Annonay

ANNUITIES

BY an annuity is meant a periodical payment, made annually, or at more frequent intervals, either for a fixed term of years, or during the continuance of a given life, or a combination of lives, as will be more fully explained further on. In technical language an annuity is said to be payable for an assigned status, this being a general word chosen in preference to such words as “time,” “term,” or “period,” because it may include more readily either a term of years certain, or a life or combination of lives. The magnitude of the annuity is the sum to be paid (and received) in the course of each year. Thus, if £1 00 is to be received each year by a person, he is said to have “an annuity of £100.” If the payments are made half-yearly, it is sometimes said that he has “a half-yearly annuity of £100;” but to avoid ambiguity, it is more commonly said he has an annuity of £100, payable by half-yearly instalments. The former expression, if clearly understood, is preferable on account of its brevity. So we may have quarterly, monthly, weekly, daily annuities, when the annuity is payable by quarterly, monthly, weekly, or daily instalments. An annuity is considered as accruing during each instant of the status for which it is enjoyed, although it is only payable at fixed intervals. If the enjoyment of an annuity is postponed until after the lapse of a certain number of years, the annuity is said to be deferred. When an annuity is deferred for any number of years, say n, it is said, indifferently, to commence, or to be entered upon, after n years,'or to run from the end of n years; and if it is payable yearly, the first payment will be made at the end of (π+l) years; if half-yearly, the first half-yearly payment will be made at the end of (ra + ∣) years; if quarterly, the first quarterly payment will be made at the end of (n + ∣) years; and so on. If an annuity, instead of being payable at the end of each year, half-year, &c., is payable in advance, it is called an annuity-due.

If an annuity is payable for a term of years independent of any contingency, it is called an annuity certain; if it is to continue for ever, it is called a perpetuity; and if in the latter case it is not to commence until after a term of years, it is called a deferred perpetuity. An annuity depending on the continuance of an assigned life or lives, is sometimes called a life annuity; but more commonly the simple term “annuity” is understood to mean a life annuity, unless the contrary is stated. A life annuity, to cease in any event after a certain term of years, is called a temporary annuity. The holder of an annuity is called an annuitant, and the person on whose life the annuity depends is called the nominee.

If not otherwise stated, it is always understood that an annuity 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 annual payment 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 course should be followed in any tables that may be published in future.

The annuity, it will be observed, is the totality of the payments to be made (and received), and is so understood by all writers on the subject; but some have also used the word 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 should therefore be carefully avoided.

The theory of annuities certain is a simple application of algebra to the fundamental idea of compound interest. According to this idea, any sum of money invested, or put out at interest, is increased at the end of a year ^by the addition to it of interest at a certain rate; and at the end of a second year, the interest of the first year, as well as the original sum, is increased in the same proportion, and so on to the end of the last year,—the interest being, in technical language, converted into principal yearly. Thus, if the rate of interest is 5 per cent., £1 improved at interest will amount at the end of a year to £1’05, or, as we shall in future say, in conformity with a previous remark, 1 will at the end of a year amount to 1Ό5. At the end of a second year this will be increased in the same ratio, and then amount to (Γ05) 2 . In the same way, at the end of a third year, it will amount to (1O5) 3 , and so on.

Let i denote the interest on 1 for a year; then at the end of a year the amount of 1 will be 1 + i. Reasoning as above, at the end of two years the amount will be (1 +i) 2 , at the end of three years (1 +t) 3 , and so on. In general, 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 of n years, is such a sum as, being improved at compound interest for n years, will exactly amount to 1. We have seen that 1 will in n years amount to (1 +t)", and by proportion we easily see that the sum which in n years will amount to 1, must be > or (1 +i)~∙. It is usual to put V for ~j~pρ so t ħat v is the value of 1 to be received at the end of a year, and tΓ the value of 1 to be received at the end of n years.

Since 1 placed out at interest produces i each year, we see that a perpetuity of i is equal in value to 1; hence, by proportion, a perpetuity of 1 is equal in value to -.. At ⅛

5 per cent., t=O5, and -=»20; or a perpetuity is worth 20 years’ purchase: at 4 per cent., it is worth 25 years’ purchase, = 25^: at 3 per cent., it is worth 33⅜ years’ purchase, (,⅛ = 33⅜) .

Instances of perpetuities are the dividends upon the public 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 the reality is the £3 a year which the Government pays by half-yearly instalments. The practice of the French in this, as in many other matters, is more logical In speaking of their public funds, they do not mention the ideal capital sum, but speak of the annuity or annual payment that is received by the public creditor. Other instances of perpetuities are the incomes derived from the debenture stocks now issued so largely by various railway companies, also the feu-duties commonly payable on house property in Scotland. The number of years’ purchase which the perpetual annuities granted by a government or a railway company realise in the open market, forms a very simple te3t 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 thus obtains 3’226 per cent, interest on his investment. Other examples are given in the subjoined tables, the figures in which are deduced from the Stock Exchange quotations of the irredeemable stocks issued by the various govern-mp∏t,R∙- » . . [table]

The following are a few other examples of perpetui« ties ·— τ . . [table]

We may mention in passing that the more usual practice of foreign governments when borrowing is not to grant the lender a perpetual annuity, but to issue to him bonds of say £100 each, bearing an agreed rate of interest, these bonds being usually issued at a discount, and redeemed at par by annual drawings during a specified term of years.

We have seen that the present value of any sum payable at the end of n years is found by multiplying it by (1+i) - "; hence the value of a perpetuity of 1 deferred n years is ^ 1+ 9~ · Nowan annuity for n years is clearly the difference between the value of a perpetuity to commence at once and a perpetuity deferred n years; its value is therefore τ - -—≠- = —; —> 0r P utt i n g « for the value of the annuity, we have— a = 1 -< 1 + i >-.

By means of this equation, having any two of the three quantities, a, i, n, we can determine the third either exactly or approximately. Thus for n we have—

log (l-tα), log (1+0 '

There is no means of determining the value of t exactly, but it may be found to any degree of accuracy required by methods of approximation which our limits will not allow us to describe.

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 reduced in the proportion of 1 to the value of 1 payable in t years, or 1: (1 + i)~ t ', and the value of the deferred annuity to continue for n years is therefore—

It Remains to find the amount at compound interest at the end of n years of an annuity payable for that term. The amount of 1 in n years being (1 +i) n , its increase in that time is (1 + i) n - 1; but this increase arises entirely from the simple interest, i, of 1 being laid up at the end of each year and improved at compound interest during the remainder of the term. Hence it follows that the amount at compound interest of an annuity of i in n years must be (1 +⅛)" - 1; and by proportion the amount of an annuity of 1 similarly improved will be -- + 9—- ·

One of the principal applications of the theory of annuities certain is the valuation of leasehold property; another is the calculation of the terms of advances in consideration of an annuity certain for a term of years. At present a large sum of money is annually borrowed by corporations and other public bodies upon the security of local rates in the United Kingdom. It is sometimes arranged in these transactions that a fixed portion of the loan shall be paid off every year; but it is more commonly the case that, in consideration of a present advance, an annuity is granted for a term of years, usually 25 or 30, but in some instances extending to 50. Landed proprietors also, who possess only a life interest in their property, have been authorised by various Acts of Parliament to borrow money for the purpose of improving their estates, and can grant a rentcharge upon the fee-simple for a term not exceeding 30 pears. These are very favourite investments with the life insurance companies of the country, as they are thus enabled to obtain a somewhat higher interest—from 4∣ to 4∣ per cent.—than they could obtain upon ordinary mortgages with equally good security; the reason for this, of course, being that these loans are not so suitable as others 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 of ordinary mortgages, the terms paid by the borrowers are higher. 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 consider each year’s payment as consisting partly of interest on the outstanding balance of the loan and partly as an instalment of the principal. The problem of determining the separate amounts of these has been considered by Turnbull, Tables, p. 128; and by Gray, Λss. 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 tables which have been published, containing values and amounts calculated for various rates of interest. An extensive set of tables of this kind was published in 1726 by John Smart; and many subsequent writers, as Dr Price, Baily, Milne, Davies, D. Jones, J. Jones, have reprinted or abridged portions of these tables. They show the amount and the present value both of a payment and of an annuity of £1 for every term of years not exceeding 100, at the several rates of interest, 2, 2∣, 3, 3⅜, 4, 5, 6, 7, 8, 9, 10 per cent.; the five most important tables giving—

(1.) The amount of £1 in any number of years, n ; or (l+⅛)*.

(2.) The present value of £1 due in any number of years, n∙, or(l+i)~".

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

(4.) The present value of an annuity of £1 for any number 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, with Corbaux, whose tables will be described below— (6.) The annuity which would amount to £1 in n years; or (i + i)∙-ι∙

Smart gives the values of these functions not only for all integral values of n up to 100, but also for the half-years. In this respect, however, he has not been followed by subsequent writers. Some (as Milne) have stated that he “has given the amounts and the present values both of sums and annuities certain, wrong for every odd number of half-years.” This remark requires some explanation. The fact is that Smart’s tables are quite correct, but are not applicable according to the conventions that are usually observed in practice. Thus, for example, he gives the amount of 1 at the end of half a year at 4 per cent, interest as equal to √Γ04 = lO1980390, this being the strictly correct theoretical value at 4 per cent, interest. But in practice the invariable rule is, when interest is calculated for a shorter term than a year, to take a proportionate part of the year’s interest, thus virtually charging a higher rate for the odd time, so that in the above case the amount at the end of half a year will be 1Ό2. This circumstance renders it undesirable to tabulate the amounts and values for half-years; and this is never now done. The course adopted by Smart of giving the values and amounts of annuities for the odd half-years is open to more serious objection. His figures in these cases are merely the values of the algebraical functions above given, corresponding to the values of n, ⅜, 1∣, 2∣, &c., and do not give the amounts and values of annuities payable half-yearly. In fact, we are unable to see that any practical use could ever be made of his results in these cases.

