The Poetry of Earth is Never Dead James S, Elston HG8065 ?K5i"mt"' '■"'""' ^'HiiMiiiiim5ii!li.i&,.P,.,fe^ olin 3 1924 032 409 579 The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924032409579 THE THEORY OF FINANCE BEING A SHORT TREATISE ON THE DOCTRINE OF INTEREST AND ANNUITIES-CERTAIN GEORGE KING, F.I.A., F.F.A. ACTUARY OF THE LONDON ASSUEANCH CORPORATION THIRD EDITION ^rfniEtr aitir ^aHisIjElr for Vcjt '^tixte.-AA ^nmij of (Ktitttlnttglr hji CHARLES AND EDWIN LAYTON, 56, FARRINGDON STREET, LONDON. 1898 PREFACE TO THE FIRST EDITION. The following treatise consists of Notes of Lectures on Interest and Annuities- Certain, delivered to tie Students for tlie Intermediate Examination of the Institute of Actuaries. The course of lectures includes also the mathematical theory of Life Contingencies; and the author's intention originEiUy was^ to have edited and published the whole of his notes under the title of the Elements of Actuarial Science. The absence of such a guide is very much felt by students ; and although it has long been the declared intention of the Institute to bring out a complete Text-Book, yet it was thought that a less pretentious work, speedily published, would in the meantime fill up the gap, while afterwards it might serve as a useful introduction to the more elaborate volume. At the close of last year the author accepted the invitation, with which he was honoured by the Council of the Institute of Actuaries, to prepare that part of their Text-Book which is to treat of the Science of Life Contingencies, and his original intention therefore fell to the ground. He thought, never- theless, that, as the five chapters on Interest and Annuities- Certain were almost finished, and could be made, with but slight alterations, complete in themselves, they might not inappropriately be published independently; and he offered iv Preface. the manascript to the Actuarial Society of Edinburgh. He feels much gratified and flattered at the cordial way in which his proposal was accepted, and he desires to express his hearty thanks to the Committee of that Society for the uniform and courteous consideration with which they have received his suggestions. He hopes that the end both they and he have in view may be secured , and that students more particularly, but also the actuarial profession generally, may derive benefit from the labour which was undertaken without the prospect of any pecuniary return. The available space being very limited, much has been omitted which, under more favourable circumstances, would have been, and with great advantage, included. An. eiiort has been made to bring this branch of actuarial science up to the latest date; but those formulas and methods which are now more of historical than practical interest have been intentionally passed over, and if fault be found on this account, lack of room must be the excuse. The work is for the most part a compilation, and it does not profess to contain much that is original. A word of explanation on this point is necessary, as, except in rare instances, it has not been thought desirable to mention other authors. A consecutive style has been adopted, which would have been broken by frequent quotations or references. The greater portion of the matter which is not given in the older text-books will be found in the Journal of the Institute oj Actuaries, to which the initials J.I. A., sometimes appearing in the following pages, refer. A competent knowledge of algebra has been assumed throughout, and no attempt has been made to bring down the demonstrations, or to explain the results, to those who do not possess such knowledge. It is unsafe for the unlearned to Preface. v deal witli subjects whiclij from the very nature of the case, are beyond their capacity. Great care has, however, been taken to make all the analysis clear and intelligible to an average mathematician ; and although some passages, especially perhaps in the third chapter, may not be found easy, it is hoped that, by attention and patience on the part of the reader, all difficulties will be overcome. In many places more fulness of explanation would have been a decided gain had space permitted. To render the book complete in itself, at least in so far as students are concerned, a few tables have been appended, so that the reader may exercise himself as he proceeds, by setting himself numerical examples. It need hardly be said that theoretical knowledge without practical skill is of little use. G. ,K. LoHDOK, Afril 188;^. PREFACE TO THE SECOND EDITION In preparing a Second Edition of this little work, it has not been thought necessary to make many changes. To render the investigation more complete, a paragraph (No. 25), with two new equations (Nos. 33 and 34), displaying the con- nection between the nominal and effective rates of interest, has been added to Chapter I. ; and in Chapter II. more attention has been given to the question of the redemption money, in the case of an annuity, to pay the purchaser one rate of interest on his investment, and to return his capital at another rate ; and Mr. G. F. Hardy's two important formulas (Nos. 42 and 43) have been introduced for finding the rate of interest in a term annuity. The numbering of the paragraphs and formulas has thus been unavoidably changed. With these exceptions, the alterations have been scarcely more than verbal. G. K. London, .iugust 1891. PKEFACE TO THE THIRD EDITION. It lias not been thought necessary for the Third Edition of the Theory of Finance to do much more than reprint tlie Second. Beyond a few verbal alterations, the only changes have been in Chapter II., article 10, to insert a new and elegant formula for the value of an annuity at simple interest, taken from the collection by Messrs. Ackland and Hardy of " G-raduated Exercises and Examples " for Actuarial Students ; and a new article, No. 43 [a) of Chapter II., dealing with the generalization (due to Mr. George J. Lidstone) of the problem to find the value of an investment, which is to pay the purchaser one rate of interest, and to return him his capital at another rate. The suggestion was made by a competent authority, so to revise Chapter III. on Varying Annuities as to render unnecessary the use of the Calculus of Finite Differences. After careful consideration, however, it was decided to adhere to the old arrangement. The use of the method of Finite Differences makes the analysis simple, and the formulas general; whereas, without recourse to that method, each case must be treated separately, and success in solving the problems depends upon knowledge of algebraical expansions, and skill in manipulation. It will be easier for tlie student to learn the few elementary formulas of finite differences required, than it will be to try to dispense with them. ISTevertheless, he will find it a useful exercise to attack the examples without that powerful aid. G. K. LoKDON, March 1808. TABLE OF CONTENTS. CHAPTER I.— ON INTEREST. 1. Definitions of Capital : Interest— 2. TTnits of Value— 3. Kate of Interest: Principal — 4. Simple and Compound Interest : Simple Interest a contra^ diction — 5. Notation for Simple Interest: Fundamental Equations — 6. Discount — 7. Fractional Interest — 8. Notation for Compound Interest— 9. Fundamental Equations — 10. Present Value — 11. Interest on 1 in « years — 12. Nominal and Effective Rates of Interest — 13. Momently Interest: Force of Interest — 14. Tabular Illustrations — 15. Symbol i used in general sense — 16. Fractional Interest — 17 and 18. Discount — 19. Rate of Discount distinguished from Rate of Interest — 20. Commercial and Theoretical Discount — 21. Discount in Fractional parts of a year — 22. Com- pound Discount, Momently Discount, Force of Discount — ^23. Force of Interest and Force of Discount illustrated by Differential Calculus — 24 and 25. Series connecting i, 4™\ v, d, and S : Tabular Illustrations — 26. In how many years will money double itself at Compound Interest ? — 27 to 29. Equated time of payment, or average due date. CHAPTER II.— ON ANNUITIES CERTAIN. 1 to 5. Definition of Annuity, Status, Perpetuity, expression " Entered-on," Annuity-due, Annual Rent of Annuity — 6. Notation — 7. Amount of Annuity at Simple Interest— 8. Present Value of ditto — 9. Formulas for approximating to present value — 10. Formulas of Difilerential Calculus and Logarithmic Series for present value — 11. Numerical Examples — 12. Amount of Annuity at Compound Interest — 13. The same without Alo-ebra 14. Amount of Annuity payable p times a year, interest convertible a times — 15. Verbal Statement of Amount of Annuity — 16. Present Value of Annuity — 17 and 18. The same without Algebra, and also by means of Perpetuities — 19. Value of Annuity payable p times a year, interest convertible q times — 20. Verbal statement of Value of Annuitv ■ 21 to 24. Continuous and other Annuities at interest convertible nioraentlv and at various periods — 25. Tabular Illustrations — 26. Constant Factors for transforming Annuities — 27. Perpetuities at Momently Interest 28. Annuitv payable for n years and a fraction — 29. Defei'red Annuities and Renewal of Leases — 30. Fines for renewing Leases — 31 to 34. Sinking Fund and th Table of Contents. ix Component Parts of a Term Annuity — 35. Amortisation — 36 and 37. Re- demption of a Loan by means of a Sinking Fund — 38. Connection between Annuity which a unit will purchase, and the Sinking Fund to redeem a unit — 39. Numerical Illustration of Sinking Fund — 40 and 41. Value of Annuity to pay one rate on purchase-money, with Sinking Pimd to accumulate at another rate — 42. Capital outstanding, and lledemption Money, in this class of Annuity — 43. NumericallUustration — 43a. General case of Remunerative and Accumulative Rates of Interest — 44 to 56. Rate of Interest in Term Annuity — 57 to 64. Government Loans redeemable by Term Annuities ; also Numerical Illustrations— 65. Examples. CHAPTER III.— ON VARYING ANNUITIES. 1 and 2. Processes of Differencing and Summing — 3 and 4. Pigurate Numbers — 5 and 6. Notation, and connection between successive orders — 7 to 9. Values of the terms of the successive orders — 10. Annuities, the payments of which are tho corresponding terms of the orders of Figurate Numbers — 11 to 14. The Amounts of such Annuities, with Numerical Example — 15 to 19. The Values of such Annuities, with Numerical Example — ■ 20 to 22. Similar Perpetuities — 23. Value of Perpetuities found by means of Differential Calculus — 24. Values of Annuities and Perpetuities found without the aid of Algebra — 25. Values and Amounts of Varying Annuities, the payments of which form series with a finite number of orders of differences— 26 to 29. Examples of the Formulas— 30 to 34. Values of Annuities, the payments of which are in Geometrical Progression — 35. Acknowledgment of the late William M. Makeham. CHAPTER IV.— ON LOANS REPAYABLE BY INSTALMENTS. 1 and 2. Statement of the Problems that occur — 3. Notation — 4 and 5. To find the Value of a Loan, bearing one rate of interest, to yield another rate — ■ 6 and 7. Modifications of Formulas to meet various circumstances — 8. Numerical Illustrations — 9. Comparison of Makeham's method with those previously in iise — 10 and 11. To determine the Rate of Interest yielded by a given Loan — 12. Numerical Illustration — 13. Formula for Second Approximation — 14. Comparison of General Formula of this Chapter with Baily's Formula, No. (35) of Chap. II. — 15 and 16. Improved Formula for the Rate of Interest yielded by a given Loan, and comparison of the same with Barrett's Formula, " No. (38) of Chap. II. — 17. Numerical Illustrations — 18. Caution as to applying Mathematical Formulas. CHAPTER v.— ON INTEREST TABLES. 1 and 2. Advantages and uses of Interest Tables — 3 to 5. Functions usually tabulated, and general form of tables — 6 and 7. Brief notice of a few of the principal tables, with their leading characteristics — 8 and 9. Extension Table of Contents. of uses of limited tables — 10 and 11. Continued method of constructing tatles — 12 to 14. To construct by multiplication a table of (l + i)", wltli numericiil examples — 15 to 17. To construct a table of log; (1 +«')", with numerical examples. Cori-ection for last place of decimals — 18. To construct by multiplication a table of «" — 19. Verification Formulas — 20. To construct by addition a table of ij — 21 and 22. And the same by multiplication; numerical illustrations — 23. Verification Formula — 24 to 26. Construction of tables of afl, and Verification Formula — 27. Verification by means of differencing — 28. Gray's Tables and Gauss's Logarithms — 29. Construction of tables of log i^| — 30. And of log a,7] — 31 to 33. Construction of tables of o^l and «^| , and of log a,7; and log s^l • TABLES. Table No. 1. Amount of 1. 2. Present Value of 1. „ 3. Amount of 1 per annum. , 4. Present Value of 1 per annum. „ 6. Annuity which 1 will purchase. THEORY OP PINANOE. CHAPTER I. On Interest. 1. Capital is wealth appropriated to reproductive employment, and, in the words of the late J. S. Mill, "the gross profit on capital may be distinguished into three parts, which are respectively the remuneration for risk, for trouhle, and for the capital itself, and which may be called insurance, wages of superintendence, and interest." It is with the last of these, Inteeest, that we have at present to do. 2. In civilised communities wealth is measured in money, and therefore in the following investigations it is of money that we shall speak ; but evidently it is immaterial, in laying down general propositions, whether the standard units are of one actual value or of another, so long as we remain consistent with ourselves, and use the same standard throughout. We shall therefore, as the most convenient course to suit all currencies, treat of units of value, withouti specifying to what currency these units belong. 3. The Kate of Interest is, in scientific investigations, the ratio between the interest earned in one unit of time, — generally a year — and the original sum invested, and it may be represented by the amount of interest earned on one unit of money in one unit of time. In ordinary commercial transactions interest is calculated at the rate_per cent., instead of at the rate per unit ; and therefore, to convert the commercial rate into the corresponding rate employed in mathematical researches, we must divide by 100. Thus, if 5 per cent, be the rate of interest charged by a banker for an advance, the corresponding mathematical rate is -05. The capital invested is called the Pbincipal. 2 Theory of Finance. [ci-iap. I. 4. It is usual to say that there are two kinds of interest, Simple and Compound. If the interest be calculated oo the original capital only, for whatever length of time the loan may bave been allowed to remain outstanding with interest unpaid, then it is called Simple interest ; but if the interest on each occasion of its becoming due be added to the original debt, and if the interest for each succeeding period be calculated on the original debt so increased by all the previous accumulations of interest, then interest is said to be Compound. It will, however, be found that the assumption of simple interest leads continually to reductio ad aisurdum, which is sufficient evidence that a fallacy somewhere lurks in the supposition. Money, whether received under the name of principal or of interest, can always be invested to bear interest, and therefore, from the very nature of the case, simple interest is impossible. It is true that borrower and lender may between themselves agree for onlj^ simple interest ; but such agreement does not prevent the borrower from investing the interest which is thereby allowed to remain in his hands, and securing interest thereon ; and it is because this interest on interest is itrnored in the doctrine of simple interest, that the mathematical formulas fail. 5. If at Simple Inteeest— P=The Principal. (S'=The Amount to which that principal wiU accumulate. ra=The Term, or the number of years the principal is under investment. «=The Kate of Interest, or the interest on 1 for one year. Then, since the interest on P for n years is evidently niF, and since S is equal to P increased by its interest, therefore S=P(l + ni) .... (J) and, by algebraical transformation, P=T:r-- • . . . (2) l + ni ^ ■' S-F '^=^P- • • . . (3) ._S~I> *-^^P" • • • • (4) We have defined P and -S as Principal and Amount respectively but equation (2) exhibits the relationship between them in a th ' light. S is seen to be a Sum Due at the end of the rie ' r! ^ CHAP. I.] On Interest. 3 P its Present Value; and if these meanings of the symbols he substituted for those already given, all the four equations still hold. 6. The difference between a sum due and its present value is called Discount ; and if we write D for the discount, we have D^S-P .... (5) niS T-r- (6) 1 + m = niP .... (7) When the sum due is unity, and the period one year, we write d for B. 7. We have not m.ade any supposition as to the value of n. If n be fractional, and equal to — , we have the interest on 1 for the m m"' part of a year equal to — . For fractional values of n, as well 111 as for integral, equations (1) to (7) hold. 8. Passing to Compotjnd Intebest, let P=The Principal ; or the Present Value. iS^The Amount to which that principal will accumulate, or the Sum Due. «=The Term. i=The Rate of Interest. « = The Present Value of 1 due a year hence. d=Th.e Rate of Discount; or the discount on 1 for one year. i'™'=:The effective rate of interest when the nominal rate is convertible m times a year. i^The effective rate of interest when the nominal rate is convertible momently. 8= The Force of Discount ; or the Force of Interest. 9. Since i is the interest on 1 for a year, the amount of 1 in a year will be (1 + i). But any other principal, P, will increase in the same proportion; and the amount of Pin a year will be P(l + ?). The amount of 1 at the end of the first year being (1 + i), if that amount be invested, its amount at the end of the second year will be (1 + i) (l + i),or (l + ?)2. The amount of (1 + «)'•* again at the end of the third year will be (1 + i)^, which is the amount of 1 in three years; and, generally, the amount of 1 at the end of n years will be (l + i)*, whence S=P (1 + i)^ . . . (8) 4 Theory of Finance. [chap. i. Ijj self-evidi3ut modifications, from this equatioa we deduce P=-A^ .... (9) -(I)" log (1 + 1 . . . (11) 10. Since (l + ») is the amount of 1 in a year, it follows that 1 is the present value of (l+«) due at the end of a year, whence the present value of 1 is—- rr , that is, (1 + i) Similarly (l + «)" being the amount of 1 in n years, 1 is the present value of (1 + *)" due at the end of n years; and^- r-,or by \ ^ J J . (l + j)« ^ equation (12), «», is the present value of 1 due at the end of n years ; whence, 8 being a sum due, and P its present value, P=.Sy» .... (13) Equation (13) is equation (9) in different symbols. It shows that Principal and Present Value, and Amount and Sum Due, re- spectively, are synonymous, and leads to the definition that the Present Value is that sum which, invested now, will, at the end of the period, have accumulated exactly to the Sum Due. 11. Because (l + »)" consists of the unit which was originally invested, together with its accumulations of interest during the n years, therefore the accumulated interest alone on 1 in ra years is {(l+0"-l}- 12. We have seen. Art. 9, that when the unit of time is a year, and when i is the interest on 1 in a year, then the amount of 1 in a year is (l + »); ^'^^ in « years (l + i)». The same principles hold if we change the unit of time. Suppose, then, we call the ■irfl'- part of a year the unit of time, and the interest on 1 in such unit — , m so that i becomes the nominal annual rate of interest. The amount of 1 in OT units of time — that is in a j'ear — will be ( 1 + — ) and in V inj n years {\-\ — j , and the interest actually realised on 1 in a year, or the effectlce rate of interest, will be ■■{ *'={(i+i)"-i} ■ • (») CHAP. I.] On Interest, 5 Also, performing obvious algebraical operations, Interest under these circumstances is said to be convertible in times in a year. Sometimes also the -gpord rests is used. Thus, for example, if interest at rate i be convertible quarterly, it is said to be at rate i with quarterly rests. The higher the rate of interest, and also the more frequently interest is convertible, the greater is the difference between the effective and the nominal rates. Thus, if interest be at the rate of 3 per cent., and, consequently, «=:-03, then i'2)_. 