11.1 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. One way to address this drawback is the utilization of alternative risk measures as explored in Chapter 10. On top of that, one can add another layer of sophistication by characterizing the risk of the portfolio not just with a single number but with a risk profile that quantifies the amount of risk contributed by each constituent asset. This refined risk characterization allows for a proper control of the portfolio risk diversification (Litterman, 1996; Qian, 2005, 2016; Roncalli, 2013b; Tasche, 2008).
This chapter introduces the risk parity portfolio from its simplest form (with a closed-form solution), through vanilla convex formulations, to the more general nonconvex formulations, providing a wide range of numerical algorithms (Feng and Palomar, 2015, 2016; Maillard et al., 2010).