# Covariance Decomposition

I have the returns of three stocks, $R_{1t}$, $R_{2t}$, $R_{3t}$, with 100 monthly observations for each return series. Lets suppose that I create a portfolio consisting of stocks 1 and 2, $P_t=w_{1t}R_{1t}+w_{2t}R_{2t}$. $w_{1t}$ and $w_{2t}$ are the relative portfolio weights for each month, which sum to 1. Now I calculate the covariance between the portfolio and stock 3, $COV(P_t,R_{3t})=COV(w_{1t}R_{1t}+w_{2t}R_{2t},R_{3t})$.

I want to know what the contribution of stocks 1 and 2 is for the covariance $COV(P_t,R_{3t})$. In other words, I want to decompose the covariance into the contribution of each stock. If $w_{1t}$ and $w_{2t}$ were constant, the solution would be $COV(P_t,R_{3t})= w_{1}COV(R_{1t},R_{3t})+w_{2}COV(R_{2t},R_{3t})$. The problem ist that the weights $w_{1t}$ and $w_{2t}$ change over time so that this exact decomposition does not apply. So how can I decompose the covariance with time varying portfolio weights? Any idea?

• I imagine portfolio weights depend somehow on stock value. Generally, your problem is intractable. But if you specify the dependence of weights on the values of the stocks (i.e. your strategy)s, then maybe there's a way to simplify. – Yair Daon Apr 14 '16 at 1:35

You would just need to include the time index $t$ in $$Cov(P_t,R_{3t}) = w_{1t}Cov(R_{1t},R_{3t}) + w_{2t}Cov(R_{2t},R_{3t})$$ and there you go. The relation would be still valid, provided $w_{it}$ were not stochastic themselves.
Indeed, since $$Cov(a X,Y) = aCov(X,Y)$$ and $$Cov(X+Y,Z) = Cov(X,Z)+Cov(Y,Z),$$ it comes out straightfowardly that $$Cov(P_t,R_{3t}) = Cov(w_{1t}R_{1t}+w_{2t}R_{2t},R_{3t}) = Cov(w_{1t}R_{1t},R_{3t}) + Cov(w_{2t}R_{2t},R_{3t}) = w_{1t}Cov(R_{1t},R_{3t}) + w_{2t}Cov(R_{2t},R_{3t}).$$