m and the values of the function and the system of differences will be as given in the following schedule: Δ, Az 60 0245351 +0100036 65 0345387 +0131208 0231244 +.0016519 70.0576631 ·0147727 - .0094756 0378971 -.0078237 75 0955602 ·0069490 0448461 80 •1404063 whence a= +.030,510,75 A= +.006,205,0 b=+.014,772,7 B= +.000,311,246,6 give by (12) c=-003,085,9 C=-000,004,114,53 d=-.009,475,6 D=-000,000,631,707 and with these values of the coefficients in (13) we obtain the hypothetical rates of mortality in column (10), between ages 60 and 80. Above this point the rates are those of the Actuaries' table. From the rates of mortality in column (8) up to age 60, and those in column (10) from 60 to the end of life, I have computed the numbers living and dying [columns (11) and (12)], starting with a radix of one million persons at age 10. The irregularities in the second differences at ages 60 and 81 might have been considerably reduced, but I did not think this worth while, as my design was simply illustration, and not the deduction of the best numerical results in this particular instance. And I wish further to state, in conclusion, that the same consideration must be my apology, if any be needed, for whatever errors may have crept into the preceding computations, as well as for some details which have reference simply to the numerical conduct of the example. It is to the general method itself that I would invite criticism. Reduction of Formula for Annuities and Assurances investigated by Mr. Sprague, on the common hypothesis of equal decrements in each year of life. By W. S. B. WOOLHOUSE, F.R.A.S., F.S.S., &c. IN 1869 Mr. Sprague read before the Institute an interesting paper “On the value of Reversionary Annuities payable half-yearly, quarterly, &c., according to the conditions which prevail in practice.” This valuable paper is given in the Journal, vol. xv, page 126. The conditions which prevail in practice, as here for the first time introduced and mathematically investigated by Mr. Sprague, are the m following:-For a yearly reversionary annuity on a life x after the death of another life y, the first payment of the annuity is supposed to be made just one year after the day of the death of y, provided that æ be then living; and for a like annuity payable by m instalments in each year, the first payment of an instalment is supposed to be made just th of a year after the day of the death of y. Also, in each case, it is further supposed that whenever x shall die, after surviving y, the annuity shall be completed up to the day of the death of x by the supplementary payment of the proportion of the annuity estimated in respect of the fractional interval that may then have elapsed since the last yearly payment or instalment was received. In the latter, and, perhaps, the most interesting portion of his paper, Mr. Sprague has, with his usual accuracy and ability, investigated a mathematical expression for the present value of the annuity of practice, on the usual assumption of equal decrements in each year of life; and, according to the same hypothesis, he has also appended expressions for what I have, in a former paper, denominated continuous assurances, that is, assurances payable at the instant of decease. Each of these expressions as they now stand, is, however, so prolix as to be wholly unfit for ordinary calculation, and Mr. Sprague himself states that the formula for the value of the complete reversionary annuity payable yearly, the most simple case, is so complicated as to be practically useless. In the present article it is proposed to examine the particular constitution of these formulæ, and the special results derivable from them, for the purpose of finally reducing the several expressions to more convenient and practical forms. For greater clearness, the several processes will be given in extenso. To begin with the formulæ for assurances, I shall take the present opportunity of