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

It will be seen that the figures in the column numbered (2) are the reciprocals of those in (1), and the figures in column (5) the reciprocals of those in (4). Also, that the figures in (4) are the sums of the first 1, 2, 3, &c., terms of (2). Again, the figures in (3) are derived by the successive addition of those in (1) to the first term, 1 ’ 000000; and the figures in (4) are equal to the product of those in (2) and (3). We have added the column (6) from Cor-baux’s tables. These figures are the reciprocals of those in (3), and are equal to the product of those in (5) and (2), while the figures in (5) are the products of those in (1) and (6).

It would perhaps be more convenient in practice if tables (3) and (6J were altered so as to relate to annuities payable in advance (or annuities-due). In that case (3) would give the amount at compound interest in n years of an annuity-due of 1, and (6) the annuity-due which would, at compound interest, amount to 1 in n years;

(l+i) n + 1 -l-i that is to say, the values of the functions and (1 + ⅞)**+ 1 1 ' ¾ > respectively. One very common application of table (3) is to find the amount of the premiums paid upon a life policy, and these premiums are always payable in advance. If that table were arranged as here suggested, the figures contained in it would be derived from those in (1), in precisely the same way as Table of Amounts, Present Values, &c., at 5 per cent. Interest, i = ∙05. [table]

(4) from (2). It would also be an improvement, for a reason to be mentioned presently, if the heading of the tables were altered, so that, for example, instead of (1) being called a table of the amounts of “£1” at the end of any number of “years,” it were called a table of the amounts of “1” at the end of any number of “terms.” It is not to be understood that the tables are arranged in the, manner here shown. Smart gives, in his First Table of Compound Interest, the values of our (1) for the various rates of interest arranged side by side; in his Second Table he gives the values of our (2) at diiferent rates of interest similarly arranged; and so for (3), (4), and (5). This arrangement has been followed by most authors, not only by those mentioned above as having copied Smart’s tables, but also by Chisholm, who states that the compound interest tables in his work (Commutation Tables, 1858) have been specially computed for it. He gives the tables (1), (2), (3), (4), and (5), at the rates of interest 3, 3⅛, 4, 5, 6 per cent., to any number of years up to 105. Hardy s Doctrine of Simple and Compound Interest, 1839, contains tables (1), (2), (3), (4), for the rates of interest ⅛, ⅜, f, 1, 1∣, 1 i, 1J, 2, 2⅜, 2⅜, 2∣, 3, 3∣, 3⅜, 3∣, 4, 4⅜, 4⅞, 4∣, 5, 6, 7, and 8 percent., for any number of years up to 100. The arrangement in our specimen is that adopted by Rance in his Tables of Compound Interest, 1852. He gives tables (1), (2), (3), and (4), at rates of interest proceeding, by steps of ⅛ per cent., from ∣ to 10 percent., for any number of years from 1 to 100. The same arrangement is also followed in the tables appended to Francis Corbaux’s Doctrine of Compound Interest, 1825; but this author gives in one opening of the book not only the six tabular results in our specimen, but also the corresponding results when interest at the nominal rate of 5 per cent, is payable half-yearly and quarterly, thus giving in one opening 24 tabular results side by side. He gives results for every quarter of a year up to 16 years, and then for each year up to 100; and his rates of interest range from 3 to 6 per cent., at intervals of ∣ per cent. The only other tables we think it needful to mention are Turnbull’s, 1863. His work gives tables (1), (2), (3), (4), and (5), for the Tates of interest 3, 3⅜, 4, 4⅜, 5, and 6, payable yearly; for 3, 3⅛, 3⅜, 3f, 4, 4⅛, 4⅜, 4J, 5, 5⅜, payable half-yearly; and 3, 3⅜, 4, 4⅛, 5, 6, payable quarterly. The results extend in every case to 80 terms of conversion—that is, 80 years, 80 half-years, and 80 quarters respectively. The arrangement of the tables differs from that cf all the others mentioned, as each page contains only the results pertaining to one table at one rate of interest. The results are given not only in decimals, but in £ s. d. in addition, which also is unusual.

A few words may be here added as to the practical method of constructing compound interest tables. The formulas we have found above are not directly used for the calculation of the greater part of the tabular results; but these are in practice deduced the one from the other by continuous processes, the values found by the formulas being used at intervals for the purpose of verification. Smart gives, on page 47 of his work, a description of the method he has employed, and the subject has been fully dealt with by Gray in his Tables and Formulée, chap. 2. Since the publication of that work, the Arithmometer of Μ. Thomas (of Colmar) has come into extensive use for the formation of tables of this kind. For a description of the instrument, and some of its uses, the reader is referred to the papers in the Assurance Magazine by Major-General Hannyngton, vol. xvi. p. 244; Mr W. J. Hancock, xvi. 265; and by Gray, xvii. 249; xviii. 20 and 123.

Hitherto we have considered the annuity payments to be all made annually; and the case where the payments are made 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, we have first to answer the question, What is the value of a sum payable in six months’ time 1 ? and, in order to find the amount of the annuity in n years, we must first determine what is the amount of a sum at the end of six months. The annual rate of interest being i, it may be supposed at first sight that the amount of 1 at the end of six months will be 1 + 5; but if this were the case, the amount at the end of a second period of six months would be increased in the same proportion, and be ^1 + , or

1 +1 + . But this is contrary to our original assump tion that the annual interest is i, and the amount at the end of a year therefore 1 + i. In fact, if we suppose the interest on 1 for half a year to be the interest on it for

. i 3 a year will not be i, but ⅛ + -. In order that the amount at the end of a year may be 1 + i, the amount at the end of six months must be such a quantity as, improved at the same rate for another six months, will be exactly 1 + i; hence the amount at the end of six months must be x ∕Γ+ι, or (1 + i) i . Reasoning in the same way, it is easy to see that, the true annual rate of interest being i, the amount of 1 in any number of years, n, whether integral or fractional, will always be (l + ⅛)*. Hence, by similar reasoning to that pursued above, the present value of 1 payable at the end of any number of years, n, whether integral or fractional, will always be (1 + i')~ n or v λ .

It is now easily seen—we omit the demonstrations for the sake of brevity—that the present value of an annuity payable half-yearly for n years (n being integral) is *∆⅛j +1 · l~( 1 +*)' —. an d that the amount of a similar annuity at the end of n years is .

It is to be observed, however, that when we are dealing with half-yearly payments in practice, the interest is never calculated in the way we have here supposed. On the contrary, the nominal rate of interest being i, the rate paid half-yearly is ∣, so that the true annual rate in praeda tice is i 4- - ; for instance, if interest on a loan is payable half-yearly, at the rate of 5 per cent, per annum, the true rate of interest is '050625, or £5, Is. 3d. per £100. Under these circumstances interest is said to be convertible into principal twice a year. Assuming that interest is thus convertible m times a year, the rate of interest for the τzι α part of a year will be —, and the amount of 1 at the end of n years, that is, at the end of mn intervals of conversion, will be fl + . Assuming the number m now to increase indefinitely, or interest to be convertible momently, the above amount becomes e i *, where ε is the base of the natural (or Napierian) logarithms.

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

It is easily seen from the above formula that the amount of 1 in mn years, at the rate of interest —, is the same as j m that of 1 in n years, at the rate of interest i convertible m times a year; and a similar property holds good of present values. Hence, the tables calculated at the rate of interest may be used to find the amounts and present values at the rate i convertible m times a year; for example, the tables calculated for interest 2 per cent, will give the results for 4 per cent, payable half-yearly. For this reason it would be an improvement, as remarked above, to use the word “terms” in the headings of the tables instead of “years.”

We pass on now to the consideration of the theory of life annuities. This is based upon a knowledge of the rate of mortality among mankind in general, of among the particular class of persons on whose lives the annuities depend. If a simple mathematical law could be discovered which the mortality followed, then a mathematical formula could be given for the value of a life annuity, in the same way as we gave above the formula for the value of an annuity certain. In the early stage of the science, De-moivre propounded the very simple law of mortaEty which bears his name, and which is to the effect, that out of 86 children born alive 1 will die every year until the last dies between the ages of 85 and 86. The mortality, as determined by this law, agreed sufficiently well at the middle ages of life with the mortality deduced from the best observations of his time; but, as observations became more exact, the approximation was found to be not sufficiently close. This was particularly the case when it was desired to obtain the value of joint life, contingent, or other complicated benefits. Demoivre’s law is now, accordingly, entirely a thing of the past, and does not call for any further notice from us. Assuming that law to hold, it is easy to obtain the formula for the value of an annuity, immediate, deferred, or temporary; but such formulas are entirely devoid of practical utility. Those who are curious on the subject may consult the paper by Charion, Ass. Mag., XV. 141. In vol. vi. p. 181, will be found an investigation by Gray of the formula for the value of an annuity when the mortality table is supposed to follow a somewhat more complicated law. No simple formula, however, has yet been discovered that will represent the rate of mortality -with sufficient accuracy; and those which satisfy this condition are too complicated for general use.

The rate of mortality at each age is, therefore, in practice usually determined by a series of figures deduced from observation; and the value of an annuity at any age is found 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 of Deparcieux, given in his Essai sur les Probabilités de la Durée de la Vie Humaine, 1746. [table]

It is to be understood from this table that the mortality among the persons observed was such that out of every 1000 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 of finding the value of a life annuity, it will be convenient, in the first instance, to establish the two following lemmas.

