# Decomposition of Contribution to Variance

$$C$$ is a $$N\times N$$ covariance matrix of stock returns. Assuming $$w$$ is a vector of positions in each asset, the total variance of the portfolio is $$w^TCw$$ The contribution to total variance of the N stocks is $$\text{contributiontoVariance} = \sqrt Cw$$ where we use the matrix square root such that $$\sqrt C\sqrt C = C$$.

$$\text{contributiontoVariance}$$ gives me a signed measure of the contribution to total variance of each asset in the portfolio, such that the sum of the squares of the contributions equals the total portfolio variance $$w^TCw$$.

How do I calculate the contribution of a group of assets to the total variance? If, say, asset $$A$$ and asset $$B$$ are perfectly correlated and are perfectly hedged against each other, I would want their combined contribution to variance to be zero.

• Is your formula for Contribution to variance (with $\sqrt{C}$ in it) correct? I thought Contribution to variance $\sigma^2_{pk}$ for asset $k$ was found with the formula here quant.stackexchange.com/questions/30844/… – noob2 Jun 18 '20 at 21:28
• yes, if you want your portfolio variance to be the same as that of RRG in the link that noob2 pointed to, then you need another $w$ on the left of $\sqrt{C}$ in your contributiontoVariance formula. – mark leeds Jun 18 '20 at 22:56