Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/96

86 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 &lt;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 .) 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