Lemma 1. To find the value of a sum to be received at a future time tn the event of the happening of a given contingency. — Suppose that the sum of 1 is to be received in n years’ time, provided that a certain event shall then happen (or shall have then happened), the probability of which is p. We have seen that the value of 1 to be certainly received in n years’ time is v*. In order to introduce the idea of probability into the problem, suppose that p = so that there are a cases favourable to the happening of the assumed event, and b unfavourable, the total number of possible cases, all of which are equally probable, being (α + δ). We may suppose, for instance, that there are (α + δ) balls in a bag, of which a are white and b black; and that 1 is to be received if a white ball is drawn. In order to determine the value of the chance of receiving 1 in consequence of a white ball being drawn, suppose that (α + δ) persons draw each one ball, and that every one who draws a white ball receives 1; then the total sum to be received is a, and the value of the expectation of all the (α + δ) persons who draw is also a. But it is clear that each of the persons has the same chance of drawing a white ball, therefore the value of the expectation of each of them is -~ =p. This is the value of the chance of receiving 1 immediately before the drawing is made in n years’ time; the value at the present time will therefore be v n p. We may also arrive at this result as follows: The same suppositions being still adhered to, the present value of the sum a to be distributed at the end of n years is αιΛ; and each of the (α + δ) persons having the same chance of receiving 1, the value of the expectation of each . α

IS τv" =pv n . a+b r Lemma 2. To find the present value of 1 to be received in n years' time, if a specified person, whose age is now x, shall be then living.— The sum to be received in this case is called an endowment, and the person on whose life it depends is called the nominee. The probability that the nominee will be alive is to be found, as already intimated, by means of a mortality table. Out of the various tables of this nature that exist, that one must be chosen which, it is believed, most faithfully represents the probabilities of life of the class of persons to which the nominee belongs. Suppose we have reason to believe that Deparcieux’s table, above given, is the most suitable in the case before us,—that the age of the nominee is 30, and the term of years 10. Then, observing that, according to Deparcieux’s table, the number of persons living at the age 30 is 734, while the number at the age 40 is 657, and the difference, or the number who die between the two ages, is 77, we conclude that the chances of any particular nominee of the age of 30 dying before attaining the age of 40 are as 77 to 734, and the chances in favour of his living to the age of 40 are as 657 to 734; or the probability of his living to 40 is .

Passing now from figures to more general symbols, we will use l x to denote the number given in the mortality table as alive at any age x; so that, for example, in the above table, Z 3o = 734, i 40 = 657; and in accordance with what we have just explained, the probability of a nominee of the age x living to the age x + n, will therefore be expressed by -γ ti . Hence, by lemma 1, the value of 1 to *jr be received if the nominee shall be alive at the end of n years, is ∙y¾". In the particular case supposed above, • * ’ x the actual value will be, taking the rate of interest at 3 per cent., × (lO3) -10 = ’ 666035. We may look at the question from another point of view. Suppose that 734 persons of the age of 30 agree to purchase from an insurance company each an endowment of £1, payable at the end of 10 years, then the probabilities of life being supposed to be correctly given by Deparcieux’s table, we see that 657 of those persons will be alive at the end of 10 years, or the engagement of the insurance company to pay £1 to each survivor amounts to the same thing as the engagement to pay £657 at the end of 10 years, and the present value of this sum is 657 (lO3)~ 10 . The sum that should be paid by each of the 734 persons, so that the company shall neither gain nor lose by the transaction, is 6ö 7

therefore — (1 ∙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 be—

S 103 )'"- 0 «· 79 ·

If now we carefully examine the reasoning of the last paragraph, we see that we have made an assumption that must not be allowed to pass without some further justification.. We have assumed, in. fact, that the lives we are dealing with will die off at the exact rate indicated by the mortality table. This, however, we know, is not necessarily the case. Even if the mortality table correctly represents in the long run the rate of mortality among the lives we are dealing with, we know that the rate of mortality will, from accidental circumstances, be sometimes greater and sometimes less than that indicated by the table. If, for example, we have 734 persons under observation all of the age 30, we have no certainty that at the end of 10 years exactly 77 will have died, leaving 657 alive. It is, indeed, within the range of possibility— firstly, that the whole 734 persons may die before the age 40; and, secondly, that none of them may die, or that the whole 734 may attain the age of 40. It appears, therefore, as if we had used the word “probability” in the second lemma in a different sense from that we attached to it in the first; for, in that case we know that if the whole of the (a + b) balls are drawn, a of them will certainly be white, and b black. But the cases will be more parallel if we suppose that each of the balls, after being drawn, is replaced in the bag. If this is done, we see it is no longer certain that when (α + b) drawings take place, a of the balls will be white, and b black. It may, under these altered circumstances, possibly happen that the balls drawn 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 of white balls drawn to the black will differ very little from the ratio of a to b, and will exactly equal it if the number of drawings is supposed to be indefinitely large. In this case we know that the probability of drawing a white ball is still ——-, and passing now to the case of lives under Λ"T" 0

observation, 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 the average of a very large number of observations, that fraction will accurately represent the number of persons surviving. We shall, therefore, be justified in basing all our reasonings on the assumption that the lives we are dealing with die precisely at the rate indicated by the figures of the mortality table.

We* are now in a position to show how the value of a life annuity is calculated. The annual payment of the annuity being 1, which is to be made at the end of each year through which the nominee shall live, the annuity consists of a payment of 1 at the end of one year if the nominee is then alive, of the same payment at the end of two years, at the end of three years, <fcc., under the same condition, and is therefore equal to the sum of a series of endowments. If a; is the age of the nominee, the value of the endowment to be received at the end of the nth year is, as we have seen in lemma 2, v n , and the total value of the annuity is therefore

⅛ + .... (1.)

<X I* lχ

By means of this formula, taking the values of l x , l x+ - l , I λ+v <fcc., from the mortality table, and calculating the values of v, v 2 , v 3 , <fcc., according to the desired rate of interest, or taking their values from the compound interest tables previously described, we can calculate the value of an annuity at any age with any degree of accuracy desired. In practice the calculations would be most readily made by the aid of logarithms.

We can arrive at the above formula more readily by availing ourselves of the supposition which we have seen to be allowable, that the lives under observation will die off exactly at the rate indicated by the mortality table.

Thus, suppose that l js persons of the· age x buy each an annuity of 1. Then the number of persons who will survive to the age x +1, and claim the first payment of the annuity, will be T t+1 . The value of 1 to be paid at the end of a year is V, and therefore the present sum that will be required to provide for all the payments at the end of the first year will be l x j rl υ. The number of persons who will survive two years, so as to claim the second year’s payment, will be ζ, +2 ; and multiplying this into the value of 1 payable at the end of two years, we get Us*, 2 as the present sum necessary to provide for. the payments at the end of the second year. Proceeding in this way, the total sum that will be required to provide the annuities to the l x persons, will be l x+λ υ + l x + i υ 2 + l x + 3 υ i + . . . Hence the value of an annuity on a nominee of the age x, or the sum that will on the average be required to provide for such an annuity, will be Z λ ÷ 1 v + k +2 v 2 + l x + 3 v 3 + i o λ

7 · · · ∖ δ )

lr x which is at once seen to be the same as (1) formula under another shape.

If we suppose money to bear no interest, or make υ = 1 in the formula for the value of an annuity, we shall obtain a quantity which is called the “expectation of life,” or the “average duration of life,” being the average number of years which persons of the given age will one with another live. Denoting this by e x , and making υ = 1 in the formula above given, we get k+1+k+2+k+3÷ · · ·

e *~ I

As in the formula for the annuity, no payment is made on account of the year .in which the nominee dies, this formula gives the average number of complete years that persons of the given age will live according to the mortality table, and makes no allowance for the portion of the year in which death occurs. The expectation thus found is called the curtate expectation; and in order to obtain the complete expectation of life, which is denoted by e κ , half a year must be added to it.

The first writer who is known to have attempted to obtain, on correct mathematical principles, the value of a life annuity, was Johan De Wit, Grand Pensionary of Holland and West Friesland. All our exact knowledge of his writings on the subject is derived from two papers contributed by Mr Frederick Hendriks to the Assurance Magazine, vol. ii. p. 222, and vol. iii. p. 93. The former of these contains a translation of De Wit’s report upon the value of life annuities, which was prepared in consequence of the resolution passed by the States General, on the 25th April 1671, to negotiate funds by life annuities, and which was distributed to the members· on the 30th July 1671. The latter contains the translation of a number of letters addressed by De Wit to Burgomaster Johan Ξudde, bearing dates from September 1670 to October 1671. The existence of De Wit’s report was well known among his contemporaries, and Mr Hendriks has collected a number of extracts from various authors referring to it; but the report is not contained in any collection of his works extant, and had been entirely lost for 180 years, until Mr Hendriks conceived the happy idea of searching for it among the state archives of Holland, when it was found together with the letters to Hudde. It is a document of extreme interest, and (notwithstanding some inaccuracies in the reasoning) of very great merit, more especially considering that it was the very first document on the subject that was ever written; and Mr Hendriks’s papers will well repay a careful perusal.

