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 .^ 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 #, 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 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. 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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 »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 >!. 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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 '-.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'-:-; 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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^ 0\ CONO OOOlO irjt^vO'-H u^OOOnOO CO Tj-t^ COO lO T-HtOOO ON On TfLoCO -r}- CVl ^>. 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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