Encyclopædia Britannica, Ninth Edition/Annuities

by Thomas Bond Sprague

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 100 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 i f 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 bo made at the end of (n+l) years; if half-yearly, the first half-yearly payment will be made at the end of ( + ) years; if quarterly, the first quarterly payment will be made at the end of (n+¼) years; and so on. If an annuity, instead of being payable at the end of each year, half-year, &, 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 de.- pending 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 1 05. At the end of a second year this will be increased in the same ratio, and then amount to (1 05) 2 . In the same way, at the end of a third year, it will amount to (1 05) 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), at the end of three years (1 + i) 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 pro portion we easily see that the sum which in n years will amount to 1, must be r- , or (1 + i) n . It is usual to (l-M)" put v for r-, so that v is the value of 1 to be received at the end of a year, and v* the value of 1 to be received at the end of n years.

{{ti|1em|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 i 5 per cent, ieq -05, and - = 20; or a perpetuity is worth 1> 20 years purchase : at 4 per cent., it is worth 25 years purchase, ( eq 25 j : at 3 per cent., it is worth 33J years purchase, (.^ eq 33^.

Instances of perpetuities are the dividends upon the public stocks in England, France, and some other coun tries. Thus, although it is usual to speak of 100 con sols, 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 speak ing 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 deben ture stocks now issued so largely by various railway com panies, 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 test of the credit of the various governments or rail ways. Thus at the present time (May 1874) the British per petual 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 governments:

No. of Years	No. of Years' Purchase.
British,	31⋅00
Dutch,	23⋅69
Swedish,	21⋅20
Russian,	20⋅40
French,	19⋅67
Brazilian,	20⋅00
Portuguese,	15⋅42
Argentine,	13⋅50
Austrian,	13⋅40
Italian,	13⋅00
Turkish,	9⋅50
Spanish,	6⋅67
Venezuelan,	3⋅50

The following are a few other examples of perpetuities:—

Interest per cent, yielded to a Purchaser. 3-23 4 22 4-72 4-90 5-08 5-00 6-48 7-41 7-46 7-69 10-53 15-00 23-57 Name, Metropolitan Board of "Works Stock, 27 50 London and N."W. Railway Deben- C1.L 1 ture Stock, North British Railway Debenture r*i_ 1 Stock, Edinburgh Water Annuities, 22 53 Interest per cent, yielded to a Purchaser 3-64 3-83 4-25 4-43

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 (l+i)"; hence the value of a perpetuity of 1 deferred n years is r Now an annuity for n years is clearly the difference between the value of a perpetuity to com mence at once and a perpetuity deferred n years ; its value is therefore - - r eq r ; or putting a for the value of the annuity, we have

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

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

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

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

us to describe.^[1]

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

[ math ]

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 -f-t )" 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 +* )" 1 ; and by proportion the amount of an annuity of 1 similarly improved will be r*.

One of the principal applications of the theory of annui ties 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 j 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 rent- charge upon the fee-simple for a term not exceeding 30 years. 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 4f 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 desir able for various purposes, and in particular for the ascer tainment 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 instal ment of the principal. The problem of determining the separate amounts of these has been considered by Turn- bull, Tables, p. 128 ; and by Gray, Ass. Mag., xi. 172.

In making calculations for these and similar purposes, it is but seldom necessary to use the formulas given above. The computer usually has recourse to one of the 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 172G 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, 3

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

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

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

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

(5.) The annuity which 1 will purchase for any num ber 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; i or j- 7 - .

, 1

The following table, on p. 75, in which the rate of in terest 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 (6) were altered so as to relate to annuities payable in advance (or annuities-due). In that case (3) would give the amount at com pound 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 ; that is to say, the values of the functions - - and 7equalsign . +1 _[_> 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

(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."

Table of Amounts, Present Values, &c., at 5 per cent. Interest.
$i=\cdot 05.$

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 different 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, 3i, 4, 5, 6 per cent., to any number of years up to 105. Hardy s Doctrine, of Simple ami Compound Iiitcrcst, 1839, contains tables (1), (2), (3), (4), for the rates of interest \, 4, f, 1, li, 14, IJ, 2, 2

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 Lis Tables and Formula, chap. 2. Since the publication of that work, the Arithmometer of M. Thomas (of Colmar) has come into extensive use for the formation of tables of this kind. For a descrip tion of the instrument, and some of its uses, the reader is refem-d 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, sup pose 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 ia six months time 1 and, in order to find the amount of the annuity in n years, we must first deter mine 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 + - ; but if this were the case, the 2i amount at the end of a second period of six months would (t\ 2 1 + - ) , or 1 + i + - . 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 . In order that the amount a year will not be i, but 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 ^fl + i, or (1 + 1)*. 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 (1 + t)". 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 frac tional, will always be (1 + i)" or v*. 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 (?4 being integral) is ^.i-l+J . 1 -(!+*)". and ttat tlie amount O f a similar 2 ^ annuity at the end of n years is %SI:f - f 1 + *)"-*. 2 i

