14.2 Euler’s Theorem and Risk Decompositions

When we used $\sigma_{p}^{2}$ or $\sigma_{p}$ to measure portfolio risk, we were able to easily derive sensible risk decompositions in the two risky asset case. However, if we measure portfolio risk by value-at-risk or some other risk measure it is not so obvious how to define individual asset risk contributions. For portfolio risk measures that are homogenous functions of degree one in the portfolio weights, Euler’s theorem provides a general method for decomposing risk into asset specific contributions.

14.2.1 Homogenous functions of degree one

First we define a homogenous function of degree one.

Definition 7.1 (homogenous function of degree one)

Let $f(x_{1},x_{2},\ldots,x_{N})$ be a continuous and differentiable function of the variables $x_{1},\,x_{2},\ldots,x_{N}$. The function $f$ is homogeneous of degree one if for any constant $c > 0,$ $f(cx_{1},cx_{2},\ldots,cx_{N})=cf(x_{1},x_{2},\ldots,x_{N}).$ In matrix notation we have $f(x_{1},\ldots,x_{N})=f(\mathbf{x})$ where $\mathbf{x}=(x_{1},\ldots,x_{N})^{\prime}$. Then $f$ is homogeneous of degree one if $f(c\cdot\mathbf{x})=c\cdot f(\mathbf{x})$.

Example 4.1 (Homogenous functions of degree one)

Consider the function $f(x_{1},x_{2})=x_{1}+x_{2}.$ Then $f(cx_{1},cx_{2})=cx_{1}+cx_{2}=c(x_{1}+x_{2})=cf(x_{1},x_{2})$ so that $x_{1}+x_{2}$ is homogenous of degree one. Let $f(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}.$ Then $f(cx_{1},cx_{2})=c^{2}x_{1}^{2}+c^{2}x_{2}^{2}=c^{2}(x_{1}^{2}+x_{2}^{2})\neq cf(x_{1},x_{2})$ so that $x_{1}^{2}+x_{2}^{2}$ is not homogenous of degree one. Let $f(x_{1},x_{2})=\sqrt{x_{1}^{2}+x_{2}^{2}}$ Then $f(cx_{1},cx_{2})=\sqrt{c^{2}x_{1}^{2}+c^{2}x_{2}^{2}}=c\sqrt{(x_{1}^{2}+x_{2}^{2})}=cf(x_{1},x_{2})$ so that $\sqrt{x_{1}^{2}+x_{2}^{2}}$ is homogenous of degree one. In matrix notation, define $\mathbf{x}=(x_{1},x_{2})^{\prime}$ and $\mathbf{1}=(1,1)^{\prime}$. Let $f(x_{1},x_{2})=x_{1}+x_{2}=\mathbf{x}^{\prime}\mathbf{1=f(\mathbf{x})}$. Then $f(c\cdot\mathbf{x})=\left(c\cdot\mathbf{x}\right)^{\prime}\mathbf{1}=c\cdot\mathbf{(x}^{\prime}1)=c\cdot f(\mathbf{x}).$ Let $f(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}=\mathbf{x}^{\prime}\mathbf{x}=f\mathbf{(x)}$. Then $f(c\cdot\mathbf{x})=(c\cdot\mathbf{x})^{\prime}(c\cdot\mathbf{x})=c^{2}\cdot\mathbf{x}^{\prime}\mathbf{x}\neq c\cdot f(\mathbf{x}).$ Let $f(x_{1},x_{2})=\sqrt{x_{1}^{2}+x_{2}^{2}}=(\mathbf{x}^{\prime}\mathbf{x})^{1/2}=f(\mathbf{x})$. Then $f(c\cdot\mathbf{x})=\left((c\cdot\mathbf{x})^{\prime}(c\cdot\mathbf{x})\right)^{1/2}=c\cdot\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{1/2}=c\cdot f(\mathbf{x}).$

