# Optimize an equity portfolio for the four central moments: problem formulation

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)?

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

• What are you are doing to optimize for mean and standard deviation? Nov 15 '13 at 14:49
• 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
– Sai
Nov 16 '13 at 3:46
• 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. Jul 3 '15 at 11:40