A unified formula of the optimal portfolio for piecewise hyperbolic absolute risk aversion utilities

Abstract

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

Keywords:

• Utility theory
• Portfolio selection
• Non-differentiability
• Asymptotic analysis
• Empirical study

JEL Classifications:

• G11
• C61

Acknowledgments

The authors are grateful to the editors and two anonymous referees for valuable comments that have greatly improved this paper. The authors thank members of the group of mathematical finance at the Department of Mathematical Sciences, Tsinghua University for their feedback and useful conversations.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Z. Liang acknowledges support from the National Natural Science Foundation of China [grant number 11871036,12271290]. Y. Liu gratefully acknowledges financial support from the research startup fund at The Chinese University of Hong Kong, Shenzhen [grant number UDF01003336].