Insurance: Mathematics and Economics 28 2001 1–11
Annuities under random rates of interest
Abraham Zaks
∗
Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel Received 1 December 1998; received in revised form 1 June 2000
Abstract
We investigate the accumulated value of some annuities-certain over a period of years in which the rate of interest is a random variable under some restrictions. We aim at the expected value and the variance of the accumulated value, and we
suggest two methods to derive these moments. In some cases both methods have similar difficulties, while in other cases one method is significantly preferable in terms of the simplicity of calculations. The novelty of our second approach lies in the fact
that we find recursive relationships for the variance of the accumulated values, and solve these relationships, whilst previous investigators first obtained the second moment. © 2001 Elsevier Science B.V. All rights reserved.
Subj. Class: IE50; IE51 Keywords: Mathematics of finance; Random rates of interest; Random variable; Independent variables; Expected value; Variance; Annuities;
Future value
1. Introduction
Many attempts have been made to investigate stochastic interest rate models and to evaluate their impact under various hypotheses: see Pollard 1971, Boyle 1976, Wilkie 1976, Westcott 1981 and McCutcheon and Scott
1986 to mention just some of the contributions. We investigate the accumulated value of some annuities-certain over a period of years in which the rate of interest is a random variable under some restrictions. We aim at the
expected value and the variance of the accumulated value, and we suggest two methods to derive these moments. In some cases both methods have similar difficulties, while in other cases one method is significantly preferable in
terms of the simplicity of calculations.
The novelty of our second approach lies in the fact that we find recursive relationships for the variance of the accumulated values, and solve these relationships, whilst previous investigators first obtained the second moment.
2. The case of a fixed interest rate
Suppose that the yearly rate of interest j is fixed through the period of n years. We recall that the yearly rate of discount d is given by
1 + j d = j 2.1
∗
Tel.: +972-4-829-4028972-4-822-3108; fax: +972-4-832-4654. E-mail address: azakstechunix.technion.ac.il A. Zaks.
0167-668701 – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 5 9 - 7
2 A. Zaks Insurance: Mathematics and Economics 28 2001 1–11
and the yearly present value v is given by 1 + j v = 1,
2.2 so that we have the equalities
d = jv, 2.3
v + d = 1. 2.4
We assume k ≤ n throughout, unless otherwise specified. The accumulated value after k years denoted ¨s
¯ k|j
of an annuity-due of k yearly payments of 1 is known to be given by the formula
¨s
¯ k|j
= 1 + j
k
− 1 d
2.5 and we note that
¨s
¯ k|j
= 1 + j
k
+ · · · + 1 + j = 1 + j 1 + ¨s
k−1|j
. 2.6
The accumulated value after k years of an increasing annuity-due of k yearly payments of 1, 2, . . . , k, respectively, is denoted I ¨s
¯ k|j
, and is known to be given by the formula I ¨s
¯ k|j
= ¨s
¯ k|j
− j d
. 2.7
We also note that I ¨s
¯ k|j
= 1 + j
k
+ 21 + j
k−1
+ · · · + k1 + j = 1 + j
I ¨s
k−1|j
+ k .
2.8 The accumulated value after k years of an increasing annuity-due of k yearly payments of 1
2
, 2
2
, . . . , k
2
, respectively, is denoted by I
2
¨s
¯ k|j
.
Theorem 2.1.
I
2
¨s
¯ k|j
= 2I ¨s
¯ k|j
− ¨s
¯ k|j
− k
2
d 2.9
Proof. By definition, I
2
¨s
¯ k|j
= 1 + j
k
+ 2
2
1 + j
k−1
+ · · · + k
2
1 + j . A straightforward calculation of I
2
¨s
¯ k|j
− vI
2
¨s
¯ k|j
leads to the following equation: dI
2
¨s
¯ k|j
= 1 + j
k
+ · · · + 2k − 11 + j − k
2
, from which we obtain
dI
2
¨s
¯ k|j
= 2[1 + j
k
+ · · · + k1 + j ] − [1 + j
k
+ · · · + 1 + j ] − k
2
and the result follows by using 2.6 and 2.8. We observe that
I
2
¨s
¯ k|j
= 1 + j
k
+ 2
2
1 + j
k−1
+ · · · + k
2
1 + j = 1 + j I
2
¨s
k−1|j
+ k
2
. 2.10
Substituting the expression I ¨s
¯ k|j
from 2.7 in 2.9, we have the following corollary.
A. Zaks Insurance: Mathematics and Economics 28 2001 1–11 3
Corollary 2.1.
I
2
¨s
¯ k|j
= 1 + v¨s
¯ k|j
+ k
2
− 2k − 2k
2
] d
. 2.11
Whenever the value of the rate of interest j is clear from the context, it is customary to omit it from the symbols and denote the corresponding values as ¨s
¯ k|
, I ¨s
¯ k|
and so on. Similar arguments may be used to show that, for given m = 3, 4, . . . , a recursive formula of type 2.9 can be
derived for I
m
¨s
¯ k|j
, which is defined as the accumulation of an annuity-due of k payments of 1
m
, 2
m
, . . . , k
m
.
3. Single payments and series of level payments