030225, ^'^' = •030339, and «'8)= -030397 ; and, if interest be at the rate of 10 per cent., and, consequently, j = 'l, then i'^' =: •1025, ^('"=•103813, and i's'^-lOMSS. 13. In theory there is no limit to the magnitude of m. If on become infinite, and interest be convertible momenth', we have still the interest on 1 in a year j(l+— 1 ~-'-|- ^^ *^^^ '■'^^® ^^^ nominal rate of interest has the special symbol S assigned to it, and we write, where i is the effective rate, •={(^+ '.)"-) • ■ (-) But by the theory of logarithms, when m increases without limit, [ 1 + — 1 has e'^ for its limit, where e is the base of the Napierian system of logarithms, and is equal to 2'71S2S18. . . . ^We there- fore have i=^e'-l) . , . _. . (10) _ loSTio (1 + »') and ^=^°^-^(l + ^') = ik^- • (1^) From this point of view 8 is called the Force of Interest, and is the nominal yearly rate of interest convertible momently, when the effective rate is i. 14. The following Tables A and B afford numerical illustration of the difference between the nominal and the effective rates of interest. Table A shows for various nominal rates the corresponding effective rates when interest is convertible half-yearly, quarterly, or momently ; and, conversely. Table B gives for the effective ]-ates the corresponding nominal rates.^ ^ A very extensive table of nominal rates of interest is given in the Journal of the InstituU of Actuaries, vol. xxiii, page 184. Theory of Finance. TABLE A. [chap. NoMINAIi Rate. EuPEOTiVE Kate. , Interest convertible Half-YearJy. •025 ■03 ■035 •04 •0i5 •05 •055 •06 •005 •07 •075 •08 •0S5 •09 ■095 ■1 ■025156 ■030225 ■035306 ■040400 •045506 •050625 •055756 •060900 •066056 •071225 •076406 •081600 •086806 •092025 •097256 ■102500 Quarterly. •025236 ■030339 •03')462 •0 J 0604 •045765 ■050945 •056145 •061364 •066602 •071859 •077136 ■082432 ■087748 ■093083 ■098438 ■103813 Momently. •025315 •030454 •035620 •040811 •046028 •051271 •056541 ■061837 •067159 ■072508 ■077884 ■083287 ■088717 •094174 ■099658 •105171 TABLE B. Nominal Rate 1 Efeeotitb Interest convertible 1 Rate. Half-Yearly. Quarterly. Momently. ■025 ■024846 ■024369 ■024693 03 ■029778 •029668 ■029559 035 ■034699 •034550 ■034401 04 ■039608 •039414 ■039221 045 •044505 •044260 ■044017 05 •049390 •049089 •048790 055 •054264 ■053901 ■053541 06 •059126 ■058695 ■058269 065 •063976 ■063473 •062975 07 •068816 ■068234 •067659 075 •073644 ■072978 •072321 , 08 •078461 •077706 •076961 085 •083266 ■082418 ■081580 1 09 •088061 ■087113 ■08617S \ 095 •092846 ■091792 ■090754 rH ■097618 ■096455 ■095310 ( CHAP. I.] On Interest. 7 15. It should be observed that in Arts 9, 10 and 11, the symbol i has been used in a general sense for the total interest earned on 1 in a year, and has not been restricted to the special ease where interest is convertrble yearly. From the definitions of the symbols it is au identity that ( 1 + — j = (l + i'"')", and unless it be desired to specially emphasise the fact that interest is convertible in some particular manner, the affix (m) may be omitted, and i may be taken to represent the interest actually realised at the end of a year hy the investment of 1. Looked at broadly from this point of view, all the equations in Arts 9, 10, and 11 remain true ; but so far we have discussed only integral values of n. 16. If i be the interest on 1 for one year, what is the interest for the OT* part of a year? This question at onetime created a great deal of warm controversy among actuaries; see J. I. A., vols, iii and iv. Some writers maintained that the answer must be — , and the m chief argument they urged in support of their view was, that— correctly represents the interest in the mP'- part of a year on the supposition of simple interest, and that under no circumstances should compound interest yield less than simple. On the other hand, rival authorities asserted that the correct I expression is {1 + *)™ — 1}, and that although this obviously gives 1 a value smaller than — , it is only risrht that it should do so. The interest is not due till the end of the year, and if the lender receive it sooner he must he content with less, because, compound interest being supposed, he can invest his interest for the remaining portion of the year and realise interest thereon. Also, the same authorities submitted that — is palpably wrong, because if 1 at the end of the »i* part of a year amount to ( H — ), it must at the end of two such parts aniount to {l-\ — ) , and at the end of a year to {\-\ — 1 , / m — 1 . ^ . . or to (1 + 1+^:; — 4^ + etc. ), which is contrary to the fundamental V 2)?i y axiom that i is the interest on 1 for a year. They therefore advocated the principle that in theoretical investigations the equation (S=P (1 + i)'* must be held to be universally true whether B S T/ieoiy of Finance. [chap. i. ■n be integral or fractional; and in recent years the majority or mathematicians have adopted their view. If - be considered to be the fractional interest, great complica- m tions must sometimes be introduced into formulas. See for instance the Treatise on Annuities by the late Griffith Davies, page 81, and Milne on Life Annuities, pp. 13 to 15, where not only are the ■exjiressions very complex, but three cases must be studied in the I solution of one problem. If on the contrary we adopt { (1 + i) ™ — 1}, these difficulties vanisli. The expressions become elegant and compact, and only one case presents itself for examination instead of three. See Baily's Doctrine of Interest and Annuities, articles 78 and 79, where the same example is discussed, to which we have already referred in the works of Davies and Milne. It must be remembered, however, that in commercial transactions the interest on 1 for the «?*'' part of a year is taken as — ; and even in actuarial formulas it is sometimes found convenient for purposes ■of numerical calculation to do the same. The effect of so doing is to omit the powers of i above the first in the expansion of (1 + z) '"', and, as the quantitj' i is always small, it is evident that there is but little practical difference between the two ways of dealing with fractional interest. That way should be adopted which may be the more convenient for the purposes in hand. 17. By definition, the Discount is the difference between a sum due and its present value ; that is d—l-v .... (18) =^. . . . . (19) = «i (20) 18. Equation (20) shows that the discount is the value at the beginning of a period of the interest to be received at the end; but this fact could have been ascertained by reasoning from first principles, without the aid of algebraical transformations. By Art 10, 1 is the present value of (1 + i) to be received at the end of a year ; that is, 1 is the present value of the original unit, together with the present value of the interest upon it ; and therefore the difference between the unit and its p]'esent value is equal to the present value of the interest. Looked at from this poiut of view ciiAr. I.] On Interest. g then, the discount is the interest paid in advance. This result is of considerable importance in connection with life annuiti'is and preniiums. Equation (20) also tells us that the discount, d, is a year's interest on the present value, v. 19. From equation (18) we immediately deduce v = l-d .... (21) l+i=^^ .... (22) Therefore, if the rate of discount as distinguished from the rate of interest, he named, the present value of a unit due at the end of a year is found hy subtracting from the unit the rate of discount. 20. In this connection it should be noticed that, when a merchant seeks to discount a bill, his hanker quotes to him the rate of discount, not the rate of interest; and when the Bank of England Directors fix their rate, it is the rate of discount ihej determine, not the rate of interest. Through failure to keep this distinction in view, confusion has sometimes arisen. It has been usual to say that commercial discount differs from theoretical discount, in that when the rate is, say, 5 per cent., the banker deducts 5 from his customer's bill of 100, instead of , as his critics say he ought to do ; and some writers have even insinuated dishonesty on the part of the banker. Bail)-, for instance — Doctrine of Interest, etc., chap. iii. — remarks that the course " is neither correct nor just." But if the banker says that his rate of discount is 5 jper cent., the merchant cannot grumble. The banker merely assigns a value to d from which i may be found. It is true that, at the same nominal rate, money improves faster under the operation of discount than of interest. If a banker can employ his funds in discounting at 5 per cent., it will not be to his profit to grant advances at 5 per cent, interest ; and if a merchant sell his bill to the hanker at 5 per cent, discount, he must remember that he is paying more than 5 per cent, for the accommodation, but with his eyes open to this fact, he suffers no wrong. 21. It is not often in commercial, as distinguished from insurance, transactions that discount has to be calculated for more than a year. In fact, the great majority of commercial bills have only a fractional part of ci yeai to run. If that fraction be denoted by — , and if d be b2 lo Theory of Finance. [chap. i. the yearly rate of discount, it is customary for the banker to give for each unit of the bill {\ \ and not (1 — i);^, thus using that which by analogy may be called "simple discount." If "simple discount" be employed for periods greater than a year, erroneous and anomalous results are produced. Thus, if the bill to be discounted have n years to run, its value at simple discount will be (1 — ndi), and, where n is large, it may very well happen that nd is greater than unity, and gives the bill a negative value. This is not because " commercial discount " differs from " theoretical discount," but because " simple discount " has been used, and to simple discount the objections mentioned in Article 4 apply equally as to simple interest. The correct formula is (1 — f?)'*, by the use of which the anomaly disappears. 22. The operation of discount is similar in its results to the operation of interest, and just as we have " compound interest," we may have, by repeating discount operations, " compound discount." When a bill matures, the banker may at once employ the proceeds in discounting a new bill. If d be the nominal yearly rate of discount, and if the process of discounting be repeated m times in a year, the discount on 1 in the «i** part of a year will be — , and m the value of 1 due at the end of the wt*^'' part of a year will be (1 J. Repeating the operation, we liave the present value of / d \^ 1 due at the end of two m'''- parts ( 1 1 , and the present value / d \™ of 1 due at the end of a whole year I 1 J . In this expression we may make m as great as we please ; but by the theory of / d \™ logarithms, when m increases without limit, (1 1 has e~'^ fol its limit. Also when discounting is performed momently, d, the nominal rate of discount, is written 8. We therefore have = n-s v^e' (23) From this point of view S is called the Force of Discount. 23. By a very simple apphoation of the differential calculus, we can form a clear idea of the meaning of the force of discount or the force of interest. The differential coefficient of a function is the measure of the CHAP. I.] On Interest. 1 1 velocity of change in the function consequent on change in the variable. Now, z>^may he considered as the function of discount, and, taking its differential coefficient, we have -— =«'" log^i'. If now we ax divide by «'", we have the measure of the velocity of change 1 diy° in the function for each unit of the function, ;— = log£,f = v^' ax -log,(l + i) = -8. In the same way (1 + »)■'' may be considered as the function of interest, and — ^^^^-^= (H-j)»^ log,(l + «), and , ^^_^ =loge (1 + ») = S. The numerical value is the same, but the sign is opposite. The difference in sign shows that the force of discount is a force of decrement, and the force of interest a force of increment. We can define that force as the annual rate per unit at which a sum is increasing by interest or diminishing by discount at any moment of time. The term " force " is a misnomer ; it should rather be "velocity." But "force" having come into general use, a change would be inconvenient. 24. The quantities ?, v, d, and S, are all mutually dependent, and can be conveniently expressed in terms of each other by means of series. Thus, since by equation (22) il + i) = (l-d)-^ i=di-d^+d^ + etc. . . (24) Also, since by equation (16), 2 = /— 1: by the exponential theorem «=8 + j^+ |-+etc. . . (25) Again, by equation (22), (1 — J) = (l + i)".' Therefore d=i — i^ + i3— etc. . . (26) and also d^]. — v 82 S3 =8-^ + ^-ete- ■ • (27) Since v—l — d v=l—i + P—i^ + etc. . . (28) and also, from (27) y=l-S+ j^-- Tg+etc. . (29) 12 Theory of Finance. [CIIAP. I. Finallj'", because S= = log,(l + 0, iberefore, by the theory of logarithms 8=.-^+ g-etc. , (30) and. since 8 = -log« V- :-l0g, (1-fO , therefore S=c? + — +— + ctc. (31) All the series given in this article are rapidly convergent, and when the numerical value of one of the functions is given, they offer great facilities for computing the values of the others. The student will find it au excellent exercise to set himself examples under this head. If we take equations (30) and (31), and combine them, wo have whence, approximately, S (i^—d:^) ii+d^ -etc. i + d (32) Equation (32) gives S correct to at least four places of decimals for all values of i not greater than '07. The following Table C gives the values of the several quantities at the rates of interest most commonly in use. TABLE C. i. V. d. S. •02 9S0392 •019608 019803 025 975610 •024390 024693 03 970874 •029120 029559 035 966184 ■033816 034401 04 06 1538 •038462 039221 015 950938 •043062 041017 05 95238 L •047619 048790 055 947867 •052133 053541 0(5 943396 •056604 05S269 0G5 038967 •061033 032975 07 931579 •065421 067659 075 930233 •069767 072321 OS 925926 •074074 076;)61 OS 5 921659 •07HHJ4 081580 00 917431 •082569 086178 095 913212 •0S675S 090754 •1 909091 •090909 095310 CHAP, I.] On Interest. 13 25. Expanding the right-hand member of equation (14) by the binomial theorem, we have .,, ., m—\ i2 (ot — 1)(ot — 2) «3 , ,„„, *'"" ^ ^ + |7, + ^ ~ • tr, + etc. (33) from which, if m become inBnite, we have again equation (25), because in that case tlie nominal rate, i, becomes S, and the effective rate, j''"-' bscomes /, wliich iu equation (25) was simply written i. -Xot only is the series in equation (33) rapidly convergent, but when m is not large there are few terms. Thus, when interest is convertible quarterly, and «i=4, all terms after the fourth vanish. The series always come to an end after m terms. From equation (14) we may also derive the relation log («j + i) = —^^^-—^ — i + log m . (34) by means of which i may be found wher. i'™' is known. 26. In how many years will money double itself at compound interest ? The answer to this question follows directly from equation (10) . Writing 2 for iSand 1 for P, we have, using Napieriau loo* 2 l'"^ i Io£rarithms, « = , f~ — ^* But lo«'e(l + 2') = i— - + -— etc., and log,(l+?) aev -r J g^ 3 if we neglect the second and higher powers of i, and write for •69 logo 2 its near value 'GO, we have approximately n-=~-r- Whence the common rule : — To find the number of years in which money will double itself, divide 69 by the rate of interest per cent. Mr. G. r. Hardy (Insurance Record, March 31, 1882) pointed out that the correction to the approximation given by this rule is very nearly a constant quantity: that is, the error ivivolved is practically the same whatever the rate of interest. Thus. log,2 -693 -693,, . ,.,,,, 693 , ^, '^ = =-^(H-J-i— J-i^+etc.) = ^-f35 logc(l + i-i2.2 + ii3_etc. t ^ ' ^- I very nearly. With this correction the rule will give a result true usually to two places of decimals. 27. It frequently happens that there are various sums of money due at different times from one merchant to another, and which it is desired to pay all at one time. That time at which they may be paid without injustice to either party is called the equated time of payment, or the average due date. H Theory of Finance. [chap. i. 28. If the various sums be -Si, S.-,, S3, etc., and the times at which thej respectively fall due iii, Hq, lis, etc., to find the equated time X, at which the total amount (Si f -Sa+zS's + etc.) may be paid, so that the parties may be on an equality as regards interest. It is evident that justice will be secured if the present value of the aggregate of the sums due at the time, x, be equal to the aggre- gate of the present values of the individual sums, and that to fiud X we have the equation Sy + So_+S^ + &io. ^ S, S.2 Ss g^^ , which may be symbolically written as ^^ s , . (i+ir "(i+i)" ■ ' ■ ■ ^' ) If we expand each side of equation (34) by the binomial theorem and neglect all powers of i above the first, we have, after reduction and division by the coefficient of x, _ Sini + S.2n.2 + S37ii+eio. """ S,+ S^ + Ss + etc. ■ ■ ■ ^'"'' The same result would be produced by assuming simple interest, and, in the expansions, neglecting all powers of i above the first. 29. The last equation gives the usual rule, which is approxi- mately correct : — To find the equated time of payment of various amounts, multij)ly each amount by the time to elapse until it will fall due, and divide the sum of the products by the sum of the amounts. The value found for a; is, however, too large. It results from taking the arithmetical mean, whereas the geometrical mean would be more correct ; and the geometrical mean of any quantities is always less than the arithmetical mean. THEOEY OF FINANCE CHAPTER II, On Annuities-Certain. 1. An Annuity is a periodical paj'ment, lasting during a fixed term of years, or depending on the continuance of a given life or combination of lives. The annuity may be payable either yearly or at more frequent intervals, but it is measured by the total amount payable in one year, which is sometimes conveniently called the annual rent. Thus, if a person be entitled to receive £25 every three months, he is said to be in possession of an annuity of £100, payable bj' quarterly instalments. When an annuity is to last during a fixed term of years it is called an annuity-certain. When it depends on the continuance of a given life or combination of lives, it is called a life annuity, or simply an annuity. 2. The word status is conveniently used to denote the period during which an annuity is payable, whether that period be a fixed term of years, or depend for its duration on the contingencies of life. We may therefore define an annuity, generally, as a periodical payment lasting during a given status. 3. A perpetuity is an annuity that is to last indefinitely. 4. Unless otherwise stated, the first payment of an annuity is supposed to be made at the end of the first year for which the annuity is to run, or, in the case of annuities payable at other intervals, at the end of the first interval. Thus, if we speak of an annuity for n years, we mean an annuity consisting of n yearly payments, the first of which is to be made a year hence : or if we speak of an annuity for ii years deferred m years, we mean an annuity consisting of n yearly payments, the first of which is to be made at the end of t6 Theory of Finance. [chap. ii. (to + 1) years. This last annuity is said to be entered on at the end of m years, altbough the first payment will not be made until the expiration of (j7i + l) years. If the first payment of the annuity be made at the beginning instead of at the end of the first interval, the annuity is called an annuitji-due. 5. For purposes of investigation we shall always consider the annual rent of an annuity to be 1. The results will be made available for other annuities by merely multiplying by the annual rent. G. Let 6'n| = the amount of an annuity for n years. rt-,i"i=the present value of an annuity for n years payable yearly. a^i=: do. half-yearly. an\= do. quarterly. Un, = do. Ill times a year. a.;i-j= do. momently, that is, a continuous annuitj', aj:| = the present value of an annuity-due for n years, so that are|=:l4-an^i|. ,„|n,"i]=the present value of an annuity for n years, deferred m years. (j!^=:the present value of a perpetuity. Note. — The symbols a«|, m\an\,. and a„, may be qualified by the aflL^es (2), (4), (jn), in the same vi'ay as an ordinary annuity. "We may also write m\a^ and dm- At pimple Interest. 