UC-NRLF GIFT OF Spreckles /'' Digitized by the Internet Archive t ^ . in 2007 with funding from ; IViicrosoft Corporation httpymww:waiim^ Stu&ie0 in ISueineea First Series, No. 3 .THE ACCOUNTANCY OF INVESTMENT. INCI.UDING A TREATISE ON COMPOUND INTEREST, ANNUITIES, AMORTISATION, AND THE VAI.UATION OF SECURITIES BY CHARLES EZRA SPRAGUE, A.M., PH.D., C.P.A., PROFESSOR IN THE NEW YORI^ UNIVERSITY SCHOOL OF COMMERCE, ACCOUNTS AND FINANCE ; PRESIDENT OF THE UNION DIME SAVINGS INSTITUTION ; CHAIRMAN OF THE SAVINGS BANK SECTION OF THE AMERICAN BANKERS ASSOCIATION : : : Published fok. the New York University School of Commerce, Accounts and Finance by the Business Publishino Company New York, 1904 ^ v^f^l SPREGKELS Copyright, 1904, by Charles E. Spragub TRUNK BROS. 85 WILL-AM STREET NEW YORK PREFACE. The following chapters embrace the substance of lectures delivered before the classes of the New York University Schooi< oe Commerce, Accounts and Finance. They have been in many places condensed, and in others expanded, with a view to their use as a text-book. I have introduced a treatise on Interest, Discount, Annuities, Sink- ing Funds, Amortisation and Valuation of Bonds, as I had not been able to find any suitable text-book which I could recommend. I hope that this will be useful to many who desire to inaugurate more scientific methods in their accountancy, but are unable to find intelligible rules for the computations. Treatises on the subject 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 made all my demonstrations arith- metical and illustrative, but, I think, none the less convincing and intel- ligible. I am indebted to Prof. Joseph Hardcastle, C.P.A., and to Walter B. Hallett, A.B., for valuable suggestions and assistance. CHARLES E. SPRAGUE. New York, November, 1904. TABLE OF CONTENTS. INTRODUCTORY CHAPTER. Thk Theory of Accounts. PAGB Balance Sheet 5 Equation of Accountancy 5 Examples 6 Ledger 8 Transactions 10 Debit and Credit 11 CHAPTER I. Capitai, and Revenue. Definition of Capital 12 Use of Capital 12 Sources of Capital 12 Investment 13 Revenue 13 Interest ; Rent ; Dividends 13 CHAPTER II. Interest. Time 15 Ratio of Increase 17 Amount 17 Present Worth 18 Abbreviated Rules 18 CHAPTER III. The Use op I^ogarithms. Nature of Logarithms 20 Rules for their use 20 Example of use in Compound Interest 21 CHAPTER IV. Amount of an Annuity. Definition and Example of Annuity 24 Abbreviated Summation '. . 25 Rule 26 CHAPTER V. Present Worth of an Annuity. Abbreviated Process 27 Example and Demonstration 28 Schedule of Annuity 28 Number of Years Purchase 29 CHAPTER VI. Rent of Annuity and Sinking Fund. PAGE Process of Finding Rent when Present Worth is Given 30 Sinking Fund 31 Relation between Sinking Fund and Rent of Annuity 31 CHAPTER VII. NoMiNAi. AND Effective Rates. Periods not Always Annual 32 Rate per Annum, in that Case, only Nominal 32 Process of Reducing Nominal to Effective Rates 32 Process of Reducing Effective to Nominal Rates 33 CHAPTER VIII. Valuation of Bonds. Definitions 34 Customary Features 34 Designation 34 Premium and Discount 35 Cash Rate and Income Rate 35 Evaluation of Bond : First Method 35 Second Method 36 Schedule of Amortisation, A 37 Schedule of Accumulation, B 39 Use of Tables 39 Schedules C and D 40 Purchases between Interest Dates 41 Schedule E 42 Balancing Periods not Coinciding with Interest Periods 42 Schedule F 43 Eliminating Residues : First Method 44 Second Method 45 Third Method 46 Schedule H 46 Short Terminals 46 Schedule 1 47 Discounting 47 Serial Bonds 48 Trust Funds 49 Justice Cullen 50 Single Column Schedule 52 Irredeemable Bonds 53 Optional Redemption 53 CHAPTER IX. Forms of Account — Generai, Principles. General and Subordinate Ledgers 55 Accounting for Interest as Fast as Earned 65 CHAPTER X. Reai, Estate Mortgages. page Definitions 57 Requisites for Accountancy 57 Form of Ledger 68 Register of Interest Due 60 Models of Mortgage Ledger 61 — 64 Mortgages Account, General Ledger 66 Classified Registers of Mortgages 67 CHAPTER XI. lyOANS ON Collateral. Directions 68 Model Account 69 CHAPTER XII. Interest Accounts. Stages in Interest 70 Daily Register of Interest Accruing 71 Monthly Summary of Interest Accruing; 72 General Ledger Accounts ; Interest Revenue, Interest Accrued and Interest Due 72—73 CHAPTER XIII. Bonds and Similar Securities. Book Value Only ; Three-Column Ledger 74 Bond Sales 75 Exhibiting Par and Cost ; Scalar Ledger 76 Model of Scalar Account 77 Amortisation Entries 78 General Ledger ; Two Plans for Keeping Par, Cost and Book Value . . 79 Irredeemable Bonds 82 Models for General Ledger Account : Plan 1 83 Plann,! 84 Plan II, 2 85 Plan II, 3 86 Plan II, 4 (Balance Column Form) 87 CHAPTER XIV. Discounted Values. Rate of Interest and Rate of Discount 88 Discount as an Offset 89 Progressive Example 90 Adjustment at Balancing Periods 90 APPENDIX I. Logarithms, to 12 Places, of Various Ratios 91 APPENDIX II. Multipliers for Finding Prices of Quarterly Bonds 92 (See note, p. 54.) APPENDIX III. Summary of Compound Interest Processes 93 ^.,x ie^. Wk THE ACCOUNTANCY OF INVESTMENT. INTRODUCTORY CHAPTER. Theory of Accounts. The Balance Sheet of a business expresses the status of that business at a certain point of time. It normally contains three classes of values: assets, liabilities and proprietorship, and expresses an equation between one of these classes, the assets, and the two others. As a fact, stripped of all tech- nicality, the assets are always exactly equal to the sum of the liabilities and the proprietorship : Assets = Liabilities + Proprietorship. This is merely an inverted way of defining proprietorship as being the excess of assets over liabilities : Assets — Liabilities = Proprietorship. In practice we generally write the equation in the form of a ledger account : Assets : [Equal Totals] LiABiiviTiES : Proprietorship As the proprietorship is simply the excess of assets over liabilities, it is evident that at any moment, if the fads are all ascertained, the same equation must hold good ; it must be per- petually true that, through all shiftings and changes : Assets = Liabilities -|- Proprietorship. This is the equation of accountancy, and all the processes of bookkeeping depend upon it. The Accountancy of Investment. The following is a condensed balance sheet of the affairs of an individual in business: Bai^anc^ Sheet of Wii.i,iam Smith. Cash 15,082.34 Bills Payable 18,000.00 Merchandise 17,082.65 Personal Creditors 5,465.35 Personal Debtors 8,123.17 Bills Receivable 7,000.00 William Smith 23,822.81 $37,288.16 137,288.16 The assets are partly composed of property, actually in possession, and partly of debts due to Smith; while the liabilities are entirely indebtedness due by Smith. The net proprietorship is designated by the name of the proprietor, although this is not universally the case. Instead of a single proprietor there may be a partnership, and the proprietorship may be represented either separately or in aggregate, as follows : Bai^ance Sheet oe Jones & Smith. Cash $8,589.09 Bills Payable 18,000.00 Merchandise 39,249.38 Personal Creditors 5,465.35 Bills Receivable 7,000.00 Mortgage Payable 4,000.00 Personal Debtors 24,095.32 James Jones 47,643.63 Real Estate 10.000.00 William Smith 23,822 81 188,933.79 188,933.79 BAI.ANCE Sheet o E Jones & Smith. Cash 18,589.09 Bills Payable $8,000.00 Merchandise 39,249.38 Personal Creditors 5,465.35 Bills Receivable 7,000.00 Mortgage Payable 4,000.00 Personal Debtors Real Estate 24,095.32 10,000.00 Jones & Smith {their joint capital) 71,468.44 188,933.79 $88,933.79 Theory of Accounts. I^et it be supposed that Messrs. Tones & Smith, instead of a partnership, had preferred to form a company, named the Jones Mercantile Company. They consider that, as the actual value of their joint proprietorship is over $71,000, it would be quite proper to capitalize it at $60,000, in 600 shares, of $100 each. Nevertheless, there is a total proprietorship of $71,468.44, as before, all of which must be represented in some form. In order to represent both the amount of the capitalization and the true proprietary value, we divide the total proprietor- ship into two parts : — Capital : par, or face value of shares |60,000.00 Surplus : excess of real value over par 11,468.44 Their sum is the real proprietorship |71,468.44 The resulting balance sheet would be : — BAI.ANCK Sheet oe the Jones Mercantii^e Company. Cash $8,589.09 Bills Payable 18,000.00 Merchandise 39,249.38 Personal Creditors 5,465.35 Bills Receivable 7,000.00 Mortgage Payable 4,000.00 Personal Debtors 24,095.32 j Capital Stock \ Surplus 60,000.00 Real Estate 10,000.00 $88,933.79 11,468.44 188,933.79 These are the usual forms of the balance sheet. As our debtors are all on the left hand side, and our creditors are on the right, it has become customary to call the left the dedttside, and the right, the credit side, notwithstanding the debit side con- tains much more than debtors and the creditor side much more than creditors. A better term for the debit side of the balance sheet is the Active ; and for the credit side, the Passive. The Passive has two widely different sets of values : the liabilities and the proprietorship, and I see no advantage in stretching the term ' ' liabilities ' ' to cover both. To call the proprietorship a liability is purely a technicality ; the part owned is precisely that part which is/ree from liability. 8 Th:^ Accountancy of Investment. The balance sheet presents the status at some certain point of time. We need also some means for recording what occurs, for the changes which take place between the balance sheets. For this purpose, the ledger is opened, being a system of accounts ; one account at least for each line of the balance sheet. It was formerly supposed that these accounts must be kept in one invariable form, regardless of their nature, such form being substantially that of the balance sheets shown above. Although this is no longer the rule we will employ the tra- ditional form in this illustration. A miniature ledger made up from the balance sheet of the Jones Mercantile Company would begin somewhat after this fashion: Cash. Balance 18,589.09 BiivivS Payabi^e. Balance 18,000.00 Mkrchandisk. Balance 139,249.38 BiiviyS Receivable. Balance 17,000.00 Theory of Accounts. "Personal Debtors" would probably comprise a number of accounts, headed by the name of each debtor. Cr, Dr. A. B. Balance ' ' Personal Creditors ' ' would likewise comprise a number of separate accounts. Dr. M. N. Cr. Balance REAi, Estate. Balance 110,000.00 Dr. Mortgage Payabi^e. Cr. Balance 14,000.00 CapitaIv Stock. Balance 160,000.00 lo Th^ Accountancy of Investment. SURPI^US. Balance |11,468.44 Thus the balance sheet has been dissected and accounts have been opened for each department of the business. A single account might have been opened for ' * Personal Debtors, ' ' but in that case it would have to be expanded into subordinate accounts, so as to give information as to each debtor. The same is true of " Personal Creditors," which is also a group account. For good reasons the balance with which each account begins is placed on the same side as it occupied in the balance sheet. Any increase of that balance will also be entered on the same side, and any decrease on the other side. Any possible transaction will increase the debit (left hand) side of the ledger, and its credit (right hand) side to exactly the same amount. Let us consider some of the possible cases : 1. If any asset is increased, we must either part with some other asset, or run into debt, or, (if neither of these is true, ) our wealth is increased. That is, an increase of assets is attended by a decrease of assets, or an increase of liability, or an increase of proprietorship, one or more of them. 2. If a liability is increased, we must either receive some asset, or pay off some other liability, or else we have lost. Therefore, an increase of liability is attended by an increase of assets, by a decrease of liability, or a decrease of proprietorship. As there are three elements in the accounts — assets, liabilities and proprietorship — and as each of these may be in- creased or decreased, there are six possible entries, at least two of which arise from every transaction, as follows : Thkory of Accounts. ii Debits. Credits. Increase of Assets. Decrease of Assets. Decrease of lyiabilities. Increase of lyiabilities. Decrease of Proprietorship. Increase of Proprietorship. The increase and decrease of proprietorship is called Profit and Loss. As it is of the utmost importance to study the causes of Profit and Loss, certain subsidiary accounts are opened for the sole purpose of classifying profits and losses according to their sources. At the time of constructing the next balance sheet, these subsidiary accounts (Interest, Rent, Expenses, Sales, etc. ) are transferred to a general Profit and Loss account, which presents an analysis of the conduct of the business for the period elapsed. This Profit and Loss account is in turn transferred to the permanent Proprietary Accounts. The foregoing is a sketch of the general theory of Double Entry Bookkeeping. There are various ways of looking at the subject, but I think this the most direct and best suited to the present purpose. For some other purposes, such as what may be called juridical accounts, the respective rights and obligations of the parties are the basis, rather than the struggle to increase proprietorship. The equation for this purpose would be : Charge = Discharge + Accountability. In general the equation comprises not two, but three, terms : Positives = Negatives -}- Resultant. 12 Th^ Accountancy of Invkstmbnt. CHAPTER I. Capitai. and Rkvsnus. 1. — Capital is that portion of wealth which is set aside for the production of additional wealth. 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 words "capital account" are often used in quite another sense to mean accounts on the credit or passive side which denote proprietorship. To prevent confusion, I will avoid the use of this expression, capital account. 2.