in the multiplication. It will be found that
the logarithm when computed has the same figures as the
number itself — a remarkable peculiarity which no other com-
bination of figures can possess.
§ 357- Different Bases
Ten is the base of the logarithmic system which we
have been explaining; it is the most useful of all systems,
because ten is also the base of our numerical system. These
are usually called common, or vulgar, or Briggsian loga-
rithms, but decimal logarithms would seem a more appro-
priate name.
Any number might form the base of a system of loga-
rithms, but the only other in actual use is one known as
the "natural" system, having for its base the number
2.718281828459+, known to mathematicians as e, which is
the sum of the series,
1+ 1 + -I_ + ___! + 1 +
■^1^1X2^1X2X3^1X2X3X4^
1
etc.
1X2X3X4X5
This is only used in theoretical inquiries, and is seldom
of utility to the accountant.
3IO
LOGARITHMS
^ Cq CO "^
o o o o
vo vo t^ 00 0\
o o o o o
O'-«CMC0»^ u-)VOt^00O\
O '-' C^ f^ Tf
CVl CN> CM OJ CV|
feo
CO r>.^ P^ "^^ fN.^ l-H Tf tV.^
Tl-" cd~ CO tC T-T vo 0~ Tf 0\ fO t^ CM VO o
oooo ooooo ooooo
^ "^^ *^. '~L ^ °o ^ ^ °0 ^^ "1
"looTrrrodcvf
VO VO r>N t^ 00
ooooo
O^ CVJ^ in On^ cm
vo" '-»" lO oC ■^'"
00 0\ On a\ o
OOOO'-"
rO vO_^ 0\ CM lO <^^ ^. "^ *^
Tf OO" Cvf t^" '-H irT o" "^"^ OO"
O"«*'00cor^ '-hvootj-on
CM CM PO fO <0
ooooo
On^ CM in 00 i-H
cm" InT '-T lO o
8 8 o o
Tf tN^ O CO vo
..,,.._ Tt 00 CO tC T-T
Tl-t^,-iTt-00 '-irfOO'-ivn
cOt^CMVOO ioOncOOOCM
vo vo t^ t>» 00
OOOOO
Tl- -* vo lO
oooo
s
Ov^ CM to C» »-H
lO O Tf' OO" CO
00 CM U-) 00 CM
\0 '-t to o\ -^
O 0\ o
o o ^
Tj- 0\ CO OC)
oC oo" oo" tvT
CM LO 00 "-I
Tf 00 CM l^
CO vo O CO
Tf 00 CO t^
O O — ' — '
CM vo^ O^ '"^^ Ol
tC VCD vo irT Tf
"^ t^ O CO vo
1-1 UT) O "^ 00
t^ O "^ t^ O
1-H vo O '^ 0\
CM CM CO CO CO
■^ b>. ,-* iq 00^ CM^ vo^ O^ Tj^ t^_
-4 CO CO CM i-H
C7\ CM to 00 — I
CM t^ T-i to O
"^ t^ ^ "^ 00
CO t^ (N vo O
Tj- Tt- to to vo
OOOO OOOOO ooooo
'-' O O 0\ 00
rj- t^ O CM to
rt- 00 CO t^ .-"
i-H "3" 00 i-i to
to Ov CO 00 CM
VO vo r>. t^ 00
ooooo
'-' "1. °Q, ^. "\
oo" *^" ^ ^' ^
00 — I "^ t^ o
to O -^ 00 CO
00 CM to 00 CM
vo r-* to 0\ ^
00 Ov 0\ On O
O O O O -H
t/3
o
H
U
<
O
w
8
^^ °0 P. °Q-
co" cm" On" i-h"
CM O CO CO
On 00 VO Tf
CM to 00 i-t
Tf 00 CM t^
CO VO O CO
rf 00 CO t^
gSSo
CM CM 00 On vo
i-h" rC On" oo" ^f
00 CO Tj- VO Tj-
—I 00 to i-H t^
"^ vo Ov CM rf
»-< to On '<1- 00
t^ O CO l^ O
T-^ vo O rf Oi
(VI CM CO CO CO
OOOOO
Ch l>.^ CM^ CM^ 00^
VO to i-h" CO T-*"
t^ VO i-t '-• l^
CM t^ CM vo ON
tx 0\ (VJ "^ VO
CM VO i-" to ON
"«^ t>x T-i Tj- r^
CO t^ CM vo O
Tt Tf to to vo
ooooo
o^ t^^ "-1, O^ to_^
tC oo" tC cm" co"
00 to 00 tN. i-H
CM to t^ ON 1—1
0\ 1— • CO to 00
CO 00 CM vo o
i-f "^ 00 —' to
to 0\ CO 00 CM
VO vo t^ t^ 00
OOOOO
vo CO vo to O
.-T vo" t^r to" o"
r-i vo t^ rf t^
CM CM CM eg T-H
O CM Tf VO 00
to C?N CO t^ i— •
00 '-' to 00 CM
vo _i to Q^ rf
00 On On o\ O
O O O O '-I
Mo
CO CM Tf O
On" i-h" o" 0\
1^ eo CM O
Tf to O 00
t^ "H CO CM
O t^ CM CO '-'
• O t^ On t^
Tf t^ Q CO
CO VO O CO
Tj- 00 CO t^
O Q i-H '-<
OOOO
iOC3\VOtOON VOOOOCOCO
VO On CO On VO
to »— I lO O CO
t^ t^ to »- > CM
'-' O O CM VO
VO 00 t^ CO VO
O On Tt- to »-<
vo r^ On o ^
vo C7\ CM vo On
»-( lo O Tf 00
CM CM CO CO CO
OOOOO
CM >— I CO O t^
00 CTn O VO ON
t>. to to CO o\
CO to CM to T^
t^ lO '-H -"^ to
CO '-' to Tf ON
T-H ^ O ON t^
CM to 00 O CO
CO t^ ^-i vo O
rf- Tj- to to vo
ooooo
CM^ ON b>j^ r^ rh
0\ rC cm" o" vo"
tt n- CM o o
CM C3n On O O
CM t^ CM 00 Tj-
Tf O to t^ 00
O t^ ON t^ .-H
VO CO O t^ -^
VO On CM rf I^
^r 00 CO r^ ^
vo vo t^ t^ 00
ooooo
On^ On t>.^ CM 00
^ vo" CO cm" oC
vo CO On ^ CO
t^ O t^ t^ VO
'T' CM to fo vo
t^ Tf On CO to
'-' t>N 00 vo o\
O lO O to ON
o CM to r^ o\
^ O "* 00 CM
00 o\ o 0\ o
o o O O --I
00
CO
coo
CM_^ vo_^ 00 CM t>.^ On^ co_^ co^ 00^ O, ON CM^ vo^ ^
• ' * * ' ' ' • ' ■ oo" tC vo" oo" to" to" oo" co" cm" ■^' co" CM* iC '-'
* . . * ■ tOrtOt^ tOtOl^Oto VOcoCM.-<r^
• • \ , , , I • ; ''-"OcovO OvocOt-hOn vot^coo^
• • • ; ; ; ; • • 'tovocMto ovcMi-Htoo \o ^ ^ ^ Zi.
' ! ! ; . . . . lOOr^toco tooOCMOO OnonKJco;?!
; ; ; , . ; ; , , ;vocmcoo cmononiovo oncmvooo?^
• * • ! I ! ■ • ' 'CMt-icooO i-iOnOOCMcO OonCMI^.'-^
• ! ; . . . , ' . lONOO-^CM Ov'-HTft^to CM»-<(MCMrt
• ; ] • \ ; ; \ • icor-iovi-" O'-'^t-cMt^ ocm"^!^^
• I I I '. '. '. I '. *. '-•<?\covo vo-^OtooO tii^CMCMr-iO
. . , . . . . . I'^t^'-'r^ t^Ocotor^ OCM'^VOOO
: : : . . . ; : . .oo'-''-' '-•cmcmcmcm cocococo^o
.-hCMCO"^ tovOt^OOON O'-iCMcO'* tovor^OOC^N p.-iCMco;J
oooo ooooo ,-1 ■^ r-i T-t wt ,-i,_tr-i^,-i cmcmcmcmc^
TABLES
311
KJ^^8Sa
to CO CO CO CO
10 NO t^ CO Ok
CO CO CO CO CO
§5^94
m NO tN. 00 ON
TJ- Tf Tf Tt Tj-
VO 0\ CO vo On
00" cvf tC ^ 10"
^ ^ CM CM
CO NO CO t^
0" Tt" On" co" tC
CO CO CO Tj- -^
o_ CO r>v Tf
cm" no" 0" 10" On"
in m NO NO NO
tN.^ ^^ "^^ I^^ "-l,
t^ l\ 00 TO ON
Tl- 00 i-H 10 00
m" on" rf 00" cm"
Tf l^ CM^ 10
10 ON (N) S ON
00 CM t^ ,-H 10
00 »-< Tt t^
00" co" rvT ^ no"
CM NO On CO NO
rf 00 CO t^
CO NO ON CM 10
0" Tj-" 00" co" rC
CM NO in On
00 i-H Tj- ts.
CO 00 CM NO '-H
543,3
977,5
411,8
846,1
280,4
^ ^ CM CM
CO CO CO Tf -^
10 10 NO NO NO
tv. t^ 00 00 C?N
»-H T-H r-H ,_( l-H
SiSia^c^
00 CM iO 00 CM
Tt Tt ro" <Nf cm"
CO VO On CM 10
tv. '-' 10 Tf
m 00 ^ ''d- t^
00 CO t^ i-H 10
co^ NO^ <> CM^
00" tC no" lo" m
CM in 00 '-H Tf
'«*• 00 CO t^
in rs.^ CO in
■^'" co" co" cm" ^"
t^ CO NO On
"-H NO rl- 00
00 0^ CO in IN.^
CM m t^ CO
CO t^ T-r NO
10 On CM NO 0\
00 CM t^ ,-H 10
^ ^ CM CM
CM NO CTn CO NO
Tj- 00 CO l>^
CO CO CO •^ '^
CO NO CO
CM NO in On
LO in NO NO NO
t^ Tf r^
CO 00 CM NO '-'
t^ t^ 00 00 C3N
»— 1 ,— 1 »-H ,_( »-H
on ON »-i
•-H .-1 CM CM CM
0_ tN.^ On^ 00^ CM^
u^ 00 00 CO ^
00 NO Tl- ^
CM On •—< On CO
in u-T co" no" tC
CM CO CM
00 -<* 10
CO CM '^
Tf" 00" 00" in" 00"
t^ t^ CO in CM
Tj- 00 CM in 00
m ^ CO CM t^
CM CO rt "^
tV.^ Tf ts.^ NO ^^
^s^ i<" co" NO no"
m CM in CO t^
m in Tt CO ^
'-< CO 10 1^
NO Tf 00 CM
10 ON CM 10 ON
00 CM CO 10
NO '-' 10 On CO
CM NO ON CM NO
NO t^ On '-'
r^ .-H in "*
On CO NO CO
CO Tt in NO t^
00 CM NO "^
NO CO t^
CO 1^ "^ t^
00 CM t>> >-< 10
^ ^ CM CM
Tt 00 CO r^
CO CO CO -^ Tj-
T-t NO in On
10 m NO NO NO
CO 00 CM NO '-'
t^ t^ 00 00 Ch
•-H ,-. CM CM CM
TO 00^ co_ CO "-^
'-<" 10 t^" On" cm"
On t^ On 10 NO
CO t^ ir> NO t^
CM^ 10 CM^ NO^ On^
in co" i-T oC rC
00 On '-• »o
t^ CM CM in r^
On CM Tt CM
t^ rj- CO in in
00_^ in in in CM^
00" 0" CO no" no"
i-H CO 00 t«v NO
ivT .-T 00" ivT co"
■^ CO t^ Tt ON
Tj- in NO NO 1-H
m CO Tt -"d-
NO NO -^ i-H t^
00 CO "^ .-< CO
Tj- lO t^ r-l 00
C7\ m NO CO t^
Tj- in in 10 Tf
CO t^ t^ CO m
CO t^ t^ CO '!i-
Q rj- ,-c CM 00
CO r^ CO Lo
CM Tf t>. On »-•
t^ T-. 10 ON Tf
t^ 00 ON
CO 10 l^ CM
00 CM NO »-• ir>
C3N 00 t^ m
rf in t^ On '-'
ON CO t>. 1-H NO
CO t^ Tf
CO in NO 00
rj- 00 CM t^
NO 1-H NO i-H in
I-" CO -^ NO t>.
»-< in ON CO t^
^ ^ ^ CM
CM CO CO T}- Tl-
Tt m in NO NO
i;^ r^ t^ 00 00
S^^?)^^
0^ 00^ 0^ CM
cm" 0" On" cm" On"
r^ t< to Tt ON
NO Ch '-' CO 00
tv.^ ^ 0\ Ol co_^
C?\" Tj-" On" tvT cm"
»-i CO T-i r^ Tf
t^ 00 CO 00
CO co_ 0^ 00_^ in
0" l^" tC no" no
in NO NO '-H CM
CO t^ NO
t^ 0\ CM
CO t^ CO m -^
co_^ vo^ tN.^ vo^ in
in T-T in" in" 00
t^ 00 CO l^ CM
i^ NO ON CO
00 r>x Ti- T-. t^
CO i^ On
Tf CO 00 On r^
vo On t^ CO '— '
CM NO On On On
Tj- Tt NO t^
Tl- (M On
m t^ m NO
:-» NO in NO
On in On in ^N,
ON 00 CM Tf NO
CO .-H t^ t^
in S 00 CM
CO CO 00 tN.
Tl- t^ NO IT) On
C?N 0\ CO i-H CO
^ T^ On CO CO
CM NO Tf .-• t^
t-i to «-• 10 •^
00 CM ^ CO Tj-
NO 00 NO
CO CM t^
On CO CTn 00 CM
m 00 -^ NO m
t>. CM rr -^
CM tN. t^ ^ NO
I-" in On -^ On
CM t^ CM »-^
t^ -^ ^ t^ CM
ON -H CO ■^ NO
CO Tt ■«;*• "^ Tf
IN. T^ ir> 00 -H
Tf Tt vo ^ S^
xn ^n \n \n *j->
CM CM CO CO CO
S S NO S NO
CO CM CM i-H Q
VO NO NO NO NO
CM CM CM CM CM
»-i CM CO Th
CO CO CO CO cO
to NO t^ 00 On
CO CO CO CO CO
-H CM CO ^
""^ -^ "* Tj- Tl-
in NO t^ 00 On
'^ "^ -t Tl- 1^
312
LOGARITHMS
Pi
O
H
u
o
<
6
o i-H rq «r> Tt
lO VO tN. 00
0\
o
^
CM CO T^
in NO
VO NO
t^ 00 ON
O 1-1 CM CO ■^
1^
to lO U-) lO lO
to to to to
to
NO
NO
NO NO NO
NO NO NO
t^ t^ t^ ^>. t>.
*
»-• »o 00 C^ lO
C7\ CM lO On
CM
vq^
C3N
CO vo O
co^ vq
O^ co^ l^^
O^ CO^ tN.^ O '^
feS
0, t>. ^ lO O -^
oo" co" tC ^
no"
o
'«t"
On" co" tC
cm'~ no"
'— r to oC
Tf" oo" cm" tvT ^"
O O t-H ^ CM
O '-' CVJ CM ro CO
CO Tj- -^ to
to
NO
^
NO t^ t^
00 00
On C3N On
1— i
• CVJ CVJ CM CM CN
CM CM CM CM
CM
<N
CM
CM CM CM
CM CM
CM CM CM
CO CO CO CO CO
tsj^ o^ CO vq^ o\
CM to 00 T-i
"it
In.
O
CO NO 00
'"1. '^^
^ O^ co_^
NO On^ CM^ to 00
$
Wb
•> - .^
rt oT CO tC i-H
no" o" •^'' oC
CO
rC
cm"
^83;
0\ co"
In." cm" no"
o T}- oT CO r>r
^ Tj- 00 '-' lO
00 CM u^ 00
CM
to
ON
CM NO
ON CO NO
O CO NO O CO
^ t^ ^ lO O Tl-
• '-H CM CM CO CO
00 CO t^ ^
NO
o
Tj-
ON CO t^
CM NO
O to On
rf CO CM tN ^
»-i
CM CM CM CM
to
NO
NO
NO t>H t^
00 00
C?N On On
O O '-I ,-( CM
CM CM CM <N1 CM
CM
CM
<N
CM CM CM
CM CM
CM CM CM
CO CO CO CO CO
$
O^ CM^ ''t t>j, 0\
»-i CO *o t^
ON
CO
Tt_^ NO^ 00
o ^
CO Tf O^
'^ c> o, *-<_ CO
rC VO to rj-" po"
Co" CM ^ O"
On"
0\
§
tC no" to"
to" rf
co" cm" '-*"
<z> on" OS oo" rC
vo CA rg to 00
'-I Tj- tv. O
CM
to
^ Tf r^
O CO
NO ON CM
to t^ CD CO NO
§
Tj- 00 CO r^ ^
NO O '<4- ON
CO
t^
NO O ^
C7N CO
tN ^ NO
O "^ On CO t^
O '-' "^ 00 -H to
00 rg lo 00
CM
to
On
CM no On
CM NO
CA CO NO
O CO NO O CO
H t^ ^ to O T»-
00 CO t^ —
NO
o
Tj-
ON ro r^
(M NO
O to ON
■^ 00 CM l^ ^
-H CM CM CO CO
CO T}- Tf to
to
NO
O
NO t^ r^
00 00
ON ON ON
O O T-H ^ CM
CM CM CM CM CN
CM CM CM CM
CM
CM
CM
CM CM CM
CM CM
CM CM CM
CO CO CO CO CO
CM O CO CO 0\
'-< On CO Tf
rl-
CO
CTn 0_ 00^
CO CO
O CO CM
00 O 00 CO Tj-
^ ^ ^ ^ -
CM lO Tt- o CM
cm'' rC o" oC
to
tC
no"
'-H Tf" cm"
oo" o"
on" t^ no"
NO 00 to
rf o" ^" O" lo"
^
t^ CM CO O CM
O CO CO t^
00
Tj-
"^
-^ l^ NO
O '-'
00 t^ ^ ,-1 NO
*
0\ t^ -"I- I-" t^
CO 00 CO r^
tO
00
^ CO »o
t^ 00
00 00 00
tN NO lO CO O
ug
o ^ ^ "* '^ "^
o c^ CO t^ 1-1 to
NO ^O t^ t^
00
^
00
On On On
On C?n
0\ On On
On On On On On
On CO r^ T-H to
ON CO t^ '-'
to
CO
l^ '-H to
On CO
t^ ^ to
O O Tf t^ ^ Tj-
t^ -H TT 00
-*
00
r-l to 00
^ lO
00 CM to
00 CM to O* CM
l^ 1-t to O Tf
00 CO t^ »-H
NO
o
Tj-
ON CO r^
CM NO
^ ^ O^
CO 00 CM NO '-t
»-< CM CM CO CO
CO Tt Tj- to
to
NO
NO
NO t^ t^
00 00
O O ^ '-• CM
CM CM CM CM CM
CM CM CM CM
CM
CM
CM
CM CM CM
<N ^
^ 04 CM
CO CO CO CO CO
On CM t^ to to
t^ 00^ Tf CvJ^
"1
°°
CO^
to CO O
00 NO
to to 00
CM^ Oi 00 O^ "^
J. i. ^
i.
^
O* 00 t^ to vo
co" InT rC On"
tC
Tj-"
,— r
to CO ON
^^
Tf cm" oo"
to" «-<" no" no" co"
VO CM r-H 00 t^
CO ON O ON
o
NO
o
Tj- CM to
CM On O
00 CO to NO NO
O O 00 '-I 00
VO ^ CO NO
CM
ON
r>^ to C3N
t^ NO
'^ NO CM
NO 00 CO Oi CO
■N'
0\ vO C?N ^ O
On 00 t^ t^
Q
to
CO
NO Tj- t^
-^ NO <N
t^ "^
On CM to
t>s O to ^ --I
*^« *
0\ «-! CO t^ '-I
to i:-5 00 NO
NO
NO
00
o o
r-l to O
t^ t^ 00 CM 00
Wo
O CM t^ t>. CO vo
rf On ON NO
CTn
00
CO
to CM NO
NO CM
Tj- CM t^
t^ Tf t^ tN C^
*-<
0\ (M to 00 O
CM CO Tl- to
to
to
to
Tj- CO ^
CTn t^
Tj- r-H t^
CO O rf CA Tj-
00 O '-H CM -^
^ VO O '^t 00
to NO r^ oo
ON
o
CM CO Tj-
Tf to
NO t^ t^
00 00 On CTn O
<N NO O '«:^
00
CO
t^
^ to ON
CO t^
^ to C7N
CO r^ '-H to o
i-H 1^ CM CM CM
CO CO -^ •*
""^
to
to
NO NO NO
1^ t^
00 00 00
0\ o\ o o zt
CM CM CM CM CM
CM CM CM CM
CM
CM
CM
CM CM CM
CM CM
CM CM CM
CM CM CO CO CO
O, Ol 00^ 00^ o^
CM CM lO C^
vO_
00
CO VO^ON^
On On
00 CH '^^
CO '-I CO to O
vcT t~>r '^^ <3^ crT
rt no" cm" cm*
CM
Co"
o"
OO" co" co"
CM <-<
O no" l^"
rf On" -^ O '-I
CO On CO O Cvl
ON O t^ NO
NO
00
ON VO 00
Tf Tf
O O CO
,-. ^ CO CM CO
CO O VO VO 00
Tf O VO to
CO
o
rf Tf ON
NO "^
t^ tN r^
O tN ■* "-I tN
, »
Tj- vo CO en On
O t^ lO CO
^
O
to
On ON CO
NO 'J^
CM CM O
o 00 ^ o c^
Q t^ "^ MD lO
S T-H CO 00 t^
OO CM ^O On
to
CO
^;%s^
to CO
M O^ o
rj- Tj- ON NO '-'
^ ;
NO O 00 CTn
o
.^
00
CO On
O CO ^ 00 tN
»
O O CO to CO
CM 00 Tf t^
CM
,_,
ON
^ O 0\
CO CO
Tf 00 ON
00 00 CM CM "-I
t^ t^ O t^ On
NO 00 l\ CM
to
to
<^
On "^ t^
T-> -rf
r^ o Tf
<^ ^ S2 £^ j:^
0\ u-> O CM CO
CO ^ 00 Tf
00
<o
CO CO »-l
ON ^n
O lO 00
O CM CO CO CM
00 t^ U3 Tl- CM
O 00 lO CO
o
00
to
CM cn NO
CM ON
NO CM 00
"^ '-' ^ 5^A
ON O '-' CM CO
rf T^ to vo
t^
t^
00
C3N On O
CM CO CO
^ K? ^ )^^B
NO t^ t^ t^ t^
t^ t^ t^ t^
t^
t^
t^
t^ t^ 00
00 S
00 00 00
00 00 00 ^H|
d
O r^ CM CO rh
to NO tN, 00
C3N
s
CM CO rh
to NO
t^ 00 On
rkkksI
to to lO to lO
to to lO to
to
NO
NO NO NO
NO NO
VO NO NO
— 1
TABLES
313
*5VOt^00O\ O'-'OJcOrJ- tovor^COOx Q'-'C^fOrJ- lovOt^00O\
i^i>.t^t^t^ 00000000 00 0000000000 a\ONONO\o\ o\o\o\o\o\
t>. »-H Tf b>.^ »-«^ '^^ 00^ '"1, "1 °0 ^^ "1 00 ^^ "1 CA ^. ^ ^ ^^ ^ ^ *^ ^^ Q
ioO"^odco rCi-Tvoo-^" oCcorvTcvfvo" oiooCcood cvfvo'-^^too
fsJcof^coTf rfioiovovo vot^t^OOOO OnOnOnOO ^»-HC^cv>ro
fO«^<^<^fO COCOCOfOCO COCOCOCOCO COCOfOTj-Tj- rJ-Tj-Tj-Tj-Tj-
»-H Tt r>.^ o CO ^ o\^ *-! "^ ^\ o CO vo ch CN "^ 00 '-H^ Tt tN^^ o fo vq 0\^ c^^
cvf vo o" in oT CO tC rg" vo" o" 10 oT crT rC csT ^ o' m" oC crT 00 cm" vo o" 10
t^QTft^o '^t^'-H'-^oo '-iTj-oO'-Hii-> ^cgiooorq u-)0\cgvoON
mO-^OOco t>.'-tvOOrf 0\cotN.cgvo Oioa\c<oOO CNJVO'-'IOOn
CvjcocooOTj- TfiOiOVOvO vot^t^OOOO CJnonOvOQ »-iT-iCVirQC^
cococococo cococococo cococococo cOcocorj--^ rJ-rf'^Tf'^
"^ to vq^ tN^ CA p^ i-H <M CM co^ Tt 10 vq^ vq^ tv.^ t^_ 00^ o\ 0\ c^ cq o^ cd '-<^ 't
vd in rt CO cvT cm" t-T o" oC oo" ivT vo" 10 rf^ ro' cm" i-T o" oC 00" cjo" tC \o tn Tf
ChCMiOOO'-H Tj-t^OCMiO 00'-i"^t^O CO»oO\T-iT|- tN.OC0VP0\
^\OOTfO\ COt^CMM^O Tj-ONCOt^CM ^O^^Onco l^CMVOO"^
t^OTft^O Ttt^T-HTfOO ^TtOO'-'iO OOCMIOOOCM iOO\CMVOON
JOO'^OOco t^t-"VOOTf ONCOt^CMVO OinONCOOO CMVO'-'inON
CMcocorOTt- "rf \n m \0 \0 vOt^t^OOOO OnonOsOO »-<t-HCMCMCM
fOrocococo cocococoeo c*ococococo cocororf'^ rfrfTt^Tj-
^_^ Lo 10^ CM_^ Tj-_^ "^ ^^ '"L ^ "1 ^> "^^ °°^ °°„ ^^ <^ ^^ ^ ^ °0 "^ *^> ^. ^> ^
tC 10 o" eg o" in vo" ltT o' »-'" ctT Tt-" irT co" 00" 0\ tC cm" CO '-T vo" tvT m" O '-h"
t^Tft^iOOs OOCO"^,— ifo OTj-coOOOO TfVO'^tN.VO QOVOOOm
tx'<^OVO'-" vO'-hioonCM inr^ONO'-' CMCMCM'-'O O\t^'^»-H00
000000t>*t^ vOVOiOrtrh coCM'-'^-'O CTvOOtvVOiO r«^CM*-HQOO
ONCOt^'-'m C7\cot^T-Hio O\rot^'-"io OOCMVOOrt OOCMVOOCO
voO\CMvOC?\ CMVOOncovO OncovOOco V00cot>.0 cot^O^tN,
tOO\«^00(M t^i-imO"^ OOcor^CMVO OiO0\C000 CMVO»-HlOO\
CMCMcorOTt Tj-ioiOVOO VOt^t^OOOO 0\C7\0\00 '-'^-^CMCMCVl
COCOCOCOCO COCOCOCOCO COCOCOCOCO COCOCO'^TJ- TfrfTj-^^
vq Tf o^ t^^ Oi o\ CO vq^ co^ -^^^ 10^ oo_^ co_ cm_^ 00^ vq^ ^ *^^ *^ "^ *"! "^ *^^ *"!. "^
'-H o" 00" o" cm" vo" CO o" 10" cm" tj-" cm" «o" cm" 10 o" 00" 00' OS t^ vo 00" •^ Tt co"
lOCOOVLOOO OOiOt^CMO OOmOOVOm tJ-OOVOtI-ON l>*Tj-tX»-<CM
CMCOCMOOVO -^ONt^VOCM '-"CMOCOt^ 0\mc00sC7\ r-iT-Hio»-H^
Tt^coOTj- mcoOvoCM OOiOTj-ioch t^OOO^Ti ONTht>^OfM
VOt^OVOTf ioOnVOioOO coCMTfONt^ OniocovoCM »-<mCMTfO\
TtCMt^t^-"* t>.voCMTl-CM r^OOiOOOOO rft^VOr-ifO riioVOrOVO
OOCMiOOO^ COiOt^OOON <:^O^O^00^^ vOTfCMOt^ ^OVOCMtX
O^^^CV, CMCMCMCMCM CMCMCMCMCM CMCMCMCM'-' r-i^OOQ.
"^OOCMVOO -"ii-OOCMvoo -"^OOCMVOO rf-QOCMVOO rtOOCMVoSv
^-H^CMCMco rocOTfrtin miovovor^ I^t^OOOOON 0\0\000
COCOCOCOCO rOCOCOCOCO COCOCOCOCO COCOCOtOCO COCO"^^Tf
t>._ oo_^ 10 "T, ■^^ o\^ vq tN.^ '-H o\^ CO vo^ vq CM^ o^ co^ '-j, vq ov i^ 00 vo cm 10 10
T-T o" cm" o o" i-h" 00" CO vo" r-T Tj^ ro" 00" o" T}-" oT ^" Lo" co" oC ocJ" os vo" cm" tC
ONOOt^ONON Ovt^OOt^vO r-HTj-^LOrt coCM-^mOV (iOcOVOONOV
coCM'-hvocm 0\OOCOcoO ^s.CMVO'-lVO Tj-cocoioiO CMOCMVOm
COCM10CM3-H VOOOCMCMVO iOt-hCMCMVO OvCMt^OOfO lOCOThiOrf
VOC3VCMOO\ OO'-HiOOvOO (Mmiol^Q OONCMrfiO OcocoIn^Ov
CMiOI>^vOO C^OOOOCM On-^CMVOO mroOOONCX) VOCMt^O«-H
'-iCOO^r^ OvincoOOOS OOOOO^CMO CM'-<r^<Mt^ CO'-h.-hVOio
>S;3;2iSf>J 9S°2;3Ji:5J^: !r2^'-'00<^ ^t-rtoooocM r<it^»N.CMco
OOO^tOvo O-^OOOCM T^TJ•^or^co CMOt^'^'-H tN.CMt^CMVO
IPQ^^i^ C000co0\rf Ov'^CAThON rhCAcooOco tvCMVO'-fiO
f^5959Sj^ S?Q!Z''~'^ cMcocoTf-*^ miovovoi^ t^ooooo\ON
0000000000 0^0\o^o^o^ onO\c?no\o\ o\ a\ o\ os a\ 0\ o\ c\ 0\ 0\
lOVOr^OOON Ot-iCMcotT mvot^OOCTv Or-iCMro'^ lOVOt^COOv
t>^t^t^t>.t^ 0000000000 0000000000 o\ o\ o\ o\ o\ 0\ 0\ 0\ 0\ 0\
314
§359.
LOGARITHMS
Table of Interest Ratios
1 + i
Logarithm
1 + i
Logarithm
1.00125
000 542 529 092 294
1.01
004 321 373 782 643
1.0015
000 650 953 629 595
1.01025
004 428 859 114 686
1.00175
000 759 351 104 737
1.0105
004 536 317 851 323
1.002
000 867 721 531 227
1.01075
004 643 750 005 712
1.00225
000 976 064 922 559
1.011
004 751 155 591 001
1.0025
001 084 381 292 220
1.01125
004 858 534 620 329
1.00275
001 192 670 653 684
1.0115
004 965 887 106 823
1.003
001 300 933 020 418
1.01175
005 073 213 063 604
1.00325
001 409 168 405 876
1.012
005 180 512 503 780
1.0035
001 517 376 823 504
1.01225
005 287 785 440 451
1.00375
001 625 558 286 737
1.0125
005 395 031 886 706
1.004
001 733 712 809 001
1.01275
005 502 251 855 626
1.00425
001 841 840 403 709
1.013
005 609 445 360 280
1.0045
001 949 941 084 268
1.01325
005 716 612 413 731
1.00475
002 058 014 864 072
1.0135
005 823 753 029 028
1.005
002 166 061 756 508
1.01375
005 930 867 219 212
1.00525
002 274 081 774 949
1.014
006 037 954 997 317
1.0055
002 382 074 932 761
1.01425
006 145 016 376 364
1.00575
002 490 041 243 299
1.0145
006 252 051 369 365
1.006
002 597 980 719 909
1.01475
006 359 059 989 323
1.00625
002 705 893 375 925
1.015
006 466 042 249 232
1.0065
002 813 779 224 673
1.01525
006 572 998 162 075
1.00675
002 921 638 279 469
1.0155
006 679 927 740 826
1.007
003 029 470 553 618
1.01575
006 786 830 998 449
1.00725
003 137 276 060 415
1.016
006 893 707 947 900
1.0075
003 245 054 813 147
1.01625
007 000 558 602 125
1.00775
003 352 806 825 089
1.0165
007 107 382 974 057
1.008
003 460 532 109 506
1.01675
007 214 181 076 625
1.00825
003 568 230 679 656
1.017
007 320 952 922 745
1.0085
003 675 902 548 784
1.01725
007 427 698 525 323
1.00875
003 783 547 730 127
1.0175
007 534 417 897 258
1.009
003 891 166 236 911
1.01775
007 641 111 051 437
1.00925
003 998 758 082 352
1.018
007 747 778 000 740
1.0095
004 106 323 279 658
1.01825
007 854 418 758 035
1.00975
004 213 861 842 026
1.0185
007 961 033 336 183
TABLES
315
Table of Interest Ratios — {Continued)
1 + i
Logarithm
1 1 + *
Logarithm
1.01875
008 067 621 748 033
1.0275
Oil 781 830 548 107
1.019
008 174 184 006 426
1.02775
Oil 887 485 452 387
1.01925
008 280 720 124 194
1.028
Oil 993 114 659 257
1.0195
008 387 230 114 159
1.02825
012 098 718 181 213
1.01975
008 493 713 989 132
1.0285
012 204 296 030 743
1.02
008 600 171 761 918
1.02875
012 309 848 220 326
1.02025
008 706 603 445 309
1.029
012 415 374 762 433
1.0205
008 813 009 052 089
1.02925
012 520 875 669 524
1.02075
008 919 388 595 035
1.0295
012 626 350 954 050
1.021
009 025 742 086 910
1.02975
012 731 800 628 455
1.02125
009 132 069 540 472
1.03
012 837 224 705 172
1.0215
009 238 370 968 466
1.0305
013 047 996 115 232
1.02175
009 344 646 383 631
1.031
013 258 665 283 517
1.022
009 450 895 798 694
1.0315
013 469 232 309 170
1.02225
009 557 119 226 374
1.032
013 679 697 291 193
1.0225
009 663 316 679 379
1.0325
013 890 060 328 439
1.02275
009 769 488 170 411
1.033
014 100 321 519 621
1.023
009 875 633 712 160
1.0335
014 310 480 963 307
1.02325
009 981 753 317 307
1.034
014 520 538 757 924
1.0235
010 087 846 998 524
1.0345
014 730 495 001 753
1.02375
010 193 914 768 475
1.035
014 940 349 792 937
1.024
010 299 956 639 812
1.0355
015 150 103 229 471
1.02425
010 405 972 625 180
1.036
015 359 755 409 214
1.0245
010 511 962 737 214
1.0375
015 988 105 384 130
1.02475
010 617 926 988 539
1.038
016 197 353 512 439
1.025
010 723 865 391 773
1.039
016 615 547 557 177
1.02525
010 829 777 959 522
1.04
017 033 339 298 780
1.0255
010 935 664 704 385
1.041
017 450 729 510 536
1.02575
Oil 041 525 638 950
1.0425
018 076 063 645 795
1.026
Oil 147 360 775 797
1.043
018 284 308 426 531
1.02625
Oil 253 170 127 497
1.044
018 700 498 666 243
1.0265
Oil 358 953 706 611
1.045
019 116 290 447 073
1.02675
Oil 464 711 525 690
1.046
019 531 684 531 255
1.027
Oil 570 443 597 278
1.0475
020 154 031 638 333
1.02725
Oil 676 149 933 909
1.048
020 361 282 647 708
3i6 LOGARITHMS
Table of Interest Ratios — (Concluded)
1 + i
Logarithm
1 + i
Logarithm
1.049
1.05
1.055
1.06
1.065
020 775 488 193 558
021 189 299 069 938
023 252 459 633 711
025 305 865 264 770
027 349 607 774 757
1.07
1.075
1.08
1.09
1.10
029 383 777 685 210
031 408 464 251 624
033 423 755 486 950
037 426 497 940 624
041 392 685 158 225
§360.
