A Unified Formula of the Optimal Portfolio for Piecewise Hyperbolic Absolute Risk Aversion Utilities

TL;DR

This paper introduces a unified formula for optimal portfolios under a new family of piecewise hyperbolic absolute risk aversion utilities, capturing diverse risk behaviors and improving financial decision analysis.

Contribution

It develops a general closed-form solution for optimal portfolios with PHARA utilities, encompassing many existing utilities and providing insights into risk behaviors.

Findings

Non-concavity increases risk-taking behavior.

Non-differentiability reduces risk-taking.

PHARA portfolios perform well on market data.

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.