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The return on investment from proportional portfolio strategies

Part of: Limit theorems Markov processes Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Sid Browne
Sid Browne*
Affiliation:
Columbia University
*
∗ Postal address: 402 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA.
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Abstract

Dynamic asset allocation strategies that are continuously rebalanced so as to always keep a fixed constant proportion of wealth invested in the various assets at each point in time play a fundamental role in the theory of optimal portfolio strategies. In this paper we study the rate of return on investment, defined here as the net gain in wealth divided by the cumulative investment, for such investment strategies in continuous time. Among other results, we prove that the limiting distribution of this measure of return is a gamma distribution. This limit theorem allows for comparisons of different strategies. For example, the mean return on investment is maximized by the same strategy that maximizes logarithmic utility, which is also known to maximize the exponential rate at which wealth grows. The return from this policy turns out to have other stochastic dominance properties as well. We also study the return on the risky investment alone, defined here as the present value of the gain from investment divided by the present value of the cumulative investment in the risky asset needed to achieve the gain. We show that for the log-optimal, or optimal growth policy, this return tends to an exponential distribution. We compare the return from the optimal growth policy with the return from a policy that invests a constant amount in the risky stock. We show that for the case of a single risky investment, the constant investor's expected return is twice that of the optimal growth policy. This difference can be considered the cost for insuring that the proportional investor does not go bankrupt.

Keywords

Portfolio theorydiffusionsstationary distributionsconvergence in distributionlimit theoremsstochastic order relationslogarithmic utilityoptimal growth policy

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60J60: Diffusion processes 60H10: Stochastic ordinary differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 216 - 238
Copyright
Copyright © Applied Probability Trust 1998 

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