Table of Sub-Reciprocals
Initial
Sub-
Initial
Sub-
Figures
Reciprocal
Figures
Reciprocal
10
90
35-36
27
11
83
37
26
12
76
38-39
25
13
71
40
24
14
66
41-42
23
IS
62
43-44
22
16
58
45-46
21
17
55
47-49
20
18
52
50-51
19
19
50
52-54
18
20
47
55-57
17
21
45
58-61
16
22
43
62-65
15
23
41
66-70
14
24
40
71-75
13
25
38
76-82
12
26
27
83-89
11
27
35
90
10
28
34
29
33
30
32
31
31
32
30
33
29
34
28
FORMING LOGARITHMS
317
§361.
Table of Multiples
1
2
3
4
5
6
7
8
9
001
002
003
004
005
006
007
008
009
002
004
006
008
010
012
014
016
018
003
006
009
012
015
018
021
024
027
004
008
012
016
020
024
028
032
036
005
010
015
020
025
030
035
040
045
006
012
018
024
030
036
042
048
054
007
014
021
028
035
042
049
056
063
008
016
024
032
040
048
056
064
072
009
018
027
036
045
054
063
072
081
010
020
030
040
050
060
070
080
090
Oil
022
033
044
055
066
077
088
099
012
024
036
048
060
072
084
096
108
013
026
039
052
065
078
091
104
117
014
028
042
056
070
084
098
112
126
015
030
045
060
075
090
105
120
135
016
032
048
064
080
096
112
128
144
017
034
051
068
085
102
119
136
153
018
036
054
072
090
108
126
144
162
019
038
057
076
095
114
133
152
171
020
040
060
080
100
120
140
160
180
021
042
063
084
105
126
147
168
189
022
044
066
088
110
132
154
176
198
023
046
069
092
115
138
161
184
207
024
048
072
096
120
144
168
192
216
025
050
075
100
125
150
175
200
225
026
052
078
104
130
156
182
208
234
027
054
081
108
135
162
189
216
243
028
056
084
112
140
168
196
224
252
029
058
087
116
145
174
203
232
261
030
060
090
120
150
180
210
240
270
031
062
093
124
155
186
217
248
279
032
064
096
128
160
192
224
256
288
033
066
099
132
165
198
231
264
297
034
068
102
136
170
204
238
272
306
3i8
LOGARITHMS
Table of Multiples — (Continued)
1
2
3
4
5
6
7
8
9
035
070
105
140
175
210
245
280
315
036
072
108
144
180
216
252
288
324
037
074
111
148
185
222
259
296
333
038
076
114
152
190
228
266
304
342
039
078
117
156
195
234
273
312
351
040
080
120
160
200
240
280
320
360
041
082
123
164
205
246
287
328
369
042
084
126
168
210
252
294
336
378
043
086
129
172
215
258
301
344
387
044
088
132
176
220
264
308
352
396
045
090
135
180
225
270
315
360
405
046
092
138
184
230
276
322
368
414
047
094
141
188
235
282
329
376
423
048
096
144
192
240
288
336
384
432
049
098
147
196
245
294
343
392
441
050
100
150
200
250
300
350
400
450
051
102
153
204
255
306
357
408
459
052
104
156
208
260
312
364
416
468
053
106
159
212
265
318
371
424
477
054
108
162
216
270
324
378
432
486
055
110
165
220
275
330
385
440
495
056
112
168
224
280
336
392
448
504
057
114
171
228
285
342
399
456
513
058
116
174
232
290
348
406
464
522
059
118
177
236
295
354
413
472
531
060
120
180
240
300
360
420
480
540
061
122
183
244
305
266
427
488
549
062
124
186
248
310
372
434
496
558
063
126
189
252
315
378
441
504
567
064
128
192
256
320
384
448
512
576
065
130
195
260
325
390
455
520
585
066
132
198
264
330
396
462
528
594
067
134
201
268
335
402
469
536
603
068
136
204
272
340
408
476
544
612
069
138
207
276
345
414
483
552
621
FORMING LOGARITHMS
319
Table of Multiples — (Concluded)
1
2
3
4
5
6
7
8
9
070
071
072
073
074
140
142
144
146
148
210
213
216
219
222
280
284
288
292
296
350
355
360
365
370
420
426
432
438
444
490
497
504
511
518
560
568
576
584
592
630
639
648
657
666
075
076
077
078
079
150
152
154
156
158
225
228
231
234
237
300
304
308
312
316
375
380
385
390
395
450
456
462
468
474
525
532
539
546
553
600
608
616
624
632
675
684
693
702
711
080
081
082
083
084
160
162
164
166
168
240
243
246
249
252
320
324
328
332
336
400
405
410
415
420
480
486
492
498
504
560
567
574
581
588
640
648
656
664
672
720
729
738
747
756
085
086
087
088
089
170
172
174
176
178
255
258
261
264
267
340
344
348
352
356
425
430
435
440
445
510
516
522
528
534
595
602
609
616
623
680
688
696
704
712
765
774
783
792
801
090
091
092
093
094
180
182
184
186
188
270
273
276
279
282
360
364
368
372
376
450
455
460
465
470
540
546
552
558
564
630
637
644
651
658
720
728
736
744
752
810
819
828
837
846
095
096
097
098
099
190
192
194
196
198
285
288
291
294
297
380
384
388
392
396
475
480
485
490
495
570
576
582
588
594
665
672
679
686
693
760
768
776
784
792
855
864
873
882
891
Part IV— Tables
CHAPTER XXXI
EXPLANATION OF TABLES USED
§ 362. Object of the Tables
Any value shown in the following tables* might have
been ascertained by the rules given in the text; but it is
convenient and time saving to have at hand, already worked
out, those results which are most frequently needed.
§ 363. Degree of Accuracy
The tables shown give each value to eight decimal places,
while the ordinary tables extend only to five or six decimals.
This allows accurate computations to be made on sums up
to one million dollars, to the nearest cent — a degree of
accuracy which will meet any ordinary requirements.
§ 364. Rates and Periods
The rates used in the tables* are as follows : 1%, 1%%,
11/2%, 1%%, 2%, 21/4%, 21/2%, 2%%, 3%, 31/2%, 4%,
4%%, 5%, and 6%. These are the rates most commonly
used, since most investments are on a semi-annual basis.
Rules for intermediate rates will be found in §§ 375, 376.
The periods given are from 1 to 50, inclusive, and also
• See Chapter XX2CII,
32Q
EXPLANATION OF TABLES USED 321
every 5th period thereafter, viz.: 55, 60, 65, YO, 75, 80, 85,
90, 95, 100. Rules for obtaining the values for periods in-
tervening above 50, and for extending above 100 periods,
will be found in § 374.
In all the following tables of compound interest, the
principal is considered to be $1; for any other principal
the tabular result must be multiplied by the number of dol-
lars in the principal.
§ 365. Tables Shown
Tables for obtaining the following results are shown in
Chapter XXXII.
Table I — Amount
II— Present Worth
III — Amount of Annuity
IV — Present Worth of Annuity
V — Sinking Fund
VI — Reciprocals and Square Roots
§ 366. Annuities — When Payable
"Annuity" in these tables signifies the ordinary annuity
where the payment is made at the end of each period. This
kind of annuity is the one most used in investment calcula-
tions. Annuities paid in advance, like the premiums in life
insurance, are sometimes called annuities due (§75). Their
amounts and present worths may be derived from the tables
of ordinary annuities.
§ 367. Table I — Amount
This gives the amount to which $1, invested now at the
rate i, will have accumulated at the end of n periods. The
rates (i) are at the top of the table and the numbers of
periods (w) are on the left-hand margin. Each term up to
50 periods is l-\- i times the term above it ; or is the term
322
TABLES
below it divided by 1 + i. Each term may also be considered
as that power oil + i, the ratio of increase, whose exponent
is on the left. Thus in the column 3%, where i = .03, the
value for 9 periods is the 9th power of 1.03, or 1.03® =
1.30477318. If this be multiplied by 1.03 it gives the 10th
term, 1.34391638; if divided by 1.03 it gives the 8th term,
1.26677008.
§ 368. Compound Interest
To find the compound interest, subtract 1 from the
amount. Thus the compound interest at 3% for 9 periods is
.30477318; for 25 periods, 1.09377793. It is unnecessary,
therefore, to give a separate table of the compound interest.
All the other tables might be derived from Table I. The
second line in each column is the ratio of increase.
§ 369. Table II— Present Worth
This gives the present worth of $1 payable n periods
from now at the rate i ; or the principal which invested now
will at the end of n periods have accumulated to $1 ; or $1
discounted for n periods at i. It proceeds in exactly the
contrary manner to Table I, diminishing instead of increas-
ing, each term being divided by l-\- i to produce the succeed-
ing one, or multiplied by 1 + i to produce the preceding.
Each term may be obtained independently from the cor-
responding term in Table I, the two terms being reciprocals
of each other. If we represent any term in Table I by (I)
and the corresponding term in Table II by (II), then we
may say: (II) =1-^- (I) and (I) =1^(11); or more
briefly with negative exponents, (II) = (I)~^ and (I) ==
(II)-'.
The second line in each column is the discount ratio or
reciprocal of 1 + f ; and each term below is the nth power of
EXPLANATION OF TABLES USED 323
that number. Thus in the 3% column the discount ratio is
1.03-^ or .97087379; the present worth for 9 periods is
.76641673, or the 9th power of .97087379, which may be
expressed 1.03"^ or 1 -^ 1.03^ Multiplied by 1.03, it gives
1.03-^ or .78940923; divided by 1.03 it gives 1.03-"'' or
.74409391. Each of these multiplied by its correlative in
Table I will give unity: (II) X (I) = 1.
All these relations should be verified by experiment until
thoroughly understood.
The compound discount is obtained by subtracting the
present worth from 1. The compound discount for 9 periods
at 3% is 1 — .76641673 = .23358327. Since this operation
is easy, it is unnecessary to give a separate table of compound
discounts.
§ 370. Table III — Amount of Annuity
This table gives the amount to which an ordinary annuity
will accumulate ; that is, if $1 be invested at the end of each
period, the total investment will, after n periods, reach this
amount.
It is formed from Table I by adding together the same
number of terms as of the periods required. The three top
lines of Table I give the third line of Table III ; the fourth
line of Table III is the sum of the first four lines of Table I ;
but for this purpose the line marked in Table I must be
counted in. Thus in the 2% column, 1 + 1.02 + 1.0404 in
Table I gives the value for 3 periods in Table III, 3.0604;
for 4 periods it is 1 + 1.02 + 1.0404 + 1.061208 =
4.121608; etc.
According to the principles laid down in § 60, we might
have proceeded in this way: Take the amount of $1 for 3
periods from Table I (not the third but the fourth line),
1.061208 ; drop the 1, giving .061208 ; divide by .02,^produc-
ing 3.0604.
324 TABLES
But where the figures have been rounded, this procedure
would leave two places indeterminate. For example, the
amount of $1 for 20 periods is 1.48594740; the compound
interest is .48594740; this multipHed by 100 and divided by
2, gives 24.297370. We have, therefore, cut down our
result from 8 decimals to 6. But by addition of the first 20
lines of Table I, we get in Table III, 24.29736980; a gain
in accuracy of two places.
It will be better, therefore, to reverse the process and test
the accuracy of the table by dividing the result in Table III
by 100 and multiplying by 2.
24.29736980 X 2 -^ 100 = .4859473960
.4859473960 + 1 = 1.4859473960 = 1.4859474| |
This not only tests the accuracy of the table, but adds two
places.
This suggests that for very accurate and extensive com-
putations we may extend Table I to 10 figures, the last of
which will be nearly accurate.
For most questions of investment 6 decimals of Tables
III and IV will be ample.
When the amount of an annuity due is required, it is
obtained as follows : subtract one from the number of periods
and subtract one from the number of dollars. Thus we have
at 2% :
Amount of ordinary annuity, 4 periods, 4.121608
Amount of annuity due, 3 periods, 3.121608
§ 371. Table IV — Present Worth of Annuity
This table gives the present worth of an annuity and is
derived from Table II, precisely as III is derived from I.
In the same way as before. Table II may be extended two
places ; but after multiplying by the rate, the result must be
subtracted from 1.
EXPLANATION OF TABLES USED 325
The present worth of an annuity of 20 periods at 2%
is, by Table IV, 16.35143334, which X .02 = .3270286668;
1—. 3270286668=. 6729713332. Table II gives .67297133,
which is correct as far as it goes.
Table IV is the one used in bond valuations for ascer-
taining premiums and discounts which, as we have seen,
are merely present worths of annuities consisting of the
difference between the cash and income rates. Any ordinary
premium or discount where the principal does not exceed
one million dollars may safely be computed by using 6 or
7 figures of the decimals.
To transform Table IV of ordinary annuities into one
of annuities due, add one to the number of periods and add
one dollar to the value.
§ 372. Table V— Sinking Fund
This table gives sinking funds. It answers the question :
What sum shall be invested at the end of each of n periods so
that the sum-total with all accumulations shall amount to $1
at the end of n periods ?
Each term of Table V is the reciprocal of the corre-
sponding term in Table III. (V) = (III)-\ Thus, to find
the sinking fund necessary to provide a total of $1 in 9
periods, we divide 1 by the total to which an annuity of $1
would accumulate in 9 periods. At 3%, the latter would
be:
10.15910613 ; 1 -^ 10.15910613 = .0984338570
which is the sinking fund required, carried two places
further than in Table V.
Another method of deriving the sinking fund would be
to divide the single interest (.03) by the compound interest
(.30477318) of $1 from Table I, which will be found to
give the same result : i-^l.
326 TABLES
§ 373' Rent of Annuity
To find what annuity has a present worth of $1, we
have only to add to the rate of interest the sum taken from
Table V. This gives the rent of an annuity which $1 will
purchase, and it is, therefore, unnecessary to provide a
table for that purpose. It might also be obtained to 8
places from Table II, dividing the single interest by th?
compound discount. It could also be derived by finding
the reciprocal of the corresponding term in Table IV.
§ 374. Extension of Time
The tables go as far as 100 periods only, but Tables I
and II may be extended to as many periods as desired by
multiplication. The values for 148 periods might be ob-
tained by multiplying together those for 100 and 48 periods
respectively. Thus, at 1%, Table I, we have:
100
2.70481383
48
1.61222608
Using contracted multiplication,
2.70481383
1.62288830
2704814
540963
64096
5410
1623
22
4.36077141
The last figure is not quite accurate, but we could have
made it more so by getting 10 figure values for 100 and for
48 periods from Table III.
EXPLANATION OF TABLES USED
327
100(170.48138294X.01=1.7048138294)+1=2.7048138294
48 ( 61.22260777X01= .6122260777)4-1=1.6122260777
2.7048138294
1.6228882976
270481383
64096277
5409628
540963
162289
1893
189
19
Correct result to 10 figures, 4.3607713911
To extend Table III, IV, or V as to time, it is easiest
to extend Table I or II and thence derive the value required.
§ 375* Subdivision of Rates
Although the rates given in these tables are those most
frequently required, yet it often happens that intermediate
rates occur, especially in bond computations. It might be
supposed that these inter-rates could be obtained by "split-
ting the difference" into as many parts as necessary. But
a trial will show that this gives only a rough approximation.
In Table I, for 10 periods at the rate 3%, the amount
is 1.34391638
and at 2%% it is 1.28008454
Midway between them is 1.31200046
but this is not the true value for 23^% ; it is. . 1.31165103
hence the error must be .00034943
and the approximation holds good for only 3 decimals. But
the correction can be very closely computed.
328 TABLES
§ 376. Interpolation
Sometimes, in compound mterest processes and also in
mathematical problems, we have a series of terms, all
formed by the same law, and based upon another series. A
familiar illustration in mathematics is the formation of
squares, for example :
Numbers, 12 3 4 5 6 etc.
Squares, 1 4 9 16 25 36 etc.
1st Differences, 3 5 7 9 11 etc.
2nd Differences, 2 2 2 2 etc.
When a series of terms such as that described above is
written down in a table opposite to certain equi-distant num-
bers called arguments, intermediate terms corresponding to
certain given arguments may be inserted by a process called
interpolation, consisting of three steps :
(1) Differencing.
(2) Multiplication of each difference by a fraction de-
pendent on the fractional distance at which the
inter-term is to be located.
(3) Application of these corrections to the preceding
term.
Differencing has already been treated to some extent in
§§250 and 276. To interpolate in Table I, 10 periods,
a value for 2%%, we first set down the two values next
greater and next less, opposite their arguments (3% and
21/2%).
3% 1.34391638 1, . ^
^ hdecreasmsf terms
21/2% 1.28008454 J ^ ^ ^ ^ "^
or
21/2% 1.28008454 1
3% 1.34391638 J
mcreasmg terms
EXPLANATION OF TABLES USED 329
The decreasing series has some advantages which make
it preferable.
Continuing the cokimn, use only equi-distant arguments,
for 4 or more lines.
3%
1.34391638
21/2%
1.28008454
2%
1.21899442
11/2%
1.16064083
1%
1.10462213
y2%
1.05114013
0%
0.00000000;
and
proceed to difference.
Dx D3
D.
z%
1.34391638
.06383184 .00274172
.00010519 etc.
21/2%
1.28008454
.06109012 .00263653
etc.
2%
1.21899442
.05845359 etc.
11/2%
1.16054083
etc.
etc.
Let this process be carried out to the 6th difference and
we have the following values, which are all that we need
to consider:
D: .06383184
D, .00274172
Ds .00010519
D; .00000355
D, .00000010
D. .00000001
From these differences any value corresponding to rates
between 3% and 2%% may be determined. Each D will
be multiplied by a certain fraction (F) according to the
330
TABLES
fractional distance from 3% where the interpoland is to be
located.
For the distance .5 (which means halfway), the F*s are
always as follows :
R .5
F. .125
Fs .0625
R .0390625
F5 .02734375
Fe .0068359375
Multiplying each D by its corresponding F :
Di X Fx = .06383184 X .5 = .03191592
D3 X F. = .00274172 X .125 = .00034271
Ds X Fa = . 00010519 X. 0625 =.00000658
D4 X F4 = .00000355 X .0390625 = .00000014
D5 X F5 = .00000010 X .02734375,
which is too small to affect the final
figure.
De X Fe, and following products are also
negligible.
Total correction, .03226535
Subtract from value at 3%, 1.34391638
Interpolated value at 23^%, 1.31165103
By using the above series of F's (.5, .125, .0625,
.0390625, etc.), any interval may be bisected. But the inter-
val may also be split into 5 parts as well as into 2. ^ = .2 ;
therefore .2 would be Fi for the first 5th, .4 would be Fi for
the second 5th; and .6 and .8 would be Fi for the third and
fourth intervals, respectively.
We will now give the proper F's for interpolating nine
values, each at one-tenth interval.
EXPLANATION OF TABLES USED
331
R
F,
F.
F.
F.
.1
.045
.0285
.0206625
.01611675
.2
.08
,048
.03360
.025536
.3
.105
.0595
.0401625
.02972025
A
.12
.064
.04160
.029952
.5
.125
.0625
.0390625
.02734375
.6
.12
.056
.03360
.022848
.7
.105
.0455
.0261625
.01726725
.8
.08
.032
.01760
.011264
.9
.045
.0165
.0086625
.00537075
To find the value corresponding to 2.60% in the same
table: Since the interval is .50, }i of the interval is .10, and
the intermediate arguments would be 2.90% at .2 distance
from .3 ; 2.80% at .4 ; 2.70% at .6 ; and 2.60% at .8. There-
fore, we must use the F's of .8 as above, multiplying by
them the same differences previously obtained.
.06383184 X. 8
.00274172 X .08
.00010519 X .032
.00000355 X .0176
The remaining terms are negligible.
Total,
Subtract from value at 3%,
.05106547
.00021934
.00000337
.00000006
.05128824
1.34391638
Interpolated value at 2.60%', 1.29262814
Had we chosen the increasing series in our differencing,
there would have been this variation in the application of
the corrections, that the first, third, fifth, seventh, and all
odd-numbered corrections would have to be added to the
preceding term and the even-numbered ones subtracted.
We should have differenced thus :
332
TABLES
D.
D.
D.
21/2
1.28008454
.06383184
.00285054
.00011260
3
1.34391638
.06668238
.00296314
etc.
3%
1.41059876
.06964552
etc.
4
1.48024428
etc.
etc.
The D's and their products would have figured thus, in
the first example :
2%% (now the basis) 1.28008464
.06383184 X. 5 + .03191592
.00285054 X .125
.00011260 X. 0625
1.31200046
— 35632
1.31164414
+ Y04
1.31165118
— 15
.00000388 X .0390625 —
234%, as before, 1.31165103
The F's already given are generally sufficient for any
practical purpose, but even if a very unusual fractional rate
requires computation, the F's may always be worked out
by the following formula :
Fi is always the distance from the first value, expressed
decimally.
Subtract Fi from 1, multiply Fi by the remainder and
divide the product by 2 ; this gives F2.
Subtract Fi from 2, multiply F2 by the remainder and
divide the product by 3, giving F3. And so on.
EXPLANATION OF TABLES USED 333
Observe that it is always the original Fi which is sub-
tracted from 1, 2, 3, etc., and that the divisor is always the
number of the F sought.
This will be plainer in symbols.
F. = F.X(1 — FO
F, = F.X(2 — FO
F. = F3X(3 — FO
F. = F.X(4 — R)
etc.
Fn = Fa_x X (n — 1 — FO - n
The F's already given should be worked out for practice
by these formulas.
As an example, we give the F's of .24.
Fx = .24
Fa = .24 X 0.76-^ 2 = .0912
Fa = .0912 X 1.76 -f- 3 = .053504
F. = .053504 X 2.76 -^ 4 = .03691776
F5 = .03691776 X 3.76 ^ 5 = .02776215552
Where the rates given in the tables are more than 1/2%
apart, interpolation is not practically useful.
§ 377. Table VI — Reciprocals and Square Roots
This table gives the reciprocals and the square roots of
120 of the most necessary ratios of increase.
The ratios begin at % of 1%, and increase by 40ths of
1% to 3% ; by 4ths of 1% to 7% ; and by 1% to 10%.
The second column, composed of reciprocals, gives the
present worth of $1 payable one period from now, like the
second line of Table II. It is used for the purpose of dis-
counting by multiplication rather than by division, the
former operation being much easier. Any reciprocal may
334 TABLES
j
be tested by multiplying it by the ratio standing opposite,
which will give as the result, unity.
The third column, composed of square roots, gives the
equivalent effective ratio for a half-period. Thus for an
obligation at 6% semi-annually the ratio of increase is 1.03.
If a quarter of a year (a half-period) has elapsed, the
amount, if scientifically treated, is not 1.015 as used in
actual business, but 1.01488916. If the loaner were to re-
ceive 1.015 as the amount after three months and reinvest
at the same rate, he would have, at the end of the half-
yearly period, not 1.03 to which he is entitled, but 1.030225
(=1.015^). But if he receives 1.01488916 and reinvests
for the other quarter at the same rate, he will have at the
end of the half-year 1.01488916' = 1.03.
In other words, if .03 is the rate for each period, the
equivalent effective rate for a half-period is .01488916. To
receive or pay 3% each half-year is exactly the same in
effect as receiving or paying 1.488916% each quarter.
Intermediate values in the second and third columns may
be readily found by interpolation, usually requiring only
one F.
CHAPTER XXXII
TABLES OF COMPOUND INTEREST, PRESENT
WORTH, ANNUITIES, SINKING FUNDS,
AND OTHER COMPUTATIONS
335
336
TABLES
^
^
COON T-HCMiOOr^ "^voo^rj-r^
VOOO CM-^t^ONON vovoooooo
u-)QC^ 00 fO u-i CN) CVJ rfvOOOT-HfO
C^O\'-i 00\000^ OOCOOO'-Ht^
VOOOOO Tj-MDvOTfOO OOOOiOON
lOOVOcO t-hOncOCOOO ocm^oocm
C^iOt^O COiOOO-— i'^ 00 '-I Tt- t^ --H
OOOr-i T-H ,— I ,-H CO <M CMfOcOcO''^
OOiOOOOOO
00 Tj-^ OS 00
Tj-00CMlOO\
■^loooom
-^OOTj-voOv
•-H 00 r^ '-I CO
00 ON '-'■^00
fOt>»C0 VOO
^ vOt^t^^OO
OOCMCOO
■^roo>otv
Tfo\oo\c
ONOOOOTfTj
iOO'^ONTj
OOOnOnOnC
rOr-HfO ^TfO'-lOO TflOONr-lr^
^ VOOCO t^tOONT-HTj- COOO\VOCO
>!. O«r)00 t^cgroro^ O'— '-^cooo
NT toiOOO ^OOiOOOt-^ CMCOQTj-Tl-
CMTtVOQN '-htJ-^OnCM '^rft^OcovO
OOOO T-H r-( r-H ,-H CM C\J CM CO CO c^
O^-^MDt^ OvOCMt>^-0 CMCMOOOni/
00 -^ ON 1— I CO CMVO'— 'COU-) covooOOnC
VO'-HCMt^O 0\OCMt-HV0 vOOMOrfts
OrO-^OOr^ OCMCMCOVO 'rtOO^-rtO
CM^f^iOr-i io\0"^CMr^ '-I CO IT) lo T)
V0I>*0nCMV0 QtO<— lOOm rfcocOTj-vr
OnCMioOnCM ^OncovOO "^OOCMvOC
co-<^-*tJ-io lOiOVO^t^ t^t^ooooo
VO OCMt^OOr^ CM'-"a\C0\O ^J- --h cm lO in. O^^tN-VOiO O\r-i00»-<0
t-H OO'^VOcotO "^cot^vot^ cot^'^CM'-" '^covOCMCM On i-" tJ- CM VC
OOCM 0(M>r>ON(N TfTt-^vOOO OOO^vOt-i t^vOONONt^-. ioOOvOtJ-'
^ VOOO'^'^^ ^ — ~ '- ^
CM
■^ VO r^ On CO
oco oovoooiooN ONt^Ttor^ NOi
SCNjTd- O'-HVONOO ONCorovOTt 001
T-iCM '<*• VO 00 T-i uo OOCOOOCOCJN >J^ ' _ _ _ _
CM'^VOOO OCM'^t^C^N "-HTt-voON^ ^t^OCMiO 00 i-h rM^ O rftN.O'^t^
OOOO ,-1 T-H ,-1 T-i ^ CMCMCMCMco co CO -^ ■<*■ tJ- '^mmxno NOvOt^t^fN
vOrfOOOa
OcovO'-Hir
fin
I
o
H
o
iO'thco VO lO u-> 00 '-• ONt^i-HOCM voirj^^o O t-i ^ VO O^ -^OvOOCV
^
^
^
^5^
lOCO'-Hf^CO Tj-f^cotoOO OOcO'-i^
VOCMC^I'-Ht^ TfOvONONVO r^OvOO
OCMiO '-lOCMOOOO r^LOCO00'-^ CM CM CM co tJ-
lOCO'^OO VOlN»^00ON Tl-CM-^^ONON CMOnO>JT^
t^Loco^ OOnOvOOOO On O '-h cm -"^ t^ ON co vo O -^ On rf o vo CMOnKi/)
^ -"^ VO OOr- (coior^ On T-H Tt VO On >— i co vo On t-h -"^voOnCV)
. CMOOt^Tj-r^ iovocoonvc
to OOvooOt-iCM QCMt^CMt^
t^coCMVOrJ- OOOOtOr-Hii-
t^iot>*coTj- On 0\ Tt Tf «
r-HCOLOt^ONOCM _ __
OOOO o '-"'-' --H '-" '-iCOCMCMCM CM co co co co "^ "^ -^ Tf- lo mmiovOvC
00 lO
CO>r)
IT) 00 CO
CMt^vo
•IT) rlCOOlOi-H irjCOOOOO'
It^ O00t^»-H00 CO irj ,— I r-H U-
- tOt^cor^CM ioc?nqcmc
CM a
cOtJ-CMON OOOOOCMvo CMioOOO
_ ,^tJ-OnOO Ttrti-^miO cOoOCM-rt-iO lOiOVOr^O Tt-O'-.-.
CM VO CO CM Tt- OO '^j- CO lOONVOmtN. CM On O co On OOOiOCOvo CTNt^OOCOO
lOOLOi-t r^coONVOco Qt^tOCOr-H oo< ~ — " ■" ~
r-(CO^vo f^ONOCM^ vSwONr-HCO ipv^ -_.. . -.
OOOO O O '-<'—' '-< 1— 1 1-H t-H CM CM CMCMCMcoco CO co CO tJ- t^- rf tJ- t}- in ir
iOOtJ- lOOOt^OOO COCMCMtOiO OOiOt^ONi-t coOOOQnij^ rJ-mCMO'-
CM^>.co r-H ^ Tt r-i r-i oOCMiOONt^ t-Hvoi-HCOVO CM t^ "^ o o oncooncocv
VOOiO CMCOOVOCM O-^-^co-^ ONON00r>.ON r^CMOQf^'-H CMiOOCMCv
lOt^Tf OOOOtOOOON t^CM^OVOtO CMOOCOr^O COVQOO'-Hm ON'^'-'ONO
lOt-HONQN OCOOO-'^CM CM-^t^CMON OOOO'-'iOCM ~ " '" --
CMiOt^O ■"^t^O'^OO CMVOOloOn -^CTnloovo
T-iCMcoiO vot-^ONOr-^ co-^vot^OO O^coiovo 00 On r-* co rj- vd 00 On '-' cr
OOOO 000'-''-< T-H ,-1 ^ ^ T-^ CMCMCMCMCM CM CM co co co CO CO co "^ rt
OOCMt^co i-hCMioOnvC
CM oo Tf o r^
COtOCf
rH IT) VO lO T-H ^N, COiOCOOO'-i VOt^COOOiO
O 0»-"Cor^CM »-HCOOCMC^l OnvO'^'^ON
•Tl-voCMm O'
'-i'^ OO^ovOiO CvJ 00 to CO '^ OOOO'^t^OO _
OO 1— iCMcotooO CMVOCMONf^ vo t^ O -^ O On On '-h VO CO coioOOnC
1— I CO VO Oto,— lOOVO vovOOOOtI" OniocO'— i»-h i— i co 1>» '— i t^ rj- CM CM CM iT
OOO '-Hi-HCMCMco -^tnvooOON OCM"^VOOO O CM '^ t^ ON CM to 00 --' -^
'-HCMCO'^ tovOt^^OOCTN O'-'CMCO'?*- vo t^ 00 On O CM co ''t to VO 00 ON O CM cc
OOOO OOOOO T-H r-H T-H T-( r-( ^ r-H r-^ ^ CM CMCMCMCMCM CMCMCOCOc:
00
CO
000
Ot-^CMco-* lOVOr^OOON OrHCMfO-* iri\ot-^COO\ O-^CMcorf »J^'Ot>oqON
^^T-trHf-t ^T-trHr-H,-l CMCMCMCMCM CM CvJ CM CM CM
f
COMPOUND INTEREST; OTHER COMPUTATIONS
337
vot^ONOO^
mor^ 00 fo
t^ O f*^ OO^o
O'—CvJCVJCO
cvj c4 csi csi cvi
a\<Mooqoo
lo m cx) Cvj Tj-
O rOr^COt>N
c4 cvj cvi cvi cvi
Tt roOCX)00
OCI-CNOO
fOOiOOOO
vOCAOnCNO
O '-' OMJ^OO
LOCVl O '— ' <^
CO"^C\J OnVO
vor^oooo On
<\i (\i c4 cvi og
OOVO fOVOO
rq CX3 '-< lo CO
fOOt^ ON vO
OiOONOOf^
ON 00 vO"^ CM
t>x CO r-. T-H ro
CO '— ' ONt^iO
p t-H T-H CM CO
CO CO fO CO fO
CMPOiOVOVO
t^Ot^CMOO
00 CO 0\ 00 CM
or^ooioo
i-.t^t^ON'-i
r^oooNt-^CM
CO 00 On t^ CO
TfoqcoONvq
COfO'^TfiO
coCM-^coio m
TtOOCM COOO CO
t^t^ rfvOco VO
O ^ vOiO O '— I
cmioonoovo r>.