$\blacksquare$

Consider a portfolio of $N$ assets with weight vector $\mathbf{x}$, return vector $\mathbf{R}$, expected return vector $\mu$ and covariance matrix $\Sigma$. The portfolio return, expected return, variance, and volatility, are functions of the portfolio weight vector $\mathbf{x}$: $R_{p}=R_{p}(\mathbf{x})=\mathbf{x}^{\prime}\mathbf{R},$ $\mu_{p} =\mu_{p}(\mathbf{x})=\mathbf{x}^{\prime}\mu,$ $\sigma_{p}^{2} =\sigma_{p}^{2}(\mathbf{x})=\mathbf{x}^{\prime}\Sigma \mathbf{x},$ $\sigma_{p} =\sigma_{p}(\mathbf{x})=(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2}.$ Additionally, the normal portfolio return $\alpha-$quantile $q_{\alpha}^{R_{p}}(x)=\mu_{p}(\mathbf{x})+\sigma_{p}(\mathbf{x})q_{\alpha}^{Z}$ and portfolio value-at-risk $\mathrm{VaR}_{p,\alpha}(\mathbf{x}) =-q_{\alpha}^{R_{p}}(\mathbf{x})W_{0}$ are also functions $\mathbf{x}.$

Proposition 14.1 (Portfolio quantites are homogenous of degree one in $\mathbf{x}$) The portfolio quantities $R_{p}(\mathbf{x})$, $\mu_{p}(\mathbf{x})$, $\sigma_{p}(\mathbf{x})$, the normal quantile $q_{\alpha}^{R_{p}}(x)$, and the normal value-at-risk $\mathrm{VaR}_{p,\alpha}(\mathbf{x})$ are homogenous functions of degree one in the portfolio weight vector $\mathbf{x}$.

The result for $R_{p}(\mathbf{x})$ and $\mu_{p}(\mathbf{x})$ is trivial since they are linear functions of $\mathbf{x}$. For example, $R_{p}(c\mathbf{x})=(c\mathbf{x})^{\prime}\mathbf{R}=c\left(\mathbf{x}^{\prime}\mathbf{R}\right)=cR_{p}(\mathbf{x})$. The result for $\sigma_{p}(\mathbf{x})$ is straightforward to show: $$\begin{align*} \sigma_{p}(c\cdot\mathbf{x}) & =((c\cdot\mathbf{x)}^{\prime}\Sigma\mathbf{(}c\cdot\mathbf{x)})^{1/2}\\ & =c\cdot(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2}\\ & =c\cdot\sigma_{p}(\mathbf{x}). \end{align*}$$ The result for $q_{\alpha}^{R_{p}}(\mathbf{x})$ follows because it is a linear function of $\mu_{p}(\mathbf{x})$ and $\sigma_{p}(\mathbf{x})$. The result for $\mathrm{VaR}_{p,\alpha}(\mathbf{x})$ follows from the linear homogeneity of the normal return quantile.

In the proposition, we stated that $q_{\alpha}^{R_{p}}(x)$ is homogenous of degree one in $\mathbf{x}$ when returns are normally distributed. It turns out that the homogeneity of $q_{\alpha}^{R_{p}}(x)$ holds generally for continuous distributions and even for the empirical quantile.

14.2.2 Euler’s theorem

Euler’s theorem gives an additive decomposition of a homogenous function of degree one.

Theorem 14.1 (Euler’s theorem)

Let $f(x_{1},\ldots,x_{N})=f(\mathbf{x})$ be a continuous, differentiable and homogenous of degree one function of the variables $\mathbf{x}=(x_{1},\ldots,x_{N})^{\prime}$. Then, $$\begin{align*} f(\mathbf{x}) & =x_{1}\cdot\frac{\partial f(\mathbf{x})}{\partial x_{1}}+x_{2}\cdot\frac{\partial f(\mathbf{x})}{\partial x_{2}}+\cdots+x_{N}\cdot\frac{\partial f(\mathbf{x})}{\partial x_{N}}\\ & =\mathbf{x}^{\prime}\frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}, \end{align*}$$ where, $$\underset{(N\times1)}{\frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}}=\left(\begin{array}{c} \frac{\partial f(\mathbf{x})}{\partial x_{1}}\\ \vdots\\ \frac{\partial f(\mathbf{x})}{\partial x_{N}} \end{array}\right).$$