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 ${\frac {i}{2}}$, so that the true annual rate in practice is $i+{\frac {i^{2}}{2}}$; 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, 1s. 3d. per. 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 $m$^th part of a year will be ${\frac {i}{m}}$, and the amount of 1 at the end of $n$ years, that is, at the end of $mn$ intervals of conversion, will be $\left(1+{\frac {i}{m}}\right)^{mn}$. Assuming the number $m$ now to increase indefinitely, or interest to be convertible momently,

the above amount becomes e *, where e 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 prac tically 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 sub ject 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 tTb 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 m 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, or 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 mortality 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 suffi ciently close. This was particularly the case when it \vas desired to obtain the value of joint life, contingent, or other complicated benefits. Demoivre s law is now, accord ingly, 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 Charlon, Ass. Mag., xv. 141. In vol. vi. p. 181, will be found an in vestigation 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 arith metical calculations. Without entering here on a descrip tion of the manner of making these observations and de ducing 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 Probcibilites de la Duree de ia Vie Humaine, 1746.

Number Number Number Age. Number living. dying in the next Age. Number living. dying in the next Age. Number living. dying in the next vear. year. year. X I, d, X I* d* X lx - d, 3 1000 30 34 702 8 65 395 15 4 970 22 35 694 8 66 380 16 5 948 18 36 686 8 67 364 17 6 930 15 37 678 7 68 347 18 7 915 13 38 671 7 69 329 19 8 902 12 39 664 7 70 310 19 9 890 10 40 657 7 71 291 20 10 880 8 41 650 7 72 271 20 11 872 6 42 643 7 73 251 20 12 866 6 43 636 7 74 231 20 13 860 6 44 629 7 75 211 19 14 854 6 45 622 7 76 192 19 15 848 6 46 615 8 77 173 19 16 842 7 47 607 8 78 154 18 17 835 7 48 599 9 79 136 18 18 828 7 49 590 9 80 118 17 19 821 7 50 581 10 81 101 16 20 814 8 51 571 11 82 85 14 21 806 8 52 560 11 83 71 12 22 798 8 53 549 11 84 59 11 23 790 8 54 538 12 85 48 10 24 782 8 55 526 12 86 38 9 25 774 8 56 514 12 87 29 7 26 766 8 57 502 13 88 22 6 27 758 8 58 489 13 89 16 5 28 750 8 59 476 13 80 11 4 29 742 8 60 463 13 91 7 3 30 734 8 61 450 13 92 4 2 31 726 8 62 437 14 93 2 1 32 718 8 63 423 14 94 1 1 33 710 8 64 409 14 95

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 attain ing 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 in the event of the happening of a given contingency. Suppose that the sum of 1 is to be re ceived 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 n . 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 (a + b). We may sup pose, for instance, that there are (a + b) balls in a bag, of which a are Avhite 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 (a + b) per sons 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 (a + b) 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 r = >. This is the value of the chance of receiving 1 immediately before the drawing is made ia n years time ; the value at the present time will there fore be v*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 av*; and each of the (a + b) persons having the same chance of receiving 1, the value of the expectation of each is 7v n =pv".

a + b f

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 be longs. 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, ^ = 734, Z 4l) = 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 . Hence, by lemma 1, the value of 1 to be received if the nominee shall be alive at the end of n years, is -y^V. In the particular case supposed above, L x the actual value will be, taking the rate of interest at 3 per cent, ^ x (1 03)- 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 (1 03) 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 CT *T therefore ^(l 03 )" 10 , as before. If we suppose the pro babilities 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 272,073 304,534 (1-03)- 10 = -664779.

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 justifica tion. 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 neces sarily 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 some times 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, there fore, 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 cer tainly 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 (a + 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 num ber 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 d i b observation, we can say, in the same sense, that the pro bability 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 frac tion 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, &c., under the same condition, and is therefore equal to the sum of a series of endowments. If # is the age of the nominee, the value of the endowment to be received at the end of the ?ith year is, as we have seen in lemma 2, -y^t B , and the total value of the annuity is therefore

By means of this formula, taking the values of l x , /, +1 , l*+v & c -> from the mortality table, and calculating the values of v, v, v 5 , &c., 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 I, 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 l, +l . 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 If+iU. The number of persons who will survive two years, so as to claim the second year s payment, will be l a+2 , and multiplying this into the value of 1 payable at the end of two years, we get l x+ ^ 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+1 v + l x+z v 2 + l a+3 iP + . . . 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

7 ... .

