Adjusted Sharpe Ratio for Higher Moments

The ordinary Sharpe ratio only uses the mean and standard deviation of returns, so it treats a strategy that occasionally crashes hard (negative skew, fat tails) the same as one with smooth, symmetric returns, as long as both share the same mean and standard deviation — a well-known blind spot for strategies like short-volatility selling that earn steady small gains punctuated by rare large losses. The adjusted Sharpe ratio (Pezier and White) corrects this by subtracting penalty terms driven by skewness $S$ and excess kurtosis $K$:

$$ASR = SR \left[ 1 + \frac{S}{6}SR - \frac{K}{24}SR^2 \right],$$

where $SR$ is the ordinary Sharpe ratio. Negative skew ($S<0$) lowers the adjusted ratio, and positive excess kurtosis ($K>0$, meaning fatter tails than a normal distribution) also lowers it — both make the score worse than the plain Sharpe ratio would suggest.

For example, a strategy with $SR = 1.0$, skewness $S = -1.5$, and excess kurtosis $K = 6$ gets $ASR = 1.0 [1 + (-1.5/6)(1) - (6/24)(1)^2] = 1.0[1 - 0.25 - 0.25] = 0.50$ — half its plain Sharpe ratio, reflecting the hidden crash risk in its return distribution that the ordinary ratio doesn't see.

By contrast, a strategy with the same $SR = 1.0$ but slightly positive skew ($S=0.3$) and near-normal kurtosis ($K=0.5$) gets $ASR = 1.0[1 + 0.05 - 0.02] \approx 1.03$ — barely different from its plain Sharpe ratio, since it has no hidden tail risk to penalize. Comparing the two strategies side by side shows exactly what the adjustment is for: two funds that look identical on mean and standard deviation alone can have very different adjusted scores once the shape of their return distribution is taken into account.