$\blacksquare$

Example 4.3 (Verifying Euler’s theorem)

The function $f(x_{1},x_{2})=x_{1}+x_{2}=f(\mathbf{x})=\mathbf{x}^{\prime}\mathbf{1}$ is homogenous of degree one, and: $$\begin{align*} \frac{\partial f(\mathbf{x})}{\partial x_{1}} & =\frac{\partial f(\mathbf{x})}{\partial x_{2}}=1,\\ \frac{\partial f(\mathbf{x})}{\partial\mathbf{x}} & =\left(\begin{array}{c} \frac{\partial f(\mathbf{x})}{\partial x_{1}}\\ \frac{\partial f(\mathbf{x})}{\partial x_{2}} \end{array}\right)=\left(\begin{array}{c} 1\\ 1 \end{array}\right)=\mathbf{1}. \end{align*}$$ By Euler’s theorem, $$\begin{align*} f(x) & =x_{1}\cdot1+x_{2}\cdot1=x_{1}+x_{2}\\ & =\mathbf{x}^{\prime}\mathbf{1}. \end{align*}$$

The function $f(x_{1},x_{2})=(x_{1}^{2}+x_{2}^{2})^{1/2}=f(\mathbf{x})=(\mathbf{x}^{\prime}\mathbf{x})^{1/2}$ is homogenous of degree one, and: $$\begin{align*} \frac{\partial f(\mathbf{x})}{\partial x_{1}} & =\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2}2x_{1}=x_{1}\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2},\\ \frac{\partial f(\mathbf{x})}{\partial x_{2}} & =\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2}2x_{2}=x_{2}\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2}. \end{align*}$$ By Euler’s theorem, $$\begin{align*} f(\mathbf{x}) & =x_{1}\cdot x_{1}\left(x_{1}^{2}+x_{1}^{2}\right)^{-1/2}+x_{2}\cdot x_{2}\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2}\\ & =\left(x_{1}^{2}+x_{2}^{2}\right)\left(x_{1}^{2}+x_{2}^{2}\right)^{-1/2}\\ & =\left(x_{1}^{2}+x_{2}^{2}\right)^{1/2}. \end{align*}$$

Using matrix algebra, we have: $$\frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{1/2}}{\partial\mathbf{x}}=\frac{1}{2}\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{-1/2}2\mathbf{x}=\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{-1/2}\mathbf{x}=\mathbf{x}\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{-1/2}.$$ Then by Euler’s theorem: $$f(\mathbf{x})=\mathbf{x}^{\prime}\frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}=\mathbf{x}^{\prime}\mathbf{x}\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{-1/2}=\left(\mathbf{x}^{\prime}\mathbf{x}\right)^{1/2}.$$

$\blacksquare$

14.2.3 Risk decomposition using Euler’s theorem

Proposition 11.1 (Additive risk decomposition for homogenous of degree one portfolio risk measures) Let $\mathrm{RM}_{p}(\mathbf{x})$ denote a portfolio risk measure that is a homogenous function of degree one in the portfolio weight vector $\mathbf{x}$. For example, $\mathrm{RM}_{p}(\mathbf{x})=\sigma_{p}(\mathbf{x})$ or $\mathrm{RM}_{p}(\mathbf{x})=\mathrm{VaR}_{p,\alpha}(\mathbf{x}).$ Then, Euler’s theorem gives the additive risk decomposition $$\begin{eqnarray} \mathrm{RM}_{p}(\mathbf{x}) & = & x_{1}\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{1}}+x_{2}\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{2}}+\cdots+x_{N}\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{N}}\tag{14.1}\\ & = & \sum_{i=1}^{N}x_{i}\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{i}}\nonumber \\ & = & \mathbf{x}^{\prime}\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial\mathbf{x}}.\nonumber \end{eqnarray}$$

