Proper use of the modified Sharpe ratios in performance measurement: rearranging the Cornish Fisher expansion

Performance analysis is a key process in finance to evaluate or compare investment opportunities, allocations, or management. The classical method is to compute the market or sub-market returns and volatilities, and then calculate the standard performance measure, namely, the Sharpe ratio. This measure is based on the first two moments of a return distribution. Therefore, a significant weakness of this method is that it implicitly assumes that the distribution is Gaussian (if it is not Gaussian, the approach may lead to a bad fit). In fact, risk comes from not only volatility, but also from higher moments of distribution such as skewness and kurtosis. The standard method to resolve this issue is to use the modified Sharpe ratio; this method replaces the classical Sharpe ratio volatility with the value at risk. The latter is computed using the Cornish Fisher expansion, a tool based on the first four moments of return distribution. This methodology, however, may present a major pitfall: in some cases, quantile functions do not stay monotone. In this paper, we show how this tool can be used effectively through a specific procedure, rearrangement. We compare various metrics using rank correlation, and demonstrate how and in which cases the proposed procedure delivers ranking different from the standard Sharpe ratio ranking. Furthermore, we show how our technique offers better distribution approximations and is therefore a more useful performance metric. Institutional investors may find the technique proposed here useful in that it allows for considering non-normality in performance analysis.