14.1 Risk Budgeting Using Portfolio Variance and Portfolio Standard Deviation

We motivate portfolio risk budgeting in the simple context of a two risky asset portfolio. To illustrate, consider forming a portfolio consisting of two risky assets (asset 1 and asset 2) with portfolio shares $x_{1}$ and $x_{2}$ such that $x_{1}+x_{2}=1.$ We assume that the GWN model holds for the simple returns on each asset. The portfolio return is given by $R_{p}=x_{1}R_{1}+x_{2}R_{2}$, and the portfolio expected return and variance are given by $\mu_{p}=x_{1}\mu_{1}+x_{2}\mu_{2}$ and $\sigma_{p}^{2}=x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}+2x_{1}x_{2}\sigma_{12}$, respectively. The portfolio variance, $\sigma_{p}^{2},$ and standard deviation, $\sigma_{p},$ are natural measures of portfolio risk. The advantage of using $\sigma_{p}$ is that it is in the same units as the portfolio return. Risk budgeting concerns the following question: what are the contributions of assets 1 and 2 to portfolio risk captured by $\sigma_{p}^{2}$ or $\sigma_{p}$?

14.1.1 Case 1: uncorrelated assets

To answer this question, consider first the formula for portfolio variance. If the asset returns are uncorrelated ($\sigma_{12}=0)$ then $$\sigma_{p}^{2}=x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}$$ gives an additive decomposition of portfolio risk: $x_{1}^{2}\sigma_{1}^{2}$ is the variance contribution from asset 1, and $x_{2}^{2}\sigma_{2}^{2}$ is the variance contribution from asset 2. Clearly, asset 1 has a higher contribution than asset 2 if $x_{1}^{2}\sigma_{1}^{2}>x_{2}^{2}\sigma_{2}^{2}$ . For example, in an equally weighted portfolio asset 1’s risk contribution is higher than asset 2’s contribution if asset 1’s return is more variable than asset 2’s return. We can also define asset percent contributions to risk, which divide the asset contributions by portfolio variance, as $x_{1}^{2}\sigma_{1}^{2}/\sigma_{p}^{2}$ and $x_{2}^{2}\sigma_{2}^{2}/\sigma_{p}^{2},$ respectively. The percent contributions have the property that they add to $100\%.$ Notice that an equally weighted portfolio may not have equal contributions to risk.

If we measure risk by portfolio volatility, $\sigma_{p}=\sqrt{x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}}$, we may be tempted to define the contributions of assets 1 and 2 to risk as $x_{1}\sigma_{1}$ and $x_{2}\sigma_{2},$ respectively. However, this would not be correct because we would not get an additive decomposition $$\sigma_{p}=\sqrt{x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}}\neq x_{1}\sigma_{1}+x_{2}\sigma_{2}.$$ To get an additive decomposition we have to define the risk contributions as $x_{1}^{2}\sigma_{1}^{2}/\sigma_{p}$ and $x_{2}^{2}\sigma_{2}^{2}/\sigma_{p},$ respectively. Then $$\frac{x_{1}^{2}\sigma_{1}^{2}}{\sigma_{p}}+\frac{x_{2}^{2}\sigma_{2}^{2}}{\sigma_{p}}=\frac{\sigma_{p}^{2}}{\sigma_{p}}=\sigma_{p}.$$

Notice that the percent contributions to portfolio volatility are the same as the percent contributions to portfolio variance.

14.1.2 Case 2: correlated assets

If the asset returns are correlated ($\sigma_{12}\neq0$) then the risk decomposition becomes a bit more complicated because we have to decide what to do with the covariance contribution to portfolio variance. However, the formula for $\sigma_{p}^{2}$ suggests a simple additive decomposition: $$\begin{align*} \sigma_{p}^{2} & =x_{1}^{2}\sigma_{1}^{2}+x_{2}^{2}\sigma_{2}^{2}+2x_{1}x_{2}\sigma_{12}\\ & =\left(x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}\right)+\left(x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}\right)\\ & =\left(x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\rho_{12}\sigma_{1}\sigma_{2}\right)+\left(x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\rho_{12}\sigma_{1}\sigma_{2}\right) \end{align*}$$ Now, $\left(x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}\right)$ and $\left(x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}\right)$ are portfolio variance contributions of assets 1 and 2, respectively. Here, we split the covariance contribution to portfolio variance, $2x_{1}x_{2}\sigma_{12}$, evenly between the two assets. Now, the correlation between the two asset returns also plays a role in the risk decomposition. The additive decomposition for $\sigma_{p}$ is $$\sigma_{p}=\frac{x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}}{\sigma_{p}}+\frac{x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}}{\sigma_{p}},$$ which gives $\left(\frac{x_{1}^{2}\sigma_{1}^{2}+x_{1}x_{2}\sigma_{12}}{\sigma_{p}}\right)$ and $\left(\frac{x_{2}^{2}\sigma_{2}^{2}+x_{1}x_{2}\sigma_{12}}{\sigma_{p}}\right)$ as the asset contributions to portfolio volatility, respectively.

14.1.3 The general case of $N$ assets

The risk budgeting decompositions presented above for two asset portfolios extend naturally to general $N$ asset portfolios. In the case where all assets are mutually uncorrelated, asset i’s contributions to portfolio variance and volatility are $x_{i}^{2}\sigma_{i}^{2}$ and $x_{i}^{2}\sigma_{i}^{2}/\sigma_{p}$, respectively. In the case of correlated asset returns, all pairwise covariances enter into asset i’s contributions. The portfolio variance contribution is $x_{i}^{2}\sigma_{i}^{2}+\sum_{j\neq i}x_{i}x_{j}\sigma_{ij}$ and the portfolio volatility contribution is $\left(x_{i}^{2}\sigma_{i}^{2}+\sum_{j\neq i}x_{i}x_{j}\sigma_{ij}\right)/\sigma_{p}$. As one might expect, these formulae can be simplified using matrix algebra.