Question 3 (a) (b) (C) ('1) Show that as the number of assets in a portfolio increases portfolio risk decreases. [5 marks] Show that the portfolio opportunity set that emerges from a combination of a risk- free asset and a risky asset is a straight line. [5 marks] Show that the beta of a portfolio (p) is a linear combination of individual asset betas. Also explain why the beta of a portfolio (,6?) is a better measure of risk than the standard deviation of a portfolio. [7 marks] The security market line can be used to try to pick underpriced or overpriced stocks. With the aid of a diagram, explain why a security with a positive alpha would be underpriced. Also explain why a security with a negative alpha would be overpriced. For both cases make a buy or sell recommendation. [8 marks] (b) The following figure shows two compound gambles, Cp and Cq. Show that the two gambles are outcome equivalent. [4] Figure 1 L,(X1, X2, X3; 1, 0, 0) q1=L(X1, X2, X3; 1/3, 1/6, 1/2) 1/2 1/3 L2(X1, X2, X3;2/8, 3/8, 3/8) Co 1/3 1/2 1/3 92=L(X1, X2, X3; 5/6, 1/12, 1/12) La(X1, X2, X3; 1/2, 0, 1/2) (c) Separate the reduced lottery derived in (b) above into a riskless component and a risky prospect. [3]Y"? 1 - r (d) An agent, whose utility function of wealth is given by: U (Y) = +2, faces a risky situation in which he can lose some amount of money with probabilities given in the following table. Table l Probabilitv 0-20 0-35 0-20 0-15 The agent's initial wealth is R10 000 and his 7 = 1.2. (i) Calculate the certainty equivalent amount of this prospect for this agent. [4] (ii) What would the certainty equivalent amount of this agent be if the agent was risk neutral? [2] (e) Consider the lottery L1 = L(50 000, 10 000: 1/2). Determine the lottery L2 = L(x, 0; 1) that makes an agent with utility function indifferent to lottery L1. That is, calculate the value of x given the above information). [4]