Long-Term Portfolio Choice Given Uncertain Personal Savings

Investors choosing a portfolio strategy, in order to secure a pension at a future date for example, are faced with many uncertainties. One major uncertainty is the amount by which their pension fund will be supplemented by personal savings from a variety of sources such as life insurance contracts, bequests, or property sales. Over long periods of time these uncertainties are likely to be large and difficult to hedge, and hence may have a significant effect on the dynamic portfolio strategy. Drawing on the results of previous literature on the reaction of investors to non-unhedgeable background risk, and on the theory of stochastic dynamic programming, this article derives optimal strategies for investors maximising the expected utility of terminal wealth, where this wealth consists of the value of a pension fund plus accumulated personal savings. Numerical results, assuming that the market portfolio and the expectation of personal savings follow (possibly) correlated geometric Brownian motions, are derived to illustrate the effects of the size and uncertainty of the personal savings, as well as the effect of the resolution of the uncertainty in them over time. The computation uses a new technique for implementing the stochastic dynamic programming. This involves a binomial approximation, in two dimensions, which ensures that the computations are feasible for relatively long-term problems