CM Q\NO00^ CO
X^OIOCMtJ- .-1
COCM^CM-^ 00
vdKodoNC) '-<
Tj- <7\ CO Tt- lO vOCMOnOnO I^ogoOO'^ 0\0'-"-'0 OOOnOOO lO O CM CM ^ O
Tj-r^OCOTf lOrt-CMCJNON ONt^CgTtTj- coonconOio rl-rl-r^OOTt OcOi-hOOCM CO
cOTfcoOON covOCMiOCM COOmO"^ vO lO co ON ^ NO r^ rt vO ,-i TfiO^rN.0N no
CMOONOTf ON'-htI-OnO OOOOOOnNO m On '— ' co O -rf CM co CM t1- t^ Tt" On lO O "^
cocM-^ONOo t^oooN'-'No «-ionOcmoo x^onvono^ oO'-'CO'-h r^'-Ho^c^^>. cd
OncoOOcoO OOl^t^ON^ iOC^nocot-i »-HCMtOONiO CMOOt^r^ vnOt^r^ON ■^
tKTncooOco I^CMr^CMOO coCOrfQNO CMOO-^Ot^ TfOO"^"^ OCOCNJCDJ^ »0
OnOnOO'-' t-iCMCMCDco tJ- rf lo VO VO r>» t^ 00 ON CTn O Tf 00 CM t^ CO On NO rt; CM CM
.-J»-^CMCMCM CMCMCMCNJCM CM'cmCMCMCM CMCMCMCMCO cococOTfTt \n\n\6t^o6 On
OOCMOnOco
lOOOuT^O
t-iCOO'-ivo
NOCO^l-cot^
CO U-) U-) CM NO
i-Hl-^ rj- CM O
i-H Tt CO C^^ NO
OOOOOOOvON
lOTj-ONCTst^
lOCOOt^t^
ONt^iO COtJ-
OOOOOOOn-'^
OOOOVOCMt^
OnOnQCMtJ-
On CO 55 CM NO
OnO O '— ' »-<
•-h' CM CM CM CM
OCMCM '-I O
00 CVI t^ CO O
O lo On Tj- On
CM CM CM CO CO
^4 CM CM CM CM
1— ( On '— I On On
CMCMiOfOl^
Tfr-H COO^
00 VOCOOOO
t^ NO NO r^ 00
CO 00 CO 00 CO
Tt r^ LO li") NO
CM CM Cvi CM CM
coi^ONf-iCM
ONorN».-HCM
OOOOCOOO
00 coco CM lO
LOI^ Ol-OtO
^^,-hCMON
On r^ 00 CM c^
NO On CM NO ON
OOrtOO'-'NO NO
u-)ioCM C0 1— 1 T:^
i-it^OOTfNO Tt
tj-oococtmo cm
CMCMCOCOCO ■^Tj-miONO t^
CO CO On CO CM
"-H T-H Tj- CM lO
OCTncoCM ■'^
OTt-HOCM
OOCM (Mt^t^
CM CM CM CM CO
OOT^^t^O
Not^r^t^oo
Ot^ON^Tf
r^ CM 00 "^ 00
Cht^NOOO T^
COOOOcot^
CMrfOcOi-i
LO t^ Q CO l^
CONOOCONO
OOOOCAONCTs
Tj-O-'^OOv
cocoCM On T-H
r^ioNOOO
ON CM NO coco
lONOCMLOTt
'-" NO CM 00 lO
O CO t^ O "^
CM CM CM CM CM
CMOOOnCMO
CM CM 00 t^t^
VOt^t^OOr-H
t^l^Tt OntT
ON^-i o^ooo
CM'-hOOnON
00 CM NO On CO
'-H CM CM CM CO
coiooo-^t^
On 00 CM t^ CM
OOt^NOiOOO
OOC^Ii-tCMOO
r^iooO'si-CM
ONO'-^OOOO
CO On CO 00 NO
COIOOOOCO
OOCMOO'^ Tt
ONONTj-OOCM C?N
O '-Ht^OCO ii->
'-H ONCOOOO u->
inco-rr coCM T-i
cONOCMOt^ 00
t^QNONOON NO
NOOCOt^'-H NO
CMCMCMCMCM CM CM CM CO co CO "* tJ- rj- lO lo
CMCMCMOOt^ CMTj-COOOiO »-« 00 CM CM CM
CM"^co.-ico covoNOCMCM "^ vo t-h 00 O
ONOtJ-OnNO ^C^nOOOO COCOr^ONCO
OOCMCMt^ON OOcot^ON'— I T-HCMrM^co
0"^fOTl-0\ OO^t^r^CM CDCMOOOOco
coNOOtI-CO coO\"^Qr^ ^r-iCONOiO
NOOO'— 'COiO OOOCOVOOO ^TtNOCTvCM
lOiOVONONO vot^t^r^t^ COOOOOOOOn
^^OOnno
OCM'-hCM"^
CONOCAOOO
r-HCMt^r^CO
CM to CM Tj- r-H
•^COCOCOTt
to 00 '-< rj- t>x
C?nOnOOO
>-; T-I CM CM CM
CMcoOni-hno
Tj- Tt- i-H Tl- lO
CMCMCMCMCM
'-iOnOO'-h'^ lO
tV.t^COlOr-1 VO
t-HCMOOOOCM VO
On NO t^-^ On
ionoOnOno
-^O-^OOrf
■ t-H ,-H CO
to (_
O CM to 00 i-H
CO CO CO CO -^
R
NOcO'-HTfON t^CMCMOOtN* VO »-* t>» to i-H
cotoiO^CM OOOO^t^OO Ttt^l^OOCM
COOOOCMNO tococONOr^ C?N-^NO00rf
T-HiococOVO COtJ-OnOOCM ^vONOCMto
NOr^'— it^to voOn-^CMco no '— ' CTn
.— lOvooNOto Tj-cocococo co'^l-T^
toNOCOOCM TfNOOOOCM Ttvo "
TfTfrflOlO tOtOLONONO NO NO
CO
OOOCM
\Ol^t>^
Tf^vVOtOON
»-iONOoot^
NOt^ CO Tj- VO
"^O-^to-rf
OnOO OCOQ
OOOCMtoOO
Tl-t^ ON T-H CO
t^t^t^oooo
t^ Oiot^l^
cot^coOO
CMCO00r-H5N
OCM '-H CMOO
y-> Ot^CMtO
NOCOOrJ-00
OOONt-hCM CO
'-h' »-; cvj CM CM
tOONONVOOO CM
corfr-Hr-
OnOOO'
r^TfNooO'
CO Ot>>toio VO
tot>s OOOCM Tt-
CMCMCMCOCO CO
CMOOOOnOn vO00t>^Tti-t coi-xOvOvr^ to to rj- 00 Tj- CMt^OONt^
ONTfNDOON t^f^-^CNlto t^coOOt^iO t^OOrJ-Oco 00 to t^ Tt CO
OOt^OONO CMOONOt^CM CO CM On t^ r^ O 00 co NO 00 '-i Tt NO NO CO
TfCM-^ONt^ ONOt^CMCM NOiOCOt^'— ' r-( to VO CM rj- CO CM ON NO NO
OOcoONVOto NOt^OtO^ OOX^t^ONCO OC^CMCMco vo to NO co l\
tv^^Tl-OOCM NOOtOON-"^ OOcoOOcoOv tT O NO CM 00 Tf 00 NO On NO
Ti-NOt^coo '-icoTfior^ ooOr-icoTi- noooont-hcm -^cM'-'OO
CO CO CO CO tJ" Tf TJ- Tt tJ- T^ Tj-tOlOtOtO tOtOLONONO VOt^OOONO
^^^^r^ ^r-ir-i^^ ^^'^^^ ^^^^^ ^^^^CM
tN.CMt^
Tt-CN] C?\
00tOC3N
CM '-'OO
CDnvoOn
O'-'CM
i—CMco
t^lO CO
NOtO 00
CMt>x CO
CO CO »-H
NOlo 00
OOco rf
Tj-t^ O
Tl- to t^
CMCMCMCMCM CM
O'-iCMCOTt tONOt^OOCN O^-hCMco
cococococo cococococo '^'^rf"^
Th tovotv.00ON OtoOtoO tOQlOOtO
-^ '^'^-^"^"^ totONONOt^ t^OOOOONON
338
TABLES
VO
00 »-" vot^N VO
Ovot^vovo
On 00 C3NtoO
t^OCM VO-^
CM VO -TfOO,
On
tOi-iCMOOv
t^to-^CM Ov
r-H vot^ '-'to
Tf VOTl-VOVO
t^ ON CM^ On
VOVO
tOOvOCO 00
r^oo VO 00 CO
CO -H CM ON On
to cot^C^'*
O CM tn \0 t-^
T-Hl^
tJ-OnOnCMO
to to t^ CO ON
CO vocOTt ro
r^ CO ^00 00
•..o
vOO "^
oocM T-^ a\CJ\
to CO t^ CO to
T— 1 lo to r^ ON
OOcoco VO CO
^
roi=H(N
OOOOcoroOv
poo CM CM O
a\Ov^-<<o VO
VO O C^J -^ to
t-. On CO On 00
1— 1 On cvj r-H 00
VO
VOCN) On VO
CO T-H OC3N00
On ''^ On to CM
OOnO— 'Tf
0\ -^ CV) Y-n-i
T-H ^ T-H T— 4 r-5
fOTftoto\q
t^OOO '-'CM
i-;r-;CMCMCM
COtovOCO o
CM CM CM CM CO
CM CO vqoq p
CO CO CO CO tJ"
CM to CO '-' ■>*
•Tt ■^■^tou-j
m
VO-^CM -^CM
CO vOcO'^O
00 On CM coo
'-' CA CM VO Tt-
^%^^%
CM
to VO -^ rj- CM
OCOCO ^ VO
'-'to CO CM CO
r^totN.t^ On
lOVO
^toOtoOO
Tl- C?N VO CTn '-'
COtI-COOnO
t^ CM O CO ON
"^CM VO CMO
CMO
00 ONOtoCM
Ovcoiorf CO
00 cooovo a^
c^j r^ ^ '-I to
On NOVO CM ON
to t^ to CM CO
^
to VOlO
OjO-HTtrO
ONOO O VO o
CV) On CM to O
COVOTtr-.^
CMt^to
voot^r^'-i
CO O to to On
00<MCMVOVO
CO to to »— 1 lO
vOto crj o VO
to
to O t.^ »— 1
r^Tj- or^to
CM— 'On 00 r^
t^OOOvOCM
to CO CM t^ CM
OOto COCM v-H
. p '"1 ^ ^.
CM CO ^ -^ to
vqt^f>.oq ch
p >-; CM ''^ to
CM CM CM CM CM
vq t^_ On O CM
CM CM CM CO CO
CO to l^ On r-i
CO CO CO CO Tf
—
:2S
rj- CM ro -H Tj-
CMtocoOCM
TttO '-<t^ r-H
(N vo^toco
vO-HVOONOv
H
^Q
Ov— loovo-^
Tj-OTfr-iav
^^cor>.co
Or-HOCOOO
Tl- O >-0 0\ Tj-
e^
lovooo
'-H O '-I Oto
OVCO ^ vO'sf
CMOVOCOO
Tf T-H cvi VO CO
Tl- On CN CJN VO
C/3
:i^
CMVO-^
OOVOVOOOV
voto CO On -^
00 1^ i-^ r^ VO
-HTt-tovo— '
CO t^ O On c-j
M
O '-•to
'-HCMOO— lO
VO CM OCM VO
a\oooo^a\
CM CO CO xf 00
r^CM VO 1— ' o
rr VO o vo o
to O CM 0\ -^
•^
tOCM r-HCM
CM CM to CM -H
to CM CO 00 I^
'— ' O COCM VO
■^ON-rfOv
Tj- O VOCM 00
to CM OM^ to
COCM "-I O O
T-H CM cotor^
O -^ OC- Cv) 00
, p O r-H ^^
CM CO CO '^ -^
to vq VO t^ 00
OnO '— ' CM CO
.-J CM CM CM CM
■^ i--> VO l^ oo
CM04CM(MCM
P .-H C^jTf to;
COCOCOCOCOi
C^
1— 1
Q
—
VO
OCM00iO»-H
OOVOCM^to
1— 'too CM 00
rJ-i>.ONThvo
COCO OOCMtOi
to
O^Ot^OOO
OlOCMtOTf
to CM to lO T— 1
r-' Ol^tO r-H
CO r^ to CO Tj-i
05
D
TfOO
c^cn^ONr-H
Tj-Tj-CM CO VO
CO '-H O VO CJN
CI CO 00 to '^
VO 0> OO CO .-1
voto
to r-H CO VO '-H
Tj- to CO t^ ^>.
^COO— 'Tt-
C\i VO ^ ^ O
CO VO VO O to^
O
•^o
VOOOCX)
VO coc?\toco
CM ^ OO VO
ONONONCOOO
r-l t^ ONt^ CO
00 "^ cor^ VO'
<
Oh
e<.
r=;-^ov
VO to to 00 CO
OON T-l to r-l
O CM i-^ to O
"-I 00 ON -^ CO
to CM c>-:i oo 00'
1
o
U
H
r^
■^OOCVJ VO
^VO-HVOCM
OOCOOVOCO
OI^ '^^CM O
CJn t^ VO VO VO
vOt^OOON'-Hi
_pp'-:-;
CM CM CO CO rj;
Tj- to vq vq tN.
oooqovp--;
y-i 1—4 1— H CM CM
"-^ CM CO rf to
CM i CM CM CM
vqt^ CO On —It
Cvi CM CM CM col
1
1 — 1
s
§88
r-i COVO "^to
VOCMVOVOCM
coT^loocM
vOt^OOOO On
1
OOVO t-i vOOOi
CO COCMO CO
t^t^ vOOto
OOOtOCM CO
OO^lOTt ^
OO^r-HTJ-OO
CJN to »— 1 On c:^
«
tot^ CO
VOtOONO\t^
00 ON CO vo^
00 VO lO On ^
Tfto vot^r^'
<:
^
CVlr-lOg
OOtot^ O Ov
On VO VO to On
'=:t- 00 1-^ CO O
CO CO ^ t-i CM
H
CM t^to
VO CM CM 00 00
to On OC^ VO
COON VOx^lO
r^-<^to t— 1 CO
CM Ovto— 1 00
fc<
^
tO'-t oor^
l-.0v(MV0C^
O CA r-H roOO
to CO '^ l-^ CM
C3NON r-i VO CO
coto— 1 O'-':
cot^o-^
00 CM l^ r-i MO
T-H to T-H \0 >-<
t^ CO ON to CM
ootocoooo
VO"^ COCM 1-'
o
CO
_ P p '-H r-J
T-< CM CM CO CO
TtTt toto vq
vq ^s.^s. CO On
Ovp r-iCM CM
'-'CM CM CM CM
co"^tovqt^
CM CM CM cvi CMi
o
<
T— 1
wor^oooo
00 1^ On '-^ CM
CM -^ COVO to
COr^ r-l ^ ^
COI^r-^OOr-
00
OCOOOO— 1
TfCM coOco
cooocot^r^
n-Tfvooo
CM to T^ to '— '
O CNl O VO to
r^ 00
VO coo CO On
r^ VO t^ CO VO
'-' ^ CO VO -^
t^ 1— ' ONl^tO
rgo
t^tot^ol^t^
^ CO VO cooo
vOO'^t coo
-hOOCOON
r^ONOOCMvo
CM ooot^r^
On CM l^ to to
CDNt^OO-^tO
'-^ CV) >-HLOt^
r^tocM 0\to
^
OCMtO
0^■^C^v0T^
co^toOOC^^
t^ -Tf CM (M CO
VO O VO COCM
COVO '— ir>. VO
1— i > I T-H T-5 T-H
toOvCM VOO
Tl-00 CM MD '-<
tOOtOOto
OVO^t^co
On to CVJ 00 to
CO
"I ""i ^] ^. *^
CO r<"\ Tf -^ to
to \q vq r>>. t-x
COCOOnCTnO
r-i r-i r-i ^ CM
O '-^ CVJ CM CO
CM eg CM Cvj C^'
—
lOtO VO
r^voC^toCM
CO -^00 CO CO
vo-^tocoO
CO^-t -^OOO
CM to to CM VO
CM to CM
CO corf too
OTft^co—i
ON-^ONt^ C3N
rfCMOr>.^
ooi^t^ooo
VOONr-l
COOOOVO VO
T-H r-H CO to -^
OOC?NtOON(>l
CO O CO CNJ VO
O to o VO VO
^
to 00 CM
t^ VOCM 00-^
to CM 00 VO ON
0\0to VOCO
CM --^ too CM
VO Tj- (M CM O
tot^l^vo
(Nt^'-Hcoto
vor^r^ CO ON
^to On to CO
x^^^ coco VO
CO »0 CM ■* CM
^
t^ to "^ "^
to VO OvCM VO
T-< t^ Tj- CM '-1
CM CO to ON -sl-
Ol^ VO vot^
O -^ Ot^ VO
CM to 00 '— <
rj-t^OTj-r^
'-I ':^oo CM VO
O -^OOCM t^
CM VO 1— 1 VO'— '
I^ CVI OO CO ON
0^
_ p p p T-J
^ r-H CM CVj CM
CO CO CO ''^f'^
to to to vqvq
i>.t^oqoooN
C?\00-^^
1-JCM CMCMCM
s-
s-^
en
00
CO
O
0'-<Mcorf
iOVOt^00O\
Oi-HCMCOTl-
iOVOt>^00ON
T-i T-i y* y—t y—t
CM CM CM CM CM
iovor^ooo\
CMCMCMCNJCM
009
Oh
COMPOUND INTEREST; OTHER COMPUTATIONS
339
OnOOOOOCNJ
roooroO'-H
t^O'-' rOrl-
OOirjoOirjo
vO t^ vo Tf CO
OO-^cOiOO
VO r-< \0 ^t^
KododcKoN
t^ '-H r^ Tf ,-H
^ VO COIOOO
CO OiOvoOO
PO 00 f5 <0 fi
00 TtT>»t^O
.-HOO'-it^O
vO "^ On OOio
'OOn VO ONt^
t^LOTf CO CO
CO Tt iri vd r>I
I^OM'JIOOO
CVI U-) 00 VO --H
'-' COVOOSON
cvjiooortr^
Tj- vo ON'-i O
OOTfCvirtON
'-HO^COTJ---
lO
O\C0O\^^
O^OrfT-H
On OnCmo
VOvOOOTf
lOONt^ vo
Onio ^'On
t^O-^OO
00 .-H
On lo
Tf O
lO CO
CO On
00ONt>.rMv,
cotJ-tJ-loOn
C^On^OOI^
•«i- CO Tf 00 T^
OnOON'-'CO
»-' 00 ■«:♦• CO CO
ca CO \o Q in
COlOt^ OCvJ
r^'^TfONio
CO '-H 0\Cvl ^
lOVO^t^ '-<
,-H ^ OI~^iO
SOOrf rrr^
!-• ONOOOO o
lo r^. p <^ t^
lotovovovd
»-<u-)VOcoOO
t^ ^10 0N<N
OOOOt^ vo o
oocooONOm
Omo vo t-<
'-H ^ ONt^
CO On vOTfio
O f^t^ '— 'lO
^s! »< t>! 00 00
8^:
On 00 On lO CO
t^.- OVO I-"
t^OO '-< Onco
8lOt^ NO CO
CVJOn<V) CO
to Tf in »-i '-H
OOCOOOCVJ
On "vj" ONrr On
OOOnOnOO
On 0^1 On vo Tf
r^ ONOOi-oio
On OioOm
ONCOOOQCq
t^lJ^ONONNO
vo<^r^co(N
Tf vovOOO-^
OnO-^
CONOlO
OOiOC^)
vom to
OTf 00
lOVO t^
NO t^ lO
COVO CM
o-^ '-^
coco o
r^ o lo
CO to Tf 00 "<*•
OOW-^O'-H
i-HtoOOcONO
coooonoco
in CO On rf NO
Tt ^ 00t>.NO
t^ONOCN-*
^vOON'-hOO
OO-^TfCviO
t>*00QON On
rft^vO'-H On
CO CO 00 (M 00
t^t^NONOtO
NO t^ On eg NO
vq 00 p CO to
Tj-TfOOOOO
tr. On -^ '— ' On
"^ QirjOO CN
NC O'—coCM
CO »-• NO Tj- ,-1
NO00»-it>vVO
»-t t^lO CO CO
oopcovqoN
io\o\dvd\d
CO 1-1 On r>. tN.
rt-V0'^i/)O
OOOnOOiO t-H
TT i-HNOml^
CVJ TfOJ Tf NO
OP Tfio 1— I CO
eg in On CM vo
t^KKodod
t^tvcOONOO
CM '—I On com
NOOCt^CMiO
cooooco
NO fO Tt t-^ »-H
CM vot^O rt
CO lO CM 00 00
On «-<■ Tt t< 1-;
,-H^^CM
OncocoOOO CO
CM Tj-i—i COVO P
NONOtOiO y-i 00
0> On too On t-i
ONp"^ '-<t^ to
Nc o 00 r^ O 00
■^coiocor^ 00
1— I 00 '-'lOTt m
K CO CM CM lO -h'
CMCOTJ-IOVO 00
*-i^ioO"^
tOTf t^ 1— I CO
t^coOO t-tNO
ONComoO'-t
CO '-• OCOCO
CO CO 00 00 'it-
Tfr^OTf On
CM CO in NO t^
fO fO CO CO CO
Oniono^oOn
CTniOOO"^ On
00 CM On CO to
00 CO 00 —I NO
OOnO ^co
NO CO 00 00 NO
-^i-OvOco^
On 1-1 CM Tf NO
CO ■ ■ * ■
COiO»-Ht^00
nOtJ-OnCMO
O »— I coioiT)
CM NO 00 ON'-"
OOt^TflO
•-H CO (MONO
Tt"'!-"^'^ Tt-'^xninm
OO^CM-^lv.
NO t^ NO CM CO
lOCMiOOO 0\
t^CM '-'CMTf
1-. 00 00"-) CO
"-•Tft^O CO
Tft^ .— t^CO
oqpcoirjoq
lo NO vdvd NO
ID CM '-'CM to
CO ON Tl- CM CO
CO NO r^ to 00
00 NO C^^ CO '-I
NO CO NO t^ NO
NO VD On CO '-'
OTf 1-hOnI-x
,-j NO lo r-x m
K CO CD CM to
vo r^ Tj- Tt Tj- 00
vOOOncoCJn t-i
Tf c?N Tj- com 00
to On O COOO -^
CM t^ NO CO CO Oi
ID On CO On '-I rf
00 CO 00 Tl- ^ O
t-iCMCMCO"^ m
OooONinco
txTftOCOCO
cO'-'tNiCMO
CJncoO-^O
t^Ot^ChOO
NOlONO r-t O
OpO-H(M
OOOnO— "CM
CM CM CO CO CO
iOi-<coCM'-<
Tf.-<TfCOr^
o
ON'
coo
CO to t^ On CM
CO -^ to NO 00
CO CO CO CO CO
CM^OnCMO
t>*OOONONO
C7\ CO t^ Cvi »— I
to CO too **
CM 00 CV) t^ CO
ONt^ '-' Onco
to On -^00 Tl-
OiOCM com
COTfTtrfTh
w^O'^OOOn
m T-H o OMO
00— 'TfODTj-
mrf OOOCVO
coONCMmp
CV) NO t^ CO NO
O vocO'-i On
tN. oqpcMco
rfTtiotom
no-^ooono
CO'-'ChVOCM
"^r^^CDOm
CM "^ O O eg
ONt-^ orv.00
TfCOOOVOCM
00 fOl^mr-H
mvq 00 f*^'-^
m VO In! On T-H
00Tj-T-.m
comco C3N
orv.oom
ID CO to t^
tOtv^OO '-I
oomtN.cM
ONt^ '-"-'
^NONOi-;
coin 00 CM
NO 00
m 0\
00 "
CM
CO
CM r-i
NO ON
CM r^
NO ^
CM CO
5
t^ to NO Tf O
""^ CO t^ CM CO
CMOCMtoto
NO 00 00 COO
CM O O CO On
r^omCM T-H
CM Ot^ to CO
Tj- tf J iq NO t>.
CM CM CM CM CM
to CO 00 00 00
tJ-cono-^On
CM 00 NO CO NO
NO t^ CM CO CM
OOCMCMt^O
cooom-^t^
'-'On octavo
OOOOOnO^
CM CM CM CO CO
ocoONrxrx
t^ ON 00 t^ CM
t^oomvocM
CO CTn CTn >— I to
OOONOmTj-
CM OnO'^'-'
NOm NO NO t^
CM CO Tt to NO
CO CO CO CO CO
TtCMCOOOTl-
oot^Ooo-«^
lo coto r-c On
On ""^ Omo .-<
m O 00 CM CM
T-imT-iCMvO
00 On*-* coto
t^OOO'-'CM
COCO-^TtTf
CMOnO^O'-i
— m,-ir^ On
00 CO CM —I
S!
CO CM '-«ONt^
COOOOCM'-i
copoqoqoN
Tt^trjvoNOtN!
CA a) tv. Tj- ,-1 NO
ooomooo 00
t^rf cootv. y~i
»-«NOCOCOiO CM
On O CM "^ vd On
coooootxvo
bx CM CO On CM
.-'00 ^t^NO
Q»OCM coto
nOOt^-OnCM
NO 00 CM t^ to
m'-'oo'^^
CMcoco'«^m
CM CM CM CM CM
'-<CMOO»0
OOtot^ '-'On
tor^ coooio
CM Ov CM to to
TfTf tOtONO
Tf to 00 cop
OOmCM OOO
lONOt^OOOO
CNJCMCMCMCM
a:
CM NO NO t^
lO''^ cot}-
cooioooo
t^ t^ O "^ CO
00 CM O* 00 --I
CM CO CO CO CO
oovocooom
t^ O Q\00 CO
xl-NOO C?N"")
NO CO r-^ 00'-'
OOOOOCgrj-
ONCOOOt^OO
oooot^t^t^
CO "^ to NO t^
CO CO CO CO CO
l^TfVOTfNO
t-^NOCOt^t^
»-< On.-! ONNO
CV) .-< to T- ' to
COCOCMOCM
CMNO CM CM CTn
CO-^CTNCOtv.
OOrfpcONO
coTf mtoNC)
CMCMOOCMTf Th
1^ O to CM 00
Tj- •^ CM CO i-O CM
CO
t^mr^ 00 T^ CM
to com .—I to Tf
onqco'-'P cm
Tf NO CO C3N vO t^
vq tv. p Tt; ,-j p
In! odo '-^ CO m
O'-'CMcoTf lOVO^x00O^ O^CMcorf u-JNOt^OOON OmoioO toOioOm O
eOCOCOCOCO COCOCOCOCO Tt •* Tj-Tf rf Tj-TfTj-TfTf mmVOVOt^ tNOOOOONON O
340
TABLES
vo 00 u->0\
vo t— I OOiO
On 0^ On C5\
"OCJOO
0\t^"^txvO
eg 00 CVl u-> CO
rl-^Ovo"O00
lO Oi O -^CNl
OO CM CMt^t^
COCN ^OO
OOMDrfCgO
00 00 00 00 00
QOOOiOOO
00 "^lo or^
ONTfioCMCM
^ CM CO to r>H
OOvOTf CNO
ooooo ooooo
"^ 0\0 0\l~^
u-5 rf to lo r^
VOCNJ 0\^ CNJ
Tf vo ^ r-c lO
ocor^x ^ to
On t^ to Tf CM
vo VOVO vo "O
oc5oc50
CVJ rOOO vo CO
Oto O VOCM
■—I On 00 vo to
vo to to to to
ooooo
OnCvI CO 00 to
to !>. r^ t>^ cvj
O "^ 0\t^ r-l
ONCOONt^VO
CO eg CO 00 vo
ONVOCOOOO
cocg ^ OOO
to to to to Tf
oooc5o
<
CM
^
^
T-HOOOO
to CVJ t^ VOtO »-H
CO cocg Tj-covO
CO CM T}- OnOO '-I
■^ ^ CM vo coog
00 t^O t^ ONto
rj- rj-ioto vo 00
y-l ONt^tOCO '-H
ON 0000000000
oc5ooo
c ) On 00 to In.
r-l ONTfOCO
O'^I^On'-h
,-( On vo '-I Ti-
to 00 vo 00 CO
8 CM to 00 CM
OOVOTfCO
oo t>s r>. tN. t^
OOOCMCO-*
(NOO '-•VOOO
vOtoCMl^rt-
CM vOtot^ CO
CMtJ-OOnCM
VO O"^ C7\to
'-1 OOO voto
t^t^ vo vo vo
r^oot^Tj-co
Tj- cotoCM vo
votOTj-i^ vo
r-H ^Cgco'-^-
OOtN. ONTj-CM
OVOCMONVO
TfCNJ r-lONOO
vo vo VO to to
OOOOO OOOOO OOOOO
CTM^t^OOco
C0 0\ '-H 00'-<
vo ON'-' CO CM
"^(N ON CM CM
cor^ CO CO to
cOOCOVOTf
t^ vOt}- COCM
to lO to to to
o o d o d
vo 00 CO CO
»-Hr^ coTj-
CM 00 CM to
owocgrj-
co^ cooo
O r-H CM CO
oovoT^r^^
On On On On
t-hOO
00 CO
O— '
coi^
t^ON
tOl^
QOO
On 00
oor^t^
.-H COCM
OOtO
vo On to
tori-r^
Oco vo
t^tO CO
CO 00 00
Orj-
coo
coco
Tfvo
COCM
OTf
CMO
00 00
00 CO CM
t-H too
CO CM to
ONCor^
Tj-OCO
oocot^
oor^to
odd
CO'-H voi^vO
t^ 00 to CO r^
Tf to eg On O
T-H T:t- vo to CO
Orj-r-( ^t:1-
cooo -^ O vo
Ttev^ t-H ooo
t^t^t^t^ vo
d'd>c:>d>d>
CO CM-* CM ON
coOOOChTj-
T-H to 0\ to 1— t
t^vt^ coto CM
ONt^oo "-ir^
CM ON vo Tf r-H
lN.tOTj-C0CM
vo vo vo vo vo
tN.00TtlO.-<
COCM Otoco
OONeg Tt e^
COt^ VOt^»-<
tOtO 00 CO »-H
ONf^tO Tj- CO
O ONOOt^vO
vo to to to to
oorN.00 ri
On t^CM'O
8 t^ to 00
On CO to
ooooe^iON
CM to ON CM
OOVOtJ-co
On ON On On
rf-rfOOt^tO
lOtO t^tO CO
CM CM CO t-H T-H
On
vo r-( to O to
'-I ooor^to
ON ON OO 00 00
t-tOOOO OOOOO
OOn
VOCO
00 CO
CM vo
t^CM
Ovo
Tt-eg
CO 00
do
ooooo
00 CM ON
r^ '-t Ti-
to ON vo
ooco
CM COTf
»-H ONCO
oot^t^
ONi-llOO'-t
tOCOOONO
Tf VOVO ON Tf
t^'-HOOt^ON
oovotor^ ^
orN.Tf »-H On
t^tO Tf CO •— I
OOOvOOt^Q
toco eg r-io
Tf t^OOOOO
eg vocMt^ CO
00 vOt^<3^Tt•
vOT^CMOON
O On 00 t^ to
t^ vo vo vovo
OOO ooooo OOOOO
VOCTntI-OOvO
ONTj-ONeg
oo\ONeg'_
00 vo >o to Tj-
Tf COCM '-lO
vo vo vo vo vo
o
w
H
Pi
o
H
w
CO
w
i>NtooNco coononcm'* cocoegcMoo o^j-vocm^
vOt^ONCM cOr-Ht^^CM CMCM^tOeg toOCMto-^
'-•'-'VOrJ- OCMvOt-iCvj t^cot^t^Ov ^^to,— it^
V..O CMvo.-<oo voTfcv)r-HON vococoeg-Ji- tocoooi-io
e^ CMVOCO'-H CVl>OOtN.to vOOnCOOOO COOCOCTnVO
Vs tOOVOCM OOTt-1-it^^ ^oOVO'^r-i ONOOVOrtCO
►X COt^tOTj- CM^-HOOOt^ vOtJ-coCM'-h CTNOOl^vOto
^CTnOnOnON CJnOnOnCOOO 00 OO 00 CO 00 t^t^tv.t>.t^
idic^dd dcSdcici cDcid>ci<d> dddcScS ddddcD
C\1 to CO 00 CM
r^C3NVOOC7\
ot^t^t^<^
I^CTnOO co-^
Tf^vO Oto
COCM Tj-iOl^
CO to r-^ CM 00
to O to On 00
OCM OOONtO
CMO OnO^O
ON On 00 On q\
OOt^ VOtO"^
vo vo vo vo vo
CM vo CO 00
CO oco CM
""t — oo^
voTJ-T^to
t^tO CO r-H
00 f^ vo to
On Ot On CTn
VOOO CO to On
O CO CDNTf vo
r >. Tt- to 00 o
l^ I^ »-i On CM
r>. ,— ( r^ CO CM
onoo votoTt-
co eg '-I o C7N
On On On On 00
covoo
ONTl-vo
ot>>.oo
coi^o
1— I CM to
COCM -H
oot^vo
00 00 CO
^OOOO ooooo OOO
CJnON
VOO
CM 00
t^ vo
00 CO
OO
to Tt-
00 00
cSd>
00 to
t-ico
covo
ONTJ-
ON ON
CM'-t
00 00
CMTtvo
Ovovo
voooo
CM CO to
vo vot^
On ON C3n
OC7N00
oot^t^
to »— I vo cor^
to 00 On too
oooor^Tti^
Ot^ voi:^ On
BcOOOTf '-'
oo^eg
00 l^ vo to -^
Tj-TfvocooO
^ CO CM to tx
■rf-^ vo OOON
COCO Tj- .-iO\
ooNoegrf
cocotovOtN.
coeg r-ioqj
ooooo ooooo ooooo
^
^oooo
On Tt to eg CM
vocMoe^ioo
to to 00 CO On
V0r^r-.00co
■^ Ot^Tf CO
'-•CM CM CO Tj-
tOTf CO CM <— I
<?N On On CA On
c? <z> <z> d d>
tOCM COOt^
ONt^CM vo On
VOCO On CM eg
OOCMr^vo vo
CM COTI- vo C?N
tOVOt>* 00 C?N
o c^oot^ vo
cjnoooooooo
t^voCTN'-^CM
rj-eg Ttcoc^N
On '— ir^r^o*
Tj-CMr^ r-H CO
cococoofN.