— Use of Capital. In active business capital must be employed — must be combined with skill and industry to pro- duce more wealth. Businesses, and consequently their ac- counting methods, vary as to the manner in which capital is used. Cash is potential capital of all kinds, as desired. In a manufacturing business it is exchanged for machinery, appli- ances, raw materials, labor, which transforms the material into the product. In mercantile business it is expended for goods, bought at one price, to sell at another, and for collecting, dis- playing, caring for, advertising and delivering the goods. To bridge over the time between selling and collecting, additional capital is required, usually known as "working capital," but which might be more appropriately styled ' ' waiting capital. ' ' Thus we may analyse each kind of business and show that its capital assets depend upon the character of the business. 3. — Sources of Capital. On the other side of the balance sheet the capital must be accounted for as to who furnishes it. Here there are two sharply divided classes : loan-capital or liability, and own capital or proprietorship. The great dis- tinction 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." Capital and Revenue. 13 •4. — While we often speak of a man's capital as being invested in the business, yet when we use the word more strict- ly, we confine it to the non-participating sense. Thus we say, he not only owns the business, but he has some investments besides. In the strictest sense, then, investment implies divest- ing one's self of the possession and control of one's assets and granting such possession and control to another. The advan- tage of the use of the capital must be great enough to enable the user to earn more than the sum which he pays to the in- vestor or capitalist. There are many cases where the surrender is not absolute, and there is more or less risk assumed by the investor. This I should call not absolute investment, but to some extent partnership. The essence of strict investment is vicarious earning, a share of the gain not dependent on the fortunes of the handler. 5. — Revenue. All investment is made with a view to revenue, which is the share of the earnings given for the use of capital. This takes three forms: interest, rent and divi- dends — the former two corresponding to strict investment, and the latter to participation. 6. — Interest and rent do not essentially differ. Both are stipulated payments for the use of capital ; but in the latter the same physical asset must be returned on the completion of the contract. If you borrow a dollar, you may repay any dollar you please ; if you hire a hous6 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 no profit nor loss in this distribution. I have more cash, but my share in the collective assets is exactly that much less. It is dis- tributed, partly because it is needed by the participants for consumption ; partly because no more capital can be profitably used in the enterprise. Some concerns, for example some banks, which can profitably use more capital, and whose share- holders do not require it for consumption, refrain from dividing, 14 I^HS Accountancy of Invkstmejnt. and the accumulation inures just as surely to the shareholders, and is realizable through increased value of the shares upon sale. Thus, dividends are not strictly revenue. Yet the share- holder may treat them as such ; the dividend may be so regular as practically to be fixed, or his shares may be prefer- ential, so that to some extent he is receiving an ascertained amount ; or, as in case of a leased railway, it may be fixed by contract. Still, legally speaking, the dividend is instantaneous, and does not accrue, like interest and rent. 8. — As all investment is really the buying of revenue, and as the value of the investment depends largely upon the amount of revenue, and as the typical form of revenue is interest, it is, therefore, necessary to study the laws of interest, including those more complex forms — annuities, sinking funds and amor- tisation. Although there is a special branch of accountancy — the actuarial — which deals not only with these subjects, but with life and other contingencies, yet it is very necessary for the general accountant to understand at least the principles of the subject. The Accountancy op Investment. 15 . • CHAPTER II. Interest. 9. — The elements of interest are rate, time and principal. 10. — The time is divided into periods ; at the end of each period a certain sum for each unit of the principal is payable. The ratio between the unit of principal and the sum paid for' its use is the rate, and is expressed in hundredths or "per centum. ' ' Thus, if the contract is to pay three cents for each dollar of principal each year, it may be expressed, .03 per annum, 3 per centum per annum, 3 per cent., or 3%. Where the period is not yearly, but a less time, it is customary to speak, nevertheless, of the annual rate. Thus, instead of 3% per half year, we say 6%, payable semi-annually. Instead of i% per quarter, we say 4%, quarterly. In our discussions of interest, however, we shall treat of periods and of the rate per period, in order to avoid complication. 11. — As the law does not recognize interest for any fraction of a day, it becomes necessary to inquire what is meant by a half year or a quarter. The Statutory Construction Law (Chapter 677, Laws of 1892, §25) solves this difficulty by pre- scribing that a half year is not 1S2}4 days, but six calendar months, and that a quarter is not 91^ days, but three calendar months. Calendar months are computed as follows : — Commence at the day from which the reckoning is made, excluding 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. One difficulty arises : Suppose we have started with the 31st and the last month has only 30 days or less. Then, the law says, the month ends with the last day. One month from January 31st, 1904, was February 29th ; one month from January 30th or January 29th would also terminate on February 29th ; in a common year, not a leap year, the last day would be February 28th. 12. — A Day also requires definition. The legal day begins and ends at midnight. In reckoning from one day to another you must not include the day from which. Thus, if a loan 1 6 The Accountancy of Investment. is made at any hour on the third of the month and paid at any hour on the fourth, there is one day's interest, and that one day is the fourth, not the third. Practically it is the nights that count. If five mid-nights have passed since the loan was made, then there is five days' interest accrued. 13. — Parts of a period. In practice any fraction of an interest period is computed at the corresponding fraction of the rate, although theoretically this is not quite just. If the regular period is a year, then the 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 is regarded as ^J^ of a year. But when the contract is for months, quarters or half years, the fractional time must be divided into months. Finally we have the odd days left over, and doubt exists as to how they should be treated. Before 1892 there was no doubt. The statute distinctly stated that a number of days less than a month should be estimated for interest 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 " interest tables are based upon this rule. But the revisers of the statutes of the State, in 1892, 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 must be computed as 365 ths of a year. In business nearly every one calls the odd days 360ths, and it is only in legal accountings that there can be any question. It would be well if the old provision could be re-enacted or re-established by the courts. If it is necessary to correct the interest on the odd days from 360ths to 365ths, it may readily be done by subtracting from such interest if^ of itself. 14. — Interest is assumed to be paid when due. If it is not so paid, it ought to be added to the principal and interest be computed on the increased principal. But as the law does not directly sanction this, simple interest is spoken of as if it were a distinct species, where the original principal remains unchanged, even though interest is in default. There really is no such thing, for the interest money which is wrongfully with- Interest. 17 held by the borrower may be by him employed, and thus com- pound interest is earned ; only the wrong man gets it. All the calculations of finance depend upon compound i7iterest, which is the only rational and consistent method. When I have occasion to speak of the interest for one period I shall call it " single interest." 15. — If we add to 1 the decimal denoting the rate, we have the ratio of increase. Thus, if the rate is .03 per period, 1.03 is the ratio of increase, or simply the ratio, or the multiplier. 16.— The Amount is the principal and interest taken together. At the end of the first period the amount of $1 . 00 at 3% interest is $1 . 03. Instead of considering the $1 . 00 and the 3 cents as two separate sums to be added together, it is best to consider the operation as the single one of multiplying $1 . 00 by the ratio 1 . 03. 17. — 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)2 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 1.03x1.03x1.03 or, 1.033 1.092727 At the end of the fourth period we have reached the amount, 1.03* or, 1.12550881 We here find that the number of decimals is becoming unwieldy, and conclude to cut it down. If we desire to limit it to seven figures we reject the 1, rounding the result off to 1.1255088; if we prefer to use only six figures, we round it «/ to 1 . 125509, which is nearer than 1 . 125508. 18. — Thus 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 by mul- tiplying $1 . 00 by the ratio as many times as the number of periods. The remainder, after subtracting the original prin- 1 8 The Accountancy of Investment. cipal, is the compound interest. Thus the compound interest for four periods is .125509. The single interest is .03. 19. — 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 that $1 . 00 will be its amount. Using the same suppositions as before we desire to find such a number as, when multiplied by 1.03, will amount to $1 . 00 in four periods. $1 . 00 must, therefore, be divided by 1 . 03 for the first period. 1.03 ) 1.00000000 ( .970873 927 730 721 900 824 760 721 390 309 81 The result, rounded up at the 6th place, is . 970874, the present worth of $1 at 3% for one period, or j^, or 1 -^ 1.03. The present worth for two periods may be obtained either by again dividing .970874 by 1.03, or by multiplying .970874 by itself, or by dividing 1 by 1.0609, each of which operations gives the same result, . 942596 = ^2 The third term is —33 = .915142, and the fourth is ^^. = .888487. 20. — If we arrange these four results in reverse order fol- lowed by $1 and by the amounts computed in article 17, we have a continuous series : .888487 .915142 .942596 .970874 1.00 1.03 1.0609 1.092727 1.125509 Intbrest. 19 21. — It may be observed that each of these numbers is an amount of every preceding number and 2, present worth of every succeeding number, and that when one number is the amount of another, the latter is the present worth of the former ; in other words, that amount and present worth are reciprocals. 22. — Each one of these numbers may be obtained from the preceding one by multiplying by 1 . 03. Hence, as multipli- cation is easier than division, if we can obtain^8487 directly, we may supply the intermediate values more readily. This brief process for finding. ^8487 will be explained in the next chapter. 23. — In the present worth, .97087 of a single period, it is evident that the original |1 . 00 has been diminished by . 02913, which is a little less than .03 ; in fact it is .03 -=- 1..03. This difference .02913 is called the discount. In the present worth for two periods the discount is 1 — .942596, or .057404. This and succeeding discounts for greater numbers of periods are compound discounts. 24. — Compound discount does not bear any such direct relation to compound interest as single discount does to single interest. It can only be found by first ascertaining the present worth and then subtracting that from 1. 25. — We can reduce the rules to more compact form by the use of symbols. Let .y represent the amount of 1 ; /> the present worth ; i the rate of interest per period ; n the number of periods, and d the rate of discount. Let the compound interest be represented by I, and the compound discount by D. 26. — Then, by article 15, the ratio of increase is (l-j-z.) By article 18, .y = (1 + i) «; and I = .y — 1. By article 19, / = 1 -^ (1 + iY ; and D = 1 — /. 27. — The method of ascertaining the values of s and / through successive multiplications and divisions is, for a large number of periods, intolerably slow. A much briefer way, by the use of certain auxiliary numbers, called logarithms, will be explained in the next chapter. 20 Thb Accountancy of Investment. CHAPTER III. The Use of IvOgarithms. 28. — For multiplying or dividing a great many times by the same number, there is no device hitherto invented which is superior to a table of logarithms. 29. — The use of logarithms does not require a knowledge of the higher mathematics. It is purely an arithmetical help. The popular prejudice to the effect that there is something mysterious or occult about logarithms has no foundation. 30. — The ordinary books of logarithms are calculated to 7 places of decimals, sometimes extended for certain numbers to 8. If you wished to multiply by 1 . 03 fifty times, the logarithm would give you the first seven figures only of the answer, but as the remaining figures are so very insignificant, the result will for most questions be near enough even if rounded off at the 6th place. 31. — All the books of logarithmic tables contain, in an in- troduction, rules for using the tables, and these should be studied, and the examples worked out. These books have the ordinary numbers on the left in a regular series, in four figures only ; the fifth figure is at the head of one of the ten columns to the right. The sixth and seventh figures are obtained by a little side-table. 