To calculate it, we take the difference of the mean of the distribution
to the target and we divide it by the l-root of the lth lower partial
moment. To calculate the lth lower partial moment we take the subset of
returns below the target and we sum the differences of the target to
these returns. We then return return this sum divided by the length of
the whole distribution.
$$Kappa(R, MAR, l) = \frac{r_{p}-MAR}{\sqrt[l]{\frac{1}{n}*\sum^n_{t=1}
max(MAR-R_{t}, 0)^l}}$$
For l=1 kappa is the Sharpe-omega ratio and for l=2 kappa
is the sortino ratio.
Kappa should only be used to rank portfolios as it is difficult to
interpret the absolute differences between kappas. The higher the
kappa is, the better.