\( \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \newcommand{\textnormal}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}} \)

10.6 Summary

The variance (similarly, the volatility) is a very simple way to measure the risk of a portfolio and was used in Markowitz’s 1952 seminal mean–variance modern portfolio theory framework. Since then, a wide variety of alternative and more sophisticated measures of risk have been proposed, such as semi-variance, downside risk, VaR, CVaR, EVaR, drawdown, and so on.

Interestingly, many meaningful risk measures can be conveniently incorporated in the context of portfolio optimization, expressed in terms of the raw returns of the assets. Some notable examples include:

  • Downside risk portfolios: The risk focuses on the downside losses, which can be formulated in convex form (parameterized by the parameter \(\alpha\)):
    • \(\alpha=1\): formulated as a linear program;
    • \(\alpha=2\): semi-variance portfolio formulated as a quadratic program;
    • \(\alpha=3\): more risk averse and formulated as a convex program.
  • Tail portfolios: The risk is measured by the tail of the distribution of the losses, which can be formulated in convex form:
    • CVaR portfolios: based on the mean of the tail and formulated as a linear program;
    • EVaR portfolios: based on a smooth approximation of the CVaR and formulated in terms of the exponential cone;
    • worst-case portfolio: extreme version of CVaR and EVaR portfolios and formulated as a linear program.
  • Drawdown portfolios: The risk is based on the drawdown, which can be formulated as linear programs:
    • maximum drawdown portfolio: based on the single worst drawdown;
    • average drawdown portfolio: based on the average of all the drawdowns; and
    • drawdown CVaR portfolio: based on the average of the tail of the drawdowns.