Encyclopædia Britannica, Ninth Edition/Annuities - Wikisource, the free online library (2024)

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ANNUITIES

BY an annuity is meant a periodical payment, madeannually, or at more frequent intervals, either fora fixed term of years, or during the continuance of a givenlife, or a combination of lives, as will be more fullyexplained further on. In technical language an annuityis said to be payable for an assigned status, this being ageneral word chosen in preference to such words as " time,"" term," or " period," because it may include more readilyeither a term of years certain, or a life or combination oflives. The magnitude of the annuity is the sum to bepaid (and received) in the course of each year. Thus, if100 is to be received each year by a person, he is said tohave "an annuity of 100." If the payments are madehalf-yearly, it is sometimes said that he has " a half-yearlyannuity of 100;" but to avoid ambiguity, it is morecommonly said he has an annuity of 100, payable byhalf-yearly instalments. The former expression, if clearlyunderstood, is preferable on account of its brevity. So wemay have quarterly, monthly, weekly, daily annuities,when the annuity is payable by quarterly, monthly,weekly, or daily instalments. An annuity is consideredas accruing during each instant of the status for which itis enjoyed, although it is only payable at fixed intervals.If the enjoyment of an annuity is postponed until after thelapse of a certain number of years, the annuity is said tobe deferred. When an annuity is deferred for any numberi f years, say n, it is said, indifferently, to commence, or tobe entered upon, after n years, or to run from the end of nyears; and if it is payable yearly, the first payment will bomade at the end of (n+l) years; if half-yearly, the firsthalf-yearly payment will be made at the end of ( + )years; if quarterly, the first quarterly payment will bemade at the end of (n+¼) years; and so on. If anannuity, instead of being payable at the end of each year,half-year, &, is payable in advance, it is called anannuity-due.

If an annuity is payable for a term of years independentof any contingency, it is called an annuity certain; if it isto continue for ever, it is called a perpetuity; and if inthe latter case it is not to commence until after a term ofyears, it is called a deferred perpetuity. An annuity de.-pending on the continuance of an assigned life or lives,is sometimes called a life annuity; but more commonlythe simple term "annuity" is understood to mean a lifeannuity, unless the contrary is stated. A life annuity, tocease in any event after a certain term of years, is called atemporary annuity. The holder of an annuity is calledan annuitant, and the person on whose life the annuitydepends is called the nominee.

If not otherwise stated, it is always understood that anannuity is payable yearly, and that the annual payment(or rent, as it is sometimes called) is £1. Of late years,however, it has become customary to consider the annualpayment to be, not £1, but simply 1, the reader supplying ​whatever monetary unit he pleases, whether pound, dollar,franc, thaler, &c. It is much to be desired that this courseshould be followed in any tables that may be published infuture.

The annuity, it will be observed, is the totality of thepayments to be made (and received), and is so understoodby all writers on the subject; but some have also used theword to denote an individual payment (or rent), speaking,for instance, of the first or second year s annuity, a practice which is calculated to introduce confusion, and shouldtherefore be carefully avoided.

The theory of annuities certain is a simple applicationof algebra to the fundamental idea of compound interest,According to this idea, any sum of money invested, or putout at interest, is increased at the end of a year "by theaddition to it of interest at a certain rate; and at the endof a second year, the interest of the first year, as well asthe original sum, is increased in the same proportion, andso on to the end of the last year, the interest being, intechnical language, converted into principal yearly. Thus,if the rate of interest is 5 per cent,, 1 improved atinterest will amount at the end of a year to 1 05, or, aswe shall in future say, in conformity with a previous remark,1 will at the end of a year amount to 1 05. At the end ofa second year this will be increased in the same ratio, andthen amount to (1 05) 2 . In the same way, at the end ofa third year, it will amount to (1 05) 3 , and so on.}}

Let i denote the interest on 1 for a year; then at theend of a year the amount of 1 will be 1 + i. Reasoningas above, at the end of two years the amount will be(1 +i), at the end of three years (1 + i) 3 , and so on. Ingeneral, at the end of n years the amount will be (1 +i)";or this is the amount of 1 at compound interest in n years.The present value of a sum, say 1, payable at the end ofn years, is such a sum as, being improved at compoundinterest for n years, will exactly amount to 1. We haveseen that 1 will in n years amount to (1 +t)", and by proportion we easily see that the sum which in n years willamount to 1, must be r- , or (1 + i) n . It is usual to(l-M)"put v for r-, so that v is the value of 1 to be receivedat the end of a year, and v* the value of 1 to be receivedat the end of n years.

{{ti|1em|Since 1 placed out at interest produces i each year, wesee that a perpetuity of i is equal in value to 1; hence,by proportion, a perpetuity of 1 is equal in value to - . Ati5 per cent, ieq -05, and - = 20; or a perpetuity is worth1>20 years purchase: at 4 per cent., it is worth 25 yearspurchase, ( eq 25 j: at 3 per cent., it is worth 33J yearspurchase, (.^ eq 33^.

Instances of perpetuities are the dividends upon thepublic stocks in England, France, and some other countries. Thus, although it is usual to speak of 100 consols, this 100 is a mere conception or ideal sum; and thereality is the 3 a year which the Government pays byhalf-yearly instalments. The practice of the French inthis, as in many other matters, is more logical. In speaking of their public funds, they do not mention the idealcapital sum, but speak of the annuity or annual paymentthat is received by the public creditor. Other instancesof perpetuities are the incomes derived from the debenture stocks now issued so largely by various railway companies, also the feu-duties commonly payable on houseproperty in Scotland. The number of years purchasewhich the perpetual annuities granted by a government ora railway company realise in the open market, forms a verysimple test of the credit of the various governments or railways. Thus at the present time (May 1874) the British perpetual annuity of 3, derived from the 3 per cent, consols,is worth 93, or 31 years purchase; and a purchaser thusobtains 3-226 per cent, interest on his investment. Otherexamples are given in the subjoined tables, the figures inwhich are deduced from the Stock Exchange quotationsof the irredeemable stocks issued by the various governments:

The following are a few other examples of perpetuities:—

Interest percent, yielded toa Purchaser.3-234 224-724-905-085-006-487-417-467-6910-5315-0023-57Name,Metropolitan Board of "Works Stock, 27 50London and N."W. Railway Deben-C1.L 1ture Stock,North British Railway Debenturer*i_ 1Stock,Edinburgh Water Annuities, 22 53Interest percent, yielded toa Purchaser3-643-834-254-43

We may mention in passing that the more usual practiceof foreign governments when borrowing is not to grantthe lender a perpetual annuity, but to issue to him bondsof say 100 each, bearing an agreed rate of interest, thesebonds being usually issued at a discount, and redeemed atpar by annual drawings during a specified term of years.We have seen that the present value of any sum payableat the end of n years is found by multiplying it by(l+i)"; hence the value of a perpetuity of 1 deferredn years is r Now an annuity for n years is clearlythe difference between the value of a perpetuity to commence at once and a perpetuity deferred n years; its valueis therefore - - r eq r; or putting a for thevalue of the annuity, we have

i-i. f l (!+*)"" l-(l + t)-" ... , ,, I/ If If

By means of this equation, having any two of the threequantities, a, i, n, we can determine the third either exactlyor approximately. Thus for n we have

log -(I- to) log/l + t)

There is no means of determining the value of i exactly,but it may be found to any degree of accuracy required bymethods of approximation which our limits will not allow

us to describe.^[1] ​

If the annuity for n years is not to be enjoyed at once,but only after the lapse of t years, its value will be reducedin the proportion of 1 to the value of 1 payable in t years,or 1: (1 +* )"; and the value of the deferred annuity tocontinue for n years is therefore

[ math ]

It remains to find the amount at "compound interest atthe end of n years of an annuity payable for that term.The amount of 1 in n years being (1 +i) n , its increase inthat time is (1 -f-t )" 1; but this increase arises entirelyfrom the simple interest, i, of 1 being laid up at the endof each year and improved at compound interest duringthe remainder of the term. Hence it follows that theamount at compound interest of an annuity of i in n yearsmust be (1 +* )" 1; and by proportion the amount of anannuity of 1 similarly improved will be r*.

