JOURNAL OF THE INSTITUTE OF ACTUARIES. AND ASSURANCE MAGAZINE. On Lubbock's Formula for Approximating to the Value ov a Life Annuity. By T. B. SPRAGUE, M.A., Manager ov the Scottish Equitable Life Assurance Society, and a Vice-President ov the Institute ov Actuaries. THE method proposed by the late Sir J. W. Lubbock ov approxi mating to the value ov a life annuity, appears hitherto to hav been very littl notist by mathematicians and not at all by practical men. It was demonstrated by him in the Cambridge Philosophical Transactions for the year 1829, in a paper entitld "On the Comparison of various Tables of Annuities"; and a short account ov it is also givn in the articl Probability, publisht by the Useful Knowledge Society, and very offen bound up with David Jones's work on Annuities: see § 56, p. 36. The former paper was reprinted in the Assurance Magazine (vol. v, p. 277). The original is disfigurd by numerous misprints, which ar all faithfully reproduced in the reprint; and when we add that very littl numerical illustration ov the application ov the method is givn, we hav probably said enuf to account for its attracting so littl attention in any quarter. Considerd as a means ov approximating to the value ov an annuity, the formula has been superseded by the simpler one givn by Mr. Woolhouse; but I believ it may still be practically useful in som cases. VOL. XVIII. Y I will therefore first proov the formula, and then giv som numerical illustrations ov its application. The following is Sir J. Lubbock's demonstration, with such amplification as seems desirabl to render it more easily intelligibl. Let yo, Yi, Yzi, ・・・・ Yni, Y(n+1)i, • ... Ymni, be successiv .... i + (n−1) (n − 1.i—1)}▲2yo + 1.2.3 {1.i—1.i—2+2.2i−1.2i−2+...+n−1(n—1.i—1) (n—1.i—2)}▲3y% Proceeding in this way for each adjacent n terms, and adding all +i{1+2+3+....+n−1}{Ayo+Ayni +....+AY(m—1) ni} + + Ayo+Ayni + Ayani+....+AY(m-1)ni=Ym-Yo. so that we get by substitution, +Ymni_i=n{Yo+Yni+Y2ni+ · +Y(m_1)ni} Yo + Yi + Y 2i+ + i.i-1.i—2+2i.2i-1.2i-2+....+n—1.i(n−1.i—1)(n−1.i—2) 1.2.3 + The coefficients ov the varios terms ar clearly those ov the 1+(1+x)2+(1+x)2i + . . . . + (1+x)(n−1)i, powers since ni=1. Now it is proovd in De Morgan's Diff. Calc. (see p. 314, § 184), that the expansion ov the last quantity is 1 Hence, substituting and putting for i, we get +Ym_i=n{yo+Y1+Y2+....+Ym–1} n Yo+Y i + Y zi+ This is the formula givn by De Morgan (p. 316), Lubbock having putting C1, C2, C3. ... for the coefficients ov the several terms after the two first. It will be seen that as far as we hav each gon, ov these quantities involvs n2-1 as a factor. The formula can be much more briefly proovd by the method ov Separation ov Symbols (or Calculus ov Operations). The problem is to find a formula for yo+Y1+Yzi+ . . . . +Ymni-i in terms ov yo+Yni+Yzni+ .... +Y(m-1)ni, (or Yo+Yi+Y2+ ··· +ym-1), and the differences ov these last quantities. ... We hav +Ymni_i=(1+D2 + D2i + . . . . +Dmni-i)yo 1 Dm-1 Di-1 It remains to evaluate the first term ov this series. The easiest method ov doing this is to make i=n=1 in the formula, which givs 1 and finally, putting for i, we get the formula (1). Prof. De n Morgan calls this (p. 315, § 185) "the most difficult instance If we hav calculated (or know otherwize) only the terms = 1 Di−1 ̄ (D—▲)—¿Di — 1 D(1 1 A D = D -i 1 -1 i+1 A 22-1 A2 22-1A3 (¿2 — 1) (19 — ¿2) ▲1 + + г +....} + + ... iA 2i 12i D Yo + Yi +Yzi+ . . . . +Ymni-i= Di_1Ym ̄ Di—1Yo |