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Basically i am confused as to which formula to use for portfolio skew and kurtosis and how to use the same in the optimization problem. I would also like to know the options available regarding the software on which the optimization can be done.

Edit

Based on this post, we are able to include portfolio skewness, using the co-skewness matrix $M_3$, using a third-order Taylor series expansion, but how to also add in the minimization of portfolio skewness, $M_4$ (fourth-order Taylor expansion)?

$$\arg \max_w w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w)$$

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  • $\begingroup$ What are you are doing to optimize for mean and standard deviation? $\endgroup$ Commented Nov 15, 2013 at 14:49
  • $\begingroup$ I think mean and standard deviation optimization can be done in excel itself, however its the other two that are difficult can you suggest some method, also i am not so clear on the formulae, it would be of great help if someone could tell me or at least show me a decent place to find the formulae $\endgroup$
    – Sai
    Commented Nov 16, 2013 at 3:46
  • $\begingroup$ The mean - variance framework can be justified via a Taylor Expansion stopping after the second step. Skew and Kurtosis can be included by expanding the Taylor approximation of the CE. $\endgroup$ Commented Jul 3, 2015 at 11:40

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