Investing for Retirement

Abstract

A frequently quoted rule of thumb for allocating assets in a pension plan is that at any time 60% should be in stocks and 40% in bonds. How can these percentages be justified? What are the criteria under which this type of dynamic investment strategy is optimal, and what are the optimal strategies, if the criteria are modified? Suppose that an investor has a certain decision horizon and has chosen an appropriate utility function for measuring his utility of wealth at that time. Then, maximizing the expected utility of wealth at the decision horizon leads to a rational compromise between risk and return. First, we consider the one-period model, in which arbitrary random payments, due at the end of the time period, are traded at the beginning of the interval and valued by a price density. To facilitate understanding of the optimal decision, we introduce the risk tolerance function that is associated with the utility function, and also the implied utility function, that is, the maximal expected utility considered a function of the initial wealth. Second, we consider continuous-time complete securities market models, in which rebalancing of the asset portfolios can take place dynamically over time. This seemingly more complex problem can be reduced to the first problem. The key is that the optimal investment strategy corresponds to the self-financing portfolio that replicates the optimal payoff in the first problem. If there are only two investment vehicles, a risky and a risk-free asset, then the optimal investment strategy is as follows: at any time, the amount invested in the risky asset must be the product of the current risk tolerance and the risk premium on the risky asset, divided by the square of the diffusion coefficient of the risky asset. This result can be restated as follows: the Merton ratio, which is the fraction of current wealth invested in the risky asset, must be the risk-neutral Esscher parameter divided by the elasticity, with respect to current wealth, of the expected marginal utility of optimal terminal wealth. In the more realistic case with more than one risky asset, equally explicit rules are given for the optimal investment strategy. For example, the ratios of the amounts invested in the different risky assets are constant in time; the ratios depend only on the risk-neutral Esscher parameters. Hence, the risky assets can be replaced by a single mutual fund with the right asset mix. In this sense, the case of multiple risky assets can be reduced to the case of a single risky asset. Throughout the paper, explicit formulas are given for the cases of linear risk tolerance utility functions.