## 14.4 Understanding Portfolio Volatility Risk Decompositions

Portfolio volatility risk reports described in the previous sub-section are commonly used in practice. As such, it is important to have a good understanding of the volatility risk decompositions and, in particular, the asset contributions to portfolio volatility. In the portfolio volatility risk decomposition (14.6), the marginal contribution from asset $$i$$ given by: $\begin{eqnarray} \mathrm{MCR}_{i}^{\sigma} & = & \text{i-th row of }\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})}=\frac{\left(\Sigma \mathbf{x}\right)_{i}}{\sigma_{p}(\mathbf{x})}.\tag{14.19} \end{eqnarray}$ The interpretation of this formula, however, is not particularly intuitive. With a little bit of algebra, we can derive two alternative representations of (14.19) that are easier to interpret and give better intuition about the asset contributions to portfolio volatility.

### 14.4.1$$\sigma-\rho$$ decomposition for $$\mathrm{MCR}_{i}^{\sigma}$$

Definition 14.1 ($$\sigma-\rho$$ decomposition) The $$\sigma-\rho$$ decomposition for $$\mathrm{MCR}_{i}^{\sigma}$$ is $\begin{equation} \mathrm{MCR}_{i}^{\sigma}=\sigma_{i}\rho_{i,p}(\mathbf{x}),\tag{14.20} \end{equation}$ where $$\rho_{i,p}(\mathbf{x})=\mathrm{corr}(R,R_{p}(\mathbf{x}))$$ is the correlation between the return on asset $$i$$ and the return on the portfolio.

Remarks:

1. Equation (14.20) shows that an asset’s marginal contribution to portfolio volatility depends on two components: (1) the asset’s return volatility, $$\sigma_{i}$$ (sometimes called standalone volatility); (2) the asset’s correlation with the portfolio return, $$\rho_{i,p}(\mathbf{x})$$.
2. For a given standalone volatility, $$\sigma_{i},$$ the sign and magnitude of $$\mathrm{MCR}_{i}^{\sigma}$$ depends on $$\rho_{i,p}(\mathbf{x})$$. Because $$-1\leq\rho_{i,p}(\mathbf{x})\leq1$$, it follows that $$\mathrm{MCR}_{i}^{\sigma}\leq\sigma_{i}$$ and $$\mathrm{MCR}_{i}^{\sigma}=\sigma_{i}$$ only if $$\rho_{i,p}(\mathbf{x})=1$$.
3. If an asset’s return is uncorrelated with all of the returns in the portfolio then $$\rho_{i,p}(\mathbf{x})=\left(x_{i}\sigma_{i}\right)/\sigma_{p}$$ and $$\mathrm{MCR}_{i}^{\sigma}=\left(x_{i}\sigma_{i}^{2}\right)/\sigma_{p}$$.
Definition 14.2 ($$x-\sigma-\rho$$ decomposition) Using (14.20) the $$x-\sigma-\rho$$ decomposition for $$\mathrm{CR}_{i}^{\sigma}=x_{i}\times\mathrm{MCR}_{i}^{\sigma}$$ is: $\begin{eqnarray} \mathrm{CR}_{i}^{\sigma} & = & x_{i}\times\sigma_{i}\times\rho_{i,p}(\mathbf{x})\tag{14.21}\\ & = & asset\,allocation\times standalone\,volatility\times correlation\,with\,portfolio\nonumber \end{eqnarray}$

Remarks:

1. Equation (14.21) shows that an asset’s contribution to portfolio volatility depends on three components: (1) the asset’s allocation weight $$x_{i}$$; (2) the asset’s standalone return volatility, $$\sigma_{i}$$; (3) the asset’s correlation with the portfolio return, $$\rho_{i,p}(\mathbf{x})$$.
2. $$x_{i}\times\sigma_{i}=$$ standalone contribution to portfolio volatility, which ignores correlation effects with other assets. In the typical situation, $$\rho_{i,p}\neq1$$ which implies that $$\mathrm{CR}_{i}^{\sigma}<w_{i}\times\sigma_{i}$$. Hence, an asset’s contribution to portfolio volatility will almost always be less than its standalone contribution to portfolio volatility.
3. $$\mathrm{CR}_{i}^{\sigma}=x_{i}\times\sigma_{i}$$ only when $$\rho_{i,p}(\mathbf{x})=1$$. That is, an asset’s contribution to portfolio volatility is equal to its standalone contribution only when its return is perfectly correlated with the portfolio return.
4. If all assets are perfectly correlated (i.e., $$\rho_{ij}=1$$ for all $$i$$ and $$j$$) then all asset contributions to portfolio volatility are equal their standalone contributions, $$x_{i}\times\sigma_{i}$$, and $$\sigma_{p}(\mathbf{x})=x_{1}\sigma_{1}+x_{2}\sigma_{2}+\cdots+x_{N}\sigma_{N}$$. This is the risk decomposition that would occur if there are no diversification effects.