'-^eg rfvor^
votoTj- roeg
00 00 CO 00 00
t^r^i— I cdnco
Tt- ,-( CM r^ T-H
T)-OVO ^ vo
Tl- CO CTn -^ vo
to ""^ CO "^ to
On '— < CO to t~>.
'-I -hOOnOO
oocooot^t^
Tj-voCMt^tO
Tf Ol C?N to r-l
00 t^ CO to CM
ooooo ooooo ooooo
O'-hCMcO'^ tovot^OOON O^CMCOTl- iovor>.00ON Oi-iCgcOTh iovOt^OOO\
,-< ,-H ,-H .-H ,-t T-( T-H ^ rH r-( CMCMCMCMCM CMCMCMCMC^
COMPOUND INTEREST; OTHER COMPUTATIONS
341
C^l Tl- O CNJ u->
■>!4- T-HI^ O O
vo 10 CO eg .—I
t^ vo >r> -rj- CO
'^ Tt Tf Tt T:t-
OCJOOO
t^ OWO 00 -^
fOOO<Nt^
vo a\oooot>s
O vO -^GO CO
CO -tT CO CO o
Tf COTJ-OO-^
CNJ CO Tj- lO l^
t^ ^O l-O Tt CO
CO CO CO CO CO
ON'-'
CMfVJ
coco
TfVONO
O.'-Ht^
Cq T-HIO
covOCNJ
COLO 00
^oo\
CO CO CN)
r-HoqoM^oo
csj 10 U-) u-j m
CM o com CO
T^m coooio
0\ '-' CMOOm
OMn CM o i^
CM CM CM CM —'
00000 00000 00000 00000
0\r^ coonco t^
Tf »o MD t^ r^ CO
T-i Tf rf irj O t^
CO o OMor^x Tj-
OM^ Lo CO r>* vo
VOOO CM <X)u-) T^
u-3 COCM O ON CO
'-"-i^^O o
odoCJCD CD
OO^COOO'-H
O O C0l0'!i-
CM CMiO ^
C/J OmJ-j 10 Ch
ON 'O OCX) CM
CM ^OCTnOn
'-' O ON 1^ o
U^LO';^r;^ Tt
CDOCDCDcrJ
OCM
VOO
ONO
CJnGO
coco
OCD
00OC50
Ot^CM
CM CM 10
ON CO CO
On coco
CO ChON
CO CM ^
000
VOTfNOlOCO
t^ m >— I CM >o
10 On CM ON NO
•<^ o r^ CM 1^
vo NO t^ ^ ^o
O '-'CMTj-io
-H OONOOt^
tT Tfcoco CO
OCJCJOO
•— • O On
0000
C?NLO00
OCM — '
Tf CO"^
NO 10 10
COCO CO
<od>d>
00 CM
10 CM
C»CM
CO NO
'^fO
coco
00 00 NO NO -,N
O CM 10 CVI O
NO iri 00 CM CO
CM r-( Tt- Tt 10
t^ T-"-H Tf NO
CO Tf CO to O
cvi On NO CO '-'
CO CM CM CM CM
1-1 coOOt^ On tJ-
On On CM On ^ 00
CO CMO C?\J>» O
t^CMt^OOtN. NO
TfNOOO ot^ o
00 00 O "^ O 00
CONOmcoCM O
00 00000 00000 O
ONr>>, ocoi^
00 CA cot^ »-•
oiocooooo
1^ Tf CO CM CM
OCM NO CM O
CM ^ 000
VO '^ CO CM '-<
10 10 10 LO 10
'-•VO CO ON CM
NO — •CA'^CM
r^ coot^ 00
CMCM '-lOO-^
o c^^ o '-' ON
O CAOO r^ NO
vO^Tj-rj-Tl-
oooo-o 00000
CM "-I CO vo^i-
rl- (N) ^ 1^ r^
oo-^ooo
ON'-^ OMDO
000 cor^T^
CM "^ 10 NO 00
vOTt COCM '-I
OOCJCDO
OCOVO^'':*-
OOt^co vo Tf
NO CO 00 r^ CO
On vo NO c > 10
r-i T-H cvi vo c:^
OCM^NOOO
-— I O On lO l^
■^ -^ t-O CO CO
c5ocDoo
00 vo
00 CM
CM O
v/o v/^
^ NO
t^ CO
CO CO
d>d>
r^ONr-H
CM NO NO
CMOt^
COLO CM
r^oo
Tf NDO
Ot^io
CO CM CM
--I COO CM vo t^
r>sr^cN -^Lo ON
tN. C7N Tf r- c On cm
LO Ot^ NO On CO
rf '--1 r^ CM CO O
NO 10 LO CO CM CO
CM O 00 NO vo CO
CMCM '-"-"-I ,-.
000 00000 O
ThNOt^cOON '-•COCMCMO OO'^OONOCh Q O O 00 rj- ro O vo NO O
NOr-iTfvoco r-HOOt^OO OnCOtTCOOn "^-t^vOTj-co OOt^CMvot^
t-^r^CMOOO cO'-i'^Ort OCOcoconO O^OOCOOn COOnOCTnno
rrCMOOr-iO r^OCTNTf^ OOnONOO O^nOvOI^ cmcmcococo
C^JOON^Tl- COloCMCMco NOOvocm^ OCM'^OOC'-J O'— i'-'J^CO
Tt^cOTt-Tj- -rt-LONOl^OO CTnt- iCMrfNO COOCM'^f^ OvococoNO
ONCOr^NOLO ■<:fcoCM'— lO C3NC7\OOt^O lolo-^coCvJ CMCOvoC^lC^
iOLOlOlOlo vovovololo -tJ- -^ tT rf ■^ "^ -"^ "^ '^ '^ Tj- CO CO CO CM
OOOCDO OCDOOO OC3CDOO O O O CJ O CJ C) CD CO CD
TfrfCMCMOO CM
'-"-I Tj- 00 r-i (N^
ON'-^CMNO'-i n-
'-^OVOTfr-^ CM
CM NOCO COrl- rf
CVl C?\ 00 On CM NO
t^rl-CM OON t^
CM CMCMCM T-H ^
CD (DO CD O CD
CO^CM OOtv.
Tj- 00 CTn VO o
CM l^ CM LO -^
N0O0n-hi>^
:-v CO On GO r^
OnOO "-i CM
CO CO CM ^ O
NO vo NO NO ^
CDOOCDCD
00 Tt ON CO NO
o i^ o 01 c^a
NO Oi CO rr .— I
NO 00 T}- CM CO
00 O'^ CTnlO
COLO NO t^ ON
On 00 r^ NOLO
LO vo vo LO LO
CM onlo cor^
CO LO CM LO NO
CM to c:^ .-I O
NO '-'00 OOC?N
CM —lOr-H CO
>— I COLOl>* ON
vo -rf CO CM '-'
vo vovovovo
00000 00000
•*1-VOCMO vo
OnnO -— ' t>.t^
Tt (M CM r-H On
"-I to O NO CM
l-^ ^ l^ CO '-'
1-1 Tl- NO ON CM
»— OONCOOO
vo O -^ Tf rt
OOCDOO
00 O I^v '-< CVI
NO O C?nCM ON
Tj- COvo CO NO
OCM CjNCOt^
OOnCMOnnO
vo o c^ chCM
t^ Tt- or^ vo
Tf -^Tj-CO CO
ooc>oo
ONvor^
CTn'-"-'
vo O ON
r^ o^. oc.
cooo o
t^ COCM
CM C'CO
CO CO CM
CD CD CD
OOOn rt
'-' On "^i-
CM NO CTn
VONO C^J
00 O NO
'— I CO vo
NOTf CM
CM CM CM
OC> CD
nOOOOOnCM
00 coTJ■r^Tt
oooo oooooN
CO CO CANOrt-
OO O --I COLO
CO 00 1^ NO LO
NO NO nO no no
OOOCDCD
I^NOCM
r^ ^ CM
'-' C7NLO
00— '
■^rf LO
t^ CTn--^
■<^ coco
NO NO NO
COO
r>»io
00 CO
t^O
CO NO
CM^
NO NO
Tj- NO CM NO NO
CO OLOio O
CO CM CO NO O
1— ' O CO LO C^l
Tf CA ''I- T-H ON
COOCONOCO
Ot^ On On On
On C3N '-• Tj- 1^
CM CO CM NOLO
t^ '— < Tj- 10 vo
t^t^t^ 000
'-"^t^OTt-
t^ NO vo vo TJ-
Vi ) to 'O to vo
00000 00000 00000
vOCMOvovo
OONNOt^O
ONCOt^i\ ON
C'l !>. NO aj CM
CO OnVOON r--
t^ Ti" Tj- 10 On
CO Ot^ rf r-l
VOVOTtTfTt-
OCDOCDCD
tv.ONNO
OOt^CM
t^NOrf
oovor>.
00'— 00
coot^
CTNl^rf
COCO CO
CDOd
VO'-l NO
CM C3N CM
Tf vo CO
CM CO CO
CTvCM t^
NOt^ 00
CMO 00
COCO CM
do d
CMvO'-*t^Tf
CTn ^ '-' Oco
CMt^ Tfcoco
Cvi t^ O O t>^
CJN vo CO '-< C?N
Ovoor^t^ Tj-»-HCMCMNO
CMOnOvconO '-I --I CJN C?n t}-
TfT^r^coON cOcoCOCJNtO
>— ' CM O LO NO LO O >— ' ON -^
C^NChO-— 'CO vo O T}- CO -^
LOOOCMLOOO »— iLOOO'— 110
O0NC?NC0r^ r^NOtoLO-'^
t^NOVONOvq VONONONONO
<Dci<z$ci<z> (zic^dc^d ddcDciiZ^
CM Tj- r-H ,-t r-t
OiNOOTj-CM
Tft^ COCDC7N
to CM NO NO .— '
01^ TfCM T-i
C3NCMNOOTJ-
COCOCMCM »-H
NO NO NO VO NO
CVIOOCM
OOONO
0000 o
COCM rf
OtOTj-
00000
O t^ to
NOLO vo
CM NO
ON 00
cOTt
co '-'
t^ CO
coco
CM 0\
vorf
ONTj-'.tOO r^
Tj- CTn CM r-l C-1 CM
or^ CO '-I o '-'
CM r-H CM ONt^ T-H
'-''-' CM CO vo l^
"«t »- ONOOOQ On
r^ vTv CM o 00 NO
■^■^ "^ Tf CO CO
00000 00000 00000 o
O— "CMco-"^ lONOl^OOOx Oi-HCMCOTf lO vo tv. 00 0\ OvoQ^OO vo '
COCOfOCOCO COCOCOCOCO Tt rj- Tj- Tj- Tj- tJ- tJ- t^ tJ- rj- vo 10 NO NO t>^ t>» (
110 0*0
I 00 ON On
342
TABLES
H
tn
W
w
H
p
o
< U
w
W
PL,
/--S
^
CO
000
fOTfOOVO
r^rr^i>Hvo
00 CONOCO NO
NO 00 CM On '-'
COOONOlO
COCOLO-^l-Tf
r^Ti-cMvo
^iOt-h COTJ-
ooot^ogco
t^LOcoOON
OCMTft^O
r>.Tf^CNjio
NOQON-Hl^
ooOr^ONO
VOvOONfO
lonOtJ-co CO
rl-LOiOt^ 00
rOCh^ O
vo NOLO r-H On
On 00 NO CO O
NO ^ NO Tf T-H
OLOOONt^
00-^vot^ON
ON -^ NO COLO
On 00 CO NOLO
^
cnonotj-oo
CO l^ On 00 CO
CM NO CO COLO
coa\o\c\)
l>^ -rl- lO t^ .-1
00 NO NO 00 Ol
t^co-HOO
»-i Tt rv. 1-1 NO
CM ONl^LOTf
CO r-H O OnCO
NO
T^00^^O^
Tf ONOCvi On
LOCNJ OnNOtI-
»-H On l^ i-O CO
»-• ONt^NO Tf
On 00 00 t>.
t^ r^ NO NOLO
lO LO -^ Tj- rt-
Tf coco coco
CO CM CM CM CM
CM CM CM '-H T-I
--^OCJOO
d>d>d>d)d>
d>d>d>d>d>
c5d>d)-d>d>
c>»d>d>d>d>
d>d>d)d)d>
loooor^
rxOcoNOCN
LO On Cvj LO lo
OCM On LO NO
oONor^T-^^
t^ CO CM ''t CM
t^t^ CO NOco
ctn rr voT^
^ rJ-COCOON
<N CNlTfCOON
»-H LO VO NO On
t4- CO 00 CO ON
OONt^sOvl
NO LOT-" ON 00
CO ON l^ r-ll^
r>» r—\ \C) o CO
C?N CM On i-H l>.
C^l O 00 CO NO
OTtrrON-^
coraoo O C3N
OOCNJ coo
Cvj •-« OOfO Q
LOCNJ NO CGNO
r-H>, COCNNO
»— < »— • On Cv) CO
OOTf tM^nO
^
coooot^
On NO CO CO O
O T-H C-l LO t^
00 ON 00 LOO
vooO'-^Loo
CM rN. coc^
S^^S^:^
CO-<* NOOLO
1-t COLO coo
»-• 00 NO -O LO
lOi-Ht^LoCM
lO
lOO ^ 0^
OOlocO'^ ON
t^LO -^CM «-•
CO CO CO CO CO
On 00 NOLO Ti-
On ON 00 00
t^ l^ t^ NO NO
vo LO LO LO LO
Tf rf Tf rf CO
CM CM CM CM CM
'-; o o d o
d>d>c5d>d>
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d><zi
vO
8^§;^
LOTtNOCOfO
00ThNO'«t^ T}-CN10Nt^0\
NOt^OONOOO Tj-cococot^
VOCOONCOt^
QOOt^ ON CM
NO -^ CO NO O
©=^
r>s o\^ '-'
r-H U-) 00 LO -rj-
r^ooco^CM
OONNOO--H
C^t^OOco
cooo^t^^
^
COCVJ 0\V0
LO OnCNJ 00 o
n-oooo^ON
CM ONNor^i^
(MNOt^OO
t^rf^f-^co
vOTfco 1. 1 CO
'^OOOLOO
Nor^t^ cot^
QNt^CvlvO
On ^ NO CM On
NO NOLO to LO
t^Tf NOLOO
CMOOTfr-HON
T^
VOJ^VOOO
cgr^'^corvj
rr NOONCOt^
lO'-tr^ CO
O NO CO Ot^
r-H On l">. LO CO
T-H ONt^NO Tj"
CO »-i O ONtN.
ON On 00 00
00 l^ I^ I^ NO
LO -^ ■<^'^ rt
Tt* CO CO CO CO
CO CO CO CM CM
'-^cJddd
d>d>d>d>d>
d>d><5d><:^
d>d>d>d>d>
<5d>d>d>d>
(ocDdxzici
vO^vOOn
^CO^T-HTt-
r^ CO LOON 00
OCOLOCMCq
LOO On CO tv.
ON NO CO CO rt
Ocot^xt^^
rlLOOOCMr^
^ONOOO
irj ^ CM '-H Tj-
00C^^LOTtTt■
OO ^ ^ '^
r^ Tj- 1^ o NO
"^ o l^ '^ "^
NO 00 On l^ t^
T^oocoooc^^
NO COLONO i-l
COLO 0\ o
<N-H^ON00
NO O t^ C^l rt
CM ON CO NO NO
00 COLO CM CM
T-H 00 T-" l^ LO
-^NOooT^No
LOONO CO o
^^5.
lOLo OnOO
On CO On NOLO
XOLOLOLO-^
CO 00 On t^^
vocM oom
-H o ON O <M
(V) ON LO COO
LT J On '^ O l^
LO coco CO rf
NOCO '-'LO O
Tf
r^ Tf CM Ot^
LO CO '-' On t^
LOCOCM O On
OS 0\ 00 CO
00 l^ !>. t^ tv.
NO NO NO NOLO
LOLOLO Tj- T^
CO CO CO CO CO
'r^d>d>d>c5
d><Dd>d><:5
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d><o
tN.O'-HCO
r>v^vovor^
»-< .-lOLOON
CM'-hOO'^On
OOOCOCOCO
C7N t^ -rl- Tt lO
lor^r^cvi
OOt^ cO'-Ht^
NOONt^-HVO
ON NO CM CO ^
COO(N1CV1
t^oS^^
OOLO COTl-,-1
O LO CO '— t LO
LO o o LO r>s
NOt^C^TfOO
rtco— 'lOTj-
^
00,-Hrf Tj-
T-H TfOOOOO
OnO ONOlo
NOl^LOOOlO
T-tiOOsrl-
ONlOON-^t^^
ONONt^Tj-t^
00 l^ CM CO 1-1
LO LO i-H CM On
r-'OOONOgg
:^
vo poor>.
»-< coloOnco
00 Tf-T-H ONt^
OOO NOCOr-H
t^ NO NO NO NO
NONOt^OOO
CM LO ON CO 1^
CMOOnOONO
■^^ 00 LOCO
On t^ LOCO CM
O 00 NO LOCO
PO
ON 0\ ON 00
00 00 t^ t^ t^
tOLOLOLOLO
LO^rl-TtTl-
Tj-Tl- cococo
T^dxdtdxzi
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d>d>
d>d>d>d><5
c5<zid>d><^
0\'-< VO"^
OOvOr-^COCO
ON CM 00 CO 00
lO tJ-lo •— • CO
LOOOOLOTj-
1^ CO NOLO NO
t>^0\^0
r> eg LO cvj t>^
ONON-rfNOO
i^ CM 'J^ t^ r>*
lot^ or^co
COlO i-il^
OOrf T-H ONNO
CO-HON-Ht^
'-'NONO^NO
NO NO --^ ON 00
CO.-I O coCvJ
lOOnCM-hco
LO-^CJNNO^
SS22oco
fN^ONTfOO
OoO OnO T-i
ONCMt^LO 1-H
r^TTONONCo
O C^l lO 00
^Tj-O^"^
NOLO 00 NO ON
CO t^ «—l NO »-H
^
f^ t^ CO On NO
^CMoSS
1-1 CO LO r^ O
r^coo t^-^
S^^5^88
NO CO r-t OOnO
Tj-cMooor^
lOCOCMOOn
t^NOLO COCNI
to
CO 00 CO t^ i^
t>»iv.r^ voNo
NO O NO LOLO
lO LO LO LO Tj-
TtTf rfTfr^
^dddd
<Z)<z> <zi ci> <d>
d>d>d>d>d)
d>d>d>d>d>
d)d>d>d>d>
d><5d>d><o
r-l COOSfO
O^OOLOlO
OOnCM cooo
^ooooo
00 -^ tT CVI NO
t^OOLO-^O
NO NO I-" 1^(30
CMOnCNC^I^O
vooor>.Lo
CO 00 CO NO
On '-' '^CMCM
r^(MLO On ON
LOONf^t^O
t^ CO -^ r^ r^
O'^'^COrl-
On I^ CO LOCO
OCOCOOOOO
^t^OOCMO
^
LOoO^oS
On On CO O ON
LO ONLO ^t^
CM CO 1-iLOCO
'-'l^O ONCO
CO On^ 00 On
NO COLO NO CM
CM NOLO 00 Tt
LOONt^OOcO
;^
CO tv. '— 1 t~>»
COONl^-rtTO
CN) T-iCMCMco
Lor^ocor>»
T-ILOOLO T-4
r^ CO o t^ LO
ooNCONOii2
t^ TfCM ON
SoSS^g?^
NO Tf-CMOOO
NO NO NO NOLO
OOnOlocoCvI
CVJ
ON On On 00
t^t^l^t^NO
LO LOLO LOLO
LTj T:^ T^ Tf Tf
y-^d>d)c6d>
d>d>d>d>d>
<5d>d><z>d>
d>d>d>d>d>
c5d>d>d>d>
d>d>d>d><::^
.2
(2) ,-1 CVl CO "^
»OvOtxOOO\
O'-'CMcOTt
to NO 1^00 ON
O-^CMCO-H-
lONot-xOOOi
»— t T-^ »-^ t— t »— •
^^^^^
CMCMCMCMNM
CNICMCMCMCVI
COMPOUND INTEREST; OTHER COMPUTATIONS
343
•-^ O) Tf c^i in
OTf rv. vo 1— I
1—1 XT) ^J-i CO r->
'-' CM ON^ 0\
t^ O lO t:!- CO
O "^ 0^ coio
OCMio C\ro
COCVI i-H OO
ooooo ooooo
C^^-hC^ICM o
o-ir^io vo o
r^ ^ o "-H r^
On ON coco t^
ooooo
c5ooc5o
OOO cOTfvO
Ot^ OOO0u->
xi~> r^ ■LD 0\ -rt
VOlO VO OMO
CVIOO Tj-Ot^
t>^. ^O VO voio
OOOOO
c>o ooo
co-^ covocrs
oot^ '^C^lt^
OO vo T-Hiocq
Cvl u-) CO VO 0\
Tj-o orgvo
lO Tf CO Cvl T-H
ooooo
OOCJOO
»-il0O<0*0 CO
T-i,-.c\]oo cq
O\<v>ro00rt tN.
rj-iovor^-^ rf
vOrt OCVJ Oi 0\
CM Qvt^voco CM
OOOOO o
OOOOC) C)
lor^^^r^o
CDN^COVOt>^
OOQCMOfO
NO^NO-* CO
\n r-i 0\ 'rt i->
^00 -H^^
POOCMOOt^
O00C?N^t^
o\
Tf-rJ- ^LO CO
cvi Tj- \o CO On
VDVO CM ^ On
CONOIONOMD
lO On '-I (?\ Tj-
t^CA^CM Tt-
ot>»ioior^
NO NO On CM CO
'-HNOONNOlO
T^
t^iovor^io
OMOCOO^
Tj-OOcoOVO
ON ON TT'^ NO
OCM COTJ-NO
iOt>^O00O
S
CO CO coco CO
i-hCMtI-nOOn
ocMcor^oo
CM CO lO On 00
t^.-H oocot^
'-hOOnOnO
CM lO 00 CM NO
'-O OOnOn
t— < f-H y—l C~) O
t^ 00 CO 1-1 CM
ioOioCMOn
t^
rOCMOCAOS
oor^^oio-^
Tf COCMCVJr-H
00 NO V^'^ CO
OOOOO
gSSS8
8
d
oc:Jc5do
<D<zS<5c5<5
c5c5c5<5c5
c5ci<=ic5c5
CMi-H^»-^0\
Ttt^r-.Tf ON
or^voTfvo
t^CMO^OO
>ntN. ,-.T^t^
ONOOrot^CA
CO
OTfO\CM C50
Tf .-H CM Tt Tf
t^O CMiOt^
coco^t^io
vOOOOnCDn
Tj-rj-Oi-HON
NO
ocv) a\ '-^lo
rfOO ONOio
OOioOOCM
"^ CO 00 t^ 1-1
OnOnCMj-)tJ-
vo
OiOTj- onoo
lOCM ONIOVO
CM CM ^ vo t^
VOCM COCTnC^N
t^OOCM^S
CO lO CM CO t>s
lO
CM O ^-T^VO
OMO-^ VO "-H
ONOCOOONO
oomr>.oCM
CM
tN. to Tl- CO C5
Tj-ioNOr^ON
r-.Tj-|^OTt
t^CM NO Om
o 00 '-it^io
NO C3n CO C7MO
CM
VO vOTt coCM
'-'OONOOt^^
t^ voioio^r
COCOCMCM'-H
T-i oor^io""^
COCMCM^^
CM CM CM CM CM
CMCM'^'-H'-'
i-OOOO
ooooo
o
CDC)C5oo
cDOCJdd
cDCDodd
(Z>cicid><5
<D<od>d>ci
(Dcici<^(^
d
t^VOTtt^CA
t^CMiOCO^
rj-COCOOiO
,-H vonOnoCM
CMi— QCMO
t^COiOOOO
g
vOCM On '— ' O
Tft>^ 00 Tj-vo
O On On Cnj co
-^OOiot^^
NO lo "^ r^ tj-
NOCOt^ONtX
00 O l^ -^ CM
lO 00 NO too
OS t^ "^ 00 NO
OOCO CM rJ-i-H
CMioOCMCA
COTfOOOO C?\
Q
y-<\0-LnO\tn
COt^t^NO"^
CTN^COCA^
T-. U-) NO CO i-H
§
CO'^OOIO
OOvOiOrfco
T}- NO CM CM ^
CM CMm 1-1 CD
coo CM lO 00
I-- NO CM r- CO
t^ NO O i-H CM
l^ CO NO CO O
cocorJ-iOvO
tv.NOmiO'^
O»nio00T}-
CM cotOCJN"^
On
OONOOt^ MD
lO rf CO CM "-I
O O O\00t^
r^1-H ONt^NO
mrfcoCMCM
T— 1
CO eg CM CM CM
CM CM CNJ CM eg
CM CM '— • '— ' '—'
»-H 1— ( 1—1 T— I r-t
-^-HOOO
OOOOO
o
cDOocrJd
dddcDo
dcDodd
d>d><5d)C>
c><5c5<:5cD
<5c>c5cS^
d
,_U-)r-l ^lO
vOCM^TfO
r^tv.00^
Tj-t^OOlOTj-
t^rJ-^OOCM
oiotxiniTj
C?nOOOO OnCO
^
TfCOt^lN. O
ooo On CM VO
OOt^VOCNlO
rf CO '-I CTnco
CMOOVOTl-CM
ONt^CANOO
cOi-H COCM^
,—1
VOCM'-Hr-HCM
CM '-< ONiOCM
CO 00 Tl-iONO
LOCM '-' cor^
S
r^ COCO -^r^
r^cocoNO 1-H
t^ cot^OC3
lo or>x00 '-'
lONO i— 'OCM
mm CO t>. 00
Noo\^CMr>»
CMCM vOCOrt-
ONCOOiO-rf
NO -^lO 00 CO
Ot^ On 00 On
t^t^t^CMO
o
VO -^CM '-I O
CM T}-iOt^ O
u-)n-coCMCM
CM U-) 00 '-' vo
CAO NO NO CTn
lOCOCOlOOO
CM
to Tl- CO CM '-H
On 00 00 l^>. NO
--^OC^nOnOO
t^iO CM OOO
t^NOlO'<^CO
CO
CO CO CO CO CO
CM CM CM CM CM
CM CM CM CM CM
CM CM '- ^ r-H
^^^^o
OOOOO
o
d>c:>d>d>cS
<5cici<5d>
c5(zi(Z>cid>
c5c5ci(5(5
c>d>cici<^
<5c^c5(5ci
d
VOlOCOlOO
Ocor^lO^O
tJ-OCMtJ-00
CMCOVOOON
OOt^CAtOVO
»-<Ot^ONCM
S
t^'-t OCM C7N
■rr Tj- ON 1— ' lO
vOiOt^OOOJ
Oi-hOCMCO
CMi— I tM^ CO
vor^t^ vo-*
CO CM CM NO CO
NO 00 On cm '— '
00 NO CO 00 o
t^t^cocor^
vnt^iorN.o
CM
00 CO CO CM T}-
00 CO 00 C^l lO
mCMiOTM^
COCOIO CJNtO
ONOCO»-H ON
""l-t^NpCMCM
CTn O CD C3n CO
CO
ONCNCOO O
coo ON CM t^
lONOCAu^ro
Tt-t^CM OnOn
i-^r^t^^CM
00 NO ON NO NO
^
lo lo •^lom
NOt^OOOCM
T^-NOCTN^Tt
CM
i-H onoo t^^
lO TfCOCM '-'
OONOOOOt^
CO CM CM CM CM
NOlOTl-Tj-CO
CM ON NO rf CM
\r>
Tj- CO CO CO CO
CO CO CO CO CO
CM CM CM CM CM
CM^^^^
Soooo
O
d>d><:5d>d>
dCDOCDCD
ddddd
cicDdcici
CDCi<DCi<zi
c5<5<z>c5<5
d
,-« ^ COOOQ
CM On O OCO
■^r^Noc?Nvo
^^^^5
CMCMCOOCM
cOi-hOnOOnO
CMCMOrhON
^
1— c CMlOlO'-H
or^i^-^CM
OO^l^^CM
CM-^'^CMVO
Tf CO r-H 1-^ CO
CO t^ 00 ON CO
C^gg^g
t^ 1— it^ On '-H
t^mNor^t^
CM rf NO CM 00
NO
rfoo "^Or^
cor^ CAoOTf
ONO'-^ CO NO
t^ Ot^^ '-•
•^
»-< CM r^ lo to
ON lO -rj- vo '-^
oooooTi-^
ON^Tl-c^NO
lO CACO-^t^
t^ i-iNO OOn
CO
CO •-< On 00 t^
NO NONo^r^
1^00 O "-I CO
rfr^ C7\'-< Tf
t^ Tj- NO I— ' On
0"^CAt^iO
NO
T^ CO 1— ' o ON
COt^NOm Tl-
CO<>JCM^O
On 00 1^ t^ NO
m CM On tv^ T:t-
COT-HONOOt^
NO
Tl- Tt Tt- Tt CO
co CO CO CO c»^
CO CO CO CO CO
CM CM CM CM <N
CMCM'""-^
'-"-.ooo
^
dddczJd
dddcDo
c^ddcDd
c5(5<z^<5<5
c5cici(5ci
ddddd
d
0«-'C<J<^"* vovOtxOOON Oi-iCMco-^ mNOt^OOON OtOQvoO lO '
cococococo cococococo '^ "^ rf ■«i- rf Tf tJ- t^ -^ tj- toioNO^t^. In.'
iOO»0
OOOnOn
344
TABLES
CO CVJfOioOO
VO lO t^ 1— t CA 00
lOto oovooiooo
CV> CO fO t-i T-i
CO l>% -^ "-H U-)
lOOO TfCOlO
CVlCOlOt^ON
CM r-i
lOiOCM
OO'-H
»-<C^cOTf »ovot>.ooo\
t^^coONt^OO
i-H\OCM ^CM
OOMDiO Tj-io
CO "^LO""^ 0\
CO COIOO CO
OOOOntJ-^
Csi "^ r^ '-' »J^
t-h' cnJ CO lo vd
V0C0»O»-tC0
VOTfOOOt^
CviCMcorJ-O
0\CNU>. CO o
COOO^O OOrJ-
C^coOOcoON
VOCM
■rj- CO
lO «-H
iot<
CM CM
OOOO
On CO C^
lOCM CO
OOtJ-O
(M ^ On
com CO
odocM
CM CO CO
coco CO "^r^
COOOO OiO
VOOQOO\
l^l^O OOCM
tN. r-«CM On^
in y-< l—^ U-) in
ripOxGOOO
CO CO CO CO rj-
Q
o
w
u
w
o
^ Q
H !^
w o
<:
o
H
o
<
CM
CM
^
CM CO
VOVO
8 CO
o
CMOOt^
CMOCO
OOt-h
t^Tj-ooooo
OM^CM '-' CO
'-hOnCM^On
»-i t^ VO '-H 0\
Ot^OONCO
corfONiOiO
CM CO -^VO 00
irJvdKocJOx
rj-t^r^vooo
OOCMt^t^ 00
o^CMr^o
t^OvCM CM t^
iOTl-(M00co
OCO^OOnCM
t-H CM CO Tj^ vd
O'-Ht>X^V0
COt-hioiO VO
'-HOOCACM On
OiOnt-h vOTf
'— I CO O t^ CO
OniocoCMu-j
OO CO ChOO
vqp'^oqco
t<C7NO >-< CO
^^CMCMCM
COCOON
OCMCO
OOnOn
CMCMTf
lOONO
r-iCMr^
^t^vo
O^T^o
r-(00
OCO
CM CO
^^
ON NO
vOco
vocoooot^
On VO li-) T-H VO
VO '-' O CVI VO
»-i cor^ CMi^
COt^OTl-00
OOO'
CM
cOTfVOOOO
COCOCOCO '(I-
\OVOC50iOfO
i-hOncoi" ■
00 OOCO(
§i2
_ rfCMOOvOCM
■NO O "-I (M CTn VO
"^OOTj-CMTt
(MMDCM
iCMCO"^ iovdKooo\
OCMCO CM irj
0"^I^IO r-(
1— I lo CTn 1— I 00
CMt-hOOcoco
t^t^Oco On
On 00 CM OCO
OOsJf^"*"^
CMiOMDOOcO
C7NCM On covO
VOLOOCM 00
'— I oor^ T—i lo
rfCM OCOiO
CO ON CM CM O
OnCO^ ^ Tj-
CM vOO->4-00
K 00 o '-H cm'
i-H^CMCMCM
OOn Tt* »— I t^
00 '-H lO CM "^
ON r^ vo CO CM
vOi-HOOVO VO
coco On 0\ 00
r^coooTf r-i
CTnOOOn'^CM
CM i^ CM CO Tl-
CMCMCOt-h T-i
^s. t^ CO COLO
ON"^ CO OTf
On OCM T-( CO
CM ON CO CM CM
CO t^rt lO 0\
O VOcoOt^
CMcouSr^cd
CO CO CO CO CO
CM CO
VOO
OCO
lOOOCM
t^CM VO
»-HlO O
oo-^
iCMCOrJ^
OWO'-HlOTj*
CO On CO Tj- CM
CMOOOOCM-
OOOOcOfH
Ot^-^LOTj-
oooooor^vo
t^ vot^ o»o
T-j CM CO lo vq
tovdt>Ioc)ON
tJ-On
On CO
^^
CM ^
OOO
CJCM
T-HCOCO
t^o»o
CO CO CM
OTfco
r-(lOlO
LO VO ON
CMiOO
CMTtr>.
corj^io
VO.-IVOC0 00
coCMiOVOt^
ONr^ VO "^lo
Tft^QCO VO
■^ VO 'O VO T-1
OOOOO'^T^
On CM VO On CO
vd 00 ON O CM*
,-1 ,-H ,-• CNJ CM
ONOOOor>.