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 v= 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 v=lin the formula above given, we get

[ math ]

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 x , 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 Hudde, bearing dates from September 1 670 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 con sidering 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 borrow ing 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 AVit 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 per petual 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 pre vailed 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 the 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 conclusion 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 Hen driks 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 tha 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 unpub lished state paper, although it contributed to its author s reputation, did not contribute to advance the exact know ledge 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 approxi mately correct mortality table (deduced from the records of the numbers of deaths and baptisms in the city of Breslaii), 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 annui ties 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. JEmilius 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:—

Age. Years Purchase. Age. Years Purchase. Birth to 20 30 45 to 46 14 20 25 28 46 47 13 25 30 25 47 48 12 30 35 22 48 49 11 35 40 20 49 50 10 40 41 19 50 55 9 41 42 18 55 60 7 42 43 17 60 and ) 43 44 44 45 16 15 upwards ) 5

Here also we have no reason to suppose that the element of interest was taken into consideration ; and the assump tion, 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 1 7th 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 tho 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 3 a for that on a life one year older, the value would be v (1 + l a). But it is un certain 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 = vp (1 + l a). This formula may also be demonstrated algebraically. We have seen that

[ math ]

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, 2 will be!07-39 &eq 68.) In the same way, we have

H H Hence a &eq v

—the same result as already proved.

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

• &eq * -r- l (i + *+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 Re versions ; 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 undoiibtedly 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 in dependently 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 Formula?,

1849, how Gauss s logarithmic table may be advantage ously employed in calculating the values of annuities by the above formula. That table gives us the value of log (1 +a) 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+a). When ordinary logarithmic tables are used,

the formulas being

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

we have to find a t by means of an inverse entry into the table before log (1 +a,) 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 con sequently 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 1G5, 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 sup posed 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 *+> its present value would be v*a f+n . But as the nominee may die before attaining the age of x + ?i, 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 x deferred annuity, t>" . -y 15 . *+ We may arrive at this result otherwise. Thus, as we have seen above, the pre sent 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 *+H n+ I years, is -y- i> n+1 . The present value of the next payment is similarly seen to be -j-v n+2 , and so on. The value It of the deferred annuity is therefore

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

(We may here mention that this formula holds good, 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

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

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

[ math ]

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

l m v

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

[ math ]

If, then, we calculate in the first instance the values of the product l x v 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 v* is placed in the column headed D, oppo site the age x, and is denoted by D,,,; while the sum ^+i v * +l + e+i y * +2 + . . . . + l x +,v* + is placed in the column headed N, opposite the same age x, and is denoted by N a ; so that the value of an immediate annuity on a life x is equal to ; the letters N and D being chosen as 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 ; whence by subtraction the value of a temporary annuity for n years on the same life is * * +n

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 by division we get the value of = 1 6 4744, which agrees L) 40 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 n -u i.- * N 40 1374058 annuity will be equal to the quotient or 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 r^ v 10 a 40 , v being equal to . But we have seen above that the value of ^-(1*03)- 10 = 664779, and that a w = 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 his 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 M = 125464, N., = 2385610;
D 40 - 83406, N 40 = 1374058.

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

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

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

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

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

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

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

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

3. For a temporary insurance for n years,

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

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, 11, 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, we 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, Cray, and De Morgan.

The earliest known specimen of a commutation table is contained in William Dale's Introduction to the Study of the Doctrine of Annuities, published in 1772. A full account of this work is given by Mr F. Hendriks in the second number of the Assurance Magazine, pp. 15–17. Dale's table, as there quoted, differs from the one above described in that it commences only at the age of 50, and that he has tabulated l x v - K instead of Ijf. Hesays, " These calculations being made for the use of the societies in particular who commence annuitants at the age of 50, it was not thought necessary to begin the tables at a younger age." He gives, however, another table based on different mortality observations, commencing at the age of 40; and in this case he tabulates l j .v* to . 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 - z - ^ c l are tabulated, and not those of l x v*; 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 w r as 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 con struction and use of the commutation table, and a sketch of the aiithor s life and writings, to which we refer the reader who desires fuller information.

The use of the commutation table w r as 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 Ijf, but l x (\ + i)" x , where 10 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 commu tation table, and this method is accordingly sometimes called Barrett s method. (It is also called the commuta tion 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 the 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. Bar rett had calculated extensive tables, and with Baily s aid attempted to get them published by subscription, but with out success; and the only printed tables calculated accord ing 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 per sons 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 arranged 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 wa?, 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 pre face to this work it is stated that " the most important dis tinction 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 con sidered more as the numerical results of algebraical expres sions ; 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 ; 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 z , &c., but omits the subscript x. Thus,

instead of the formula r&eq ^- he would write N.

In some respects this notation is perhaps preferable to tha t 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 x might be mistaken for the number in the column N" oppo site the age 1. We may, however, consistently with the principles of the notation adopted by the Institute of Actuaries, write the formula _rj^ s&eqrnj. The notation at present commonly used is due to David Jones, whose work (mentioned below) was the first that contained an extensive series of commutation tables.

On a general review of the whole evidence, we 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 +i) 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 Magazine, xii. 348-9 ; Gray s Tables and Formulae, chap. viii. ; the preface to Davies s Treatise on Annuities ; also Hend- riks s papers in the Assurance Magazine, No. 1, p. 1, 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, K ; Carlisle Table, interest 3, 3J, 4, 4, 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, 3

Jenkin Jones, New Rate of Mortality, 1843, Seventeen Offices Experience, 2^, 3, 3J per cent., columns D, N, S, M, R.

