Problem 2 a) In the mean-variance problem with a riskless asset, show that the tangency portfolio: 2-107 R1) B AR is a member of the minimum-variance set (of all the securities without the riskless asset). Wt = b) Consider the (0,u) plane. Consider the line o = a + bu which passes through the riskless portfolio and the tangent portfolio. Prove that this line is tangent to the minimum-variance set. Prove that the point of tangency is at the tangency portfolio. Problem 3 Consider N firms. The returns on their stocks are independent and identi- cally distributed, with mean m and variance v. Describe the minimum variance set. What happens when N 00? Suppose that investors want to minimize variance for a given expected return. What is the market portfolio? Can firms have different market capitalizations in this model? Problem 2 a) In the mean-variance problem with a riskless asset, show that the tangency portfolio: 2-107 R1) B AR is a member of the minimum-variance set (of all the securities without the riskless asset). Wt = b) Consider the (0,u) plane. Consider the line o = a + bu which passes through the riskless portfolio and the tangent portfolio. Prove that this line is tangent to the minimum-variance set. Prove that the point of tangency is at the tangency portfolio. Problem 3 Consider N firms. The returns on their stocks are independent and identi- cally distributed, with mean m and variance v. Describe the minimum variance set. What happens when N 00? Suppose that investors want to minimize variance for a given expected return. What is the market portfolio? Can firms have different market capitalizations in this model