One of the principal applications of the theory of annuities certain is the valuation of leasehold property; anotheris the calculation of the terms of advances in considerationof an annuity certain for a term of years. At present alarge sum of money is annually borrowed by corporationsand other public bodies upon the security of local rates inthe United Kingdom. It is sometimes arranged in thesetransactions that a fixed portion of the loan shall be paidoff every year j but it is more commonly the case that, inconsideration of a present advance, an annuity is grantedfor a term of years, usually 25 or 30, but in some instancesextending to 50. Landed proprietors also, who possessonly a life interest in their property, have been authorisedby various Acts of Parliament to borrow money for thepurpose of improving their estates, and can grant a rent-charge upon the fee-simple for a term not exceeding 30years. These are very favourite investments with the lifeinsurance companies of the country, as they are thusenabled to obtain a somewhat higher interest from 4to 4f per cent. than they could obtain upon ordinarymortgages with equally good security; the reason for this,of course, being that these loans are not so suitable asothers for private lenders. In this case, as in all others,the price is determined by the laws of supply and demand;and the number of lenders being less than in the case ofordinary mortgages, the terms paid by the borrowers arehigher. When a loan is arranged in this way, it is desirable for various purposes, and in particular for the ascertainment of the proper amount of income-tax, to considereach year s payment as consisting partly of interest on theoutstanding balance of the loan and partly as an instalment of the principal. The problem of determining theseparate amounts of these has been considered by Turn-bull, Tables, p. 128; and by Gray, Ass. Mag., xi. 172.

In making calculations for these and similar purposes,it is but seldom necessary to use the formulas given above.The computer usually has recourse to one of the tableswhich have been published, containing values and amountscalculated for various rates of interest. An extensive setof tables of this kind was published in 172G by JohnSmart; and many subsequent writers, as Dr Price, Baily,Milne, Davies, D. Jones, J. Jones, have reprinted orabridged portions of these tables. They show the amountand the present value both of a payment and of an annuityof 1 for every term of years not exceeding 100, at theseveral rates of interest, 2, 2, 3, 3

(1.) The amount of 1 in any number of years, n; or

(2.) The present value of 1 due in any number of years,n; or (1 +i) n .

(3.) The amount of an annuity of 1 in any number of(l+i)-l years, n; or s?

(4.) The present value of an annuity of 1 for anynumber of years, n; or *. -

(5.) The annuity which 1 will purchase for any number of years, n; or - - ^

The scheme would be more complete if we add, withCorbaUx, whose tables will be described below

(6.) The annuity which would amount to 1 in n years; i or j- 7 - .

, 1

The following table, on p. 75, in which the rate of interest is 5 per cent., will serve to illustrate the nature of thetables in question, as reprinted by Baily, D. Jones, andothers.

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Table of Amounts, Present Values, &c., at 5 per cent. Interest.

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Hitherto we have considered the annuity payments tobe all made annually; and the case where the payments aremade more frequently now requires attention, First, suppose that the annuity is payable by half-yearly instalments;then, in order to find the present value of the annuity, wehave first to answer the question, What is the value of asum payable ia six months time 1 and, in order to findthe amount of the annuity in n years, we must first determine what is the amount of a sum at the end of sixmonths. The annual rate of interest being i, it may besupposed at first sight that the amount of 1 at the end ofsix months will be 1 + -; but if this were the case, the2iamount at the end of a second period of six months would(t\ 21 + - ) , or1 + i + - . But this is contrary to our original assumption that the annual interest is i, and the amount at theend of a year therefore 1 + i, In fact, if we suppose theinterest on 1 for half a year to be -, the interest on it for. In order that the amounta year will not be i, butat the end of a year may be 1 + i, the amount at the endof six months must be such a quantity as, improved at thesame rate for another six months, will be exactly 1 + i;hence the amount at the end of six months must be ^fl + i,or (1 + 1)*. Reasoning in the same way, it is easy to seethat, the true annual rate of interest being i, the amount of1 in any number of years, n, whether integral or fractional,will always be (1 + t)". Hence, by similar reasoning tothat pursued above, the present value of 1 payable at theend of any number of years, n, whether integral or fractional, will always be (1 + i)" or v*.It is now easily seen we omit the demonstrations forthe sake of brevity that the present value of an annuitypayable half-yearly for n years (?4 being integral) is^.i-l+J . 1 -(!+*)". and ttat tlie amount O f a similar2 ^annuity at the end of n years is %SI:f - f 1 + *)"-*.2 i

In consequence of the above-mentioned practice as tohalf-yearly interest, the values given in Smart s tables forthe odd half-years, though theoretically correct, are practically useless, and they have been superseded by the othertables above mentioned. It is important, however, alwaysto bear in mind that when interest is thus payable half-yearly or quarterly, the true rate of interest exceeds thenominal. From want of attention to this point, the subject has become involved in much confusion, not to sayerror, in the works of Milne and some other writers.

It is easily seen from the above formula that the amountof 1 in mn years, at the rate of interest , is the same astTbthat of 1 in n years, at the rate of interest i convertible mtimes a year; and a similar property holds good of presentvalues. Hence, the tables calculated at the rate of interestmay be used to find the amounts and present valuesmat the rate i convertible m times a year; for example, thetables calculated for interest 2 per cent, will give the resultsfor 4 per cent, payable half-yearly. For this reason itwould be an improvement, as remarked above, to use theword "terms" in the headings of the tables instead of" years."

We pass on now to the consideration of the theory oflife annuities. This is based upon a knowledge of therate of mortality among mankind in general, or among theparticular class of persons on whose lives the annuitiesdepend. If a simple mathematical law could be discoveredwhich the mortality followed, then a mathematical formulacould be given for the value of a life annuity, in the sameway as we gave above the formula for the value of anannuity certain. In the early stage of the science, De-moivre propounded the very simple law of mortality whichbears his name, and which is to the effect, that out of 86children born alive 1 will die every year until the lastdies between the ages of 85 and 86. The mortality, asdetermined by this law, agreed sufficiently well at themiddle ages of life with the mortality deduced from thebest observations of his time; but, as observations becamemore exact, the approximation was found to be not sufficiently close. This was particularly the case when it \vasdesired to obtain the value of joint life, contingent, orother complicated benefits. Demoivre s law is now, accordingly, entirely a thing of the past, and does not call for anyfurther notice from us. Assuming that law to hold, it iseasy to obtain the formula for the value of an annuity,immediate, deferred, or temporary; but such formulas areentirely devoid of practical utility. Those who are curiouson the subject may consult the paper by Charlon, Ass.Mag., xv. 141. In vol. vi. p. 181, will be found an investigation by Gray of the formula for the value of anannuity when the mortality table is supposed to follow asomewhat more complicated law. No simple formula,however, has yet been discovered that will represent therate of mortality with sufficient accuracy; and those whichsatisfy this condition are too complicated for general use.

The rate of mortality at each age is, therefore, in practiceusually determined by a series of figures deduced fromobservation; and the value of an annuity at any age isfound from these numbers by means of a series of arithmetical calculations. Without entering here on a description of the manner of making these observations and deducing the rate of mortality, and of the construction of" Mortality Tables," we append, for the sake of illustration,one of the earliest tables of this kind, namely, that ofDeparcieux, given in his Essai sur les Probcibilites de laDuree de ia Vie Humaine, 1746.

NumberNumberNumberAge.Numberliving.dying inthe nextAge.Numberliving.dying inthe nextAge.Numberliving.dying inthe nextvear.year.year.XI,d,XI*d*Xlx -d,3100030347028653951549702235694866380165948183668686736417693015376787683471879151338671769329198902123966477031019989010406577712912010880841650772271201187264264377325120128666436367742312013860644629775211191485464562277619219158486466158771731916842747607878154181783574859997913618188287495909801181719821750581108110116208148515711182851421806852560118371122279885354911845911237908545381285481024782855526128638925774856514128729726766857502138822627758858489138916528750859476138011429742860463139173307348614501392423172686243714932132718863423149411337108644091495

It is to be understood from this table that the mortalityamong the persons observed was such that out of every1000 children alive at the age of 3, 30 died before attaining the age of 4, leaving 970 alive at 4; 22 died between;the ages of 4 and 5, leaving 948 alive at the age of 5;and so on, until one person is left alive at the age of 94,who died before attaining the age of 95.