It appears that it had long been the practice in Holland for life annuities to be granted to nominees of any age, in the constant proportion of double the rate of interest allowed on stock; that is to say, if the towns were borrowing money at 6 per cent., they would be willing to grant a life annuity at 12 per cent.; if at 5 per cent., the annuity granted was 10 per cent.; and so on. De Wit states that “annuities have been sold, even in the present century, first at six years’ purchase, then at seven and eight; and that the majority of all life annuities now current at the country’s expense were obtained at nine years’ purchase;” but that the price had been increased in the course of a few years from eleven years’ purchase to twelve, and from twelve to fourteen. He also states that the rate of interest had been successively reduced from 6∣ to 5 per cent., and then to 4 per cent. The principal object of his report is to prove that, taking interest at 4 per cent., a life annuity was worth at least sixteen years’ purchase; and, in fact, that an annuitant purchasing an annuity for the life of a young and healthy nominee at sixteen years’ purchase, made an excellent bargain. It may be mentioned that he argues that it is more to the advantage, both of the country and of the private investor, that the public loans should be raised by way of grant of life annuities rather than perpetual annuities. It appears conclusively from De Wit’s correspondence with Hudde, that the rate of mortality assumed as the basis of his calculations was deduced from careful examination of the mortality that had actually prevailed among the nominees on whose lives annuities had been granted in former years. De Wit appears to have come to the conclusion that the probability of death is the same in any half-year from the age of 3 to 53 inclusive; that in the next ten years, from 53 to 63, the probability is greater in the ratio of 3 to 2; that in the next ten years, from 63 to 73, it is greater in the ratio of 2 to 1; and in the next seven years, from 73 to 80, it is . greater in the ratio of 3 to 1; and he places thé limit of human life at 80. If a mortality table of the usual form is deduced from these suppositions, out of 212 persons alive at the age of 3, 2 will die every year up to 53, 3 in each of the ten years from 53 to 63, 4 in each of the next ten years from 63 to 73, and 6 in each of the next seven years from 73 to 80, when all will be dead. This is the oonclusion we have drawn from a careful study of the report; but, in consequence of the inaccuracies above mentioned, some doubt exists as to De Wit’s real meaning; and Mr Hendriks’s conclusion is somewhat different from ours (see his note, Ass. Mag. vol. ii. p. 246). The method of calculation employed by De Wit differs much from that described above, and a short account of it may interest the reader. Suppose that it were desired to apply it to deduce the value of an annuity according to Deparcieux’s mortality table given above, then we assume that annuities are bought on the lives of 1000 nominees each 3 years of age. Of these nominees, 30 will die before attaining the age of 4, and no annuity payment will be made in respect of them; 22 will die between the ages of 4 and 5, so that the holders of the annuities on their lives will receive payment for 1 year; 18 attain the age of 5 and die before 6, so that the annuities on their lives are payable for 2 years. Reasoning in<the same way, we see -that the annuities on 15 of the nominees will be payable for 3 years; on 13, for 4 years; on 12, for 5 years; on 10, for 6 years; and so on. Proceeding thus to the extremity of the table, 2 nominees attain the age of 93, 1 of whom dies before the age of 94, so that 90 annuity payments will be made in respect of him; and the last survivor dies between the ages of 94 and 95, so that the annuity on his life will be payable for 91 years. Having previously calculated a table of the values of annuities certain for every number of years up to 91, the value of all the annuities on the 1000 nominees will be found by taking twenty-two times the value of an annuity for 1 year, eighteen times the value of an annuity for 2 years, fifteen times the value of an annuity for 3 years, and so on,—the last term being the value of 1 annuity for 91 years,—and adding them together; and the value of an annuity on one of the nominees will then be found by dividing by 1000. Before leaving the subject of De Wit, we may mention that we find in the correspondence a distinct suggestion of the law of mortality that bears the name of Demoivre. In De Wit’s letter, dated 27th October 1671 (Ass. Mag., vol. iii. p. 107), he speaks of a “provisional hypothesis” suggested by Hudde, that out of 80 young lives (who, from the context, may be taken as of the age 6) about 1 dies annually. In strictness, therefore, the law in question might be more correctly termed Hudde’s than Demoivre’s.

De Wit’s report being thus of the nature of an unpublished state paper, although it contributed to its author’s reputation, did not contribute to advance the exact knowledge of the subject; and the author to whom the credit must be given of first showing how to calculate the value of an annuity on correct principles is Dr Edmund Halley, F.R.S. In the Philosophical Transactions, Nos. 196 and 198 (January and March 1693), he gave the first approximately correct mortality table (deduced from the records of the numbers of deaths and baptisms in the city of Breslau), and showed how it might be employed to calculate the value 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 requires a series of laborious calculations, he says that he had sought in vain for a more concise method. His papers, which are full of interest, are reprinted in the eighteenth volume of the Assurance Magazine.

Previous to Halley’s time, and apparently for many years subsequently, all dealings with life annuities were based upon mere conjectural estimates. The earliest known 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, and which declared that a testator should not give more than three-fourths of his property in legacies, so that at least one-fourth must go to his legal representatives. It is easy to see how it would occasionally become necessary, while this law was in force, to value life annuities charged upon a testator’s estate. Aemilius Macer (230 a.d.) states that the method which had been in common use at that time was as follows :—From the earliest age until 30 take 30 years’ purchase, and for each age after 30 deduct 1 year. It is obvious that no consideration of compound interest can have entered into this estimate; and it is easy to see that it is equivalent to assuming that all persons who attain the age of 30 will certainly live to the age of 60, and then certainly die. Compared with this estimate, that which was propounded by the Praetorian Prefect Ulpian— one of the most eminent commentators on the Justinian Code—was a great improvement. His table is as follows :— [table]

Here also we have no reason to suppose that the element of interest was taken into consideration; and the assumption, that between the ages of 40 and 50 each addition of a year to the nominee’s age diminishes the value of the annuity by one year’s purchase, is equivalent to assuming that there is no probability of the nominee dying between the ages of 40 and 50. Considered, however, simply as a table of the average duration of life, the values are fairly accurate. At all events, no more correct estimate appears to have been arrived at until the close of the 17th century. Fuller information upon the early history of life annuities will be found in the article “Annuities on Lives, History of,” in Mr Walford’s Insurance Cyclopaedia.

Demoivre, in his Treatise on Annuities, 1725, showed that it was unnecessary to go through the whole of the calculation indicated by the formula (1) or (2) for each age, and that the value of an annuity at any age might be deduced by a simple process from that at the next older age. This may be demonstrated as follows :—If it were certain that a person of any age, say 39, would live for a year, then the value of an annuity on his life would be such a sum as would increase at interest in a year to the value of an annuity on a life one year older, say 40, increased by a present payment of 1; that is, putting a for the value of an annuity and 1 α for that on a life one year older, the value would be v (1 + 1 α). But it is uncertain that the life will exist to the end of a year, and the value of the annuity must therefore be reduced in the proportion of this uncertainty, or be multiplied by the probability that the given life will survive a year. Putting then p to denote this probability, we have a = υp (I + l α). This formula may also be demonstrated algebraically. We have seen that h fl fl .’I a = v ~J rv 'Z-J r ' t P- + where z is the difference between the age of the given life and that of the oldest in the mortality table. (Assuming the present age to be 39, then in the English Table No. 3, Males, z will beI07-39 = 68.) In the same way, we have . 2 l fl 1 ∕Z

⅛=≈τ i +≈⅞+*⅞+ + ’·T .

ττ 1 i∕ f - l 2 i , 1 'A

Hence a = v A 1 + v-^ + + + υ t ~ 1 i ∣ J

= ⅛l+⅛)

= ty>(l + 1 α)

—the same result as already proved.

If we suppose the present age to be x, we may put the formula in the shape β .= J⅛ ’ (l+¾+l)i but it will be found preferable to omit the subscript x whenever 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 is applicable to annuities on the joint duration of two or more lives, the first discovery of it is undoubtedly due to Demoivre. (See Farren’s Historical Essay on the Use and early Progress of the Doctrine of Life Contingencies in England, p. 46.) The formula appears to have been independently discovered by Euler, and was given by him in a paper in the Memoirs of the Royal Academy of Sciences at Berlin, for the year 1760.

Mr Peter Gray has shown in his Tables and Formulae, 1849, how Gauss’s logarithmic table may be advantageously employed in calculating the values of annuities by the above formula. That table gives us the value "of log (1 +α) when that of log a is known. In other words, the argument of the table is log a, and the tabular result is log(l + α). When ordinary logarithmic tables are used, the formulas being log a a = log vp a + log (1 + ¾, +1 ), log α,-ι = log ‰ι + log (1 + α,); we have to find a a by means of an inverse entry into the table before log (1 + α,) can be found; but when Gauss’s table is used (as recomputed and extended by Gray), “All the entries of the same kind—direct and inverse—are brought together, the whole of the logarithms being found before a single natural number is taken out. We consequently proceed right through the table; and as we proceed, we find two, three, four, and even as many as six and 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, when the common tables are used, direct and inverse entries alternate with each other, and involve likewise a continual turning of the leaves backwards and forwards, by which the process is rendered exceedingly irksome.”—Page 165, second issue, 1870.

When the only object is to form a complete table of immediate annuities, the above is the simplest and most expeditious mode of procedure; but when it is desired to have the means of obtaining readily the values of deferred and temporary annuities, it is better to employ a wholly different 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 the age of x + n, the value of the annuity on his life being then a x+n , its present value would be v*a x+κ . But as the nominee may die before attaining the age of x + n, the above value must be multiplied by the probability of his living to that age, which is , and we thus get the present value of the deferred annuity, v n . -y- 5 . α x+n . We may arrive at this result otherwise. Thus, as we have seen above, the present value of the first payment of the annuity, that is, of 1 to be received if the nominee shall be alive at the end of ∙+ 1 i n +1 years, is -γ-√* +l . The present value of the next payment is similarly seen to be — v n + 2 , and so on. The value <of the deferred annuity is therefore

«+1Z b +S7 w+37

— ^+ι + __%-+î + ^+3 + " z n∕" +li ,"^.2,^3, 1

-r I v≈' + v , ' + v" + ∣

≈∕ t, ^ ^ a ' (or ⅛^ α '+∙)

(We may here mention that this formula holds good, not only for ordinary annuities, but also for annuities payable half-yearly or quarterly, and for continuous annuities; also for complete annuities.)

A temporary annuity is, as explained above, an annuity which is to continue for a term of years provided the· nominee shall so long live. Hence it is clear that if the value of a temporary annuity for n years is added to that of an annuity on the same life deferred n years, this sum must be equal to an annuity for the whole Continuance of the same life; the value of a temporary annuity for n years will therefore be equal to the difference between the value of a whole term annuity and that of an annuity deferred n years, or to a - ~υ n .. n a, ^or a x - y h -^ .

We are now in a position to explain the method of calculating the value of annuities above referred to. We have seen that the value of an annuity for the life of a nominee whose age is x, is k÷ιfl+⅛H-∑v 2 + .... +lχ- ∖ - t v g lχ ’ which, multiplying both numerator and denominator by the same quantity v“, becomes k + ι¾∙* +1 ÷k +a ι^+ i8 + .... -⅛-ζ, + ,¾∙+*

l x v*

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

⅛±→>+∙ + ⅛⅛+V+∙ + .... + ⅛,.

_ ^a+H+lf w ^*^ 1 + ∕ a - ∣ -n4-2'ι'* t + 2 + .... +Z¾4, j ,V*

l x = ⅛ + , +1 ^+"+ 1 ÷? g+ » +2 ^+”+ a + + ⅛¾*÷ ’ e

If, then, we calculate in the first instance the values of the product l x υ x for all values of x, and then form their sums, beginning at the highest age, we shall have the means of obtaining by a single division the value of any immediate or deferred annuity we wish.