32. — Briefly, the rules of logarithms are as follows : By adding logarithms you multiply numbers. By subtracting logarithms you divide numbers. By multiplying logarithms you raise numbers to powers. By dividing logarithms you extract roots of numbers. 33. — The last two of these rules are the only ones necessary to be employed in the calculations of compound interest. 34. — In the common system of logarithms 10 is the base ; that is, the logarithm of 10 is 1. The logarithm of 100 (being The Use of Logarithms. 21 two tens multiplied together) is 2. The logarithm of 1,000 (in which 10 is three times a factor) is 3. We may thus go on in- definitely, saying in abbreviated language, log. 10,000 = 4 ; log. 1,000,000 = 6. In all these cases, the logarithm is the number of zeroes used to express the number. What is the meaning of these zeroes ? Each of them means that ten, the base of numeration, enters once as a factor. 1,000,000 : 1 followed by 6 zeroes means that 1 is multiplied 6 times by 10 ; or it may be written 1 x (10) \ Log. 1,000,000 = 6 Similarly Log. 100,000 = 5 • To multiply these numbers together you really add the logarithms, and write 1 followed by 11 zeroes. Thus there is a kind of logarithmic method in ordinary arithmetic. 35. To demonstrate the use of logarithms in compound interest, let us take an example and work it out, illustrating each Step. We will take the same rate as before, .03, but endeavor to find the amount for 50 periods. 36. — The ratio of increase is 1.03. We look for the log- arithm of this ratio. At the top of page 192 (Chambers' or Babbage's tables) we find this line: No. 12 3 4 5 6 7 8 9 10300 0128 3722 4144 4566 4987 5409 5831 6252 6674 7096 7517 37. — The meaning of this is that the logarithms are as follows : log. 1.03 . .01283722 '• 1.03001 .01284144 " 1.03002 .01284566 •♦ 1.03003 .01284987 " 1.03009 .01287517 The first figures are given once only, which saves space and time in searching. 38. — Since 1.03 is to be taken as a factor 50 times, we must multiply its logarithm by 50. This gives : 50 X .01283722= .6418610. 22 The Accountancy of Investment. This result is the logarithm of the answer, for when we have found the corresponding number we shall know the value of 39. — 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. This we find on page 73, and w^e find that the nearest approach is . 6418606 which is the logarithm of 4.3839. We now, from our logarithm, .6418610 subtract the above approximation .6418606 and have a remainder 04 In the margin is a little difference-table, reading thus : 99 1 10 2 20 3 30 4 40 5 50 6 59 7 69 8 79 9 89 The left hand column represents the 6th figure of the answe . If the remainder were 10, instead of 4, the next figure would be 1 ; if it were 69, the next figure would be 7. But is less than 10, therefore the 6th figure is 0. The 7th figure is 4, because 40 would give 4 for the 6th figure. 40.— Thus we have obtained our result. $4.383904 is the amount of $1 .00 compounded for 50 periods at 3%. This result is slightly inaccurate in the last figure, for the reason that two places were lost by multiplying. Had we taken the 10 figure logarithm on page xviii of Chambers' , . 0128372247 this multiplied by 50 w^ould give 641861235 or rounded off at the 7th place 6418612 From this subtract 6418606 and we have the remainder 06 which gives the more accurate result 4.383906 41. — As it is necessary, for problems involving many periods, The Use of I^ogarithms. 23 to use a very extended logarithm, I give in Appendix 1 a table of twelve-place logarithms for a number of different ratios of increase (1 -{- i). These are at much closer intervals than any table previously published, and, with a 10 figure book of logarithms, will give exact results to the nearest cent on $1,000,000. 42. — We will further exemplify the advantage of the logarithmic method by solving a, present- worth problem. Taking 50 periods at 3% for $1 . 00, we discount it as follows : Multiply the logarithm of 1.03 by 50, just as in Article 40, giving .641861235. But it is the reciprocal of 1.03^^ or 1 -f- 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 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. 43. — Neglecting the 1, we search in the right hand column for .358138765. On page 31 we find that .3581253 is the logarithm of 2.2810. From 3581388 Subtract 3581253 and we have a remainder 135 From the marginal table we find that 133 corresponds to 7, hence the 6th figure is 7, giving so far the result 2 . 28107. There is still a remainder of 2 which by the table is equivalent to 1 for the 7th figure. Hence, we have the full result . 2281071, the decimal point being moved one place to the left, as directed by the 1. 24 Tun Accountancy of Invkstment. CHAPTER IV. Amount of an Annuity. 44. — We have now investigated the two fundamental problems 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 pay- ments are irregular as to time, amount and rate of interest, the only way is to make as many separate computations as there are sums and then add them together. But if the sums, times and rate are uniform, we can devise a method for finding the amount or present worth at one operation. Annuity. A series of payments of like amount, made at regular periods, is called an annuity, even though the period be not a year, but a half year, a quarter or any other length of time. Thus, if an agreement is made for the following payments : On Sept. 9 1904 $100. On March 9 1905 100. On Sept. 9 1905 100. and on March 9 1906 100 . this would be an annuity of $200 per annum, payable semi- annually; in other words, an annuity of $100 per period, term- inating after 4 periods. It is required to find on March 9, 1904, assuming the rate of interest as 6% per annum, payable semi- annually {3% per period) : First, what will be the total amount to which the annuity will have accumulated on March 9, 1906; second, what is now, on March 9, 1904, the present worth of this series of future sums. It is evident that the answer to the first question will be greater than $400, and that the answer to the second question will be less than $400. 45. — It is easy, in this case, to find the separate amounts of the payments, for the number of terms is very small, and we have already computed the corresponding values of $1 . 00. Amount of an x\nnuity. 25 The last $100 will have no accumulation, and will be merely $100. The third $100 will have earned in one period, $3.00, and will amount to 103 . The second $100 will amount to 106.09 The first $100 (rounded off at cents) will amount to 109.27 and the total amount will be $418.36 46. — If, however, there were 50 terms instead of 4, the work of computing these 50 separate amounts, even by the use of logarithms, would be very tedious. 47. — lyet us write down the successive amounts of $1 . 00 under one another: a Amounts of $1. 1.00 1.03 1.0609 1.092727 48. — Now, as we have the right to take any principal we choose and multiply it by the number indicating the value of $1.00, let us assume one dollar and three cents, and multiply each of the above figures by 1 . 03, setting the products in a second column : a. b. c. Amounts of $1.00 Amounts of $1.03 Amounts of $0.03 1.00 1.03 1.03 1.0609 1.0609 - 1.092727 ■ 1.092727 ^ 1.12550881 49. — Our object in doing this was by subtracting column a from b to find the amount of an annuity of three cents. Be- fore subtracting, we have the right to throw out any numbers which are identical in the two columns. Expunging these like quantities, we have left only the following : a. b. c. Annuity of $1.00 Annuity of $1 . 03 Annuity of $0.03 1.00 1.12550881 less 1.00 1 . 12550881 Amount . 12550881 That is, an annuity of three cents will amount, under the conditions assumed, to twelve cents and the decimal 550881. Therefore, an annuity of 07ie cent will amount to one-third of 26 The Accountancy op Investment. .12550881 or .04183627. An annuity of $1.00 will amount to 100 times as much, or $4.183627, which agrees exactly with the result obtained by addition, in Article 45. 50.— The number .12550881 (obtained by subtracting 1.00 from 1.12550881) is actually the compound interest for the given rate and time, and the number .03 is the single interest; the amount of the annuity of $1.00 is .12550881-^ .03 = 4 . 183627. This suggests another way of looking at it. The compound interest up to any time is really the amount of a smaller annuity, one of three cents instead of a dollar, con- structed on exactly the same plan, and used as a model. 51. — Rule. To find the amount of an annuity of $1.00 for a given time and rate, divide the compound interest by a single interest, both expressed decimally. 52. — Let S and P represent the amount and the present worth, not of a single $1.00, but of an annuity of $1.00, then S = I -f- /. 53. — To illustrate, let us take the case worked out in Article 40, 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. Thk Accountancy of Investment 27 CHAPTER V. Present Worth of an Annuity. 54. — To find the present worth of an annuity, we can, of course, find the present worth of each payment and add them together ; but it will evidently save a great deal of labor if we can derive the present worth immediately, as we have learned to do with the amount. 55. — The like course of reasoning will give us the result. Take the four numbers representing the present worths of $1 . 00 at 4, 3, 2 and 1 periods respectively, and multiply each by 1.03. a. b. Present Worth of Present Worth of Annuity of |1.00 Annuity of $1.03 .888487 .915142 .915142 .942596 : .942596 .970874 .970874 1.000000 Canceling all equivalents, we have .888487 ....... 1.000000 Present Worth of Annuity of .03 1.000000 less .888487 .111513 Annuity of $1 . 00 = . 111513 -^ . 03 = 3 . 71710 This is the same result (rounded up) as that obtained by adding column a, 56.— But . 111513 is the compound discount of $1 . 00 for four periods, and we therefore construct this rule : 57. — Rule. To find the present worth of an annuity of $1.00 for a given time and rate, divide the compound discount for that time and rate by a single interest. Symbolically P = D -f- /. We might give this the form P = D x ^, but in practice this would not be so convenient. 58. — It may assist in acquiring a clear idea of the working of an annuity, if we anal3^se a series of annuity payments from the point of view of the purchaser. 28 Thb Accountancy of Investment. 59.— He who invests $3.7171 at 3%, in an annuity of 4 periods, expects to receive at each payment, besides 3% on his principal to date, a portion of that principal, and thus to have his entire principal gradually repaid. Principal. 60. — His original principal is 3 . 7171 At the end of the first period he receives 1 . 00 consisting of 3% on 3.7171 1115 and payment on principal 8885 .8885 leaving new principal 2 . 8286 (or present worth at 3 periods). In the next instalment 1 .00 there is interest on 2.8286 0849 and payment on principal 9151 .9151 leaving new principal 1 . 9135 Third instalment 1.00 Interest 0574 on principal .9426 .9426 .9709 I^ast instalment , 1 . 00 Interest 0291 Principal in full .9709 .9709 61. — Thus the annuitant has received 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 corroborated. 62. — It is usual to form a schedule showing the com- ponents of each instalment in tabular form. Date '^°^^^ Interest Payments principal Instalment Payments principal Outstanding 1904 Mar. 9 3.7171 1904 Sept. 9 1.00 .1115 .8885 2.8286 1905 Mar. 9 1.00 .0849 .9151 1.9135 1905 Sept. 9 1.00 .0574 .9426 0.9709 1906 Mar. 1 1.00 .0291 .9709 0.0000 4.00 .2829 3.7171 Had the purchaser re-invested each instalment at 3%, he would have, at the end, $4.1836 (Article 45), which is equivalent Present Worth oe an Annuity. 29 to his original $3.7171 compounded (3.7171 X 1.1255 = 4.1836). 63. — In Article 43, using logarithms, we found the present worth of a single $1 . 00 at 50 periods, at 3%, to be . 2281071 Subtracting this from 1.0000000 we have 7718929 which is the compound discount. Dividing this by .03 we have 25 .72976 which is the present worth of an annuity of $1 . 00 for 50 periods. Thus we see that the process of finding the present worth of an annuity, or, as it is termed, evaluation, is rendered very easy, no matter how long the time, by using logarithms. 64. — The present worth of an annuity of $1.00 is some- times called the number of years' purchase. Thus we would say, in the example just given, that a 50 year annuity, at 3%, is worth nearly 26 years' purchase ; meaning that one should pay now nearly 26 times a year's income, whatever that may be. In Hardcas tie's " Accounts of Executors," page 27 and following, will be found several examples of the evaluation of leases for years, which are a species of annuity. It will be useful to work these out by logarithms to as many places as possible. 30 The Accountancy of Investment. CHAPTER VI. Rent of Annuity and Sinking Fund. 65. — The number of dollars in each separate payment of an annuity is called the rent of the annuity. 66. — We saw that 3.7171 is the present worth of an annu- ity composed of 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 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 invested. Or $1 is the present worth at 3% of an annuity of . 26903. This may be illustrated by making up a schedule : Rent. Beginning of first period End of first period 26903 End of second period 26903 End of third period 26903 End of fourth period 26903 1.07612 .07612 1.00000 67. — Rule, To find the rent of an annuity valued at $1, divide $1 by the present worth of an annuity of $1 for the given rate and time. Rent = 1 -^- P. 68. — This may be also called finding how much per period for n periods at the rate t can be bought for $1. A borrower may agree to pay back a loan in instalments, which comprise prin- cipal and interest. Suppose a loan of $1,000 were made under the agreement that such a uniform sum should be paid annually as would pay off (amortise) 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 contribution would be $269 . 03. In countries imposing an income-tax it is usual to incorporate in the bond a schedule with columns like those in Article 66, showing what part of the instalment is interest, as that alone is taxable. TEREST. Reduction. Value. 