# Show that the variance of the portfolio market portfolio is function of the betas of its consituents [closed]

Let us assume that the market portfolio consists of n assets. Given that the return of the market portfolio can be written as $$r_m = \sum_{j=1}^{n} w_jr_j$$, we have that $$\sigma^2_m = E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2$$, but how do I show that $$E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2 = \sum_{j=1}^{n} w_jCov(r_j,r_m)$$? If I show that the equation above is true, than I can claim that $$E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2 = \sum_{j=1}^{n} w_jCov(r_j,r_m) = \sum_{j=1}^{n} w_j\beta\sigma^2_m$$

• Please put this together with your other question and put some more effort in that question itself. What did you try? Please show your work. – Bob Jansen Apr 10 '19 at 13:51