Delay, feedback and variability of pension contributions and fund levels

Abstract In this paper, we construct a general model for studying the variability of contribution rates and fund levels, when there are time delays and feedback into a pension fund. Time delays are introduced into the model using a parameter q , which takes the values 0, 1, 2, 3, …. the time delay of q years is introduced at each time t , when the contribution rate is fixed by considering the information or the data we have for the pension fund at time t − q [e.g. the fund level at time t − q, F ( t − q )]. Real rates of retur on the fund are assumed to be represented by independent, identically distributed random variables. Other assumptions are made but these assumptions are not essential and we use them in order to simplify our mathematical formulae. We obtain expressions for the expectations and the variances of fund and contribution levels for finite t and in the limit as t goes to infinity (for all values of q ). Then we investigate their behaviour and how they change as the parameter q takes the values 0, 1, 2, 3, … up to infinity.