G. Davies, Treatise on Annuities, 1825 (issued 1855). Equitable Experience, 2^,3, 3, 4, 4i, 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, 3, 4, 44, 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.

Jardiue Henry, Government Life Annuity Commutation Tables, 1866 and 1873, single lives male and female, 0, 1, 2, 2, 3, 3, 4, 4, 5, 5J, 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, 3J, 4, 4

R. P. Hard"y, Valuation Tables, 1873, gives the same table at 4J per cent, for H M W.

The Sixth Report of tlie 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,,,, N^y , 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

The explanations of the tables in the last four works are by Dr William Fair, 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 column N is shifted down one year, so that in them the ratio N&eqp gives, not the value of the ordinary annuity, but the -L a; value of the annuity increased by unity, or the annuity-due. It is very needful to bear this in mind for the prevention of error ; and the existence of a difference of this kind is ex tremely 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 w denote the payment to be received at the age x + m, and A 1? A 2 , A,, denote the lead ing differences of V , V,, V,,, Vy, the value of the

annuity is approximately

A, + _ 24u 2 720n 3

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

As an example, we will apply this formula to 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 =1-0000 - 2654 V 7 = 7346 + -0521 2133 - -0115 V 14 = -5213 -0406 +-0029 1727 -0086 V.,= -3486 -0320 1407 V 88 = -2079 V 35 = -0990 V 42 = -0318 V 49 = -0055 V= 0004 Sum=2-9491

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

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

Next, taking =11, we have

V =1-0000 - -3924 V u = -6076 +-1115 2809 - -0310 V 2a = -3267 -0805 + 0453 2004 + -0143 V 33 = -1263 -0948 1056 V 44 = -0207 V 56 = -0006 2-0819

Hence, the value of the annuity is approximately

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

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

An improved form of this method was given by Mr W. S. B. Woolhouse in the Ass. Nag., 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 con tingencies has rendered of great importance tJie force of mortality at a given age. This may be defined as the pro portion 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 obser vation 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. M. Makeharn, 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 _ x the number dying between the ages x 1 and x, then the number dying between the ages x-- and x + - will be approximately x i + d, , and the force of mortality is approximately a -^ -. Thus, in the English Table, No. 3, Males, the * 3465 + 3529 value of the force of mortality at age 40 is - = 012853.

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

[ math ]

where it will be noticed that, since V = 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

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

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

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

Making n = 11, we have the value

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

Comparing the two processes, we see that when we have

the values of p. and 8 already computed, Woolhouse s L> 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., v. 277), or to the short account of it given in the Treatise on Probability, issued by the Useful Know ledge Society, and often bound up with D. Jones s work on annuities, will see that the terms involving A 2 , A 3 , A 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 r- r; -, Diff. Calc., p. 314, 184. Lubbock not only considered it unnecessaiy to calculate the terms involving A 2 , A 3 , &c., but thought that the value of the term containing A 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 correc tions 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 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 value of i, payable at the instant of the nomi nee s death ; in the case of half-yearly annuities, by adding the value of \ ; and in the case of quarterly annuities, the value of

The previous remarks refer almost exclusively to annui ties 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 so 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 sur vivors ; 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 gene rally, 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 ij, 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 im 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 copyhold 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 care fully study the various articles contributed to the Journal of the Institute of Actuaries, particularly those of Woolhouse and Make- ham. The Institute was founded in the year 1848, the first sessional meeting being held in January 1849. Its establishment has con tributed 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 wholo 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 im provements, 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 after wards 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 18o2, 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. Thf> name was again changed in 1866, the words Assurance Magazine" being dropped ; but in the following year it was considered desir able 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 many original papers which were unsuitable for reading, together with correspondence, but also reprints of many papers published else where, 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 Com panion 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 placet!, on record, successive writers have been able, one after the other, to

take it up and develop it, cacli 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 anti quated ; and until a new one shall be produced, bringing the treat ment of the subject down to the present time, a complete know ledge 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 em ployed for the calculation of single life annuities are easily extended to the case of joint life annuities. The funda mental equation

a &eq vp(l + *a)

is true of annuities on two, three, or any number of joint lives, if we consider/) to denote the probability that they will all survive for one year ; and l 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 de pends. Thus we have x, y, 2, being the ages of the nominees—

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

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

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

where n T) xy is used to denote T) x+n . y+n . An improved form of the table was suggested by De Morgan, according to which we should have D xy &eq x+y IJyV 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 X+y+l advantageous form would bs D^, = ljl t v 3 ; but the ex tent 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 Formidce, pp. 122-137, and Ass. Mag., xviii. 26. Mr Jardine Henry i.a3 described in the Ass. Mag., xiv. 212, a mechanical method of computing the values of D xy = IJyV 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 + l ci) 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 lasfi survivor of two lives, x fo^ a x + a y - a I}> and y, An annuity on the lasO survivor of three lives, a^ji eq a x + a y + a,- a a , - a, x - a xy + a xy , x, y, and z, ) An annuity on the last two \ survivors of the three \a x j t eq a 1/ , + a 1JC + a xy -2a X!/ , lives, x, y, z, )

