Dynamics and Optimal Control of Stochastic Pension Plan Models
Andrew J.G. Cairns
This paper discusses the modelling and control of pension plans.
A continous-time stochastic pension plan model is proposed in which there are two risky assets as well as randomness in the level of benefit outgo. We consider Markov control strategies which optimise over the contribution rate and over the range of possible asset-allocation strategies.
A quadratic loss function is used which extends that of Boulier et al. (1995). Using this loss function it is shown that both the optimal contribution rates and the asset-allocation strategies are linear functions of the fund size.
Boulier et al (1995) proved the perhaps surprising result that under the optimal asset-allocation strategy as the level of surplus in the pension plan increases the proportion of the fund invested in high-return, high-risk assets, such as equities decreases. This paper demonstrates that this result applies to a much larger class of models and loss functions.
Dynamic optimal solutions are compared with their static counterparts.
It is also found that the optimal strategies do not depend upon the amount of uncertainty in the level of benefit outgo. This means that, for quadratic loss functions, small pension plans should operate in precisely the same way as large pension plans even though the relative level of variability will be higher for a small pension plan.
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; optimal control; contstraints