  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.
