On simple approximate calculations appropriate to populations with random growth rates

A three-parameter second-order autoregressive process is suggested as a suitable discrete-time stochastic model for the instantaneous growth rate of a population whose mode of growth is basically exponential. Rather complicated formulae are derived for the moments of the population size, but these formulae do have simple asymptotic forms which are shown to be reasonably accurate for smaller values of the time parameter. The relevant equations are numbered (25), (26) and (27). Several numerical examples are given. The model may be of use in dealing with certain actuarial problems when the rate of interest is a random variable. It is necessary, however, to take into account age-specific mortality and the correlation between the present values of payments made at different points of time. Because of these added complications and the fact that actuarial notation is necessary, the details of these methods are not included here. The author hopes to publish a paper on these techniques in an actuarial journal in the not-too-distant future (J. H. Pollard, 1970). The present model has many desirable features, but we cannot claim that it is appropriate in all circumstances. We may be interested in a discrete-time process in which the instantaneous growth rate in year t is anticipated to be δ_t, for example. The actual instantaneous rate δ_t will be different from this, and it might be appropriate to assume that δ_t - δ_t = 2k(δ_{t - 1} - δ_{t - 1}) - k(δ_{t - 2} - δ_{t - 2}) + e_t. The methods described above are still applicable, and only a few of the formulae require rather minor alterations.