An Approximate Solution for the Unknown Rate of Interest: For an Annuity Certain
推导了确定年金问题中未知利率的近似公式，给出了两个近似解，并讨论了其精度，对精算和金融实践有实用价值。
This article develops approximations for the unknown rate of interest in an annuity problem, ani= kThe approximate solutions i = (nk)2 and i = (+ (k~)2 1-(k)2 1 + n [n) n+l)f 1 ( n )(n+l) I k (1 + ) are derived. The results are useful in practice. Introcluction. On page 60 of [ 1], Kellison considers the problem of finding the unknown rate of interest determined by the equation a= k. Three different solutions are offered: 1. Interpolation in the tables, 2. Treating a1 = u + u2 + Un = k as an nth degree polynomial whose roots are to be found, and 3. Iterat&n by means of i = 1 (1+ij)n. The purpose of this paper is to derive approximate analytical formulae and discuss their accuracy. Derivation a= U + U2 + *..un = k. If we multiply by (1 ,u) and rearrange, we obtain =Vn+l (l+k) v + k (1) f'(v) = (n+ 1) vn(1+k) (2) f(v) = (n+ 1) n vn-I > 0 for v> 0 (3) The graph of this function on the interval (0, 1) is obtained from elementary calculus. The following facts about the graph on (0,1) is obtained from elementary calculus. The following facts about the graph on (0,1) are stated without proof: 1. The graph is concave upwards. 2. The graph passes through the points (O,k),(, k)n+l) and (1,0). Murray Silver is an Assistant Professor of Actuarial Science in the Department of Insurance and Risk at Temple University. He holds the Ph.D. degree and is an Associate of the Society of Actuaries.