7. To find SSI When the paymentr. of an annuity are not taken as they fall due, but are allowed to remain to accumulate at interest, the annuity is sometimes said to be forborne, and the sum to which the paj'ments accumulate is called the amount of the annuity. Thus the amount of an annuity is the sum of the amounts of its several payments. There- fore, the last payment having just been made, the last but one having been made a year ago, and so on, we have, commencing with the last payment, s,;ii=i+(i-hi) + (i+2/;)-f . . . +(i-i-;;iii/) n { ) = -|2 + m-Ii| . . , . . (1) . ciiAr. II.] On Anmnties-Cerlain. 17 8. To find arv,. The present value of an annuity is tlie sum of the present values of its several payments. We therefore have, beginning with the first payment, 111 1 /„^ There is no direct ■ means of summing this series, and to obtain exactitude, each term must be calculated separately. A very close approximation may, however, be obtained from the formula of the differential and integral calculus given in Art. 10. 9. Several other so-called approximations have been proposed, based on plausible reasoning, which, however, fails when simple interest is assumed. Some of these are as follows : — a. It is evident that the present value of an annuity and the present value of the amount of the same annuity should be equal, ibr, as regards present value, it should be a matter of indifference whether a person is to receive the annuity during the n years, or the equivalent of the annuity at the end of the n years : therefore n3 = - -.J and, substituting for 5,71 its value found above, x + m n{2+(n~l)i} y8. Again, the value of the annuity should be equal to the difference between the values of a perpetuity to be entered on at once, and another to be entered on at the end of n years. But 1 invested now will yield i at the end of each year for ever : that is, 1 is the value of a perpetuity of i, and, therefore, the value of a perpetuity of 1 is - , and the value of a perpetuity of 1 defended n years is — :— ., whence «'^=-fi-rz-0 • • W J V l+nU y. Again, from the nature of this case, each payment of the annuitv must contain interest on the purchase monej', together with a return of part of that purchase money.-' The purchase money being o^ the interest on it is ia:,;], and the return of capital is the balance of the yearly payment, namely (1 — iaji]). This last quantity 1 Tlie statement here made will be amxilified and proved in Arts. 31 to 34 of this cliapter. i 8 Theory of Finance. [chap. il. constitutes the rent of an annuity which must be accumulated at interest in order to replace the capital at the end of the period. That is a^ = s^(l — Jffni): or (5) ra{2+(« — 1)7} 2(l + nj) + «(m — i)i 10. The formula of the calculus referred to in Art. 8 is as follows : — where V is any function of x and 2", its finite integral. It is skilfully used by Mr. Woolhouse, J. I. A., vol. xv. p. 100, in the solution of problems in life contingencies, and there he also demonstrates the formula. It is also discussed in the Institute of Actuaries' Text-Book, Part II, Chapter xxi\% Arts. 31 and 32. In the pi'esent case F^^ , and the formula becomes loge(l + mi) I 'If, + ,4.(1—^—1-^^ ^-(IT^) • (°^ 12V (l + ni)y 720V Remembering that the modulus J/ of the common system cf logarithms is -4342945, and that log, (l + wO= ''"°'° ^V^'"-* , the formula is very easily applied. Lastly, a good approximation may be obtained by moans of a formula given in Algebras under the heading "Exponential and Logarithmic Series." We have h ^Oi-lc , \f}i. — h\^ ^ \ l°S^;,=2{,;^^ + 3(^J+etc.} Writing, now, 7j = l+(m+i)j, and h = \+{n — \')i, so that h—h—i, and /i + /(; = 2(l + 5iO ; then l + (m + i)i 'H-()i- 1)l-K24i^- + 3(2T2^-)'+'^^°-} CHAP. II.] On Annuities-Certain. 19 But i is always a small fraction. Therefore the terms inyolving its- third and higher powers may be neglected, and we have, approxi- mately, after dividing by i. rloa; % ^^60 \^-(ji — i)i 1 + ni' Whence, approximately. If, 1 + Ui l + 2\i l + 3ij i + ln + \)i "ge i(. ° l + i* 1 + liJ 1 + 2*1 l+(»i 11. As an illustration of the foregoing formulas, we give, at 5 per cent, interest, the values they bring out for an annuity for twenty years. flio"! = 13'616068, by direct summation (true value). = 14-750000, by formula (3). = 10-000000, „ „ (4). = 11-919192, „ „ (5). = 13-6160G8, „ „ (6). = 13-617542, „ „ (G«). It will be seen tbat formula (6) gives the result correct to six places of decimals, and that formula (6a) gives a good apjjroximation, but that the others are very wide of the mark. Annuities at simple interest have no practical importance, and the analysis is inserted here purely as a matter of curiosity. Passing to Compound Imtebest — 12. To find Sn\ The amount of the several payments form a geometrical progression, with common ratio (1 + i), and we have, beginning as before with the last payment of the annuity, s^ = l+(l + i) + il + iy+ . .. +(1 + 0"-' ^(l +0"-l . , , , . (7) 13. This result can be easily obtained without having recourse to series. A unit invested produces an annuity of i per annum. The amount of 1 in « years consists of the original unit and the- 20 Theory of Finance. [chap. II. amount of the annuity of i for n years which the unit produced ; and therefore the amount of the annuity of i is {(l-f*)'* 1}' ^'■^"^i ■ (1 + 0"-! by simple proportion, tlie amount of an annuity oi 1 is -. as before. 14. If -bhe annuity be payable -p times in a year, and the interest " be convertible g times, to find the amount of the annuity. The amount of 1 in a year is (1 + -) : (Chapter i. Art, 12), and in the IH — Y , (Chapter i. Art. 16). The amount of the annuity, beginning Avith the last payment, is therefore a geo- metrical progression of ^:m terms, with common ratio llH — I*", and we have, each payment of the annuity being - , ](-(-D= " + n^+^ 1+-J -1 when q^2^ becomes '+^/ i\' . m J i 1 J , when we write for J its value {(l + j)m— ]}. W((l + 4)m_l} Therefore, in order to find o.f[' from a;^, we must multiply by i — ■ i , the same constant factor as before. m{(l + i)m-l} CHAP, ii.j \Jn Anmiities-Certai; in. Y — * QJ 1 r^ ^\% (a !Ll 1 •— 1 S 1 ■J" ^ 1 K SI" ■' ^ •c* 1 s + - IS + + s^ -f' I— f —1 1 — I 1 ^ IM ^1 = I— 1 f 1 S ■^ 1 1 .„ ■* 1 •» l^ ■» ^ ■SJ '^ ■*_iS_^ -ra l'^ + -' ^ + ■^ ^ + "tS" + ■» ■-!? + 1 + 1 + ^— ' I + r-t 1 P— 1 rH N ^ + s r— 1 .= l' T— ( =7 1 'l c. ^ v* + I 4- ■?a + + + + ^ + 1— i N ^ + ' + ^"^ HH 1 1— i 1 I— 1 1 1— f ^ [ I-I 1 t-0 o ri S rH c 1 I 1 P + ■l!i + -I-. -l|T + c^ + 6 1—1 + + "Y" V^ 1 1 r-i 1 rH ■-— ' r-\ ^~-^ r->i (N T-1 '^ ^ ■^ c S 0) o .*j ^ o +3 A n H o rf r^ ■4-3 -*-' -M Cl :^ CPJ i o ^ p; .',-. CIJ feD 1-^ i=! :3 ?1 o CO O Cl 4-1 (U d o aj ^ GJ rH Ed 2 ** r^^ fcn o ^ BO rfl ■f .SO I— 1 M c o 3 H II E-t OJ iW rO ^ S rt 1=1 11 ^. S +3 OJ o n Tl 1-. «3 q:3 ff! o ■^ n g ?- g -S Uj o rn" O *-w 1 rH rH 1-i rH nH rH 1^ ? xr'rT.OU;)LQJ>.rHCr:OO uDOiocooicOLra^tra^cDOOcoio x--.-r)HrHoo^T-ij>.coaju:)0»oo»jD 10. Bedemption Money. Formula ^8. iCi lO Ci lO -# Cq (M O to ift Ol 00 i>. xo O J^ O Cvj CD 1-1 GO LO CD CO O i-H i-H iC ID O Ipr-itpiOCiOCOGO-^CCrHCOOOCCIO O O i^ ^ Oi (M i>4 r-H O) i-H fM O '^ -T Qococor-((joocooc;cocniooo "■^'■^■^■^CQCOCOCOCMCsli-.i— (i— 1 9. Redemption Money. Formula a. cs(MO"ticx)a5-Tf<«)C;ocO'~'Ocoo QOCicoaJTr-O 00 4fi 6:) r^ di CO M r^ lb lo 00 Ah I'o oD o cDCicco^c^'-'oo-^aoOi-HcJiio X-^ lO N O I> rji ^ 1-- -^ O 1^ CO CO '^ -^■^■^-c0(McoioocoxioaO(M cpOipi:sasc£)vpi>(Nn-(tMOOl^OOO r-Hi— ICQ0:iCO-^iCVDCOi>00a3 Interest at 5 per cent, earned on repaid Trin- eipal at end of previous year. O >b -^ "^ iQ i>- rH (7J CO -^ 'X' J> 05 rH M Tfi CD 00 O (M iH i-H tH M tH DJ CM 6. Interest at 7 per cent, on out- standing Principal at end of previous year (pro- vided by Annuity). oocn^o»0"^co»orHai05i>-ai'^ pi;-.C--C»p«iOrH>.b6:)T^'r-icN lOCOr-HCDOOCDCO — Cil-^'T'rHCiCDCO COC0C0(MCvlC^5(MCSIr-'r-lrHr-H 5. Principal still out- standing. OWOlOOOCri'^COOOCDi-IOCOO oDc:icoai-^--iOcDcvi:DiN!MOi>-o CMGJu:)(MCD0i^C0Ol0^0DX>i.--O aO-^CDrHOicbcOi-'-UtiVCCOr-IVOOOO CCOICDO.-l'Mf-IQO-TC000"^ -^xi^-^'^COCOCOtMtMC'lr— *iH Principal repaid to date. ^Q0Ou:)Cl.-ICD-t^OO-#CsOt^O l>O-dHi-^C0OiOCDC^'7'00rH03G^O rHiboaOOii5=icNI"rTi-^THOO'^r-IO coi-*coo:iODi-^oOt-Hirai-icjGOO-^0 Gsj-^l-^OS(M1000C(M»0J>-05l0rHC0O'*t0rHi>C0 ,HascDco^csi^"*-*^coow2:ji:- -^OTiboorHibocbtfqGit-'i'cbcbaD CO^lOCOOOOii— IGvi-^lC-t^Clr-HCOvO (NCICQiMiMMCOCOCOCOCOCOTjl-T^Tli 2. Annual Rent of Annuity. ) WD |I pH(MC0-*iCCDi>C0CiO^tMC02J5 38 Theory of Finance. [chap. ii. 43«. In Arts. 40 to 43 the value has been considered of an annuity to pay a purchaser one rate of interest, i, on the purchase money, and to permit of the accumulations of the sinking fund at another, and lower, rate,y. But the annuity is only a special case of a larger problem ; and we may now consider the general question of the value of a series of payments of any amounts, to be made at any times, the value to be so calculated as to yield the purchaser the remunerative rate, i, on his whole investment throughout the longest of the periods, and to return him his capital at the accumulative rate, _/. First, let it be required to find what sum should be paid t years hence in consideration of a present advance of 1 ; it being stipulated that, at the eij'iration of n years from now (n>t), the lender is to be in the same position as if he should receive interest at rate i throughout the n years, although he can otAj reinvest his capital at the accumulative rate^. At the end of t years, The sum originally advanced is . . . . 1 The amount of the annual interest, i,^ accumulated for t years at rate r . „ , is'fi i, is ^ The present value at rate / of the % difference between the interest at rate i which is to be realised, ( and at rate y, which only can be / ' • \ J) «- obtained for the remaining n — 1\ years, is The amount therefore to be paid to) the lender at the end of t years is j ^^ + **'*l) + (*-i)«'S^ =1 + «■ («'(! + «'iir;?|) —j a':,c:rt\ = l + is'n\ v'>^ + i—(l — v'»-t) =v'«-*(l + is':;^) where the functions with the accent are all taken at rate J. This is the sum receivable at the end of t years in consideration of a present advance of 1. Therefore, the present value of 1, receivable at the end of t years on like conditions, is the reciprocal of the last expression, namely: — (a) 1 + i s'si (P) CHAP. 11.] On Annuziies-Certain. 39 Considering now expression (a) , it will be seen that the only factor involving t is (!+_;■)«-*, wtioh is equal to the aecumukted amount at the end of n years of a payment of 1 to be made t years hence. Similarly, in expression (^), the only factor involving t is »'*, which is equal to the present value of a payment of 1 to be made t years hence. Therefore the present value under the conditions named, n remaining constant, of any series of payments, may be found in either of the following ways, viz. : — (i.) By multiplying the amount accumulated at rate j to the end of the n years of the series of payments by r-^ . (ii.) By multiplying the present value at rate j of the series of payments by ^ t-^ . 1 + i fi Til These results are perfectly general. Applying them to the ordinary annuity, the present value of the series of payments is a'^ , and, under the special conditions, ~\ + is'i\ which agrees with formula 30. 44. In connection with annuities, various quantities come under our notice, namely, the term, the present value, the amount, the annual rent, and the rate of interest, and in the majority of cases simple relations subsist between these quantities, so that, certain of them being given, the others can at once be found by the ordinary operations of algebra. Thus we have seen that «S(= ■■ > so that, having given n and i, we can find a. There is one case, however, which presents difficulties. When the term, and the amount or the present value, of an annuity are given, and it is required to ascertain the rate of interest, we have to solve an equation of the n*^ degree, and rm approximate solution. is alone possible. This case we now proceed to investigate. 45. Having given s, the amount of an annuity, and n, the term, to find i, the rate of interest. Using Table 3, which shows the amounts of annuities at various rates of interest, we find, in the 40 Theory of Finance. [chap, il table opposite the given number of years, the value nearest to s. Let that value be denoted by s\ and let the rate of interest under which it is found be denoted by j. If s', the amount found in the table, be exactly equal to s, then the rate under which s' is found must be the rate sought ; that is, «'=/. If, however, s' differ from s, then j differs from i. Let i=zj + p, and the problem resolves itself into finding p, which is necessarily a very small quantity. We have s= ^=^ _^ {(l+i)+p}"-l J+P Therefore s(,/ + p) = { (1+i) +p}''-l. If we expand the right-hand member of the equation by the binomial theorem, and neglect all powers of p above the first, we have s(;+p) = (l+y)» + M(l+y)«.-ip_l. whence, remembering that (1 +_/)»— !=;«', we have '^ s~n (i+y)«-' ' By means of formula (31) the value of p, and thence of i, cai be found, generally with sufficient accuracy ; but should a closer approximation be desired, the process may be repeated. The amount of the annuity may be calculated at the rate of interest found by the first application of the formula, and inserted in the equation instead of s', by which means a result very accurate indeed may be obtained. 46. Other formulas for the rate of interest in s^ may be deduced, analogous to those given in the succeeding articles for the rate of interest in ag ; but, as the problem is not tne of great importance, we do not prolong the ii»yestigation here. 47. Having given a, the value of an annuity, and n, the term, to find i, the rate of interest. By the help of Table 4, giving the values of annuities, we must find a value of a near to the given value. Let that near value be denoted by a', and let the rate of interest under which it is found be denoted by^, and let the rate sought, i=j+p. (31) CHAP. II. J On Annuities-Certain. 41 We have a= ^^ — —^ — i whence a(j,Vp;=l-{(l+y) H-p.}-« . . . (32) and „=[i_|(i+j)+p}-.](y+p)-i . (33) From these last two equations we can deduce as follows the four approximate formulas A, B, C, and D, for p. •18. If in equation (32) we expand by the binomial theorem, neglect the powers of p above the second, and collect the terms, we have — writing v for (l+y)~S and remembering that (1 — v'"'^)=:ja' — - the quadratic equation in p, ''l^±}l^n+Y+{a-nv^'ri)p^j(^a'-a)=Q . (34) If we neglect the second power of p, we have merely a simple equation to solve, from which we obtain A. p= ~— : ..... (.rfOj The value just found in A may be used in conjunction with equation (34) to obtain a closer approximation. If in the equation we write 0^=0 x — -, and then solve for p, we have '^ '^ a — nv"'+^ B 0= -^'^"'-^ -^ . (36) 49. Returning now to equation (33), if we espand both the factors of the right-hand member by the binomial theorem, and retain only the first and second powers r>i p, we have another equation ii«'_,j,;»+i_!i^!!±l)««+2^-}p2..-l.(«'_„y«+i)p+(«'-«) = (37) and if in that equation we solve for p, neglecting the second power, we have d2 42 Theory of Finance. [chai-. n. Lastly, if in equation (34), or equation (37), we write i)''^=p X -4-^ — , and solve for p, we have D. p= i(«'-" ) Agg-v 50. By means of the Calculus of Finite Differences two very useful formulas may be obtained, which are more easy to apply numerically than any of the others, and which at the same time usually give better results. Let a, as before, be the given value of the annuity, and let asi, a^, a%, be the tabulated annuity-values for the same term, at the rates j, j-\-li, j-\-1h; where a lies between aj, and 0121 and consequently, i between y and y -HA. Let i=;-f p ; and let A, A^, represent the successive differences of a^, a^, etc. Then, stopping at second differences. =.,+ f(A_Si=)+g4" . (10) In the last equation, neglecting the second power of j- we have, , p a — a-i approximately, y =• _^^ ; and, making use of this value, p2 A2 p a — ai A2 . , , p • -^ = T • A _ 1.A2 ' "9" ' ^.pproximately : whence, inserting this value in equation 40, and solving for j , p a — Oi A-iA2 (41) _j A-|A^ ^ jA^ \-i I a — a, ■ A — iA^i and E. (42) CHAP. II.] On Annuities-Certain. 43 51. The second formula referred to may be very easily deduced from equation 41. If we multiply numerator and denominator by , and if we omit the term (■s-A^)^ wherever it appears in the result, that term being very small, we have p_ A + AA2 li (A )^ Ax-i-A^ » — «i A — iA2 If, again, we multiply numerator and denominator of the right- hand expression in the denominator by A + -|-A^, and in the result omit the terms involving (i-A^)^, we have finally, after multiplying up by A, -'(^-- • • <*^) ■^^+iA2 a — ffii 52. Formulas E and P possess one great advantage over the others. Instead of the annuity-values themselves, the reciprocals, or the logarithms, of the annuity-values may be employed ; and as the differences of both these functions are smaller than those of tbe annuity- values, a greater degree of accuracy may frequently be obtained. 53. A numerical example will enable us to gain a clearer idea of the formulas given in the six jsreceding articles. An annuitj'"-certain for 30 years is bought at 19 years' purchase. What rate of interest is made on the investment ? If in Table 4 we look against 30 years, we find that 3 per cent. rt80i=19-60044, and at 3^ per cent. 030!= 18-39205. We therefore conclude that «', the rate of interest sought, lies between '03 and '035, and we assume that_y=:'03. Then, using formula A, ffl'= 19-60044 ;)3i- -399987 a = 19-00000 gOy3i = ll 99961 «'-«= -60044 o_30z!3i= 7-00039 j(a'~-a)= -018013 - p=0025732 log -018013=2-25559 ^:=.0326732 log 7-00039=0-84512 log p= 3-4 1047 44 Theory of Finance. [chap. ii. rormvila B has the appearance of being lengthy and intricate, but it is not really so. The second term of the denominator consists merely of the value of p found by formula A multiplied by the factor '^^''^'^^' . ««+2; and in finding the value of p by formula A, 2i the values have already been found of all the other terms of the expression in B. In the example in hand we have log (1x30x31) = 2-66745 log 1)32 , =1-58921 log ^^"'-"'^ =3-41047 1-66713 vrhence the second term of the denominator is -46465. Adding to this 7-00039, the value of a — 30y3i already found, we have the denominator =7-46504. The numerator is -018013 as before: whence p= -0024130 and i= -0324130 Formula C is the same as formula A, except that a! occurs instead of a in the denominator. From it p= -0023699 and i= -0323699 Formula D is the same as formula B, except that al occurs instead of a in the second term of the denominator. From it p= -0024249 and »•= -0324249 Applying formula E by means of Table 4, at the end of the volume, we have A = -005, because the part of the table to be used sjives annuity-values at intervals of one-half per cent, in the rate of interest. A A2 ffi=:19-60044 fl,= 1839205 «3= 17-29203 — 120S39 -1-10002 -F -10837 -«,= --60044; JA'-i- -05419; A- iA2= —1-20258. CHAP. II.] On Anmiities- Certain. 45 „„^f 126258 5419 ) -' 005 p=00o|-^ S- wlienee and 60044 126258 ) p= -0024273 i= -0324273 20599 ■ By formula F, changing the signs as in this case both numerator and denominator are negative, p = -005 1-20839 --05419 (1-20839)^ -60044 •05419 and = 0024271 2=-0:S24271 The true value of i, correct to seven places of decimals, is ■0324252. Collecting together for comparison the results of the various formulas, we have By formula A, i= B, *•= C, i= E, «■= 0325732 0324130 0323699 0324249 0324273 0324271 errors +1480 -=-10' = - 122 = - 553 = + 21 = + 19 54. Formula C gives a better value than formula A, and, if we examine the processes by which the formulas were deduced, the reason becomes apparent. Equation (32) is obtained fi-om equation (33) by multiply- ing by (y + p) with loss of accuracy. If in the expansion of 1 — {(1+y) -|-p}~'* we denote the coefficients of the ascending powers of p by Tc, I, andw, so that ^={1 — (l-h,/)-'*}, Z=?2(l -!