T-i COCM C7NCO
'-I On vOtT VO
•-"OOCMio T-i
VO CO On VOO
^-H VOlo O •-<
O T- 1 u-j CM '— '
t^y-nnCDm
CO lo vo' 00 On
CM CM CM CM CM
lOONOOt^tN.
T- 1 vocoi^ VO
C?NOnCM CTnCM
t^ O Ot^ CO
CM t^ Tl- ro VD
O LO '-I Tn. CO
»-J (Vi r!^ LO t<
CO CO CO CO CO
00 COCOON r-iC3N
CO C?N Ov "-I '-' VO
irjco VO O -^ On '-t
VO CMOVOloOncoco
©^ CM On CMloOnOOco
vc< roLOO (MOnCMCMON
kJs, l-HTl-ON lOCMCMCOLO
OOO riCMco^iO
JcMfO-^ irjvdt<odO\
1>.OncoOlo
vOTfTfvoO
1-HCMr-H On(N
<MVO'-iCM00
t^ CM CM 00 CO
CM cOt-( VO O
O VO "Tf COLO
t>. 0C3 O CM ^
Cz5'-< corfLfi
00"^OnCMVO
t^ 00 CO l^ CO
r^ ONLOLo VO
CO VOLOt^ y-t
T-H CO CO CO t^
CMCM t-i On VO
00 COO CO On
VOOnCM Tft-^
CiS'-i'^Tl-O
i-H T-H On vooo
r^ CM On CO O
VOCM t^Tj-CM
VO LO LO 1— I LO
CO Of^LO CO
CM t^ CO CM CO
T-H Tf OOCM VO
coTfiot<od
CM CM CM CM CM
'-Hvoor^to
NO OnlovOOO
CO 00 00 OOO
CM voiN.r^ o
O On vo-^t^
CO CO VO '-^ 00
VO'-iOOCOCJN
CqiOONTj-ON
C> t-J CM -^ LO
CO CO CO CO CO
tOiO
CM ON
VO VO
iHvovo
Sjt^LO
Scot^
CO"*
^<N
0"<^CMC7nO
CM^VOO(M
CM-^r^oo^t
r^ LOCO 001^
LO^OOCO CO
vooooooco
CM ONVOlo VO
^t-hCMCO-^
irjvdKodON
t>sOCM"*00
CO CM Tf On 00
NO t^ »— I lO ON
VO CO VO '-I t^
VO On CO 1— I CO
'-I CO O '-I ^
00 '-H VOCM On
LOt^ COO '-H
O '-H cm' "^ LO
CO CM "OCO CO
VO 00 COLO On
rt- coco T-t 00
CO NOLO CJN VO
COr-HO'-Hr^
VO t-H r-H VO VO
00 ON^ -^ ON
COLO 00 OCM
vdt<odo r-i
^^^CMCM
Tj-txt^toco
OOlooO vo^
t^ ^l^ VO oo
On O O CO O
CMLo cor^oo
VO -^ TfLO CO
LO 00 1— 1 -^ t^
CM COLO vdr<
CM CM CM CM CM
LOOO cOTl- VO
cocMr^oo^N,
"^VO 00 CO CO
LO c:a o CMo
COONCO t^ On
T-HTt00<Nvo
OnO'-I cotJ^
CM CO CO CO CO
o
♦rjcovo
poo
!CMcO"^
,-HVO'-H vot^
OOCMiOCM
LO LO LO Ot^
o i-H cor^cM
O OLO vOlo
.-H CM COLO 00
OLO'— 100 VO
.-I 1-1 CM CM CO
io\dt>^odoN
r^^^^T^cM
LO VO O O CO
CMr^coCOr-l
i-H coo CM CM
CM COLO CO -^
CM VOC^l ONt^
vovooootj-
rJ-LOVOOOC^
C5 ^' CM CO Tf
m 'Tt r-cin o
lotj-cox^lo
On VO -^ "^ On
00 00 -^t^ 00
VOX^O T}-0
Onlo CO '— < '— <
OCM "^^00
vdt<oc)ONC)
*-H ,-H t-4 T-4 CM
OCO 00 COLO
CTnO OnOOCO
CO -^ LO T— I -rf
O ONOOO VO
O '— I LO CO Tf
CTnOn^voco
l-H COt^ T-Hf^
pCMTf 1-xOn
CM corfLOvd
CM CM CM CM CM
OO^Onvo
lOLO CO vO VO
On T-Ht^ VO l^
On CO 00 On 00
'-• vocooco
COLO O On O
■^CMCM CMiO
CM LO 00 r-H Tj-
OCJOnOCM CO
CM CM CO CO CO
O
00
CO
coo
P4
iCMcOTj" iOVOtxOOO\ O'-HCNjcOrf »ovOt>.00ON O >-< CM CO -^ lOVOt-^OOON
^^r-i^^ ^^^^^ CMCMCMCMCM CM CM CM CM 04
COMPOUND INTEREST; OTHER COMPUTATIONS
345
t^og CM o CO
CMOOTfCM
O OlOiO I— 1
OnO'-; CO vo
<ri vd 00 o cvi
TfCOTfTfVO
■ VOO
t^ CV) t^ ^O C7\
Tj-VOO\'
CM "^On eg On
cx) '-I CO t^ eg
rgoforgoo
ONCotN.cgt^
Ttt^'-<O00
ID CO CO o o
cor^ t^ fo CO
U-) t-H o org
to vo 00 00 CO
CN[r^o\Ocg
O CO CO vOiO
rro CO vqu-j
r< o eg' lo 00
\OcOOfO On
,-. Tl- CO Ti- 0\
CO CO 00 00 r^
»-H ocoioo
vO-^I^ On ^
,— (lO VOlO CO
lOiO VOCO '— •
»-<' Tt r< o "^
00 CO CO On ON
0\vomt>.^
Tt-CM CO cOr-H
coOnioco^
'^O^OOTt
OOio On^-h 00
■^looN^cg
t~>; LTJ lo On u-j
On^coiOOO
lOLOlO
o voio
t^cgoN
ON^-t vo
cgr^io
oorgoo
oocot^
00 CO eg
-^odvd
'-I'^OO
eg eg eg
0000 Tj-
e^j ON o
coco Tf
into vo
eg^-^ vo
'^^ 00
lovo -^
^ vo lO
On in! eg
egt^ CO
coco "«}•
eg vovooo CO
QTtege^ivo
•^t^ egioio
VOlO »-" T-H t^
eg voonoon
lo Tj-t>^ VO On
OntT cofNU-)
»-< ^ r-i ^ og
eg* rf vo" 00 o
OO-^vOiOTj-
ovoococo
lOOOlOt^CO
eg ^ cor^t^
00 vOTfcoiO
O On t^io Tj-
On VO ONr^ O
COlOtNO-^
eg" Tt^ vd On 1-H*
lOlOliOlO vo
Tj- 1-H CM »-l T-H
CM eg ON CM VO
vo lO ITi »— ' .-<
t^ NO -"^LO Tf
1— I CO CO CO ^
VO'-H i-nr^O
OOCM 1— iio vo
t^ eg t^ CM CO
CO vd 00 ^* CO
VOVO votNtN.
lOCOrfvo vo
O-^co VOCM
voegoO'-H^
O vOiotN. ,— t
iJ^CM CM OOiO
CMrt-r^CM CM
egrt egt^oo
locM ooqtN.
vd On CM* Tf t<
l^t^COOOOO
vOOco
r^vo On
i-iCOco
vOOOtI-
IT) VO"^
tN VO'"^
Ovdrf
ONOCM
t^OO
On^
lO vo
eg CO
com
vOt^tN
cor^io
lOOOOO
covo CO
t^m On
r^r^CM
vo r-lCO
CO r-l '-H
^*ONcd
OV-HlO
^egeg
lo^ eg
eg CO o
.-H vo lO
ooeg vo
OntJ- Tf
tNiO 00
"Tt CO vd
ooeg vo
CM CO CO
'-hON'-'OO
CMt^ voeg vo
OnO OnO >— •
tN.Tj-CMt^O
o^O^ooo
00 ON t^'-' CO
vot^eg T-( ro
lOCOCM ^ O
cdeg-^vdod
TJ- Ti- rt tT '^
^oc:^coeg^-^
VOr-HlOVO-^
t^r^n-ONOO
t^ voioco CO
Tfcoevj oncm
•^ Tf Tj- Ti- 1^
On On CO '-I CO
OnOnOt-hCM
ON^rf vdod
rJ-LTjiOiOiO
ooTfot^^eg
'-HOOcot^T-H
COCM cot^t^
COCMCM VOIO
ONoeg Tt-vo
t-hOCMOnCM
o »-• voloo
TfVOOO^iO
CD eg T}^ f< ON
vo vo vo vo vo
tNfNVOI^vo
CMT^t^egvo
O-^ioOnCTn
^-1 vot^ '—I 00
t^irj T-iioio
CM OtN coo
On CO '—I to Tl-
00 COOO COC7N
T-HTtvdON^
t>»tNt>»t^OO
to to eg .-«Tj-
Tj- vo Tj- Tt T-H
1^ coOnio»-i
O CO coto »-i
Tj-Loio »-H On
C3n vo i-H vo tN.
t^ oo»oegt>x
loto O '— ' On
""^ 0C5 rf ,-; ON
00 ON'-' CO Tj-
VOOtN
t^OO Os
CMt^OO
tNto CO
tN<7NC?N
'-H .-H CO
ONX>»Tt
OCOON
t^ON^
^^eg
CMOO »-•
cor^ ctn
vo O vo
loto O
VOON CO
vo'51- eg
VOOO CO
r-HO eg
t<od CM
eg CM CO
ONCMiOVOI^
eg-^xoocM
O OOncOvo
toto ON^ ^
^ ON'-i T^^
r^cnegTi-r^
i-H Oi '— I vo CM
O NO Tf; ^_ ON
CJn O cm' rt vo
OnOnvovovO
I^Ti-t^vOO
0\ ONVOCOCM
CO CM CO CM VO
CX) '-I vo VO ON
OVOcocovO
CO vo CO CO vo
!>. to T^ CO CM
tv! CJN 1-h' CO vo
^Ttmvovo
cDr^Tt•oo^^
o\eg vot^ vo
cOi— I VOCM CO
CO CO VOCM vo
^r^ COVO '-I
Tt vo CM rj- CO
CO cot^ rf vo
eg CM CNi CO Tj-
tN On '-h' CO vo
lOio vo vo vo
VOOOtNVOOO
OOOCOCM On
coONVOTfCM
00 vo CO 00 00
vo vor^r^ CO
00 '-H eg eg eg
OnOOOVOVO
vor^ocM vo
l<ONCM'<^vd
vo vo t>. t>. t>.
CJOCTnOOOVO
vovooot^eg
TfCM VOCOOO
eg vo r-. ,-ito
CM '-hCM vot^
CMOVOOOO
O CO t^ CO CO
cjNeg vq coco
00 '-< Tf CJnvo
t>.a\o»-"co
'-'Tfvooovo On
QCMt^T-H ON 00
voegt^ ^ 00 vo
Oooegvo-H t^
eg COON Tj- Q '-"
^ ~ - vo •^ vo
t^ On CO "
tv. tv. vo ,-j 00 tN.
CM ^-ICMiOOv vd
lOtNOi^co vo
^r-i,-.(Meg csi
t^ On '-t CO CM
CO voO CO VO
00 vo 00 '-'ON
vor^eg VOO
CO '-'00 00 CO
COOCO Onco
to,-; vqCM On
t< C?nOCM CO
COCO-<*'<f^
ONr^'^OOvO
ooegt^co vo
tN. ONOO voco
oovoooooo
OC7n,-i00V0
CM vovo OnO
ON r^ 00 T-" 00
VOCM On t^-^
vot<odoe4
Tf^Tj-VOVO
i-^egQCMTj-
0\C0O'-i On
coCM >— I OOt^
On "-i-^ COVO
00 On '-I On 00
t^ '-I co^ 00
VO 00 CM On 00
CMo or^ o
Tt vd K On »-<
lOlOVOVO VO
vo 00 00 On 00
C?N On r-i CMvo
OCOOONt^
O'-i'-^'-'ON
CMtI-OnCM vo
tJ-00t-«vo00
1—1 vovo VOO
VO vovovo VO
CO lots! o\.-<
vovovovot^
TfCO'-llOlO
O ONt^'— I On
oooO'-iegcM
eg CTwotN. vo
OOvovot^lN.
vovo eg 00 CO
coThvdodeg
t^oooNOeg
*OtN,-(
VO'«!l-lO
OCMvo
00 VOCM
t^OOCM
CMOvo
tN'-i VO
ONt^VO
vdCM On
covo vo
OOVO O
cotJ- CO
8 CM CO
On^ O
ON vo CO
CMO O
o\vo 00
^s!^s; 00
oeg coiovoooo
'-"-' ,-1 ,-1 T-c ,-. CM
OVOC^nO"*
CO '-I VOCM CO
OOCM O^co
vOOOrft^O
O VO""^ vo CO
OnOOOOvO
CM OOvocOTl-
i-t voOvOO
COOCMCO^
VO voco"^eg
O* vo c?\ CM On
VOO-^TJ-CM
00 vo'rj- OnCM
o vooveg VO
t^ -— 't^ vo vo
vo r-H vo CM 00
covovdcd o\
"* "* Tf Tj- Tj-
00TfiO»-i|^
OvoCMOOO
t^VO'-'OO vo
votN T^o CO
VO'-' CO CO CO
CDNCor^CMOO
OOco C?NOOOO
1-1 CO "vt- vd 00
lovomiovo
00 '-I 00*^0
oeg'-^egr-H
>— I tN to 00 CO
ONCO'^OOtJ-
VOVOtJ- cot^
VOTf- VOOOCO
•-"vocoeg T^
ONvqTtcM lO
On '-I CO vo t<
vovo vovo vo
On eg vovovo
ciOvoi^CMrN
ONvorwor^
ooiooegoN
t>.TfVOT-l o\
T-' CM Tt r^ '-'
ooe^irNtNt^
00 Tf to CO 00
odododcKo
vorNOOONT-'
Tj-VOOCM'-I Q
TfCM '— 1 NOVO O
vovocot^ ^ eg
O0C3N VOO CO ■^
TttN'-Hoeg CO
COOOOOvovo CM
O'-'VOOOO t^
»-"-" ONtNCO O
fo' vd cjn Tt^ CD i<
cgcoTfvooo ON
COiOVOCOCNJ
VOtJ-COvO vo
'—or^oooo
OntJ- VOO On
00 r^ OOVO
Ti-eg^j-ONt^
00 CO On vovo
1^ '-< Tf 00 eg
TtvdtvIodcD
CO CO CO CO rt
O VO'?t '-•vo
VO CO'^ VOOO
voOOtN CO O
t^r^Ti-cM vo
CMOOVOt^CM
Qvor^CM CM
vOt^O vo ^
VO OvoOntJ-
vOOnVOvOio
coOrfcO'-i
cot^ ONONt^
t>^ cooot^ vo
CO eg ovt^r^
NOVO CO tN '—I
OOt^t^ Onco
oocooococdn
oc5c3'-<cot^
•^ to vovovo
CMt^CMt^vo
tN.-^cotNOO
Tj-VOTtfN CO
tNOOTtOco
OOOco VOOO
»— ' vo VO CM Tt
00 Tj- eg eg CO
Ti-o voeg 00
vd 00 ON '-h' eg*
lOvo vo vo vo
00 vovo CM -^
'-'CO 00 00 00
CMtNCTNOOvO
OOvoVOTj-co
T-H Tj- VOVOco
CO CM On VOVO
vovovocot^
Tj;oqvqoNvq
VOt^OOONO
M-c^vovO'-i Th
OOrN^-^O On
VO'-<tNtNVO CM
rJ-CM On vovo 00
oovocJNegr^ co
CMt-iOOCOCO '-<
'-it^tN^ vo 00
OnvOOnOOco Tf
o''-^'CM'^t< CD
T-'egcoTj-vo *^
O'-'CgcOTf lOVOt^OOON O'-iCMCO'^ iOVOtN,OOON OiOOiOO lOOi'^OiO Q
cocococOCQ COCOCOCOCO rj-"^'*"^"^ ■^ Tl- Tt Tt Tj" mvovovOt>. 1>»0000OnO\ o
346
TABLES
o
w
Oh
W
u
<
W
o
H ^
O
W
►J
H
l-H
o
12;
o
<
o
00
CO
000
VO-^lOi-HOO
TfrJ-Ot^CO
OOOOvOioO
OOO 00 On ii->
OCMOOCMCM
O f~x NO NO CO
CM oot^r^io
OnvoCMVo On
ooot>»u->t^
CM NO CM NO CO
VO
Tf CM 1— it^io
On 00 On eg T^
1— iNo pf^r^
C?NCM 0\CMt^
0\^ rovo I— 1
ONTi-TtCOVO
VOCMt^lOON
^OOVO^^
^§
ocooc^co
t^voONi-iO
OnioOOVOON
vot^CMOom
lOCO t^ T-H t^
^
l»>»ti-}COX^ r-H
O i-i OnCMio
lOCMCMiOON
mcMCMiom
"ifvOiOOOCTN
vOto OCMco
vOCOt-^
CO t^ Oi On CT\
oor^vooO'— 1
^>.t^ 1— 1 o^o
OOOnOnOni-h
VO
O'-i CO
vOONCOOqrl;
i-iONOOOOO
CMVOCMONt^
iv. ON CO On 00
OOi-nr^lOVO
'-^CvJcOTt
irj vd 00 cK »-5
corf vdoc5 1-5
£PifiaJoco
vd ON CO vd o
Tf oncoocJco
T— t
,_(^,_,,_iCM
CM CM CM CO CO
COCOTf Tf lO
ir>ii-}NOVot^
lO l-H to 00 CM
rJ-VOCMiOON
ONl^VOtN.r-*
OT-iOcMr>.
CM NO u-> tN. T-(
CgoOrl-OOCO
lO »—<io 00 On
ir)tN.covoON
i-<00^-hOO
OOl^Tf t^ON
00 CO VOCM T-H
ID
CO-HOO^
CMt^VOCMi-1
COi-iVOtJ-co
rfi^irfioSo
oq
0\ 00 CM 00 CO
voONVOOOQ
lOlO'-'t^ON
OnioCM 00 1— "
irji— 1
VOONO^IO
OOt^i-JoNVO
mrJ-cocoO
On CM CM Tf ON
OTf ^lOt^
^
CMO
u-)-hCMO\vO
OOf^OCMON
mo\"^pT-H
NOt-h OcOO
t^ CO On CM CM
ir)io.-i
^?^^o
t->, o 1— < I—" On
t>slOTt-COCO
CM 1-H NOOCM
t^-HVO^co
lO
O'-' CO
lOCM ONtVjlO
novo 00 1-1 in
pi>.LOT4-U-)
i-J cm' CO Tl^
irjvd odes'-"
CM Tt lo I< C>
,-5coioodo
CM CM CM CM CO
COCOCO^r^
t< ^' Tf 00 CnJ
»— < »— 1 1-H 1— < T—t
'«f lOlOlDNO
CO
CO VO On CM CO
CO r^ CO cvi CO
^^^^^
t^omvoi-i
ITJOi-Hl^NO
t^vot^ VOCM
t^ 00 ON On CO
if^?
0\ .-C T-^ CO T}-
CDnOO-hCOOn
Tf NOVO CO CM
CM NO l^ On NO
OTf cococo
CM 0^
O OMO »— • 1— 1
or^ cor^o
in CO o 00 VO
CM CO IN. CM On
T-H Tf (NJ CO CO
O^
t^OO 1-1 O '-I
O voOnOCM
OO ^ ^ONCM
o cor^oio
Tf t^COOt-h
CM NO CO coo
^
lot^oo
Tt CDNi-HlOCO
T-H coco t^ On
lOOi-HCOco
Tj-cor^
ts^^^OOO
00 ''^ NO "^ CO
COt^tJ-iovO
r^ooocooo
NO t^ ^ On CM
^
O^C<l
Tfr^ocooo
CM 00 Tj- 1-; 0\
i^t^i>.ooo
CO r^ CO On NO
mior^ OvTf
^
,-H'OJcOTf
lo vd 00 ON o
CM* CO lO l^ 0(5
O CM T*^ vd ON
t-h' CO vd 00 ^
Tf t<o cor<
CM CM CM CM CM
CO coco CO Tf
Tf Tf lOVOlO
VOVOOOVO'-I
CM^VOOOCJn
Tf tJ-CTnOOO
NO T-H CO 00 Tj-
00CMC3N00CM
OnCMOOOO
\0'<:^r^(Nco
T-H rf Tf VO T-H
iorN.t^iOT-H
CM NO Tf On CO
Th
CMu-j TJ- \OiO
tN. ,— ItOt^ 1-1
t^ T-H CM CM On
00 CO T-H ^ CM
CM i^ On (N ON
OvOOCOt-(
r>> o NO 00 o
O^^OO^
VOrJ-
CO On C^J C^l r^
^co 00 00 On
lomm "^ CM
OCMO^OOVO
ON r^ CM to CM
^
VOCMOOn-CM
vovoiovo ^
CO Tt l^ in i-H
CO o\t^r^CM
lO T-H Ttt^ NO
T^c^JT^
I-' CO On '-H 00
O OOCM CM CJn
CM CM On-* tN.
t^ NO Tf-H 00
Tf ^00 NO NO
"^
Or-i(Nl
rf VO 00 CM lO
cz5Ti-pvqcM
poovqvovo
t>. On CM NO p
vOcoOCTnO*
i-n'oicO"^
ir> vd (< On O
CM coiovdod
Oi-I co'iot^
On'i-h" Tf Nd On
T-I Tf K CTs cm'
TfTf Tf TflO
1— 1 1— 1 »— < 1— 1 »— 1
CMCMCMCMC<I
CM CO CO CO CO
00
OOOOr-lt^T-l
VOCM"^OVO
ooi-iiooo
T-H 00 lO CO T-H
OOVOT-Ht^C^
CTvOOTf Tf O
00
OO^intr^OO
^Os^roco
oor-st^coio
NO VOCM coco
in CM
loCMt^^io
O OMO i-H o
VO I—* O ^^ On
CM-^
VOiOOCOON
CMOOn^OO
On On VO CO 00
00 <M T-H ON 00
00 t~x O I—* CM
ioonocmon
C^On
COONC3NOON
VOOOVO^
VOTf ONTf to
oot-i o Not^x
lOVOTj-
to T-H U-) On t^
C3^COO^OO
COO'-'
NOlOt^lONO
COTtOT-lt^
t^^VO^NO
On r^ o ONin
t~>» VO CM VO VO
Tf T-ltOONi-H
.Ot=;(M
coiot^oco
CMONt^TfCO
CMCMcOTf VO
CACOt^ CMO*
CO
'-'Cvico'^
»i-Jvdx<ONO
1-1 CO Tj^ vd r>I
0\ p CM Tf Nd
00 O CM Tf vd
O6y-^rri\oc6
T-H
1-hCMCMCMCM
CM CO CO CO CO
COTf Tf Tf Tf
1-1 00 00 in CO
i-< OnVOiovo
ONOrfr^Tt-
OnCNOOCM
CM to CM CMO
OOOO-HO^
CONOIO-^T-.
00 CO t^ CO "^
Tf r^fot^cs)
CO CM to >0 CM
t^
lO On CM VO NO
ONloONOT:^
CO >— 1 t^lO 00
Tf lOOCOO
Tf CM CO CMO
CM
coovocoo
t^ OnCM C^nCM
T-H 00 00 CO NO
t^ 00 00 00 t^
NOTf CO CMtO
^
00 t^ O t^ CO
ONOOlOTfOO
coTf r^ooTf
S^^^8§8
O CO
ON 00 CM CM On
cOt^CMtv.vo
OOVO^rf ^O
OVO VOCM VO
CO
CO 0\ 00
ONOVO Onu^
VOO CTn^OO
ONIONO^T-*
t^t^coioCM
lO too CO T-H
O O '"'
coTf vqoO'-^
Tj;oOi-^VOO
lO T-H t>^ Tj- T-H
oqvqioTf Tf
Tf lO tN. C?N CM
'-^CqCOTl-
lO NO ts! 00 o
1-1 CM "^ lO t<
oc) o T-H* CO lo
vdodocMTf
vd OO' O CM to
1-1 CM CM CM CM
CM CM CO CO CO
COCOTf Tf Tf
lOO
voovoioo
o-^r^CMoo
cmijooni^ c^
CMOOCMI^Q
lOTf OnOOvO
OCOIO-hON
OCMt^CMTf
<M00
OOOOCMOco
mCA^ CM On
T-HONVOTf CM
VO^O
■^ VOt^ T— NO
§^§8^^
1-^ U-) NO On T-i
OOOOTf to(M
lO-^
VO-^OcOi-i
VO l-H CO (Moo
OnCnJ nOt-h (M
Tf otOt^P
5S|
IDCM O
VO ONt^OOCM
CM Tf CM VO On
r^ cjNTfTfo
COCOIOONCM
r^rooo
CM t^ ''^ CO VO
^cot^coco
lN.f^iO^ 00
oo^ooSon
CMOOVO
OOCVJQr-HlO
CM Tj- VOOOO
COrtONt^ON
CO VO ONCOt^
NO VOO On CM
ON-HOOOVO
^PO'-H^
CM <^ CO 00 lO
T-H On NO lo CO
CM CM CVj CO -"^
1-5 (M* CO ''I"
lO NO t< 00 o
1-1 CM CO lO MD
00 On ^" <N Tf
vdf^ONi-I CO
to x< On 1-H CO
i-H^CMCMCM
CM CM CM CO CO
COCOCOTf Tf
-a
l-H CM CO Tf
iovot^00 0\
Oi-iCMCOtJ-
in NO tv. 00 ON
P i-HCMcOTf
CM CM CM CM CM
in^i>20S^
.2
Ph
,_, T-H r-l T-H ,-(
CM CM CM CM CM
COMPOUND INTEREST; OTHER COMPUTATIONS 347
CMOsfO — O
rsjVOVOOOro
CMNOt>.T^oo
0\CM OCMtO
COt^O vOto
t^tNCMOO'-^
t^
CNlcoOtv^vo
oovovor^T^
On VO 00 10 00
votntoCM 00
tv. vo '— 1 OOto
totoCOtN vo
r^coTj-i— ivo
to
VOt^OOrr-^
to co-<^t^^
vooOTj-r^co
CO -^(M COO
rf VOOOON
On t— 1-1 r^ ON
CO
00 t^t^vO 10
t^NO'-H OlO
^CM-HCMO
OCM OONONO
I>-ONTtOOTf
to
^ vOI^t-t^
^000^^
On ^ to 10
tOt— 1 VOtO -^
Nooor^r-HCM
00 •-' On COCO
"-^t^otNoo
CO CO 00 Tf 00
tOCM^CM CM
OC ONT-HtOTj-
00
»0 00 rj- CO
COCM VOOLO
NOTttOOtO
Tf On voto
CO t^ CM CO CO
CO ^ -^, C7N
^ONOOtNO
vo
OOOOOfO^
Tf »-H CM On
tNOCTNtotN.
tN. to to On
On to ON ^
CO
On Tj; IN.* Tj:
»-h' C>\ t< 10 10
Tttotot< On
CM* vd '-< vd CM
O*^ coc7\rN
ONONCO^NO
Q vd cvi ^ o\
OTt-TfTl-O
CO
tN.00 0\0\O
r-t .-1 c\j CO T^
tOVOtN.00O\
<— < CM tT to tN.
^
T— ( »— 1 1— 1 r-l T-H
CM CM CM CM CM
CM cOtot>^C?N
CO tN CO '-CM
'-I »-H CM CO rf
^
oootN.r^oo
lOCMVOOON
Tftoo vO On
t^ votoT^ ON
CMCOOOOOt-h
ooTtc^ooCM
CO
1000 CO 00 CO
CO t^ 00 000
r^cM oovo CO
CNl ON '-H NO to
CO NO 00 ON to
r^ COOO '-'In.
rrco 00ONC3N
On
t^ONONOON
Tf CM -H 00 >0
to CO '— ' CM CM
to00tN.-HO
00 »-H 00000
NO
TtCOfMt^LO
0<Mcort(M
t^ votocoo
to vo CM On NO
a\T-ir—'^y-<
to
oor^ 00t>^ CN
CO CO '-H 100
o.r^i^ CO
l-H-HTJ-CONO
CjNNOt^OtO
t^OOOcotO
00000 fO^
OCOONtO
ON CJN r-< CO CO
to On to NO
00 '-' CM CM
^^S^8S
COOOt^t^CO
to
CO^ONVOg
CM coCvIOOn
Onco coONrJ•
vqcM pNONO
CM
CO 00 vq rN.
W 00 CM C3N -^
t^ VO i-H "^
CO r>. to ^>; to
vdotoo"S
vot>.r>»oooo
uS »-h' t< Tt^
In." to CM '^
ggggg
cK CM* CO vd 00
Ot^ to to CO
CM CM CO ri- to
vd '-h' to Tt CJ
cd
OnOnOO'-h
CM CM CO"*- to
to r^ rj- <7n Tl-
tx On CM too
NO
CV4
voa\'<*-oovo
000 CM r^ ON
VOO-^CMO
NOtrjOCM
cot^oorfcg
cor}- coo
00 00 00 NO to
CO Tt CM 00
t>.
voi^cg (N Tt
r-llO ^ VOCM
CM cOCTvONto
ONto^torf
CNl
On tv, U-) NO "O
00 10 Tl--^ CO
to CO CO '-itN
00 CTncoOOn
^00 NO NO On
NO
NO '-/J ^ CNl !/->
ro CM CM eg
^NC^OCM
NO '-H CO OtO
CM to toco r^
NOON COQtO
to
VOON coCM -^
VOcO'-^OOtJ-
coNOt^^OO
OvCMNOC^NCO
OC7NC3NOOVO
CO vo CO OCM
CTn
I^ (VJ NO VO
O^TtNOCO
ONCOCM t^ ON
CO tN, t^tN On
Tft^CMOvrf
TftO CO NO 00
to
>0 ^ u-j CO
c^ NO"^coMD
co"^ CMtN --H
ooqoNCMC?N
rJ- ON t^ CO to
OOOVOtoOO
O'- ON CO NO
to
ot^^ot^
^-hO-^^
to ON Tj- CM 00
to NO CvJ vo
00
—I -^ CO rg t<
1-h' vd '-H* vd —J
X< CM* 00 to r-I
covdco'-I cdn
00 tN." c?N vd .-h'
'-^' On rf to cvi
OOOOONC3NO
O'-"-'CMc0
COTftOVONO
t^ CM 00 NO NO
1-HCMCMCOTt-
00 CM '-'Tf CO
lOt^ON'->rl-
^
'^'~"~"^'~'
lOVOt^CM-H
voioovo T-H
ocoooovo
Tj-CM coTh On
VOC3NOtONO
CMiOtorJ-iO
t— 1
t^CM \OTj-io
CO 00"^ CM IN.
tN cot^ ^ On
NOC7NCMT^t^
lOtN'rtiOTl-
tN,"-) OOt^ 00
rOco^ScMQ
0\f^ Tt-iOOs
TfcovOvOO.
to vor^ '-I NO
CM t~x NO T}*
coCMtoOOO
NO NO CO CO 00
^
CM'-"^cOTt
1-" coONOOt^
On vo ONO CO
00 t^ 00 00 to
NCtNCM co^J-
Q
CM cr5CMCO-H
to to to CO 00
CO to CO CVJ t^
0--VOCOT}-
COOn-hcovO
fv.
Tj-OO'-'ONt^
CMOOCMOOn
to vo On CM CM
On Oto CO CO
t^ONQOCO
NO to On NC ON
^ Tj-Oco^i-
CO
00 CN LO
u^ en i-N
CMCM—i — "-H
CM r^rt- vo CO
co rf 00 00
CM
Ocot^CMCO
NOlOt^ONTj-
qooocjoTj^
p 00 ON CM 00
vq 1-1 ON On CM
NOCMOC^^x
NO
vd 0< CM vd 0^
CO i< ^' 10'
tOCJNTf OtO
r-<" vd CM* On to
CM* »-H* ^s.■ Tt Tt
to On CO On NO
00 '-<' vd ^s! CM
I<
irjio VOVO vO
tN.rxoooooN
ONONO^J-"
CM CM coco Tt
■^^tot^CM i-"
Ti- to NO 000
CO
CM
rHr-Hr-4
^^^^^
1-H »— 1 CM CM CO
00 oor^ioO
Tj-00O(M"<1-
to t^ CO 000
QtOtOONt^
On tJ- to to to
^OT^oooON
cotooo^io
NO
CNONTj-O'"^
tN ^ cotN
tNTf CMCMCO
'—I 00 00 to to
00 00 rl- On
ONOrrt^to
NO
r^ocMOCM
CM CO On rj- NO
rN.J>^^ ONr-l
CM .-iCM VOto
OnCM VO^t-
NO
t^r^ "-I u-3
OvSoOOOCTN
I^ CO l^ CM CO
8167
8403
5097
8825
0184
r=.00ONCMr-H
to
vo-^iOCM -H
CVJ to CO vo CO
OtN coo 00
ONO tN to
NO
CM ON '"^-H CO
Tj-r^t-^00''^
OOnI^OO 00
lN NO VOCM t^
CM CM CO -^ 10
r^o>ocM CM
tOOOrJ-co
CTNTf^NO CO
COOVOOOO
T— 1
VOrl-cOcO'*
vOO^Ot>N
to to NO 00 CM
tNTf cocovq
CJNONtorN; ON
to CO CO CM t^
NO
'-H* rf t< CO
lOLOlT) VOVO
vdocotvio
COOOONCAO
to to to
CD Q Nd 00* 0(5
CO vo ON CO 00
0(3 0\* CO* CO «-•
CM*
vot>.t^t^oo
^ ^ CM (N)
TJ-^OOCM
NO
'— '
1-Hr-H^CMCM
corfto vot^
00
T-^ 00 CM 00 CM
'-ir^ONtNio
CO CO to rf ■r->
c:^ coto ooto
OnCMOCM^J-
ONtOlOtOCh
I^^iOCM>0
00 CM to C^4 t^
In. to Tf CO '—1
COCM O\C?n00
CMfNOO OOtN
to to NOVO NO
i<
ii-> 00 00^ VO
^""^ CO On CO
Osr^NOCM On
T- 't^otor^
VOlOOONTf
tN CA vot^ CO
to 00 '— ' CO to
CM
»-• r-x 10 Tf r>x
00 -^ CM "^ CO
to On On On
vo»-H coto vo
tO'-HtoOCM
CO
rf VOt^OO—'
OONCMTt-CM
CM(M -hOO^
OO^t toco NO
00 vo^ttNO
NO i-«l CO CM rj-
00 C?n<7nO
t>:
ID CM CM r^
CMLOTj- On"^
»-i COCO COCO
CTni-h VOOO
CO NO 00 CM
tN.
t^QOr^co
m 10 r~^
VOt^t^tO CO
NO CM 00 rf
-HOONOrf
On t^ to CO 0\
CONOtOT}-!^
00
TfCMT-H^CM
rj-NOOTT
t>xtOCOTt-tO
tNjOp coto
NO CO 00 CO CM
CM
t< c5 CM 10 t<
CO vd On CM*
to 00 CM to ON
SJ^SSS
CM* vd CO Tt^
CM '-< K CO On
^s[
Tj-LOU->tOLO
VONONOVOt>»
tNt^OOOOOO
'-H CO vo C7n CO
tv. rq tN rj- ,-.
