This is probably an easy question but i'm confused, it's presented in a finance textbook (often proofs aren't really rigorous in such), but for me it makes absolutely no sense.
Edit: (To those guys who understand finance).
The claim is exactly that $X$ is the return of the minimum-variance portfolio of $Y$ the return on some other efficient portfolio. The assignment states: There is a formula of the form $Cov(X,Y)=A\cdot Var(X)$ [Hint: Consider the portfolios $(1-a)X+aY$, and consider small variations of the variance of such portfolios near $a=0$.] I promise I'm not keeping any information from you, I just tried to take all the finance out, since I expected it to creep you guys out.
I got two stochastic variables $X$ and $Y$ with all relevant moments. I want to look at the variance of a convex combination of them to conclude something about their covariance, here's how:
$Var(aX+(1-a)Y)=a^2Var(X)+(1-a)^2Var(Y)+2\cdot a\cdot (1-a)Cov(X,Y)$
I can then evaluate:$\frac{\partial}{\partial a}Var(aX+(1-a)Y)=2aVar(X)-2(1-a)Var(Y)+2Cov(X,Y)-4aCov(X,Y)$
If I evaluate this at $a=1$ and set equal to zero I get:
$2Var(X)-2Cov(X,Y)=0\implies Var(X)=Cov(X,Y)$
I pretty much don't get the conclusion, I got the feeling that I would always be able to do this trick. So after this proof what would be fair to conclude and what not?
Hope someone is willing to help.