The partial derivatives in (14.1) are called asset marginal contributions to risk (MCRs): $$\begin{align} \mathrm{MCR}_{i}^{RM} & =\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{i}}=\text{ marginal contribution of asset } i \tag{14.2} \end{align}$$ The asset contributions to risk (CRs) are defined as the weighted marginal contributions: $$\begin{align} \mathrm{CR}_{i}^{RM} & =x_{i}\cdot\mathrm{MCR}_{i}^{RM}=\text{ contribution of asset } i \tag{14.3} \end{align}$$ Then we can re-express the decomposition (14.1) as $$\begin{eqnarray} \mathrm{RM}_{p}(\mathbf{x}) & = & x_{1}\cdot\mathrm{MCR}_{1}^{RM}+x_{2}\cdot\mathrm{MCR}_{2}^{RM}+\cdots+x_{N}\cdot\mathrm{MCR}_{N}^{RM}\tag{14.4}\\ & = & \mathrm{CR}_{1}^{RM}+\mathrm{CR}_{2}^{RM}+\cdots+\mathrm{CR}_{N}^{RM}\nonumber \end{eqnarray}$$ If we divide both sides of (14.4) by $\mathrm{RM}_{p}(\mathbf{x})$ we get the asset (PCRs) $$\begin{eqnarray*} 1 & = & \frac{\mathrm{CR}_{1}^{RM}}{\mathrm{RM}_{p}(\mathbf{x})}+\frac{\mathrm{CR}_{2}^{RM}}{\mathrm{RM}_{p}(\mathbf{x})}+\cdots+\frac{\mathrm{CR}_{N}^{RM}}{\mathrm{RM}_{p}(\mathbf{x})}\\ & = & \mathrm{PCR}_{1}^{RM}+\mathrm{PCR}_{2}^{RM}+\cdots+\mathrm{PCR}_{N}^{RM} \end{eqnarray*}$$ where $$\begin{align} \mathrm{PCR}_{i}^{RM} & =\frac{\mathrm{CR}_{i}^{RM}}{\mathrm{RM}_{p}(\mathbf{x})}=\text{ percent contribution of asset } i \tag{14.5} \end{align}$$ By construction the asset PCRs sum to one.

14.2.3.1 Risk decomposition using $\sigma_{p}(\mathbf{x})$

Let $\mathrm{RM}_{p}(\mathbf{x})=\sigma_{p}(\mathbf{x})=(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2}$. Because $\sigma_{p}(\mathbf{x})$ is homogenous of degree 1 in $\mathbf{x},$ by Euler’s theorem $$\begin{align} \sigma_{p}(\mathbf{x}) &= x_{1}\frac{\partial\sigma_{p}(\mathbf{x})}{\partial x_{1}}+x_{2}\frac{\partial\sigma_{p}(\mathbf{x})}{\partial x_{2}}+\cdots+x_{n}\frac{\partial\sigma_{p}(\mathbf{x})}{\partial x_{n}}\\ &=\mathbf{x}^{\prime}\frac{\partial\sigma_{p}(\mathbf{x})}{\partial\mathbf{x}}. \tag{14.6} \end{align}$$ Now, $$\begin{eqnarray} \frac{\partial\sigma_{p}(\mathbf{x})}{\partial\mathbf{x}} & = & \frac{\partial(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2}}{\partial\mathbf{x}}=\frac{1}{2}(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{-1/2}2\Sigma \mathbf{x}\nonumber \\ & = & \frac{\Sigma \mathbf{x}}{(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2}}=\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})}.\tag{14.7} \end{eqnarray}$$ Then, $$\begin{equation} \frac{\partial\sigma_{p}(\mathbf{x})}{\partial x_{i}}=\mathrm{MCR}_{i}^{\sigma}=\text{i-th row of }\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})}=\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x})},\tag{14.8} \end{equation}$$ and $$\begin{eqnarray} \mathrm{CR}_{i}^{\sigma} & = & x_{i}\times\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x})},\tag{14.9}\\ \mathrm{PCR}_{i}^{\sigma} & = & x_{i}\times\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}^{2}(\mathbf{x})}.\tag{14.10} \end{eqnarray}$$ Notice that: $$\sigma_{p}(\mathbf{x})=\mathbf{x}^{\prime}\frac{\partial\sigma_{p}(\mathbf{x})}{\partial\mathbf{x}}=\mathbf{x}^{\prime}\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})}=\frac{\sigma_{p}^{2}(\mathbf{x})}{\sigma_{p}(\mathbf{x})}=\sigma_{p}(\mathbf{x}).$$