For the purpose of explaining more fully the method offinding the value of a life annuity, it will be convenient, inthe first instance, to establish the two following lemmas.

Lemma 2. To find the present value of 1 to be receivedin n years time, if a specified person, whose age is now x,shall be then living. The sum to be received in this caseis called an endowment, and the person on whose life itdepends is called the nominee. The probability that thenominee will be alive is to be found, as already intimated,by means of a mortality table. Out of the various tablesof this nature that exist, that one must be chosen which,it is believed, most faithfully represents the probabilitiesof life of the class of persons to which the nominee belongs. Suppose we have reason to believe that Deparcieux stable, above given, is the most suitable in the case beforeus, that the age of the nominee is 30, and the term ofyears 10. Then, observing that, according to Deparcieux stable, the number of persons living at the age 30 is734, while the number at the age 40 is 657, and thedifference, or the number who die between the two ages,is 77, we conclude that the chances of any particularnominee of the age of 30 dying before attaining the ageof 40 are as 77 to 734, and the chances in favour ofhis living to the age of 40 are as 657 to 734; or theprobability of his living to 40 is - .

Passing now from figures to more general symbols, wewill use l x to denote the number given in the mortalitytable as alive at any age x; so that, for example, in theabove table, ^ = 734, Z 4l) = 657; and in accordance withwhat we have just explained, the probability of a nomineeof the age x living to the age x + n, will therefore beexpressed by . Hence, by lemma 1, the value of 1 tobe received if the nominee shall be alive at the end ofn years, is -y^V. In the particular case supposed above,L xthe actual value will be, taking the rate of interest at3 per cent, ^ x (1 03)- 10 = "666035. We may look atthe question from another point of view. Suppose that734 persons of the age of 30 agree to purchase from aninsurance company each an endowment of 1, payable atthe end of 10 years, then the probabilities of life beingsupposed to be correctly given by Deparcieux s table, wesee that 657 of those persons will be alive at the end of10 years, or the engagement of the insurance company topay 1 to each survivor amounts to the same thing as theengagement to pay 657 at the end of 10 years, and thepresent value of this sum is 657 (1 03) 10 . The sum thatshould be paid by each of the 734 persons, so that thecompany shall neither gain nor lose by the transaction, isCT *Ttherefore ^(l 03 )" 10 , as before. If we suppose the probabilities of life to agree with those of the English Table,No. 3, Males, which is printed at the end of this article,the value of the same endowment will be272,073304,534(1-03)- 10 = -664779.

If now we carefully examine the reasoning of the lastparagraph, we see that we have made an assumption thatmust not be allowed to pass without some further justification. We have assumed, in fact, that the lives we aredealing with will die off at the exact rate indicated by themortality table. This, however, we know, is not necessarily the case. Even if the mortality table correctlyrepresents in the long run the rate of mortality among thelives we are dealing with, we know that the rate ofmortality will, from accidental circ*mstances, be sometimes greater and sometimes less than that indicated bythe table. If, for example, we have 734 persons underobservation all of the age 30, we have no certainty thatat the end of 10 years exactly 77 will have died, leaving657 alive. It is, indeed, within the range of possibilityfirstly, that the whole 734 persons may die before the age40; and, secondly, that none of them may die, or that thewhole 734 may attain the age of 40. It appears, therefore, as if we had used the word "probability" in thesecond lemma in a different sense from that we attachedto it in the first; for, in that case we know that if thewhole of the (a + b) balls are drawn, a of them will certainly be white, and b black. But the cases will be moreparallel if we suppose that each of the balls, after beingdrawn, is replaced in the bag. If this is done, we see itis no longer certain that when (a + b") drawings take place,a of the balls will be white, and b black. It may, underthese altered circ*mstances, possibly happen that the ballsdrawn at each of the (a + b) drawings will all be white,or on the contrary all black. But when a very large number of drawings are made, we can prove that the ratio ofwhite balls drawn to the black will differ very little fromthe ratio of a to b, and will exactly equal it if the numberof drawings is supposed to be indefinitely large. In thiscase we know that the probability of drawing a white ballis still -, and passing now to the case of lives underd i bobservation, we can say, in the same sense, that the probability of a person of the age of 30 living for 10 years is,according to Deparcieux s table, -, and that on theaverage of a very large number of observations, that fraction will accurately represent the number of personssurviving. We shall, therefore, be justified in basing allour reasonings on the assumption that the lives we aredealing with die precisely at the rate indicated by thefigures of the mortality table.

We* are now in a position to show how the value of alife annuity is calculated. The annual payment of theannuity being 1, which is to be made at the end of eachyear through which the nominee shall live, the annuityconsists of a payment of 1 at the end of one year if thenominee is then alive, of the same payment at the end oftwo years, at the end of three years, &c., under the samecondition, and is therefore equal to the sum of a series ofendowments. If # is the age of the nominee, the value ofthe endowment to be received at the end of the ?ith yearis, as we have seen in lemma 2, -y^t B , and the total valueof the annuity is therefore

By means of this formula, taking the values of l x , /, +1 ,l*+v & c -> from the mortality table, and calculating thevalues of v, v, v 5 , &c., according to the desired rate ofinterest, or taking their values from the compound interesttables previously described, we can calculate the value ofan annuity at any age with any degree of accuracy desired.In practice the calculations would be most readily madeby the aid of logarithms.

7 ... .

which is at once seen to be the same as (1) formula underanother shape.

If we suppose money to bear no interest, or make v= 1in the formula for the value of an annuity, we shall obtaina quantity which is called the " expectation of life," or the" average duration of life," being the average number ofyears which persons of the given age will one with anotherlive. Denoting this by e x , and making v=lin the formulaabove given, we get

[ math ]

As in the formula for the annuity, no payment is madeon account of the year .in which the nominee dies, thisformula gives the average number of complete years thatpersons of the given age will live according to the mortalitytable, and makes no allowance for the portion of the yearin which death occurs. The expectation thus found iscalled the curtate expectation; and in order to obtain thecomplete expectation of life, which is denoted by e x , half ayear must be added to it.

The first writer who is known to have attempted toobtain, on correct mathematical principles, the value ofa life annuity, was Johan De-Wit, Grand Pensionary ofHolland and West Friesland. All our exact knowledgeof his writings on the subject is derived from two paperscontributed by Mr Frederick Hendriks to the AssuranceMagazine, vol. ii. p. 222, and vol. iii. p. 93. The formerof these contains a translation of De Wit s report upon thevalue of life annuities, which was prepared in consequenceof the resolution passed by the States General, on the 25thApril 1671, to negotiate funds by life annuities, andwhich was distributed to the members- on the 30th July1671. The latter contains the translation of a number ofletters addressed by De Wit to Burgomaster Johan Hudde,bearing dates from September 1 670 to October 1671. Theexistence of De Wit s report was well known among hiscontemporaries, and Mr Hendriks has collected a numberof extracts from various authors referring to it; but thereport is not contained in any collection of his worksextant, and had been entirely lost for 180 years, untilMr Hendriks conceived the happy idea of searching for itamong the state archives of Holland, when it was foundtogether with the letters to Hudde. It is a document ofextreme interest, and (notwithstanding some inaccuraciesin the reasoning) of very great merit, more especially considering that it was the very first document on the subjectthat was ever written; and Mr Hendriks s papers will wellrepay a careful perusal.

De Wit s report being thus of the nature of an unpublished state paper, although it contributed to its author sreputation, did not contribute to advance the exact knowledge of the subject; and the author to whom the creditmust be given of first showing how to calculate the valueof an annuity on correct principles is Dr Edmund Halley,F.R.S. In the Philosophical Transactions, Nos. 196 and198 (January and March 1693), he gave the first approximately correct mortality table (deduced from the records ofthe numbers of deaths and baptisms in the city of Breslaii),and showed how it might be employed to calculate thevalue of an annuity on the life of a nominee of any age.His method of procedure exactly agrees with the formula(1) above given; and while he confesses that it requiresa series of laborious calculations, he says that he hadsought in vain for a more concise method. His papers,which are full of interest, are reprinted in the eighteenthvolume of the Assurance Magazine.

