Assume that there are two risky assets and one risk-free asset in the investment universe. Denote by1and1(2and2) the mean return and the standard deviation of returns ofthe first (second) risky asset. The covariance between the returns of the two risky assets isdenoted by12, and r denotes the risk-free rate of return.

We consider an investor that constructs a portfolio of all available assets. Recall that themean return and the variance of returns of the investors portfolio are given by:

where w1 and w2 are the weights of the first and the second risky asset, respectively, in theportfolio.

The investors goal is to select the optimal risky portfolioby maximizing the mean-varianceutility function:

Where A is the investors risk aversion coefficient.

a) Without using linear algebra, derive the analytical solutions for the weights ofthe optimal risky portfolio and show that the weights are given by:

Hp=w1(#1 - r) + wy(12-1)+, o = wo? +20, 2012 + wc, max U = Hp A04, AO ?u? - r}) - fun =ryu) 1 22 A qu - di 1(Pl? -r}di-(li-r}di? A qu - 12 Hp=w1(#1 - r) + wy(12-1)+, o = wo? +20, 2012 + wc, max U = Hp A04, AO ?u? - r}) - fun =ryu) 1 22 A qu - di 1(Pl? -r}di-(li-r}di? A qu - 12