If we have the values of annuities on the last survivor of two lives tabulated, as is the case in the Institute of Actuaries Life Tables, we may find the value of an annuity on the last of three lives by means of the formula a= rt + ( i* rt r. where w is found by means of the relation = *-,; see Ass. May., xvii. 266, 379. The methods of approximation given by Lubbock 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, Ass. 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 Ass. Mar/., 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 de duced 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 ar s death. A more correct formula, as given by Sprague (Ass. Mag., xv. 126), is *. x (. If the annuity is payable half-yearly, 1 4. f 1 4- f\* the value will be approximately (a x - a xu ] - ; and if jj quarterly, (a z -a^) / 1 , -a , I n practice, it is often sufficient to deduct half a year s interest from the value found by the formula a x 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 in

volved, we are met by the difficulty that no tables exist iu 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, there fore, we desire to find the value of an annuity on the joint lives of say x, y, and 2, 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 a suppose, and lastly, to find the value of a wi , and use it as an approximation to that of rt^,. De Morgan, in a paper written for tho Philosophical Magazine for November 1839, and reprinted in the Ass. Mag., x. 27, proved that the value of or,,, 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 q . 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 inde pendent (if not prior) discovery of the formula has been put forward by Mr T. lv. Edmonds; but this claim, respecting which many communications will be found iu the Assurance Magazine, is generally repudiated by competent judges. De Morgan further showed (Ass. Mag., viii. 181) that if the above property holds good, or a xy , = a vl , then the mortality must follow Gompertz s law; and Woolhouse gave inde pendently 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 dg^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 Mortalitdts- verhciltnisse und Hire Ursaclie (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^/fW*; and by Gompertz (see Ass. Mag., xvi. 329),

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

If l x represents the number living at any age in the mortality table, the force of mortality, or the instantaneous rate of mortality, mentioned above (see p. 83), is equal to --T-logJr Hence, in Gompertz s original law the force of mortality at any age x is proportional to <f, or is equal to a(f t where a is a constant; in Makeham s law the force of mortality is equal to atf + b, where a and b are constants; and in Lazarus s law the force of mortality is equal to aq* + b + cp x , where a, b, and c are constants, or to ae"* + b + ce* 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, o 1 t*j*+a s i * ( ** )B + a 3 * 8 *j 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 Bum of three terms of the form dg qX s*.

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 1 /"*oo -I y-oo *- ., of the age x is j- I l x ifdx eq j I l x e Sx dx, putting m^/ x IxJ x I 8 eq log e (l+t). 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 GO . 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 cg^ e "*, the value of a continu ous annuity is equal to where n eq + log q 10- 10 *.e- r 10- and z eq x Iog 1( tf + log? - ; and he has 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 gamma- function. As Lazarus has pointed out in his above-men tioned 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 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 com panies 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 pre sent of saying exactly what is the total sum assured ; but it is probably about 330000000, the average premium for insurance being about 3 per cent, per annum. The actual 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 annu ally paid to insurance companies for the purchase of annui ties, and that the aggregate amount of their liabilities under that head is nearly 420,000 a year. The Govern ment competes with the companies in the grant of annui ties ; 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 Fries- land 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 time, the Manchester Cor poration grants annuities on favourable terms for the pur pose of obtaining funds to defray the expense of the water works 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 bor rower, 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 liberty to make such terms with each other as they

may think best.

(t. b. s.)

TABLE (1). Showing out of 1,000,000 Children lorn, tlie Number of Males and Females Surviving at each Age, and the Number Dying in each Year of Life. English Table, No. 3.

