Combining covariances?

Consider an economy with assets with return processes $A$, $B$, $C$, $D$. Consider a weighted index with return process $I=aA + bB + cC + dD$ where $a,b,c,d$ are coefficients, and $a+b+c+d = 1$.

Suppose I want to find $cov(I,A)$. Is this possible given that I know the covariance between all possible pairs of $A,B,C,D$?

Also, suppose I have some asset $E$. Suppose I know $cov(A,E),cov(B,E),cov(C,E),cov(D,E)$. How do I find $cov(I,E)$.

-

1 Answer

$\sigma(x,y) = E[xy] - E[x]E[y]$

and

$\sigma(ax+by,cz) = ac\, \sigma(x,z) + bc\, \sigma(y,z)$

(paraphrasing the $\sigma(ax+by,cW+dV)$ rule).

So

$\sigma(I,A) = \sigma([aA+bB+cC+dD],A)$ $\sigma(I,A) = a\,\sigma(A,A) + b\,\sigma(B,A) + c\,\sigma(C,A) + d\,\sigma(D,A)$ $\sigma(I,A) = a\,\sigma^2(A) + b\,\sigma(B,A) + c\,\sigma(C,A) + d\,\sigma(D,A)$

Since you know the covariances between all the pairs and presumably the variance ($\sigma^2$) of A, you can thus calculate $\sigma(I,A)$.

The same holds for $\sigma(I,E)$, only you won't get a variance term: $\sigma(I,E) = a\,\sigma(A,E) + b\,\sigma(B,E) + c\,\sigma(C,E) + d\,\sigma(D,E)$

-