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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?

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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.

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  • $\begingroup$ The way you "define" the optimization problem in Latex notation is not the most intuitive form I've ever seen haha. $\endgroup$ – SRKX Oct 28 '15 at 9:19
  • $\begingroup$ You are right .. let me fix it a bit ... $\endgroup$ – Richard Oct 28 '15 at 9:19

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