Previous to Halley s time, and apparently for manyyears subsequently, all dealings with life annuities werebased upon mere conjectural estimates. The earliestknown reference to any estimate of the value of life annuities rose put of the requirements of the Falcidian law,which (40 B.C.) was adopted in the Roman empire, andwhich declared that a testator should not give more thanthree-fourths of his property in legacies, so that at leastone-fourth must go to his legal representatives. It is easyto see how it would occasionally become necessary, whilethis law was in force, to value life annuities charged upona testator s estate. JEmilius Macer (230 A.D.) states thatthe method which had been in common use at that timewas as follows: From the earliest age until 30 take 30years purchase, and for each age after 30 deduct 1 year.It is obvious that no consideration of compound interestcan have entered into this estimate; and it is easy to seethat it is equivalent to assuming that all persons who attainthe age of 30 will certainly live to the age of 60, andthen certainly die. Compared with this estimate, thatwhich was propounded by the Praetorian Prefect Ulpianone of the most eminent commentators on the JustinianCode was a great improvement. His table is as follows:—

Age.YearsPurchase.Age.YearsPurchase.Birth to 203045 to 461420 252846 471325 302547 481230 352248 491135 402049 501040 411950 55941 421855 60742 431760 and )43 4444 451615upwards )5

Here also we have no reason to suppose that the elementof interest was taken into consideration; and the assumption, that between the ages of 40 and 50 each addition of ayear to the nominee s age diminishes the value of theannuity by one year s purchase, is equivalent to assumingthat there is no probability of the nominee dying betweenthe ages of 40 and 50. Considered, however, simply as atable of the average duration of life, the values are fairlyaccurate. At all events, no more correct estimate appearsto have been arrived at until the close of the 1 7th century.Fuller information upon the early history of life annuitieswill be found in the article " Annuities on Lives, Historyof," in Mr Walford s Insurance Cyclopaedia.

Demoivre, in his Treatise on Annuities, 1725, showedthat it was unnecessary to go through the whole of thocalculation indicated by the formula (1) or (2) for eachage, and that the value of an annuity at any age mightbe deduced by a simple process from that at the nextolder age. This may be demonstrated as follows: If itwere certain that a person of any age, say 39, would livefor a year, then the value of an annuity on his life wouldbe such a sum as would increase at interest in a year tothe value of an annuity on a life one year older, say 40,increased by a present payment of 1; that is, putting afor the value of an annuity and 3 a for that on a life oneyear older, the value would be v (1 + l a). But it is uncertain that the life will exist to the end of a year, andthe value of the annuity must therefore be reduced in theproportion of this uncertainty, or be multiplied by theprobability that the given life will survive a year. Puttingthen p to denote this probability, we have a = vp (1 + l a).This formula may also be demonstrated algebraically. Wehave seen that

[ math ]

where z is the difference between the age of the given lifeand that of the oldest in the mortality table. (Assumingthe present age to be 39, then in the English Table No. 3,Males, 2 will be!07-39 &eq 68.) In the same way, we have

H H Hence a &eq v

—the same result as already proved.

If we suppose the present age to be x, we may put theformula in the shape

but it will be found preferable to omit the subscript xwhenever this can be done without risk of confusion.

This formula has been commonly attributed to Simpson,who in 1742 published his Doctrine of Annuities and Reversions; but, although he certainly showed that it isapplicable to annuities on the joint duration of two ormore lives, the first discovery of it is undoiibtedly due toDemoivre. (See Farren s Historical Essay on the Use andearly Progress of the Doctrine of Life Contingencies inEngland, p. 46.) The formula appears to have been independently discovered by Euler, and was given by him ina paper in the Memoirs of the Royal Academy of Sciencesat Berlin, for the year 1760.

log a, &eq log vp x + log (1 + a -+1 ),
log a_, &eq log />_! + log (1 + a.);

we have to find a t by means of an inverse entry into thetable before log (1 +a,) can be found; but when Gauss stable is used (as recomputed and extended by Gray), "Allthe entries of the same kind direct and inverse arebrought together, the whole of the logarithms being foundbefore a single natural number is taken out. We consequently proceed right through the table; and as weproceed, we find two, three, four, and even as many as sixand eight entries on the same opening. At the close,moreover, the taking out of the numbers may, if necessary,be turned over to an assistant. On the other hand, whenthe common tables are used, direct and inverse entriesalternate with each other, and involve likewise a continualturning of the leaves backwards and forwards, by whichthe process is rendered exceedingly irksome." Page 1G5,second issue, 1870.

When the only object is to form a complete table ofimmediate annuities, the above is the simplest and mostexpeditious mode of procedure; but when it is desired tohave the means of obtaining readily the values of deferredand temporary annuities, it is better to employ a whollydifferent method.

The value of a deferred annuity may be found as follows:—If it were certain that the nominee, whose age is supposed to be now x, would survive n years, so as to attain theage of x + n, the value of the annuity on his life being thena *+> its present value would be v*a f+n . But as the nomineemay die before attaining the age of x + ?i, the above valuemust be multiplied by the probability of his living to thatage, which is - , and we thus get the present value of thexdeferred annuity, t>" . -y 15 . *+ We may arrive at thisresult otherwise. Thus, as we have seen above, the present value of the first payment of the annuity, that is, of 1to be received if the nominee shall be alive at the end of *+Hn+ I years, is -y- i> n+1 . The present value of the next payment is similarly seen to be -j-v n+2 , and so on. The valueIt of the deferred annuity is therefore

B+37 -_V+3 + ........ n? / 7 &eq 7 v" . "a, (or ^ . v n . v \ *

(We may here mention that this formula holds good, notonly for ordinary annuities, but also for annuities payablehalf-yearly or quarterly, and for continuous annuities; alsofor complete annuities.)

A temporary annuity is, as explained above, an annuitywhich is to continue for a term of years provided the-nominee shall so long live. Hence it is clear that if thevalue of a temporary annuity for n years is added to thatof an annuity on the same life deferred n years, this summust be equal to an annuity for the whole continuance ofthe same life; the value of a temporary annuity for nyears will therefore be equal to the difference between thevalue of a whole term annuity and that of an annuitydeferred n years, or to

o- j-tr.. a, (ora a -^ v n a* ijC

We are now in a position to explain the method ofcalculating the value of annuities above referred to. Wehave seen that the value of an annuity for the life of anominee whose age is x, is

[ math ]

which, multiplying both numerator and denominator bythe same quantity v*, becomes

l m v

In the same way, the value of an annuity on the samelife, deferred n years, is

[ math ]

If, then, we calculate in the first instance the values ofthe product l x v x for all values of x, and then form theirsums, beginning at the highest age, we shall have themeans of obtaining by a single division the value of anyimmediate or deferred annuity we wish.

It is convenient to arrange these results in a tabularform, as shown in the appended tables (3) and (4). Thequantity l x v* is placed in the column headed D, opposite the age x, and is denoted by D,,,; while the sum^+i v * +l + e+i y * +2 + . . . . + l x +,v* + is placed in thecolumn headed N, opposite the same age x, and is denotedby N a; so that the value of an immediate annuity on alife x is equal to; the letters N and D being chosen asthe first letters of the words Numerator and Denominator.Then it is easy to see that the value of an annuity on xdeferred n years is equal to; whence by subtractionthe value of a temporary annuity for n years on the samelife is * * +n

If, for example, we wish to find the value of an annuityon a male life of 40 according to the English Table No. 3,with interest at 3 per cent., we find from table (3)appended to this article, N 40 = 1374058, D 40 = 83406, andby division we get the value of = 1 6 4744, which agrees L) 40with the value contained in the table (5), also appended tothis article.

Next, suppose we wish to find the value of a deferredannuity on a life of 30 to commence at the end of 10years. From what precedes, we see that the value of thisn -u i.- * N 40 1374058annuity will be equal to the quotient orwhich will be found to be equal to 10 9518.

If we wish to find the value of this deferred annuitywithout using the D and N table, the formula for it willbe r^ v 10 a 40 , v being equal to . But we have seenabove that the value of ^-(1*03)- 10 = 664779, and thata w = 16*4744; and multiplying these together, we get thevalue of the deferred annuity, 10 9518, as before.

