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Chapter 10 Portfolios with Alternative Risk Measures

“I try all things, I achieve what I can.”

— Herman Melville, Moby Dick

Markowitz’s mean–variance portfolio optimizes a trade-off between expected return and risk measured by the variance. The higher the variance, the more uncertainty, which is undesired, and vice versa. In principle, this makes sense and follows our intuitive expectation of a measure of risk.

However, as already indicated by Markowitz, the variance and volatility are very simplistic measures of risk. To start with, they penalize both the unwanted losses and the desired gains. In addition, the shape of the distribution function of the returns is being ignored. Rather than focusing on the width of the middle part of the distribution (as the volatility does), it is the tail of the distribution that characterizes the big losses.

This chapter explores a variety of alternative and more sophisticated measures proposed over the past seven decades (such as downside risk, semi-variance, value-at-risk, conditional value-at-risk, expected shortfall, and drawdown) and, more importantly, how to incorporate such measures in the portfolio formulation in a manageable way.

This material will be published by Cambridge University Press as Portfolio Optimization: Theory and Application by Daniel P. Palomar. This pre-publication version is free to view and download for personal use only; not for re-distribution, re-sale, or use in derivative works. © Daniel P. Palomar 2025.