-»-'"+", and -m:= or say (1 -Hi) -'"+2', equation (32) will take the form mp^-J- {a — V)p— (/c — a;')=0, iro--|-yp — s = .... (44) where x^^m, y=^{.ci — T), and e^=(Jc — aj), and equation (33) will take the form ja=(Jc+lp — mp t-'j-9 46 Theory of Finafice. [chap, il or which is seen to be the same as {-j(y+j)}p=+(y+7)p--=o • (*5) Equations (4J<) and (45) are simply equations (34) and (37) so displayed that they may be compared, and they show that in equation (34) we neglect part of each power of p. Seeing then that formulas A and are identical in form, and therefore equally easily applied, while C is the more accurate of the two, that is the one which should be used. 55. Formula A was first published by Mr. Francis Baily in the appendix, page 129, of his work on the Doctrine of Interest and Annuities. In the same place he investigated other formulas which do not require the aid of interest tables for their application, but at the present day these formulas have no practical importance, owing to the number and extent of tlie interest tables which are available. Formula is due to Mr. George Barrett, who, in writing to Mr. Baily, suggests it as an improvement on formula A. (See the extract of a letter from Barrett to Baily, J.I.A., vol. iv., p. 189). To Professor De Morgan we owe formula B. It is given by him in a paper, J. I. A., vol. viii., p. 67; and Mr. M'Lauohlan has supplied formula D, J.I.A., vol. xviii., p. 295. Formulas E and F were devised by Mr. G. F. Hardy, and pub- lished by him in a paper, J.I.A., vol. xxiii., p. 266. 56. The principles by which equation (84) was reduced from a quadratic into the simple equation (36) may obviouslj^ be extended. We may retain all terms of the original expansion which invohe powers of p not greater than the third, and write p^=o \— ^l f^ ^ \ a—iiv"+^ f' means and p^=p \- — — i , and then solve for p. Bv this (. a—nD"+^ J '^ Professor De Morgan {J.I. A., vol. viii., p. 67) deduced a formula which produces very accurate results, but it is too complicated fef common use. CHAP. 11.3 Oil Anmiities-Certain. 47 57. A very frequent instance of the application of the forecoino- formulas is the case of Foreign Government loans. The Eno-lish {jovernment has usually horrowed in exchange for perpetual annuities — that is, in consideration of a sum advanced, a promise has been given to pay a certain yearly interest, but without stipulation as to repayment of the principal. 58. Foreign Governments, however, frequently find it more con- venient to undertake to repay the principal vpithin a limited period. The arrangement is to issue numbered bonds, with coupons attached representing the annual interest, and these coupons are •cut off by the lender and presented for payment as the interest falls due. Besides providing for the interest yearh', the Govern- ment also sets aside a fixed sum, called the sinking fund, to redeem the bonds; and bonds to the amount of the sinking fund are drawn by lot and cancelled. The yearly interest, however, on the can- •eelled bonds is still provided by the Government, and applied, in addition to the sinking fund, to liquidate the loan. In fact the Government simply contracts to pay a fixed yearly instalment, including principal and interest, for a limited number of j-ears till the whole loan is paid off', or, in other words, it raises money in exchange for a terminable annuity. The sinking fund under these conditions is called an accumulative sinking fund. 59. If such a rate of interest were offered that the public would take the bonds at par, the transaction would not be complex. The rate of interest paid by the Government would merely be the nominal rats of interest borne by the bonds. But loans of this ■class generally are issued at a discount, while the bonds on being drawn are paid off at par, so that, in addition to the yearly interest, the lending public are offered the inducement of a bonus on repay- ment. Under these conditions we must make use of one of the approximate formulas to find the true rate of interest paid by the Government. 60. Let the price at which the bonds are issued be Ic per unit: let the nominal rate of interest paid by the Government be i' ; and let the sinking fund per unit of the loan, that is the amount to be repaid at the end of the first year be z: — then h is simply the present value of an annuity of («" + «) per annum. We therefore have h=(j!-\-z) ^-^ — , and 4.S Theory of Finance. [chap. il i-a+i)- (i'+«) (46) where i is the actual, as distinguished from the nominal rate of interest paid by the borrower, and n is the number of j'ears for which the loan will run. 61. In the above equation both i and n are unknown quantities, but 11 is easily found. The sinking fund, being employed to pay off the bonds, accumulates at the nominal rate of interest, «', and at the end of n years amounts to the total loan. We therefore have the equation \-=-e- i; whence z + l=(l + i')» logg+l) and n— -^ . ., . . . (47) log(l + /) "^ ^ Haring found n we can then apply one of the formulas, A. to F, to find i. Usually n will not be an integer, but it will be sufiicient for practical purposes if we take the nearest integral value which equation (47) brings out. Instead of using equation (47), it will most often be convenient to refer to an interest table to find the value of n. 62. As an example, let us examine the Russian 1864 loan of £6,000,000. The issue price was 85 per cent., the nominal rate of interest 5 per cent., and the accumulative sinking fund 1 per cent. That is, in equation (46), 7(;=-85, ji'='05, and «='01; therefore Ic loo- 6 =14-1667. Also in equation (47) m= , — =36-71 or 37 % +z log- l-Oo nearly: whence a8yi = 14'1667. If we apply formula C, using Gi per cent, for the approximate rate/, we find » = -063297. It need hardly be remarked that we can use interest tables to give an approximate rate when the tables do not contain minute subdivisions of the rate of intei-ost, ami even when the required rate lies beyond the limits of the table. For instance, in the present case, if the table employed contains annuities at only intep-ral rates CHAP. II.] On Anmdties-Ceriam. 49 of interest, we shall find that at 5 per cent. ffl3fi = lG-7ll29, and at 6 per cent. as7J = 14-73678. We see therefore that the required rate must lie between 6 per cent, and 7 per cent., and must be about 6i per cent., and we then calculate a^\ and v^ at 6i per cent., and insert the results in formula C. 63. An interesting case occurs where a loan, redeemable by an accumulative sinking fund, is quoted in the market at other than par some years after issue, and we require to ascertain tlie rate of interest which it yields.. Let us take the i'ollowing illustration. A Government 5 per cent, loan was issued eight years ago, repayable by an accumulative sinking fund of 1 per cent. The eighth annual payment is just due but not paid, and the market price of the loan is 102 per cent. What rate of interest does it yield ? The full period of the loan was, as we have seen by the last article, approxi- mately 37 years, and tlie annual payment made by the Government is 6 per cent, of the original amount of the loan. The capital at present outstanding is that which was outstanding when the seventh payment was just made, that is, by formula (28), Gxaio"!, or 92-2347 for each 100 of the original loan (the annviity being taken of course at 5 per cent, interest) . The market value of the capital still out- standing is, at 102 per cent., 94'07939, and this is the value of an annuity of 6 per annum of thirty payments, first payment due at once, that is of Q(\ + a^'^, whence a29l = 14'6799 ; and applying formula C with 5j- p)er cent, forj, we find the rate of interest to be 5"28008 per cent. In this case the remunerative rate comes out greater than the nominal rate, and this may seem to be anomalous, the loan being quoted at a premium. It is, however, that the premium is nominal, because a payment of the annuity is just due. Deducting that payment, the loan stands at a discount. 64. We must be careful not to confound the rate of interest incurred by the Government with the rate or rather rates of interest realised by the lenders. If the loan stand at a discount and be repayable at par, it is evident that the holders of the bonds which happen to be drawn early for repayment will realise a higher rate of interest than will the holders of the bonds that Happen to remain undrawn till a late period. Thus, in the case of the loan discussed in Art. 60, a £100 bond will cost a purchaser £85, and if the bond happen to be drawn at the end of the first year, the holder will then receive £5 for interest and £100 for principal, that is for an 50 Theory of Finance. [chap. ii. advance of £85 for a year, lie will receive in prin.dp&l and interest £105, and thus realise interest at the rate of more than 231 per cent. If, however, the bond he not drawn till the end of the thirty- seven years, the holder will realise interest at the rate of only very little more than 5f per cent. In Art. 62 we spoke of the rate of interest yielded by a loan, hut that must be understood to mean the rate yielded to a purchaser provided he take up the whole loan, or at least a sufficiently large part of it to ensure a fair average of bonds drawn. It cannot be held to refer to the rate yielded by any particular bond. 65. It will be convenient to close this chapter with one or two ■examples. a. What is the value of an annuity-certain, taking interest at i for the first n years and/ thereafter? Let us assume that the annuity has to run for (n + m) years in a,ll. The value of the annuity for the first n years is evidently 5^-: — - — . When n years have expired, the value of the remaining portion of the annuity will, by the conditions of the question, be 1— (l + i)-m ^^ — r-^ — . But at the present time this latter portion is deferred J , , . , ■n years, during which period interest is at rate i, and its present 1_(1 4-y)-m value is therefore (! + «)"" r-^ . The whole value of the J . ■ , 1— (H-i)-« ,^ .^ 1—(1— ;■)-'» annuity is thus ^^- — + (l + »)~" /3. A person holds a lease at a rent of £20 per annum, with the option of renewing it every seven years by paying a fine of £100. What is the equivalent uniform annual rent? Interest 5 per cent. This question is likely to occur to the tenant at one of the periods for renewal, when he is debating in his mind whether or not he should continue his lease. We may therefore assume that one of the lines is just due. By Art. 30 the value of the future fines is 100 X , and this we must convert into a perpetual annuity by dividing by a„, or by multiplying by i. The annual rent 100 X i equivalent to the fines is therefore — , which we shall find to 1 — ?v CH-4P. II,] On Anmdties-Certain. 51 be £17'282, making the total uniform annual rent £37-282, or £37. hs. %A. If the next fine be not due for seven years, we shall find that the total annual rent will be £32. 5s. Si^. y. A loan of £P is to be discharged by (_p + 9+r) annual instal- ment.s compounded of principal and interest. The p instalments are to be of £a each, the q of £/?, and the r of £y. What is the rate of interest, «? We can only approximate to this rate of interest, and it will be sufficient if in our approximation we neglect all powers of i above the first. We have If now we multiply out, still neglecting all powers of i above the first, and arrange the terms, we have . fa + qP + ry—P 8. The following very interesting question appeared in the exami- nation paper set to the candidates at the intermediate examination, of the Institute of Actuaries held at Christmas 1874: — A Parochial Union has obtained certain loans upon the under- mentioned terms, and it now desires to consolidate the debts and redeem them by a single terminable annuity running from 31st. December, 1874 — 1st Jany. 1850, £5000, by 60 equal half-yearly payments, int. G % , 1st July 1870, 3000, „ 80 do. do. „ 5 % 1st Jany. 1872, 2000, „ 60 do. do. „ 41 7„. Required, 1st, the terminable equal annuity payable half-yearly for thirty years from 31st December, 1874; and, 2nd, the rate of interest (approximate) returned upon the debt when consolidated as above. 5 2 Theory of Finance. [chap. ii. Here it will be proper to assume that the half-yearly instalment of each of the three original annuities which falls due on 31st December, 1874, will be paid, and that the consolidation of the debt will take place immediately subsequent to such payment. The first step is to ascertain the annuities at present payable by the Parochial Union. By Art. 37 we find the half-yearly charge for the first debt to be ^^ at 3 %, or 180G65. «oo| „ second „ ^522at2i%,or 87-078. oso] ., third „ ?^at2i%, or 61-071. Immediately after 81st December, 1874, of 1st debt 38 payments will have been made, and 22 will remain. „2nd „ 9 do. do. 71 do. „ 3rd „ 6 do. do. 54 do. There will therefore remain outstanding, by Art. 33, Of 1st debt, 180-665 xaaal at 3 %, or 2879-240 „ 2nd „ 87-078 x afil at 2^-%, or 2879-765 „ 3rd „ 61-071 X cc^ at 2i%, or 1897-995 Total amount outstanding, 7657-000 We must now take each debt separately, and find — at the rate of interest proper to that debt — the half-yearly annuity of sixty payments which the amount outstanding will purchase. This we shall find to be, by Art. 87, ^STQ^'^^O For 1st debt . " . _ — at 3 %, or 104-035 «60| 00170. Yfir; „ 2nd „ . . ^^_ at 21%, or 98-170 „ 3rd „ . . ^:^^— at 21%, or 57-956 «ool Total consolidated half-yearly payment, 255-161 The Parochial Union has now to pay in each year, for thirty years, £510822, in order to liquidate a debt of £7,657, and we have to find the rate of interest. CHAP. II.] On Annidiies-Certain. 53 Using formula C, Art. 51, and h\ per cent, for our approximate rate of interest, we have a'=14-9439 «'=:14i-9439 a = 15-0043 30 y3i= 6-1410 - -0604 "8-8029 - °^° log -00317=8-50106 302 log 8-8029 =0-94463 12 ^ 3 log (-p) = 4-55643 ^ — ^= 5-214% It -(rill be noticed that in solving this question we have treated the annuity as one of £510-332, payable annually, and so found the annual rate of interest. This we may consider to be the nominal annual rate convertible half-yearly. It would have come practically to the same thing had we taken the case as it really stands in the question, viz.: an annuity of 60 payments of 255-161 €ach, and assumed an approximate rate of interest 2|- per cent. We should thus have obtained the half-yearly rate of interest. The course we have adopted is the more convenient, as being better adapted to the majority of published interest tables. Another method of finding the rate of interest in the foregoino- question might suggest itself; but though at first sight it seems correct, it nevei-theless gives an erroneous result. Of the whole debt of £7,657, there is 2879-240 at 6 %, 2879-765 at 5 %, and 1897-995 at 44- % , and therefore the average for the whole debt is 5-252 %, a higher rate than that already found. To take the average would be the correct course if the three portions of the debt were repayable in the same proportions at the same times. But this is not so. Those portions of the debt at the higher rates of interest are repaid by the sinking fund more slowly in the earlier years and more rapidly in the later years than those at the lower rates, and thus, as time advances, the average is destroyed. The only correct way to look at the question is to treat the consolidated debt as the present value of an annuity of the total annual sum payable by the Parochial Union. THEOKY OF FINANCK CHAPTER III. On Varying^ Annuities. 1. In the direct process of the calculus of FinLte Differences, we form one set of functions from another by the operation called difi'erenoing; that is, if we have a series whose terms are %, v^, it^, etc., we form another series whose terms are (112 — ^1), («3 — «2)j etc., and which we represent by Az«i, Awa, etc.; the successive terms of the second series being the differences between the successive terms of the first series. The operation of differencing may be repeated, the series of differences being itself diiferenced. 2. It is evident that this process may be inverted. Instead of forming a series of differences, we may form a series of sums. We may pass from the original series to another series, of the terms of which the terms of the original series are the differences. Thus, if we have the series Uy, Ui, M3, etc., we can form another series, 5^1, 7^, ^3) etc., such that F^— Fi=Mi, F3— ^2=^2, etc., or, put- ting the same operation in another form, such that F^=F"i+Mi, ^3=1^2+««2i etc. Further, like the process of differencing, the inverse process of summation may be repeated. We may pass to another series from T^i, Y^, V%, etc., in the same way that we passed to F], Fs, Vz, etc., from Ui, u^, Us, etc., and so on without limit 3. The successive orders of the figurative numbers are series con- nected with each other in the way described in Art. 2, and they are represented in the following scheme : — Term 1st 2nd 3i'd 4th 5th 6th 7th etc. ('») order. order. order. order. order. order. order. 1 etc. 2 1 etc. 3 2 1 etc. 4 3 3 1 etc. 5 4 6 4 1 etc. 6 6 10 10 5 1 etc. 7 6 15 20 15 6 1 etc. 8 7 21 35 35 21 7 etc. 9 8 28 56 70 56 28 etc. 10 9 30 84 126 126 84 etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. CHAP. III.] On Varying Annuities. 55 4. The first order is a series of constants, which, for the sake of convenience, we may assume to he unity. Each term of the second order is the sum of all the preceding terms of the first order, so that the m* term of the second order is the sum of the first (m— 1) terms of the first order, the value of which is therefore (m — 1). In the same way the «i*'' term of the third order is the sum of the first (jii — 1) terms of the second order ; and generally the m"' term of the r*'' order is the sum of the first {m — 1) terms of the (r — 1)* order. 5. If by i!^ fl we denote the ot* terms of the r*''' order, and use X? as a symbol of summation of terms from 1 to m inclusive, the fundamental relation between the orders may be represented by the equation fm| ~r\~~^i ^mj r— li • • (1) 6. Since by the last equation, t^ ;:|i=:5T"^ ^m\ 'r-i\, and ^Ilfn| + (l + z)»/o|;^l ' See InsiUute of Actuaries' 2'e.rt-Soo/.; Piivt 11., Chap. XXII., Arts. 20 and 21. CHAP. III.] On Varying Anmdties 57 Adding the two equations together by the help of formula (2), we have =^s+ii ;i + (i + »)^^Ti+ (i + 0%i=i| 7i+etc. + (i + i)''-i<2;r. ^(H-j>:^r| + fiHri;]F| (6) Therefore, after arranging the terms, „-.-,_ ^ «■] '''^1 ~ '^ «+l I rr fn. Sn\ r\— ^ • . (/j Bjf means of this formula we can find s:^ 7| having given Sn\r^\- 12. We have seen, Art. 8, that an annuity of the 0* order con- sists of but one payment, or rather that the annuity is reallj^ only a unit in possession at the beginning of the period. "We may therefore infer that the amount of such an annuity is (! + «)", and its present value unity, and we may use the conventional symbols «n|oI=(l + *)" • • • (8) a-^-^ = l . . . . (9) 13. By formula (7), s:^T|=— °'~^"^'"^ ■ But ^^+11^ = 1, and (1+ i)"' 1 by formula (8), s7q-ol=(l + »)"- Therefore s^^ = ^^ { . This result agrees with formula (7) of Chapter II, and show* that we have correctly interpreted the symbol s^~o\. 14. From formulas (5), (7), and (8) we have (l+i)»_l Snl'Tl—'^ ^•"il ■ n(n — ±) ■s^Tl 3| ^ n(n— -l)(«-~2) 6 *«l *|— ^• «tc. = etc. E 2 58 Theory of Finance. [chap. in. And, to take a numerical example, when « = 5. and j = '05, ^-6l-o[= 1-2762815625 •2762815625x20 S6lTi= 5-52563125 •52563125 '"x20 'x20 "x20 «=5 s-6| ^1=10-5126250 n{n-V)^^^ •5126250 sjl ^1=10-252500 6 •252500 S5|^i= 5-05000 «(7l-l)0l-2)(H-3) 24 =5 •05000 X 20 «'5l"6|= 1' The last result proves the correctness of the work, hecause an annuity for 5 years of the S*'*- order consists of but one payment of unity, made at the end of the fifth year. 15. To find a^ 7], the present value of an annuity for n years of the r*'' order. Following the method adopted in Art. 11, «u| V^l = «°!5"o| 7^ + « if i] 7^ + vH-^^ 73i( + etc. + vH-^ -^cii] an\ 1{^''tl\ ~\-\-vt^ l\ + vH-^ 7| + etc. + i)»i;j+i| 7] = (i + i)«5r+r( V( = (1 + «■) a^n Ti + v'^-i^iT+II ^ Therefore *rtl r- :i|=i«^l 7( + «"!fi+r( 7( and o^ 71= -^^ —. -^ . . . (10) 16. By formula (10), ff^ 7|=-^i4 since ^;r+ii i]=l. This result agrees with formula (10) of Chapter 11. if for a^ -^ we write 1, in accordance with formula (9) of the present chapter. 17. From formulas (5), (9), and (10) we have r.HAP. III.] Oit Varying Annuities. 59 On) I>| = 1 1 — «» d7~l TTI — «51-^. ^ «k| 4|— ^. 11(11— V) (n—2) (n- -3) "■"■' ■^ 24 18. As an illusti'ation, let it be required to determine the values of the first five orders of annuities for 40 years at 5 per cent, interest. ««= -14204568 1- •14204568 •85795432x20 «46l 11= !?• 1590864 40«^"= 5^6818272 40 X 39 -i>«= 110-795630 ffiol Tl= 17^1590864 i^2^^,.o= 1403^41132 ^•6818272 2x3 11-4772592 x20 *«-^^^^^.^o=12981-5547 "^"^l ^^ f'^'f^^' 2x2x4 110 795630 118-749554 x 20 O4"oil= 2374^99108 1403-41132 971^57976 x 20 ejoll]- 19431^5952 12981-5547 64500405 X 20 «Jol -51= 129000-810 6o Theory of Finance. [chap. iii. 19. It will be seen that ia passing to higher orders, the amounts and values of annuities increase for a while with great rapidity when the term n is at all oonsiderahle, and that the result of the division by i at each stage is to diminish accuracy by reducing the number of decimal places. If, therefore, we wish for extreme accuracy in the higher orders, we must commence with a great many decimal places. 