The derivation of (14.20) is straightforward. Recall, the vector of asset marginal contributions to portfolio volatility is given by $\frac{\partial\sigma_{p}(\mathbf{x})}{\partial\mathbf{x}}=\frac{\Sigma \mathbf{x}}{\sigma_{p}(\mathbf{x})}$ Now, $\mathbf{\Sigma x=}\left(\begin{array}{cccc} \sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1n}\\ \sigma_{12} & \sigma_{2}^{2} & \cdots & \sigma_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{1n} & \sigma_{n2} & \cdots & \sigma_{n}^{2} \end{array}\right)\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)$

Without loss of generality, consider the first row of $$\Sigma \mathbf{x}$$ $x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}+\cdots+x_{n}\sigma_{1n}$ This expression is the covariance between the return on asset 1, $$R_{1}$$, and the return on the portfolio, $$R_{p}(\mathbf{x})$$: $\begin{eqnarray*} \mathrm{cov}(R_{1},R_{p}(\mathbf{x})) & = & \mathrm{cov}(R_{1},x_{1}R_{1}+R_{1}x_{2}R_{2}+\cdots+x_{n}R_{n})\\ & = & \mathrm{cov}(R_{1},x_{1}R_{1})+\mathrm{cov}(R_{1},x_{1}R_{1})+\cdots+\mathrm{cov}(R_{1},x_{n}R_{n})\\ & = & x_{1}\sigma_{1}^{2}+x_{2}\sigma_{12}+\cdots+x_{n}\sigma_{1n}\\ & = & \sigma_{1,p}(\mathbf{x}) \end{eqnarray*}$ Then we can re-write (14.19) for $$i=1$$ as $\begin{equation} \mathrm{MCR}_{1}^{\sigma}=\frac{\left(\Sigma \mathbf{x}\right)_{1}}{\sigma_{p}(\mathbf{x})}=\frac{\sigma_{1,p}(\mathbf{x})}{\sigma_{p}(\mathbf{x})}\tag{14.22} \end{equation}$

Define the correlation between the return on asset 1 and the return on the portfoio as $\begin{eqnarray} \rho_{1,p}(\mathbf{x}) & = & \mathrm{corr}(R_{1},R_{p}(\mathbf{x}))=\frac{\mathrm{cov}(R_{1},R_{p}(\mathbf{x}))}{\sigma_{1}\sigma_{p}(\mathbf{x})}=\frac{\sigma_{1,p}(\mathbf{x})}{\sigma_{1}\sigma_{p}(\mathbf{x})}\tag{14.23} \end{eqnarray}$ Then we can write $\begin{equation} \sigma_{1,p}(\mathbf{x})=\sigma_{1}\sigma_{p}(\mathbf{x})\rho_{1,p}(\mathbf{x})\tag{14.24} \end{equation}$ Substituting (14.24) into (14.22) gives the $$\sigma-\rho$$ decomposition for $$\mathrm{MCR}_{1}^{\sigma}$$ $\mathrm{MCR}_{1}^{\sigma}=\frac{\sigma_{1}\sigma_{p}(\mathbf{x})\rho_{1,p}(\mathbf{x})}{\sigma_{p}(\mathbf{x})}=\sigma_{1}\rho_{1,p}(\mathbf{x})$

Example 3.4 (Adding $$\rho_{i,p}(\mathbf{x})$$ to the portfolio volatility risk report)

Once you have the volatility risk report, you can easily calculate $$\rho_{i,p}(\mathbf{x})$$ from (14.20) using $\rho_{i,p}(\mathbf{x})=\frac{\mathrm{MCR}_{i}^{\sigma}}{\sigma_{i}}$ For example, to calculate $$\rho_{i,p}(\mathbf{x})$$ for the volatility risk report for the equally weighted portfolio of Microsoft, Nordstrom, and Starbucks use:

rho.x = MCR.vol.x/sig.vec
rho.x
##       [,1]
## MSFT 0.567
## NORD 0.644
## SBUX 0.735

Here we see that all assets are positively correlated with the portfolio, Microsoft has the smallest correlation (0.567), and Starbucks has the largest correlation (0.735). In this respect, Microsoft is most beneficial in terms of diversification and Starbucks is least beneficial. If all correlations were equal to one, then each asset MCR would be equal to its standalone volatility.

The expanded portfolio risk report is:

$$\blacksquare$$

### 14.4.2$$\sigma-\beta$$ decomposition for $$\mathrm{MCR}_{i}^{\sigma}$$

Definition 14.3 ($$\sigma-\beta$$ decomposition) The $$\sigma-\beta$$ decomposition for $$\mathrm{MCR}_{i}^{\sigma}$$ is $\begin{equation} \mathrm{MCR}_{i}^{\sigma}=\beta_{i,p}(\mathbf{x})\sigma_{p}(\mathbf{x})\tag{14.25} \end{equation}$ where $\begin{equation} \beta_{i,p}(\mathbf{x})=\frac{\mathrm{cov}(R_{i},R_{p}(\mathbf{x}))}{\mathrm{var}(R_{p}(\mathbf{x}))}=\frac{\mathrm{cov}(R_{i},R_{p}(\mathbf{x}))}{\sigma_{p}^{2}(\mathbf{x})}\tag{14.26} \end{equation}$