CM CO CO Tj- to
^^^^CM
NO
Oco— tONiO
^CM-^^rt
cjnooonocm
ONt^COVOTt-
ON^oooNo
C7\tOtxOr:t
t^
CO Q CO VO vO
ONIN.CM OnO
ON On to CTn CO
t^tovotort-
t^ ^ cooOTi-
Orj-COWIN.
VO
CO ON t^O On
Tj- 00 -rl- to
CO 00 ""^ to to
1700
0714
9140
9808
8427
tNjfOOO-HVO
00
ScM^cScM
OOQO COOO
T^^NT^ONT^
Tf ^00 ON CO
t^coCMCTNOO
Tf
'-'OCvj— .0
'-<0'-' OnCM
rC 00 to to NO
CM CM cotN ^
Tj-
ONin vo»o
looNiocor^
00 00 On Tf to
rt- -tM^n to CO
^
»-i ONiOOOOO
NO CM NO ONO
p On In. to CO
On CM t^ coo
^C^OOt-"
'—1 On ONO
CM
vq o\ CM ^ ^
vo^ootofo
CM CM CM CO VO
00 CO 001^ to
00 CM rt rt- CM
t^
10 l< CM 10
t< CM to 00
:-^* Tj^ K CO
vd CO K t-I
rftoodtovd
'-H CM* 0(3 »-; CM*
]
•^T}- 10 10 to
10 vo NO ^ vo
I^t^t>s00 00
00 ON On OnO
OCMTftxO
rJ-OOCMOOrJ-
,-H
»— I
CM CM CO CO r^
to
O'-'CM COT^
10 NO tN. 00 On
O^CMcorh
to NO r^ 00 On
^r "^ Tf Tt >*
Oto Oto
too»ooto
t^OOOOCTvON
CO^OCOCOCO
CO CO ro c*3 CO
^^^^^
iOiOVONOt>.
348
TABLES
VOIOVO'-"
ovoor^co
co^Ot^t>^
ONf^ voOn^
COVOCOCO'^
OM^"^00 CO
COTft^NO'-H
t^,-iii-)CM
IOCOVQt-hIO
t^ NO r^ NO CO
r^i LO CO -^ 00
rfr^OONOON
^
ONTTCOVf
ooio orv^io
CO 00"^'^ CO
VOOVOOO-H
t>^CO t^ CO ^
COOOcOOLO
NO t-H OOOON
OCNlCNt^
<M<NONCONO
t^OONVOON
NOrf^r-lOO
t^ T-I r-l 00 Tj-
^
VO-^OCA
OO^co— 'OO
ocor^ '-HON
coo --H CO 00
,-iiocO COOO
^ in -rt y-> 0\
CONO OOOlO
irjt^vO'-H
u^OOOnOO
CO Tj-t^ COO
lO T-HtOOO ON
On TfLoCO -r}-
CVl
^>. CN) uo VO
Tj-O'^t^r^
OOlO ^LOt>^
00 00 NO CO 00
ONO\oqt>»
vq lo CO i-; On
t^ LOCO On NO
COOt^ COON
LO T-I ^s. CO 00
Tf ONTl-ONTf
O'-Jcvi CO
■rfiovdr>lt^
OOOnOO^
CO CO CO ^ Tj:
LO NO NO t< in!
oooooNONd
.-HlOt^ ^
COOVO '— 'CN
lo-^ ror^rj-
^ T-I fONOO
t^ lO C^l LO Tj-
fOONOTf t^
^mooc<j
cocoooooco
lOCOrfOON
CO t^ CO LOCO
NO NO t^ NO l-^
Q
^
lO 0\V0O
C^l VO VO ■'l" 'O
vo r-H COt^ON
lO '— ' CO '— 1 LO
C4 t^ CO ON NO
CO CO CO 1^ On
owooN-*^
lOt^Tt 00 o
T-( T-Ht^ON CO
VOCOOO NO On
T-H COLO 00 CO
CO T-H oco -^
O
CM
ONTj-OOt^
'^ Tf c^i '-' r^
00 '-Ht^ LOON
T-H NO NO NO 00
r^ rf cot^ o
CO rHLO 00 CO
t— i
t^rt-ON"^
ONTfOi^LO
NO On "^ COLO
CO rsif^ i^ CO
COOCOCOON
CO CO r-l l^ CO
«
t^co^oo
r^ii-).— ( Tfvo
NO-"^ '-I voOn
'-H »-• OnNOCO
NOOnOOOO
NO CO r^ o CO
M
ON ON 001^
NOlOTTCVlO
CO vorf-H 00
NO CO ON vq CO
ON LO CO CO CO
On LOONO r-l
Or-; CM* CO
'"i^ioNdt^od
ooonOt-^t-;
CO coco •^LO
lONoKtNiod
^2^^^
vOrH^O
^ONt^-"^^
»— 1 lO C^J lO tv.
O-HOOLOrl
TfNOOCOO
I^NOONONO
<
T-H ONCV) t^
loooo-^t^
oocoi^r^
LOCOOOCO o
CO --H CO "-H NO
T)-t^OO CO NO
w
rgocooo
ONO-^-^^
lo 00 '-' CO 00
CO On '-H .-1 CO
CO ONOOrfio
NOLor^co Tf
^
ON^OOCNl
COLO 00 r^
LOCO On 00 CO
Noor^coNO
CO OTt- OC4
LO COONt^OO
■^"^ Os-^CM
LO 00 CO CO (N)
CN) t^ OO O rj-
Tf CO OCO On
"^OOO CO CO
o
fNI
ro »-H ^lOCM
(N) NO LO 00 NO
ONt^ T-H CO 00
-Hr-IOOCv] CO
coT^vo-HTi-
ONONCOS
^ O r>. CNj o
oooot^^o
Tt l^ On On t^
lO '-H lo On 1—1
CvJ CO O 00 Tf
OM>.LOCOr-H
00 LO CO On NO
CO O NO CO ON
LO rHr->co 00
cJ^csico
Tj^iovdtNlod
00 ON O -h' CO
CO CO T^ Tt LO
vot<t<odod
On d d r-H* T^*
T— 1 T— t T— t
T— ( T-I ,— 1 r-l ,— 1
T— 1 T-H ,— 4 T-I T— 1
•-H CO CO CO CO
T-i
ooioco'5}-
00 CO ON tN. CM
T-i r-l ONt^t^
NO t^ CO CO CO
OOnOO^lo
NO NO -^Tt r-H
H
^
ONt^OlO
O vOcoOnco
On CO NO ON 00
TM^oor^r^
CO '-H xt-NONO
On NO 'sT t^ X^
s
<
^
OOOTj-CNJ
lOl^ r-lCM -^
Oq t-H onOlo
ONOCO CO NO
r-l ONONt^LO
r-H T-H NO rj- T-H
OOsOOrf
LOONTl- vo On
COON^Tt-Q
CO ON 00 NOLO
00 '^ NO rj- 00
CO CO CO LOO
<
^
OOMDONON
OOONMDOTf
CO Tl-LONOO
CO OlononO
CO LOCO CO NO
r^t^t^ ONNO
PLH
tH
I>nOO '^LOO
t-it^ONt^CO
CO t^O '-H o
OOLOr-HVO^
^
T— t
00 '^ 0\ CO
^TfcoOvO
oco«^roco
On LO On CO '^
lO"^ coo NO
OTj-r^oooN
1
ON On 00 00
t^ VO lO Tj- CN)
T-H On r^ LOCO
O CO to CO o
t^rt-— lOO'^
T-H !>, CO On LO
p[4
o
O t-h" CN) CO
Ttiovdt<od
On On d -^ C4
COCOTtLONO
NO tN.' 00 00 On
O o ^' -h' CO
>
»— 1 1— 1 1— 1
T-I 1—« T-I T-H 1—1
»— 4 T— < T-t T-I r-4
cococqcNco
t^CVJCVJiO
l^tN.vO00C<l
IOONt-ICO O
r-ILOi— iOnI^
ONCOt^lOt^
OCOnOi-hOO
H
VO^-*^
On^OnOco
LOJ>.C0 COLO
oo<^oo<r)
co^OnOOO
t^st^ cO"^co
coi^vTfr^io
M
r^t^ cou^^>.
Tl-I>vlOC0 '-H
OOVO'^-HLO
T-H T-H t>^ NOLO
^
(\j 00 O 00
-"tOOr-HCM^
VO r-H CO OMO
ao >-< o CO 00
CO NO T^ NO NO
CO coco NO O
T-H CO l-H 1-H t^
<
f\j 00 OJ CO
,— 1 ,-.lOLO CO
CO CO NOLO r-l
NO NO NO t^ O
ONOor^No^
H
S
irj»J-)CM Tj-
CO t^ 00 too
CO-Ht^^co
CO r-l l^ CO NO
00 o o o O
15
00 lOt-HU-)
00 On ON 00 vo
CO t^O fO '^
CO pONt^LO
"^ CO Ot^CO
NO O CO CO CO
T-H On NO CO t^
ON ON ON 00
t^ vOvoTt CO
CO T-H ON NO Tf
T-H On NO CO p
tN. CO p tN. CO
<
o
Or-Ic\ico
Ti^iovot^od
On d d i-H CO
COTf TfLONO
tN.' r< 00 On d
^-Hr-H^CO
d r-I CO co' CO
CO CO CO coca
oq oO'-'oo
Tj- CO in ON 00
r-l t>^t^ NOLO
CO t^ On CO On
COLO'-HTfr-l
LOON LOCO NO
W
CO<OC>»ON
O On 00 CO On
ONCO ONVOt^
On CO CO On LO
T-H OnOntJ-io
NO ONCor^Lo
Tj-iocor>.
lO ONlO -^ -^
coQcocJT^
OOOt^ T-( CO
LOCOr-nrtCq
LOCO 00 00 r^
NOTj-COt^rf
OOCOONt^tN.
H
^
lO'-H COlO
CNJ O ^OQLO
TtON'-H'^O
rH On NO coco
NO LOON T-H T-H
VO'-<LOO
t>^co^ooo
LOOO COr-iLO
LO CO On LO CO
CONOLO OCJ
CO CO CO LOO
t^ r-l NO CVl O
«
::5t
t^ NO CO 00 CO
LOt^ONOO
SR^R)2^
ONOnOCO'^
o
00MDCV)t>.
"-H Tf voNo vo
rj- t-H t^ CO t^
ON NO CO 00 Cv)
LOOOONOO
1— t
ON On ON 00
oqt^^u^rt;
coco pONtN.
NO "^ CO p CO
LO COr-^ CONO
CO O t». LO CO
CJT-Jcsico
Tfiovotvlod
On d T-; r-! CO
COTfLOVO^
K 00 On Ovd
l-Hr-I^T-HCO
T-H* CO CO CO rj^
CO CO CO CO CO
,-lVO'-llO
Tft^COVOOO
COLOt^l^rJ-
C0 00t»»00O
r^ CO "^ CO NO
OnO OnNOO
M
OOCvjiO
ONlOlOlT)
COrfiOt^iO
LOCO T^oo
LO r^ Cv) LO LO
ON r-H CO r-(CO
X^NOLO i-H CO
w
'-' vOtM:>»1>.
rJ-OOt^Oco
C^l CO r-H CO OO
CO COONt-hW
LO CO l^ COLO
W
On ON 00 vO
COt^ ONt^ t-"
OCsIt^Tf O
LOr^ LO NO O
iooot>,cx) CO
LOOO-^OO
^
OCOONON
SOOr-H
rt rl- ^ VOO
coNO or^r^
O 00 Cvl CO o
LO On coooco
"-H CO NO '^^ r>.
i^
COiOOO '-H VO
T-H t^ IJ-) CO CO
LO l>^ C^l CO NO
LO NO OlO CO
CO lo On NO LO
Qj
ONt^-^O
lO O^, CO to \o
t^ NOLO coo
NO ^ NO ON CO
■^ LO NOLO Tf
Cq ONLOr-H NO
HH
ON ON On On
00 In. tN; NOLO
Tf CO CO '-H p
00 ^s. LO CO CO
p CO NO Tt CO
or^Locoo
O '-< 04 CO
■^uSvdtNlod
On d t-h" CO CO
CO '?}-LO NOl^
00 00 On d r-H*
CO CO CO ^ LO
1^
T— 1 T— 1 T—l T-H T— 1
»-• T-H T-H CO CM
CO CO CO CO CO
M
00
en
.2
rHCVlfOTj-
io\ot>»ooo\
OrMCOco-^
ioNor>»oooN
COCOCOCOCsl
lONOtN^OOON
CO
009
«— 1 1— H T— 1 T— 1 1— t
CO CO CO CO CO
COMPOUND INTEREST; OTHER COMPUTATIONS
349
CM Ti- .-H oor^
o >^o o\ '— I CO
ON CO 00 eg r^
O ^' -h" (VJ (NJ
COO'-' o-^
t^t-H OOco-^
.-'Ol COVOCO
lO vot^ooo
Tj- lO >r) Tl- CO
'— '1^ On cot^
lOOcO Omo
OOC»MDtJ-
lOCM nO^^On
l>. CM O -^ ■'^j-
r^ .-I NO "^00
CM NO O ^ CO
ONOCMNOO
,-, -rf CO T-* xn
u-j to trj NO Nc5
CM CM CM CM CM
VOCMiO^-it^
OONOvot^T^
CO C3n CM COON
CMNOCX)iONO
0.-H Tt'-l CO
C0'<^t^c0'-'
COLONO t^t^
OOOOCMOOO
NOCMTfOON
T-H ONNOt^NO
CO On NO IT) 00
CM CO 00"^ t^
NO '-H ONOC^N
cOt^C7\ON00
00 On c5 ^ cm'
CM CM CO CO CO
TfCMVOOOON NO
"^tCMOOTfOO CM
Ot^TfOO O in
Tt r-Hr-H «Ot^ O
r^oocMt^'-' 1-H
CM T-i NOm On rl"
CM voOnNONO t-H
r>. Tt p NO T-<^ NO
COTj^lOU-JNO NO
COfOCOCOCO CO
in NO On t^ 00
COOO'-'t^'-'
On r-i r^ONNO
CM CM r^ CM CM
coo NOLO 00
u-jr^t^ t^ NO
Tj- -"^ CO 1— I 00
Vq T-H NO T-H U-3
'-H CM CM* CO CO
CM CM CM CM CM
t^C?\00 00NO
irjiOOO '-I NO
ONNO»J^O\t^
l^ NO NO On 00
lOTfcoCM CM
Tt On CO NO 00
O'^ONCOt^
^TfOVOON
CMt>^C7N^NO
CM r-tCOCOON
CMCOOCOO
Lo T-i CA Ot^
COiOt^CMt^
O\ON00t^ ■'^
NONONCJKt^
CM CM CM CM CM
OO0Ntv»0\
inirjLot^ ,-t
a\rf CM t^o
^Tj-NO'^t^
'— ' rr OOiO NO
lOTj-lOONlO
'-it^CM^O
T-( Tj- 00 '-I lo
t^OOiOvoCM
CMcooOQOn
NOTfCM^Ti-
ONlOlOTt 00
CO vo On COO
""^CMOOOCM
cor^Ti-00 00
00 CO t^N. On p
On ^ cm" CO "S
CM COCO COCO
lOCMCMtOCM
lOLOCOiOt^
li-) ,— I rl- CO »— I
lO Ocovo 0>
rj-r^ comvo
NO t^ 00 00 00
On CO NO 00 On
coCTn-^On"^
cm' CM CO CO "^
CM CM CM CM CM
COOO '-" Oco
cOTfrfNOOO
c^cM coooo
OOOQOnOCM
c^oONO'^o
On Tj- On Tj^ On
•^XOlONONO
CM CM CM CM CM
rtmoocor^
CM -^u-) coo
On On CO CM CO
t^ 00 On NO NO
lOONTl-r-i On
»J-J ONCONOt^
cot^CM vo O
t^QVOl^»-<
OOOOnioo
OncOi-hCTnOO
lO T-H OOt-T^
r-H COlJ^T-l O
OCMNOCOCM
On O\00l^"^
TfOOCMNOO
ONCMOOmON
OOiONOiOCM
to t^ NO lO On
OOOOONOt-h
NO t^ 00"^ NO
COTj-Ot^CX)
CMr^NOC7\C7\
'^^ T-H W ,-H Tf
'-I corf NO tv!
CO CO CO CO CO
COO\C?N
com CA
TfCOON
1-H^OO
T-HinCM
t^Tt ,-H
NOt^t>.
odo\CJ
cocoTt
NOTf Tt
T— iLO NO
C?\CM T-H
CM CM IT)
On O CO
NCO 00
00 00 On
mco O
t-h'cM CO
<«1- O t^ T-H CTn
rO»0 Onu-)00
C?\NO00C?Nt^
■<^t^VONOt^
00 00 00 ON CO
lOONCOl^CM
CO 'O Tt O NO
1-H t^ CO CA Tj-
CO CO Tf ^' vo
CM CM CM CM CM
OCOtOt^t^
ooomtoLo
T-HCMTl-T^OO
tOiOrJ-OOCM
CM t^ QCM NO
t-^CM 0\NOrt-
Si
NONOt^t^OO
CM CM CM CM CM
ioC?nnOCMt-i
lO 00 CO CM CM
CNt^i-HtoCM
CMcoOMDt^
OCOnOnOO
CM CM CM CO CO
T-H T-H T-H On ro
NO CO 00 CM NO
CMTfCMT-H o
NOt^TfOOO
CM Tf On 00^
NO NO 00 CO T-H
NO T-H lO ON CM
0^ '<:*• 00 CM t^
O r-J ,-H* CM CM
CO CO CO CO CO
VOOCMOOl^
rj-mmt^NO
ONIOIONOCM
OTf oo<?\0
CM "^CMOOn
T-HirjcoOt^
Tj-CONO'^t^
^T-HChNOT-H
roiONOodo
CO CO CO CO"*
t-HTj-ONt^lO Tf
t>vt^t^CONO O
t^'^'^OCM CO
t^coOT-HCo t>x
-^OnOVOOn Tl-
t^ CTmo T— I tN. ,_
OOt^NOlOTj- NO
lo 00 p 1-; T-H o
t-h' CM rf lo NO »<
Tf "t rt Tj- Tf Tj-
T-H CM Tt CM ON
OGOI^rfTj-
00 vo 00"* 00
CO "* covo CM
00 T-H ^ ONt^
lo Nor>> 00 T-H
T— < r:J- NO r^ 00
p NO CM 00 Tf
Tt Tt lOVONO
CM CM CM CM CM
OOT-tOcoOO
lOCO"* VOOO
r^r^r^^CM
OnOOCMvoOO
vo NO "-H O vo
VOOt^ VOTf
t^ NO CO O NO
ONOCNOO CO
tvlt^ododoN
CM CM CM CM CM
OOntM^CO
CMt^OvoCM
voOOt-hCM
Tt NOVO CO CM
OOONOCMNO
VOOOTfi-HO
OOcoioioOn
T-H 00 On NO CO
t^ ON T-H CO CO
COOO OvvoOO
CO Tj- T-H vo NO
CM NO CO CM Tl-
lovovorf CM
lO Ovrjo vo
CM* CO CO Tt^ ■'^
CO CO CO CO CO
t^r;^c7N00co
opooo^oo
CO ,^ 00 NO T-H
OO^NOOOt^
vo'^CMrv.oo
Chj-^Ot^'^
On CNj crj CO T-H
rt t^ C3N T-H* CO
co^^coTf*
■«tO\00
COrl-T-H
8 CO CM
CM CM
NOcot^
cmtj- r^
vo vo NO
5^
TfO
vno
CO
r-)
r^
g^
;^
CM Tj-
vo
ONO
,_!
Tl-VO
lO
lOTl-
NO
COTt
CO
Ot^
NO CM
0\
CO
CO
NO T-H
S:^
o
coOootOTh
CM ^T-H '-HTj-
nOOtJ-CM NO
OONt^ NOVO
ON CM CM ON Tf
OOOn^tI-O
OONOrfONO
(50 vOCM ONvrj
Tj- vo NO M3t>I
CMCMCNlCMCM
CMt^CTNCOCO
CM CO vo CO 00
00 t^ CM T-H On
vo NOOOO T-H
00 CM t^ vo to
t>^t^ OOCM 00
OTfr^O'-H
CM 00 Tj- ,-H t^
OCJodONOO
CM CM CM CO CO
NOCMrJ-ONO
T-H CM rococo
CO vo oco vo
CO CO T-H t^ ON
On 00 CO ■* CO
NO t^ T-H t^ NO
CMCM CM OOO
CO On vo >-H NO
T-H '-H* CM* CO CO
CO CO CO CO CO
»OCMCMt-hO
CMCM -^ONt^
OOCMTj-ONO
NO 00 CM 00 CO
T-H OONO-^vo
OOCM O^vo
VOCM 00 cot^
CM 00 CO On Tf
Tf Ti^ vrj vn NO
coco CO coco
Tff^NONCMCNj
t^ NO t^ CM NO
vo NOt-h ovo
t^ 00 ON 00 t^
00 MDvo On NO
^^^IZJ'^OCJN
T-H p CO CM NO
pvqocoTf
InI C?N CM Tf NO
CO<*^TfTtTl-
r^ NO 001
CM Ovo'
OI^Oni
t^ VOVO'
CTnNOO'
rf CO T-H 00 Tf C7\
00 O CM* CO lo NO
TflOlOlOlO lO
CMt^r^toON
CM CO-* vo 00
00"^ On CM vo
OOO 00 ON NO
t^CM VONONO
t^ CM On On CM
O*N0 00O
00 vo CM ov ^>:
toNot<i^od
CM CM CM CM CM
{Tn^-^-OOO
SOCTncoOn
vo On CO CM
00 O ONO CO
vo vo VONO O
00 t^ On* CO
OOOnOOno
*-Ht^rl-^
On O O T-H* CM
COCOCOCO
i-HCM-^NOCM
T-HCM i-H Ovo
NO On 00 00 CO
00 00 OOvo
nOn3t-hO*
Tf OnOOOvo
CO ON vot-h vo
OO^-r-HOO*
CM
TfOOONO^
*OOTft^
00 NO On On 00
O CO On vo t^
voCMNOOnO
■*t^ CO CO 00
OnCM vot^xOO
Ot^COC^NVO
NO NO t»i ^s.■ 00
CO CO CO CO CO
COVOr-Ht^lO
»or-;"*t^co
I^CMoOt^*
T-H On CO O T-H
T-H T-H ovo vo
NO fi VONO 00
(?\*vocM NO
T-H T-H C7N NO T-H
C^NCMrttviO
CO**-* in
O'-'C^cO"* lONOtxOOOx Q'-hCMcO* iOV0tN,00O\ 0»00i00 lOOtOOiO o
fococococo COCOCOCOCO '*'*•*■*'* *"*■*"*"* iOvovOVOt>. t>. 00 00 On CTn O
350
TABLES
Q
O
Ph
w
u
o
Q
"A
< <
Ph ^
^ S
W l-H
H ^
<
o
W
O
W
00
CO
009
CO tV.lO 1— <
ON CO TfT-Ht^
looorfvoco
ONt>.ON00C3N
CM CM CM 00 CO
VO ON Tj- 00 rj
CM voOnvo
t^ CO Tf 00 CVl
CDiOOvON 0\
c^cM vo-*"^
OOuoOncovo
CM VOIRON to
»-H ,-H ^ (Nl O
VOCM '-IVO
CO Tj- r-H CO CM
t^-^ CO CM CO
— * VO i-H 00 t^
vovOrt-rf>-H
0\0n— <o
vo(^^ooc:^ON
OOt^rtcOOO
'^CMOO'-i
CM IN. 00 t^ to
to voco VOCM
^
cOfOO'-'
cococor>N VO
OCOOOVOON
O vOcoCM Tt-
VOOOOO»JO On
CM 00 CM VO l-H
OOtOCOCO
CO^-i tOT-il^
CO CO com
CMt^CM 0> r-i
CMiOI^t-xOO
ONTfr-H CO O
S3S28^
VO
Tt-cor>xVO
--ht-hOOOO
'-I Ot^CMtO
VO VO-* Oto
ON00VO-=t
C^l On Lo CM 00
CO 00 CO 00 CM
t^—'TfOO'-i
tM>vO coto
t^ p CM ""^ to
Ot-h'cMCO
rt TtiOVOvd
KKociodcK
OnOOCDi-;
»-5 T-H cm' cm cm'
CM CO CO CO CO
lOCOCO O
r^vo o vooo
CO CM-* ON Tf
TtVOiOOVO
Th^OOOOCTN
tv.OCMVOOO
ON'^OiO
vOO-^t^* VO
ON CVl VO C3N On
O i-O <M ON 00
OOCTNVOVOO
CO r^ to 00 IN.
to CO VO CM tr)
ooooo
VOCM roCM ^
'^l-M-'-'CNO
O CM CVI CO '-H
TttOCOt^CO
in
OOi-HTfLO
r^ONi^ '-' CM
CO'-HtOt^-"^
irjVO VOOOCM
CM '-'Otovo
■^OOCOCMt^
COtJ-CMOn
TfvocoCMOO
^^T;^CMlo VO
VOt^OiOCO
ON-HOr-HO
CM ONCOiO
CMO VOCOI>^
T-H VOCOCOC50
ONt>v^CMO
CM -H CO 00 00
CO to CO 00'-'
iOU-)CM Tj-
CMt-xOO VOO
CM O VO OOn
tN. cot^OOOO
VOCM VOOOOi
CM>. Tf C?N ■*
0\00t^>0
COOt>^Tfr^
t^ CO 00 CO 00
COOOCM VOO
't^OO'-H'^I^
OCOVOOO'-H
O^'CMCO
TtlOlTJvdK
KOCJOCJONON
O O t-I i-h' CM
CM* CM CO CO CO
rf Tf rt^ Tf to
Ovomo
-■^oo^t^o
00 CM 00 CM 00
COIOCOOON
tOOO VOCDnI^
VO to CV1 '-' CO
^
oorN.coi^
t^ -^ C7N O VO
-HONt^rl-(N
t^O^OOiO
rf 00 1^00 CO
CTNTf OOtoto
tN.r^rl-iO
VOOJOVOO
OOVOOCMiO
U-5 to .— 1 T— 1 CO
VOCO^'^OO
00 '— ' CM CO 00
:^
COVOVOCM
l^ t^ O CO 0^
r-H ,-1 ooiocg
;?;s2^8^^
COCM CMl^tN
Or-HOt^OO
CM voco 00 00
On VOOMO
o\ooi^ cor^
rv. On to 00 00
OM^ ■* tv. Tj-
Tj-
vocMoor^
On t^ C^4 lO 00
CM 00 00 CM CM
CATft^ONCO
t^Tl-Tft^lO
OOVO^CN'-'
irjt^Tj-CO
OOLO On On VO
CO CO O to On
OO00T^C7^
C^Tj-tOTf CM
C3N00t^iO
CO .-; 00 "^ CM
ONio^^vqcM
t^CMt^ .-H to
O'^t^^'^:!-
00 i-i rt t>. p
CDi-h'cMCO
T}-' Lo liS vd tN."
tN^odcSoNO
O ^ -^ CM CM
CO CO CO TJ-" Tt
rfiotoiovd
\
VoKcoCM
covot^t>>'-.
O0r-< voioco
co»-Htof^O
rttocot^rf
Tf 00 to CM CO
■^NOOCM
COOO vooovo
r^t^x^oo c^
TfVOOOONTj-
coOnco VO i-H
ONr-Ht^CMVO
CMvOt1-tJ-^
lONOcot^CM
t^voOO VOOn
VO Omo '-H CO
CJN <?N to CO -rt
COiONONON
CM CO lO-^ CO
ONt^l^TfCM
OOONVOCAco
(N) 1/-J ^ Tt VO
t^ VOOOVO T-H
^
lolOOCO
OO'-i Ol>. CO
OO-'^fOVOT-H
cO(M VOCM On
CO ,-.^00 ON
Otv.toOtN
^KO"-JO\
'-I CM C^l CM U-)
O O VO lO CO
CO cvi to On CO
oc^ l-H VO VO
CM CM On CO CO
-*
VOPOt^CM
lOT^ococo
'-CVOOOOOVO
1— ito votoco
ON CM to to Tf
CVJ 00 CM VO 00
Tt CM o i>: "^
T-Ht^COONlO
^NO—iVO'-H
to p Tt 00 CM
vqcAcovqoN
Oi-TCMCO
Tt»J^vdvdt>I
0C5 00 CTn On O
r4 -r^ CM (Ni CO
cO'^'^rfto
T-H T— ( 1— 1 1— ( 1— t
toio vd vd vd
»— I .— 1 .— 1 l-H 1— 1
tN.OOOO'-"
OOCM 00"*T-H
CM-* CO ON 00
O^OncoCM
oo-^t^o
On VO »— 'to O
tOCM tOOOO
lO CM On CM
COOO\LOIO
cOOCOTfCM
lOr-HTJ-OOO
c?N 00 to r>. rt-
CO CM 00 ■* VO
rOTf vOC?\
CM CO coi-O VO
OvoOi-Ht^
cOrt-rfOt^
Ti- cv) -^ 00 r^
^
. 00O\cot^
O '-0 Lo On VO
OVOCOCOCM
i-^ l-H CM 00 CO
or^CMr-ivo
'— 1 to VO »— ^ NO
NOlOCOt^lO
Tj-T-H fOVOOO
rJ-ON^Tt-co
tOCOCOOt-^
^
voOi'-i CO
lOOO-^ cot^
VO -^ CO CM O
r^n-^cTNON
CM t^r^ooo
'-I OtOt-NtO
vOONOt^
r-HCM^t^O
;^8^^S^
i— ' ONtoOOO
^ONVOCMtO
00 On 00 voco
CO
ON 00 00 VO
lOCOr-i OOVO
to O NO T-H t^
(N| NO i-H vq p
Tj- 00 CM VOO
CD t-h' o4 CO
TtirjvdvdtNl
OCJONONOO
'-< CM CM CO CO
Tt Ttiotrivd
vdvd In! t<OC)
0\0i00
Ont:1-voCACM
Tt l-H On CO Tf
00 '-I ONCOr-i
ONcor^oO'-t
vOTtTj-ONCM
c^NCMtNcocn
r^t^ coTj-
-HTtC3N-HON
OO-^O'-H
00 '-I voco '-'
vOrf rt CMto
co ON l-H oO
l^ t-HCM CM 00
CM ■* CO to CO
to CM COCOON
rf Tt VOOOCM
r^CM '-' OOrf
OC^nOOOnO
O CM O lO t^
coo-^-^c:jn
t^CM-^O-^
^''t COO|^
00 ^ *o o
t^ r-(CM VO'-'
CM VOOC3NO
on^ r-(tor^
Tt-OONVOtO
^
ocooot^
0^r^0 ctnvo
OCM -<^rfvo
t^i-H VOCOCO
t^tovo coto
covor^s'^oo
l^ ,-H ro ^ 00
coiou^ CO Ov
CO NO VO to CM
f^i— 1 CO-^co
r-H r>>. CNl VO CO
CO
ON ON 00 r^
lO Tj- CM p t^^
u-5 CM ON vq CM
ONtOi-jt^CO
OOTfCJN-^ON
T^oocot^'-'
Ot-4cmco
"^ lO vd iNi t^
OOOnOnO T-H
'-< CM CO CO rt
Tt toio vd vd
t<K0d0C3ON
T— 1 •^ COl^
VOOO VO '-' VO
vovovo VO-*
CM to On '-'t^
COCMr^t-nO
tOCM COOOO
Oco^ 00
OOt^O^fM
.-^CMVO00.-H
C3N r-H NO VO to
T-H-HOOVOtN
C^NON^rf O
CM
vC^CMt^
.-HVOOOTfCO
VOOCOO 00
OOCOt^ VOr- 1
(M VOOnOOVO
X^ to ■rfvot^
CO CM VO CM
OOVOO '— 't^
r^ voo '-' o
CM^ OVOQ
to'^ C7\ t-i On
1— Ito r^ CM to
CM "^ CM "^
IJ^ CO "^ CO VO
OOCM OO
VO to >—< l"^ CZi
CM C^N^t cotN
CO CM On 00'-'
COOCM On
CM CM On -^t^
O C^4 Tf t^ 1— c
"<|-00OO On
vo-rf toco VO
t^CMcoONO
OOCMCMOVg
OpOOtoCD
C3N •g; OC) CO 00
t^CMTfco
.— 1 VOOO ONt^
too co-^ Tf
CNl On -"^ t^ O
On On 00 t^
vq^cMOoq
vOCOr=;00^
i-; 00 Tf p vq
CM !>. CO CO "^
c5'-H CM CO
rj- irj vd t>^ tN^
00 ON CD O -^
CM cm' CO Tt ""^
totovdvdt^
bvlodoCJONOv
.2
0)
Ph
i-tCMCO'^
lOVOt^OOON
0»-'CMCO'«1'
to NO t^ 00 ON
o^cMco:^
tOVOt^00O\
CM CM CM CM CM
CMCMC<ICMCM
COMPOUND INTEREST; OTHER COMPUTATIONS
351
lO o\ 0\ »-i tJ-
t-H Onco VO'-"
,-Hioroa\.-H
coOOrfCM Tt
OOOOCN »-•
VO C^^00 CO vo
t^ONOCVJ CO
CVJ ONt^OO
OOOVOVOO\
CA CM CO rf n-
Tj- vot^OO On
CM ON 00 VO CM
covoCNCNt^
OOCOOOi^
m'Tf CTn Ot^
trj IT) lO irj lO
vooNr^t^oo
OCNt>*CMCO
VOrf CMCMTf
00i-OTf«— <iO
^O^Ov"*
VO ONVOCX)00
t>; On '-H_ CM CO
Ot^OC^""^ CO
t^ CM CM "-^ CM
OOCTncoOnO
VO w^ COCN (M
ONi-HTft^OO
CMOlOr-H^
VO 00-^ NO CO
^COCvJ ooo
O'-'CM coco
coco'Tt-Tt-Tf ■^Tt'^TfTt- u-JLTJioiOLo toirjioirju-j ii-3irj\dvdvd vdNOvdvONd vd
'-H r^ NO Tt Ti- CM
OOONOOCM NO
T^ cOTf On CO Tf
00 '—I On NOON m
itjOOOOOO t>.