1.00000 .03 .23903 .76097 .02283 .24620 .51477 .01544 .25359 .26118 .00785 .26118 0. Rent of Annuity and Sinking Fund. 31 Annual ^%^^^^^ Payment principal Balance Principal Instalment «Jil^^ vri^nir^at Outstanding Jan. 1 1901 1,000.00 Dec. 31 1904 269.03 30.00 239.03 760.97 Dec. 31 1905 269.03 22.83 246.20 514.77 Dec. 31 1906 269.03 15.44 253.59 261.18 Dec. 31 1907 269.03 7.85 261.18 0. 1076.12 76.12 1,000.00 69. — It may be required, also, to find such an annuity as will, at the end of a certain number of periods, have accu- mulated to $1.00 or any other vSum^ This is called a sinking fund, when it is intended to provide for a liability not yet matured. In the case exhibited in the schedule, in Article 68, the debt was amortised, with the assent of the creditor, by gradual payments. Let us suppose that the creditor prefers to wait till the day of maturity, and receive his $1,000 at once. He must be paid his interest of $30 each year, but the debtor, to provide for the principal, must also transfer from his general assets to a special account (or into the hands of a trustee, if he doubts his self-control), where it will draw interest at S%, such a sum as will accumulate to $1,000. This is the sinldng fund. Since $1.00, set aside^ annually .. amounts,, after 4 years, to $4.183627, to find what sum will amount to $1,000, we must divide 1,000 by 4.183627, giving for the contribution to the sinking fund $239.03. 70.— Rule. 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. Sinking fund contribution ^ 1 -^ S. 71. — If you observe the two results obtained by the two preceding rules: .26903 and .23903 you will see that they differ by . 03 , which is exactly the period- ical interest on the original loan. Hence the amount paid in the second case, if interest be included, is just the same as in the former. This is as it should be, for in the latter case we are investing in some other 3% security, while in the former, we are investing in part of this very obligation. Gradual pay- ment, or gradual accumulation for a single payment, come to the same thing. These two forms of practically the same pro- cess, amortisation and sinking fund, will be useful to guide us when we study the subject of premiums on securities. 32 Thk Accountancy, of Investmtjnt. CHAPTER VII. NoMiNAi, AND Effective Rates. 72. — We have reduced all our operations to so many periods, and such a rate per period, but it is usual to speak of such a rate per aiinum, payable so many times a year, or ' ' con- vertible half yearly or quarterly." Where the interest is pay- able otherwise than annually, the rate per annum is only nominally correct. For example: if we take 3% per half year, this would be nominally 6% per annum, but efFectively it would be 6.09% per annum, because 1.03 X 1.03 = 1.0609. If paid quarterly, the effective rate per annum, 6 . 1364% would correspond to the nominal rate 6%. Evidently, the more fre- quent conversion results in more rapid accumulation. If paid monthly, the effective rate would be, 6. 1678%, while if paid daily it would be 6. 1826%. But there is a limit beyond which this acceleration will not go ; 6% compounded every minute, or every second, or every millionth of a second, or constantly, could never be so great as 6 . 184%. 73. — The process of finding the effective rate follows naturally from the ordinary rule for compound interest. If the nominal rate is 6% per annum, and it be paid quarterly, the actual ratio is 1.015, and the fourth power of 1.015 is the amount at the end of the fourth quarter, which by multiplica- tion or by adding logarithms is found to be 1 . 061364, or interest . 061364. It will well exemplify the logarithmic process if we apply it, finding an effective rate for daily compounding. Let the nominal interest be 6%, then the actual rate per period of a day will be . 06 -f- 365. We will first perform this division by logarithms: Ivog. .06 2.7781513 — Log. 365 2.5622929 Difference 4.2158584 4.2158584 is, we find, the logarithm of .0001643835, hence the daily ratio of decrease is 1.0001643835; and we again find the logarithm of this number, which is .00007138; 365 times .00007138 being .02605370, which is opposite the number NOMINAI. AND KfFKCTIVK RATKS. 33 1.061826. If m represent the number of payments per year, andy the effective rate, we have/ = (1 + ^) *^ — 1. 74. — It may also be necessary to solve another problem, in order to produce an effective rate of 6% per annum, what nominal rate is required, conversion being half-yearly, quarterly, etc. In this case we have to find the ratio of increase for the lesser unit of time, which we do by dividing the logarithm of the effective rate by the number of conversions. Thus, if we are required to find the nominal rate, which, compounded quarterly, will be equivalent to the effective rate 6%, we divide the logarithm of 1.06 02530587 by 4, giving 00632647 The number opposite this is ... . 1 . 0146738 This being the quarterly ratio, the nominal annual rate will be 4 times this rate 0586952 75. — There are some other problems in compound interest, such as finding the time or the rate, when the other elements are given. But the foregoing rules will suffice for most of the purposes of investment accounts. 34 ^HS Accountancy of Investment. CHAPTER VIII. Vai^uation of Bonds. 76. — Investments of loan-capital are usually made by means of written instruments, known as debentures or bonds. These are promises to pay: first, a principal sum at a certain date in the future ; this principal sum is the par value of the bond; secondly, to pay at the end of each period, as interest, a certain percentage of the principal. The bond also contains provisions as to the time, place and manner of these payments, and usually refers, also, to the security obligated to insure its fulfillment, and to the law (in case of a corporation, public or private) which authorizes the issue. In Prof. Frederick A. Cleveland's "Funds and their Uses" will be found furthei particulars as to the various descriptions of bonds and similar securities. 77. — The rate of interest named in the bond is usually aD integer per cent., or midway between two integers: as 2%, 2>^%, 3%, 3^%, 4%, 4^, 5%, 6%, 7%. Occasionally, such odd rates occur, as 3}(%, 3.60%, 3.65%, 3%%, but these are unusual and inconvenient. Most bonds pay interest semi- annually, and on the first day of the month. Here again there is some deviation. A considerable number of issues pay interest quarterly, and a very few annually. A very few have the interest fall due on some other day than the first of the month. These vagaries are of no benefit, and slightly injure the value of the bond. It would in some respects be better, however, if interest were payable on the lasi day of the half year, thus bringing the entire transaction inside of a calendar period. 78. — Bonds are usually designated according to the obligor, the rate of interest, the date or year of maturity, adding, if requisite, the initials of the months when interest is payable. Thus, " Manhattan 4's of '90, J & J," indicates the bonds of the Manhattan Railway Company, bearing 4% interest per annum, payable semi-annually on the first days of January and July, and payable in 1990. VAI.UATION OF Bonds. 35 79. — Bonds very frequently are bought and sold at a differ- ent price from par. This has its effect on the income derived from the investment. The amount invested being greater, the percentage of fixed income is less; beside this, the excess or premium will not be repaid at maturity, but will be sacrificed; hence, a bond purchased above par earns less than the con- tractual interest. Similarly, if the purchase is below par, the percentage of fixed income is greater; besides, at maturity the owner will receive not only all that he invested, but also the discount bringing it up to par. Hence a bond purchased below par earns more than the contractual interest. 80. — ^There are thus two rates of interest relatively to the par and the price : a nominal rate, which is so many hundredths of par; and an effective rate, which is so many hundredths of the amount invested and remaining invested. The words, nominal and effective, are as correctly applied in this case as in relation to frequency of conversion; but for the sake of dis- tinction we shall prefer to call them the cash rate and the income rate, and designate them, when desirable, by the symbols c and i respectively. 1 + / is the ratio of increase as heretofore. 1 + ^ is not required, as c is not an accumulative rate, but merely an annuity purchased with the bond. The difference of rates is c — / or i — c. 81. — The following are some of the expressions used to denote an investment made above or below par : * ' 6% bond to net 5%;" ''6% bond on 5% basis;" "6% bond yielding 5%;" '' 6% bond pays 5%. " 82. — In a bond purchased above or below par, we have, therefore, the following elements: the par principal payable after n periods ; an annuity of c per cent, of par for 71 periods, and a ratio of increase, 1 -j- i. Given these, there are two dis- tinct methods for finding the value of the entire security, and these must give the same result. 83. — First Method. Separate Evaluation of Principal and Annuity. Let us suppose a 1% bond, interest semi- annually, 25 years to run (50 periods), for $1,000. The present value is composed of two parts: (1) the present worth of $1,000 in a single payment, 50 periods hence; (2) an annuity of $35 36 Thk Accountancy of Investment. for 50 terms. We can only value these when we know what is the income-rate current upon securities of this grade. Let us assume 3V as the income- rate per period, or what is usually called a 6% basis. The ratio of increase is 1.03. 84. — The first part of the solution is to find the present worth of $1,000 at 3% in 50 periods. In Article 42, we have found the present worth of $1.00 on the same conditions, which is . 2281071; hence, the value of the $1,000 is $228 . 1071. It will be noticed that this result has not the slightest reference to the 1% rate of the bond. As a matter of compound interest, the 1% does not exist. 85. — We next have to value an annuity of 50 terms at $35. In Article 63, we valued a similar annuity of $1.00 and found it to be worth $25.72976. If each term be $35, the value will be $900 .5417. Adding this to the value of the $1,000, we have the value of the bond, 228.1071 +900.5417=$1128.6488. The ordinary tables, which give the values of a $100 bond only, read 112 . 86, which is the same, rounded off. The above com- putation gives a result which is correct to the nearest cent on $100,000, viz., $112,864.88. 86. — Second Method. Division of Income and Eval- uation of Premium or Discount. Each semi-annual pay- ment of $35 may be divided into two parts: $30 and $5. The $30 is the 3% income on the $1,000; we may disregard this and consider only the $5, which is surplus interest, and, in fact, is an annuity which must be paid for in a premium. Having devoted $30 to the payment of our expected income- rate on par, we have $5, the difference of rates per period, as a benefit to be valued. 87. — We have found the present value of an annuity of $1 . 00 to be 25 . 72976. Multiplying this by 5 to get the present worth of a $5 annuity, we have $128. 6488, which is the premium, agreeing with the result of the previous method. The second method is not only quicker, but it often gives one more place of decimals. 88. — In the case of a bond sold below par, the cash-rate being less than the income-rate, the same procedure is followed for finding the present worth of $5, but the result, $128.6488, Valuation of Bonds. 37 is subtracted from the par, giving $871 . 3512 as the value of a 5% bond earning 6% per annum. 89. — As this second method is superior to the first, we will adopt it as the standard. 90. — Rule. The premium or discount on a bond for $1 . 00 bought above or below par, is the present worth of an annuity of the difference of rates. 91.— We have found the value of a 7% bond for $100, paying 6% (semi-annual), 25 years to run, to be $1128.65 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 same half year, the holder must receive 3% interest on this $1128 . 65, which is $33 .86. But he actually collects $35, and after using $33.86 as revenue, he must apply the remainder, $1.14, to the amortisation of the premium. This will leave the value of the bond at the same income-rate, $1127.51. If our operations have been correct, the value of a 7% bond to net 6%, 24:}4 years or 49 periods to run, will be $1127 .51. To test this, and to exemplify the method, we will go through the entire operation: 92.— The logarithm of 1.03 is 01283722 This X 49 = 6290238 This subtracted from = 1.3709762 We find that the logarithm of .23495 is 1.3709754 Remainder 8 which gives the figures 02. Hence, . 2349502 is the total discount at 3% per period for 49 periods on $1.00. Subtracting the .2349502 from 1, we have .7650498, which is the present value of an annuity of .03 for 49 periods. Dividing by . 03 we have the present value of an annuity of $1.00 per period, viz., 25.50166. But the surplus interest (35 — 30) is $5; hence, we must multiply 25.50166 by 5, giving $127,508, or, rounded off, $127.51, as the premium, at 49 periods. Adding this to the 1,000 we have $1127 .51, the same result as in Article 91. 93. — When bonds are purchased as investment, a Schedule' of Amortisation should be constructed, showing the gradual extinction of the premium by the application of surplus interest. 38 Th^ Accountancy of Investment. The following is the form recommended, but it should be con- tinued to the date of maturity, and at intervals corrected in the last figure by a fresh logarithmic computation. SCHEDUI.E OF Amortisation. 7% bond of the , payable Jan. 1, 1954. Net 6%. J J. Total Net ^^ , Date Interest Income Amortisation f.