Example 2.5 (Two asset example)

For a two asset portfolio we have: $$\begin{eqnarray*} \sigma_{p}(\mathbf{x})=(\mathbf{x}^{\prime}\Sigma \mathbf{x})^{1/2} & = & \left(x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}+2x_{1}x_{2}\sigma_{12}\right)^{1/2},\\ \Sigma \mathbf{x} & \mathbf{=} & \left(\begin{array}{cc} \sigma_{1}^{2} & \sigma_{12}\\ \sigma_{12} & \sigma_{2}^{2} \end{array}\right)\left(\begin{array}{c} x_{1}\\ x_{2} \end{array}\right)=\left(\begin{array}{c} x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}\\ x_{2}\sigma_{2}^{2}+x_{1}\sigma_{12} \end{array}\right),\\ \frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})} & = & \left(\begin{array}{c} \left(x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x})\\ \left(x_{2}\sigma_{2}^{2}+x_{1}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}) \end{array}\right), \end{eqnarray*}$$ so that $$\begin{eqnarray*} \mathrm{MCR}_{1}^{\sigma} & = & \left(x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}),\\ \mathrm{MCR}_{2}^{\sigma} & = & \left(x_{2}\sigma_{2}^{2}+x_{1}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}). \end{eqnarray*}$$ Then $$\begin{eqnarray*} \mathrm{MCR}_{1}^{\sigma} & = & \left(x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}),\\ \mathrm{MCR}_{2}^{\sigma} & = & \left(x_{2}\sigma_{2}^{2}+x_{1}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}),\\ \mathrm{CR}_{1}^{\sigma} & = & x_{1}\times\left(x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x})=\left(x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}),\\ \mathrm{CR}_{2}^{\sigma} & = & x_{2}\times\left(x_{2}\sigma_{2}^{2}+x_{2}\sigma_{2}\right)/\sigma_{p}(\mathbf{x})=\left(x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}\right)/\sigma_{p}(\mathbf{x}), \end{eqnarray*}$$

and $$\begin{eqnarray*} \mathrm{PCR}_{1}^{\sigma} & = & \mathrm{CR}_{1}^{\sigma}/\sigma_{p}(\mathbf{x})=\left(x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}\right)/\sigma_{p}^{2}(\mathbf{x}),\\ \mathrm{PCR}_{2}^{\sigma} & = & \mathrm{CR}_{2}^{\sigma}/\sigma_{p}(\mathbf{x})=\left(x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}\right)/\sigma_{p}^{2}(\mathbf{x}). \end{eqnarray*}$$ Notice that risk decomposition from Euler’s theorem above is the same decomposition we motived at the beginning of the chapter.

