10.1 Introduction
Markowitz’s mean–variance portfolio (Markowitz, 1952) formulates the portfolio design as a trade-off between the expected return \(\w^\T\bmu\) and the risk measured by the variance \(\w^\T\bSigma\w\) (see Chapter 7 for details): \[ \begin{array}{ll} \underset{\w}{\textm{maximize}} & \w^\T\bmu - \frac{\lambda}{2}\w^\T\bSigma\w\\ \textm{subject to} & \w \in \mathcal{W}, \end{array} \] where \(\lambda\) is a hyper-parameter that controls the investor’s risk aversion and \(\mathcal{W}\) denotes an arbitrary constraint set, such as \(\mathcal{W} = \{\w \mid \bm{1}^\T\w=1, \w\ge\bm{0} \}\).
Nevertheless, it has been well recognized over decades of research and experimentation that measuring the portfolio risk with the variance \(\w^\T\bSigma\w\) or, similarly, the volatility \(\sqrt{\w^\T\bSigma\w}\) may not be the best choice for out-of-sample performance. Markowitz himself recognized and stressed the limitations of the mean–variance analysis (Markowitz, 1959). As a consequence, academics and practitioners have explored alternative risk measures that satisfy desirable properties, notably the family of coherent risk measures (Artzner et al., 1999).
This chapter explores a variety of risk measures alternative to the variance, namely, the downside risk, semi-variance, semi-deviation, value-at-risk (VaR), conditional value-at-risk (CVaR), expected shortfall (ES), and drawdown. Particular emphasis is placed on how to incorporate such alternative risk measures in the portfolio formulation itself.