11.3 Risk contributions
The whole idea of 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 precisely 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.50 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 contributions 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: \[ \begin{aligned} \textm{MRC}_i &= \frac{(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}},\\ \textm{RC}_i &= \frac{w_i(\bSigma\w)_i}{\sqrt{\w^\T\bSigma\w}},\\ \textm{RRC}_i &= \frac{w_i(\bSigma\w)_i}{\w^\T\bSigma\w}. \end{aligned} \]
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 satisfied by the variance, for example).↩︎