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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.
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