Question 1 a. The expected return and standard deviation of return on two stocks are given below, together with the correlation coefficient. Expected Return (%) Standard Deviation (%) Stock 1 11 18 Stock 2 10 13 Correlation = -1 Suppose that it is possible to borrow at the risk-free rate Fr . What must the value of the risk-free rate be? Explain and show all your workings. (8 marks) b. If you were to combine Stock 1 and Stock 2 from a) into various portfolios, what would be the shape of the efficient frontier on which those portfolios would plot? Explain. (4 marks) c. Under which circumstances is the two-asset portfolio standard deviation equal to the weighted average of standard deviations of its assets? Explain, using equally weighted portfolio where two assets have equal variances as an example. (5 marks) d. “The more stocks in a portfolio, the greater degree of diversification”. Is this statement always true, only sometimes true or never true? Justify your answer, explaining the impact of the number of stocks on portfolio diversification. (8 marks) Page 3 of 7
Question 2 a. When risk free borrowing and lending at different rates is allowed, a more and a less risk averse investor can end up having the same optimal portfolio on the efficient frontier. Is this statement true or false? Describe this efficient frontier and discuss. (10 marks) b. Investors are only allowed to lend at a risk-free rate. Show the efficient frontier in such an environment and explain whether all investors have to invest a proportion of their funds in the risk-free asset. (7 marks) c. Assume that risk free lending and borrowing is allowed. The optimal portfolio of risky assets (tangency) is composed of Debt and Equity, which have the following characteristics: Expected return (%) Standard Deviation (%) Equity 15 18 Debt 7 10 Covariance Equity, Debt 18 The value of the risk-free rate is 4% i. What is the weight of equity and debt in the tangency portfolio? (3 marks) ii. What is the expected return and standard deviation of the tangency portfolio? (3 marks) iii. If an investor holds a portfolio with 13% standard deviation on the efficient frontier in this case, what is his expected return? (2 marks) Page 4 of 7
Question 3 a. You are presented with the following graph: Why is portfolio A plotted below the Capital Market Line? Sketch this graph on your answer paper and mark on the graph portfolio A’s level of systematic and unsystematic risk. Explain your answer. (6 marks) b. You are presented the following graph: Why is security B plotted below the Security Market Line? Derive the SML equation and explain why we use beta as a measure of risk in the CAPM. (9 marks) c. Assume that the following assets are correctly priced according to Security Market Line (SML): Security 1 Security 2 E(R1) = 5% E(R2) = 10% Beta1 = 0.5 Beta2 = 1.5 What is the risk free-rate, market risk premium and the equation of the SML? (5 marks) Page 5 of 7 d. Portfolio A has beta of 0.76 and the correlation with the market portfolio 0.82. If the standard deviation of the market is 12%, what is the standard deviation of Portfolio A? Answer explaining the link between beta and correlation coefficient. (5 marks)
Question 4 a. You are given the following characteristics for two portfolios: Portfolio Characteristics Portfolio 1 Portfolio 2 Expected return 5% 8% Standard deviation 13% 9% Beta 1.1 -0.2 R2 of 60 months of portfolio return regressed on the market return 84% 61% Assume that the model used to determine these characteristics is the Single Index model. The standard deviation of the market return is 11%. i. Explain which portfolio is more diversified. (2 marks) ii. What is correlation between portfolio 2 and the market? (2 marks) iii. What is the proportion of systematic risk for each Portfolio? (2 marks) iv. What is the covariance between Portfolio 1 and 2? (2 marks) v. What does R-squared suggest the level of unsystematic risk in Portfolio 1 is? (2 marks) vi. What is the difference between the Single Index Model and the Market Model? (4 marks) b. Describe Zero-beta CAPM model and outline characteristics of the zero-beta portfolio. Do portfolios that contain a risk-free asset plot on the Zero-beta line? Explain and show graphs where appropriate. (11 marks) Page 6 of 7
Question 5 a. Consider a single factor APT. Stocks A and B have expected returns of 15% and 18% respectively. The risk-free rate is 6%. Stock B has a factor beta of 1. If arbitrage opportunities are ruled out and APT holds, what would be the factor beta of stock A? (5 marks) b. Discuss the application of APT in practice and whether it is more used than the CAPM model. (10 marks) c. Assume that stock returns in a particular market should be affected by the following factors: a stock’s beta with the bond index, its dividend yield and the firm size. Furthermore, assume a riskless rate of 5%. Find the expected return on stock ABC if the following annualised data is given: Factor Factor beta Risk premiums (in %) Bond Beta 1.2 5.36 Dividend Yield 6 0.24 Size 0.8 -5.50 (3 marks) d. If the observed return of stock in c) is 10% is that stock under or overvalued? (2 marks) e. What can you infer about factors in the APT model? (5 marks) Page 7 of 7
Question 6 a. Analyse the following utility function: 𝑈ሺ𝐴ሻ ൌ െ𝐴ି.ହ i. Is marginal utility increasing, constant or decreasing? (1 mark) ii. What is the function’s absolute risk aversion (ARA)? Define ARA. (4 marks) iii. What is the function’s relative risk aversion (RRA)? Define RRA. (4 marks) iv. Is this a function of a risk-averse investor? (2 marks) b. Why are the indifference curves of risk-averse investors not linear? (4 marks) c. Compare the indifference curves of more and less risk averse investors. Explain why they are different. (5 marks) d. Why do risk averse investors refuse to accept the ‘fair bet’? (5 marks)