Here, $$\beta_{i,p}(\mathbf{x})$$ is called asset $$i$$’s beta to the portfolio.93 This decomposition follows directly from the $$\sigma-\rho$$ decomposition (14.20) since $\beta_{i,p}(\mathbf{x})=\frac{\mathrm{cov}(R_{i},R_{p}(\mathbf{x}))}{\sigma_{p}^{2}(\mathbf{x})}=\frac{\rho_{i,p}(\mathbf{x})\sigma_{i}\sigma_{p}(\mathbf{x})}{\sigma_{p}^{2}(\mathbf{x})}\Rightarrow\rho_{i,p}(x)=\frac{\beta_{i,p}(\mathbf{x})\sigma_{p}(\mathbf{x})}{\sigma_{i}}.$ It follows that: $\begin{eqnarray} \mathrm{CR}_{i}^{\sigma} & = & x_{i}\times\beta_{i,p}(\mathbf{x})\times\sigma_{p}(\mathbf{x})\tag{14.27}\\ \mathrm{PCR}_{i}^{\sigma} & = & x_{i}\times\beta_{i,p}(\mathbf{x})\tag{14.28} \end{eqnarray}$

Remarks:

• By construction, the beta of the portfolio is $$1$$ $\beta_{p,p}(\mathbf{x})=\frac{\mathrm{cov}(R_{p}(\mathbf{x}),R_{p}(\mathbf{x}))}{\mathrm{var}(R_{p}(\mathbf{x}))}=\frac{\mathrm{var}(R_{p}(\mathbf{x}))}{\mathrm{var}(R_{p}(\mathbf{x}))}=1.$
• When $$\beta_{i,p}(\mathbf{x})=1$$, $$\mathrm{MCR}_{i}^{\sigma}=\sigma_{p}(\mathbf{x}),\mathrm{\,CR}_{i}^{\sigma}=x_{i}\sigma_{p}(\mathbf{x})$$, and $$\mathrm{PCR}_{i}^{\sigma}=x_{i}$$. In this case, an asset’s marginal contribution to portfolio volatility is portfolio volatility and its percent contribution to portfolio volatility is its allocation weight. Intuitively, when $$\beta_{i,p}(x)=1$$ the asset has the same risk, in terms of volatility contribution, as the portfolio.
• When $$\beta_{i,p}(\mathbf{x})>1$$, $$\mathrm{MCR}_{i}^{\sigma}>\sigma_{p}(\mathbf{x})$$, $$\mathrm{CR}_{i}^{\sigma}>w_{i}\sigma_{p}(\mathbf{x})$$, and $$\mathrm{PCR}_{i}^{\sigma}>x_{i}$$. In this case, the asset’s marginal contribution to portfolio volatility is more than portfolio volatility and its percent contribution to portfolio volatility is more than its allocation weight. Intuitively, when $$\beta_{i,p}(\mathbf{x})>1$$ the asset has more risk, in terms of volatility contribution, than the portfolio. That is, having the asset in the portfolio increases the portfolio volatility.
• When $$\beta_{i,p}(\mathbf{x})<1$$, $$\mathrm{MCR}_{i}^{\sigma}<\sigma_{p}(\mathbf{x})$$, $$\mathrm{CR}_{i}^{\sigma}<x_{i}\sigma_{p}(\mathbf{x})$$, and $$\mathrm{PCR}_{i}^{\sigma}<x_{i}$$. In this case, the asset’s marginal contribution to portfolio volatility is less than portfolio volatility and its percent contribution to portfolio volatility is less than its allocation weight. Intuitively, when $$\beta_{i,p}(\mathbf{x})<1$$ the asset has less risk, in terms of volatility contribution, than the portfolio. That is, having the asset in the portfolio decreases the portfolio volatility.
Example 3.5 (Adding $$\beta_{i,p}(\mathbf{x})$$ to the portfolio volatility risk report)

Once you have the volatility risk report, you can easily calculate $$\beta_{i,p}(\mathbf{x})$$ from (14.25) using $\beta_{i,p}(\mathbf{x})=\frac{\mathrm{MCR}_{i}^{\sigma}}{\sigma_{p}(\mathbf{x})}$ For example, to calculate $$\beta_{i,p}(\mathbf{x})$$ for the volatility risk report for the equally weighted portfolio of Microsoft, Nordstrom, and Starbucks use:

beta.x = MCR.vol.x/sig.px
beta.x
##       [,1]
## MSFT 0.747
## NORD 0.886
## SBUX 1.367

Here, both $$\beta_{MSFT,p}(\mathbf{x})$$ and $$\beta_{NORD,p}(\mathbf{x})$$ are less than one whereas $$\beta_{SBUX,p}(\mathbf{x})$$ is greater than one. We can say that Microsoft and Nordstrom are portfolio volatility reducers whereas Starbucks is a portfolio volatility enhancer.

$$\blacksquare$$

1. It can be thought of as the population regression coefficient obtained by regressing the asset return, $$R_{i},$$on the portfolio return, $$R_{p}(\mathbf{x}).$$↩︎