u-> O "^ t^ O y—
rt-iniomNO
NO
coOt^'-' '-•
OiONOCM O
'-I OnOOn-^
to r-H r^ T^t O
Tj-oONO»o On
CM CM CM CM CM
t^ONOCDCh
coiooqo r-H
vo irj u-j NO NO
On »— ' -^ '— ' t^
CMt^corv.NO
rf^r^CMO
Onio 00 On'^
^ OOCMOOO
■"^ NO I— I t~^ t^
t^Tf^NO'-i
CO lo tv. 00 o
NONdvCNdlNl
IT) NO OO 00^
CO OMO ONCO
NOt^ t^ '— I CO
ooNOOr-r^
OCOCM ONt^v
On"^ CO to cm
ioOnCM -^NO
^ CM "^ lO NO
i< ^N.' ^sI t< t<
CMOr-HCMCO
oovnr^oot^
ONNOlOI^ .— I
NO NO »-i in CN)
TfO'-Ht^OO
1^ CO 001^ NO
t^oqoNO»-t
t<t<t<odoc)
vo NO CM CO in
■^ ONin CO NO
in T-i onono
CM t^ 00 IN. t^
CJNrf-CM ONO
incoc?\^CM
mcoCM NC't-
CM NO CJn T- 1 CO
odododc?NON
inoOcoTfCTN O
CTN'^CMt^in CM
OnOnO'-hO O
NONOt-iNOON y->
ONTfOOCMOO On
rj- NOcoCMin t^
OOONOOinO Tf
Tf in NO t^ 00 oq
On On On On On On
Tj-lONOtN VO
inONOOO ON
OOOOCMtN
OQ OnOnNOio
00 CO 00 00 tN
00 Tt 00 CM NO
00 Tt OOCM "^
CMint^OCM
QOOCTncoCM
CMOONOm
'— I Tf CO ONin
OOCM OnNO
T-HNOCM OnCJn
NO NO NO -rf CM
"<:*■ NO 00 O Csj
CM OMnONin
TfTj-r^CMO
"^OnOnOco
OOOrl-T-HOO
in »— iinCM CO
^NOCOTj-OO
ONOCMtN '-I
"^ini^ooo
VONOVOt^tN tN.tNtNOO0O OOOOOOOOOn
CM rJ- n- rr CO
Ttt^OOmi-H
t^OOONCOO
^t^OOON
CO CO In NC CM
NCOO-^inT-i
in 00 »-i com
T-; CM -?t in NO
On On On On On
OOtNTfCOtN
tNinO'-i 00
tNOCM ON'-i
OCMCM tN ^
O OO Ont-1
CMOOOOOCM
NO Tt com o
tN CM NO On CM
On O O CD T-I
'-I CM CM CM CM
Ocoin
NO On CO
corto
CO"-^^
NOCOi-H
comin
OVOOn
TfmNO
mo "^
tNCO tN
OO CM
■^O in
CM 00 00
C?nCM On
CTnOO -*
tNOO On
CMCMCMCMCNJ CM
ONOOtNNO
comm NO^N,
CO co»-«mi^
coChm-"^ ON
Or^mNO'-'
CM OOcot^ T-i
CjNOOtNTt^
CMinCO'-H rf
KtNtNOcJod
comocoTt
CVl OnOOCMOO
CO '-H 00 -"^f "*
'-^OOtNNOOO
NOCMmOO-^
Tt-OOCMtNT^
NOO'^MDOO
NOON r-;COm
00 00 On On On
00 1-1 -^Tf On
OOOOtNOCM
CO '-< NO '^^ »-H
tNincM On""^
tNOVOWOO
CM comoOO
c?NONCorNTf
i^C?nt-h com
cKonooo
T-H,-iCMCMCM
tNin
OnconOOCM
comco cotN
ONOC7\'-iTt
O'^CMint-H
CM00r^ON■«^
tNOOO'-'CO
O o i-J -^ 1-h'
CM CM CM CN) CM
CM00t>.C7N00
NO »-' ON CTn On
TfCMON'-'Tl-
OC^OOOO'-i
T-1 NO ■«:t NO in
CM 00 CO NO -^
00 O Cvi Tt On
-rf T-H NO O CO
'-' OQ CM CO CO
CM CM CM CM CM
Tj- in NO On
CO 00 '-'in
oOT-i^r>.
O On cotN
rJ-comCM
OmoOlN.
OO'-'ONO
NO On r-; CM
COCOTt^-'Jt
CM CM CM CM
ON O
in o
in On
m On
CM CM
.-HNOtNOOco
"* tN T:t- t-H CM
lominoo "^
'>?^tNNoooo
OCMOOCMNO
CMNOOOOQ
OncOnoCTnO
COtNO cotN
00 0(5 CJN On On
O '-'CM NOtN
'-' 00"^co00
.— I comt^C^N
NOOnCMOO C7\
NO-^moTj-
OOO'-'CM
80NtNTfO
CMmoO'-'
rJ-'-i'-'OCM
cotNOotNO
C<l coCM 00'-'
t^ O 00 CO On
Ot-i CONOt^
in On -^ CM CM
inCTN CONO 00
comOO OCM
^' ^' r-^ CM cm'
CM CM CM CM CM
NO CO omo
cvi T-H 00 CM m
OOOtN"^"^
m ,-c co"^NO
Tf o-rJ-CMin
On O ON On In
""^tNOOOCM
CM CM CM CO CO
CM CM CM CM CM
IN CO CM NO"*
00 CM '-"-•NO
tN CO Tf On NO
^incoTfCTN
NOOtNOOco
inrf Tft^O
mNOTi-^O
Tf-CM ONino
OOtNCO
NONOtN
COmco
OOl^O
NO IN 00
NO 00 NO
O'^co
Tfin o
inrj- m
i-iO
com
C3n CO m
tNOO in
CMTf NO
CM
mONcooooo
COrJ-intN NO
'-'OOin '-'NO
"sl- CM NO C?\ CO
rJ-TftNt^OO
OQOOm'-'
OOOONOCO
NO O COIN T-H
tNO'^CTNCO
O m "^in »— I
OCMm T-Hio
CMincoNO'-'
CM CM CM"* CM
tNCMt^CMOO
OOconoOnO
Tf 00 '-; * 00
'-^ '-I cm' CM CM
CM CM CM CM CM
t>.tN.OCOCM
0\ 0\CM '— ' On
T-f OnOnCM CO
tN ONinotN
tN CO CO ON CM
rj-CM '-^ '-'*
'-"-'OOOm
'-'Tft^ONCM
Ttr^co"*co
inooOMDON
CMCTvtNNONO
i-^^OOm
tNTtrNtNNO
OOmrJ-NO'-'
'-'tNCM ^O
mtNOCMin
NOmco
"*t^CO'-' '-1
NOCM NOOCM
tNTt-mOOrf
a\"*mcM CO
CMlNlNinCM
tNtNNOrj-'-'
inNoKodON
CM CM CM CM CM
00 in CO Tt
CVjTfO'-'
NOCO'-'t^
CMNomo
00 t^'-'*
^O'-'CM
OOCOO
tNCMNOO
ON CD d-H*
CM CO CO CO
CM "*
On CO
in in
in o
NO ON
CM 00
OO CO CM CM
CO CM CM CO'-"
r- 1 in NO CO IN.
OOOCM coo
comcooOTj-
OnOOOO NO
TfCOOOO
CM NO r-cin ON
CMinONO
OtNNomoo
•* '-"-'O*
CO CJN* '— ' CM
coCTnnOcoO
<7\ NO CO CTn *
CM NOO COtN
VO'-' CTnCOI^
OONONOO
T-H ,— I omin
O '-' CM NOtN
'-' CTn O CO rf
00 NO NO 00'-'
tN OCM CO*
0*tNOco
ON'-irf*in
OOOmCTN'-i
CM CO 1—1 OCM
^>.tNc:^ CO On
*m ctn CTNin
NO COCM * CTn
COCM OtN CO
NO 0\ CM * In
mm NO NO NO
CM CM CM CM CM
ciOON'-'mt^
C^Nt^OO*
OnOOCvJ NOCM
NO NO NO NO In
'-I CVI NO ■* CO
tNin CM C50 On
On OC CM CM '-'
On T-i CM '-' CA
noocJoncdo
CM Cn) CM CO CO
OOOOOnOnt-i cm
inOsOOOtN CO
mOO*NO CM
m*rj-min •*
ctn C3N * '— 'in o
C?N(M ON OnQ r->
O >— ' CO On O in
NOCMInt-^nO CTn
O'-hCMcO'* mvotNOOON Q'-"CMcoTf m NO In. 00 C7N omo^no inomcm o
cococococo cococococo ^■*"**'* ■***■■*■* mmNONOtN t>.ooocoNON o
352
TABLES
Q
O
i
w
o
Q
w
w
w
H
H
O
H
H
O
3.^
H
»-4
o
Q
o
t-t
«
00
CO
COO
^
:^
^
t-ht-hOO COO\
00 '-'00 CNJu-3
cou-jo O VO
Ttrooo
m t^ MD
On vOiO
-^-^^
CO'— I o
»-«ooo ooooo
VO vo cot^ CO
t~^ Ov T— c CM U-)
00"^ t^oo VO
»J^ O 00 Tf CO
CM "-H Tf OiO
0\ O CM VO O
OOOOI^VO VO
ooooo
ooooo
VOOO t^
t^u-) On
lO r-H l^
lOlO ^
ooo
ooc5
OOCNJ
O VO
OO
OO
do
cOcOt-HOOCNl
•-H CO VOCOOO
r^ t^ vo vocN
Tj- 00 rfON ^
'— it^ vo vo On
On VO TiJ-Cvj O
CO CO CO CO CO
OOOOO
<d) (Z> <:5 d> d>
c^^lotN»co^
ONt^xOOONC
lO 00 ^t^ T-
l^NOt^OOC
CM t^ COO
C^ t^ \OLO c
CM CM Cvj CV) r
poooc
OOOOco r-cvOiJ^CMO
tOiOON CMOnCM'OI^
t>vTj-oo oxfo^'-H
co-rfr-i OcoOCOOO
TtONt^ CMiOiO-^Tj-
Tj-U-).— ( r-H t^ CO LO -— <
ONCM-^ ONmco^O
TfCOCM T-H r-l ^ r-H ,-(
icJdd ddc5cDc5
ooonovoo
VD -^-^ 00 CO
t^ NOt^ VOCM
00 CO ^I>s NO
CM '— ' tn Oio
O »— I cot^ "-H
ON 001^ NO NO
ooooo
CM CO
lO NO
00 NO
t^NO
VOOJ
U-J LO
OO
(DO
ONOCM
CO CM 00
or^-H
Tft^NO
c?NNor^
CO ""3 CM
'^'^^
ooo
ddcD
t^CM '-ir^co
O t^ CVl On CM
Cvi lO 00 o o
rl-r^CM t^OO
»-H t^NONOOO
Ol^iO CO 1— I
Tj- coco CO CO
poooo
cDoddczJ
ONTf OOiJ^T-
ONCOOOCMO
CO CM CM C^l O
CM r^ CO oc
ooor>xNOT
CO CM CN) CM r
ooooc
cDodcJc
Ot^to On^noO"^
lONOt^ coOOOnOOtT
On Tt CO COLOr-HONlO
Tl-LOCM LOCM'—iO'-t
or^NO .— iioioLOLo
lONOCM CMOO'^NOCM
OnCM-^ OMO CO '-H o
T^COCM ^T-^T-^r-^r^
ic)cDd d)d>d)di<:o
COTj-OtOt^
m ON NO CO On
Nor^CJNOO ^
CM t^iOT-iO
CO '— tlO T-i NO
'-'CMTl-OOCM
On CO t^ NO NO
ooooo
dddcDO
rv. CO
^^
lO o
CMiO
00 NO
t^co
mm
OO
dcD
Tf OI>»
00 '-it^
OnCM r-H
NOOOO
ONt^t^
ON NO CO
ooo
odd
CMt^
NOTf
lOOO
'-H t^
l-HOO
ss
ooo
T:f '-' r-1
l-HOO-H
coNor^
NO NO 00
nOtI-CM
CO CO CO
ooo
rt roCJNtxV
Tl-CMONO^
OOn coOnO
CM ovON oo^
CM NO Cvi On b
'-C OnOOnOu
CO CM CNJ CM C
ooooc
ooo ooooc
lONOtN. CM
On-^co rf
CMt^CM '-H
NONOCO CM
NOLOiO '— <
u-jl^ CO CO
ONCM'* C7\
TfcoCM '-'
NO ON CM CO
i_oio On '-H
CMOOJOO
CM CO TfiO
Loioiom
C?\iOt^ CO
lOCO "-I O
'-•OOO OOOOO
rf00t>xCOCM
CO CO r^ CO NO
too CO CM lO
t^ CO '— it^lO
CO CM NO'— I NO
C^J CO lO On CO
On 00 1^ NO NO
ooooo
OnOOcoCM ^
CO to CM On NO
t^ONNO'^O
t^ CTn '-I Tj- CM
OONOOt^OO
OOTj-^t^Tt
I o m u-3 Tj- Tj-
ooooo
CM "^00 NO lO
CM NO CO On NO
»— ' "^NOt^lO
Ont-(U-)00 00
'-'OO NO NO 00
CM ONf^mco
rf CO CO CO CO
ooooo
CM OON '-•^
m Not^mo
ONCMO'-''"^
cmoonoon:
CM t^cM ONr~
CM 00\WNC
CO CO CM CM C^
ooooc
ooooo ooooc
CM NO 00
ONONt^
t^CMTf
t^oo-^
CMCOTJ-
NO 00^
ONCM'^
TfcoCM
id do
CM -H NO (N CM
COCM '-' ooo
On LO NO T}- On
OOCMmOOO
CO lOLOlO NO
'^ONOOOTf
Onno cO'-i O
00 Tt On NO CM
»—• 00 On coco
Tf- coOnOco
CO On t^*-^ CM
rl-CM NOCM t^
cOTfNOO'^
OOOOO OOOOO
NO CO
COO
rt-NO
CJNt^
CMO
vom
OO
Nooot^
NOt^^'^f
ONIJ^OO
t^ ot^
ococo
CM oom
OOO
rro
lOtO
n-No
CM 00
COO
OO
COt^ T— I
coOrJ-
OCOCM
OO^'^
CO CO CO
OOO
lO NO I\ 00 Oj
"^ On CM O M
CO .— t IT) r-l 0>
NO CO '-I O bi
CM t^ coOb
co^ OONb
CO CO CO CM C
OOOOC
.-•t^CM ^'-iCM'^iO
rJ-'-«0 •-• CO r^v-* ID
Tf'-i'-i CMcoOOcoO
OO^ NOcoOOcot^
OOCMCO OiOiONONO
VOCTnlo uO'— tt^ONLO
ONCM-^ CJNNOcO'-iO
TfCOCM ^r-lT-H,-!,-!
ococo _
CO 00 00 T— iiO
ONOiOCM O
to CO r^ CO 00
■^ LO r>» T-H IT)
ONCOt^t^NO
ooooo
NO CM CO On 00
Tj-r^CM t^rr
NO NO O ■'^LO
CM '^noOOlo
ooo T-toO On
,-H NO COCTNNO
NOLOLO Tj- tT
OOOOO
On 00 00 NO lO
CO -^ CO NO NO
or^ CM NO NO
CM CO t^ On 00
CO ONt^r^ On
"Tf »— ' ONt^LO
rl" -^ coco CO
OOOOO
r^ONt^fOo
■<*CMt>.VOC*
CMivsNor
CM 00 NO'
cor^ coOo
'^CM'-'OO
CO CO CO ""^
ooo<
ddd<5c
Tj-'-iO\ Ot^COONNO
Tt '-< O OOCOCMCMCO
CMCM-H OnOOOOOO
^ CM 00 co-^CMC^NTj-
voOCM OLONONOt^
t^ONO NOCMOOONO
Onco-"^ OiNOcoCMO
rj- CO CM ,-1 r-H ,-4 »-H ,— !
iddd <od)<Z)d>d>
00 00 ON CM l^
o ot^oo »-"
CM T^ 00 '^j- '-<
COLO-'^'-'O
LO "^ 00 -^ On
LO NO 00 CM NO
ONOor^t^NO
OOOOO
<Z) cS CD <d> d)
OOONOlolo
tN^NOOOt^
CO'^OOCM r-H
CM -^moOLO
'—1 ON CM On O
CMt^'^OCO
nOlololo rf
ooooo
c5d)<:6<od>
'-iioCMTj-t^
cor^t^oorj-
LOO COLO CO
'—I CO NO 00 t^
""cfOOOCOO
LO coOCOt^
rl- -rf rf CO CO
ooooo
5i
loooco , ^
r^ooLo-^c
Nooom^ij
ONOTj-NC
'^OOTfiFHC
LO COCM '-'C
CO CO cococ
OOOOC
mNOt^00C>> OOCMCO"* lONOt^COON O'-iCMcOrJ- iT) no t^ 00 C
^^^^^ ^^^^^ CMCMCMCMCM CM CM 04 CM C
COMPOUND INTEREST; OTHER COMPUTATIONS
353
txON00O\M3
t^ CO ^iT) O
iv. t^t^ 00 o
ooooo
000C)0
oooooc^in
to U-) On r-H ,-(
LO»— I OOVO
oor>. vovoio
covo vooot^
CVJOOt^OOCO
COVOOq r-H CO
00 CM t^ CM r^
TfTfCOCOCNJ
CVJVOON
VOCMO
^^^oO'<:^
On 00
ON'*
to CO
OCM
r-to
ooooo
doodo
ooooo
ooo
oo
dd
vOONOrOCVl
O '— ' "^ ^ »— '
oOTj-coTj-r^
lomioooON
CNJ VO COCNJ CO
ooot^vom
•-H OOOO
ooooo
ooioo
tOOr-H
covoco
lOCM On
vOOtT
Tf ■* CO
888
odd
0\V0 00
ooo 00
OOt^ '-H
CO-"* T-l
O^O CO
COCN CN
OOO
ooo
Tfoiorgio
coco^cviio
ONCN -^r^vo
Onu-j t^vo On
NO vo vor^ 00
COCM r-H O On
CM eg (N eg ^
ooooo
'-H CM CO CO CO
CO CM rj- ir> Tj-
rN.io vot^to
00 CM OCMOO
O COVO ONCM
0\ 00 t^^^
ooooo
d>d>d>cid)
OOt^Tj-Tt T-H
COOONOVOO
tN.Oco C0C7\
rv. OLO coco
<0 '-tu^ OiO
lO lO Tj- Tj- CO
ooooo
lO'-it^NCOON
OCMOcot^
OOChi-HCM T-H
vor-Ha^ooc^
o vD '-<r>»co
coCMCMi-n-H
ooooo
cf>d>cidic>
vOOrooOOO
cooocot^io
00 "* to 00-*
i-ir^cocM o
O coo CAO
'-•C^NCO vovo
t-H OOOQ
ooooo
lOt^OO
lO cot^
CM ^0\
CMiO C7\
lorj- CO
ooo
ooo
VOOO Th
CMt^ ON
1— I o to
T-i Ov Cvj
too r^
coco CM
ooo
ooo
d>ci d
CM to 1-1 CO t^
CTncovOiovO
OnvOO'OOO
Tf On —I 00 >-•
VOiO^ VOOO
rJ-coCM '-lO
CM CM CM CM CM
ooooo
(DcSddxD
T-<iO00t^'«*
CM 00 i^"^'-'
CM CM VOO'-"
OcoOCMt^
OCM lOOO '— '
OC7\00t^t^
ooooo
c5d>cicic:>
lOOOC^iCO^
t^OOCM OOn
lo T-Hi^ On t-^
lOt^ '-H 00 00
to ON Tj- 00 CO
vo^J^toT^ rj-
ooooo
ddddd
CMCMCMTfvo
VO'* ON 00 ON
On cot^ T-' CO
QtO 1— I O O
0\Tt OVOCM
CO CO CO CM CM
OOOOO
»— • t^ tN. Tf in
CM coC^vCM NO
CO CO r^ NO t^
CMtTNOCM vo
00 '-'t^ NO NO
i-iOOOt^^
r-H r-H ooo
OOOOO
00^^
Ot^CM
to O CO
to NO NO
oO'-^in
lOlOTf
ooo
ooo
odd
CMCM Tj-
o o t^
NOVO CM
"^ 0\ o
OLO CM
Tj- CO CO
OOO
OOO
OO
lOioCM On CO
r^O'-<t^ NO
ONOoor^co
CMt^t^Ttt^
NO to to NO t^
lOTtcoCM '-H
CM CM CM CM CM
OOOOO
<:5d>d>c>c>
CMt^t^OON
00 OtoONC7\
OtoCM ONCO
to t^ Tl- "Tj- C3N
On— irM^O
OOOnOOOO
CMCM ^ '-ht-i
OOOOO
c>>ot^Noo
Ot^iOVOt-i
CM'-hOvoOO
r^oocvi 00 1^
Tj-OOcot^CM
t^ NO NO to to
ooooo
'-I CO NOON Tj-
CM Tt CO NO CM
coOOOto 1— •
ON CO 00 NOVO
t^coOOTfO
'et Tj- CO CO CO
ooooo
dic5ci<oci
^ONOCMO
CTNCMCOtoco
CO'-' CO ON ON
t^ NO to r^ 00
NO CJn to CO CO
CM O C7\CX)I%.
ooooo
Ocoto
t~v.ONr^
to O CO
Tt-CM CTn
tOOO'-H
NO to to
ooo
ooo
ooooo ooo
Orf O
NOrf (50
f^ON 00
Tl-NO Tf
NO'-H tX
'^Tl- CO
ooo
ooo
d>c5 d
OnOO-^On
'-^co^Tj-OO
cor^t^Tt-NO
NO to to NO t^
NO to Tj- COCM
CM CM CM CM CM
ooooo
coot^coco
NO^t COi— I NO
COCM '^ NO"*
CO to t-H r-H to
(?nt-h Tj-r^ o
'-"-iOOnOn
CMCMCM r-i ,-H
ooooo
ONONOIOOO
r-l OCM NO CO
t^r- Tt TtO
CM CONOCM >-•
Tt-OOCM t^CM
OOt^t^NONO
NOtoCMOCO
f^CMTl-Ot^
ON^HCOtO-*-
r-H tOOt^NO
r^ CM 00 CO ON
lOtO"*""* CO
OOOOCOtMO
NO '-H ■* Onco
ooooo
d>CDd><0<D
ooooo
to CX3 CO >— I »— I
cO'-i O O\00
ooooo
CMCMVOcO'-H r^
t^ COON ^-H 00 to
OOOCO'-H NO o
O'^C^NCM '-H t^
cotoOOcoOO CO
r>vNO tOlO Tj- Tj-
88888 8
dxz^dc^d d
"^CMt-inOI^
to T}- ON 00 00
00 ON l^ t^ CO
t^oo^oo
NO NO NO NO t^
t^ NO to ""^ CO
CM CM CM CM CM
ooooo
^COO
»-H cor^
i-ctoCM
to NO CM
CA ^ rf
CMCM '-H
CM CM CM
ooo
CO to
00 NO
On CO
r-H to
t^O
oo
CM CM
oo
d>ci
»-<CONONOt^
T*NO ONOtO
r-lOON-^tO
CMCM Tl-OOO
n-oocMt^'-i
ONOOCOt^t^
CM to NO to CO
'-Ht^Ot^NO
Q^-* OtO
0\ ^ NO CO'— I
NO CVl t^ CO ON
NONOtOtO "*
CO to CO 00 "-I
NO Tj- On NO"*
t^'-i C7NCM c^
i-itoooNo '-"
^^^^^ ^^^^^ ^^^^o
ooooo OOOOO OOOOO
ooooo
d>dd>cic>
ooooo
(Oc5d><5c>
ooooo
lOCMOOVONO 00
CMtoOTt NO CM
CO NO 00 >-< CO Tj-
CM Tt-Nor^Tj- t^
'-I CO NO Oto O
001^ NONOtO to
88888 8
ddddd d
»-< CO C?\ Tl- t^
'-I t>. 00 Tj- ON
OOtOOt^ON
Tft^t^CM CO
t^ NO NO t^ 00
cor^ NO to Tf
CM CM CM CM CM
ooooo
ddddd
OO^iOOQ
NOcoOtO VO
CO-^OO'-H t-H
O '-•NO NO ON
OCM'^-t^O
Tl- COCM '-"-"
CM CM CM CM CM
ooooo
ddddd
OCM cot^'-i
NOONOCOTf
to »-• tot^ ■*■
to to t^ CM O
n- 00 CM t^ CM
OONONOOOO
CM -H^^^
ooooo
ddddd
lOtOl-lrt■r^
ot^'-ioor^
tOt^ '— I CO ■*"
OCMl^co '-"
r^CM t^ CO On
t^ t^ vo NO to
ooooo
ddddd
CO t^ to t^ CM
t^ CO-* NO 00
CM NO Tj- NO CM
'-I CM -^CTn CO
tot^ CM On C^
to COCM o c^
'-I —t '-"-•O
ooooo
ddddd
On to 00
ooQC^
NO 00 ON
OCM to
ONCX)t^
Soo
oo
odd
NOl-l Ti-
CO to to
OtO NO
ON CO 00
NO NO to
OOO
oo o
dd d
Oi-iCMCOTf lONOr^OOCTN O'-<CMC0Tf
CO CO CO CO CO CO CO CO co co ^ ^ ^ ^ '^
tONOt^00C3N OtoOtoO tootooto
'*'*'^'*'«* totoONotv. ^N0000ONON
354
TABLES
ON'-hON
OcO(M'<:f -^
VO rl- CO r-H ,-4
vOTj-O-rfvO
mdlo r^oo o
CM LO r>«. to 1— t
oooo-=i-
Tf vOOON<M
OnOn Oi-i On
r^ l-H OOLO 00
lOLO LO Tf o
Tf-rfLOOO On
t^ CO '-^ LO NO
VOOn^
VO CN) to lO C<l
i>.rsir^O'^
CMCMTf-voo
NC^t^OslON
COOC^
C>>VOCOCOCNl
VO OM^ MD CO
voLorrLocg
OO O TM^t^
CM o OA ON r>s
^
Tj- i-HLO
CO CO T-H oo
oor>.CM ONLo
On ONTj-co^O
1-hOOCM^
CM On NO LOlO
cT
10-^00
t^co 0\'— 1 1^
LOVO OnCNII^
CM OOLO CM On
r^Lo CO l-H c?\
00 NOLO Tf CO
VO
00 ^CNl
tv.^T-|O00
t^ ^OLO LO -rj-
"^ CO CO CO CM
CM (M CM CM '-H
o
.Tj-cocvi
OOOOO
OOOOO
OOOOO
OOOOO
a
^ooo
d>d>d>d>d>
<5d>d><:5d>
d d c3 d o"
<:5d>d>c5<5
d>d>d>c^d>
P4
OOiOOO
ot>>.rvi^oo
00 o^l-Hr^^s.
C?M-H Tj- CM »-•
On ^ l-H CM O
NO CM NO CO T-t
00 Tf ooooo
Lo00Tj-r>.ON
CM On ^ Cvi o
LO l-H LOCO On
Tl- CO OO LO lO
■^ 00 1 — 1
T|-l^O\-HO
Tj-OOLOLO CO
CM OnOn^lo
CM VOONO O
CM "^ 1— ' CNJ VO
O O '— '
r^ »-< '-H c\j c^
OOOCMloCV)
Tf VO ON Tf -rt-
-^^ ONt^cot^
LO NO On CM "«}■
<:5
OOCNIO
OnOOOI^vO
iOCOOOtJ-O
CO CM NOLO r^
<M ON c:tn T-H Tt
On LO CNj ,-H o
w
t^t^CM
Ot^CN^O
0\OCM VOi-H
MD CM OOLO CM
Of^LO-^CM
O On X t>»NO
OO'-HCO
00 Tl- (MOON
t^r^ vololo
■rt-^ coco CO
CO Cvi CM CM CM
Tj- cocg
r-i^^^O
OOOOO
OOOOO
OOOOO
OOOOO
o
T-^dod
d>d>d><:5d>
d>d>d>d>d>
c5d>d>d>d>
<:Dd>d>d>d>
cid>d>d>(6
Q
vovoio
TfONt^lOt^
<N00 ONLOCNI
1— it^OO Or}-
■^r^inoNco
cor^NO^»-M
w
^
lOCOMD
^cOTj- vo-^
00 '-H l-H coco
CC CO LO ON CO
i-HioNOrJ- O
O CO Tf-OOVO
t^ COCO
-•OO-^CATJ-
CO 00 VO LOO
CO LO i-^ vo r^
vooLoc<ir>.
On^OnO^
;?2
ONt^Tj-
OM^ OOt^
t>»cox^\ou^
t^Tf\OJ>sC\l
l-H l-H l-H coo
r>. O"^ooco
CO CM ^Cg l-H
w
ONt^r^
COCVI VOCM OO
l-H OTj-CM Ti-
00 VDlovOOn
^ Ot^LOrJ-
^i-
00 00 CO
C^OOtJ-vOCN
-H CM rf 00 <N
ooTfor^^
l-H CTn t^lO CO
CM -H ON00t>x
w
00-^ CO
OO^i-CvlOON
001>. VOLOLO
Tt- Tj- Tj- CO CO
CO CM CM CM CM
CM CM '-' ^ '-I
^ _
.Tt tOC^l
^^^^o
OOOOO
ooooo
OOOOO
OOOOO
"-^diQd)
o c5 o" c5 d
d>d>d>d>d>
c>c5c5d)d>
d>d>d>cSd>
d)d>d>d>d>
c^
o5)0
^0-Hf0 0\
Tj- rj- t>^ CO t^
OOCM coCM
lOl-H l-H VOCO
vooOTj-ooeo
-hONVO00O\
onOi— ir^ON
l-H OLOCONO
t-^i-HOOOCO
■■-H OCOOnMD
ON COLO OnCJN
H
^^s
t^ r-l ONl^CNI
l-H OOOCOOO
l-H t^ 00 CM On
<
CNJ VO OCM On
ON ^ LO ^ VO
Tj- CM On On CO
COCOOnOOO
l-H NO CO l-H tN,
{fX r.
^
1—* CO '^
vOt>^ vQiO-^
O\00 ^ On T-H
lOCN '— 1 COLO
OLoCslOOO
l-H O
OOlo
'f o ^oo"^
CO-^OO"^
ONLOCM OOMD
CO^ ONt^LO
Tf-CM^OOO
P-.