$\blacksquare$

14.2.3.2 Risk decomposition using $\mathrm{VaR}_{p,\alpha}(\mathbf{x})$

Let $\mathrm{RM}(\mathbf{x)}=\mathrm{VaR}_{p,\alpha}(\mathbf{x})$. Because $\mathrm{VaR}_{p,\alpha}(\mathbf{x})$ is homogenous of degree 1 in $\mathbf{x},$ by Euler’s theorem $$\begin{align} \mathrm{VaR}_{p,\alpha}(\mathbf{x})&=x_{1}\frac{\partial\mathrm{VaR}_{p,\alpha}(\mathbf{x})}{\partial x_{1}}+x_{2}\frac{\partial\mathrm{VaR}_{p,\alpha}(\mathbf{x})}{\partial x_{2}}+\cdots+x_{n}\frac{\partial\mathrm{VaR}_{p,\alpha}(\mathbf{x})}{\partial x_{n}}\nonumber\\ &=\mathbf{x}^{\prime}\frac{\partial\mathrm{VaR}_{p,\alpha}(\mathbf{x})}{\partial\mathbf{x}}.\tag{14.11} \end{align}$$ Now, $$\begin{align*} \mathrm{VaR}_{\alpha}(\mathbf{x}) & =W_{0}\times q_{1-\alpha}^{R_{p}}(\mathbf{x})=W_{0}\times\left(\mu_{p}(\mathbf{x})+\sigma_{p}(\mathbf{x})\times q_{\alpha}^{Z}\right)\\ & =W_{0}\times\left(\mathbf{x}^{\prime}\mu+\left(\mathbf{x}^{\prime}\Sigma \mathbf{x}\right)^{1/2}\times q_{\alpha}^{Z}\right), \\ \frac{\partial\mathrm{VaR}_{\alpha}(\mathbf{x})}{\partial\mathbf{x}} & =W_{0}\times\frac{\partial}{\partial\mathbf{x}}\left(\mathbf{x}^{\prime}\mu+\left(\mathbf{x}^{\prime}\Sigma \mathbf{x}\right)^{1/2}\times q_{\alpha}^{Z}\right)\\ & =W_{0}\times\left(\mu+\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right). \end{align*}$$ Then $$\begin{align} \mathrm{MCR}_{i}^{\mathrm{VaR}} & =W_{0}\times\left(\mu_{i}+\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right),\tag{14.12}\\ \mathrm{CR}_{i}^{\mathrm{VaR}} & =x_{i}\times W_{0}\times\left(\mu_{i}+\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right),\tag{14.13}\\ \mathrm{PCR}_{i}^{\mathrm{VaR}} & =x_{i}\times W_{0}\times\left(\mu_{i}+\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right)/\left(W_{0}\times\left(\mu_{p}(\mathbf{x})+\sigma_{p}(\mathbf{x})\times q_{\alpha}^{Z}\right)\right).\tag{14.14} \end{align}$$

It is often common practice to set $\mu=0$ when computing $\mathrm{VaR}_{p,\alpha}(\mathbf{x})$ especially for short time horizons such as a day. In this case, $$\begin{eqnarray*} \mathrm{MCR}_{i}^{\mathrm{VaR}} & = & W_{0}\times\left(\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right)\propto\mathrm{MCR}_{i}^{\sigma},\\ \mathrm{CR}_{i}^{\mathrm{VaR}} & = & x_{i}\times W_{0}\times\left(\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x)}}\times q_{\alpha}^{Z}\right)\propto\mathrm{CR}_{i}^{\sigma},\\ \mathrm{PCR}_{i}^{\mathrm{VaR}} & = & x_{i}\times\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}^{2}(\mathbf{x)}}=\mathrm{PCR}_{i}^{\sigma}, \end{eqnarray*}$$

and we see that the portfolio normal VaR risk decompositions above give the same information as the portfolio volatility risk decompositions (14.8) - (14.10).

