To compute the [[Variance]] of a portfolio of assets, we can extend the [[Variance of Sum of Random Variables#Multiple Random Variables|Variance of Sum of Random Variables]] by a weighted sum, where each asset has a weighting. No assumptions on the distributions of individual [[Random Variable|Random Variables]] need to be taken, as long as a mean and a variance exist. Assume we have a portfolio with $n$ assets, and each asset has: - $R_i$: Return of the asset (random variable) - $\mu_i$: [[Expectation]] of $R_i$. - $w_i$: Relative weight in the portfolio The portfolio return is defined as: $ R_p = \sum_{i=1}^n w_iR_i$ The portfolio variance can be derived as follows $ \begin{align} \mathrm {Var}(R_p)&=\mathbb{E}\left[\Big(\sum_{i=1}^n w_i(R_i-\mu_i) \Big)^2\right] \tag{1}\\[4pt] &=\mathbb E\left[ \sum_{i=1}^n w_i^2(R_i-\mu_i)^2\right] + \mathbb E\left[\sum_{(i,j): i \neq j}^n w_i w_j(R_i-\mu_i)*(R_j-\mu_j) \right] \tag{2}\\ &=\sum_{i=1}^n w_i^2*\mathbb E\Big[ (R_i-\mu_i)^2\Big] + 2*\sum_{(i,j): i < j}^nw_i w_j* \mathbb E\Big[(R_i-\mu_i)*(R_j-\mu_j) \Big] \tag{3}\\ &=\sum_{i=1}^nw_i^2*\mathrm{Var}(R_i) + 2*\sum_{i < j}^n w_i w_j\mathrm{Cov}(R_i, R_j) \tag{4}\\ &=\sum_{i=1}^nw_i^2*\mathrm{Var}(R_i) + 2*\sum_{i < j}^n w_i w_j \rho \,\sigma_i\sigma_j \tag{5} \end{align} $ where: - (2) Expanding the quadratic formula into the quadratic terms and the cross terms. - (3) Factoring out constants out of the expectation (e.g. $w_i$) - (4) Recognizing the variance and [[Covariance]]. - (5) Express the covariance in terms of [[Correlation]] $\rho$ and standard deviations $\sigma_i$ and $\sigma_j$. ## Special Cases Correlation of $0$: $ \mathrm{Var}(R_p)=\sum_{i=1}^nw_i^2*\mathrm{Var}(R_i)$ Correlation of $1$: $ \mathrm{Var}(R_p)=\sum_{i=1}^n w_i^2*\mathrm{Var}(R_i) + 2*\sum_{i < j}^n w_i w_j \,\sigma_i\sigma_j=\left(\sum_{i=1}^n w_i \sigma_i \right)^2 $ Correlation of $0$ (with equal weighting and variance): $ \begin{align} \mathrm{Var}(R_p) &=\sum_{i=1}^n w_i^2*\mathrm{Var}(R_i) \\[4pt] &=\sum_{i=1}^n \left(\frac{1}{n}\right)^2*\sigma^2 \\[4pt] &=\left(\frac{1}{n}\right)^2n*\sigma^2\\[8pt] &=\frac{\sigma^2}{n} \end{align} $