Downside Performance Measures and the Sharpe Ratio

In a mean-downside risk framework, portfolio lines that combine the market portfolio and the risk-free asset (i.e., passive benchmark strategies) are non-linear if the target differs from the risk-free rate. Here, downside risk-based performance measures assign different performance levels to passive strategies depending on the composition of the passive portfolio. The same applies for the evaluation of mutual funds. Therefore, we analyze the shape of portfolio lines for arbitrary targets above the risk-free rate in detail. We show that these portfolio lines can be characterized by the Kappa ratio with a target equal to the risk-free rate and, surprisingly, the Sharpe ratio. In addition, we show that minimum downside risk portfolios with varying target can be positioned on a straight line. This allows us to apply target-independent performance measures, even if portfolio lines depend on the target set by the investor. We substantiate our analytical results with an empirical study.