20. To find fir^ 7], the value of a perpetuity of the r*'' order. We have seen, formula (10), that a^ •;;:]=: -. ^'. If n increase without limit, then a^| 73T| Avill become «^ r^; «" will diminish without limit ; and J^iHTT] 7i will increase without limit except in the case where r=l. When r=l, then vH^"^ 7| will vanish when n increases without limit; and as a;^ o]=:l for all values of n, the value of the perpetuity of the first order is -^ , the same result as that given in Chapter II., formula (11). 21. Taking the general case, where «(«—!) . . . («— r + 2) , „ t)«i';r+r|7] = f"X j , wc shall now prove that for all finite values of r, t;'VijrFi] 1\ vanishes when n becomes infinite. „ ■ ., n(.n — V) . . . (n—r + 2) r or conciseness we may write ^"X =jy. When n increases to (ra + l), then JV changes from ^n X 1: that is, as long as >(l + t); n—r + 2 (1 + *) n — r + 2 n+1 or, in other words, iV'=iV" when '■ — =(1 + *); that is, iV'=iV n — r + 2 ^ •" when n— ^ ■ —B say, a finite quantity, and when n still CHAP. III.] On Varying AnwMties. 6i increases N begins to diminish. If M. represent the value — which is finite — of IS when n=^R, the value of N becomes successively, as n still increases, . ^^±1^ ^' ^' (igl'^+2)(H^+ 3J ^' '*°' ' *^"^ is, M is continuously multiplied by a series of factors of the form V r r^ , each of which is less than the preceding one, and le— r+(m + l)' ^ =■ less than v -=, , which is less than unity. Therefore, when n J?— r + 2 •' ( B+l )»» becomes (H + m), iV becomes less than If \v — r . But the ( -ti — • r + 2 -" coefficient of Jif being less than unity, diminishes as m increases, and may be made as small as we please by sufficiently increasing m. When therefore m, and consequently n, becomes infinite, iV vanishes, and we have Ocoj r,= — ■■ — • . (.J-i; The value of the jDerpetuifcy of the r*^ order is therefore finite when that of the (r—iy^'- order is finite. But the value of a perpetuity of the first order being finite, so will be that of the second order, and of the third order, etc. ; and generally the value of a perpetuity of the r* order will be finite as long as r is finite. 22. From formula (11) we have _ 1 _ 1 _ 1 and generally o^ 71= > • ■ ■ ^^^^ 23. The reasoning of Art. 21 may appear to the student to be somewhat laboured. That is only because we have avoided the use of the Differential Calculus. If we call in the aid of the Differential Calculus, we can very easily prove that ti^i^iT+i] 7] vanishes when n becomes infinite. The term which we are in- vestigating is '-—^^T^^^-^-^^rr^ (l+0~". the numerator and denominator of which both increase without limit with the increase of n. To determine the limit of the value of such a fraction, we must substitute for the numerator and denominator their respective differential coefficients, repeating the process, if neces- sary, until a fraction is obtained of a form exhibiting its ultimate 62 Theory of Finance. [chap. ni. or limiting value. In the case in hand we shall find that the (^_l)a differential coefficients will answer our purpose. Sub- stituting these differential coefficients for the original numerator and denominator, we have -; r- .. ,^. ,-., , .,„ , which evidently {loge(l + «)}'-'(l + 0" vanishes when n becomes infinite, and we therefore conclude that vH:;;^ ^ also vanishes. 24. In Art. 17 of Chapter II., we found the value of an annuity of the first order by means of reasoning without having recourse to series, and we may now extend the reasoning to annuities of higher orders. The mP^ payment of an annuity of the (r — 1)*'' order, if it be in- vested immediately on its receipt, will produce at the end of each succeeding year the sum of «X im\ ~r~i\-, and it will itself still remain in hand at the end of the n years for which the original annuity was granted. If, then, each of the jDayments of the original annuity be thus forborne, at the end of the first year there will be entered on an annuity arising from interest, for (« — 1) years of i X ifi) irzji, at the end of the second year another annuity for (» — 2) years of « x ii] 7Z\\, and so on, each of these annuities, which we may call secondary annuities, running concurrently with all those that have preceded it. Thus, at the end of the nil-^ year the payments of all these seoondar}' annuities will aggregate i{t\\ 7:Zi| + ^2i 7=i] + etc., 4-^;;^^ V^i]), which by formula (1) is equal to iy.i^i:\. We therefore see that an annuity of the (r— 1)^'' order forborne, Avill produce for n years an annuity of i of the ?■''' order, and in addition there will of course remain in hand at the end of the n years the aggregate of all the n payments of the original annuity, the aggregate of these payments being, by formula (1), ^i^+i] ■;:i,and the present value of the aggregate being y"^,T+i| 7| . Thus we reach the result, a^] 7zr| = Ja::;^ 7] + y"fi;ij:il V, ; whence, as in formula (10), t Similarly for perpetuities. If the payments of a perpetuity of the (r—iy^ order be forborne, each of them will produce another perpetuity, the m"^ payment producing a perpetuity of «!!5r]V3i], and the aggregate payment of the secondary perpetuities to li", received at the end of the «?/'» year will be KfT\ r^ + tii, r-^n + etc. + Z,;;^!^! v:=i|) , CHAP. III.] On Varying Annuities. 63 or bj formula (1), J^m^. Therefore a!»|7^=io^71; whence as in formula (11), a^i^= '" T^ . It will be noticed that in the argument in this article we have escaped the difficulty which occupied us in Arts. 21 and 23. 25. To fiud the amount and the present value of an annuity whose successive payments are Wi, M2, ««s, etc. If forM2) M3, etc., we substitute their equivalents in terms of Ux and its differences, we have* ih — ih «(,3= jji + 2 Ami + ^^i«i 2^4=;«i + 3A2«i + 3A2i«i + A3i(i ; and generally (m — 1) (ot — 2) „ z(„j=z«i+(fli — 1)AmiH '-^ A%i + etc. Therefore the given annuity is equivalent to an annuity of u-^ of the first order, together with an annuity of A Ui of the second order, and an annuity of A^i of the third order, etc. Let s and a be the amount aud the present value respectively of the annuity Mi, %, M3, etc., and let the series Mj, u^, ih-, etc., have (« — 1) orders of differences : then s = s^ II z«i + SnW A2fi+ s^i 3; A^iti + etc. + «m|7| A^'-'m, (13) fl = a^i]2fi + a^-2j Am, + a,T:3 A%i + ete. + a;;;]VI A'— 'zfi (14) If the given annuity be a perpetuity, then a!=«^ i|Mi4-a^2] A2«i + etc. + a!^7, bT~^ui = !i^ + ^+...+^ . . . (15) 26. The great power of the formulas demonstrated in the last article may best be illustrated by some examples. a. Let it be required to find the value of an annuity certain for n years whose several payments are 1, 2, 3, etc., n. Here ?«i = l, Ami = 1, and all the higher orders of differences of m, vanish. Therefore «= «» i| + «;ri 2] _ {\-\-i')an\ i\—nv^ i * See Institute of Actuaries' Text-Boole, Part II., Cli. XXII., Arts. 20 and 21. 64 Theory of Finance. [chap. hi. /3. Similarly, if it be required to find the amount of the same annuity, y. If the annuity be a perpetuity increasing 1 per annum for ever, I 1 «=-. + - « t' 8. Let it be required to find the value of an annuity for fortv 3'ears, whose several payments are 4, 7, 12, 19, etc. ; interest 5 per cent. Here 2fi=4, Ai(| = 3, and A^;«i = 2. Therefore (3!=4aioj il + Saiol 2; + 2aioj a] Taking the values of the annuities as given in Art. 18, n=5507-2.33. e. Find the value of an annuity for fifteen years whose several payments are 21, 39, 54, 60, etc. Here z«i=21, Ami=18, A^z^j^— g. Therefore fl!=21«i5| i\ + ISais] "21 — 3ai5] si This is an annuity which increases till the maximum payment is 84, and which then decreases till the fifteenth payment vanishes. 27. A lease is granted for ten years at an annual rent, with pover to the tenant to renew ten times for a like period on payment of a fine of 1 for each year of the lease expired. What is the value of the fines ? Interest 5 per cent. Here the fines are respectively 10, 20, etc., 100, payable at the end of 10, 20, etc., years. A sum payable periodically at the end of 10, 20, etc., 100 years, may, in accordance with Chapter II. Art. 30, be looked upon as an annuity at rate of interest {(l + z)'"— 1}==/ say. If the annuity at rate j be denoted by ffl'jo], we have the value of the fines 10«'io| 1] + lOa'jo) 'si • Calculating the values of these annuities when «=05, and consequently j=; '0288946, we shall find the value of the fines to be 39'00. 28. The following example is an instructive one : — An annuity-certain deferred twenty years, and after that to run twenty years, is to be paid for by an annual premium, the first payment tt, to be paid down now, and afterwards, at the beginning of each year, a regularly diminishing amount, the last premium being paid at the beginning of the twentieth year, after which the premium becomes extinct. Find tt, having given «;-"= -456387 at 4 per cent, interest. CHAP. III.] On Varying Annuities. 65 Here the benefit to be received is an annuity for twenty years deferred twenty years. Its present vakie is therefore tPa^j^: The consideration for this benefit is an annuity for twenty years, commenc- 77* ing at TT, and decreasing — each year. If this annuity were payahlo at the end of each year, its value would be Tf[a^ i]— ;^«20] 1;) but as it is payable at the beginning of each year, its value is 7r(14t) ( «2ol Tl Pi "^201 "2| ) • The benefit and the consideration for the benefit must he equal in present value: whence 7r(H-«) U^; H— ^«2oi 2|J='y%(ro| 1] and (1 + (^"2i511l-^^'20l 2|j «2o= -456387 fl261T|=13-5903 1— •y2o= -54,3613-^ -04 f^" reversed = 783654 54861 ffiaol T|= 13-5903 20 v'^— 9-1277 4-4626 -^ -04 6795 815 41 ajo] 2= 111'565 ^Q m 1]= 13-5908 - 1 lr^20i'2i= 5-5783 f2»«2ol 11= 6-2023 20 ' 8-3325) 0-2023 (-74435 8-0120x1-04 3695 3205 362 29 4 ,r=: -74435 l + i)(«20|Il-^«2-5 2|)=8 3325 29. In former days, -when De Moivre's hypothesis as to the law of life (see Institute of Actuaries' Text-Boole, Part II. Ch. VI. Arts. 2 to 6, and Ch. VII. Arts. 104 to 109) was of greater importance in the science of life contingencies than it now is, attention was much directed to the annuities certain the payments of which are the powers of the natural numbers. Mr. Baily devoted two chapters of his worli to this subject. By means of the formulas of this chapter, the amounts and present 66 Theory of Finance. [chap. III. values of suoli annuities can be readily found. Thus, let it te required to find the value of an annuity, the payments of which are l^ 24, 3^ etc. Here 1 IG 81 256 5^= 62.5 6^=1296 1^= 2"= 4^= A A2 A3 A^ 15 50 60 24 65 110 84 24 175 19 i 108 369 302 671 Ai- Therefore, the value of the annuity is «^ T| 4' ISffiui 2l + SOa;;] s| + 60«,i;| i] + 24«,3 5] • To find the amount of an annuity for n years, the successive payments of which are n^, (n — 1)^, (m — 2)^, etc. Here Ui^^n^ Ami= — (3m2— 3m + l) A2Mi=(6m— 6) A%i=— 6 and s=ra3s;^T|— (3j»5— 3ra + l)sj[2l+(6«— 6)s^^— 6s^ij. 30. Formulas (13), (14), and (15) apply only to annuities where the differences of the payments vanish after a finite number of orders, and it is not sufficient if the differences merely rapidly diminish but do not vanish. The (r— 1)"' difference is multiplied into an annuity of the j-*'' order, and we have seen that the amounts and values of annuities of the higher orders are very large. There- fore, although the (r — 1)* difference may be very small, its coefficient will be very large if r be large, and the product will be a quantity which cannot be safelj^ neglected. 31. Let us take the following case : — A A2 A3 A4 1st payment 1-000000 •010000 ■000100 -000001 •OCOGOO 2nd " „ 1-010000 •010100 •000101 3rd 1-020100 -010201 etc. 4th 1-030301 etc. etc. etc. These terms are in geometrical progression, the ratio being l^Ol, and the differences diminish with considerable rapidity, the fourth having no significant figure in the seventh decimal place. Let it ha CHAP. III.] On Varying Annuities. 67 required to find the value of the perpetuity at 5 per cent, by means of formula (15). A%i= -000001 -f- -05 •000020 b:hi^— -000100 ■000120^-05 •002400 Az/.i= -010000 -01 2400 -^- -05 •248000 Mi= 1-000000 1-248000-^-05 24-960000 The true value of the perpetuity is 25-0, so that although the first difference neglected is only -00000001, there is a considerable error in the result. This is accounted for by the fact that the coefficient of the first neglected difference is — or 3,200,000. Of course, theoretically, any required degree of accuracy could be obtained by computing a sufficient number of terms to a sufficient number of decimal places, provided that the differences of the series of payments diminish faster -cnan tne increase in the coefficients ; but more convenient formulas for such cases we now proceed to find. 32. From the example in the last article we infer that formulas (13), (14), and (15), are not apjslicable to annuities which increase or decrease in geometrical progression; but it is easy to give a general demonstration of the fact. Let there be a geometrical series, the first term of which is K, and the common ratio M. The following scheme shows the series and its differences: — Term A A^ A^ A< K K{B-V) K{li-1Y E(B~iy KiE-iy KB KB{B-1) KB(B-iy KB(B-iy etc. KB^ KB\B~V) XB%B-iy etc, KB3 KB^{B- 1) etc. KB^ etc. etc. 68 Theory of Finance. [chap. hi. It therefore appears that if the common ratio of the original series be It, the successive orders of differences, because A"=A"~'(J2— 1)^ form another geometrical series with the common ratio {It, — 1), and the diJferenoes can never vanish. Therefore, by Art. 30, formulas (13) to (15) are inapplicable. 33. To find the value of an annuity, the payments of which are in geometrical progression with common ratio H,. If a bo the value of the annuity, we have ffi=(l + i)-' + ig(l + i)-2 + i22(-i^i) -3 _). etc. + iJ»-i(l -!-»■)-» 1 1(1+^)/ 1-fi ^_ R (1 + ^) 1- B I" (1 + J (H-»)-iB (16) Ijet ..,,., = ^ , so that ?= -^ --; (1 + (1+j) ■' H Then 1 1 -(!+./)-« a= li 3 (17) If the annuity be a perpetuitj^, we can find its present value only — — . <1, that is, when i^<(l + ^■). In other cases tho 1 + ^ present value will be infinite. When i2<(l + i), and the annuity is a perpetuity, then fl= y^ ;r ^ (18) (1 + i)— i2 ^ ' In the e.xample in Art. 31, i2=l'01, and the value of the perpetuity at 5 per cent, is, therefore, by formula (IS), -r- or 25. Thus the value of the increasing perpetuity is equal to the value of a fixed perpetuity of 1 calculated upon the assumption that the rate of interest is diminished by the rate of increase of the payments. 34. From formula (17) we see that \)j changing the rate of interest, we can substitute for an annuity, the payments of which are in geometrical progression, another annuity, the payments of which are uniform. The changed rate of inibrest may, however, CHAP. III.] On Varying Anmiities. 5g present anomalous features. If R be greater than unity, then j the substituted rate, will be less than i\ and if i2 be greater than (1 + i), then J will be negative. If B, be less than unity, then j will be greater than i; and there is no limit to the magnitude which _; majr assume when JR, is diminished. 35. It is to the late William M. Makeham that we owe those formulas in this chapter, by which, with the aid of the figurate numbers, we can deal so effectively with varying annuities. Previous writers had investigated particular cases, but Makeham furnished the general theory. He published it in a remarkable paper in J.I.A., vol. xiv. p. 139. THEOKY OF FINANCE. CHAPTER IV. On Loans Repayable by Instalments. 1. In Chapter II we gave consideration to loans wliich are repaj'able within a limited term by equal periodical instalments, including principal and interest. It is the object of the present chapter to extend the investigation to cases in which the capital is repayable in any other manner whatever. The formulas of Chapter III will be of great value in the inquiry. The analysis divides itself naturally into two branches which are of equal importance ; namely, first, knowing the conditions of the loan, to find that value for it which will secure to an investor a given rate of interest ; and secondly, to ascertain the rate of interest which the loan yields at a given market price. 2. When a corporation or a foreign government contracts a loan, it generally undertakes obligations of a twofold nature. It agrees to pay the lender interest at a fixed rate as long as his advance remains outstanding, and it promises to repay at stated periods the capital itself. Frequentlj'- the borrower holds out to investors inducements beyond the stipulated rate of interest, by issuing the loan at a discount and repaying it at par, or by issuing it at par and giving a bonus on repayment. Sometimes, on the other hand, where the credit of the borrower is good, he may find it to his advantage to raise the loan at a higher rate of interest than the public demand from him, and therefore to issue his bonds at a premium. These complications, however, will not render the investigations much more intricate. 3. Let (7= the capital repayable by the borrower. y=the nominal rate of interest thereon paid by the borrower. »=the actual rate of interest realised by the lender, whom we may also call the investor, or purchaser. ^=:the present value of the capital at rate i. .4= the purchase money, or the value of the loan. CHAP. IV.] On Loans Repayable by Instalments. 71 To illustrate the symbols, let us suppose a loan to be contracted for £10,000,000 at 3 per cent., by ten thousand bonds of £1,000 each, the bonds to be paid off at maturity with a bonus of 25 per cent. Here the capital repayable by the borrower is really £12,500,000, which sum we therefore represent by O, and the nominal rate of interest is not 'O.S, as would appear from the stated conditions, but -j-f^' or '024, which we represent by j. The loan being nominally issued at par, we have ^=10,000,000. The p)oints to be noted in connection with the symbols are, that G represents the capital repayable by the borrower, including any bonus he may contract to pay along with it, and that j represents the ratio between the annual interest contracted for and the capital repayable as thus defined. We must therefore be careful to dis- tinguish, as in the foregoing example, between the nominal rate of interest as stated in the conditions of the loan, and the nominal ratey which concerns us in our investigations. These rates will in many cases be identical, but they are not necessarily so. 4. To find the value of a loan, repayable by instalments at stated periods of time, with interest in the meantime at rate j, so as to yield the purchaser a given rate of interest, i. The value of the loan consists of two parts, the value of the capital, and the value of the interest. The value of the capital at rate i being i', the value of the interest is evidently {A— SI). Had the borrower contracted to pay interest; at rate i, then the required value of the loan would have necessarily been par, and A would have been equal to C ; and therefore the value at rate i of the interest in this particular case would have been (JJ—K). The annual interest on the loan would, on the same supposition, have been iO, and the value of each annual unit of interest pay- Q ]^ able by the borrower is therefore — -r- . But the borrower has actually contracted to pay interest at rate j instead of i, or jO jO j annually, and the value of this interest is -rp^{G—K), or - {C—K). Adding to this the value of the capital we have the value of the whole loan, A=K+'^G-K) . . . (1) 5. It will perhaps enable us better to understand the demon- stration of the last article if we confine om' attention for a moment y2 Theory of Finance. yZiiKv. iv. to a single unit of the loan. Suppose that unit to be represented bj' Oi, and to be payable at the end of iii years, and let its value be Ki, so that ^i=(H-i)~"i. The interest payable on the unit is an annuity of j per annum for n^ years ; but the value of 1 per annum for 111 years is ^^ — ; — , or — ^-^ — ^ , and therefore the value of/ per annum is 4(Ci— -Zi). Adding to this the value of the unit of the capital itself, we have j -£'i+ 4(Ci— Xi) k the total value of that particular loan-unit under consideration. If now vre take into account the other units Ca, G^, etc, due at the end of «2, %, etc., years, we have corresponding formulas |Z2+|(C2-X2)J, jis+^CCa-Xs)}, etc., and adding together the values of all these portions of the loan, we have for the value of the entire loan (X, + X2-|-X3+eto.) + |{(Ci+t72+C3+etc.)