How to optimize a portfolio using skewness?

Question

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.

Can anyone please help on how to go about it citing the formulae used for portfolio optimation including the objective function?

Answer 1

You can not account or skewness in the mean-variance framework as skewness is the third central moment. Thus what I would do is

• 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.