D M = 125464, N., = 2385610;
D 40 - 83406, N 40 = 1374058.

Transpose the numbers opposite each age to the other age; then whatever may be the present age (less than 30)—

A person might now give up £83,406, due at the age of 30, to receive £125,464, if he live to be 40.

A person might now give up an annuity of £1,374,058, to be granted at the age of 30, to receive in return another of £2,385,610 to be granted at the age of 40, if he should live so long.

"These proportions are independent of the present ageof the party, and show that the most simple indication of the tables is the proportion in which a benefit due at one age ought to be changed, so as to retain the same valueand be due at another age. They might, therefore, withgreat propriety, be called Commutation Tables."

It is clear that this property will not be altered if all thequantities in the D column, and consequently those in the N column, are increased or diminished in a constant ratio.

A " D and N table" may be used, not only to find the valueof annuities, immediate, deferred, and temporary, but alsoto find the annual premium that should be paid for a given number of years as an equivalent for a deferred annuity.If the annuity is deferred n years, and the annual premium of equal value is to be paid for m years, it will be ^T i_ vT* The table may also be used to find thesingle and annual premiums for insurances, immediate,deferred, or temporary. The single premiums are—

1. For an ordinary insurance, - -^y;

2. For an insurance deferred n years, ^-Ty;

3. For a temporary insurance for n years,

The annual premiums payable during life for the samebenefits are found by substituting N. e _ 1 for D x in thedenominator; and the annual premiums payable for myears, by putting N,_, - N xini _ l in the denominatorinstead of D.,..

Before quitting this subject, we should mention that inpractice other columns are added to the table besides theD and N columns. A column, S, is given for the purposeof calculating the values of increasing annuities; a column,M, for calculating the values of assurances; and a column,11, for calculating the values of increasing assurances. Anexplanation of the M column belongs to the subjectINSURANCE; for an account of the S and R columns, werefer the reader to the works and papers on life insurancecontingencies, in which the D and N (or commutation)method is described; particularly to those of David Jones, Cray, and De Morgan.

The earliest known specimen of a commutation table is contained in William Dale's Introduction to the Study of the Doctrine of Annuities, published in 1772. A full account of this work is given by Mr F. Hendriks in the second number of the Assurance Magazine, pp. 15–17. Dale's table, as there quoted, differs from the one above described in that it commences only at the age of 50,and that he has tabulated l x v - K instead of Ijf. Hesays, " These calculations being made for the use of the societies in particular who commence annuitants at the age of 50, it was not thought necessary to begin the tables at a younger age." He gives, however, anothertable based on different mortality observations, commencing at the age of 40; and in this case he tabulatesl j .v* to . His table also differs from the common form inthat it is adapted to find the values of annuities payableby half-yearly instalments.

The next work in which a commutation table is foundis William Morgan s Treatise on Assurances, 1779. Inthis work the values of - z - ^ c l are tabulated, and notthose of l x v*; but, as above mentioned, the properties ofthe table are not altered by the change. Morgan gives thetable as furnishing a convenient means of checking thecorrectness of the values of annuities found by the ordinaryprocess. It may be assumed that he was aware that thetable might be used for the direct calculation of annuities;but he appears to have been ignorant of its other uses.

The first author who fully developed the powers of thetable w r as John Nicholas Tetens, a native of Schleswig, whoin 1785, while professor of philosophy and mathematics atKiel, published in the German language an Introductionto the Calculation of Life Annuities and Assurances. Thiswork appears to have been quite unknown in Englanduntil Mr F. Hendriks gave, in the first number of theAssurance Magazine, pp. 1-20 (Sept. 1850), an account ofit, with a translation of the passages describing the construction and use of the commutation table, and a sketchof the aiithor s life and writings, to which we refer thereader who desires fuller information.

The use of the commutation table w r as independentlydeveloped in England apparently between the years 1788and 1811 by George Barrett, of Pet worth, Sussex, whowas the son of a yeoman farmer, and was himself a villageschoolmaster, and afterwards farm steward or bailiff. .Inthe form of table employed by him, the quantity tabulatedis not Ijf, but l x (\ + i)" x , where 10 is the last age inthe mortality table used. It has been usual to considerBarrett as the originator in this country of the method ofcalculating the values of annuities by means of a commutation table, and this method is accordingly sometimescalled Barrett s method. (It is also called the commutation method and the columnar method. ) Barrett s methodof calculating annuities was explained by him to FrancisBaily in the year 1811, and was first made known to theworld in a paper written by the latter and read before theRoyal Society in 1812.

By what has been universally considered an unfortunateerror of judgment, this paper was not recommended by thecouncil of the Royal Society to be printed, but it wasgiven by Baily as an appendix to the second issue (in1813) of his work on life annuities and assurances. Barrett had calculated extensive tables, and with Baily s aidattempted to get them published by subscription, but without success; and the only printed tables calculated according to his manner, besides the specimen tables given byBaily, are the tables contained in Babbage s ComparativeView of the various Institutions for the Assurance of Lives,1826. It may be mentioned here that Tetens also gaveonly a specimen table, apparently not imagining that persons using his work would find it extremely useful to havea series of commutation tables, calculated and printedready for use.

In some respects this notation is perhaps preferable totha t now used, as it is certainly better, when there is norisk of confusion, to omit the subscript x. But Davies snotation cannot be adopted without alteration, as N xmight be mistaken for the number in the column N" opposite the age 1. We may, however, consistently with theprinciples of the notation adopted by the Institute ofActuaries, write the formula _rj^ s&eqrnj. The notation atpresent commonly used is due to David Jones, whose work(mentioned below) was the first that contained an extensive series of commutation tables.

On a general review of the whole evidence, we cannothelp thinking that Barrett s merits in the matter have beensomewhat exaggerated. The first idea of a commutationtable was not due to him, but (leaving Tetens out of view)to Dale and Morgan; and it is certain that he was familiarwith the latter s treatise. The change he introduced intothe arrangement of the table, namely, multiplying by apower of (1 +i) instead of by a power of v, is the reverseof an improvement; and accordingly, his form of table hasnever been in practical use by any person but himself,excepting only Babbage. It is, of course, not to bedenied that great credit is due to him as a self-educatedman, for perceiving more clearly than his predecessors thegreat usefulness of the commutation table; but in ouropinion he does not stand sufficiently out from those whopreceded and followed him, to justify the attempt toattach his name to the columnar method of calculating thevalues of annuities and assurances. Those who desire to.obtain further information on the matter, and to learn theviews of other writers, can refer to the appendix to Baily sLife Annuities and Assurances, De Morgan s paper " On theCalculation of Single Life Contingencies," Assurance Magazine, xii. 348-9; Gray s Tables and Formulae, chap. viii.;the preface to Davies s Treatise on Annuities; also Hend-riks s papers in the Assurance Magazine, No. 1, p. 1, andNo. 2, p. 12; and in particular De Morgan s " Account ofa Correspondence between Mr George Barrett and MrFrancis Baily," in the Assurance Magazine, vol. iv. p. 185.The principal D and N tables published in this countryare contained in the following works:—

The explanations of the tables in the last four works areby Dr William Fair, F.R.S.

Very unfortunately, these tables are not all arrangedupon the same principle, but those contained in the Reportsof the Registrar-General, in the English Life Table, inChisholm s and in Henry s tables, are so arranged that thecolumn N is shifted down one year, so that in them the ratio N&eqp gives, not the value of the ordinary annuity, but the -L a;value of the annuity increased by unity, or the annuity-due.It is very needful to bear this in mind for the prevention oferror; and the existence of a difference of this kind is extremely perplexing. For information upon the subject ofthis confusing change, see De Morgan s paper " On theForms under which Barrett s Method is represented, andon Changes of words and symbols," Ass. Mag., x. 302.

A, + _ 24u 2 720n 3

Here V = 1, V, = -j-v*, V 2)l = -y v- n , &c. 480?i 3.

As an example, we will apply this formula to calculatethe value of an annuity on a nominee of 40, according tothe English Table, No. 3, Males, at 3 per cent, interest.