-(Xi+X2+i3+etc.)} or 1-^+ 4 (G-^)| as before. 6. In Art. 4 we have treated the capital of the loan as a whole. FoE" purposes of calculation it will often be convenient to represent the entire capital by unity, so that we write 0=1. Under these circumstanced formula (1) takes the elegant form t =1-(1-X)(^1-^) . . (2) 7. In formulas (1) and (2) we do not limit the repayments ot capital to any particular conditions. We simply find the present value, K, of the capital, under whatsoever arrangements it may be repayable, and insert it in the formulas. The formulas will take various shapes according to the various methods by which loan« may be liquidated. Thus, if the loan of unity is to be repaid in one sum at the end of n years, we have ^=1- (!-««) (i- 4) =:1— a!n|(i— i) ... (3; where aiA is taken at rate i. CHAP. IV.] On Loans Repayable by Instalments. 12) -i). . w Again, if the capital of I be repayable by n equal annual iustal- ments of - each, then \ n . Also, if the capital of C be repayable by annual instalments, Ui at the end of the first year, M2 at the end of the second year, and so on, then in formula (1), j£'=yMi + 'a%2+ w^Wa + etc, and if the payments Ui, U2, etc., form a series, the differences of which vanish after a finite number of orders, tlien by Chapter III., formula (14), -2^= rt-S] T| «i + «^ 2] Ami + a;^ -3] A^mj + etc. (0) 8. The following examples will elucidate the subject : — ■ (a.) A bond for £1000 is to be sold. It bears interest at 3 per cent., and will be repaid at par in twenty years. An intending purchaser desires to make 5 per cent, on his investment. What price can he afford to give for the bond ? Here formula (3) is applicable, writing «ii] = «2ol at 5 per cent., and » = '05 and j= Q'i. ^=1000{l-«2ol(-05-03)} =1000{1 — 12-462 X -02} =:::750-7G. (/3.) A loan of £1000 is to be repaid as follows, with a bonus of 25 per cent : — One twenty-seventh at the end of four years, „ ^ » five years, and so on. Finally, one twenty-seventh at the end of thirty years, interest being payable in the meantime at the rate of six per cent. What price must a purchaser give so as to realise 5 per cent, on his outlay ? Here in formula (1) we must write 0=1250 andy= — ' °^ '^'^^ ' also, K= ^^ X 1250 at 5 per cent. log 1-3 =i-93643 C-X=664-39 logflMfi =1-16563 __840 log 1250 = 3-09691 26576 4-19897 J'W^, log 27 = 1-431.36 -05)3 1-S9l" iQ^Jt =2-76761 i.'O-X) = 637-82 ./t = 585 61 i^ • l(G-K)- 637-82 ^=1223-43 !■ 2 74 Theory of Finance. [chap. iv. (y.) A loan of £10,000 is repayable as follows : — £94 at the end of the first year, 102 ,, second year, 110 „ third year, and so on till the whole loan is liquidated, interest at the rate of 3 per cent, being allowed in the meantime on the outstanding capital. Eequirod the value of the loan at 5 per cent. We must first find the term of the loan. By the well-known formula for summation in the calculus of Finite Differences {Inst, of Actuaries' Text-Booh, Part II., Ch. XXIV., Art. 17), if there be a series of n terms, the first of which is u^ , the sum of the series is ^liti 1 ) nu\^ 7^ Am, + etc. In the present case the sum of the series If is 10,000, U\ is 94, and A««i is 8. Making use of these values, we have a quadratic equation in n, namely, 4?i^ + 90;i— 10,000=0; whence, n being necessarily positive, we have m=:40. The capital repayments are therefore of the nature of an increasing annuity for forty j'ears — first payment 94, second payment 102, third payment 110, etc., and by formula (5) we have ^=94«io] "I] + Saio] 21 • In Chapter III., Art. 18, we have already calculated the value of aio] ~i\, and employing those figures, we have X=3449-316. Making use of this value in formula (1), ^=3449-316+ % X 6550-684 5 =7379726, or £73. 15s. ll\d. per cent. 9. In the foi-egoing investigation we have divided the loan into two parts, the capital and the interest, and we have expressed the value of the interest in terms of the value of the capital. We might have pursued a different, and apparently a more direct course, and valued the capital and interest separatelj-, but the effect of the method of solution which we have followed, and which is due to the late W. M. Makeham, is generally to reduce the series to be valued by one order of differences. Thus, suppose we take the case of the capital of unity repayable by n equal annual instalments of - each, — formula (4) — and value the capital and interest independently. The value of the capital is — . The interest payable at the end of the first year is j, at the end of the second year ( 1 )/, at the end CUAP. iv.j On Loans Repayable by Instalments. 75 of tlie third year M — -U", and so on. The value of the interest is therefore >^ H — ^ a^ -51 , and the value of the whole loan ^~"^\^~j) '^- '^^^^ formula, although it involves an annuity of the second order, is really identical with formula (4). Thus, n \ ij I = 1-^1- ^) {1- 4") , as in formula (4). 10. We pass now to the converse problem: — Having given the value of the loan, lo determine the rate of interest. 11. We have, by formula 1, A=K-\- 4 {O—K), whence, by simple algebraical transformation, . . 0-K '-^A^K ■ • • (6) In this equation, seeing that i is the unknown quantity, and that X. is calculated at rate «', we cannot assign the true value to K, but if we find an approximate value for K, and insert it in the formula, we shall obtain an approximate value for i ; and the nearer the assumed value is to the true value of IL, the nearer will the resulting approximate value be to the true value of i. 12. As an illustration of the formula, let us take the following example. What rate of interest does a Government 3 per cent, loan issued at 73 per cent, yield when it is redeemable at par by uniform annual drawings of 2 per cent. ? Here the 3 per cent, paid by the Government yields a little over 4 per cent, on the issue price of 73, and, in addition, the lender will on repayment get a bonus of 27 on each 73 invested, and this 76 Theory of Finance. [chap. iv. is evidently equal to fully 1 per cent, per annum additional interest. We may therefore assume, to begin with, a rate of 5 per . r.^ 100- 2^501 , cent. .The formula becomes 4=03 X—- — -,r^r ■, where ajol is 7o — Aaf,(i\ (33-488 ^^„^„ taken as a trial at 5 per cent. The result is, i = '03 X _ = •05219. The assumed rate turns out to be too low, and to get a closer approximation, we might insert in the formula the value just found for i, and work it out again. We can, however, proceed in another way, which will often, by the help of interest tables, be more easily , applied. Assuming a higher rate, say 5i- per cent , we find i to be '05070, and from the two approximate rates for i we can find a third more near than either. Thus 5 gives 5'219 per cent. 5'5 gives 5'070 per cent. Diff. -SgivesDiff. — -IIU •149 whence 5+a'=5'219— — 77"^, nearly, ' li whence a;:=^169, and the rate which we seek is 5'1G9 per cent very nearly. 13. The rationale of the method of approximation above illus- trated is easily seen. If the trial rate, which we may denote by Ji, ^vere the true one, then that rate itself- would be the result of making use of it in the formula, but, seeing that .Zi is only approximately true, our result, which may be written J^ , is also only approximate. If now we assume another trial rate,, J2, differing from Ii, by A, say, we get another approximate result, J^, differing from Jj by 8, say. Since a change of A in the trial rate produces a change of 8 in the resulting rate, therefore a change of x in the trial rate will approximately produce a change of —a; in the resulting rate. AVe seek a rate which, when used in the formula, will reproduce itself, and if that rate be Ti + x, we must therefore have -Zi + a;=J'iH — x, Ax(J-,- J,) , wnence *■= : r and A — S Ax (J-,- J,) «_J,+ —-5 . ., (y) CHAP. IV. j Oil Loans Repayable by Instalments. 77 This method of approximation may often be advantageously resorted to with other formulas than that to which we have just now applied it. 14. It is very easy to show that formula (6) of the present chapter is an extension and generalisation of Baily's, No. 35 of Chapter II., which applies only where the loan is repayable by equal annual instalments, including principal and interest. In the ease of such a loan we have, by Chapter II., Art. 34, the capital included in the «»''' payment equal to ii'i-'i'+i. The present value of the capital in the to* payment is «)™y»-™+i, or «"+', or (l + »)~'"+" ; and there being n such payments, the present value of the whole of the capital is to(1 + i)^'"+>'. Now, if an approximate rate I be obtained by inspection of the tables, we have in formula (6) 1— ('14- J")"" C= ^^ — ' ^'^^ i'=ra(l + -Z') "'"+", while A is the value of the loan. Therefore l-iX + iy- ^^^^^ ^ ^^_,„^„ i=Ix - ^-«(l + 7)-'«+i' l-(l + 7)-_^ ^ ^1 -""^ .4-m(l + /)-'"+i' I 1-(1 + J)-'' ^ whence ^--^=-^X ^_^^^_^^^_(„^„ . . . (8) In Chapter II., formula (35), we have denoted (i—I) by p, I by 1 — ("l + J"^"" j, ~ — ^ ^ ^ — by a', A by «, and w(l + J)-l»+i' by ««»+'. Making these substitutions in the above formula (8), we at once have formula (85) of Chapter II. 15. By the following reasoning, we may devise a formula which is moi-e accurate in its results than formula (6). We have seen, Art. 4, that {0—K) is the value of the interest when it is payable at rate i. Therefore - (0—K) is the value of the interest for each unit of the i rate and — is the annual rate of interest for which a payment (C— -£) yS Theory of Finance. [chap. iv. Q ^ of 1 down will provide. Hence i— — =. is the extra annual rate of interest for which the discount {C — A) will provide, and this extra rate which we may denote by A, must be added to j, the rate actually payable, in order to get i, the rate realised by the lender. Seeing that i is unknown, we may take a near value, and denoting that by I, we have approximately 16. We have shown that formula (6) is a generalisation of Baily's, No. (35) of Chapter II. ; and novf we can similarly show that formula (9) is a generalisation of Barrett's, No. (38) of Chapter II. As in Art. 14, we have C equivalent to a', A equivalent to a, and K. equivalent to »«"+', while we have now denoted by h that which we formerly wrote p, and by I that which we formerly wrote j. Formula (9) therefore becomes p=y which is identical with Barrett's formula. 17. Following precedent, we shall close this chapter with a few examples. (a.) A loan of £2,000,000 is issued, repayable as follows : 50,000 at the end of 5 years. 60,000 „ „ 6 „ 70,000 „ „ 7 „ and so on till all is repaid, interest on the outstanding amount being allowed at the rate of 5 per cent. The issue price of the loan is 93 per cent. What does the loan cost the borrower ? Here, by a process similar to that followed in example y of Art. 8, we find that the loan will all be paid off in 16 years after the repayments commence. The repayments are in the form of an annuity, and as the first payment of that annuity will take place at the end of five years, the annuity is deferred four years. We therefore have, for the value of the capital, after dividing by 10,000 for the sake of brevit3% -^=w''(5aie] + ai^2;), while C=200, and ^=186. If we use formula (9), and take for a trial rate 5^ pel cent., we have CHAP. IV.] On Loans Repayable by Instalments. 79 aio| = 10-46216 16«'s= 679330 •055) 3-66886(66-706=aie)2i 868 388 360 5ai6]= 52-311 ;i-05'; 200- 186 gi6i2|= 66-706 ~ ^ ,200— 96 0725 119-017 _ r 1* «< reversed 712708 — -055 x j^^g.ggys log -055=2-7404 „ 14 =1-1461 952136 8331 238 12 8 X= 96-0725 1-8865 log 103-9275= 2-0166 log A=3-8699 /t= 007401 i=-05 »•= -057401 =5-7401 per cent. The trial rate of 05 per cent, is evidently too low. If we try again at 6 per cent, we have aie] 2] = 63-459 and X=90-2899, and the rate brought out by the calculation will be 5-7656. Applying now formula (7) we have jri^5-5 per cent., J'i = 5-7401 per cent., A = -5, and S=-0255 : whence the final rate ? = 5-753 per cent. (/8.) Colonial 5 per cent. Government bonds repayable at par in 19 years are quoted in the market at 107|- per cent, after making allowance for the interest accrued since the last payment of dividend. What rate of interest do they yield? and, to yield the same return, what should be the price of 4 per cent, bonds repayable at par in 25 years ? For the first part of the question, using again formula (9) and trying 4 per cent., we have {K being equal to 47-464), «=4-410 per cent. Trying again 4|- jjer cent, we have (^ being equal to 43-330), 4=4-385 per cent. Interpolating by means of formula (7), we finally have for the rate yielded by the bonds 4-390 per cent. For the second part of the question we may conveniently employ formula (2) where y=-04 and ?=-0439, and where K\% v^^ taken at rate i, or -34161. The formula becomes ^=1— -65839 x \\— — — j or 94-151 per cent. So Theory of Finance. [chap. iv. 18. In practically applying the formulas of tUs chapter and of those that precede, circumstances may render necessary various modifications ; and it may also frequently happen that there is no formula given which will directly meet the case in hand. The IJrinciples will however remain constant, and the actuary who has. fully mastered the principles will find no difiicul*y in adapting the formulas to special conditions. In the affairs of life, mathematical rules cannot be made rigidly to apply, and the actuary, having made himself thoroughly acquainted with the mathematical rules, must never fail to eserclse a sound judgment when he comes to make use of them. THEOEY OF FINANCE, CHAPTER V. On Interest Tables. 1. In the preceding chapters we have given formulas by means of which all the values to be found in interest tables could be independently calculated as required, but it is evident that such a process would be tedious, and tliut the convenience is great of having those values that will be commonly wanted ready prepared and presented in tabular form. The solution of many problems too — such, for instance, as finding the rate of interest involved in a term arlnuity — is rendered very much more easy by a reference to tables, and to the skilful actuary they are at all times invaluable auxiliaries in his work. It will be freauentlv noticed that where a novice goes through an intricate calculation in answering a question, the adept produces the same result seemingly without thought or effort ; and on enquiry it will usually be discovered that tables are the tools he uses to shorten his labour and save his time. 2. An intimate acquaintance with the nature of the tables he may find in his hands is essential to the actuary, for without that knowledge he cannot turn them to the best account. It is ther®. fore very desirable for the student to practise the construction of tables for himself, although those he will require may already be in print, as by actual experience in their manufacture he will much more readily obtain a clear knowledge of the properties of his tools, than by only theoretical study. The principal object of the present chapter is therefore to explain the best methods of constructing and verifying interest tables, and it will for the most part be left to the reader to gain for himself practical skill in using them, by studying the first four chapters of the work with the tables in his hands. Sufficient tables are given at the end to enable him to do so without going beyond the pages of the book itself. 82 Theory of Finance. [chap. v. 3. In interest tables the functions most commonly tabulated fai eacli rate of interest are the following : — (i.) (1 + »)'', the amount of 1 in n intervals, (il.) »», the present value of 1 due at the end of n intervals, (iil.) s^, the amount of an annuity of 1 for n intervals, (iv.) ffl^, the present value of an annuity of 1 for n intervals. (v.) — =s-\ the sinking fund which will redeem a debt of 1 in n intervals; or, in other words, the annuity which will accumulate to 1 in « intervals. Cvi.') — =«^^, the annuity for n intervals which 1 will purchase. The values under each heading are arranged in columnar form, commencing with the v.alue for one interval, and finishing generallv with that for one hundred intervals. 4. The following is an example of the form which the interest table sometimes assumes : — Interest 5 per cent. (i-) (ii.) (iii.) (iv.) (v.) (vi.) {y^iy «» «S| «S1 a--^ n 1 1-050000 •952381 1^0000 •9524 1000000 1-050000 2 1102500 •907029 2^0500 1^8594 •487805 •537805 3 1-157625 •863838 31525 2-7232 •317209 •367209 etc. etc. etc. etc. etc. etc. etc. 5. All tables are not arranged in the same manner. Frequently the functions, all at one rate of interest, are placed in parallel cohimns as in the specimen in Art. 4, so that there is a distinct table for each rate of interest. Sometimes each function is kept by itself, the rates of interest being side by side. Such are the tables which are given at the end of the book. 6. Corbaux in his work, published in 1825, Doctrine of Compouni Interest, supplies very complete tables. He gives aU the columns (i.) to (vi.) mentioned in Art. 4, for each rate of interest rising by \ per cent, from 3 per cent, to per cent. ; and not only does he do so, CHAP, v.] On Interest Tables. 83 but for each rate of interest he also gives the values of all the functions when interest is convertible either yearly, half-yearly, or quarterly. Some tables, instead of supplying the values for interest con- vertible half-yearly or quarterly, very minutely sub-divide the rate of interest, and also begin with very small rates. In this way they answer the same purpose as Corbaux's tables. Thus, an annuity of 100 per annum payable quarterly for twenty years, at 4^- per cent., interest convertible quarterly, is equivalent to an annuity of 25 per annum for 80 years at 1-|- per cent., interest convertible yearly. We therefore see that, in speaking of interest tables, it is more appropriate to name internals, as we have done, rather than years. The tables of Colonel Oakes, published in 1877, are of the last- named description. Thej' furnish columns (i.) to (iv.) at each rate of interest, rising i per cent, from -f per cent, to 10 per cent. Such also are the tables of the late P. Hardy, F.K.S., published in 1839. They contain columns (i.) to (iv.) for rates beginning at \ per cent., and rising by \ per cent, to 5 per cent., and also for 6,. 7, and 8 per cent. Eance's, published in 1852, are o£ the same kind, and give cols, (i.) to (iv.) for rates beginning at \ per cent, and proceeding by steps of \ per cent, to 10 per cent. 7. It may he noted here that although some published tables give all the sis columns (i.) to (vi.), it is not really necessary that columns (v.) and (vi.) should both appear. We saw. Chapter II. formula (29), that a-^ — i=^s^. If therefore ffl",S the annuity which 1 will purchase, be given, we can at once, by deducting the rate of interest, find the sinking fund. We therefore, in the specimen tables at the end of the book, have not included column (v.) of Art. 4. 8. We have said that it is usual to tabulate the functions for all values of n from 1 to 100, but for ordinary purposes a table of such extent is not essential. It is not often in practice that the values and amounts of annuities are required for so long a period M 100 years. A table, too, may be effectually used for longer terms af years than it actually includes. Thus, suppose there is a table which goes up to 50 years only, and it is required to find the values of the functions for 60 years. We have (l4-i;)(iii=(l-|-t)60x (l + i)^", ■y»=-y™X«"', «60l = «5ol+«™»io|, and S6ol=(l l-i)i»S601 + siol. It is, however, convenient to possess exten- sive tables, especially if they are of the description of those of Oakes, and not of Corbaux, and we must use them for intervals instead of years ; because, for example, an annuity for 25 years at 84 Theory of Finance. [chap. v. 5 per cent., payable quarterly, is equivalent to one at ];| per cent, for 100 intervals. 