Portfolio Optimization Rules beyond the Mean-Variance Approach

Résumé : In this paper, we revisit the relationship between investors' utility functions and portfolio allocation rules. We derive portfolio allocation rules for asymmetric Laplace distributed $ALD(\mu,\sigma,\kappa)$ returns and compare them with the mean-variance approach, which is based on Gaussian returns. We reveal that in the limit of small $\frac{\mu}{\sigma}$, the Markowitz contribution is accompanied by a skewness term. We also obtain the allocation rules when the expected return is a random normal variable in an average and worst-case scenarios, which allows us to take into account uncertainty of the predicted returns. An optimal worst-case scenario solution smoothly approximates between equal weights and minimum variance portfolio, presenting an attractive convex alternative to the risk parity portfolio. Utilizing a microscopic portfolio model with random drift and analytical expression for the expected utility function with log-normal distributed cross-sectional returns, we demonstrate the influence of model parameters on portfolio construction. Finally, we address the issue of handling singular covariance matrices by imposing block structure constraints on the precision matrix directly. This comprehensive approach enhances allocation weight stability, mitigates instabilities associated with the mean-variance approach, and can prove valuable for both short-term traders and long-term investors.