„,,_ Par 7% 6% V^l"« 1904 Jan. 1 Cost 1,128.65 1,000.00 July 1 35.00 33.86 1.14 1,127.51 1905 Jan. 1 35.00 33.83 1 17 1,126.34 July 1 35.00 33.79 1.21 1,125.13 etc., etc., etc. ' ' Book Value ' ' might also be termed * * Investment Value. ' ' 94. — This schedule is the source of the entry to be made each half year for "writing off" or "writing up" the premium or discount, so that at maturity the bond will stand exactly at par. We give two more examples continuing them to maturity, one being above par and the other below par. As the formation of schedules is the basis of the accountancy of amortised securities, we .shall present the same materials in various forms, lettering them (A), (B), etc. SCHEDUI.E OF Amortisation (A). 5% Bond of the , payable May 1, 1909. M N. Total Net Date Interest Income Amortisation tt„i..^ Par Book 5% 4% Value 1904 May^l ^ Cost 104,491.29 100,000.00 Nov. 'l 2,500 2,089.83 410.17 104,081.12 1905 May 1 2,500 2,081.62 418.38 103,662.74 Nov. 1 2,500 2,073.26 426.74 103,236.00 1906 May 1 2,500 2,064.72 435.28 102,800.72 Nov. 1 2,500 2,056.01 443.99 102,356.73 1907 May 1 2,500 2,047.13 452.87 101,903.86 Nov. 1 2,500 2,038.08 461.92 . 101,441.94 1908 May 1 2,500 2,028.84 471.16 100,970.78 Nov. 1 2,500 2,019.42 480.58 100,490.20 1909 May 1 2,500 2,009.80 490.20 100,000.00 25,000 20,508.71 4,49129 Valuation of Bonds. 39 Schedule of Accumulation (B). 3% Bond of the , payable May 1, 1909. M N. Total Net Book Interest Income Accumulation vaii,* . 3% 4% ^^^"* 1904 May 1 Cost 95,508.71 100,000.00 Nov. 1 1,500 1,910.17 410.17 95,918.88 1905 May 1 1,500 1,918.38 418.38 96,337.26 Nov. 1 1,500 1,926.74 426.74 96,764.00 1906 May 1 1,500 1,935.28 435.28 97,199.28 Nov. 1 1,500 1,943.99 443.99 97,643.27 1907 May 1 1,500 1,952.87 452.87 98,096.14 Nov. 1 1,500 1,961.92 461.92 98,558.06 1908 May 1 1,500 1,971.16 471.16 99,029.22 Nov. 1 1,500 1,980.58 480.58 99,509.80 1909 May 1 1,500 1,990.20 490.20 100,000.00 15,000 19,491.29 4,491.29 .9;i. — It will be observed in these two schedules that the one is exactly as much above par as the other is below it, and that the ' ' accumulation ' ' and ' ' amortisation ' ' are exactly the same in both, being added in one case and subtracted in the other. In one line the net income is apparently in error 1 cent, but this is on account of the roundings of the fractions of a cent, and would disappear if the operation were carried to one place further. 95. — The figures in the column " Book Value " might be taken from the tables of bond values, published in book form. The column of amortisation would, in this case, be derived from the Book Values, and the Net Income from the Amorti- sation. The schedule would then be roughly accurate, unless the table used were carried to a greater number of places than is usually done. Sprague's Bond Tables will give eight places instead of four, and from them schedules (A) and (B) can be obtained to the nearest cent. 40 1 rni J Accountancy of : Invkstment. (C) Total Net Book Date Interest Income Amortisation Value 5% ^% Approximate 1904 May 104,490 Nov. 2,500 2,090 410 104,080 1905 May 2,500 2,080 420 103,660 Nov. 2,500 2,080 420 103,240 1906 May 2,500 2,060 440 102,800 Nov. 2,500 2,060 440 102,360 1907 May 2,500 2,040 460 101,900 Nov. 2,500 2,040 460 101,440 1908 May 2,500 2,030 470 100,970 Nov. 2,500 2,020 480 100,490 1909 May 2,500 2,010 490 100,000 96. — It will be observed that in schedules (A) and (B) the entire interest is accounted for, both the interest on the par and that on the premium. We may easily construct the schedule so as to eliminate the par and its interest at the rate i, and deal only with the surplus interest or the deficient interest, according to the theory in Article 86. As this may be prefer- able for some forms of accounts, we again work out the schedule for ''5% bond net 4%, 5 years, semi-annual": (D) Surplus Interest on Date Interest 1% Premium Amortisation Premium 1904 May 4,491.29 Nov. 500 89.83 410.17 4,081.12 1905 May 500 81.62 418.38 3,662.74 Nov. 500 73.26 426.74 3,236.00 1906 May 500 64.72 435.28 2,800.72 Nov. 500 56.01 443.99 2,356.73 1907 May 500 47.13 452.87 1,903.86 . Nov. 500 38.08 461.92 1,441.94 1908 May 500 28.84 471.16 970.78 Nov. 500 19.42 480.58 490.20 1909 May 500 9.80 490.20 5,000 508.71 4,491.29 VAI.UATION OF Bonds. 41 ■97. — We have hitherto assumed that the purchase of the bond took place exactly upon an interest date. We must now consider the case when the initial date differs from the interest date. I^et us suppose the purchase to take place on July 1, when one-third of the period has 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 is also considered as vanishing by an equal portion each day, so that one- third of the half-yearly amortisation takes place by July 1. The amortisation from May 1 to November 1 being $410.17, that from May 1 to July 1 must be $136.72, and the book value on July 1 is $104,354.57, with accrued interest, $833 . 33— in all $105,187 . 90. This last number is the flat price, that is to say, it is inclusive of interest. It might have been obtained in the following manner : To the value on May 1 $104,491.29 add simple interest thereon, at 4%, for 2 months . , 696 . 61 giving the flat price $105,187.90 In buying bonds, there is usually a stipulation that the price should be so many per cent. * ' and interest, ' ' otherwise the price named is understood to be " flat. ' ' 98. — This practice of adjusting the price at intermediate dates by simple interest is conventionally correct, but is scien- tifically 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 readily be seen that the buyer does not net the effective rate of 4% semi- annually on his investment of $105,187.90. The true price would be $105,183.31, giving both buyer and seller, not 4% nominal, but the equivalent effective with bi-monthly and four- monthly conversion. In practice, however, for any time above six months, simple interest is generally used, to the slight dis- advantage of the buyer, who may claim, and probably legally, that the November value -f interest due should have been dis- counted at 4%; 106,581.12-^1.01/3 ==105,178.74; which is al- most exactly as much too low as the $105,187.90 is too high. 99. — The schedule would, therefore, in practice, read as follows : 42 Thb Accountancy of Investment. (E) Total Net Date Interest Income Amortisation 5% i% Book Value 1904 July 1 Cost 104,354.57 100,000.00 Nov. 1 1,666.67 1.393.22 273.45 104,081.12 1905 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 1906 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 1907 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 1908 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 1909 May 1 2,500.00 2,009.80 490.20 100,000.00 24,166.67 19.812.10 4,354.57 100. — The interest dates may not always be the most con- venient epochs for periodical valuation. There may be many kinds of bonds, the interest on some falling due in every rponth in the year, and yet on a certain annual or semi-annual date the entire holdings must be simultaneously valued. It will then be convenient if we can arrange our schedules so that without recalculation every book value will be ready to place in the balance sheet. Fortunately, this is easier than would be supposed. 101. — We again take a 5% bond, payable on Nov. 1, 1904, on a 4% basis, but we 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. As this is between the May and the November periods, we must adjust the price as in Article 96, so that the August price is midway between 104,491.29 and 104,081.12, namely: $104,286.20 and interest, being the customary, not the theoretical, method. The No- vember value need not enter into the schedule, but we must locate the December 31 value, just as we found the July 1 value in Article 96. One-third 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 reads: Aug. 1 Cost 104,286.20 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, 1905, from those of May 1 and November 1, we get $103,520.49. Valuation of Bonds. 43 ' 102.— But 6 months interest at 4?^ on $103,941.66 is $2,078.83, which, subtracted from $2,500, gives the amorti- sation $421.17, and this, written off from $103,941.66, gives $103,520.49, precisely the same as obtained by interpolation between May and November. Hence we have two ways of continuing the schedule : interpolation and multiplication. In this respect the commercial practice is much more convenient than the theoretical one. Having once adjusted the value at one of the balancing periods, we can derive all the remaining by subtracting the net income from the cash interest and re- ducing the premium by the difference, completely ignoring the values on interest days (M & N). f/wn.. 103. — No diflBculty arises until we reach the broken period, l^ttiy^l — May 1, 1909. Here the computation of the second column. Net Income, is peculiar. The par and the premium must be treated separately. The net income on $100,000 is taken at fs of 2% for the fs 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, not $326.80, is the conventional premium, on which 4% is to be computed; hence, instead of taking 3/2 of $326.80 for ^ of a period, we take $326 . 80 itself for a whole period, two- thirds of three-halves being unity. (F) Total Net Date Interest 5% Income 4% Amortisation Book p Value ^^^ 1904 Aug. 1 Cost 104,286.20 100,000.00 Dec. 31 2,083.33 1,738.79 344.54 103,941.66 1905 June 30 2,500.00 2,078.83 421.17 103,520.49 Dec. 31 2,500.00 2,070.41 429.59 103,090.90 1906 June 30 2,500.00 2,061.82 438.18 102,652.72 Dec. 31 2,500.00 2,053.05 446.95 102,205.77 1907 June 30 2,500.00 2,044.12 455.88 101,749.89 Dec. 31 2,500.00 2,035.00 465.00 101,284.89 1908 June 30 2,500.00 2,025.70 474.30 100,810.59 Dec. 31 2,500.00 2,016.21 483.79 100,326.80 1909 May 1 1,666.67 1,339.87 326.80 100,000.00 23,750.00 19,463.80 4,286.20 44 Tii:Bi Accountancy of Invkstme^nt. 104. — 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 datum 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 difi&cult problem. 305. — 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 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 re- quired. Sprague's Tables give, by the use of auxiliary figures, values for each one-hundredth of one per cent. 106.— I.et us suppose that the $100,000 5% bonds, 5 years to run, MN, are ojffered at the round price of 104>^ on May 1, 1904. 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 ^^ 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 unwieldy 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 maturity, and on a 4% basis it will be even larger at maturity than now. Three ways of ridding ourselves of it may be suggested. 107. — First Method of Eliminating Residues. Add the residue $8.71 to the first amortisation, thereby reducing the value to an exact 4% basis at once. In our example (A), instead of $410.17, the first amortisation would be $418.88. f OF THE "^ \ OF VAI.UATION OF Bonds. 45 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. 108.— Second Method. Divide $8.71 into as many parts as there are periods. This would give . 87 for each period, except the first, which would be .88 on account of the odd cents. Set down the 4:% amortisation 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: (G) Date Total Interest Net Income Amort isatiot Book p ^ Value ^^^ 5% *% (-) 1904 May 1 104,500.00 100,000.00 Nov. 1 2,500.00 2,088.95 411.05 104,088.95 1905 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 1906 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 1907 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 1908 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 1909 May 1 2.500.00 2,008.93 491.07 100,000.00 25,000.00 20,500.00 4,500.00 109. — In example (G) the income rate varies from 3 .99798 to 3.99828; hence the approximation is suflSciently close for any, except large holdings for long maturities. 110. — Third Method. For still greater accuracy, we may divide the $8 . 71 in parts proportionate to the amortisation. The amortisation on the A% basis runs off $4,491.29, and we have an extra amount of $8 . 71 to exhaust. Dividing the latter 46 The Accountancy of Investment. by the former, we have as the quotient . 00194, which is the portion to be added to each dollar of amortisation. With this we form a little 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 amortisation, writing down, for example, to the nearest mill: 410 .17 .400 .776 10.019 .100 .070 410.97 418.38 419.19 426.74 427.57 435.28 400.776 400.776 400.776 10.019 20.039 30.058 8.016 6.012 5.010 .301 .701 .200 .080 .040 .080 436.12 The result, in schedule (H), varies at the most 5 cents. (H) Total Net Interest Income Amortisation 6% 4% (-) Book Value Par 1904 May 1 Nov. 1 1905 May 1 Nov. 1 1906 May 1 Nov. 1 1907 May 1 Nov. 1 1908 May 1 Nov. 1 1909 May 1 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 2,089.03 2,080.81 2,072.43 2,063.88 2,055.15 2,046.25 2,037.18 2,027.93 2,018.49 2.008.85 410.97 419.19 427.57 436.12 444.85 453.75 462.82 472.07 481.51 491.15 104,500.00 104,089.03 103,669.84 103,242.27 102,806.15 102,361 30 101,907.55 101,444.73 100,972.66 100,491.15 100,000.00 25,000.00 20,500.00 4,500.00 100,000.00 111. — 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 author has discovered a very simple method of obtaining the present value in this case. It will not be necessary to demonstrate it, but an example will test it. Vai^uation of Bonds. 47 ■ 112. — Suppose the 5% bond, M N, yielding 4%, bought May 1, 1904, were payable October 1, instead of May 1, 1909, or 10| periods. The short period is |. The short ratio (4%) will be 1.0166^. The short interest (5%) will be .02083^. 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 follow- ing schedule: /j\ SHORT TKRMINAI,. Total Net ■D^^Ay Date Interest 5% come 4% Amornsation ^^^^^ Par 1904 May 1 104,827.50 100,000.00 Nov. 1 2,500.00 2,096.55 403.45 104,424.05 1905 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 1906 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 1907 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 1908 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 1909 May 1 2,500.00 2,017.84 482.16 100,409.83 Oct. 1 2,083.33 1,673.50 409.83 100,000.00 27,083.33 22,255.83 4,827.50 48 Thk Accountancy ov Invkstmknt. 113. — Rule for Short Terminals. Ascertain the value for the full periods, disregarding the terminal. To this value add the short interest and divide by the short ratio. 