A popular asset allocation strategy for managing equity risk during the accumulation phase of a defined contribution (DC) pension plan is deterministic lifestyling. At the beginning of the plan, the contributions are invested entirely in equities. Then on a predetermined date prior to retirement (e.g., ten years), the assets are switched gradually into bonds at a rate equal to the inverse of the length of the switchover period (e.g., 10% per year). By the date of retirement, all the assets are held in bonds, which are then sold to purchase a life annuity that provides the pension. The aim of the strategy is to reduce the impact on the pension of a catastrophic fall in the stock market just before the plan member retires. Although deterministic lifestyling is a simple strategy to explain to plan members and to implement, there is no evidence that it is optimal in the sense of maximising the plan member’s expected terminal utility.
The purpose of this paper is to find the optimal dynamic asset allocation strategy for a defined contribution pension plan, taking into account the stochastic features of the plan member’s lifetime salary progression as well as the stochastic properties of the assets held in his accumulating pension fund. Of particular importance is the fact that salary risk (the fluctuation in the plan member’s earnings in response to economic shocks) is not fully hedgeable using existing financial assets. To illustrate, wage-indexed bonds could be used to hedge productivity and inflation shocks, but such bonds are not widely traded. The paper builds on Blake, Cairns & Dowd (2001) which developed a pension plan accumulation programme designed to deliver a pension in retirement that is closely related to the salary (and hence standard of living) that the plan member received prior to retirement. We call the optimal dynamic asset allocation strategy stochastic lifestyling and we compare it against various static and deterministic lifestyle strategies in order to calculate the cost of adopting suboptimal strategies. Despite the apparent increase in complexity in comparison with deterministic lifestyling, stochastic lifestyling is still a relatively easy strategy to implement in practice.
The solution technique makes use of the present value of future contribution premiums into the plan. This is not a new idea and has been used, for example, by Boulier et al. (2001), Deelstra et al. (2000) and Korn & Krekel (2001), building on the original work of Merton (1969, 1971). Liu (2001) examines ways in which the Merton framework can be generalised in different ways to include, for example, stochastic interest rates and stochastic risk premia, but only for the case where utility is a function of the cash lump sum at the beginning of the retirement period.
Where our approach differs from these studies is in:
- the use of a salary-related numeraire or argument in the plan member’s
utility function; and - assuming that the purpose of the pension plan is to deliver a pension (life annuity) in retirement rather than a cash lump sum at retirement
Although these differences do not affect the basic form of the optimal solution derived in these earlier studies, we find that the optimal proportions invested in each of the key asset classes, cash, bonds and equities, are very different. More significantly, we also find that these optimal proportions, in general, differ substantially from those implied by deterministic lifestyling (which ignores both the plan member’s attitude to risk and any correlation between his salary and the returns on assets held in the fund), so that the cost of the latter strategy can be considerable in terms of the additional premiums into the plan needed to match the expected utility of the optimal strategy. Indeed we show that a strategy with a fixed asset allocation throughout the life of the plan that takes into account the correlation between the plan member’s salary and the return on risky assets can, if the plan member’s degree of risk aversion is sufficiently low, have a higher level of expected utility than deterministic lifestyling.
However, the optimal stochastic lifestyle asset allocation, unlike that for deterministic lifestyling, is sensitive to certain underlying assumptions, e.g., concerning the process determining interest rates. Hence, we first develop our results using a simple stochastic model in which the interest rate is deterministic (Section 2). We then extend the model to a more general stochastic setting (Section 3). This allows us to analyse separately (a) the effect of the salary-related numeraire in the utility function and (b) the pension purchased at retirement and its dependence on uncertain interest rates.
We show that in the former case the optimal asset allocation can be replicated using two efficient mutual funds, whereas the latter case needs three efficient mutual funds. One mutual fund (which is heavily dominated with equities) is designed to satisfy the risk appetite of the plan member. The second fund (which is heavily dominated with cash) is designed to hedge the salary risk within the pension plan. The third fund (which is heavily dominated with bonds) is designed to hedge interest rate risk (and hence annuity risk) in the case where interest rates are stochastic.
