Assume that there are two risky assets and one risk-free asset in the investment universe. Denote by 1 and 1 (2 and 2) the mean return and the standard deviation of returns of the first (second) risky asset. The covariance between the returns of the two risky assets is denoted by 12, and r denotes the risk-free rate of return. We consider an investor that constructs a portfolio of all available assets. Recall that the mean return and the variance of returns of the investors portfolio are given by p = w1(1 r) + w2(2 r) + r, 2 p = w 2 1 2 1 + 2w1w212 + w 2 2 2 2 , where w1 and w2 are the weights of the first and the second risky asset, respectively, in the portfolio. The investors goal is to select the optimal risky portfolio by maximizing the mean-variance utility function max w1,w2 U = p 1 2 A2 p , where A is the investors risk aversion coefficient.

Question:

Determine the weight of the risk-free asset in the investors overall portfolio.