9. Insurance premiums are payable in advance, and thus. are of the nature of annuities-due. It might therefore be thought usijful for the purposes of life offices to tabulate the values and amounts of annuities-due instead of those of ordinary annuities : but the more usual form of tables, which is that we have adopted, provides for every object. Thus, the amount of an annuity-due for n years is evidently si^| — 1, which can be found by inspection from a table of tlie amounts of ordinary annuities. For instance, to find the amount of an annuity-due for 25 years at 4 per cent, interest, we enter table 3 with 26 years, and, deducting unity, the result is 43'31174. Again, to find the sinking fund payable at the beginning of each year to redeem a debt of I in n years, we can find the sinking fund payable at the end of each year and multiply it by v. For example, to find at 4 per cent, interest the sinking fund jiayable at the beginning of each year which will redeem 1 in 25 years, we use table 5, and find first the annuity which 1 will purchase for 25 years. This is ■064012. Deducting the rate of interest, '04, we have •024012, the sinking fund payable at the end of each year to redeem 1 in 25 years. Multiplying now by -961538, the value of v taken from table 2, we have '023088, the quantity required. This might be obtained equally easily in another way, namely, by finding the amount of an annuity-due for 25 years, and taking the reciprocal by means of Barlow's or Oakes's table of reciprocals.' 10. To construct a table, the most obvious course is to c.ilcuhite independently each value that is to be tabulated; but that course would very seldom be the best to pursue. It would usually he very laborious ; and, besides, in order to insure accuracy, the whole work would have to be done in duplicate. It is generally much preferable to take some formula connecting the consecutive values of the function, and by means of it to compute them one from the other in succession. In this way each value i.s made to depend on all that go before it, with the consequence that it' an eri'or occur in one it is carried on to those that succeed, and we can thei'cfore feel confidence that if any particular value be correct, all those that go before it are correct also. This method of constructing tables is 1 By means of Orchard's Tables, the sinking fund payable at the beginning o£ each year to redeem a debt in n years, can be found i)y inspection. We have only to enter the table of annual premiums with a,iri|, and the result is the sinking fund required. It is beyond the scope of the present work to describ* Orchard's Tables. See Inst. Aotuaries' Te.vt-Sook, Part II., Ch. VIII. CHAP. V.J On Interest Tables. 85 called the " continued method," and when it is used, a periodica] check, say at everj"- tenth value, is all that is required. 11. To employ the continued method, three things are necessary. We m.ust have a convenient worhing formula connecting the value of the function for n years with that for (« + l) years : we must know the initial value on which all the others are to be built: and we must have a verification formula by which to apply periodical checks. In interest tables these three requisites are simple and easily obtained. In many other tables they do not present themselves so obviously, but the computer will find it to his advantage to seek them in order to construct his tables continuedly, on account of the great facilities which the continued method gives for insuring accuracy. > 12. To construct a table of (1 + i)". This table for the majority of the rates of interest in use, can best be formed by direct multi- plication. The values in the column are connected by the relation (1 + j)"+'=(l + »)(l + *)"! and i being a small quantity not usually consisting of many digits, multiplication by (1 + ?) is easy. We add to (1 + i)" the result of its multiplication by i, and so obtain (1 + ?)"+'. This is the working formula. The amount of 1 in one year is (l + «), the initial value. To cheek the work we must calculate by means of logarithms, say every tenth value ; or, if we construct the tenth value by logarithms, we can form the twentieth, thirtieth, etc., by raising the tenth to the second, third, etc., powers by ordinary contracted multiplication. 13. When i consists of but one significant figure — for instance, when «=-04 — the work of making the table is very (i_|_2)m easy. A type of the operation is given in the margin. 4 pg^, (,g,j^ The number of decimal places will go on increasing — _ indefinitely unless the increase be checked. When we have obtained as many as we require, we must, as in the example, cease to allow them to extend, merely 2. 1-0S16 taking account of the proper carriage from the neg- 432G4 lected figures to the figures which we retain. Thus, ^ 1-121S64 in the marginal example, the result of multiplying the 44995 last two figures of (1' + *)^ by '04 is 256 : we neglect _ the 56, and as the figures neglected are greater than 4. l-1698t)9 49, we increase the next figure by unity, carrying 3 ^^'^^^ instead of 2. In order to ensure accuracy in the last 5 1.216653 decimal place of the tables, we must work to two * # * 86 Theory of Finance. fcii. AP. V. pkces more than we mean finally to keep, and when the work is finished we must cut down the results to the required limit, taking care always, when the value of the rejected figures is greater than 49, to increase by unity the last place retained. Thus, if we wish to cut down 2'4647155, to five places of decimals, we should write 2'46472, whereas to cut down 2'4647145 we should write 2-46471. 4ii per cent. 1. 1-0425 41700 2606 14. Where i consists of more than one significant figure, we can generally find a short method of multi- plication. Thus, if «=:'0425, we can divide hy 4, instead -of multiplying by 25, of course correctly placing the result as regards the decimal point. Again, if the rate be 4|- per cent., we can divide by the result of the multiplication by 4. A few lines of each of these examples are given in the margni. 15. Where i is such a number that multiplica- tion becomes troublesome, recourse must be had to logarithms. This leads us to the next problem. 16. To construct a table of log (1-J- »)«■. Because log (l-|-i)"=w log (1 + *) the table con- sists of the successive multiples of log(l + «), and these may be formed most conveniently by addition. The value of log (H-») should be written at the top of the column, and again at the foot of a card, which is moved down as the additions are performed. A verification is naturally obtained at every tenth value, the tenth being ten times the first, the twentieth ten times the second, etc. ; the figures of each pair, there- fore, being the same, the decimal point isnly being moved. 2. 1-086806 43472 2717 3. 1-132995 45320 2832 4. 1-181147 47246 2953 5. 1-231346 * •* # (i+O'' 4f per cent. 1. 1-046667 41867 6978 2. 3. 5. 17. The last figure of log(l + i') is only approxi- mately true, and the error in it is progressively multiplied as the work proceeds, so that a correction must- duced to counteract the accumulation of error. Thus at 4 1-095512 43820 7303 1-146635 45865 7644 1-200144 48006 8001 1-256151 * * * be intro- per cent. CHAP. V. On Interest Tables. 87 log (1 + *) = '01703334. If in the operation we retain only six places of decimals, and work with the number •017033, the value of log (l + i)'" will come out -170330, and of log (l + i)'»", 1-703300, whereas they should be -170333 and 1-703334 re- spectively. We may keep correct the last place to be retained in one or other of two ways — either by working with two places more figures than are to appear in the final table, or by applying a correction as the work progresses. This second method is as follows: — The seventh and eighth figures of log (l + i), at 4 per cent., are 34, or almost exactly a third of a unit in the sixth place. If, therefore, we work with only six figures, and increase by a unit in the sixth place the second, fifth, eighth, etc., values, our results will be accu- rate. The upper specimen in the margin shows the first of these methods of procedure. Eight figures are there used, although only six are to be retained, and the two that are to be cut ofl' are separated by a space from the others. When the final cutting- down process is effected, the usual correction must be made when the figures neglected are greater than 49. It will also be observed that if we had used only six places of figures in the operation, and, applying the second method of procedure above named, if we had, to form the second, fifth, eighth, etc., values, added 017034, instead of 017033, which is used to form the other values, the result would have been also correct. By examining the first two figures of log (1 + which are to be neglected'^ we can always see at what intervals in the work the last place which we retain is to be increased or diminished by a unit. Thus, at 6 per cent., log (\^i) is 02530587, or to six figures, •025306 This last quantity is -13 in excess in the last place, and therefore when we use it for continued addition, we must diminish the results by a unit at the fourth value (because 4 X 13 = -52 > •49), and thereafter at every eighth value. The necessity of going over the work to correct the last figure will be obviated if, at the top of the column, lut not on the moveable card, we increase the first figure rejected by 5. In the ma'-gm the operation is repeated with this adjustment, and it will be noticed that without further alteration the six figures to be retained are accurate. lo ■g (! + «■)« 4 per cent. 1. 017033 34 2, 034066 68 3! 051100 02 4. 068133 36 5. 085166 70 6. 102200 04 7. 119233 38 8. 136266 72 9. 153300 06 10. 170333 40 * * * log(l + »)» 4 ! per cent. 1. 017033 84 2. 034067 18 3. 051100 52 4. 068133 86 5. 085167 20 6. 102200 54 7. 119233 88 8. 136267 22 9. 153300 56 10. 170333 90 * * * 88 Theory of Finance, [chap. v. 18. To construct a table of ««■. Since ij"+'=«x«", we could form the table by beginning -with « and multiplying continuedly by v ; but this would not be convenient. The quantity v has generally many significant figures, and the multiplications would therefore be lengthy. By changing the working formula into «''=(14- «')«'"■'"', we can reduce the labour, and make it no greater than that involved in preparing a table of (l + i)". Commencing then with the last value of «* to be tabulated, we work backwards ; but in other respects we proceed exactly as if we were preparing the column (1 + »')"'. All that we have already said will therefore apply, and it is unnecessary to repeat illustrations here. 19. It is very useful, when we have completed an entire column of a table, to be able to verify by one operation the whole work, or to be able at once to check a printed table with which we are not familiar. In the case of the majority of columns of interest tables, excellent formulas for this purpose are easily found. We have ss|=l+ (1 + + (! + «)'+ +(l + i)»-'- If, therefore, we add up our column of (l + »)" and increase the sum by unity, the result, if our work is correct, will be sj+H. Similarly, «;g='y + ?)2+ . . . »», so that the sum of our column of »«• should be a^. 20. To construct a table of g^ . Since si^ = si)+ (l + f)» it follows that the column (l + i)» con. sists of the differences between the successive values in column s^, and we can therefore, if we have already formed a table of (1 + »■)", construct a table of sj) by mere summation. Commencing with 1, the amount of 1 per annum for one year, we add succes- sively (1 + 0, (1 + *T' etc., so forming sa], sj), etc. At any stage the work may be checked by calculating independently the amount of the annuity by means of the formula sii\-=. —.-^- . As the values in the table of (l + «)" are only approximately true in the last place, the errors, although they are in both directions, and so will in general tend to neutralise each other, may sometimes at certain points in the table of s:^ fall in such a manner as to produce a sensible inaccuracy in the last place. We must therefore, to insure exactitude, work to at least one place more than we mean finally to retain. ■CHAP. v-J On Interest Tables 89 21. If we do not already possess a table of (l-f-j)", theless form our table of 5,7] with great facility for the majority of rates oE interest, by simple multiplication. The relation to be used is s;j+ii = (l + »)*«i + l- A-t ■each step we multiply our previous result by (1 + j), •exactly as we did in Arts. 13 and 14, the only diflier- •ence in the operation being that now we add a unit ■at the same time that we make the multiplication. In the margin we show the construction of the ■table at ■i per cent, interest. It is needless to add further examples. 22. At each stage, for our addend, we multiply ■our previous result by i and add unity : that is, "to form s^H^ ■^^s ^■•^d to s^ the quantity «s,i| + l. we can never- 4 per cent. 1. 100 104 2. 204 1-0816 3. (1 + 0'^ 31216 1-124864 4-246464 1-169859 5-416323 1-216653 6. 6-632976 4. 5. it follows But from the equation Sn\'=^ that 2S,i| + l=(l + 0", and this shows, if proof be needed, that oi^r -addends are the successive powers of (l + ^). Therefore, in con- structing our table of s,7i by this method, we at the same time form -a table of (1 + «')", and that without greater labour than when we •construct the table of (l + i)"- alone. It is thus desirable, even if we want only a table of (1 + »)"•, to form, at least when the rate of interest is integral, — the table of si) , as by so doing we obtain two tables in one operation. 23. A very useful formula is available to check our complete •column of s^l, or to verify a printed table of this function — ( 1 + - 1 ^ {\+iy-i ^ (1 +^)^-1 i -+' - + -^ + etc.-4- (l + j-)«^l (! + «•) + (l + 0'+(l- + ^)' + etc.+ (l + i)»-»_(l + »>.;r|-« If, therefore, we multiply'- the last value in the column by (l-|-«), •deduct TO, and divide by «', we have as result the sum of the column. Should the equation hold we know that all our work is correct, unless indeed there be an exact balance of errors — an unlikely ■event ; but, should the equation not hold, we may be sure that there is an error somewhere, and we must set to work to discover ■and eliminate it. "We may divide our table into sections, and apply ■our formula again, and so localise the error. Thus, for example, the =sum of the column under 4 per cent, in Table III. is 4687-75784, G 2 90 Theory of Finance. [chap. v. 1-04 X 237-990 69-60 . and, applying the formula, we find that .q^ " 4687-75744, differing from the sum of the column by 40 in the last two places, which is due to the fact that the figures in the last place of the tabulated function are only approximately true, and that in multiplying s^\ by 1-04 we have not made allowance for the carriage from the figures beyond the fifth. Had we used six places of decimals in sm, the result of the formula would have been 4687-75781, or only 3 out in the fifth place. Suppose, however, that the fiftieth value had by mistake been printed 15366708, the result of addition would have differed from that given by the application of the formula by "99960, thus showing that there must be a mistake somewhere. Splitting the column into three equal sections, we find the sum. of the first twenty values to be 274-23005, while ^^^-t^l^^tl?? =274 23008, thus showing that the error is not in that section. Summing the second section, we have 1196-43339, which, added to the sum of the first section, gives 1470-66344, and ^^^i^^$3ll!2= 1470-66352. The error is therefore not in the second section, and it must be in the last ; and examining the values in it, we find sjoj to be wrong. 24. To construct a table of a^. This table may be formed from a table of v^ exactly as a table of s^ may be formed from a table of (! + «)". If, however, we have not already a table of »", we can construct that of a^ directly by means of aSl the relation fl!^:ri]=(l + »)ffi^|— 1. 4 per cent. We must proceed as in Art. 21, except that 60. 22-623490 we must begin at the end of the table and work 904940 backwards, and that we must deduct, instead of gg 22-628430 add, unity at each step. The example in the 901137 margin shows the construction at 4 per cent. ■ ^ 58. 22-429567 25. From the relation aa= — : — it follows that 897183 I ■ i ffa^l-z)". Therefore the addends that we form 57. 22326750 are the arithmetical complements of d", from which 8930/0 t!"- can be derived very easily, almost by inspection. 56. 22-219820 Therefore, if we wish to construct a table of t^"-, 888793 we may, without much additional labour, do so by __ inQfliQ first forming one of nia, and so secure both tables at once. CHAP. V.J 071 Interest Tables. 9 1 26. A very similar relation to that given in Art. 23 is available to check the column of aj] ■ We have «i] + «2i + «8i+etc.4-a^i- \ — v l~i>^ , 1—v^ , 1—1)" « — nil = — I — I .-— H . 1- etc. H : — = : — - . I Z Z Z I This formula may he applied exactly as in Art. 23 to detect errors. 27. We may here mention that for tables of all kinds a very useful check can be applied by simply differencing. The differences •of a table almost alvva^'s follow some sufficiently defined law to render apparent any irregularities produced by errors. Thus, in the case of the supposititious error discussed in Art. 23, we should find the differences of the column at that point to run as follows : — 47. 6-31782 48. 6-57052 49. 7-83335 50. 610669 51. 7-39095 52. 7-68659 The difference between the forty-ninth and fiftieth values is evidently too great, and that between the fiftieth and fifty-first too small, and it is thus seen that the fiftieth value is -wrong. If we make the .proper correction the differences v?ill run smoothly. It -will very often happen that the differences need not be actually taken out and ■entered on paper. The operation can be performed mentally by a ■careful inspection of the table. 28. Logarithms may very conveniently be employed to construct tables of s.,g, and ffl^, if they are arranged in the form first given by 'Gauss. In Gauss's logarithmic table the number -with which the table is entered, called the "argument," is log cc, and the result is log (l-|-a;). ' If by the letter T prefixed to a quantity, we denote the result of entering a table with that quantity, then the property above mentioned of Gauss's tables may be represented by the ■equation, Tlog .r = log (l + cc). The late Peter Gray was the first to apply, in his book, Tables ■and Formula for the Computation of Life Contingencies, Gauss's loo-arithmic tables to the calculation of life annuities and assurances, and of annuities-certain, and for that purpose he recomputed and ■e.xtended Gauss's tables. In Gray's tables we have log (1 + x) given to «ix places of decimals for all values of log x, from a;=-001 to a;=100. 92 Theory of Finance. [chap. v. Since Gray's tables were issued, Witfcstein's more extensive ones- of the same kind have been published. These give to seven places log (1 + a;) for all values of x, from j;= -0000001 to a;= 1000000. A small five-place table has also been published by Messrs. Galbraith & Haughton, giving to five places log (1 + a;) for all values of a;,, from x = \ to 37=10000. 29. To construct by Gauss's logarithms a table of si| We have s^ = (l + i) «,7iri] + l ; whence log«i| = log{(l + *) SiTTTj+l}. Now, if we know log {(1 + i) ft;;;:::!]}, that is {log (l + «) +log s;j3i)},. we can, by Gauss's tables, without finding the natural numbers,, obtain log{(l + ») s;^3t|H-l}. We have only to enter the table with the sum of the two given logarithms, and the result is the logarithm which we require. The equation may therefore be written log 6-,T| = T {log Sii^Ti + log (1 + i) } . We commence with log S\\ , which is 0, since si]^ 1, and write below it,, and on every succeeding third line, the constant quantity log (l + j). log 4 per cent. log «Tj log (1 + loc los «2) Adding the first two lines together, and placing the sum on the third line, we have {logstl + log (1 + «')}, with which we enter the table, and have for result log S2I, which we place on the fourth line. Adding to this again, log (14-«), which we have already placed on the fifth line, and entering the table with the sum, we have log S3), and so on. In the example in the margin, a few values of logs;;;), at 4i per cent, are worked out, and Gray's tables of Gaussian logarithms are used. If we write once for all log (1 + 1) at the foot of a card, to be moved down as the work proceeds, we shall save ourselves the trouble of repeating it on every third line, and our calculations will not occupy so much space. Seeing that the seventh figure of log (l + »), at 4 per cent., is 3 — or one-third of a unit in the sixth place — ■ we must, at every third value, increase by a unit the sixth figure. This we have done in the example. * * * When sg becomes greater than 100, Gray's Gaussian logarithms are no longer available, and the table must be completed in some logs 31 log(l + ») loa Si) 000000 01703a •017033- •309630- ■017033 •320663- ■494377 ■017034 •511-111 ■G28027 log (1 -I- i) ■017033. ■6450G0 logsji ■733704 log(l + i) -017033 CHAP, v.] On Interest Tables. 