114. — It may be remarked that the same process applies to short initial periods, or even to bonds originally issued between interest dates, and also maturing between interest dates. In the latter case it would be applied twice. 115.— Discounting. Hitherto we have calculated the longest period, and then obtained the shorter ones by multipli- cation and subtraction. We can also work backwards, obtain- ing each value from the next later by addition and division. Thus, beginning at maturity with par 100,000.00 and adding to it the coupon then due 2,500 . 00 102,500.00 We discount this by dividing by 1.02, which gives 100,490.20 the value one period before maturity. To obtain the next value add the coupon 2,500 . 00 102,990.20 Divide again by 1 . 02, giving the next previous value 100,970 . 78 Thus successive terms may be obtained as far as desired. 116. — In the last half year of a bond, its value should be discounted, and not found as in Article 97. Thus, if the bond in question were sold, when only three months remained to maturity, $102,500 would be divided by 101, 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. The theoretically exact value (recognizing effective rates, which is never done in business) would be $100,240.12. To ''split the difEerence " would be an easy way of adjusting the matter and would be almost exact. 117. — Serial Bonds and Various Maturities. Bonds are often issued in series. For example: $30,000, of which $1,000 is payable after one year, another $1,000 after two years, and the last $1,000 30 years from date. Other series are more complex; as, $2,000 each year for 5 years, and $4,000 each year thereafter for 5 years. After finding the initial value for each instalment and adding these together, the aggregate may, for purposes of deriving successive values, be treated as a unit, and multiplied down in one process. The principles in Article Valuation of Bonds. 49 103 must be observed as to that part of the par value, if any, which comes due between balancing periods. 118. — Investment of Trust Funds. A bond which has been purchased by a trustee at a premium is subject to amorti- sation in the absence of testamentary instructions to the con- trary. (Hardcastle on Accounts of Executors, p. 49.) The trustee has no right to pay over the full cash interest, because he must keep the principal intact for the remainder man. If, for example, he were to invest $104,491.29 in a 5% bond having five years to run, and the life tenant were to die at the end of five years, the fund would be depleted by $4,491.29, to the injury of the remainder man. Since this is a 4% basis, he should pay over at the end of the first half year only 4% of $104,491.29 (=$2,089.83), not 5% of $100,000 (=$2,500). He then has $410 . 17 cash to re-invest, and the fund, including this, is still $104,491.29. It may be difficult to invest the $410 . 17 at as favorable a rate as the bonds, very small and very large amounts being most difficult to invest. He can, at least, deposit it in a trust company and receive interest at some rate or other. 119. — At the end of the second half-year the bond interest is only $2,081.62; but the beneficiary is entitled, also, to the interest on the $410 . 17. If this has been re-invested at exactly 4%, the interest thereon is $8 . 20, and the total payable to the beneficiary is $2,081.62 + 8.20 = $2,089.82, practically the same as before, and $418.38 is deposited or invested as before. He now has in the fund $103,662.74 + 410.17 + 418.38 = 104,491 . 29. He has paid over all the new interest earned, and he has kept the corpus or principal intact. Suppose, however, he was not able to get 4% for the $410.17, but only 3%, so that from this source would come only $6.15, making the total income $2,081.62 + 6.15 = $2,087.77. There is a slight falling off in income, but that is to be expected when part of an investment is returned and re- invested at a lower rate. If the re-investment had been at 4>^%, the income would have been $2,090.74, slightly more than the first half year, owing to the improved demand for capital. It might be urged that the beneficiary ought to receive $2,089.83 periodically — no more, no less — being 4% on 50 The Accountancy of Investment. $104,491.29. This would leave $410.17 each half-year to be invested in a 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%. If less, the original fund would be to some extent depleted, and the remainder man wronged; if more, there is too much in the fund, and the life tenant has received too little. It seems, therefore, that the sinking fund principle is not correct in a case like this, and that we should rather recognize a gradual disappearance of capital than constitute a fictitious sinking fund. 120. — Prof. Hardcastle's example (p. 50) expanded to $10,000 instead of $100 — would be scheduled thus: Coupon. Income. Cash. Bond. 200.00 200.00 200.00 200.00 1800.00 152.94 152.13 151.47 150.74 607.28 47.06 47.87 48.53 49.26 192.72 10.192.72 10,145.66 10,097.79 10,049.26 10,000.00 The life tenant would receive, at the end of the first half- year, $152.94; at the end of the second, $152.13 + whatever the $47 . 06 cash had earned ; at the end of the third, $151 . 47 ■+- whatever $94.93 had earned; at the close, $150.74 + whatever $152.46 had earned; and if the cash balance was constantly deposited in the trust company at 3%, the life tenant would receive a uniform income of $152 . 94. 121. — In a case reported in the State of New York (38 App. Div. 419), Justice CuUen 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 impairment of the principal of the trust, to pro- tect 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^^% on the amount invested, nor 5% on the face of the bond, but 2^\% on the investment, or d^^^Q% on the face of the bond. The matter is simply one Vai^uation of Bonds. 51 of arithmetical calculation, and tables are readily accessible, showing the result of the computation. ' ' 122. — The learned judge's example would work out in a schedule as follows, with a slight correction in the initial figures, and applying it to a par of $100,000: Total Interest Income Paid Over Re-invested. Present Value 120,039.00 ' 2,500.00 1,620.52 879.48 119,159.52 ' 2.500.00 1,608.66 891.34 118,268.18 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 20,039.00 123. — This 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 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 j ustice 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 till the very last payment has he overtaken his true share. Thus, if he dies before the maturity of the bonds, it is certain that 52 Thb; Accountancy of Investment. "equal justice" will not have been done, but the remainder man would have altogether the best of it. 124. — To particularize, the ''referee's plan" would be scheduled as follows: Trust Fund. "Referee's Plan." Total Interest Income Paid Over Re-invested Present Value 120,039.00 2,500.00 1,498.05 1,001.95 119,037.05 2,500.00 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 2,500.00 1,498.05 1,001.95 110,019.45 &c. &c. &c. &c. It is unnecessary to continue this further, but by com- paring it with our schedule (K) it will be seen that, if the remainder man received the fund after five years, it would be at such a valuation on the bonds that he could enjoy an income of over 2 . 80 per cent, for the other five years, and yet keep the principal intact. The method of the referee is false and arbitrary. 125. — Single Column Schedule. Instead of distributing the figures of the schedule into four columns, as in our examples, it is frequently easier to ignore the amortisation column and simply add the net income, then subtract the cash interest. Thus, Schedule A would begin as follows: 104,491.29 plus 2,089.83 106,581.12 minus 2,500.00 104,081.12 plus 2,081.62 106,162.74 minus 2,500.00 103,662.74 etc. The Book Values may then be set down at once; the amorti- sations will be their differences; and the Net Income will be the Cash Interest minus the Net Income. By using red ink for Vai^uation op Bonds. 53 the subtrahends (which we indicate by italic figures) setting them down in advance on the proper lines, the addition and subtraction can be performed at one operation. For example: 104,491.29 2,089.83 2,500.00 104,081 .12 2,081. 62 2.500 00 103,662, .74 2,073.35 2,500 00 103,236.09 etc. It will be noticed that the computation of the interest is done without using any other paper. Even with a fractional rate, like that in Judge CuUen's example, 2.7%' per annum, or 1 . 35% per period, the 1%, the .3% and the .05% can be successively written down direct: 120,039.00 1,200.39 360.117 60.019 2.500 00 119,159.526 1,191.595 357.479 59.580 2,500.00 118,268.180 etc. 126. — Irredeemable Bonds. Sometimes, as in the British Consols, there is no right nor obligation of redemption. If the government wishes to pay off it has to buy at the market price. There is, then, no question of amortisation; the invest- ment is simply a perpetual annuity. 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; the book value is ^96. As an investment of ^"96 produces /^ 4, the rate of income is 4 -f- 96 = . 04 J . 127. — Optional Redemption. Sometimes the issuer has the rtg-ht to redeem at a certain date earlier than the date at 54 I^HK Accountancy of Investment. which he must redeem. It must always be expected that this right will be exercised if profitable to the issuer; hence, bonds bought at a premium must be considered as maturing, or reaching par, at the earlier date. Bonds bought at a discount must be considered as running to the longer date. The option of redemption is sometimes attended by a premium. For example: the issuer of a thirty-year bond re- serves the right to redeem after twenty years at 105. Unless you buy at such a basis that after ten years the book value will be 105 or more, this redemption right is a detriment. You must, in that case, consider that you are buying a twenty-year bond, and that the par is 1 . 05 times the nominal par. There is also a form of bond-issue, not uncommon in Europe, where a certain or indefinite number of bonds is drawn by lot each year to be paid off. As they are usually issued at a dis- count, the earlier drawn bonds are the more profitable. The investor, in estimating his income, must assume that his bonds will be the last ones drawn. If drawn earlier, there is a profit exactly the same as that arising from a sale above book value. Note. — When quarterly bonds are offered in competition with those on which interest is paid half-yearly, it is desirable to know how much is added to the value by the fact of quarterly conversion. For this purpose Appendix 2 gives a series of multipliers — decimal fractions by which the premium or discount is multiplied to give the increment thus added to the value of a semi-annual bond. At the resulting price the quarterly bond will pay an effective income equivalent to that of the semi-annual bond. Thus, a 3>^% bond 35 years, yielding 2>^% income, payable semi- annually, is worth 1 . 23234838; the premium is . 23234838. In Appendix 2, on the line 2.50, in the column 3^, we find the multiplier .0109035, by which we multiply .23234838, giving .00253341 as the increment. This, added to 1.23234838, makes 1.23488179 as the value of a quarterly bond, which will be exactly as profitable as the semi-annual bond at 1 . 23234838. If the same bond were to yield 4^%, we have the value for semi-annual interest .82458958, or a discount of .17541042, which is multiplied by the decimal .019578, giving the increment .00343419, and the value of the quarterly bond, .82802377 {= .82458958+ .00343419). Appendix 3 gives in condensed and progressive form the processes explained in the previous chapters. ' B R Af? ,-^^ OF THE OF Thk Accountancy of iNVESTfelfclSSfiS-- CHAPTER IX. Forms of Account — Generai, Principles. 128. — In any system of accountancy on an extensive scale, in order to fulfill the opposite requirements of minuteness and comprehensiveness, 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 sum- marizes the entire contents of one Subordinate Ledger. The General Ledger accounts deal with whole classes of like nature; the Subordinate Ledger with each individual asset or liability, or with 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; it is the function of the Subordinate Ledger to give every desired information as to details even beyond the figures required for balancing — facts not of numerical accountancy only, but descriptive, cautionary, auxiliary. Thus the General Ledger may contain an account, ** Mortgages," which will show the increase and 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, and any thing useful or necessary to be known. 129. — We shall assume that a General Ledger exists with Subordinate or Class Ledgers. We shall also assume that the accounts are to be so arranged as to give currently the amount of interest earned and accruing, the amount which has accrued up to any time, and the amount outstanding and overdue at any time. It hardly seems necessary to argue this point, were it not that many important investors pay no attention to interest until it matures, and some do not carry it into account 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. 56 The Accountancy of Investment. The Profit and lyoss 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. 130. — 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 direct mortgages and loans upon collateral security. Both of these classes of investment are for comparatively short terms, and are usually the result of direct negotiation between borrower and lender, 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. Thk Accountancy of Investment. 57 CHAPTER X. Rkai. Estate Mortgages. 131. — 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 mortgagee being a trustee for all the bondholders. The instruments 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 mort- gage rather than on the bond. Therefore, the mortgagee must be vigilant in seeing that his margin is not reduced to a haz- ardous 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, extensions are made from time to time; or, even more frequently, without formal extension, it is allowed to remain ''on demand," either party having the right to ter- minate the relation at will. A large proportion of outstanding mortgages are thus "on sufferance," or payable on demand. The market rate of interest seldom causes the obligation to change hands at either a premium or a discount, 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 investment must likewise be treated as a unit, both as to principal and income. 