11.3 Risk Contributions
The whole idea of the risk parity portfolio hinges on quantifying the decomposition of the portfolio risk into the sum of risk contributions from the individual assets: \[ \textm{portfolio risk} = \sum_{i=1}^N \textm{RC}_i, \] where \(\textm{RC}_i\) denotes the risk contribution (RC) of the \(i\)th asset to the total risk. The choice of risk measure depends on the portfolio designer, with common options being volatility, value-at-risk (VaR), or conditional VaR (CVaR). For a list of performance measures, refer to Section 6.3, and for a detailed discussion on portfolio design based on alternative performance measures, see Chapter 10. Euler’s theorem offers the solution for the desired decomposition of portfolio risk, cf. Litterman (1996), Tasche (2008), and references therein.
Theorem 11.1 (Euler's homogenous function theorem) Let a continuous and differentiable function \(f:\R^N \rightarrow \R\) be a positively homogeneous function of degree one.51 Then \[\begin{equation} f(\bm{w}) = \sum_{i=1}^N w_i \frac{\partial f}{\partial w_i}. \tag{11.1} \end{equation}\]
From Theorem 11.1, for a given risk measure \(f(\w)\), we can interpret the risk contribution from the \(i\)th asset as \[ \textm{RC}_i = w_i \frac{\partial f(\w)}{\partial w_i}. \] In addition, it is convenient to define the related marginal risk contribution (MRC) as \[ \textm{MRC}_i = \frac{\partial f(\w)}{\partial w_i}, \] which evaluates the portfolio risk’s sensitivity with respect to the \(i\)th asset weight, and the relative risk contribution (RRC) as \[ \textm{RRC}_i = \frac{\textm{RC}_i}{f(\w)}, \] which satisfies \(\sum_{i=1}^N \textm{RRC}_i = 1\).
The following measures of risk do satisfy Euler’s requirement in Theorem 11.1 and the decomposition in (11.1) can be employed.
For the volatility, \(\sigma(\w) = \sqrt{\w^\T\bSigma\w},\) Euler’s theorem can be used with \[ \textm{RC}_i = w_i \frac{\partial \sigma(\w)}{\partial w_i} = \frac{w_i(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}}. \]
For the VaR, defined as \(\textm{VaR}_{\alpha}(\w) = \inf\left\{\xi_0 \mid \textm{Pr}(-\w^\T\bm{r}\leq\xi_0)\geq\alpha\right\}\) with \(\alpha\) the confidence level (e.g., \(\alpha=0.95\)), \(\bm{r}\) representing the random market return vector, and \(-\w^\T\bm{r}\) denoting the (random) loss of portfolio \(\w\), it follows (Hallerbach, 2003) that \[ \textm{RC}_i = w_i \frac{\partial \textm{VaR}_{\alpha}(\w)}{\partial w_i} = \E\left[-w_i r_i \mid -\w^\T\bm{r}=\textm{VaR}_{\alpha}(\w)\right]. \]
For the CVaR, defined as \(\textm{CVaR}_{\alpha}(\w) = \E\left[-\w^\T\bm{r} \mid -\w^\T\bm{r}\geq\textm{VaR}_{\alpha}(\w)\right]\), it follows (Scaillet, 2004) that \[ \textm{RC}_i = w_i \frac{\partial \textm{CVaR}_{\alpha}(\w)}{\partial w_i} = \E\left[-w_i r_i \mid -\w^\T\bm{r}\ge\textm{VaR}_{\alpha}(\w)\right]. \]
In practice, the VaR and CVaR risk contribution expressions are not easily computable. Interestingly, if the returns \(\bm{r}\) follow a Gaussian distribution, then the VaR and CVaR can be expressed explicitly as (McNeil et al., 2015) \[ \begin{aligned} \textm{VaR}_{\alpha}(\w) &= -\w^\T\bmu + \kappa_1(\alpha)\sqrt{\w^\T\bSigma\w},\\ \textm{CVaR}_{\alpha}(\w) &= -\w^\T\bmu + \kappa_2(\alpha)\sqrt{\w^\T\bSigma\w}, \end{aligned} \] where \(\kappa_1(\alpha) \triangleq \Phi^{-1}(\alpha)\), \(\kappa_2(\alpha) \triangleq \frac{1}{(1-\alpha)}\frac{1}{\sqrt{2\pi}}\textm{exp}\left(-\frac{1}{2}\Phi^{-1}(\alpha)^2\right)\), and \(\Phi(\cdot)\) is the cumulative distribution function of a zero-mean unit-variance Gaussian variable.
Volatility Risk Contributions
In the rest of the chapter, we will focus on the portfolio volatility, \(\sigma(\w) = \sqrt{\w^\T\bSigma\w},\) which admits the decomposition \[ \sigma(\w) = \sum_{i=1}^N w_i \frac{\partial \sigma}{\partial w_i} = \sum_{i=1}^N \frac{w_i(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}}, \] leading to the following expressions: \[ \textm{MRC}_i = \frac{(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}},\qquad \textm{RC}_i = \frac{w_i(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}},\qquad \textm{RRC}_i = \frac{w_i(\bSigma\w)_i}{\w^\T\bSigma\w}. \]
References
A function \(f(\w)\) is a positively homogeneous function of degree one if \(f(c\w)=cf(\w)\) holds for any \(c>0\). This condition is satisfied by the volatility, VaR, and CVaR, among others (but not by the variance, for example).↩︎