First, taking n = 7, we find

V =1-0000- 2654V 7 = 7346 + -05212133 - -0115V 14 = -5213 -0406 +-00291727 -0086V.,= -3486 -03201407V 88 = -2079V 35 = -0990V 42 = -0318V 49 = -0055V= 0004Sum=2-9491

Hence A x = -2654, A 2 = 0521, A 3 = - 0115, A 4 - 0029; and thevalue of the annuity is approximately

4 2= 7 + 2-9491- 4-yx-2654-yX 0521- -1808 x -0115- 1283 x -0029.=20-6437-4-0000- -1517- -0149- -0021- -0004= 20-6437-4-1691= 16-4746.

Next, taking =11, we have

V =1-0000- -3924V u = -6076 +-11152809 - -0310V 2a = -3267 -0805 + 04532004 + -0143V 33 = -1263 -09481056V 44 = -0207V 56 = -00062-0819

Hence, the value of the annuity is approximately

10 5=11 x 2 -0819 - 6 - x -3924 - x -1115 - 2878 x -0310= 22-9009-6-0000- -3567- -0507- Od89- -0093= 22-9009-6-4256= 16-4753.-2044x -0453

The value of the annuity calculated ia the ordinary way is, as wehave seen (page 80), 16-4744.

An improved form of this method was given by MrW. S. B. Woolhouse in the Ass. Nag., xi. 321. In orderto explain this, we must introduce the reader to a termwhich is of recent origin, but which the application ofimproved mathematical methods to the science of life contingencies has rendered of great importance tJie force ofmortality at a given age. This may be defined as the proportion of the persons of that age who would die in thecourse of a year, if the intensity of the mortality remainedconstant for a year, and the number of persons under observation also remained constant, the places of those who diebeing constantly replaced by fresh lives. More briefly,it is the instantaneous rate of mortality. A very fullexplanation of this term is given by Mr W. M. Makeharn,in his paper "On the Law of Mortality, "Ass. Mag., xiii. 325.The value of the function can be approximately found bydividing the number of persons who die in a year by thenumber alive in the middle of the year. Thus, if l x denotethe number of persons living at the age x, d x the numberdying between the ages x and x+1, and d x _ x the numberdying between the ages x 1 and x, then the number dyingbetween the ages x-- and x + - will be approximately x i + d, , and the force of mortality is approximately a -^ -. Thus, in the English Table, No. 3, Males, the * 3465 + 3529value of the force of mortality at age 40 is - = 012853.

This quantity is usually denoted by the Greek letter p.,while 8 is used to denote the quantity log (l+i), whichWoolhouse has called the force of discount. This beingpremised, Woolhouse s formula for the approximate valueof an annuity is

[ math ]

where it will be noticed that, since V = 1, the two first termsare exactly equal in value to those in Lubbock s formula.

Taking the same example as above, we have seen that

^o= -01 285 3also 8 =-029558^40 + 8 = 042411

Making n = 7, we have the value of the annuity

= 16-6437 -4 x -042411= 16-6437- -169644= 16-4741.

Making n = 11, we have the value

= 16-9009- 10 x -042411= 16-9009- -4241= 16-4768.

So long as we consider the annuity to be payableyearly, no allowance being made for the time which elapsesbetween the death of the nominee and the last previouspayment of the annuity, it is, as we have seen, a verysimple problem to calculate its value. But in practiceannuities are generally payable by half-yearly instalments,and it is the custom to pay a proportionate part of theannuity for the odd time that elapses between the lasthalf-yearly payment and the death of the nominee; andthe value found by the methods described above thereforerequire to be corrected before they are strictly applicablein practice. Approximate values of the necessary corrections are very easily found; but the strict investigation oftheir correct values is a problem requiring a considerableknowledge of the higher mathematics, and it would bequite beyond our present purpose to consider it.

When an annuity is payable half-yearly, the commonrule for finding its value is to add -25, or a quarter of ayear s purchase, to the value of the annuity payable yearly.When it is payable quarterly, 375 is added; and when byinstalments at n equal periods throughout the year (or bythly instalments), the addition is The values thusfound are sufficiently correct for most purposes. Morecorrect methods of finding the values of annuities payablehalf-yearly, quarterly, &c., are investigated in papers inthe Assurance Magazine, by Woolhouse, xi. 327, and bySprague, xiii. 188, 201, 305. Some authors have assumedthat when an annuity is payable half-yearly, interest isalso convertible half-yearly, overlooking the circ*mstancethat the true rate of interest is thereby changed, as wehave explained in the earlier part of this article. In fact,as we showed, 5 per cent, interest convertible half-yearlyis equivalent to a true rate of interest, 5, Is. 3d. per cent.If, then, we have found the value of an annuity whenpayable yearly at 5 per cent, interest, and require, perhaps,in the course of the same investigation, the value of anannuity payable half-yearly, it is clear that that valueshould be computed, not at .5, Is. 3d. per cent, interest,but at 5 per cent.; or if we prefer the rate 5, Is. 3d.,then the value of the annuity payable yearly should alsobe calculated at that rate.

The approximate value of an annuity payable up to theday of the nominee s death, or of a " complete " annuity, asit is now usually called, is found in the case of annuitiespayable yearly by adding to the value of the ordinaryannuity the value of i, payable at the instant of the nominee s death; in the case of half-yearly annuities, by addingthe value of \; and in the case of quarterly annuities, thevalue of

The previous remarks refer almost exclusively to annuities which depend on the continuance of one life, or to" single life annuities," as they are commonly called. Butan annuity may depend on the continuance of two or threeor more lives. It may continue so long as both of twonominees are alive, in which case it is called an annuityon the joint lives; or it may continue as long as either ofthem is alive, in which case it is called an annuity on thelast survivor. Again, if it depends on the existence ofthree nominees, it may .either continue so long only asthey are all three alive, when it is called an annuity onthe joint lives; or so long as any two of them continuealive, when it is called an annuity on the last two survivors; or so long as any one of them is alive, when it iscalled an annuity on the last survivor. In addition tothese, we have "reversionary" annuities, which are tocommence on the failure of an assigned life, and continuepayable for the life of a specified nominee; or, more generally, to commence on the failure of a given status, orcombination of lives, and continue payable during theexistence of another status. There are also "contingent"annuities, which depend on the order in which the livesinvolved fail. Thus, we may have an annuity on the lifeof x, to commence on the death of ij, provided that takeplace during the life of z, and not otherwise, and tocontinue payable during the remainder of the life of x.Reversionary annuities are of considerable practical importance, but contingent annuities are rarely met with.Lastly, we may mention annuities on successive lives,These are of importance in the calculation of the values ofadvowsons, and of fines on copyhold property. It doesnot fall within the scope of this article to treat at anylength of annuities on more than one life, and we mustrefer the reader who wishes for further information with,regard to them to the works of Baily, Davies, and DavidJones, already mentioned, and Milne s Treatise on theValuation of Annuities and Assurances, 1815.

As intimated above, our remarks on annuities involvingmore than one life will be very brief. The methods employed for the calculation of single life annuities are easilyextended to the case of joint life annuities. The fundamental equation

a &eq vp(l + *a)

is true of annuities on two, three, or any number of jointlives, if we consider/) to denote the probability that they willall survive for one year; and l a the value of an annuityon the joint continuance of lives which are severally oneyear older than those on which the required annuity depends. Thus we have x, y, 2, being the ages of the nominees—

and a,, Jt &eq vp m p t p, (1 + l a xyi } .