132. — It is desirable to know at any time how much is due on principal, allowing for any partial payments. It is also de- 58 The Accountancy of Investment. sirable to know what interest, if any, is due and payable, and to be able to look to its collection. An account of principal and an account of interest are, therefore, requisite. It is better, however, that these two accounts for the same mortgage be adjacent. 133. — Accrued interest need not be considered as to each mortgage. It should be treated in bulk, as the revenue of the aggregate mortgages, as will be explained hereafter. The In- terest account here referred to is debited on the day when the interest becomes a matured obligation, and credited when that obligation is discharged. 134. — 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 containing anything but the bare figures that will make the 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. 135. — The form of mortgage-ledger which seems to the author to be the best, contains four parts : 1. Descriptive. 2. Account of principal. 3. Account of interest. 4. Aux- iliary 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 should be 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. 136. — Mortgages should be numbered in chronological order, and every page or document should bear the number of the mortgage loan to which it refers. 137. — The account of principal (Part 2) may be in the ordi- nary ledger form; but what is known as the balance column, or three column form, will be found more convenient. It contains but one date column, so that successive transactions, whether Real Estate Mortgages. 59 payments on account, or additional sums loaned, appear in their proper chronological order. 138. — The mortgage usually contains clauses which permit the mortgagee, when the mortgagor fails to make any payment for the benefit of the property, like taxes and insurance premiums, to step in and advance the money, which he has the right to recover with interest. It will be useful to have columns, also, for these disbursements and their reimbursements. The complete Part 2 will then take the form shown on page 62. 139. — The Interest account (Part 3) 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 on page 63 has been found advantageous. 140. — Experience shows that the safest way to ensure attention to the punctual and accurate collection of interest is to charge up, systematically, under the due date, every item, and 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 afterward to verify the state of the accounts on any given date. 141. — It is not proposed in this treatise to prescribe the forms of Posting Medium (Cash Book, Journal, etc.) from which the postings in the ledger are made, because these forms are, in recent times, so largely dependent upon the peculiarities of the 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. 142.— 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. 6o Tnn Accountancy of Investment. 143. — In the General I^edger the entry will be simply as above: Interest Due / Interest Accrued $... and this may be a daily, a weekly, a 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 I^edger (Article 130), but the debit entry (Interest Due / ) must be somewhere analysed 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. 144. — The following headings will suggest the require- ments for such a list, the form to be modified to conform to the general system. REGISTER OF INTEREST DUE. MORTGAGES. Date No. Principal Rate Time Interest Total 145. — Part 1 of the Mortgage account is descriptive. Its elements may be placed in various orders of arrangement. It is believed that the form on page 61 combines all the particulars ordinarily required in the State of New York. 146. — Part 4 is not an essential feature, and may be re- placed by card-lists, if preferred. Yet, if there be space, there are advantages in having 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 arranged under the headings on page 64. 147. — The card-form of Mortgage Ledger is also very con- venient in many respects, and the forms here given may be re- arranged so as to suit different sizes of cards. Both in cards Rkai. Estat:^ Mortgages. 6i V. ^ a> C<3 t Ph ^ « -> M ^ WP d .^'^ o ^ ^ ^ a o >^ 'o O N 00 OS O iH iH rH rH CM 03 Q O 0) G) rH CI W rjl to rl iH i-H r-l iH 03 0) OS 0> O) © N 00 OS O o O O O >H 0) OS OS OS OS iH rH fH r-t tH i-i 01 CO t|< lO O O O O O O A OS OS OS /OF THE \ 62 Thk Accountancy of Investment. i > 3 M > s > a n w a pi o > d !^ w o g ^^ 2! H Rkai. Estatb Mortgages. 63 64 The Accountancy op Investment. > > I i o pi W § d 1— I i o o 1 - > w 03 o o t— ( o 1 o p > o a !^ o 8 o > Rkai. Estate Mortgages. 65 and loose leaves it will be helpful to use different colors for pages of different contents. Where interest on different mort- gages falls due in different months, tags marked "J J," "FA," "MS," "AO," "MN," " J D," may project from the interest- sheet like an index, the tags of each month at the same dis- tance from the top. This will greatly facilitate the compiling of the Register of Interest Due. 148. — The Interest Register should always be made up and proved (subject to modifications) in advance. In doing this, instead of making the computations in the register and posting thence to the ledger, a surer way is by "reverse posting"; that is, making the computation from the data in the ledger and entering it there at once, in pencil if preferred; then copying the items into the register, where the total can be proved. When this has been done, we can be sure without further check that the ledger is correct. 149. — It is desirable, also, to have receipts prepared in ad- vance ready for signature. The correctness of these receipts may be assured by introducing them into the ' ' reverse posting ' ' process, as follows : Having made the computation on the ledger, prepare the receipts /^-^m the ledger, copying down the figures just as they appear; from the receipts make up the register, which prove as before. This method may be extended to the notices, if any are sent to the mortgagors, the notice being derived from I^edger account, the receipt from the notice, and the register from the receipt; if the register proves correct, the correctness of its antecedents is established. These interest notices may be made of assistance in the book-keeping, if their return is insisted upon and made convenient. Below the formal notification of the sum falling due on such a date, with all par- ticulars, is a blank form something as follows : * 'In payment of the above interest I inclose check on the ...for $ and request you to acknowledge receipt as below. [Signature) Address " 66 Th^ Accountancy of Investment. The notice upon its being received, together with the check, becomes a ** voucher- with-cash," and the credits on the cash book, and the interest page of the Mortgage I^edger are made directly from the documents. Book-to-book posting, which formerly was the only method of re-arranging items, is becoming obsolete, being superseded in many businesses by voucher or document posting. By the carbon process the notice and the receipt may be filled in simultaneously in fac- simile. 150. — General Ledger, Mortgages Account. The Class account * ' Mortgages ' ' in the General I^edger is simply kept to show aggregates. Its entries are, as far as possible, monthly, the posting-mediums being so arranged as to give a monthly total of the same items which have already been posted in detail to the Mortgage Ledger. The standard form of Ledger account may be used, or the three column. In the former, the debits and credits of the same month should be kept in line, even though one line of paper be wasted. Dr. [Form 1.] MORTGAGES. Cr. 1904 Jan. Feb. March April May June July 1-31 1-29 1-31 1-30 1-31 1-30 Balance Total loaned Balance 169.000 12,000 10,000 50.000 20.000 5,000 10.000 276,000 182.000 00 Jan. March April May June 1-31 1 31 1-30 1-31 1-30 30 Total paid in Balance 7,000 32,000 40,000 12 3 000 182,000 276,000 [Form 2.] MORTGAGES. 1904 Dr. Cr. Balance Jan. Feb. March April May June Transactions for month Transactions for half year 12,000 10,000 50 000 20.000 5,000 10,000 00 00 00 00 00 00 7,000 32000 40.000 12,000 3,000 00 00 00 00 00 169,000 174.000 184,000 202,000 182,000 175,000 182,000 00 00 00 00 00 00 00 107,000 00 94,000 00 4- 13,000 00 July 1 i 182 000 00 Rkal Estate Mortgagks. 67 151. — In order to keep the fullest control of the interest accruing and falling due periodically, it is useful to keep tab- ular registers, classifying the mortgages, first, by rates of interest; and second, by the months in which the interest comes due. Those investors who require all interest to be paid at the same date can dispense with the latter. The two presentations or developments may be on opposite pages, both proved by the same totals. Mortgages CivAssieied by Rates of Interest. Date Total Changes zy^% 4% ^%% 5% 6% 1904 Jan. 169,000 11,000 43,000 7,000 262 — 7,000 162,000 12,000 984 + 50,000 12,000 60,000 5,000 Feb. 174,000 11,000 36,000 38,000 60,000 5,000 Mortgages Classified by Dates. Date Total Changes J J FA MS ; AC MN JD 1904 Jan. 169,000 23,000 30,000 4,000 8,000 7,000 262 -~ 162,000 12,000 984+ 12,000 90,000 14,000 7,000 Feb. 174,000 23,000 42,000 4,000 8,000 90,000 7,000 The numbers in the column headed ' ' Changes ' ' are the serial numbers of the respective Mortgages. 68 Thb Accountancy of Investment. CHAPTER XI. Loans on Coi.i 1 6 S 70 Thb Accountancy of Investment. CHAPTER XII. Interest Accounts. 156. — Interest is earned and accrues every day; then, at convenient 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 book-keeping formulas: 1. Interest Accrued / Interest- Revenue. 2. Interest Due / Interest Accrued. S. Cash I Interest Due. Frequently we see the three accounts, Interest-Revenue, In- terest Accrued and Interest Due, are confused under the one title ' ' Interest, ' ' although they have three distinct functions. 1. Interest- Revenue (which alone may be termed simply " Interest ") shows how much interest has been earned during the current fiscal period. 2. The balance of Interest Accrued shows how much of those earnings and of previous earnings has not yet fallen due. 3. The balance of Interest Due shows how much of that which has fallen due remains uncollected. 157. — The first of the three entries in Article 149 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. 158. — 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 be the amount of the entry * ' Interest Accrued / Interest- Revenue. " Or a daily rate for the entire investments (or entire class) may be established, and this is used without change, day after day, until some change in the principal or in the rate causes a variation of the daily increment. The most complete and accurate method is to keep a double register of interest earned: Jirst, by daily additions; second, by monthly aggregates, classified under rates and time. Interest Accounts. 71 159. — To exemplify 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 are $100,000 running at 4%, $60,000 at 4>^%, 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 4>^%, and one of $6,000 at 5%. 160. — We begin by establishing the daily rate as follows: One day at 4% on $100,000 11.11,11 One day at 4)4% on $ 60,000 7 .50 One day at 5% on $150,000 .20.83,33 Daily rate 39.44,44 161.— The decimals are carried out two places beyond cents, and only rounded in the total. The Daily Register will then be conducted as follows: Daii,y Register of Interest Accruing. Date No. Principal I,ess Principal More Rate 1 2 3 4 5 6 7 647 453 981 982 10,000 5,000 15,000 6,000 4 5 5 39.44,44 1.11,11 38.33,33 69,44 37.63.88 1.87.5 .83,33 39.44.44 39.44,44 38.33,33 38.33,33 38.33,33 37.63,88 37.63,88 8 9 10 40.34.72 40.34,72 40.34,72 15.000 21,000 390.21 Balances at Close 90,000 75,000 151,000 4 5 Proof of Rate One day 10. 9.37,5 20.97,22 316,000 40.34,72 162. — 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. 72 Thk Accountancy of Invkstmknt. Monthly Summary of Interest Accruing. Date No. Paid off Remaining Rate Days 2 5 7 10 647 453 981 982 10 000 5,000 15,000 6,000 90,000 60 000 145,000 4 5 ^% 5 4 ■ W2 5 2 5 3 3 10 10 10 2.22,22 3.47,22 5.62.5 2.50 100.00 75.00 201.38,88 316,000 390.21 163. — 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 for the daily, and one-half for the monthly. If an accurate daily statement of affairs is kept, probably the Daily Interest Accrued will form part of that system. Again, the interest on Mortgages, on Bonds, on lyoans, on Discounts, may be separated or be all thrown together. In all such respects the individual circum- stances must govern, and no precise forms can be prescribed. Our main contention is that in some manner interest should be accounted for When Earned rather than When Collected, or W^hen due. 164. — The General I,edger accounts of Interest, Interest Accrued and Interest Due will now be exemplified 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. Interest Revenue. Mortgages. 1904 June 30 Carried to Profit and I^oss 4270 60 1904 Jan. Feb. March April May June 1-31 1-28 1-31 1-30 1-31 1-30 Total Earnings ^ a V CIS (U « Jz; I I CO ^ ^ ^ ^ ^ (^ I >Q ^ Ji (U - p. ^ ^ ^ 6 ,2i «1 CO t>^ CO cd Ph ^ ^ a O s W ;z; S 8 § i 3 § SS 50-* Of* 35 <* (N to 8^ is £•§ 5£ M a Q .2 a S o r- w< B Cfl < ^^ ^ 78 Ths Accountancy of Investment. in Article 141; in fact, they are but sub-divisions of the same Register. Of course, the cash interest is alone considered. 173. — The Interest pages of the Bond lycdger are also similar to those of the Mortgage Ledger (Article 136), but the dates of interest due may be printed in advance, there being but little chance of partial payments disturbing their orderly arrangement. 