Formulating Portfolio Optimization with Skewness and Kurtosis

Summary

The document asks how to extend mean-variance portfolio optimization to include skewness and kurtosis. It presents an objective with expected return, a variance penalty, and a third-order co-skewness term based on the portfolio weights and co-skewness matrix. The question is how to incorporate a fourth-order term to account for kurtosis, including whether the goal should be to minimize it, and what software can solve the resulting problem.

No solution, derivation, empirical evidence, or software recommendation is provided. The material is therefore a problem statement rather than a complete optimization method. Any implementation would need to define the desired treatment of skewness and kurtosis, specify constraints and parameter estimates, and address the computational complexity of higher-order moments; those details are not developed here.

Key ideas

• The proposed objective combines expected return, variance, and a third-order co-skewness contribution.
• The question asks how to add a fourth-order moment term to portfolio optimization.
• The document does not provide a derivation, solver recommendation, or empirical results.
• Implementing the idea would require additional choices about moment estimates and portfolio constraints.

Tags

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


# 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)?

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.