Pitfalls of downside performance measures with arbitrary targets

The Sharpe ratio has been criticized with regard to the assumptions of mean-volatility portfolio selection. Downside performance measures were developed to resolve this critique; they are consistent with expected utility under less restrictive assumptions. The most prominent family of downside performance measures is known as Kappa ratios and puts above target returns into relation to lower partial moments. While the Sharpe ratio of a mutual fund examines whether portfolios of mutual fund and risk-free asset dominate risk-adjusted passive portfolios of benchmark and risk-free asset, this characteristic cannot be transferred to downside performance measures with arbitrary targets. We show that Kappa ratios assign different values to passive strategies with varying fractions of benchmark and risk-free asset if the target differs from the risk-free rate. This effect can lead to reverse rankings of inferior and superior performing mutual funds. In addition, even the ratio of excess return and excess downside risk of passive portfolios is not constant in general. Therefore, downside performance measures turn out to be only applicable in asset management if the target is set equal to the risk-free rate.