Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans

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Abstract

We investigate asset-allocation strategies open to members of defined-contribution pension plans with a model that incorporates asset, salary (labour-income) and interest-rate risk. We propose a novel form of terminal utility function, incorporating habit formation, that uses the member's final salary as a numeraire. The paper discusses various properties and characteristics of the optimal asset-allocation strategy both with and without the presence of non-hedgeable salary risk. Finally, we compare the performance of the optimal strategy with some popular alternatives used by pension providers and we conclude that it significantly enhances the welfare of a wide range of potential plan members relative to these other strategies.

Introduction

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, beginning on a predetermined date (e.g., 10 years) prior to retirement, 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 aims of the strategy are to reduce the impact on the pension of a catastrophic fall in the stock market just before the plan member retires and to hedge the interest-rate risk inherent in the annuity-purchase decision. Deterministic lifestyling is a simple strategy to explain to plan members and to implement, and is widely used as the default strategy or as one option offered by many UK DC pensions providers. Similar deterministic strategies have also been recommended in other countries (for example, in a US context, Malkiel (2003), recommends a mix of bonds and equities which changes over time in a similar way to the deterministic lifestyle strategy). However, there is no evidence that it is an optimal strategy in an objective sense.

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 (or labour-income 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 both productivity and inflation shocks, but such bonds are not widely traded. The paper builds on Blake et al. (2001) which developed a pension plan accumulation programme designed to deliver a retirement pension that is closely related to the salary (and hence standard of living) that the plan member received immediately prior to retirement. We call the optimal dynamic asset allocation strategy stochastic lifestyling and compare it against various static and deterministic lifestyle strategies to calculate the cost of suboptimal strategies. Moreover, stochastic lifestyling is still a relatively easy strategy to implement in practice, despite the apparent increase in complexity compared to deterministic lifestyling.

The solution technique uses the expected present value of future contribution premiums into the plan. This is not a new idea and has been used by Boulier et al. (2001), Deelstra et al. (2000) and Korn and Krekel (2002) and others, building on the original work of Merton, 1969, Merton, 1971. Liu (2005) examines ways in which the Merton framework can be generalised 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 date of retirement.

Where our approach differs from these studies is in

  • the use of a salary-related numeraire (or utility numeraire) as an argument in the plan member's utility function;1 and

  • assuming that the purpose of the pension plan is to deliver a pension (i.e. life annuity) in retirement rather than a cash lump sum at the date of retirement.2

Although these differences do not alter 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 often 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.

However, unlike the case for deterministic lifestyling, the optimal asset allocation under stochastic lifestyling is sensitive to certain underlying assumptions, e.g., concerning the process determining interest rates. To clarify key issues, we therefore first derive 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 (and hence annuity) risk in the case where interest rates are stochastic.3

Section snippets

The structure of the model

This section develops the optimal asset allocation strategy using a simple model with deterministic nominal interest rates. The aim of the simple model is to highlight the following features:

  • the use of the plan member's salary as a numeraire in the utility function;

  • the treatment of a stream of contribution premiums linked to salary; and

  • the consequences of salary not being fully hedgeable.

The first two features are straightforward to deal with and do not cause particular problems when salary is

The structure of the model

We will now incorporate three extensions to the problem:

  • the introduction of a stochastic risk-free nominal rate of interest, r(t);

  • the extension of the investment opportunity set to N risky assets rather than 1;

  • the introduction of the replacement ratio15 as an argument in the terminal utility function.

The components of the model are as follows:

  • The risk-free rate of interest is a one-factor diffusion process

Conclusions

Stochastic lifestyling has at least two advantages over deterministic lifestyling in respect of defined contribution pension plans: it takes into account both the plan member's attitude to risk and the correlation between his salary and asset returns. It is implemented using three efficient mutual funds, resembling investment in cash, bonds and equities, respectively. The equity fund is regarded as high risk and its purpose is to satisfy the risk appetite of the plan member. The cash and bond

Acknowledgements

The authors gratefully acknowledge support from the BSI Gamma Foundation. They also wish to thank participants at the “Risk Theory” and “Stochastic Analysis in Finance and Insurance” meetings in Oberwolfach in September 1999 and May 2000, respectively, for their helpful comments. This includes, in particular, Griselda Deelstra and Ralf Korn. Finally, they wish to thank the two referees of the paper for their extremely helpful comments.

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