Rates of Return as Random Variables

Abstract

An attempt is made to develop a theory of life contingencies and compound interest using a stochastic model of one year returns. Some particular results are obtained using the lognormal distribution. Actuaries traditionally have used a deterministic approach in most of their calculations. In particular it is usual to assume a constant rate of interest. Occasionally varying rates of interest have been used as, for example, when it is assumed that interest rates will drop steadily from 7 percent to 5 percent over 20 years. In these cases the approach is still deterministic. The proposals to introduce variable life insurance generated a number of papers in the Transactions of the Society of Actuaries which used statistical models of investment returns. Kahn [1] assumed that the returns on the underlying investment portfolio are lognormally distributed while Di Paolo [2] generated a probability distribution of returns using historical statistics. Writing in this Journal, Ziock [3] has used time series techniques and simulation in connection with a stochastic model of bond yields. Most of the significant recent advances in the theory of investment have been published in North American academic journals. The pioneering paper in this area was written by Markowitz [4] in 1952. One of the basic ideas in this paper is that the return on a security (and the return on a portfolio of securities) may be represented by a random variable. It is perhaps surprising that this idea has not been more widely exploited within the framework of actuarial science. In the present paper the assumption is that the one year returns are generated by a stochastic process and that the returns are independent from year to year. An additional assumption is that the distribution of the rates of return remains unchanged over time.