**DISCUSSION PAPER PI-9811**

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

*Andrew Cairns*

*Heriot-Watt University, Edinburgh*

**ABSTRACT**

This paper discusses the modelling and control of pension funds.

A continuous-time stochastic pension fund model is proposed in which there

are *n* risky assets plus the risk-free asset 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.

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 familiar 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.

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.

** ISSN 1367-580x**.