93 log a^ 4 per cent, log ax = log V T log aj. 982967 292597 log ffl2l 275564 other way. Gray shows how to do so by means of a table of log (1 — x), which he also gives, and those who wish to pursue the subject further may cousult his book. With Wittstein's edition of Gauss's logarithms the difficulty does not arise. Galbraith & Haughton's table is too restricted in extent to be of use in the present connection. 30. To construct by Gauss's logarithms a table of ffl^ . Prom the equation «,lj = «)(l + a,;3i|) we pass to log«^ = log« + log (l + oiTiZli), which may be written log ffi,T| =log u + jT log a^ri) . As each value of log a is formed, we enter the table with it, and to the result add log v, thus obtaining the next value of log a. The example in the margin gives a few lines of the work at 4 per cent. Because we continuedly add log v, we must make the usual correction for the last place as we proceed. At 4 per cent, log v is -3 in excess in the sixth place, and we must therefore deduct a unit every third time it is used. As before, we may write log v on a moveable card to save trouble. 31. There is no continued formula for the construction of the columns s^^ and «=^ We must take the reciprocals of «„) and «^| respec- tively. In order to insure accuracy, it will therefore be necessary carefully to verify each tabulated value. This may best be done, not by performing the work in duplicate, as the same error might thus be repeated, but by again taking out the reciprocals of the values in the newly-formed columns, and these should be the values of s^ and rt,i| respectively, with which we started. 32. If both the columns s:^^ and a'^ are wanted, one may be formed from the other in such a way as to check them both. Since s-'^—a~-^ — i it follows that the two columns have the same differences. If, therefore, we first form the column s:^ by means of a table of reciprocals, and difference it, we shall so produce the differences (which are negative) of the column «r' log V T log a2) 982967 460311 logai] 443278 log V 982966 T log as] 576928 log ai] 559894 log V 982967 T log ail 665571 log asl 648538 Star tins 94 Theory of Finance [chap. v. with the first value of the column a-^, namely, (1+J), and adding (algebraically) continuedly the differences, we complete the column. To check the whole work, we take the reciprocals of the values in the last column, and these should be the successive values of a^ . The following is an example at 4 per cent. Year 1 1000000 — A 490196 1^040000 2 •490196 830153 •530196 3 •320349 915141 •360349 4 •235490 949137 •275490 5 •184627 966135 •224627 6 •150762 975848 •190762 7 •126610 981918 •166610 8 •108528 985965 •148528 9 •094493 988798 •134493 10 •083291 •123291 Where i consists of but one significant figure, as in the example this method of construction does not jjossess any advantages ; but if there be several significant figures in «', then considerable benefits are experienced by its adoption. 33. The column log s^ may be conveniently formed by means of the logarithms of s^, because, omitting the characteristics of the logarithms, log sq^=l — log s^. "We have only to take the arithmetical complements of the logarithms of the successive values of s^j, which can be done by inspection. In the same way log a'^ may be formed from log «j] . TABLE 1. Amount of 1: — viz. (l + i)". n 3 7= sr/o 47o 1 447o s7o 6 7o n 1 1 £•030000 1-035000 I -040000 [-045000 [-050000 [-060000 2 [■060900 1-071225 1-081600 [-092025 [-102500 [-123600 2 3 I"092727 1-108718 1-124864 1-141166 1-157625 1-191016 3 4 1-125509 i'i47533 I-I69859 [■192519 1-215506 1-262477 4 5 1-159274 1-187686 1-216653 i'246i82 rz76282 1-338226 5 6 [-194052 1-229255 1-265319 1^302260 [■340096 1-418519 6 7 1-229874 1-272279 1-315932 i'36o862 [■407100 1-503630 7 8 1-266770 1-316809 1-368569 1^422101 1-477455 I-593S48 8 9 1 "3047 7 3 1-362H97 1-423312 1-486095 1-551328 1-689479 9 10 1 '3439 16 [-410599 [-480244 1-552969 [-628895 1-790848 10 1 1-384234 I '45997° ••539454 1-622853 1-710339 1-898299 1 2 [•425761 1-511069 [-601032 1-695881 1-795856 2-012196 2 3 1-468534 i's63956 1-665074 [-772196 1-885649 2-132928 3 4 1-512590 1-618695 1-731676 1-851945 1-979932 2-260904 4 15 1-557967 I -675349 1-800944 1-935282 2-078928 2-396558 15 6 1-604706 i'7339*^6 I-87298I 2-022370 2-182875 2-540352 6 7 1-652848 1-794676 1-947901 2-113377 2-292018 2-692773 7 8 r7o2433 1-857489 2-025816 2-208479 2-406619 2-854339 8 9 i'7535o6 1-922501 2-106849 2-307860 2-526950 3-025600 9 20 1-806111 1-989789 2-I9I123 2-41 J 714 2-653298 3-207135 20 1 1-860295 2-059431 2-278768 2-520241 2-785963 3-399564 1 2 [-916103 2-131512 2-369919 2-633652 2-925261 3-603537 2 3 i'973587 2-206114 2-464716 2-752166 3-071524 3-819750 3 4. 2-032794 2-283328 2-563304 2-876014 3-225100 4-048935 4 25 2-093778 2-363245 2-665836 3-005434 3-386355 4-291871 25 6 2-156591 2-445959 2-772470 3-140679 3-555673 4-549383 6 7 2-221289 2-531567 2-883369 3-282010 3-733456 4-822346 7 8 2-287928 2-620172 2998703 3-429700 3-920129 5-111687 8 9 2-356566 2-711878 3-118651 3-584036 4-116136 5-418388 9 80 2-427262 2-806794 3-243398 3-745318 4-321942 5-743491 30 1 2-500080 2-905031 3-373133 3-913857 4-538039 6-088101 1 2 2'575°83 3-006708 3-508059 4-089981 4-764941 6-453387 2 3 3'6S2335 3-111942 3-648381 4-274030 5-003189 6-840590 3 ; 4 2-731905 3-220860 3-794316 4-466362 5-253348 7-251025 4 35 2-81:5862 3-333590 3-946089 4-667348 5-516015 7-686087 35 6 2-898278 3-450266 4-103933 4-877378 5-791816 8-147252 6 7 2-985227 3-571025 4-268090 5-096860 6-081407 8-636087 7 8 3-°74783 3-696011 4-438813 5-326219 6-385477 9-154252 8 9 3-167027 3-825372 4-616366 5-565899 6-704751 9-703508 9 -40 3-262038 3-959260 4-80I02I 5-816365 7-039989 10-285718 40 1 3'359899 4-097834 4-993061 6-078101 7-391988 10-902861 1 2 3-460696 4-241258 5-192784 6-35161S 7-761588 11-557033 2 3 i'S(>\S^1 4-389702 5-400495 6-637438 8-149667 12-250455 3 4 3-671452 4-543342 5-6165IS 6-936123 8-557150 [2-985482 4 45 3'78iS96 4-702359 5-841176 7-248248 8-985008 13-764611 45 6 3-895044 4-86694T 6-074823 7-574420 9-434258 14-590487 6 7 4-01 1895 5-037284 6-317816 7-915268 9-905971 15-465917 7 8 4-132252 5-213589 6-570528 8-271456 10-401270 16-393872 8 9 4-256219 5-396065 6-833349 8-643671 10-921333 17-377504 9 50 4-383906 5-584927 7-106683 9-032636 11-467400 18-420154 50 1 4-515423 5-780399 7-390951 9-439105 12-040770 19-525364 1 ' 2 4-650886 5-982713 7-686589 9-863865 12-642808 20-696885 2 3 4-790412 6-192108 7-994052 10-307739 13-274949 21-938698 8 4 4'934I2S 5-082149 6-408832 8-313814 10-771587 13-938696 23-255020 4 55 6-633141 8-646367 11-256308 14-635631 24-650322 55 6 5'2346i3 S'39i65i 5-553401 6-865301 8-992222 11-762842 15-367412 26-129341 6 7 7-105587 9-351910 12-292170 16-135783 27-697101 7 , 8 7-354282 9-725987 12-845318 16-942572 29-358927 31-120463 32-987691 8 9 60 5-720003 5-891603 7-611682 7-878091 10-115026 10-519627 13-423357 14-027408 17-789701 18-679186 9 60 TAIiLK 11. Present Value of 1 ; — viz. ti". n 3 7o 3JV0 4% 4h% 5 7o 6 7.. n 1 ■970874 ■966184 ■961538 •956938 ■952381 ■943396 1 2 ■942596 ■9335" ■924556 ■9'S730 •907029 •889996 2 3 •915143 ■901943 ■888996 "876297 •863838 ■839619 3 4, ■888487 ■871442 ■854804 •838561 •822702 ■792094 4 5 •862609 ■841973 "831927 •803451 ■783526 ■747258 5 6 •837484 ■8I350I "790315 ■767896 ■746215 ■704961 6. 7 •813093 ■785991 ■7599'8 •734828 •710681 "665057 1 8 •789409 ■759412 ■730690 ■703185 •676839 ■627412 8 1 ^ •766417 ■733731 "702587 •672904 •644609 ■591898 9 |io •744094 ■708919 ■675564 •643928 •6I39I3 ■558395 10 1 1 •723421 •684946 '649581 ■616199 •584679 •526788 2 ■701380 ■661783 ■624597 •589664 •556837 •496969 2 3 ■680951 •639404 ■600574 ■564272 "530321 •468839 3 4 ■66III8 •617782 ■577475 ■539973 •505068 •442301 4 15 •641862 •596891 ■555265 ■516720 •481017 ■417265 15 6 •623167 •576706 ■533908 "494469 •458112 ■393646 6 7 •605016 •557204 ■513373 ■473176 ■436297 ■371364 1 8 ■587395 •538361 ■493638 "453800 ■415521 ■350344 8 9 •570286 ■520156 "474643 ■433302 ■395734 ■330513 9 20 •553676 ■502566 ■456387 •414643 ■376889 ■311805 20 1 ■537549 ■485571 ■438834 •396787 ■358942 ■294155 1 2 ■521893 •469I5I •42195s •379701 ■341850 ■277505 2 3 ■506693 ■453286 "405736 ■363350 ■325571 •261797 3 4 ■491934 ■437957 ■390121 ■347703 ■310068 ■246979 4 25 ■477606 ■423147 •375117 ■332731 ■295303 •232999 25 6 ■463695 ■408838 •360689 •318402 ■281241 ■319810 6 7 •450189 ■395012 ■346817 •304691 ■267848 ■207368 7 8 "437077 ■381654 •333477 •291571 ■255094 •195630 8 9 •424346 ■368748 ■330651 •279015 ■242946 ■184557 9 30 ■4II987 •356378 "308319 •267000 ■231377 •174110 30 1 ■399987 ■344230 ■396460 "255503 ■220359 "164255 1 2 •388337 ■332590 "285058 •244500 ■309866 ■154957 2 3 •377036 '321343 •274094 •233971 ■199873 •146186 3 4 ■366045 '310476 •263552 •223896 ■190355 •137912 4 35 '355383 •299977 •253415 •214354 "181290 •130105 35 6 ■345032 ■389833 •343669 •205028 "173657 •I2274I 6 7 ■334983 '280032 ■234297 'X96199 '164436 ■"5793 7 8 •325226 ■370563 ' ■225285 ■187750 ■156605 •109339 8 9 •315754 ■26I4I3 ■316621 •179665 "I49I48 •103056 9 40 '306557 ■252572 •308389 ■'71929 "142046 •097222 40 1 •297628 ■244031 •300278 ■164525 •'35282 •091719 1 2 ■388959 •235779 ■'92575 ■'57440 ■128840 •086527 2 ■ 3 ■280543 ■227806 "X85168 "150661 •123704 '081630 3 4 •272372 ■320102 "1 78046 ■144173 •I 16861 ■077009 4 45 ■264439 "213659 "171 198 ■137964 •III397 "072650 45 fi ■256737 ■305468 ■ 1 64.6 1 4 "(^2033 ■105997 •068538 6 7 •249259 '198520 •158283 ■126338 •100949 ■064658 7 8 •241999 •I9I806 ■152195 "120898 •096143 •060998 8 9 ■234950 •185320 ■14634' ■'15692 •091564 ■057546 9 50 •228107 •'79053 ■'40713 •110710 "087204 •054388 50 1 •221463 •172998 ■135301 ■105942 "083051 •051215 1 2 ■2I50I3 ■I67I48 •130097 '101380 "07909r) •048316 2 3 •208750 ■I6I496 ■125093 ■097014 ■075330 •045582 3 4 •202670 ■156035 "120283 "092837 ■071743 •043001 4 55 •196767 ■150758 •115656 "08S839 •068326 •040567 55 G "19(036 "145660 "11 1207 •085013 •065073 •038271 6 7 ■185473 ■140734 ■106930 ■081353 •061974 ■036105 7 8 •180070 ■"35975 •103817 ■077849 ■059023 •034061 8 •17482s ■131377 •098863 ■074497 ■056213 "032133 9 60 ■169733 "136934 ■095060 '071289 ■053536 "030314 60 Table III. Amount of 1 per annum: — viz. s^. n 3°./o 3i7o 4 7. 4j7o 5 7o 6 7o n 1 1 I'OOOOO I '00000 I -00000 I-QOOOO 1-00000 I -00000 2 2"03000 2-03500 2-04000 2-04500 2-05000 1-06000 2 3 3-09090 3-10623 3-12160 3''3703 3-15250 3-18360 3 4. 4'iS363 4-21494 4-24646 4-27819 4-31013 4-37462 4 5 5"309i4 5'36247 5-41632 5-47071 5-52563 5-63709 5 6 6-46841 6-S50'S 6-63298 6-71689 6-80191 6-97532 6 7 7-66246 7'7794i 7-89829 8-01915 8-14201 8-39384 7 8 8-89234 9-0516'^ 9-21423 9-3 8c 01 9-549" 9-89747 8 9 10-15911 10-36850 10-58280 I0-I-021I 11-02656 II-49I32 9 ' 10 I [-4(5388 i'73i39 I2006I1 12-28S2I 12-57789 13-18079 10 1 12-80780 1314199 13'48635 13-84118 14-20679 14-97164 1 2 14-19203 14-60196 15-02581 15-46403 I5-9I7I3 16-86994 2 3 15-61779 16-11303 16-62684 17-15991 17-71298 18-88214 3 4 17-08632 17-67699 18-29191 18-93211 i9-59'-'63 21-01507 4 15 18-59891 19-29568 20-02359 20-78405 21-57856 23-27597 15 6 20-15688 20-97103 21-82453 22-71934 23-65749 25-67253 6 7 21-76159 22-70502 23-69751 24-74171 25-84037 28-21288 7 8 23'4I444 24-49969 2S'6454i 26-85508 28-13238 30-90565 8 9 25-11687 26-35718 27-67123 29-06356 30-53900 33-75999 9 20 26-87037 28-27968 29-77808 31-37142 33-06595 36-78559 20 1 28-67649 30-26947 31-96920 33-78314 35-71925 39-99273 1 2 30'S3678 32-32890 34-24797 36-30338 38-50521 43-39229 2 3 32-45288 34-46041 36-61789 38-93703 41-43048 46-99583 3 4 34'42647 36-66653 39-08260 41-68920 44-50200 50-81558 4 25 36-45926 38-94986 41-64591 44-56521 47-72710 54-86451 25 6 38'5S304 41-31310 44'3ii74 47-57064 51-11345 59-15638 6 7 40-70963 4375906 47-0S421 50-71132 54-66913 63-70577 7 8 42-93092 46-29063 49-96758 53-99333 58-40258 68-52811 8 9 45-21885 48-91080 52-96629 57-42303 62-32271 73-63980 9 30 47'57S42 51-62268 56-08494 61-00707 66-43885 79-05819 30 1 50-00268 54-42947 59-32834 64-75239 70-76079 84-80168 1 2 52-50276 57'33450 62-70147 68-66625 75-29883 90-88978 2 3 55-07784 60-34121 66-20953 72-75623 80-06377 97-34316 8 4 Sy73oi8 63'453iS 69-85791 77-03026 85-06696 104-18375 4 35 60-46208 66-67401 73-65222 81-49662 90-32031 1 1 1-43478 35 6 63'27594 70-00760 77-59831 86-16397 95-83632 II9-I 20S7 6 7 66-17422 73-45787 81-70225 91-04134 101-62814 127-26812 7 8 69'i594S 77-02889 85-97034 96-13820 107-70955 135-90421 8 9 72-23423 80-72491 90-40915 101-46442 114-09502 145-05846 9 40 75-40126 84-55028 95-02552 107-03032 120-79977 154-76197 40 1 78-66330 88-50954 99-82654 112-84669 127-83976 165-04768 1 2 82-02320 92-60737 104-81960 118-92479 135-23175 175-95054 2 3 85-48389 96-84863 110-012^8 125-27640 142-99334 187-50758 3 : 4 89-04841 101-23833 115-41288 13''9I384 151-14301 199-75803 4 45 92-71986 105-78167 121-02939 13884997 159-70016 212-74351 45 6 96-50146 110-48403 126-87057 146-09*21 168-68516 226-50812 6 7 100-39650 iiS'35097 132-94539 153-67263 178-11942 241-09861 7 8 104-40840 120-38826 139-26321 161-58790 188-02539 256-56453 8 9 108-54065 125-60185 145-83373 169-85936 198-42666 272-95840 9 50 112-79687 130-99791 152-66708 178-50303 209-34800 290-33590 50 1 117-18077 136-58284 159-77377 187-53566 220-81540 308-75606 1 2 121-69620 142-36324 167-16472 196-97477 232-85617 328-28142 2 3 126-34708 148-34595 174-85131 206-83863 245-49897 348-97831 3 4 131-13749 i54'538o6 182-84536 217-14637 258-77392 370-91701 4 55 136-07162 160-94689 191-15917 227-91796 272-71262 394-17203 55 6 141-15^77 167-5^003 199-80554 239-17427 287-34825 418-82235 444-95169 6 7 7 146-38838 174-44533 208-79776 250-93711 302-71566 8 15 1-78003 157-33343 181-55093 218-14967 263-22928 318-85144 472-64879 8 9 188-90520 227-87566 276-07460 335-79402 502-00772 9 60 i63'05344 196-51688 237-99069 289-49795 353-58372 533-12818 60 Taule IV. Present Value of 1 per annum: — viz. a:;!]. n 3 7o 3i7o 4 7o 47o 5% 6% n 1 1 •97087 -96618 -96154 -95694 ■95238 -94340 2 I "01 347 1-89969 1-88609 1-87267 1-85941 1-83339 2 3 2-82861 2-80164 2-77509 2-74896 2-72325 2-67301 3 ' 4 3-71710 3-67308 3-62990 3-58753 3-54595 3-465 1 1 4 5 4-57971 4-51505 4-45182 4-38998 4-32948 4-21236 5 6 5'4i7i9 5-32855 5-24214 5-15787 5-07569 4-91732 6 7 6-23028 6-11454 6-00205 5-89270 5-78637 5-58238 7 8 7-01969 687396 6-73275 6-59589 6-46321 6-20979 8 9 7-78611 7-60769 7-43533 7-26879 7-10782 6-80169 9 10 8-530Z0 8-3I66I 8-11090 7-91272 7-72173 7-36009 10 ■ 1 9-25262 9-00155 8-76048 8-52892 8-30641 7-88687 1 2 9-95400 9-66333 9-38507 9-11858 8-86325 8-38384 2 3 10-63496 TO-30274 9-98565 9-68285 9-39357 8-85268 3 4 11-29607 10-92052 10-56312 10-22283 9-89864 9-29498 4 15 '>"93794 "-5I74I II-I1839 10-73955 10-37966 9-71225 15 6 12-56110 12-09412 11-65230 11-23401 10-83777 10-10590 6 ■ 7 I3'i66i2 12-65132 12-16567 11-70719 11-27407 10-47726 7 . 8 i3'753Si 13-18968 12-65930 12-15999 11-68959 10-82760 8 9 14-32380 13-70984 13-13394 12-59329 12-08532 11-15812 9 20 14-87748 14-21240 13-59033 13-00794 12-46221 11-46992 20 ■ 1 15-41502 14-69797 14-02916 13-40472 12-82115 11-76408 1 2 I5'93692 15-16713 14-45112 13-78442 1 vi6^oo 12-04158 2 3 16-44361 15-62041 14-85684 14-14777 i3-4»857 12-30338 3 4 16-93554 16-05837 15-24696 14-49548 13-79864 12-5503O 4 25 i7'4i3i5 16-48152 15-62208 14-82821 14-09394 12-78336 25 6 17-87684 16-89035 15-98277 15-14661 14-37518 13-00317 7 18-32703 17-28537 16-32959 15-45130 14-64303 13-21053 7 ^ 8 18-76411 17-66702 16-66306 15-74287 14-89813 13-40616 8 9 19-18646 18-03577 16-98372 16-02189 15-14107 13-59072 9 30 19-60044 18-39205 17-29203 16-28889 15-37245 13-76483 30 1 20*00043 18-73628 i7-58?49 16-54439 15-59281 13-92909 1 2 20-38877 19-06887 17-87355 16-78889 15-80268 14-08404 2 ; 3 20-76579 19-39021 18-14765 17-02286 16-00255 14-23023 3 ■1 4 21-13184 19-70068 18-41120 17-14676 16-19290 14-36814 4 35 21-48722 20-00066 18-66461 17-46101 16-37419 14-49825 35 6 21-83225 20-29049 18-90828 17-66604 16-54685 14-62099 ' 7 22-16724 2°'57o53 19-14258 17-86224 16-71129 14-73678 7 8 22-49246 20-84109 19-36786 18-04999 16-86789 14-84602 8 9 22-80822 21-10250 19-58449 18-22966 17-01704 14-94907 9 40 23-11477 21-35507 19-79277 18-40158 17-15909 15-04630 40 1 23-41240 21-59910 19-99305 18-56611 17-29437 15-13802 1 2 23-70136 21-83488 20-18563 18-72355 17-42321 15-22454 2 3 23-98190 22-06269 20-37080 18-87421 17-54591 15-30617 8 4 24-25427 22-28279 20-54884 19-01838 17-66277 15-38318 4 45 24-51871 22-49545 20-72004 19-15635 17-77407 15-45583 45 6 24-77545 22-70092 20-88465 19-28837 17-88007 15-52437 6 7 25-02471 22-89944 21-04294 19-41471 17-98102 15-58903 7 8 25-26671 23-09124 21-19513 19-53561 18-07716 15-65003 8 9 25-50166 23-27656 21-34147 19-65130 18-16872 15-70757 9 . 50 25-72976 23-45562 21-48219 19-76201 18-25592 15-76186 50 ; 1 25-95123 23-62862 21-61749 19-86795 18-3389S 15-81308 1 2 26-16624 23-79577 21-74758 19-96933 18-41807 15-86139 2 : 3 26-37499 23-95726 21-87268 20-06634 18-49340 15-90697 3 ; 4 26-57760 24-11330 21-99296 20-15918 18-56515 15-94998 4 55 26-77443 24-26405 22-10861 20-24802 18-63347 15-99054 55' 6 26-96546 24-40971 22-21982 20-33303 18-69854 16-02881 6 : 7 27-15094 24-55045 22-32675 20-41439 18-76052 16-06492 7 ' 8 27-33101 24-68642 22-42957 20-49224 18-81954 16-09898 8 9 27-50583 24-81780 22-52843 20-56673 18-87575 16-13111 9 60 27-67556 24-94473 22-62349 20-63802 18-92929 16-16143 60 Table V. Annuity which 1 ivill pwchase : — vig. Qt:^)' n 3% 3^/0 4%, 4i7o 5 7o 6 7o }J. ' 1 I '030000 f035ooo i^o400oo 1^045000 fo5oooo 1^060000 1 2 0"S226lI o^5264oo o'53oi96 •533998 •537805 0-545437 2 3 "35353° ■356934 ■360349 ■363773 •367209 •374110 3 4 ■269027 ■272251 •275490 •278744 •282012 •288591 4 • 5 •2183SS ■221481 •224627 •227792 •230975 •237396 5 6 •184598 •187668 •190762 •193878 •197017 ■203363 6 7 •160506 •163544 •166610 •169701 •172820 ■I79I35 7 8 ■142456 ■145477 ■148528 •151610 •154722 •161036 8 9 ■128434 •131446 •134493 ■137574 •140690 •147022 9 10 ■117231 ■120241 ■123291 ■126379 •129505 •135868 10 1 ■108077 ■111092 ■114149 ■II7248 •120389 •126793 1 2 ■100462 ■103484 ■106552 ■109666 •II2825 •119277 2 3 ■094030 ■097062 ■100144 ■103275 ■106456 •II2960 3 4 ■088526 ■091571 ■094669 ■C97820 •101024 ■107585 4 15 ■083767 ■0S6825 ■089941 ■093II4 ■096342 •102963 15 6 ■079611 ■082685 ■085820 ■089015 •092270 •098952 6 7 ■075953 ■079043 ■082199 ■0854IS •088699 ■095445 7 8 ■072709 ■075817 ■078993 •082237 ■085546 ■092357 8 9 ■069814 ■072940 ■076139 ■079407 •082745 •089621 9 20 ■067216 ■070361 ■073582 •076876 •080243 •087185 20 1 ■064872 ■068037 ■071280 •07460) •077996 •085005 1 2 ■062747 ■065932 ■069199 •072546 ■075971 •083046 2 3 ■060814 ■064019 ■067309 •070682 •074137 •081278 3 4 ■059047 ■062273 ■065587 •068987 ■072471 ■079679 4 25 ■057428 ■060674 ■064012 •067439 •070952 •078227 25 6 ■055938 ■059205 ■062567 •066021 ■069564 ■076904 6 7 ■054564 •057852 ■061239 •064719 ■068292 ■075697 7 8 ■053293 ■056603 ■060013 ■063521 ■067123 ■074593 8 9 ■052115 •055445 ■058880 •062415 ■066046 •073580 9 30 ■051019 •054371 ■057830 ■061392 ■065051 ■072649 30 1 ■049999 •053372 ■056855 •060443 ■064132 ■071792 1 2 ■049047 ■052442 •055949 ■059563 ■063280 ■071002 2 3 ■048156 •o5'572 •055104 ■058745 '062490 •070273 3 4 ■047322 ■050760 •054315 •057982 ■061755 ■069598 4 ■ 35 ■046539 ■049998 •053577 •057270 ■061072 •068974 35 6 ■045804 ■049284 ■052887 •056606 ■060434 •06839s 6 7 ■0451 12 ■048613 •052240 •055984 ■059840 •067857 7 8 ■044459 ■047982 •051632 '055402 ■059284 •067358 8 9 ■043844 ■047388 ■051061 •054856 •058765 •066894 9 40 ■043262 •046827 ■050523 •054343 •058278 •066462 40 ' 1 ■042712 •046298 ■050017 ■053862 ■057822 ■066059 1 3 •042192 •045798 ■049540 •053409 ■057395 ■065683 2 3 ■041698 •045325 ■049090 052982 ■056993 ■065333 3 4 ■041230 •044878 ■048665 ■052581 ■056616 ■065006 4 45 ■0407H5 ■044453 ■048262 ■052202 ■056262 ■064701 ■064415 ■064148 ■063898 ■0C3664 45 6 7 8 9 6 ■040363 •04405 1 •047882 ■051845 ■055928 7 ■039961 •043669 ■047522 ■051507 ■055614 8 •039578 •043306 ■047181 ■051189 ■055318 9 ■039213 •042962 ■046857 ■050887 ■055040 50 ■038865 ■042634 ■046550 ■050602 ■054777 ■063444 50 1 ■038^34 ■042322 •046259 ■050332 ■054529 ■063239 ■063046 ■062866 ■062696 •062537 ■062388 •062247 •062116 •061992 •061876 1 2 3 4 55 6 7 8 9 60 ■ 2 3 4 55 6 ■038ii7 ■037915 ■037626 ■037349 ■037084 ■036831. ■036588 ■036356 ■036133 ■042024 ■041741 ■041471 ■041213 ■040967 ■045982 •045719 ■045469 ■045231 ■045005 ■050077 ■04983s ■049605 ■049388 ■049181 ■054294 •054073 ■053864 ■053667 ■053480 7 ■040732 ■044789 ■048985 ■053303 8 ■040508 •044584 •048799 ■053136 9 60 ■040294 ■040089 •044388 •044202 ■048622 •048454 ■052978 ■052828