14.2.4 Interpreting marginal contributions to risk

The risk decomposition (14.1) shows that any risk measure that is homogenous of degree one in the portfolio weights can be additively decomposed into a portfolio weighted average of marginal contributions to risk. The marginal contributions to risk (14.2) are the partial derivatives of the risk meausre $\mathrm{RM}_{p}(\mathbf{x})$ with respect to the portfolio weights, and so one may think that they can be interpreted as the change in the risk measure associated with a one unit change in a portfolio weight holding the other portfolio weights fixed: $$\mathrm{MCR}_{i}^{\sigma}=\frac{\partial\sigma_{p}(\mathbf{x})}{\partial x_{i}}\approx\frac{\Delta\sigma_{p}}{\Delta x_{i}}\Rightarrow\Delta\sigma_{p}\approx\mathrm{MCR}_{i}^{\sigma}\cdot\Delta x_{i}.$$ If the portfolio weights were unconstrained this would be the correct interpretation. However, the portfolio weights are constrained to sum to one, $\sum_{i=1}^{N}x_{i}=1$, so the increase in one weight implies an offsetting decrease in the other weights. Hence, the formula $\Delta\sigma_{p}\approx\mathrm{MCR}_{i}^{\sigma}\cdot\Delta x_{i}$ ignores this re-allocation effect.

To properly interpret the marginal contributions to risk, consider the total derivative of $\mathrm{RM}_{p}(\mathbf{x})$: $$\begin{align} d\mathrm{RM}_{p}(\mathbf{x}) & =\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{1}}dx_{1}+\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{2}}dx_{2}+\cdots+\frac{\partial\mathrm{RM}_{p}(\mathbf{x})}{\partial x_{N}}dx_{N}\tag{14.15}\\ & =\mathrm{MCR}_{1}^{RM}dx_{1}+\mathrm{MCR}_{2}^{RM}dx_{2}+\cdots+\mathrm{MCR}_{N}^{RM}dx_{N}.\nonumber \end{align}$$ First, consider a small change in $x_{i}$ offset by an equal change in $x_{j},$ $\Delta x_{i}=-\Delta x_{j}$. That is, the small increase in allocation to asset $i$ is matched by a corresponding decrease in allocation to asset $j$. From (14.15), the approximate change in the portfolio risk measure is $$\begin{equation} \Delta\mathrm{RM}_{p}(\mathbf{x})\approx\left(\mathrm{MCR}_{i}^{RM}-\mathrm{MCR}_{j}^{RM}\right)\Delta x_{i}.\tag{14.16} \end{equation}$$

When rebalancing a portfolio, the reallocation does not have to be limited to two assets. Suppose, for example, the reallocation is spread across all of the other assets $j\neq i$ so that $$\Delta x_{j}=-\alpha_{j}\Delta x_{i}\text{ s.t. }\sum_{j\neq i}\alpha_{j}=1.$$ Then $$\sum_{j\neq i}\Delta x_{j}=-\sum_{j\neq i}\alpha_{j}\Delta x=-\Delta x_{i}\sum_{j\neq i}\alpha_{j}=-\Delta x_{i},$$ and $$\begin{align} \Delta\mathrm{RM}(\mathbf{x}) & \approx\mathrm{MCR}_{i}^{RM}\cdot\Delta x_{i}+\sum_{j\neq i}\mathrm{MCR}_{j}^{RM}\cdot\Delta x_{j}\tag{14.17}\\ & =\left(\mathrm{MCR}_{i}^{RM}\cdot-\sum_{j\neq i}\alpha_{j}\cdot\mathrm{MCR}_{j}^{RM}\right)\Delta x_{i}.\nonumber \end{align}$$

In matrix notation the result (14.17) be written as $$\begin{equation} \Delta\mathrm{RM}(\mathbf{w}) \approx\left((\mathbf{MCR}^{RM})^{\prime}\mathbf{\alpha}\right)\Delta x_{i},\tag{14.18} \end{equation}$$ where $$\begin{align*} \mathbf{MCR}^{RM} & =(\mathrm{MCR}_{1}^{RM},\ldots,\mathrm{MCR}_{n}^{RM})^{\prime}, \\ \mathbf{\alpha} & =(-\alpha_{1},\ldots,-\alpha_{i-1},1,-\alpha_{i+1},\ldots-\alpha_{n})^{\prime}. \end{align*}$$