introducing and elucidating the use of a new quantity, which, it appears to me, ought henceforth to be admitted as a recognised element of assurance investigations appertaining to two lives, viz., the value of an assurance payable at the end of a year, and which is contingent upon both lives failing in that year. As the contingency here stated is of the second order, the value of an assurance of this kind, which, for brevity, may be appropriately termed a Conjoint Assurance, must always be a small quantity. The probability of the life x failing in the nth year, is Pr,n-1 Px-1, n -Pr,n= -Pr,n; the same in respect to the life y, is Px-1 Py-1, -Py,n; and if Wry denote the present value of the conjoint Pyassurance on the two lives x and Y, then = + axy 1+ axy + axy P:-1,n Py_1,n Py,n Py Px Pr. , Px-l Py-1 ax-1.4 ax. 9-1 Po-1.4-1 P.- Py-1 ax-1.4 @x y-1 1+i Px-1 Py-1 1+(2+i)axy dx-1.9 1+i Px-1 Py-1 From this and the well known forms Ax-1.9 ax.y-1 Px-1 Py-1 1/1-iaxy Ax-1.7 Ax.y-1 + Px-1 Py-1 (1) (2) the following are obtained, viz., 1+i 1 2 1 1 + axy xy xy (4) lti 2 Referring now to Mr. Sprague's paper, at page 137 he gives for the value of a contingent assurance payable at the instant of the death of x provided y survive, on the supposition of equal decrements, the formula AL = ху ¿- + Ax-1.4 (5) Py-1 When the life y is replaced by certainty, the fraction Px-1 Ox-1 becomes ; and hence, in this particular case, (5) Px-1 gives or 1+i + art 8 + ax ax = (6) аху xy xy II Way. i8-i +8 2i8-2i+28-22 is-i +8 i-81+as Ax(1+i)82 (1+1) 8 82 1+i i8 2i8-i +8-22 id-i +8 (1+i)82 (1+i) 82 i į l-iax Az (1+i) 8 (1+i)8 8 Again, from (2) and (5) we obtain generally, i8-2i+28 488—41+48-212 + 128 SĀL-iAl = + 2(1+i) 2(1+i)8 i8-2i+28 Ax.y-1 28-2i+28 Qx-1.7 28 Py-1 28 Pr-1 18–2i+28 < 1+(2+i)dry _ 0x0.9-1 – 0x=1.9} { Py-1 Px-1 28 Here we may observe that the coefficient i8-2i+28 1 1 1 + + =1 12 720 38 is, with great practical precision, represented by 12 since the terms rejected cannot affect even the seventh place of decimals. Hence, as Wry is a symmetrical function as regards x and y, there will result the two relations, 28 (7) 12 i8 (8) ) (9) The reductions here made in the formulæ for assurances will now enable us to reduce Mr. Sprague's complicated formula for the complete reversionary annuity, according to the conditions which prevail in practice. Before proceeding, however, we may observe that, whenever recourse is had to the relations (7) and (8), the i8-2i+28 id exact coefficient will be retained in place of Thus 28 12: xy xy XY Wxy ary ху . Xy Oxy (1+1)82 0x {1 { . no error whatever will be permitted to enter beyond what belongs to the hypothesis, and any simplification of an approximate nature will be deferred to a final step. The expression investigated by Mr. Sprague for the present value of the annuity when payable yearly, and incomplete, is the following, viz., 28-+8 (10) (1+i)82 82 Py-1 For the correction of this formula, in order to make the annuity complete—that is, for the present value of the addition to be made on account of the payment of the annuity up to the day of deathMr. Sprague determines separately two component portions, viz., 21-82-28 u 2(1+2)83 Px-1 (1+2)82 +28-2i -iar—Qxy+(1+1) 9.2 --- (12) 2(1+2)83 Py-1 the sum of which gives, for the required correction, the expression, 2i-82-28 22 (21-82-28) (2+i) -282 2(1+i)83 2(1+i)8 2(1+1) 83 21–82–28 dx-1.9 (1+i)82 +28–2i 2x.y-1 + (13) 283 Px-1 283 Py-1 Adding (10) to (13), Mr. Sprague's formula for the value of the complete reversionary annuity payable yearly is therefore 23-82-28 (2-8)72 21—28+22-2182—282 2(1+1)88 * 2(1+i)83 4x+ (1+1) 83 21-82-28 0x-1.9 (3+1)82+28—21–2i8 .0.x.y-1. . (14) + 283 Py-1 To effect a reduction of this expression, we proceed to reduce separately the component expressions (10), (11), and (12), with the aid of the foregoing results. The formula (10) reduces as follows, viz., i 1-iar Py-1 1 A 1 W 2 i8 W W 238 Ax + . + day |