Journal of Risk

A variational derivation of risk-adjusted performance measures

George Xiang, Jiangyang Liu and Qi Wang


We derive the risk-adjusted performance measure (RAPM) Omega and the more general Kappa by applying the variational principle to the utility function with respect to the investment choice, which is comprised of how an investment is funded, and its composition, risk and return attributes. These risk measures are shown to be valid for both funded and unfunded investors, but they may fail to distinguish between portfolios with different leverage ratios. In addition, we propose a method for optimizing the RAPMs in order to yield optimal investment choices.

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