Institute of Actuaries' Text Book of the Principles of Interest, Life Annuities, and Assurances, and Their Practical Application Volume 2 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...lead to a modification of Simpson's rule, whereby, when the requisite annuity-values are tabulated, it is much improved. 48 We have seen, Art. 29, that in substituting a life, (to), for the joint lives, (y) and (z), Gompertz's law of mortality is assumed in Simpson's rule. But Makeham's formula represents the law of mortality much more accurately than Gompertz's formula does; and we have seen, Art. 43, that, by altering the rate of interest at the same time that we substitute the single life, (wj), for the joint lives, (y) and (z), we replace Gompertz's formula by Makeham's. Therefore, if i be the given rate of interest, and iw the altered rate; (in the case of the table in this volume -( =+-0065); and if ax, aj, &a., denote the values of annuities at the rate -, and a'x, a'xy, &c, the values at the rate t(5); to find axyl, we must take to such that a'w=ays, and we shall have very nearly a'xw=axyz (14) 49 The values of annuities at rate im are not tabulated, but yet formula 14 may frequently be effectively used by resorting to a simple process of interpolation. 50 We may take an example, again from the "Institute of Actuaries' Life Tables," and in so doing we may safely assume that -"(2)=- + '0065. We cannot find by formula 14 the value of an annuity at 4 per-cent interest, because in the Institute volume joint-life annuities are tabulated only at 3, 3J, and 4 per-cent, and im is greater than ." Let it, however, be required to find by formula 14 the value of an annuitj' at 3 per-cent on three joint lives aged 30,40, and 50. Assuming first that?,," = -+-005=035, we have -- jo=ll'8177=a'M.jM, and a3o;40:5o=-'30:34a-=107SS4. Taking now i'(2)=t +-01=04, we have -&#171...