When all risk-adjusted performance measures are the same: in praise of the Sharpe ratio
证明，若投资收益率服从Q-径向分布，则多种基于下行风险的绩效指标（如索提诺比率、欧米伽统计量）均等价于夏普比率，反之亦然，从而刻画了夏普比率作为唯一风险调整绩效指标的分布类。
The classical mean-variance investment model is simple, elegant, and popular. As such, it is also subject to criticisms. One unsatisfactory feature of the model is that variance treats the upside and downside equally as risks. In this regard, the downside Lower Partial Moments (LPM) are more attractive as alternative risk measures, since they only penalize the downside. In the meanwhile, considerable amount of recent research efforts have been paid to the so-called Q-radial distributions, which are capable of better modeling the investment returns. In this paper we show that if the investment return rates follow a Q-radial distribution, then the LPM related Risk Adjusted Performance Measures (RAPM), such as the Sortino ratio, the Omega Statistic, the upside potential ratio, and the normalized LPM, are all equivalent to the ordinary Sharpe ratio, which is easy to compute and optimize. Conversely, if all normalized LPM’s are equivalent to the Sharpe ratio, then the underlying distribution must be Q-radial. Therefore, this property characterizes the class of distributions in which the Sharpe ratio is essentially the only risk adjusted performance measure. 1