Optimal Portfolio with Power Utility of Absolute and Relative Wealth

TL;DR

This paper introduces a new portfolio optimization model combining absolute and relative wealth using power utility functions, providing explicit solutions and applications within the CAPM framework.

Contribution

It develops an explicit solution for a combined absolute and relative wealth utility maximization problem, extending the classic Merton model to multiple benchmarks.

Findings

Explicit solutions for combined wealth utility optimization.

Comparison with the classic Merton solution.

Application within the CAPM setting.

Abstract

Portfolio managers often evaluate performance relative to benchmark, usually taken to be the Standard & Poor 500 stock index fund. This relative portfolio wealth is defined as the absolute portfolio wealth divided by wealth from investing in the benchmark (including reinvested dividends). The classic Merton problem for portfolio optimization considers absolute portfolio wealth. We combine absolute and relative wealth in our new utility function. We also consider the case of multiple benchmarks. To both absolute and relative wealth, we apply power utility functions, possibly with different exponents. We obtain an explicit solution and compare it to the classic Merton solution. We apply our results to the Capital Asset Pricing Model setting.