Investment Choices, and Risk-adjusted Performance Measures with Skewness

Asset returns are involved with multi-dimensional risks and therefore any
good risk-adjusted performance measure (RAPM) should incorporate these
risks. We extend the risk-adjusted performance measure theory developed
by Xiang, Liu and Wang (2012) to include multiple risks. The following
models are derived in this paper: a RAPM with skewness under individual
equilibrium for an individual investor, a RAPM that comprises the skewness
under market equilibrium for all investors using a three-moment asset
pricing model. Both the individual and market RAPMs show that investors
prefer a positive skewness to a negative skewness, ceteris paribus. We
prove that the RAPM is the Sharpe ratio when the return doesn’t show any
skewness. Finally this paper provides a theoretical justification to the
relationships between investment choices, an asset pricing model, an
individual RAPM, and a market RAPM.

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