The columnar method of calculating annuities admitsalso of being extended to annuities on joint lives. In theextensive tables contained in D. Jones s work,

Djcy " IJ-yV", y being the older of the two ages,

where n T) xy is used to denote T) x+n . y+n .An improved form of the table was suggested byDe Morgan, according to which we should have D xy &eqx+y IJyV 2 . This would simplify the formulas for the valuesof contingent annuities, but no tables have as yet beenpublished calculated on this principle. The same methodmight be extended to three lives, in which case the mostX+y+ladvantageous form would bs D^, = ljl t v 3; but the extent of the tables when three lives are involved renders itextremely improbable that such will ever be published.The practical construction of a D and N table for jointlives has been considered by Gray, Tables and Formidce,pp. 122-137, and Ass. Mag., xviii. 26. Mr Jardine Henryi.a3 described in the Ass. Mag., xiv. 212, a mechanicalmethod of computing the values of D xy = IJyV 9 , by meansof which he has calculated the values in his extensivetables mentioned above.The values of annuities on the last survivor of two ormore lives cannot be calculated by the ordinary methodsthat apply to annuities on joint lives; thus, for example,the equation a = vp ( 1 + l ci) does not hold good with regardto them. Their values must be found from those of jointlife annuities by means of the following formulas:—

An annuity on the lasfisurvivor of two lives, x fo^ a x + a y - a I}>and y,An annuity on the lasOsurvivor of three lives, a^ji eq a x + a y + a,- a a , - a, x - a xy + a xy ,x, y, and z, )An annuity on the last two \survivors of the three \a x j t eq a 1/ , + a 1JC + a xy -2a X!/ ,lives, x, y, z, )

If we have the values of annuities on the last survivorof two lives tabulated, as is the case in the Institute ofActuaries Life Tables, we may find the value of an annuityon the last of three lives by means of the formula a=rt + ( i* rt r. where w is found by means of the relation= *-,; see Ass. May., xvii. 266, 379.The methods of approximation given by Lubbock andWoolhouse also apply to the calculation of annuities onthe joint existence of any number of lives; see the latter sexplanation of his method, Ass. Mag., xi. 322, and for anillustration of its application to complicated cases, xvii.267. They may also be applied to find the value of anannuity on the last survivor of any number of lives; seeAss. Mar/., xvi. 375.The formula usually given for the value of a reversionaryannuity on the life of x to commence on the death of yis a x - a xy . But this is not sufficiently correct, being deduced frbm suppositions that do not prevail in practice.It assumes the first yearly payment of the annuity to bemade at the end of the year in which y dies, and the lastat the end of the year before that in which x dies; whereasin practice the annuity runs from the death of y, the firstyearly payment being made one year after such death, anda proportionate part being paid up to the date of ar s death.A more correct formula, as given by Sprague (Ass. Mag.,xv. 126), is *. x (. If the annuity is payable half-yearly,1 4. f 1 4- f\*the value will be approximately (a x - a xu ] -; and ifjjquarterly, (a z -a^) / 1 , -a , I n practice, it is oftensufficient to deduct half a year s interest from the valuefound by the formula a x a xy , when the annuity is payableyearly, a quarter of a year s interest when it is payablehalf-yearly, and an eighth of a year s interest whenquarterly.

The fact that Gompertz s law does not correctly representthe mortality throughout the whole of life, proves that theabove-described practical method of finding the value ofan annuity on three joint lives is accurate only in certaincases. Makeham has shown (Ass. Mag., ix. 361, and xiii.355) that when the mortality follows the law indicated byhis modification of Gompertz s formula, the value of anannuity on two, three, or any number of joint lives, canbe readily found by means of tables of very moderateextent. Thus the value of an annuity on any two jointlives can be deduced from the value of an annuity at thesame rate of interest on two joint lives of equal ages; thevalue of an annuity on any three joint lives, by means ofa table of the values of annuities on three joint lives ofequal ages; and so on; and Woolhouse has shown (vol.xv. p. 401) how the values of annuities on any number ofjoint lives, at any required rate of interest, can be found bymeans of tables of the values of annuities on a single lifeat various rates of interest. These methods, we believe,have not hitherto been practically employed to any extentby actuaries, and it would perhaps be premature to saywhich of them is preferable.

As the reader will have observed, neither Gompertz snor Makeham s formula represents correctly the rate ofmortality for very young ages. Various formulas havebeen given which are capable of representing with sufficientaccuracy the number living at any age from birth toextreme old age, but they are all so complicated that theyare of little "more than theoretical interest. They are,however, likely to prove of increasing value in the problemof adjusting (or graduating) a table of mortality deducedfrom observations, an important subject, which does notfall within the scope of this article. We may mentionin particular those given by Lazarus in his Mortalitdts-verhciltnisse und Hire Ursaclie (Rates of Mortality and theirCauses), 1867, of which a translation is given by Spraguein the eighteenth volume of the Assurance Magazine,namely, CK^/fW*; and by Gompertz (see Ass. Mag., xvi.329),

l, eq const. A*B / *-"C*D p , where P eq 0^ X(x -^\

If l x represents the number living at any age in themortality table, the force of mortality, or the instantaneousrate of mortality, mentioned above (see p. 83), is equalto --T-logJr Hence, in Gompertz s original law theforce of mortality at any age x is proportional to <f, oris equal to a(f t where a is a constant; in Makeham slaw the force of mortality is equal to atf + b, wherea and b are constants; and in Lazarus s law the force ofmortality is equal to aq* + b + cp x , where a, b, and c areconstants, or to ae"* + b + ce* x . Dr Thiele has shown(see Ass. Mag., xvi. 313) how to graduate a mortalitytable, by assuming the formula for the force of mortality,o 1 t*j*+a s i * ( ** )B + a 3 * 8 *j and Makeham has explained(Ass. Mag., xvi. 344) a very convenient practical methodfor adjustment, which results in assuming that the numberliving at any age x can be accurately represented by theBum of three terms of the form dg qX s*.

The employment of formulas such as those given in thelast paragraph, and the application of the differentialcalculusfto the theory of life contingencies, have naturallyled to an improvement in the theory which is probablydestined to become of very great importance we refer tothe introduction of the idea of " continuous " annuitiesand assurances. If the intervals at which an annuity ispayable are supposed to become more and more frequent,until we come to the limit when each payment of theannuity is made momently as it accrues, the annuity iscalled continuous. Strictly speaking, of course, this is animpossible supposition as regards actual practice; but ifan annuity were payable by daily instalments, its valuewould not differ appreciably from that of a continuousannuity; and if the annuity be paid weekly, the differencewill be so small that it may be always safely neglected.The theory of continuous annuities has been fully developedby Woolhouse (Ass. Mag., xv. 95). Assuming the numberliving in the mortality table at any age x to be representedby l x , the value of a continuous annuity on a nominee1 /"*oo -I y-oo *- .,of the age x is j- I l x ifdx eq j I l x e Sx dx, puttingm^/ x IxJ x I8 eq log e (l+t). From the nature of the case, l x must bea function that is never negative for positive values of x;and as x becomes larger, l x must continually diminish, andmust vanish when x becomes infinite. It will be noticedhere that the superior limit of the integral is GO . This isnecessary if l x is a continuous mathematical function; forin that case, however large x be taken, l x will never becomeabsolutely zero. Makeham has shown (Ass. Mag., xvii.305) that when the number living, l x , can be correctlyrepresented by the formula cg^ e "*, the value of a continuous annuity is equal towhere n eq +log q 10- 10 *.e- r10-and z eq x Iog 1( tf + log? -; and he hasgiven (pp. 312-327) a table, by means of which the valueof the annuity can be found when the values of n and zare known. This table requires a double interpolation,and is therefore rather troublesome to use. Mr EmoryM Clintock has shown in the eighteenth volume of theAssurance Magazine, how the value of an annuity may befound by means of the ordinary tables of the gamma-function. As Lazarus has pointed out in his above-mentioned paper, when mortality tables are given in theordinary form, it is difficult to compare them and defineprecisely their differences; but if they can be accuratelyrepresented by a formula containing only a few constants,it becomes easy to show wherein one table differs fromanother; and the methods of Makeham and M Clintockenable us to compare the values of annuities, for any agesdesired, according to different tables as determined by suchconstants, without the labour of computing the mortalitytables in the usual form. They can therefore scarcelyfail to grow in popularity as they become better known.