It is convenient to arrange these results in a tabular form, as shown in the appended tables (3) and (4). The quantity l x υ* is placed in the column headed D, opposite the age x, and is denoted by D λ j while the sum ζ, 4 √w* +1 + ζ. +ΐ ·υ' +2 + .... + l x+ ,v" +, is placed in the column headed N, opposite tfie same age x, and is denoted by Ν Λ ; so that the value of an immediate annuity on a N

life x is equal to ~; the letters N and D being chosen as ∙L×χ ~^ . _

the first letters of the words Numerator and Denominator. Then it is easy to see that the value of an annuity on x deferred n years is equal to ⅛± n ; whence by subtraction the value of a temporary annuity for n years on the same life is ^ ~ n ≈ + ".

•L'a:

If, for example, we wish to find the value of an annuity on 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, and 1ST

by division we get the value of = 16 ’ 4744, which agrees -L , 4o with the value contained in the table (5), also appended to this article.

Next, suppose we wish to find the value of a deferred annuity on a life of 30 to commence at the end of 10 years. From what precedes, we see that the value of this . 11 , ι x xκ x∙ x n 40 1374058

annuity will be equal to the quotient or y25464^ ’ which will be found to be equal to 10 ’ 9518.

If we wish to find the value of this deferred annuity without using the D and N table, the formula for it will be ~ √ 0 α 4o , y being equal to 777-. But we have seen t 3 o ΓUo above that the value of ⅛ (1O3)~ 1 ° = *664779, and that α 40 = 16*4744; and multiplying these together, we get the value of the deferred annuity, 10 ’ 9518, as before.

We have, in conformity with popular usage, called our auxiliary table a “D and N table.” It is also called a “commutation table,”—a name proposed by De Morgan in hiä paper “On the Calculation of Single Life Contingencies," which appeared in the Companion to the Almanac for the year 1840, and which is reprinted in the Assurance Magazine, xii. 328. His explanation of the term is to the following effect :—Taking any two ages, say 30 and 40, we have, according to the English Table No. 3, Males-— see appended table (3),—

D 30 = 125464, N 30 = 2385610;

D 40 = 83406, N 4o = 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 age of the party, and show that the most simple indication of the tablée is the proportion in which a benefit due at one age ought to be changed, so as to retain the same value and be due at another age. They might, therefore, with great propriety, be called Commutation Tables.”

It is clear that this property will not be altered if all the quantities 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 value of annuities, immediate, deferred, and temporary, but also to 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 w—Γ⅛Γ · The table may also be used to find the single and annual premiums for insurances, immediate, deferred, or temporary. The single premiums are—

∙υN _ i — N

1. For an ordinary insurance, — -; -L'j?

2. For an insurance deferred n years, ~~*~~jj —;

3. For a temporary insurance for n years,

'V(N x _ 1 — N 1 -∣-n - l)-N x +N j ..f.n

⅜ *

The annual premiums payable during life for the same benefits are found by substituting N 1-1 for D x in the denominator; and the annual premiums payable for m years, by putting N x . 1 - N x+m . 1 in the denominator instead of D 1 ∙

Before quitting this subject, we should mention that in practice other columns are added to the table besides the D and N columns. A column, S, is given for the purpose of calculating the values of increasing annuities; a column, M, for calculating the values of assurances; and a column, R, for calculating the values of increasing assurances. An explanation of the M column belongs to the subject Insurance; for an account of the S and R columns, w e refer the reader to the works and papers on life insurance contingencies, in which the D and N (or commutation) method is described; particularly to those of David Jones, Gray, 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 Zχ~ w instead of l x v t . He says, “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, another table based on different mortality observations, commencing at the age of 40; and in this case he tabulates Z x v x-40 . His table also differs from the common form in that it is adapted to find the values of annuities payable by half-yearly instalments.

The next work in which a commutation table is found is William Morgan’s Treatise on Assurances, 1779. In this work the values of l fv t ~ 1 are tabulated, and not n those of l x v t ; but, as above mentioned, the properties of the table are not altered by the change. Morgan gives the table as furnishing a convenient means of checking the correctness of the values of annuities found by the ordinary process. It may be assumed that he was aware that the table 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 the table was John Nicholas Tetens, a native of Schleswig, who in 1785, while professor of philosophy and mathematics at Kiel, published in the German language an Introduction to the Calculation of Life Annuities and Assurances. This work appears to have been quite unknown in England until Mr F. Hendriks gave, in the first number of the Assurance Magazine, pp. 1-20 (Sept. 1850), an account of it, with a translation of the passages describing the construction and use of the commutation table, and a sketch of the author’s life and writings, to which we refer the reader who desires fuller information.

The use of the commutation table was independently developed in England—apparently between the years 1788 and 1811—by George - Barrett, of Pet worth, Sussex, who was the son of a yeoman farmer, and -was himself a village schoolmaster, and afterwards farm steward or bailiff. .In the form of table employed by him, the quantity tabulated is not l x v e , but Z x (l +i) w ~ x , where v) is the last age in the mortality table used. It has been usual to consider Barrett as the originator in this country of the method of calculating the values of annuities by means of a commutation table, and this method is accordingly sometimes called Barrett’s method. (It is also called the commutation method and the columnar method.) Barrett’s method of calculating annuities was explained by him to Francis Baily in the year 1811, and was first made known to the world in a paper written by the latter and read before the Royal Society in 1812.

By what has been universally considered an unfortunate error of judgment, this paper was not recommended by tho council of the Royal Society to be printed, but it was given by Baily as an appendix to the second issue (in 1813) of his work on life annuities and assurances. Barrett had calculated extensive tables, and with Baily’s aid attempted to get them published by subscription, but without success; and the only printed tables calculated according to his manner, besides the specimen tables given by Baily, are the tables contained in Babbage’s Comparative View of the various Institutions for the Assurance of Lives, 1826. It may be mentioned here that Tetens also gave only a specimen table, apparently not imagining that persons using his work would find it extremely useful to have a series of commutation tables, calculated and printed ready for use.

In the year 1825 Griffith Davies published his Tables of Life Contingencies, a work which contains, among other tables, two arrauged on the plan we have above explained, the idea of them having been confessedly derived from Baily’s explanation of Barrett's tables. The method was, however, improved and extended by the addition of the columns (M and R) for finding the values of assurances. Davies’s treatise on annuities, as issued by his executors in 1855, with the explanation that it is an uncompleted work, but that the completed portion had been in print since 1825, contains several other tables of the same kind. In the preface to this work it is stated that “the most important distinction between the two methods is, that Mr Davies’s method is much simpler in principle than that of Mr Barrett, as the columnar numbers given by the latter must be considered more as the numerical results of algebraical expressions; whereas in Davies’s arrangement it will be found, on reference to age 0, that the number in column D represents the number of children just born, and those opposite ages 1, 2, 3, 4, &c., to the end of life, the present sums which would be required for the payment of £1 to each survivor of such children at the end of 1, 2, 3, 4, &c., years to the extremity of life; and the sum thereof inserted in column N, opposite age 0, represents the present fund required to provide the payment of annuities of £1 each for life to all the children given in column D at age 0; and from this method very considerable amount of labour is avoided by multiplying the number living at each age by a fraction less than a unit; but by Barrett’s method, the number living at each age has to be multiplied by the amount of £1 improved for as many years as are equal to the difference between that age and the greatest tabular duration, as already stated, which makes each product a large multiple of the number living.” This passage, we are informed, correctly represents Mr Davies’s own views on the subject. It may be noticed that Davies does not employ the notation used above, D x , N t , <fcc., but omits the subscript x. Thus, instead of the formula v? ⅛r he would write

N,- i -N x+w - l

•N

N i -"N i *

In some respects this notation is perhaps preferable to that now used, as it is certainly better, when there is no risk of confusion, to omit the subscript x. But Davies’s notation cannot be adopted without alteration, as N 1 might be mistaken for the number in the column N opposite the age 1. We may, however, consistently with the principles of the notation adopted by the Institute of •N

Actuaries, write the formula —==τ<φ. The notation at l NN

present commonly used is due to David J ones, whose work (mentioned below) was the first that contained an extensive series of commutation tables.

On a general review of the whole evidence, we cannot help thinking that Barrett’s merits in the matter have been somewhat exaggerated. The first idea of a commutation table was not due to him, but (leaving Tetens out of view) to Dale and Morgan; and it is certain that he was familiar with the latter’s treatise. The change he introduced into the arrangement of the table, namely, multiplying by a power of (1 + ι) instead of by a power of v, is the reverse of an improvement; and accordingly, his form of table has never been in practical use by any person but himself, excepting only Babbage. It is, of course, not to be denied that great credit is due to him as a self-educated man, for perceiving more clearly than his predecessors the great usefulness of the commutation table; but in our opinion he does not stand sufficiently out from those who preceded and followed him, to justify the attempt to attach his name to the columnar method of calculating the values of annuities and assurances. Those who desire to .obtain further information on the matter, and to learn the views of other writers, can refer to the appendix to Baily’s Life Annuities and Assurances, De Morgan’s paper “On the Calculation of Single Life Contingencies,” Assurance Maga zine, 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, and No. 2, p. 12; and in particular De Morgan’s “Account of a Correspondence between Mr George Barrett and Mr Francis Baily,” in the Assurance Magazine, vol. iv. p. 185.

The principal D and N tables published in this country are contained in the following works :—

David Jones, Value of Annuities and Reversionary Payments, issued in parts by the Useful Knowledge Society, completed in 1843, which gives for the Northampton Table, 3 per cent, interest, columns D, N, S, M, R; Carlisle Table, interest 3, 3⅜, 4, 4⅜, 5, 6, columns D, N, S, M, R; and interest 7, 8, 9, 10, columns D, N, S. Volume ii. contains D and N tables for all combinations of two joint lives, according to the Northampton Table, 3 per cent., and the Carlisle, 3, 3⅜, 4, 4⅜, 5, 6 per cent.

Jenkin Jones, New Rate of Mortality, 1843, Seventeen Offices’ Experience, 2⅜, 3, 3⅜ per cent., columns D, N, S, M, R.