Males. Females. Males. Females. Males. Females. Age. ^Number Number Age. Number Number Age. Number Number Number | dying in Number dying in Number dying in Number dying in Number dying in Number dvingin alive at the fol alive at the fol alive at the fol alive at the fol alive at the fol alive at the fol each age. lowing each age. lowing each age. lowing each age. lowing each age. lowing each age. lowing year. year. year. year. year. year. X I, d, I* d, X I* d x h d x X h d x l x d x 511745 83719 488255 65774 37 282296 3352 276563 3326 74 83416 7639 93071 7724 1 428026 27521 422481 26159 38 278944 3406 273237 3350 75 75777 7483 85347 7653 2 400505 14215 396322 14023 39 275538 3465 269887 3376 76 68294 7268 77694 7521 3 386290 9213 382299 9243 40 272073 3529 266511 3402 77 61026 6990 70173 7329 4 377077 6719 373056 6596 41 268544 3596 263109 3431 78 54036 6655 62844 7071 5 370358 5033 366460 4866 42 264948 3668 259678 3459 79 47381 6266 55773 6755 6 365325 3953 361594 3815 43 261280 3746 256219 3490 80 41115 5832 49018 6382 7 361372 3310 357779 3249 44 257534 3826 252729 3522 81 35283 5361 42636 5959 8 358062 2734 354530 2724 45 253708 3912 249207 3555 82 29922 4862 36677 5496 9 355328 2297 351806 2328 46 249796 4001 245652 3591 83 25060 4349 31181 5003 10 353031 1983 349478 2045 47 245795 4095 242061 3627 84 20711 3834 26178 4490 11 351048 1776 347433 1861 48 241700 4192 238434 3665 85 16877 3328 21688 3972 12 349272 1666 345572 1765 49 237508 4292 234769 3705 86 13549 2840 17716 3458 13 347606 1637 343807 1745 50 233216 4395 231064 3746 87 10709 2384 14258 2962 14 345969 1679 342062 1789 51 228821 4626 227318 3788 88 8325 1965 11296 2494 15 344290 1781 340273 1888 52 224195 4758 223530 3832 89 6360 1590 8802 2063 16 342509 1928 338385 2029 53 219437 4885 219698 3876 90 4770 1260 6739 1673 17 340581 2112 336356 2205 54 214552 5013 215822 4246 91 3510 979 5066 1331 18 338469 2320 334151 2400 55 209539 5144 211576 4439 92 2531 744 3735 1037 19 336149 2541 331751 2609 56 204395 5281 207137 4628 93 1787 553 2698 790 20 333608 2764 329142 2819 57 199114 5428 202509 4817 94 1234 401 1908 588 21 330844 2801 326323 2867 58 193686 5584 197692 5009 95 833 285 1320 428 22 328043 2836 323456 2912 59 188102 5752 192683 5206 96 548 196 892 304 23 325207 2868 320544 2952 60 182350 5929 187477 5409 97 352 132 588 210 24 322339 2S97 317592 2989 61 176421 6118 182068 5619 98 220 86 378 142 25 319442 2926 314603 3024 62 170303 6314 176449 5835 99 134 55 236 92 26 316516 2954 311579 3055 63 163989 6515 170614 6057 100 79 33 144 59 27 313562 2981 308524 3084 64 157474 6720 164557 6282 101 46 21 85 36 28 310581 3009 305440 3112 65 150754 6921 158275 6509 102 25 11 49 22 29 307572 3038 302328 3138 66 143833 7115 151766 6731 103 14 7 27 12 30 304534 3068 299190 3163 67 136718 7297 145035 6947 104 7 3 15 7 31 301466 3100 296027 3187 68 129421 7458 138088 7149 105 4 2 8 4 82 298366 3134 292840 3209 69 121963 7593 130939 7332 106 2 1 4 2 33 295232 3171 289631 3233 70 114370 7695 123607 7489 107 1 1 2 1 34 292061 3211 286398 3255 71 106675 7756 116118 7613 108 ... 1 1 35 288850 3254 283143 3279 i 72 98919 7770 108505 7698 109 30 285596 3300 279864 3301 73 91149 7733 100807 7736

L

TABLE (2). - Showing tlie Probability of a Male or Female of any Age Dying within a Year. English Talle t No, 3

Age. X. Probability of Dying in a Year 2* Age. X. Probability of Dying in a Year. 9* Age. X. Probability of Dying in a : Year. f. Males. Females. Males. Females. Males. Females. 163597 134714 37 011873 012025 73 084840 076748 1 064298 061918 38 012212 012262 74 091570 082984 2 035494 035383 39 012575 012508 75 098758 089668 3 023850 024178 - 40 012968 012766 76 106412 096812 4 017820 017683 41 013392 013038 77 114544 104430 5 013590 013278 42 013845 013320 78 123154 112526 6 010820 010554 43 014334 013620 79 132256 121112 7 009160 009080 44 014858 013936 80 141844 130192 8 007636 007684 45 015418 014268 81 151926 139774 9 006465 006618 46 016018 014618 82 162500 149855 10 005616 005853 47 016660 014985 83 173564 160440 11 005060 005358 43 017343 015373 84 185116 171528 12 004768 005108 49 018072 015780 85 197148 183115 13 004710 005074 50 018844 016210 86 209654 195196 14 004854 005232 51 020220 016666 87 222626 207767 15 005173 005548 52 021222 017142 88 236050 220814 16 005630 005998 53 022263 017646 89 249914 234332 17 006203 006553 54 023364 019673 90 264203 248302 18 006854 007184 55 C24548 020980 91 278900 262710 19 007558 007865 56 025838 022344 92 293987 277543 20 008285 008563 57 027260 023788 93 309442 292778 21 008468 008788 58 028830 025338 94 325243 308397 22 008645 009004 59 030575 027018 95 341367 324373 23 008820 009210 60 032518 028850 96 357787 340687 24 008990 009413 61 034676 030862 97 374479 357309 25 009160 009610 62 037074 033070 98 391411 374210 26 009333 009805 63 039733 035500 99 408556 391353 27 009507 009998 64 042672 038178 100 425883 408738 28 009688 010190 65 045910 041123 101 443358 426301 29 009878 010378 60 049470 044354 102 460953 444021 30 010073 010570 67 053370 047898 103 478631 461863 31 010283 010764 68 057626 051772 104 496361 479793 32 010504 010962 69 062256 055994 105 514109 497777 33 010740 011163 70 067278 060586 106 531839 515777 34 010994 011368 71 072708 065563 107 549520 533760 35 011265 011580 72 078556 070946 108 567116 551688 36 011558 011798

