Some Notes on the Dynamics and Optimal Control of Stochastic Pension Fund Models in Continuous time

This paper discusses the modeling and control of pension funds. A continuous-time stochastic pension fund model is proposed in which there are risky assets plus the risk-free asset as well as randomness in the level of benefit outgo. We consider Markov control strategies which optimize over the contribution rate and over the range of possible asset-allocation strategies. For a general (not necessarily quadratic) loss function it is shown that the optimal proportions of the fund invested in each of the risky assets remain constant relative to one another. Furthermore, the asset allocation strategy always lies on the capital market line l'amitiar from modern portfolio theory. A general quadratic loss function is proposed which provides an explicit solution for the optimal contribution and asset-allocation strategies. It is noted that these solutions are not dependent on the level of uncertainty in the level of benefit outgo, suggesting that small schemes should operate in the same way as large ones. The optimal asset-allocation strategy, however, is found to be counterintuitive leading to some discussion of the form of the loss function. Power and exponential loss functions are then investigated and related problems discussed. The stationary distribution of the process is considered and optimal strategies compared with dynamic control strategies. Finally there is some discussion of the effects of constraints on contribution and asset-allocation strategies. Keywords: Continuous time; stochastic differential equation; asset-allocation; contribution strategy; Bellman equation; optimal control; constraints.