G. Davies, Treatise on Annuities, 1825 (issued 1855). Equitable Experience, 2⅜,3, 3⅜, 4, 4⅛, 5, 6 per cent., columns D, N, S, M,’R; 7, 8 per cent., columns D, N; also three tables relating to joint lives for the differences of age 19, 20, 21 years, and one relating to three joint lives of equal ages, all giving D and N columns at 3 per cent, interest: Northampton, 3 per cent., columns D,N, S,M, R; 4 per cent., columns D, N; also tables for two joint lives similar to those above mentioned.

David Chisholm,. Commutation Tables, 1858; Carlisle, 3, 3∣, 4, 5, 6 per cent., columns D, N, S, C, M, R; also columns D, N, for joint lives, and M, R, for survivorship assurances.

Neison’s Contributions to Vital Statistics, 1857. Mortality of England and Wales (males), 3, 3⅛, 4, 4⅛, 5, 6, 7, 8, 9, 10 per cent., columns D and N, with logarithms; two joint lives, males, 7 per cent., columns D and N; also D and N columns relating to the mortality of master mariners, and to that among friendly societies, and in particular the Manchester Unity.

Jardine Henry, Government Life Annuity Commutation Tables, 1866 and 1873, single lives male and female, 0, 1, 2, 2⅛, 3, 3⅜, 4, 4⅛, 5, 5⅛, 6, 7, 8, 9, 10 per cent.

Institute of Actuaries Life Tables, 1872. New Experience,(or Twenty Offices), males and females separately, H m and H f , 3, 3 J, 4, 4⅜, 5, 6, columns D, N, S, M, R, also logarithms; and similar tables at 3, 3⅜, and 4 per cent, for the table H m < 5 >.

R. P. Hardy, Valuation Tables, 1873, gives the same table at 4⅛ per cent, for H m < 6 >.

The Sixth Report of the Registrar-General, 1844, contains the English Table (No. 1), 3 and 4 per cent., columns D, N, S, C, M, R, for males and females separately; also D, y , N cy , 3 and 4 per cent., for x male and y female; also five tables for joint lives, one male and one female, differences of ages — 20, -10, 0, 10, 20.

The Twelfth Report, 1849, contains the English Table (No. 2), males, 3, 4, 5, per cent., columns D, N, S, C, M, R.

The Twentieth Report, 1857, contains the English Table (No. 2), females, 3 per cent., columns D, N, S, C, M, R.

The English Life Table, 1864, contains columns D, N, at 3, 3∣, 3⅛, 3⅜, 3⅞, 4, 5, 6, 7, 8, 9, 10 per cent., also the logarithms; columns S, C, M, R, at 3, 4, 5 per cent.; special tables for half-yearly and quarterly payments, columns D, N, at 3 per cent.; joint lives at all combinations of ages, columns D, N, at 3 per cent., (1) male and female, (2) two males, (3) two females.

The explanations of the tables in the last four works are by Dr William Farr, F.R.S.

Very unfortunately, these tables are not all arranged upon the same principle, but those contained in the Reports of the Registrar-General, in the English Life Table, in Chisholm’s and in Henry’s tables, are so arranged that the col umn N is shifted down one year, so that in them the ratio N

— i gives, not the value of the ordinary annuity, but the •A-'X

value of the annuity increased by unity, or the annuity-due. It is very needful to bear this in mind for the prevention of error; and the existence of a difference of this kind is extremely perplexing. For information upon the subject of this confusing change, see De Morgan’s paper “On the Forms under which Barrett’s Method is represented, and on Changes of words and symbols,” Ass. Mag., x. 302.

All the preceding methods require a considerable amount of calculation in order to obtain the value of an annuity on a life of any particular age. We will now explain some methods of approximation, by means of which we can calculate with much less labour the value of an annuity at a single age, when we do not require a complete table of annuities. The following method was demonstrated by Mr Lubbock (afterwards Sir J. W. Lubbock) in a paper “On the Comparison of Various Tables of Annuities” in the Cambridge Philosophical Transactions for the year 1829. Instead of calculating the value of each payment of the annuity to be received at the ages x + 1, x + 2, to the extremity of life, it will be sufficient to calculate the values of the payments to be received at a series of equidistant ages, say at the ages x + n, x + 2n, x + 3n, Then, if V m denote the payment to be received at the age x + m, and ∆ 1 , Δ 2 , ∆ t , denote the leading differences of V o , V,, V 2 ,, V y , the value of the annuity is approximately n(V 0 +V. + V,+V s .+ )-2±1 + ^- 1 δ , n 2 -l. (n 2 -1) (19n 2 -1) (n 2 -1) (9n 2 -1)

^ 24n δ * + 720n 3 3 480n 3

»7 2λ∕

Here V o = 1, V. = -fr, V 2b = j% 2 ", &c.

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

First, taking n=7, we find.

V o = 1'0000

- ∙2654

V 7 = 7346 + ∙0521

•2133 - ∙0115

V 14 = ∙5213 ∙0406 + ∙0029

•1727 ∙0086

V 21 = ∙3486 ∙0320

•1407

V s8 = ∙2079

V 35 = ∙0990

V 4a = ∙0318

V 4β = ∙0055

V 66 = ∙0004

Sum=2∙9491

Hence ∆ 1 = - 2654, Δ a ='0521, Δ 3 =- ∙0115, Δ 4 - ’ 0029; and the value of the annuity is approximately

=7 + 2∙9491 - 4-y × ∙2654-y× ∙0521- ∙1808× *0115

- ∙1283 × ∙0029. -20∙6437-4∙0000

- ∙1517

- ∙0149

- ∙0021 - ∙0004

= 20∙6437-4Ι691 = 16'4746.

Next, taking n=ll, we have

V o =ι∙oooo

- ∙3924

V 11 = ∙6076 +∙1115

•2809 - ∙0310

V 22 = ∙3267 ∙0805 +‘0453

•2004 + ∙0143

V 33 = ∙1263 ∙0948

•1056

V 44 = ∙0207

V w = ∙0006

2∙0819

Hence, the value of the annuity is approximately

10 5

=ll×2∙0819-6-jj× ∙3924- i j× ∙∏15 - ∙2878 × ∙0310

- ∙2044 × ∙0453 =22 *9009-6 *0000

- ∙3567

- ∙0507

- ∙0089

- ∙0093

= 22∙9009-6∙4256

= 16∙4753.

The value of the annuity calculated in the ordinary way is, as we have seen (page 80), 16 ∙4744.

An improved form of this method was given by Mr W. S. B. Woolhouse in the Ass. Mag., xi. 321. In order to explain this, we must introduce the reader to a term which is of recent origin, but which the application of improved mathematical methods to the science of life contingencies has rendered of great importance— the force of mortality at a given age. This may be defined as the proportion of the persons of that age who would die in the course of a year, if the intensity of the mortality remained constant for a year, and the number of persons under observation also remained constant, the places of those who die being constantly replaced by fresh lives. More briefly, it is the instantaneous rate of mortality. A very full explanation of this term is given by Mr W. Μ. Makeham, in his paper “On the Law of Mortality,”Ass. Mag., xiii. 325. The value of the function can be approximately found by dividing the number of persons who die in a year by the number alive in the middle of the year. Thus, if l x denote the number of persons living at the age x, d x the number dying between the ages x and x+ 1, and d x ~ 1 the number dying between the ages x - 1 and x, then the number dying between the ages a;-- and a; + - will be approximately and the force of mortality is approximately

Thus, in the English Table, No. 3, Males, the i r xi. i e ιu χ x<n · 3465 + 3529

value of the force of mortality at age 40 is - r - j -, √

= 012853. 544146

This quantity is usually denoted by the Greek letter μ, while δ is used to denote the quantity log e (l + i), which Woolhouse has called the force of discount. This being premised, Woolhouse’s formula for the approximate value of an annuity is n(V.+V 1 . + V s .+ . . . + where it will be noticed that, since V o = 1, the two first terms are exactly equal in value to those in Lubbock’s formula.

Taking the same example as above, we have seen that ∕⅞ 0 =O12853 also δ = Ό29558

μ i0 + δ =∙042411

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

= 16∙6437-4× Ό42411

= 16*6437-Ί69644

= 16∙4741.

Making »=11, we have the value

= 16*9009 -10 × Ό42411

= 16 ’ 9009 - ’ 4241

= 16*4768.

Comparing the two processes, we see that when we have the valuee of μ and δ already computed, Woolhouse’s is decidedly the shorter. On the other hand, it is easy to see that Lubbock’s formula applies, not only to annuities, but to other benefits; and that it will be applicable to find the values of such quantities as contingent annuities, the values of which cannot be found exactly except by a very long series of calculations. (See Davies, p. 354.) The reader who refers to Lubbock’s paper (which is reprinted in the Ass. Mag., n. 277), or to the short account of it given in the Treatise on Probability, issued by the Useful Knowledge Society, and often bound up with D. Jones’s work on annuities, will see that the terms involving Δ 2 , Δ 3 , Δ 4 are not given there; and it may assist the student who is desirous of working out the formula fully, to be referred to De Morgan’s expansion of . ∖ i —τ, Diff. Calc., p. 314, § 184. Lubbock not only considered it unnecessary to calculate the terms involving Δ 2 , Δ 3 , &c., but thought that the value of the term containing ∆ 1 , as calculated for one mortality table, might be used without material error in finding the values of annuities by other tables. The above examples show that the formula, as now completed, is capable of giving the values of annuities (and of course of other quantities) with very great accuracy.

So long as we consider the annuity to be payable yearly, no allowance being made for the time which elapses between the death of the nominee and the last previous payment of the annuity, it is, as we have seen, a very simple problem to calculate its value. But in practice annuities are generally payable by half-yearly instalments, and it is the custom to pay a proportionate part of the annuity for the odd time that elapses between the last half-yearly payment and the death of the nominee; and the value found by the methods described above therefore require to be corrected before they are strictly applicable in practice. Approximate values of the necessary corrections are very easily found; but the strict investigation of their correct values is a problem requiring a considerable knowledge of the higher mathematics, and it would be quite beyond our present purpose to consider it.

