# How to optimize a portfolio using skewness?

I am trying to do portfolio optimization for 5 stocks taking into account skewness of the portfolio but I am unable to incorporate skewness to the mean variance model.

• formulate the skewness in terms of the asset returns. I.e. for each time-step you have $$r_t = \sum_{i=1}^5 w_i r^i_t,$$ where $r_t^i$ is the return of asset $i$ at time $t$, $w_i$ is the weight and $r_t$ the portfolio return at $t$.
Then you can use the empirical estimator of skewness: $$skew = \frac{ 1/T \sum_{t=1}^T (r_t-\mu)^3}{ \sigma^3},$$ where you need the portfolio variance $$\sigma^2 = w \Sigma w$$ and the expected value $$\mu = 1/T \sum_{t=1}^T r_t,$$ where the above is the sample estimator and $$\mu = \sum_{i=1}^5 w_i \mu_i$$ is the expression in terms of individual expectations. Then you can use this skewness above, $\sigma$ and $\mu$ to define the problem. E.g. $$\mu - \lambda \sigma^2 \rightarrow Max$$ under the constraint $skew \ge x$ for some desired level $x$. Or you use the definition of Cornish-Fisher-VaR in the constraint.