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The Johnson-Omega Ratio is a parametric version of the Omega ratio and employs Johnson distributions instead of the empirical distribution.

The ratio is calculated as:

\Omega_{Johnson}(r,\mu,\sigma,sk,ku) &=& \frac{\int_{r}^\infty (1-J(\mu,\sigma,sk,ku,x))\,dx}{\int_{-\infty}^r J(\mu,\sigma,sk,ku,x)dx}\\[0.3cm]

Where J is the cumulative Johnson distribution, r the threshold and partition defining the gain versus the loss. \mu, \sigma, sk and ku represent asset characteristic or state dependent statistics mean (e.g. see ARIMA or factor models), variance (e.g. see GARCH or stochastic volatility), skewness (asymmetry) and kurtosis (tail fatness), respectively. Only the first four moments uniquely determine Johnson-Omega, whereas moments of order 5 and higher are endogenously determined and cannot be biased by estimates.

Comparisons can be made with the commonly used Sharpe ratio which considers the ratio of return versus volatility and the (empirical) Omega ratio, which by construction considers all moments.