When an annuity is payable half-yearly, the’common rule for finding its value is to add ∙25, or a quarter of a year’s purchase, to the value of the annuity payable yearly. When it is payable quarterly, *375 is added; and when by instalments at n equal periods throughout the year (or by 1 71—1

--thly instalments), the addition is · The values thus found are sufficiently correct for most purposes. More correct methods of finding the values of annuities payable half-yearly, quarterly, &c., are investigated in papers in the Assurance Magazine, by Woolhouse, xi. 327, and by Sprague, xiii. 188, 201, 305. Some authors have assumed that when an annuity is payable half-yearly, interest is also convertible half-yearly, overlooking the circumstance that the true rate of interest is thereby changed, as we have explained in the earlier part of this article. In fact, as we showed, 5 per cent, interest convertible half-yearly is equivalent to a true rate of interest, £5, Is. 3d. per cent. If, then, we have found the value of an annuity when payable yearly at 5 per cent, interest, and require, perhaps, in the course of the same investigation, the value of an annuity payable half-yearly, it is clear that that value should 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 also be calculated at that rate.

The approximate value of an annuity payable up to the day of the nominee’s death, or of a “complete” annuity, as it is now usually called, is found in the case of annuities payable yearly by adding to the value of the ordinary annuity the valiιe of ⅜, payable at the instant of the nominee’s death; in the case of half-yearly annuities, by adding the value of ∣; and in the case of quarterly annuities, the value of ⅜, similarly payable. The more exact value of the correction is investigated by Sprague, Ass. Mag., xiii. 358.

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. But an annuity may depend on the continuance of two or three or more lives. It may continue βo long as both of two nominees are alive, in which case it is called an annuity on the joint lives; or it may continue as long as either of them is alive, in which case it is called an annuity on the last survivor. Again, if it depends on the existence of three nominees, it may .either continue so long only as they are all three alive, when it is called an annuity on the joint lives; or so long as any two of them continue alive, when it is called an annuity on the last two survivors; or so long as any one of them is alive, when it is called an annuity on the last survivor. In addition to these, we have “reversionary” annuities, which are to commence on the failure of an assigned life, and continue payable for the life of a specified nominee; or, more generally, to commence on the failure of a given status, or combination of lives, and continue payable during the existence of another status. There are also “contingent” annuities, which depend on the order in which the lives involved fail. Thus, we may have an annuity on the life of X, to commence on the death of y, provided that take place during the life of z, and not otherwise, and to continue payable during the remainder of the life of x. Reversionary annuities are of considerable practical ini portance, 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 of advowsons, and of fines on copy hold property. It does not fall within the scope of this article to treat at any length of annuities on more than one life, and we must refer the reader who wishes for further information with, regard to them to the works of Baily, Davies, and David Jones, already mentioned, and Milne’s Treatise on the Valuation of Annuities and Assurances, 1815.

The student who wishes to pursue the subject more thoroughly, and to become acquainted with all the improvements in the theory of annuities that have been introduced of late years, should carefully study the various articles contributed to the Journal of the Institute of Actuaries, particularly those of Woolhouse and Makeham. The Institute was founded in the year 1848, the first sessional meeting being held in January 1849. Its establishment has contributed in various ways to promote the study of the theory of life contingencies. Among these may be specified the following:— Before it was formed, students of the subject worked for the most part alone, and without any concert; and when any person had made an improvement in the theory, it had little chance of becom ing publicly known unless he wrote a formal treatise on the whole subject. But the formation of the Institute led to much greater interchange of opinion among actuaries, and afforded them a ready means of making known to their professional associates any improvements, real or supposed, that they thought they had made. Again, the discussions which follow the reading of papers before the Institute have often served, first, to bring out into bold relief differences of opinion that were previously unsuspected, and afterwards to soften down those differences,—to correct extreme opinions in every direction, and to bring about a greater agreement of opinion on many important subjects. In no way, probably, have the objects of the Institute been so effectually advanced as by the publication of its Journal. The first number of this work, which was originally called the Assurance Magazine, appeared in September 1850, and it has been continued quarterly down to the present time. It was originated by the public spirit of two well-known actuaries (Mr Charles Jellicoe and Mr Samuel Brown), and was carried on by them for two years, we believe, at a considerable loss. It was adopted as the organ of the Institute of Actuaries in the year 1852, and called the Assurance Magazine and Journal of the Institute of Actuaries, Mr Jellicoe continuing to be the editor,—a post he held until the year 1867, when he was succeeded by Mr Sprague. The name was again changed in 1866, the words “Assurance Magazine” being dropped; but in the following year it was considered desirable to resume these, for the purpose of showing the continuity cf the publication, and it is now called the Journal of the Institute of Actuaries and Assurance Magazine. This work contains not only the papers read before the Institute (to which have been appended of late years short abstracts of the discussions on them), and manv original papers which were unsuitable for reading, together with correspondence, but also reprints of many papers published elsewhere, which from various causes had become difficult of access to the ordinary reader, among which may be specified various papers which originally appeared in the Philosophical Transactions, the Philosophical Magazine, the Mechanics' Magazine, and the Companion to the Almanac; also translations of various papers from the French, German, and Danish. Among the useful objects which the continuous publication of the Journal of the Institute has served, we may specify in particular two :—that any supposed improvement in the theory was effectually submitted to the criticisms of the whole actuarial profession, and its real value speedily discovered; and that any real improvement, whether great or small, being place! on record, successive writers have been able, one after the other, tσ take it up and develop it, each commencing where the previous one had left off. The result has been, as stated above, that great advances have lately been made in the theory. It may be truly said that the recent advances and improvements in the theory of life contingencies have rendered all the existing text-books antiquated; and until a new one shall be produced, bringing the treatment of the subject down to the present time, a complete knowledge of it can only be gained by a diligent study of the Journal of the Institute of Actuaries and Assurance Magazine.

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

α=vp (l + 1 α)

is true of annuities on two, three, or any number of joint lives, if we consider p to denote the probability that they will all survive for one year; and 1 a the value of an annuity on the joint continuance of lives which are severally one year older than those on which the required annuity depends. Thus we have— x, y, z, being the ages of the nominees—

3 ,⅛.⅛(i +σ )(

; ≡ijp∙p y (i+‰), and α,,, = vp a p y p t (1 + 1 c⅛ 1 ).

The columnar method of calculating annuities admits also of being extended to annuities on joint lives. In the extensive tables contained in D. Jones’s work,

D zj , lj> y v y , y being the older of the two ages, and N zy = 1 D jcj , + 2 D zy + 3 D zy + , where *D zy is used to denote D x+n . y+w .

An improved form of the table was suggested by De Morgan, according to which we should have D zj , = »+y lf y v 2 . This would simplify the formulas for the values of contingent annuities, but no tables have as yet been published calculated on this principle. The same method might be extended to three lives, in which case the most advantageous form would be D xy , = lf, j l,v 3 ; but the extent of the tables when three lives are involved renders it extremely improbable that such will ever be published. The practical construction of a D and N table for joint lives has been considered by Gray, Tables and Formulae, pp. 122-137, and Am. Mag., xviii. 26. Mr Jardine Henry has described in the Ass. Mag., xiv. 212, a mechanical method of computing the values of Ω zy = l x l y υ 9 , by means of which he has calculated "the values in his extensive tables mentioned above.

The values of annuities on the last survivor of two or more lives cannot be calculated by the ordinary methods that apply to annuities on joint lives; thus, for example, the equation a = vp (1 + 1 α) does not hold good with regard to them. Their values must be found from those of joint life annuities by means of the following formulas :— An annuity on the last)

survivor of two lives, x (a xy =a x + a 9 - a xi and y, *

An annuity on the last!

survivor of three lives, r a^i =a x +a y + a,-a y , -a, x - a xy + a xy , x, y, and 2, )

An annuity on the last two )

survivors of the three >a^,--a y , + a tx + a xy -2a xy , lives, x, y, z, ) '

If we have the values of annuities on the last survivor of two lives tabulated, as is the case in the Institute of Actuaries Life Tables, ∖ nq may find the value of an annuity on the last of three lives by means of the formula a— = where ιo is found by means of the relation 0 ιe = o⅛5 see Am. Mag., χvii. 266, 379.

The methods of approximation given by Lubbock and Woolhouse also apply to the calculation of annuities on the joint existence of any number of lives; see the latter’s explanation of his method, Am. Mag., xi. 322, and for an illustration of its application to complicated cases, xvii. 267. They may also be applied to find the value of an annuity on the last survivor of any number of lives; see Am. Mag., xvi. 375.

The formula usually given for the value of a reversionary annuity on the life of x to commence on the death of y is 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 be made at the end of the year in which y dies, and the last at the end of the year before that in which x dies; whereas in practice the annuity runs from the death of y, the first yearly payment being made one year after such death, and a proportionate part being paid up to the date of x’s death. A more correct formula, as given by Sprague (Am. Mag., XV. 126), is a * .- x ∖. If the annuity is payable half-yearly,

√ 1+⅛

the value will be approximately (α x - a x ,f) 1 -; and if Xi quarterly, (α x -α⅛) {∙ practice, it is often sufficient to deduct half a year’s interest from the value found by the formula α z - a xy , when the annuity is payable yearly, a quarter of a year’s interest when it is payable half-yearly, and an eighth of a year’s interest when quarterly.