174. — The paging of the Bond Ledger will probably be geo- graphical, as far as possible, in respect to public issues, and alphabetical in respect to those of private corporations. The loose-leaf plan permits an indefinite number of classifications to choose from. The date-tags suggested in Article 144 are especially useful for pointing out dates for interest falling due, as "J J," '*FA," etc. 175. — The entries of amortisation are made directly from the schedules of amortisation, the preparation of which has been fully taught in Chapter VIII. But it is necessary, also, to make up a list of these several amortisations in order to form the General Ledger entry: Amortisation / Bonds, or^ Amortisation / Premiums, 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 complete list of the holdings, which may give the par, cost, book and market values, the titles of the securities being written but once. Bond Statement for the Hai,e Year Ending. Name and Description Amorti- sation Book Value Par Value Original Cost Market Value The total of the second column will form the entry for amorti- sation. The next three columns will corroborate the General Ledger balances. Bonds and SimiIvAr Securities. 79 We have provided in this form for amortisation 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 : ' ' amortisation ' ' and * * accumulation. ' ' 176. — While the book value is the proper one to be intro- duced into the General lycdger, the par is very necessary, and sometimes the cost, and these requirements inevitably introduce some complexity. There are two modes of effecting the purpose : I. By considering the par and cost as extraneous infor- mation and ruling side columns for them beside the book value. II. By dividing the account into several accounts, by the proper combination of which the several values may be obtained. 177. — Plan I. will preserve the conformity of the Bonds account with the Bond I^edger 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. 178.— Plan II. 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 personalistic 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 we may adopt whichever is most suitable to our purposes. 179. — If original cost is disregarded, or deemed easily obtainable when required, the accounts may be 1. Bonds at Par. 2. Premiums. 3. Discounts. or, 1. Bonds at Par. 2. Premiums and Discounts. 8o Ths Accountancy of Investment. If premiums and discounts are kept separate, Premiums account must always show a debit balance, being credited for amorti- sation ; Discounts account must show a credit balance, being debited for accumulation. If the two are consolidated, the net amortisation only will be credited (see Art. 173); or, if the greater part of the bonds were below par, the net accumulation only would be debited. The choice between one account and two for premiums and discounts would be 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 on pages 83 and following, an example of each. We shall hereafter confine the discussion to premiums, leaving the cases of discount to be determined by analogy. 180. — Where it is deemed necessary to keep account of cost also, as well as of par and book value, the diflSculty 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 amortisation. This carrying of a dead value, which is somewhat artificial, necessitates the carrying, also, of an artificial annulling or off- setting account, the sole function of which is to express this departed value. We may call this credit account ' 'Amortisation Fund." It is analogous to Depreciation and Reserve Funds. The part of the premiums which has been extinguished by the Amortisation Fund may be designated as ' * Premiums Amor- tised," or "Ineffective Premiums," while the live premiums may be styled " Effective Premiums," being what in Art. 177 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 rejection of ineffective premiums upon redemption or sale. 181. — There are two ways of carrying on these accounts, differing as to Premiums. We may keep two accounts : ' 'Effec- tive Premiums" and "Amortised 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. Amortisation Fund. Bonds and Simii^ar Securities. 8i or^ a. Bonds at Par. c. Effective Premiums. d. Amortised Premiums. e. Amortisation Fund. **a*' will in both schemes be the same; "e" will also be the same, "b" is the sum, c+d. In the former, the cost is a-f-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. 182. — Account a. Bonds at Par, is debited for par value of purchases and credited for par value of sales. Its two only entries are : Bonds at Par/Cash. Cash/Bonds at Par. 183. — 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 premium at this time. 184. — When premiums are written off, on the first plan, there is but one entry : crediting the Amortisation Fund and debiting the Profit and I^oss account or its sub-division. Amortisation/ Amortisation Fund. 185. — The second plan involves not only this process, but a transfer from Effective to Amortised 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 : Premiums Amortised/ Effective Premiums. Amortisation/Amortisation Fund. 186. — The word "Amortisation" has been used in the specimen entries as the title of an account tributary to Profit and I,oss. At the balancing period it may be disposed of in either of two ways'. : it may be closed into Profit and lyoss direct ; or it may be closed into Interest account, the balance of which will enter into Profit and Loss at so much lessened a figure. In the former method the Profit and I^oss account will show, on 82 Thk Accountancy of Investment. the credit side, tlie gross cash-interest, and on the debit the amount devoted to amortisation ; the second method exhibits the net income only. Whether it be preferable to show both elements, or only the net resultant, will be determined by expediency. 187. — In Articles 169 and 171 we discussed two methods of keeping account of amortisation : the first (169), where any in- cidental amortisation occurring in the midst of the period is at once entered; the second, exemplified in 171, where all such entries are deferred to the end of the period, and comprised in one entry in the General I^edger. If the latter method be adopted, the Amortisation account may be dispensed with altogether, and the total amount amortised (which is credited to Bonds, or to Premiums, or to Amortisation Fund) may be debited at once to Profit and I,oss or to Interest, without resting in a special account. A single item, of course, needs no machinery for grouping. 188.— Irredeemable Bonds (Art. 126) merely lack the element of amortisation, and require no special arrangement of accounts. The par is purely ideal, as it never can be demand- ed and is merely a basis for expressing the interest paid. What the investor buys is a perpetual annuity. If he buys such 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 be $100 par at 50% premium, or $200 par at 25% discount. The par value is really non-existent ; and this illustrates the absurdity of re- ducing even redeemable securities to par, which is practised by some investors, par being, except at the moment of maturity, an unreal sum. 189. — We will now give examples of the two plans for the General Ledger outlined in Articles 176 to 187. We will sup- pose that on Jan. 1, 1901, the following lots of bonds are held: Bonds and Similar Securitibs. 83 Par 100,000 100.000 10.000 210,000 January 1, 1901. Book Vai^ue 5% Bonds, J. J., due Jan. 1, 1911, net 2.7%; value 120,039.00 original cost, 124,263.25 3% Bonds, M. N., due May 1, 1904, net 4%; value 96,909.10 original cost, 93,644.28 4% Bonds, A. O., due Oct. 1, 1902, net 3%; value 10,169.19 original cost, 10,250.00 Totals 227,117.29 The premiums on the 5% and 4% bonds amount to $20,208.19. The discount on the 3%s is $3,090.90. The net premium is $17,117.29. The total original cost was $228,157.53, Dr. P1.AN I (Art. 176) For Genkrai. I^bdgkr. Bonds Account. Cr. 1901 Par Cost 1901 June 30, A mortisation Dpc. 31, 1902 June 30, " Oct. 1, Redeemed Dec. 31, Amortisation 1903 June 30, Dec. 31, " Balances Par Cost Jan. 0, Balances 210,000.00 228,157.53 227,117 29 10,000.00 200,000.00 10,250.00 217,-907.53 488.76 492.59 496 40 10,000 00 475 21 453.63 456.68 214.254.02 1904 210,000.00 228,157.53 227.117.29 210,000.00 228,157.53 227,117.29 Jan. 0, Balances 200,000.01; 217,907.53 214,254.02 84 Thk Accountancy of Investment. PI.AN II, 1 (Art. 179), For General IvEdger. Dr. Bonds at Par. Cr. 1901 Jan. 0, Balance. .210.000.00 1902 Oct. 1, Redeemed 10,000.00 Dr. Premiums. Cr. 1901 Jan. 0, Balance. .20,208.19 1901 June 30, Amortisation 926.94 Dec. 31, •' 939.54 1902 June 30, " 952.28 Dec. 31, ♦* 940.21 1903 June 30, " 927.93 Dec. 31, ** 940.47 Dr. Discounts. Cr, 1901 1901 June 30, Accumulation.. . . . .438.18 Jan. 0, Balance 3,090.90 Dec. 31, ({ ....446.95 1902 June 30, (( ....455.88 Dec. 31, (( ....465.00 1903 June 30, <( ....474.30 Dec. 31, <« ....483.79 Bonds and Simii^ar Skcuriti^s. 85 P1.AN II, 2 (Art. 179), For Gknkrai. I^kdoer. ORIGINAI, COST OMITTED. Dr. Bonds at Par. Cr. 1901 Jan. 0, Balance. .210,000.00 1902 Oct. 1, Redeemed. .10,000.00 Dr. Premiums and Discounts. Cr. 1901 Jan. 0, Balance. ,17,117.29 1901 June 30, Amortisation. . . .488.76 Dec. 31, t( . .492.69 1902 June 30, t( ...496.40 Dec. 31, ^% Bond 4% Bond 4>^% Bond 5% Bond 6% Bond 7% Bond 2.50 .0186918 .0109035 .0083075 .0070094 .0062306 .0053405 .0048460 2.55 .0211827 .0117062 .0087653 .0073325 .0064845 .0055259 .0049982 2.60 .0242963 .0125981 .0092557 .0076725 .0067490 .0057168 .0051538 2.65 .0282994 .0135948 .0097825 .0080309 .0070247 .0059133 .0053129 2.70 .0336369 .0147161 .0103498 .0084092 .0073124 .0061158 .0054758 2.75 .0411092 .0159869 .0109604 .0088091 .0076128 .0063245 .0056424 2.80 .0523175 .0174392 .0116261 .0092325 .0079269 .0065397 .0058131 2.85 .0709980 .0191148 .0123475 .0096815 .0082556 .0067617 .0059867 2.90 .1083586 .0210697 .0131344 .0101586 .0085999 .0069909 .0061668 2.95 .2204401 .0233800 .0139962 .0106665 .0089610 .0072275 .0063501 3.00 .0261523 .0149442 .0112081 .0093401 .0074721 .0065381 3.05 : 2278844 .0295406 .0159919 .0117871 .0097387 .0077249 .0067308 3.10 .1158030 .0337759 .0171560 .0124075 .0101582 .0079864 .0069284 3.15 .0784424 .0392212 .0184570 .0130737 .0106003 .0082571 .0071311 3.20 .0597619 .0464815 .0199206 .0137912 .0110670 .0085374 .0073392 3.25 .0485535 .0566458 .0215794 .0145661 .0115604 .0088279 .0075528 3.30 .0410812 .0718922 .0234750 .0154055 .0120827 .0091292 .0077721 3.35 .0357438 .0973026 .0256622 .0163178 .0126367 .0094418 .0079975 3.40 .0317407 .1481232 .0282139 .0173131 .0132253 .0097664 .0082291 3.45 .0286271 .3005843 .0312196 .0184031 .0138518 .0101037 .0084672 3.50 .0261361 .0348482 .0196021 .0145201 .0104545 .0087121 3.55 .0240981 ! 3092587 .0392710 .0209273 .0152344 .0108195 .0089640 3.60 .0223997 .1567976 .0447993 .0223997 .0159998 .0111998 .0092234 3.65 .0209625 .1059770 .0519071 .0240452 .0168217 .0115963 .0094905 3.70 .0197306 .0805666 .0613841 .0258964 .0177069 .0120097 .0097656 3.75 .0186629 .0653202 .0746517 .0279944 .0186629 .0124419 .0100493 3.80 .0177287 .0551559 .0945530 .0303920 .0196985 .0128936 .0103417 3.85 .0169043 .0478956 .1277216 .0331662 .0208242 .0133662 .0106435 3.90 .0161715 .0424503 .1940585 .0363860 .0220521 .0138613 .0109549 3.95 .0155159 .0382150 .3930687 .0402002 .0233969 .0143805 .0112766 4.00 .0149260 .0348273 .0447780 .0248767 .0149260 .0116091 4.05 .0143920 .0320549 ! 4029757 .0503720 .0265116 .0154991 .0119525 4.10 .0139065 .0297444 .2039616 .0573642 .0283280 .0161022 .0123080 4.15 .0134631 .0277893 .1376231 .0663540 .0303580 .0167379 .0126758 4.20 .0130567 .0261134 .1044536 .0783402 .0326418 .0174089 .0130567 4.25 .0126830 .0246613 .0845532 .0951223 .0352305 .0181185 .0134516 4.30 .0123379 .0233906 .0712855 .1202944 .0381887 .0188696 .0138611 4.35 .0120183 .0222693 .0618086 .1622475 .0416019 .0196664 .0142859 4.40 .0117216 .0212726 .0547008 .2461535 .0455840 .0205128 .0147271 4.45 .0114453 .0203807 .0491724 .4978708 .0502900 .0214138 .0151856 4.50 .0111874 .0195780 .0447497 .0559371 .0223748 .0156624 4.55 .0109462 .0188517 .0411310 ; 5089964 .0628391 .0234021 .0161586 4.60 .0107200 .0181914 .0381154 .2572791 .0714664 .0245028 .0166755 4.65 .0105075 .0175886 .0355637 .1733731 .0825586 .0256849 .0172143 4.70 .0103075 .0170359 .0333765 .1314200 .0973481 .0269579 .0177766 4.75 .0101188 .0165274 .0314808 .1062679 .1180532 .0283328 .0183638 4.80 .0099407 .0160581 .0298221 .0894664 .1491106 .0298221 .0189777 4.85 .0097722 .0156257 .0283585 .0774795 .2008728 .0314410 .0196201 4.90 .0096125 .0152198 .0270575 .0684893 .3043969 .0332069 .0202931 4.95 .0094610 .0148441 .0258934 .0614968 .6149681 .0351410 .0209989 5.00 .0093171 .0144933 .0248457 .0559028 .0372685 .0217400 The Accountancy of Investment. 93 APPENDIX III. Summary of Compound Interest Processes. To find the Ratio of Increase Add 1 to the Rate of Interest. To find the Amount ^$1 Multiply 1 by the Ratio as many times as there are periods. To find the Present Worth of%\,or to discount $1 Divide 1 by the Ratio as many times as there are periods. To find the Total Interest Subtract 1 from the Amount. To find the Total Discount Subtract the Present Worth from 1. To find the Amount of an Annuity ^$1 Divide the Total Interest by the Rate of Interest. To find the Present Worth of an Annuity of%\ Divide the Total Discount by the Rate of Interest. 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. To find what Annuity (Sinking Fund) will produce $1 Divide 1 by the Amount of the Annuity. To find the Premium or Discount on a Bond Consider the Difference of Interest as an Annuity to be valued, and find its Present Worth. BY THE SAME AUTHOR : EXTENDED BOND TABLES GIVING ACCURATE VAI^UBS TO EIGHT PI^ACES OF DECIMAI.S OR TO THE NEAREST CENT ON $1,000,000 : : : New York, 1905. BUSINESS PUBLISHING CO. ^■^■' >^^^:^i^ii^^-^W^\>^^ ' ■' '^h'i?\v:'i'*ih^'''/U :';ir;> i^-s''"?''' v!- A^P^ LIBRARY USE RCTUKN TO DESK FROM WHICH BORROWED MAIN LIBRARY CIRCULATION DEPARTMENT THIS BOOK IS DUE^BEFORE CLOSING TIME * ON LAST DATE STAMPED BELOW tKmrw-j^^^ im- ffiCQlR. fEB2576 LD 21-100wi-8,'34 YC 6'^37l