TABLE (3). Auxiliary (D and N) Table for finding the Values of Annuities at 3 per cent. Interest. No. 3. Males. English Table,

Age. X. D* NX Age. X. D* Nj Age. X. D* N* 511745 9288491 37 94564 1635186 74 9360-4 50726-5 1 415559 8872932 38 90720 1544466 75 8255-5 42471-0 2 377514 8495418 39 87002 1457464 76 7223-5 35247-5 3 353510 8141908 40 83406 1374058 77 6266-9 28980-6 4 335028 7806880 41 79926 1294132 78 5387-4 23593-2 5 319474 7487406 42 76559 1217573 79 4586-3 19006-9 6 305953 7181453 43 73300 1144273 80 3863-8 15143-1 7 293828 6887625 44 70145 1074128 81 3219-2 11923-9 8 282657 6604968 45 67090 1007038 82 2650-6 9273-3 9 272329 6332639 46 64132 942906 83 2155-2 7118-1 10 262688 6069951 47 61267 881639 84 1729-3 5388-8 11 253605 5816346 48 58491 823148 85 1368-1 4020-7 12 244972 5571374 49 55803 767345 86 1066-4 2954-30 13 236703 5334671 50 53198 714147 87 818-28 2136-02 14 228726 5105945 51 50676 663471 88 617-58 1518-44 15 220987 4884958 52 48205 615266 89 458-06 1060-38 16 213441 4671517 53 45808 569458 90 333-58 726-80 17 206057 4405460 54 43483 525975 91 238-30 488-50 18 198815 4266645 55 41230 484745 92 166-83 321-67 19 191701 4074944 56 39047 445698 93 114-35 207-323 20 184711 3890233 57 36930 408768 94 76-668 130-655 21 177845 3712388 58 34877 373891 95 50-225 80-430 22 171203 3541185 59 32885 341006 96 32-117 48-313 23 164780 3376405 60 30951 310055 97 20-025 28-288 24 158570 3217835 61 29072 280983 98 12-161 16-1271 25 152567 3065268 62 27247 253736 99 7-1856 8-9415 26 146767 2918501 63 25472 228264 100 4-1261 4-8154 27 141162 2777339 64 23748 204516 101 2-2999 2-5155 28 135748 2641591 65 22072 182444 102 1-2429 1-2726 29 130517 2511074 66 20446 161998 103 6505 6221 30 125464 2385610 67 18868 143130 104 3293 2928 31 120583 2265027 68 17341 125789 105 1610 1318 32 115867 2149160 69 15866 109923 106 0759 0559 33 111310 2037850 70 14445 95478 107 0345 0214 34 106907 1930943 71 13080 82398 108 0151 0063 35 102653 1828290 72 11776 70622 109 0063 0000 36 98540 1729750 73 10535

60086-9

4\ Auxiliary (D and N) Table for finding the Values of Annuities at 3 per cent. Interest. English Talle t No. 3, Females.

Age. X. D, Nj Age. X. D* Ns Age. X. D* .NX 488255 9203701 37 92644 1647581 74 10444 60281-5 1 410175 8793526 38 88864 1558717 75 9298-2 50983-3 2 373571 8419955 39 85218 1473499 76 8217-9 42765-4 3 349858 8070097 40 81701 1391798 77 7206-1 35559-3 4 331456 7738641 41 78309 1313489 78 6265-6 292937 5 316112 7422529 42 75036 1238453 79 5398-6 23895-1 6 302830 7119699 43 71880 1166573 80 4606-6 19288-5 7 290907 6828792 44 68836 1097737 81 3890-1 15398-4 8 279870 6548922 45 65900 1031837 82 3248-9 12149-5 9 269630 6279292 46 63068 968769 83 2681-6 9467-9 10 260044 6019248 47 60336 908433 84 2185-8 7282-1 11 250993 5768255 48 57701 850732 85 1758-1 5524-0 12 242377 5525878 49 55159 795573 86 1394-4 4129-6 13 234116 5291762 50 52707 742866 87 1089-5 3040-10 14 226143 5065619 51 50343 692523 88 838-00 2202-10 15 218408 4847211 52 48062 644461 89 633-94 1568-16 16 210870 4636341 53 45862 598599 90 471-25 1096-91 17 203501 4432840 54 43741 554858 91 343 92 752-99 18 196279 4236561 55 41631 513227 92 246-18 506-81 19 189193 4047368 56 39571 473656 93 172-68 334-13 20 182238 3865130 . 57 37560 436096 94 118-56 215-574 21 175415 3689715 58 35598 400498 .95 79-611 135-963 22 168809 3520906 59 33686 366812 96 52-221 83-742 23 162417 3358489 60 31821 334991 97 33-427 50-315 24 156234 3202255 61 30003 304988 98 20-858 29-457 25 150256 3051999 62 28230 276758 99 12-672 16-7848 26 144478 2907521 63 26501 250257 100 7-4882 9-2966 27 138894 2768627 64 24816 225441 101 4-2985 4-9981 28 133501 2635126 65 23174 202267 102 2-3942 2-6039 29 128292 2506834 66 21573 180694 103 1-2924 1-3115 30 123262 2383572 67 20016 160678 104 6752 6363 31 118407 2265165 68 18502 142176 105 3410 2953 32 113721 2151444 69 17033 125143 106 1663 1290 33 109198 2042246 70 15611 109532 107 0782 0508 34 104835 1937411 71 14238 95294 108 0354 0154 35 100624 1836787 72 12917 82377 109 0154 0000 36 96562 1740225 73 11651 70726