​

Males.Females.Males.Females.Males.Females.Age.^NumberNumberAge.NumberNumberAge.NumberNumberNumber | dying inNumberdying inNumberdying inNumberdying inNumberdying inNumberdvinginalive atthe folalive atthe folalive atthe folalive atthe folalive atthe folalive atthe foleach age.lowingeach age.lowingeach age.lowingeach age.lowingeach age.lowingeach age.lowingyear.year.year.year.year.year.XI,d,I*d,XI*d xhd xXhd xl xd x5117458371948825565774372822963352276563332674834167639930717724142802627521422481261593827894434062732373350757577774838534776532400505142153963221402339275538346526988733767668294726877694752133862909213382299924340272073352926651134027761026699070173732943770776719373056659641268544359626310934317854036665562844707153703585033366460486642264948366825967834597947381626655773675563653253953361594381543261280374625621934908041115583249018638273613723310357779324944257534382625272935228135283536142636595983580622734354530272445253708391224920735558229922486236677549693553282297351806232846249796400124565235918325060434931181500310353031198334947820454724579540952420613627842071138342617844901135104817763474331861482417004192238434366585168773328216883972123492721666345572176549237508429223476937058613549284017716345813347606163734380717455023321643952310643746871070923841425829621434596916793420621789512288214626227318378888832519651129624941534429017813402731888522241954758223530383289636015908802206316342509192833838520295321943748852196983876904770126067391673173405812112336356220554214552501321582242469135109795066133118338469232033415124005520953951442115764439922531744373510371933614925413317512609562043955281207137462893178755326987902033360827643291422819571991145428202509481794123440119085882133084428013263232867581936865584197692500995833285132042822328043283632345629125918810257521926835206965481968923042332520728683205442952601823505929187477540997352132588210243223392S97317592298961176421611818206856199822086378142253194422926314603302462170303631417644958359913455236922631651629543115793055631639896515170614605710079331445927313562298130852430846415747467201645576282101462185362831058130093054403112651507546921158275650910225114922293075723038302328313866143833711515176667311031472712303045343068299190316367136718729714503569471047315731301466310029602731876812942174581380887149105428482298366313429284032096912196375931309397332106214233295232317128963132337011437076951236077489107112134292061321128639832557110667577561161187613108...113528885032542831433279 i7298919777010850576981093028559633002798643301739114977331008077736

L ​

Age.X.Probability of Dying in aYear2*Age.X.Probability of Dying in aYear.9*Age.X.Probability of Dying in a:Year.f.Males.Females.Males.Females.Males.Females.163597134714370118730120257308484007674810642980619183801221201226274091570082984203549403538339012575012508750987580896683023850024178- 40012968012766761064120968124017820017683410133920130387711454410443050135900132784201384501332078123154112526601082001055443014334013620791322561211127009160009080440148580139368014184413019280076360076844501541801426881151926139774900646500661846016018014618821625001498551000561600585347016660014985831735641604401100506000535843017343015373841851161715281200476800510849018072015780851971481831151300471000507450018844016210862096541951961400485400523251020220016666872226262077671500517300554852021222017142882360502208141600563000599853022263017646892499142343321700620300655354023364019673902642032483021800685400718455C245480209809127890026271019007558007865560258380223449229398727754320008285008563570272600237889330944229277821008468008788580288300253389432524330839722008645009004590305750270189534136732437323008820009210600325180288509635778734068724008990009413610346760308629737447935730925009160009610620370740330709839141137421026009333009805630397330355009940855639135327009507009998640426720381781004258834087382800968801019065045910041123101443358426301290098780103786004947004435410246095344402130010073010570670533700478981034786314618633101028301076468057626051772104496361479793320105040109626906225605599410551410949777733010740011163700672780605861065318395157773401099401136871072708065563107549520533760350112650115807207855607094610856711655168836011558011798

Age.X.D*NXAge.X.D*NjAge.X.D*N*511745928849137945641635186749360-450726-51415559887293238907201544466758255-542471-02377514849541839870021457464767223-535247-53353510814190840834061374058776266-928980-64335028780688041799261294132785387-423593-25319474748740642765591217573794586-319006-96305953718145343733001144273803863-815143-17293828688762544701451074128813219-211923-98282657660496845670901007038822650-69273-3927232963326394664132942906832155-27118-11026268860699514761267881639841729-35388-81125360558163464858491823148851368-14020-71224497255713744955803767345861066-42954-30132367035334671505319871414787818-282136-02142287265105945515067666347188617-581518-44152209874884958524820561526689458-061060-38162134414671517534580856945890333-58726-80172060574405460544348352597591238-30488-50181988154266645554123048474592166-83321-67191917014074944563904744569893114-35207-32320184711389023357369304087689476-668130-65521177845371238858348773738919550-22580-43022171203354118559328853410069632-11748-31323164780337640560309513100559720-02528-28824158570321783561290722809839812-16116-12712515256730652686227247253736997-18568-941526146767291850163254722282641004-12614-815427141162277733964237482045161012-29992-515528135748264159165220721824441021-24291-2726291305172511074662044616199810365056221301254642385610671886814313010432932928311205832265027681734112578910516101318321158672149160691586610992310607590559331113102037850701444595478107034502143410690719309437113080823981080151006335102653182829072117767062210900630000369854017297507310535

60086-9 ​

Age.X.D,NjAge.X.D*NsAge.X.D*.NX488255920370137926441647581741044460281-51410175879352638888641558717759298-250983-32373571841995539852181473499768217-942765-43349858807009740817011391798777206-135559-34331456773864141783091313489786265-62929375316112742252942750361238453795398-623895-16302830711969943718801166573804606-619288-57290907682879244688361097737813890-115398-48279870654892245659001031837823248-912149-5926963062792924663068968769832681-69467-91026004460192484760336908433842185-87282-11125099357682554857701850732851758-15524-01224237755258784955159795573861394-44129-61323411652917625052707742866871089-53040-10142261435065619515034369252388838-002202-10152184084847211524806264446189633-941568-16162108704636341534586259859990471-251096-91172035014432840544374155485891343 92752-99181962794236561554163151322792246-18506-81191891934047368563957147365693172-68334-13201822383865130. 573756043609694118-56215-5742117541536897155835598400498.9579-611135-96322168809352090659336863668129652-22183-74223162417335848960318213349919733-42750-31524156234320225561300033049889820-85829-45725150256305199962282302767589912-67216-784826144478290752163265012502571007-48829-296627138894276862764248162254411014-29854-998128133501263512665231742022671022-39422-603929128292250683466215731806941031-29241-3115301232622383572672001616067810467526363311184072265165681850214217610534102953321137212151444691703312514310616631290331091982042246701561110953210707820508341048351937411711423895294108035401543510062418367877212917823771090154000036965621740225731165170726

Age.X.Value of Annuity.a xAge.X.Value of Annuity.a xAge.X.Value of Annuity.a xMales.Females.Males.Females.Males.Females.18-150618-85023418-061818-4807687-25397-6842121-351821- 43853517-810518-2539696-92847-3469222-503622-53913617-553818-0218706-61007-0162323-031623-06683717-291817-7841716-29936-6928423-302223-34743817-024517-5405725-99716-3773523-436723-48073916-752117-2910735-70366-0702623-472423-51064016-474417-0353745-41935-7721723-441023-47424116-191616-7733755-14455-4832823-367423-39994215-903716-5047764-87955-2039923-253623-28864315-610816-2293774-62444-93461023107123-14704415-312915-9471784-37934-67531122-934722-98184515-010215-6576794-14424-42621222-742922-79874614-702615-3608803-91924-18721322-537422-60324714-390215-0563813-70403-95831422-323422-40004814-073014-7439823-49863-73951522-105222-19334913-751114-4233833-30273-53071621-886721-98675013-424214-0942843-11623-33151721-671021-78295113-0925137562852-93883-14191821-460421-58445212763613-4090862-77032-96171921-256821-39235312-431513-0522872-61042-79042021-061221-20935412-096012-6852882-45872-62782120-874321-03425511-757012-3279892-31492-47372220-684120-85735611-414511-9699902-17882-32772320-490420-67825711-068711-6107912-05002-18942420-292920-49655810-720311-2505921-92812-05862520-091320-31205910-369710-8892931-81291-93502619-885320-12446010-017610-5274941-70421-81812719-674819-9334619-665010-1653951-6014170782819-459619-7387629-31259-8037961-50431-60362919-239419-5401| 638-96129-4431971-41261-50523019-014319-3374648-61199-0844981-32611-41233118-784019-1303658-26578-7284991-24441 -32453218-548618-9187667-92338-37581001-16711-24153318-307818-7022677-58588-0275