## 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 \(\sigma_{p}\)?

### 14.1.3 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.