I have the following risk adjusted portfolio which I optimise,
where gamma is the risk return trade off, $r$ are the returns and $C$ is the covariance matrix which considers scenarios, so it is not defined as $r^\top r$, but as shown in the following Markowitz paper (page 3, $C = D + GPG'$): https://www.jstor.org/stable/2327552?seq=1
$P$ is a diagonal $SxS$ matrix with the probability
$G$ is an $NxS$ matrix whose entries are given by $𝑔𝑛𝑠=𝜇𝑛𝑠−𝜈𝑛$. Where $𝜇𝑛𝑠$ are the returns of the assets and $𝜈𝑛$ are the returns of the nth asset class weighted by the probabilities of the scenarios. $N$ total numer of assets
$D$ is a diagonal $NxN$ matrix whose entries are given by $𝑑𝑛𝑛=Σ^S_s 𝑝𝑠*(𝜎𝑛𝑠)^2$. Where $𝜎𝑛𝑠$ is the standard deviation of the nth asset for the sth scenario
Now I want to add also the third moment thus the skewness to this optimisation function, but I do not really know how, and if I have to include the scenarios in this skewness and how.
Can you guide me pls? Thanks