-*
OnCVJ CO
00 LO C^l o o\
oor^ vo voLo
rj-rj-^coco
CO CO c^j CM r^i
C^J CVl CM CM »-"
^TffOCM
^^^^o
OOOOO
OOOOO
OOOOO
OOOOO
r-^d>d>d>
ddddd
d><:5d>d>ci
d>d><zSd)d>
<d>d>d><5d>
d>c5<z^d)d)
>
w
K <
OnoO'*
1>»^ONLOt-<
tv.t^iot^ro
t^ ^Ti tn -rt- t^i
OOONt^Oro
'^O'-'inco
>_J
u
coCNTt^O
coONONtor^
O 00 '-H CO CO
OLOOOOOO
O'^'Tj-NOCO
pq
•^ J
o com
^OOTif^'O
^ ,—1 CO l-H o
LO '^ CO vo O
l-H \OCMOOCM
rJ-LOCMCMLO
K -]
^
oovc^r^-^
Ti-ONOO^r^
(M 00 Tf r-H T:f
NO CO CO l-H t^
t^OLOOTf-
<
TfONCNJ
■^ \0 >-0 "'^ "^
CNJ 0"^OLO
LOMD OOCN VO
OOt^ VOVOLO
OO^OCO On
cOOOnO CM
NO CNl 00 NO ^
H
^^
^
VOCM OOO VO
-Ht^Tj-Ot^
lOCOOCM^
LOTi-CM-HO
On CO CO
OOLOCM '-' 0\
LO Tj- Tt rt CO
CO CO CO CM CM
CM CM CM CM CM
CO
-^COCM
1—i T— I ,—( l-H O
OOOOO
OOOOO
OOOOO
OOOOO
r-^d>d>d>
d>d)d>c5d>
d><zic^d>d>
d>d><:5cid>
<z>c5>d>d>d>
d>c5d>d)d>
'^^O^JO
t>sOlOO^ vo
'-iLOONTh^
LOTtOLOCO
OOLOCOOOQ
LOOOLOt^OO
r-HOOONOCM
!z;
00 coo
lO to CO CO 00
t^ t^ CO On •^
00CN|CM(MVO
tv.OOrfcO'^
^
oot-^
tM^vo^co
Ot^CN On^
VOOCM 00 CO
LO^ t^ COt^
'-I co(M
lO ONOLO CO
cor^ vocM eg
VO-^LOO-H
l-H r^ xt- l-H -^
CM CO NO ON »-H
<
vOloo
COIOIO'^''^
CM O-Tj-OLO
t^vo ONi^co
CM OOt^ OOO
Tt ONLOCM»-t
^
C<lcoON
OO^OCMCO
r^ooo'^oo
CO ON LO CM ON
t^TTCMOON
t^uT^coCM
0\CN> CO
OOiOcO'— 1 ON
oor^t^ vOLo
LO "^ -rf T}- CO
CO CO CO CO CM
CM CM CM CM CM
«
CO
.^COCNl
^^^^o
OOOOO
OOOOO
OOOOO
OOOOO
o
Q
y-*d><z^d>
d)d>d>d)d>
d>d)d>d>d>
d><:5d>(od>
<D<5d>d>d>
d>d>d>d)0
ipcOON
Cv) CO t^ LO m
CMONr-^CMt^
t^OvOcoCM
cOt-<OOco
t^ NO NO 00 to;
CMT:^lO
cO00rfO\ON
t^CMt^LOLO
tv.TtM--HMD
CJN l-H t^ CO CO
Ph
^
00 (N O
ooot^r^o
OnVOOOCM -^
On t~>« l-H o 00
'-HONNOTJ-OO
On 1-Ht^r^ON
l-H COCN)
ONt^ ONLO t:}-
cooo^cocg
LO 0\ COCOf^
cOTfLOt^OG
^
rqcoON
CMuT^^rf
CM O^OLO
t^LO On ^t^
1-HOOVOt^ON
coco-^-Hc:rN
o
COT;J-a\
Onlo 1— 1 coo\
00 On^loOn
oot^r^ voLo
■"^ O ^coO
OOLO CO '-H ON
OONOLOTj-CM
^
C<1
0\(M CO
OOLO CO^ o\
LOLO T^ Tj- Tj-
CO CO CO CO Og
CM (M CM CM CM
Tf coCN
OOOOO
ooooo
OOOOO
OOOOO,
/"N
g
in
'-^dcJd
cDd>d>d>d>
d>d>d>d><z^
d>d>d>d>d>
d><5d>d>c^
d><od>d><^\
00
en
O
^CSICO"^
lOVOtxOOON
OT-iCMrOrJ"
lOVOI^OOOv
O^CMCO'^t
lovorvooON
CO
T— t t— < T-< T— 1 T-<
»— 1 1— 1 »— 1 1— ( »— t
CM CM CM CM CM
CMCMCMCvlCM
609
p^
COMPOUND INTEREST; OTHER COMPUTATIONS
355
OnCVI roONTt
OOCN (VI (Nl 00
CVl ^ — <OC7N
ooooo
OC>OOCJ
ooooTf^r>^
coTt-r^ooco
t^ OMOu^ On
On coco ro 00
oooot^r^ vo
ooooo
ooooo
OOCJOO
TtvOCMCM vo
inoo-^ ^ o
'-H OOcorO VO
VOLOOO ro O
Tto^<^0
,VO ^ LO VO lO
ooooo
OOOOO
ooooo
Omoovo vo
to 00 VOvQiO
o -^t^t^ fO
O'-HTt-ON^
ooooo
oworvi vofo
CVJ Ost^vO'-i
Tf \Ovo OfO
tT cot^ ONfO
TfiOOOcoO
roCM ^ T-._
ooooo
ooooo
ooooo
r^rf^vooo vo
voiooofom fo
oocvj vooot^ r>^
VOt^C^ T-H CO t^
80Q00
ooo ~
oooo
ooooo
Tf CM rvj Tjr to
»-iCMOOiO
»Oco00O\»O
O^CVjTtt^
lO Tf CO eg >-•
'-"<oo\<^rM
t^Tft-xCVl\0
'-1 ''I- ON -"^ Tj-
IN» corOOO ^
OTfCOCvit^
'-1 OOnOnOO
OOOOO ooooo
ooooo
ocJooo
voOn'— irom too T-ir^\n rfvoootoio »— i rvj vo »- ' fO
^CN)r>vco(vj t>v(Nirvj'rfvo r>.00'-i^^ vo^i— "-'O
OOCM-^roNO '-'OO'^OOOn vo nO 00 ON On ^ On ro rv Q
r^CMONON^ vocMt-H^co t^NOCMOOON cgcMOCMOv
CVJOOcoOnvO CVJONVOcoO t-^VOOO'-'^ fOOOOvOTt
OOt^t^NONO "^ \J-) xn xn in tj- ro CN CM »- 1
ooooo ooooo ooooo
ooooo ooooo ooooo
ooooo ooooo ooooo ooooo ooooo o
TfvoOfO'-H
tOTt CMioON
'-'fororr^
ONTj-NO'<^00
COTflOt^ 0\
VOlOTf COCVJ
ooooo
OOOCJC)
lOOOCMONt^
''^•t^O vo NO
OiO -^ .— iirj
i>.O00OiO
CM VOONTf 00
CM -hOOOn
ooooo
c> o o o o*
lOOOOOiOi-t
»-H lO VC fO t^
CO '-I 00 CV) o
'^NOOOOOO
ro CO Tj- On «-0
On oocot^r^
OOOOO
CM '-H"«^OOCM
O r^ foio CM
CM rft^OOt^
OTf ooooo
CM OOiO— '00
t^ vo NO ^«J^
8 oooo
oooo
dddoo
iOTj-vorx'-<
r-HloCM Tfr^
CM t^ -"^ OiO
O OO^^fO^
NO fO"^tN. 1— (
uo Tl- fOCM (M
ooooo
Tj-O\rj-V0 a\ o\
O VO CO '-' On CO
'-C O CO fOt^ 00
CM l^ ONt^ ON lO
r>Nfoooovo lo
ooooo o
ooooo o
c5<ocSc^(zi d
OiOO\t>.t^
.-. coioirjt^
O^OOO cOTj-
coiOTj- O '— <
00 00 On ^ CO
t-^ NO VOIO -^
ooooo
<:5<od)d><:D
CM 00 tN. CM fO
rooOioO\00
r^ voon •-I o
t^OO fO CO NO
lOOOCM no O
CO CM CM '-"-I
ooooo
ONOOOONTf
Tj-fOCM OOiO
fOt^ O OntJ-
CM '-I n-ooNo
to 0>J-) C NO
OOONONOO
'—'-^ OQ O
ooooo
VOIO OnioCM
TfOOONO'-H
CM CM r-OI^
NO 00 CM 00 vo
CM OO^O T-i 00
oor^i^r^No
ooooo
ooooo
OTfioOvVO
CM CM 00^ O
0'-"^OiO
lo fO O On Ti-
lO CM CM CO tN.
VOiO Tl- coCM
qoooniooo q
Q O Ot^ CO O
OvTj-csr^t^ 00
CM '-HtN.ooO O
CMOOtI-CM On 00
CM '-H ,-H ^ O O
O QOQO O
ooooo o
cSd>'^<:5<zi c5cid>ci<5 d
—•CM '-'CM ON
t^ r>. •'I- 1^ lo
CO CO "^ to t^
On 001^ NO to
ooooo
lovotOTj-m
COt-iCM "-H t^
00 '^ CO CM t^
ON 00 r-HOOOO
OnOvJ NO On CO
■«:1- Tf CO CM (M
ooooo
00 CM 00 On 00
CM CM CM CO NO
t^ 00 00 to t^
CM On ON CM t^
00 CM t^ CO 00
;ij;i^22o
ooooo
CO 00 NO NO t^
TfO-^o^NO
CO "-I owe-*
ti-5 »0 NO O NO
rfO NO CO 0\
On On 00 00 t^
OOOOO
ooooo
^-HCOCMvOtO
t^ CM NO CM On
CO CO 00 00 o
CO^OOOONO
NocM o^-i T^
tN» NO tOTf CO
~ " " o
o
<ziocD<od>
o oncm '— ' vo rx
'-'OONOOOTf CM
C?N rf NO t^to On
NOOOOOtOOO to
00 CO On NO CO •-I
CMCM '-"-H ^ ^
88888 8
ddddd d
vOfOCMCMVO
CM On NO'-" ON
On 00 NO ^'—'
'-' ONTt tDcq
O OnO '— I CO
^ On On 00 t^
(M— .^r-^-H
ooooo
On On CM '^ to
CM t^ NO CO 00
C3n CO »-' On CO
coo '—I to Tj-
tOOO '-H "^00
NO to to "^ CO
OO^t^T-ito
CO rj- NO '-H 00
CMCM '-'OOCA
NO'-H OnCTnCM
CMt^'-iNOCM
COCMCM '-H »-<
ooooo
ooooo
,—1 lO to l^ i-H
to CM Ot^ CO
OONO NOt^ r-l
r^ CO On to CM
O O On On ON
ooooo
or^ VD "-H CO
to O On 00 vo
to ON CM to VO
VO"^ corf CO
00 CO '-H r-H CO
oo^^ NOtoT^
88888
NOioovor^ r^
ON tN. to tOt^ vo
tN» ,— 1 vototo vo
NO '-1 Tl-IOCM Tf
vo '-H VOCM On vo
COCOCMCM i-H ^
88888 8
ooooo ooooo o
CM CO CO CO to
^ to NO to t>»
Tj".*- CM CM 00
00 to On On -^
OOOOOOON'-"
— 'OONGOOO
CM CM •-' ^ '-'
ooooo
to CM CO
■^ COLT)
NO'— ON
to T- ' o
CO NO ON
t^ NO to
ooo
Tj-NO
NO to
r^CM
CM NO
to Tf
So
d>cS
T-HQtO'-i O
to or^i^ O
'-' Cvj '-' 00 '-H
coi^ Tf CO vo
O'^ONTfC^
Tf coCM CM '-'
ooooo
CO CO 00
On On to
NOTf CO
or^ NO
to O vo
ooo
00 CO
tot^
t^ 0\
C^lOO
O 0\
'-' o
o o
CM CO CM
ONtO O
OOnO
Tj-t^CM
to ONt^
ONt^ NO
ooo
ooo
OOO
(M -H
^CM
ON'rf
NO 00
to Tf
88
QCMOtO'-H CJO
NOTf CM CM Tf 1— 1
tOCOTf ^ ,— I Tf
CO Tf Tf CM NO to
'-HtoOvOCM
Tf CO CO CM CM
8 oooo o
oooo o
d>d>d>cid> d
ON
O'-'CMCOTt iONOt^00C?N O^CMCOT^ to vO t^ 00 C?\ O to O to O to O to O to
COCOCOCOCO COCOCOCOCO T^TfTfTfTf TfTfTfT^fTf lOtONONOt^. t^ 00 00 On ON
356
§383.
TABLES
Table VI
Reciprocals and Square Roots
Square
Square
Ratio
Reciprocal
Root
Ratio
Reciprocal
Root
(1 +
(Discount
(Quarterly
(1+0
(Discount
(Quarterly
Multiplier)
Ratio)
Multiplier)
Ratio)
1.005
.99502488
1.00249688
1.02
.98039216
1.00995049
L00525
.99477742
1.00262156
1.02025
.98015192
1.01007425
1.0055
.99453008
1.00274623
1.0205
.97991181
1.01019800
1.00575
.99428287
1.00287088
1.02075
.97967181
1.01032173
1.006
.99403579
1.00299551
1.021
.97943193
1.01044545
1.00625
.99378882
1.00312013
1.02125
.97919217
1.01056915
1.0065
.99354198
1.00324474
1.0215
.97895252
1.01069283
1.00675
.99329526
1.00336932
1.02175
.97871299
1.01081650
1.007
.99304866
1.00349390
1.022
.97847358
1.01094016
1.00725
.99280218
1.00361845
1.02225
.97823429
1.01106380
1.0075
.99255583
1.00374299
1.0225
.97799511
1.01118742
1.00775
.99230960
1.00386752
1.02275
.97775605
1.01131103
1.008
.99206349
1.00399203
1.023
.97751711
1.01143462
1.00825
.99181751
1.00411653
1.02325
.97727828
1.01155820
1.0085
.99157164
1.00424101
1.0235
.97703957
1.01168177
1.00875
.99132590
1.00436547
1.02375
.97680098
1.01180532
1.009
.99108028
1.00448992
1.024
.97656250
1.01192885
1.00925
.99083478
1.00461435
1.02425
.97632414
1.01205237
1.0095
.99058940
1.00473877
1.0245
.97608590
1.01217588
1.00975
.99034414
1.00486317
1.02475
.97584777
1.01229936
1.01
.99009901
1.00498756
1.025
.97560976
1.01242284
1.01025
.98985400
1.00511193
1.02525
.97537186
1.01254630
1.0105
.98960910
1.00523629
1.0255
.97513408
1.01266974
1.01075
.98936433
1.00536063
1.02575
.97489642
1.01279317
1.011
.98911968
1.00548496
1.026
.97465887
1.01291657
1.01125
.98887515
1.00560927
1.02625
.97442144
1.01303998
1.0115
.98863075
1.00573356
1.0265
.97418412
1.01316336
1.01175
.98838646
1.00585784
1.02675
.97394692
1.01328673
1.012
.98814229
1.00598211
1.027
.97370983
1.01341008
1.01225
.98789825
1.00610636
1.02725
.97347286
1.01353342
1.0125
.98765432
1.00623059
1.0275
.97323601
1.01365675
1.01275
.98741052
1.00635481
1.02775
.97299927
1.01378006
1.013
.98716683
1.00647901
1.028
.97276265
1.01390335
1.01325
.98692327
1.00660320
1.02825
.97252614
1.01402663
1.0135
.98667982
1.00672737
1.0285
.97228974
1.01414989
COMPOUND INTEREST; OTHER COMPUTATIONS
357
Reciprocals and Square Roots — (Conclvuled)
Square
Square
Ratio
Reciprocal
Root
Ratio
Reciprocal
Root
(1+0
(Discount
(Quarterly
(1+0
(Discount
(Quarterly
Multiplier)
Ratio)
Multiplier)
Ratio)
1.01375
.98643650
1.00685153
1.02875
.97205346
1.01427314
1.014
.98619329
1.00697567
1.029
.97181730
1.01439637
1.01425
.98595021
1.00709980
1.02925
.97158125
1.01451959
1.0145
.98570725
1.00722391
1.0295
.97134531
1.01464279
1.01475
.98546440
1.00734800
1.02975
.97110949
1.01476598
1.015
.98522167
1.00747208
1.03
.97087379
1.01488916
1.01525
.98497907
1.00759615
1.0325
.96852300
1.01612007
1.0155
.98473658
1.00772020
1.035
.96618357
1.01734950
1.01575
.98449422
1.00784423
1.0375
.96385542
1.01857744
1.016
.98425197
1.00796825
1.04
.96153846
1.01980390
1.01625
.98400984
1.00809226
1.0425
.95923261
1.02102889
1.0165
.98376783
1.00821625
1.045
.95693780
1.02225242
1.01675
.98352594
1.00834022
1.0475
.95465394
1.02347447
1.017
.98328417
1.00846418
1.05
.95238095
1.02469508
1.01725
.98304252
1.00858812
1.0525
.95011876
1.02591423
1.0175
.98280098
1.00871205
1.055
.94786730
1.02713193
1.01775
.98255957
1.00883596
1.0575
.94562648
1.02834819
1.018
.98231827
1.00895986
1.06
.94339623
1.02956301
1.01825
.98207709
1.00908374
1.0625
.94117647
1.03077641
1.0185
.98183603
1.00920761
1.065
.93896714
1.03198837
1.01875
.98159509
1.00933146
1.0675
.93676815
1.03319892
1.019
.98135427
1.00945530
1.07
.93457944
1.03440804
1.01925
.98111356
1.00957912
1.08
.92592593
1.03923048
1.0195
.98087298
1.00970293
1.09
.91743119
1.04403065
1.01975
.98063251
1.00982672
1.10
.90909091
1.04880885
INDEX
(References are to sections unless otherwise noted.)
Account,
Amortization, §§ 198, 205, 208-212, 214.
Insurance, § 179.
Principal, §§ 164, 169, 179, 186-188, 202.
Taxes, § 179.
Accounts,
Amortization, installation, §§ 334-336.
Bond and mortgage loans, §§ 162-184.
Bonds, §§ 197-214.
Collateral, §§ 185-188.
Discounts, §§215-221.
Interest, §§ 161, 164, 165, 171, 172, 174-176, 179-181, 183, 186-196,
203, 204, 220.
Accumulation,
Dual rate for, §§ 326, 327.
Schedule of, § 122.
Amortization, §§249, 348.
Account, §§ 198, 205, 208-212, 214.
Accounts, installation, §§ 334-336.
Definition of, § 70.
Development of series of, §§ 72, 249.
Interest-difference, §271.
Of premiums, §§ 328-330.
Relation to present worth, § 71.
Relation to sinking fund, § 90.
Schedules of, §§ 121, 122, 126, 130, 134, 139-141, 281.
Amount, §§25, 26, 28, 30, 31.
Of ordinary annuity, §§ 53-61.
Of prepaid annuity, § 75.
Proof of by reciprocal, § 227.
Amounts of $1, table of, § 378; comment, § 367.
Amounts of annuity of $1, table of, § 380; comment, § 370.
359
360
INDEX
Annualization, §§ 301-310.
Annuities, Chs. IV-VII.
Amount of ordinary, §§ 53-61.
Finding, §§60, 61.
Analysis of payments, § 68.
As sinking funds, §§ 87-90.
Deferred, §§ 11, 78.
Deferred payment, § 86.
Definition of, § 54.
Due, §§ 75, 76.
Immediate or ordinary, §§ 74-76.
Instalments of, § 69.
Present worth of ordinary, §§ 62-73, 304.
Problems, §§ 232, 233.
Rents of, §§ 83-86, 373.
Problems, §§ 234, 235.
Tables,
How formed, §§ 56-58, 257.
Of amount, §380; comment, §370.
Of present worth, §381; comment, §371.
To four periods, § 63.
Varying, §82.
B
Base in logarithms, § 391.
Bond problems, Chs. XXI-XXVII.
Broken initial and short terminal bonds, §§255, 256.
Cash and income rates, § 243.
Compound discount, § 261.
Compound interest, § 262.
Discounting, §§ 270, 271.
Income rate, accurate, §§ 263-269.
Initial book values, § 247.
Nominal and effective rates, §§ 236-240.
Premium and discount, § 243.
Present worth, finding, §§258-261.
Redemption of bonds, §§ 287-294.
Semi-annual basis, § 300.
Serial bonds, §§ 274-286.
Successive amortizations, § 249.
Successive method, §§ 244, 318.
Tabular methods, § 283.
Varying rates, §§ 298, 299.
Varying time basis, § 312.
INDEX
Bonds, Chs. IX-XI, XVII, XXI, XXII, XXV-XXVIII.
Accounts, §§ 197-214.
As trust fund investments, §§ 148-154.
Cullen decision, §§ 152-154.
Broken initial, problems, §§255, 256.
Cost and par of, § 104.
Cost of, relation to net income, § 99.
• Discounts on, §§ 207, 214, 258.
Earning capacity of, § 103.
Elements of, § 110.
How designated, § 98.
Interest. (See "Interest.")
Investments in, § 119.
Investment value of, §§ 106, 108.
Irredeemable, §§ 146, 213.
Last half-year of, § 144.
Ledger for, §§ 200, 202.
Loans on, §§ 162-181.
Premiums. (See "Premiums.")
Present worth of, §§ 101, 103, 106.
Problems. (See "Bond Problems.")
Provisions of, § 96.
Purchase of, §§ 102, 128, 129.
Adjusting errors, § 129.
Rates,
Annual and successive, §§ 295-318.
Income, §§ 134, 136-140, 263-269, 287.
Interest, §§ 97, 100, 109.
Varying, § 150.
Redemption of, §§ 146, 147.
Problems, §§ 287-294.
Repayment and reinvestment, §§319-336.
Replacement, §§ 148, 322, 323.
Residues, eliminating, §§ 138-140, 280.
Schedules,
Accumulation, § 122.
Amortization, §§ 121, 122, 126, 130, 134, 139-141, 281.
Checks on accuracy, § 124.
Serial, § 145.
Problems, §§ 274-286.
Short terminal, §§ 141, 142.
Problems, §§ 255, 256.
Tables, § 155; comment, § 156.
Of differences, § 276.
361
362 INDEX
Bonds (Continued)
Tables (Continued)
Use in compound interest problems, §§ 257-262.
Use in determining accurate income rate, §§ 263-269.
Valuation of, Chs. X, XI.
First method, §§111-114.
Interpolation method, §§ 131, 132.
Multiplication method, §§ 133, 134.
Periodic, §§ 130-133.
Problems, §§ 243-318.
Rule for, §§305, 316.
Schedules, §§114, 122, 126, 127, 130.
Second method, §§ 115-120.
Values of, § 127.
Book, §§ 123, 125.
Found by discounting, §§ 143, 144.
Initial book, § 247.
Intermediate, § 105.
Market, § 107.
Various, §§ 104-108.
C
Capital,
Account, §§ 1, 3.
Cash, § 2.
Definition of, § 1.
Potential, § 2.
Sources of, § 3.
Use of, § 2.
Working, § 2.
Card records for mortgages, § 178.
Cash, § 2.
Characteristic, §41.
Collateral, loans on, §§ 185-188.
CompoundMiscount, §§ 33-35, 261.
Compound interest, §§ 13, 17-19.
Amount of, § 28.
Problems, §§ 225, 226.
Interpolated values, § 376.
Use of tables, §§ 262, 368.
Rules and formulas, §§35, 157-159.
Tables, §§ 359, 360, 378.
Use of logarithms in computing, § 49.
Contracted division, § 248.
Contracted multiplication, § 228, 270.
INDEX 363
Conversions of rates, § 92.
Coupons, §§ 113, 296.
CuUen decision, §§ 152-154.
Day as time unit, §§ 20, 24.
Day basis, 360 and 365 methods, §§ 23, 24.
Days, odd, how reckoned, § 23.
Deferred annuities, §§ 11, 78.
Differences, table of, § 276.
Differencing,
Discovery of errors by, §§250-254.
Present worth by, § 259.
Discount, Chs. XVIII, XIX.
Compound, §§ 33-35, 261.
Formulas, § 35.
On bonds, §§ 207, 214, 258.
Single, §§33,216-218.
Discounting,
Contracted methods for, §§ 270-273.
To find bond values, §§ 143, 144.
Dividends, § 7.
Division,
By logarithms, §45.
Contracted, § 248.
Dual rates, §§ 326-333.
E
Effective rates, §§ 91-95.
Problems, §§236-240.
Errors, discovery of, by differencing, §§ 250-254.
Evaluation,
Method by logarithms, § 73.
Of a series of payments, § 53.
Exponents, §§ 27, 38, 39.
Fractional, § 48.
F
Factors, logarithmic, table of, §358; comment, §§342-357.
Forms,
Bond accounts, §§ 197-214.
Discount accounts, §§ 215-221.
Interest accounts, §§ 176, 179, 188, 192, 193, 195, 220.
Loans on collateral accounts, §§ 185-188.
Mortgage accounts, §§ 176-183.
3^4
INDEX
Formulas for interest calculations, §§ 157-159.
Fractional exponents, § 48.
Frequency, definition of, in interest computations, § 10.
G
General ledger, §§ 160, 180, 182, 195, 200.
Gray's tables of logarithms, § 341.
H
Half-year, legal definition of, § 22.
Immediate annuities, §§ 74-76.
Income rate on bonds, §§ 134-140, 263-269, 287, 297.
Dual, § 326.
Relation to cost, §93.
Rule for determining, § 293.
Use of tables, §§ 257-259.
Increase, ratio of, § 17.
Initial book values of bonds, § 247.
Insurance account, § 179.
Interest, Chs. II, XVI, XVIII, XIX, XXI.
Accounts, §§ 161, 164, 165, 171, 172, 174-176, 179-181, 183, 186-196,
203, 204, 220.
Calculations, formulas for, §§35, 157-159.
Compound. (See "Compound Interest.")
Computations, §§ 15, 19.
Constant compounding, § 238.
Contract, essentials of, § 10.
Definition of, §§ 6, 9.
Equivalent rates of (annual, semi-annual and quarterly),
§§241, 242.
Laws of, § 8.
One per cent method, § 223.
Periods, §§ 20-24.
Punctual, §§ 14, 17.
Rates,
Coupon (cash), §§ 100, 109.
Diminishing, §§ 323, 324.
Dual, §§ 326-333.
Effective, §§ 91-95.
Nominal, §§91, 97.
Usual, on bonds, §97.
Varying, § 325.
INDEX
Interest (Continued)
Ratios, table of, §359.
Receipts and notices, § 181.
Register, §§ 180, 183, 192, 193.
Simple (single), §§ 13, 16, 19.
Problems, §§222-224.
Interest-difference, present worths of, § 271.
Intermediate dates, bonds purchased at, § 128.
Interpolated method of valuation of bonds, §§ 131, 132.
Interpolation in interest problems, §§ 339, 376.
Inter-rates, §§ 275, 375, 376.
Investment, §§4, 119, 150.
Absolute, §4.
Investment value of bonds, §§ 106, 108.
Investments, trust fund, bonds as, §§ 148-154.
Irredeemable bonds, §§ 146, 213.
Ledger,
Bond, §§ 200, 202.
Books auxiliary to, § 173.
Forms, §§ 182, 195, 200.
General, §§ 160, 180, 182, 195, 200.
Loose-leaf, § 178.
Modern, § 166.
Mortgage, §§ 167, 182.
Subordinate, § 160.
Life tenants, payments to, §§ 148-154.
Little's table of multiples, §361; comment, §356.
Loans,
On collateral, §§ 185-188.
On bond and mortgage, §§ 162-184.
Periodic payment, §319.
Uneven, § 282.
Logarithms, Chs. Ill, XXIX, XXX.
Accuracy of results, §§ 50, 51.
Application of.
To amount of annuity, §61.
To present worth of annuity, § 73,
Bases of, §357.
Characteristic, §41.
Division by, § 45.
Factoring method, §§ 343-346.
Finding numbers from, §§337-347.
365
366 INDEX
Logarithms (Continued)
Forming, §§ 348-357.
Multiplying up, §§ 353-356.
In connection with effective rates, § 95.
In connection with valuation of bonds, § 120.
Mantissa, §§41, 42.
• Multiple method, §347.
Multiplication by, § 44.
Powers, finding, § 46.
Problems, §§229-231.
Roots, finding, §47.
Rules for use of, §§ 40, 44-47.
Tables,
Gray and Steinhauser, § 341.
Standard, §339.
To fifteen places, §359; comment, §51.
To four places, § 43.
To twelve places, §358; comment, §§342-357.
United States Coast Survey, § 340.
Two parts of, § 41.
Use of, §§ 36, 37.
In compound interest computations, §49.
In present worth computations, § 52.
Loose-leaf records for mortgages, § 178.
Mantissa, §§41, 42.
Market value of bonds, § 107.
Month as time unit, § 21.
Mortgages, §§ 163-184.
Multiples, Little's table of, §361; comment, §356.
Multiplication,
By logarithms, § 44.
Contracted, §228.
Tabular, §248.
N
Net income, problems, § 293.
Nominal rates, § 91.
Problems, §§ 236-240.
Notes, §§217-220.
Notices of interest, § 181.
INDEX 367
O
Optional redemption, bonds with, § 147.
Problems, §§ 287-294.
Ordinary annuities, §§ 74-76.
Payments, periodic, §§ 319-336.
Periods,
Annuity and interest compared, § 81.
Interest, §§ 20-26, 364, 374.
Perpetuities, §§ 79, 80.
Powers, §§ 27, 38.
Finding, by logarithms, § 46.
Premiums on bonds.
Accounts, §§ 207, 209, 210, 214.
Amortization of, § 328.
Analysis of, § 317.
Deferred, §315.
Immediate, §314.
Valuation of, §§ 329, 330.
Prepaid annuities, §§ 75, l(i.
Present worth, §§29-31, 261.
By differences, § 259.
By division, § 260.
Formulas for, § 67.
Logarithmic computation of, § 52.
Of annuities due, §§ 75, Id.
Of bonds, §§101, 103, 106.
Of coupons, §§ 113, 296.
Of deferred annuities, §§ 11 , 78.
Of ordinary annuity, §§ 62-73, 304.
Of perpetuities, §§79, 80.
Of principal, § 112.
Proof of by reciprocal, § 227.
Relation to amortization, § 71.
Short method for, §§ 64-67.
Present worths,
Of $1, tables, §§63, 379; comment, §369.
Of annuity of $1, table, § 381 ; comment, § 371.
Of interest-diflerence, §271.
Principal, § 12.
Account, §§ 164, 169, 179, 186-188, 202.
Change in, § 290.
Definition of, § 10.
368 INDEX
Problems,
Annuities, §§ 232, 233.
Bonds at annual and successive rates, §§295-318.
Bonds with optional redemption, §§287-294.
Interest,
Compound, §§ 225, 226.
Simple, §§ 222-224.
Logarithms, §§229-231.
Nominal and effective rates, §§ 236-240.
Rent of annuity, §§ 234, 235.
Serial bonds, §§ 274-286.
Sinking funds, §§ 234, 235.
Valuation of bonds, §§243-318. (See also "Bond Problems.")
Punctual interest, §§ 14^ 17.
Q
Quarter-year, legal definition of, § 22.
Rates,
Annual and other, §§295-318.
Conversions of, § 92.
Coupon (cash), §§ 100, 109.
Definition of, § 10.
Dual, §§ 326-333.
Effective, §§91-95, 100.
Problems, §§ 236-240.
Equivalent (annual, semi-annual and quarterly), §§ 241, 242.
Income, on bonds, §§ 109, 134, 136, 137, 138-140, 263-269, 287, 291.
In interest tables, § 364.
Inter-rates, §§ 275, 375, Z76.
Logarithmic method, § 95.
Nominal, §§91, 97.
Problems, §§ 236-240.
Trial, §§ 264-269.
Ratios of increase, § 17.
Logarithms of, fifteen-place table, §359; comment, §51.
Real estate mortgages, §§ 163-184.
Receipts and notices, interest, § 181.
Reciprocals,
Amount and present worth as, § 227.
Meaning of, § 52.
Table of, § 383; comment, § Z77.
INDEX 36^
Redemption of bonds, §§ 146, 147.
Problems, §§ 287-294.
Register,
Collateral, §§ 186,-188.
Interest, §§ 180, 183, 192, 193.
Rent,
Definition of, §6.
Of annuity, §§ 83-86, Z73.
Of deferred payments, §86.
Problems, §§ 234, 235.
Repayment and reinvestment, §§319-336.
Replacement, §§ 148, 322, 323.
Residues on bonds, eliminating, §§ 138-140, 280.
Reussner's tables, § 273.
Revenue, forms of, § 5.
Roots, § 38.
Finding, by logarithms, § 47.
Rules and formulas, §§ 35, 157-159.
Security, collateral, §§ 185-188.
Serial bonds, § 145.
Problems, §§ 274-286.
Series, §§ 30-32.
Amortization, §§ 72, 249.
Of annuity amounts, §56.
Short terminal bonds, §§141, 142.
Problems, §§255, 256.
Simple (single) interest, §§ 13, 16, 19.
Problems, §§ 222-224.
Single discount, %%ZZ, 216-218.
Sinking funds, §§ 87-90.
Problems, §§ 234, 235.
Relation to amortization, § 90.
Table, §382; comment, §372.
Square roots, table of, §383; comment, §377.
Steinhauser's tables of logarithms, § 341.
Subordinate ledgers, § 160.
Sub-reciprocals, table of, § 360.
Successive method, § 277.
Problems, §244.
Successive rates, bonds at, §§313-318.
Symbols, explanation of, page xviii.
370 INDEX
T
Tables,
Amounts of $1, §378; comment, §367.
Amounts of annuity of $1, §380; comment, §370.
Bonds, § 155; comment, § 156.
Differences, § 276.
Four-place, § 43.
Interest ratios to fifteen places, §359; comment, §51.
Multiples, §361; comment, §356.
Present worths of $1, §§63, 379; comment, §369.
Present worths of annuity of $1, §381; comment, §371.
Reciprocals, §383; comment, §377.
Reussner's, comment, § 273.
Sinking funds, §382; comment, §372.
Square roots, §383; comment, ^377.
Sub-reciprocals, § 360.
Twelve-place logarithmic factors, §358; comment, §§342-357.
Tabular multiplication, § 248.
Time units, §§ 10, 23.
Day, §§ 20, 24.
Half and quarter years, § 22.
Month, §21.
Trust funds, §§ 148-154.
Cullen decision, §§ 152-154.
Twelve-place logarithms, tables, §358; comment, §§342-357.
U
United States Coast Survey tables, § 340.
Valuation of bonds, Chs. X, XI.
First method (two operations), §§111-114.
Interpolation method, §§ 131, 132.
Multiplication method, §§ 133, 134.
Periodic, §§ 130-133.
Problems, §§243-318.
Rule for, §§305, 316. ■
Schedules, §§ 114, 122, 126, 127, 130.
Second method (one operation), §§115-120.
Valuation of premiums, §§ 329. 330.
INDEX 2^j
Values,
Discounted, §§ 215-221.
Of bonds, § 127.
Book, §§ 123, 125.
Found by discounting, §§ 143, 144.
Initial book, § 247.
Intermediate, §§ 105, 376.
Market, § 107.
Various, §§ 104-108.
Varying annuities, § 82. /
y ^
Year, legal, §§23, 24.
14 DAY USE
RETURN TO DESK FROM WHICH BORROWED
LOAN DEPT.
This book is due on the last date stamped below, or
on the date to which renewed.
Renewed books are subject to immediate recall.
2lOct6lsL
r.« O
00^'^^'
LD 2lA-50m-8,'61
(Cl795sl0)476B
General Library
University of California
Berkeley
UNIVERSITY OF CAUFORNIA UBRARY