Consider a portfolio choice problem in a world with risk free rate rf and two risky assets i = 1; 2. Assume that one asset has both a higher expected return and volatility than the other, so that 1 > 2 and 1 > 2, and that the returns to each are uncorrelated 12 = 0. Let denote the portfolio weight on stock 1. (a) Find the weight ~ that yields the minimum variance portfolio. (b) Let be the weight that de nes the tangency portfolio. Write the condition that de nes and, without solving for it, determine how it compares to ~. (c) We now wish to consider how the composition of the tangency portfolio shifts as the risk free rate increases. Without solving the optimization problem, how do you expect to change to change as rf increases? (d) Using the envelope theorem, determine how the slope of the tangent line changes as the risk free rate increases. Once again, do not solve the optimization problem.