TABLE (5). Showing the Value of an Annuity, at 3 per cent., on the Life of a Male or Female of any Age, English Table, No. 3.

Age. X. Value of Annuity. a x Age. X. Value of Annuity. a x Age. X. Value of Annuity. a x Males. Females. Males. Females. Males. Females. 18-1506 18-8502 34 18-0618 18-4807 68 7-2539 7-6842 1 21-3518 21- 4385 35 17-8105 18-2539 69 6-9284 7-3469 2 22-5036 22-5391 36 17-5538 18-0218 70 6-6100 7-0162 3 23-0316 23-0668 37 17-2918 17-7841 71 6-2993 6-6928 4 23-3022 23-3474 38 17-0245 17-5405 72 5-9971 6-3773 5 23-4367 23-4807 39 16-7521 17-2910 73 5-7036 6-0702 6 23-4724 23-5106 40 16-4744 17-0353 74 5-4193 5-7721 7 23-4410 23-4742 41 16-1916 16-7733 75 5-1445 5-4832 8 23-3674 23-3999 42 15-9037 16-5047 76 4-8795 5-2039 9 23-2536 23-2886 43 15-6108 16-2293 77 4-6244 4-9346 10 231071 23-1470 44 15-3129 15-9471 78 4-3793 4-6753 11 22-9347 22-9818 45 15-0102 15-6576 79 4-1442 4-4262 12 22-7429 22-7987 46 14-7026 15-3608 80 3-9192 4-1872 13 22-5374 22-6032 47 14-3902 15-0563 81 3-7040 3-9583 14 22-3234 22-4000 48 14-0730 14-7439 82 3-4986 3-7395 15 22-1052 22-1933 49 13-7511 14-4233 83 3-3027 3-5307 16 21-8867 21-9867 50 13-4242 14-0942 84 3-1162 3-3315 17 21-6710 21-7829 51 13-0925 137562 85 2-9388 3-1419 18 21-4604 21-5844 52 127636 13-4090 86 2-7703 2-9617 19 21-2568 21-3923 53 12-4315 13-0522 87 2-6104 2-7904 20 21-0612 21-2093 54 12-0960 12-6852 88 2-4587 2-6278 21 20-8743 21-0342 55 11-7570 12-3279 89 2-3149 2-4737 22 20-6841 20-8573 56 11-4145 11-9699 90 2-1788 2-3277 23 20-4904 20-6782 57 11-0687 11-6107 91 2-0500 2-1894 24 20-2929 20-4965 58 10-7203 11-2505 92 1-9281 2-0586 25 20-0913 20-3120 59 10-3697 10-8892 93 1-8129 1-9350 26 19-8853 20-1244 60 10-0176 10-5274 94 1-7042 1-8181 27 19-6748 19-9334 61 9-6650 10-1653 95 1-6014 17078 28 19-4596 19-7387 62 9-3125 9-8037 96 1-5043 1-6036 29 19-2394 19-5401 | 63 8-9612 9-4431 97 1-4126 1-5052 30 19-0143 19-3374 64 8-6119 9-0844 98 1-3261 1-4123 31 18-7840 19-1303 65 8-2657 8-7284 99 1-2444 1 -3245 32 18-5486 18-9187 66 7-9233 8-3758 100 1-1671 1-2415 33 18-3078 18-7022 67 7-5858 8-0275

---

1. A similar difficulty meets us in many other cases, as, for instance, when we wish to determine the rate of interest, having given the amount of an annuity for a given term, or the value of an annuity 01 perpetuity deferred for a certain number of years. The reader who wishes to pursue this subject is referred to Francis Baily s Doctrine of Interest and Annuities, 1808, in the appendix to which the formulas of previous authors are examined, and new ones, which are at once simpler and more correct, are demonstrated. Particular cases of the problem are considered in Turnbull s Tables of Compound Interest and Annuities, p. 132, and in various papers in the Assurance Maga zine, among which may be specified De Morgan s paper " On the Deter mination of the Rate of Interest of an Annuity," vol. viii. p. 61, and a letter by J. M Lauchlan in the eighteenth volume. The analogous problem of determining the rate of interest in the bonds of foreign governments above mentioned, debentures, and similar securities, has been fully treated of by Gray, Ass. Mag., xiv. 91, 182, 897 ; and by Makeham, xviii. 132.