In dealing with annuities in which three lives are involved, we are met by the difficulty that no tables exist in which the values of such annuities are given to the extent required in practice. Such tables as those computed for the Carlisle 3 per cent, table by Herschel Filipowski are of too limited extent to be of any practical utility; for the values being given only for certain ages differing by multiples of five years, a considerable amount of labour is required to deduce the values for other ages. When, therefore, ∙we desire to find the value of an annuity on the joint lives of say x, y, and z, it is usual to take the two oldest of the lives, say x and y, and find the value of a xy , then to look in the table of single life annuities for the annuity which is nearest in value to this,— a v suppose,—and lastly, to find the value of a w ,, and use it as an approximation to that of a xyz . De Morgan, in a paper written for the Philosophical Magazine for November 1839, and reprinted in the Am. Mag., x. 27, proved that the value of a xyt thus found would be strictly accurate, if the mortality followed the law known as Gompertz’s; that is to say, if the number of persons living according to the mortality table at any age, x, could be represented by means of the formula dg∕ l *. Gompertz proved, in the Philosophical Transactions for 1825, that by giving suitable values to the constants, the above formula might be made to represent correctly the number living during a considerable portion of life, say from age 10 to 60; but in order to represent by the same formula the numbers living at higher ages, it is necessary to give fresh values to the constants; and the discontinuity thence resulting has always been a fatal obstacle to the practical use of the formula. It has, however, from its theoretical interest, attracted a great deal of attention from actuaries; and numerous papers on the subject will be found in the Assurance Magazine. A claim to the independent (if not prior) discovery of the formula has been put forward by Mr T. R. Edmonds; but this claim, respecting which many communications will be found in the Assurance Magazine, is generally repudiated by competent judges. De Morgan further showed (Am. Mag., viii. 181) that if the above property holds good, or a xy , = a κ ,, then the mortality must follow Gompertz’s law; and Woolhouse gave independently a simple algebraical demonstration of the same property, x. 121. Makeham removed the above mentioned objection to Gompertz’s formula by introducing another factor, and showed (Ass. Mag., xii 315) that the formula dgi x s * will correctly represent the number living at any age x from about the age of 15 upwards to the extremity of life; and this formula has been found very serviceable for certain purposes.

The fact that Gompertz’s law does not correctly represent the mortality throughout the whole of life, proves that the above-described practical method of finding the value of an annuity on three joint lives is accurate only in certain cases. Makeham has shown (Ass. Mag., ix. 361, and xiii. 355) that when the mortality follows the law indicated by his modification of Gompertz’s formula, the value of an annuity on two, three, or any number of joint lives, can be readily found by means of tables of very moderate extent. Thus the value of an annuity on any two joint lives can be deduced from the value of an annuity at the same rate of interest on two joint lives of equal ages; the value of an annuity on any three joint lives, by means of a table of the values of annuities on three joint lives of equal ages; and so on; and Woolhouse has shown (vol. XV. p. 401) how the values of annuities on any number of joint lives, at any required rate of interest, can be found by means of tables of the values of annuities on a single life at various rates of interest. These methods, we believe, have not hitherto been practically employed to any extent by actuaries, and it would perhaps be premature to say which of them is preferable.

As the reader will have observed, neither Gompertz’s nor Makeham’s formula represents correctly the rate of mortality for very young ages. Various formulas have been given which are capable of representing with sufficient accuracy the number living at any age from birth to extreme old age, but they are all so complicated that they are of little more than theoretical interest. They are, however, likely to prove of increasing value in the problem of adjusting (or graduating) a table of mortality deduced from observations,—an important subject, which does not fall within the scope of this article. We may mention in particular those given by Lazarus in his Mortalitätsverhältnisse und ihre Ursache {Rates of Mortality and their Causes'), 1867, of which a translation is given by Sprague in the eighteenth volume of the Assurance Magazine, namely, CK ’ *ΛΗ p *j and by Gompertz (see Ass. Mag., xvi. 329),

ζ=const. A'*B zx ∙*C<D p , where P = 0ω 7r *<*->.

If l x represents the number living at any age in the mortality table, the force of mortality, or the instantaneous rate of mortality, mentioned above (see p. 83), is equal to — ^log e l x , Hence, in Gompertz’s original law the force of mortality at any age x is proportional to ç*, or is equal to aq x , where a is a constant; in Makeham’s law the force of mortality is equal to ag x + b, where a and b are constants; and in Lazarus’s law the force of mortality is equal to aq x + b + cp x , where a, b, and c are constants, or to ai~ mx + b + c∈" x . Dr Thiele has shown (see Ass. Mag., xvi. 313) how to graduate a mortality table, by assuming the formula for the force of mortality, α 1 e~*** + af^⅛ x ~ e > l + a.f ,3x ; and Makeham has explained (Ass. Mag., xvi 344) a very convenient practical method for adjustment, which results in assuming that the number living at any age x can be accurately represented by the sum of three terms of the form dg^ c s t .

The employment of formulas such as those given in the last paragraph, and the application of the differential calculusfto the theory of life contingencies, have naturally led to an improvement in the theory which is probably destined to become of very great importance—we refer to the introduction of the idea of “continuous” annuities and assurances. If the intervals at which an annuity is payable are supposed to become more and more frequent, until we come to the limit when each payment of the annuity is made momently as it accrues, the annuity is called continuous. Strictly speaking, of course, this is an impossible supposition as regards actual practice; but if an annuity were payable by daily instalments, its value would not differ appreciably from that of a continuous annuity; and if the annuity be paid weekly, the difference will be so small that it may be always safely neglected. The theory of continuous annuities has been fully developed by Woolhouse (Ass. Mag., xv. 95). Assuming the number living in the mortality table at any age x to be represented by l x , the value of a continuous annuity on a nominee of the age x is l x υ x dx = ^ ljΓ iχ dx, putting

δ = log e (l+⅛). From the nature of the case, l x must be a function that is never negative for positive values of x; and as x becomes larger, l x must continually diminish, and must vanish when x becomes infinite. It will be noticed here that the superior limit of the integral is ∞. This is necessary if l x is a continuous mathematical function; for in that case, however large x be taken, l x will never become absolutely zero. Makeham has shown (Ass. Mag., xvii. 305) that when the number living, l x , can be correctly represented by the formula cgi*e~ ax , the value of a continu- 1 1 /"»®

ous annuity is equal to -—z ∕ 10~ 10 '. e~* ’ .dz, log q where n=^ + ⅛-, and z = x log 10 <7 + log? 0 i; and he has tθg io? 9 given (pp. 312-327) a table, by means of which the value of the annuity can be found when the values of n and z are known. This table requires a double interpolation, and is therefore rather troublesome to use. Mr Emory M‘Clintock has shown in the eighteenth volume of the Assurance Magazine, how the value of an annuity may be found by means of the ordinary tables of the gammafunction. As Lazarus has pointed out in his above-mentioned paper, when mortality tables are given in the ordinary form, it is difficult to compare them and define precisely their differences; but if they can be accurately represented by a formula containing only a few constants, it becomes easy to show wherein one table differs from another; and the methods of Makeham and M i Clintock enable us to compare the values of annuities, for any ages desired, according to different tables as determined by such constants, without the labour of computing the mortality tables in the usual form. They can therefore scarcely fail to grow in popularity as they become better known.

The principal application of the theory of life annuities is found in life insurance. (See Insurance.) At the present time there are upwards of one hundred companies of various kinds transacting the business of life insurance in the United Kingdom. It is only since the passing of the Life Assurance Companies Act, 1870, that it has been possible to form an accurate estimate of the extent of the business transacted by these companies; but, from the returns made under that Act, it appears that the total assets of the companies amount to about £110,000,000, which are invested so as to produce an annual income of about £4,000,000, and that the total premiums received annually for insurance amount to about £10,000,000. There is no means at present of saying exactly what is the total sum assured; but it is probably about £330 000.000, the average premium for insurance being about 3 per cent, per annum. Theactual transactions at the present time in the purchase and grant of immediate annuities, although small in comparison with the life insurance transactions, are yet of considerable amount. It appears from the returns made under the above-mentioned Act, that upwards of £250,000 is annually paid to insurance companies for the purchase of annuities, and that the aggregate amount of their liabilities under that head is nearly £420,000 a year. The Government competes with the companies in the grant of annuities; and although its terms are on the whole very much less favourable than the companies’, still in consequence of the greater security offered, the business transacted by the Government is much in excess of that transacted by the whole of the insurance companies. It appears from recent returns (see Ass. Mag., xv. 23), that the life annuities annually paid by the National Debt Office amount to about £1,000,000, and that about £600,000 is on the average annually invested with the Government for the purchase of fresh annuities. The purchase and grant of life annuities have been carried on to a very considerable extent, apparently at all times. We learn from De Wit’s above-mentioned report, that the Governments of Holland and West Friesland had granted annuities systematically for one hundred and fifty years before any correct estimate was formed of the value of a life annuity. The British Government has at various times granted life annuities, more especially on the Tontine principle, for the purpose of raising money when it was difficult to obtain the sums required for the public service by the ordinary methods. Various local bodies have at different times raised money on the security of the local rates in consideration of the grant of life annuities; and, at the present tiιhe, the Manchester Corporation grants annuities on favourable terms for the purpose of obtaining funds to defray the expense of the waterworks belonging to the city. During the existence of the usury laws, it was very common for persons borrowing money upon the very best security to grant annuities upon their lives in consideration of a present advance. Thus, for example, if a country gentleman of the age of 40 wished to borrow £10,000 upon a landed estate, the law forbade him to pay, or the lender to receive, more than 5 per cent, interest, say £500 a year; but the law did not forbid his granting an annuity of £1000 for his life, secured upon the estate. Speaking roughly, an annual payment of £300 would be required to insure £10,000 upon the borrower’s life, and the annuity would therefore return the lender about 7 per cent, interest, in addition to the premium on the insurance necessary to return his capital. In this way the law, which was intended as a protection to the borrower, to enable him to obtain a loan at a fixed moderate rate of interest, very often had the directly opposite effect of greatly increasing the cost of borrowing. The usury laws being now repealed, borrowers and lenders are left at full Eberty to make such terms with each other as they may think best. (τ. b. s.) Table (1).— Showing out of 1,000,000 Children born, the Number of Males and Females Surviving at each Age, and the Number Dying in each Year of Life. English Table, No. 3. [table]

Table (2). - Showing the Probability of a Male or Female of any Age Dying within a Year. English Talle, Eθ∙ 3 [table]

Table (3).— Auxiliary (D and Λ r ) Table for finding the Values of Annuities at 3 per cent. Interest. English Table, Λo, 3. Males. [table]

Ta3LB (4).— Auxiliary (D and N) Table for finding the Values of Annuities at 3 per cent. Interest. English Table, No. 3, Females. [table]

Table (5).— Showing the Value of an Annuity, at 3 per cent., on the Life of a ⅛Iale or Female of any Age, English Table, No. 3. [table]