Question 1 (Mean-variance optimization - 10 marks)
You have been approached by a client who is interested in investing in equities. After some questionnaires, you discover that you client's utility function can be approximated by a quadratic utility,
U (E[rd,Var[rq]) = E[rq] — 21 AVa r[rq] , A 0.
(a) Based on the monthly closing prices provided, estimate the expected return E[ri,] and variance Var[rp] of ASX200 per annum (p.a.). Note that you should first compute the continuously compounded returns by using ln(Pt±i/Pt), then compute the sample average and sample variance of the monthly returns. Finally, to annualize the estimates, multiply the sample average and sample variance by 12. [1 mark]
(b) Given that the 1-year term deposit rate is 2.3% p.a. and the risk aversion coefficient is given by A = 4, suppose your client only invests in ASX200 and the 1-year term deposit, compute the portfolio weights (wp*, 1 — w;), expected return E[rq] and standard deviation aq for the optimal portfolio q*. What is the maximum utility (or risk-adjusted return) that can be achieved? Then, briefly explain why is it not optimal to allocate 100% of capital to either the risk-free security or the equity index. [1 mark]
(c) Plot the capital allocation line (CAL) in the (a, E[r]) space for a in the range of 0%30%, you should also indicate the position of the optimal portfolio q*. Furthermore, on the same graph, also plot the indifference curve that corresponds to the maximum utility achieved by the optimal portfolio q*. Based on the indifference curve plotted, briefly explain why a higher utility cannot be attained by a portfolio different from q*. [2 marks]
(d) Based on the quarterly returns provided for the ASX200 and the Australian Semi-Government Bond ETF, estimate the expected return and variance p.a. using the same approach as in Part (a). Moreover, also estimate the correlation coefficient. Then, suppose your client only invests in ASX200 and the Bond EFT, by using the Solver add-in in Excel, work out the portfolio weights (w*E, wt,), expected return E[rq] and standard deviation crq for the optimal portfolio q* (assume A = 4). What is the maximum utility that can be achieved? Briefly explain why is it not optimal for any risk-averse investor to allocate all her capital to debt (Hint: you may want to compute the expected return of the minimum variance portfolio.). [2 marks]
(e) Plot the investment opportunity set (I0S) in the (a, E[r]) space that represent all the possible portfolios attainable by investing in ASX200 and the Bond ETF. You should also indicate the positions of ASX200, the Bond ETF and the optimal portfolio q*. On the same graph, also plot the indifference curve which corresponds to the maximum utility that can be achieved by q*. [2 marks]
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(f) Suppose now your client invests in the ASX200, Bond ETF and the 1-year term deposit, work out the portfolio weights (wi, w*D), expected return E[rT] and standard deviation QT for the tangent portfolio that maximizes the slope of the CAL when combined with the risk-free security. Plot the CAL together with the IOS from Part (e) in the (a, ]E[r]) space, also indicate the positions of ASX200, Bond ETF and the tangent portfolio T. Suppose your client has a target expected return of E[rq] = 0.06 p.a., what is the minimum standard deviation aq? How do you construct this portfolio? Finally, indicate this portfolio q on the graph. [2 marks]
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Question 2 (Statistical Arbitrage - 5 marks)
You are now discussing statistical arbitrage by investing in long-short portfolios with another client.
(a) Let rk be the monthly (continuously compounded) return of stock k and rM the monthly return of ASX200. Based on the single index model (SIM),
rk = ak + Okrm + Ek,
estimate the ak and /3k for BHP, CBA, TLS and WOW respectively. [1 mark]
(b) Compute the return variance p.a. a for each individual stock, and decompose it into systematic risk °Lys and idiosyncratic risk 0E2,k. Also, estimate the expected stock returns p.a. [1 mark]
(c) Construct an arbitrage portfolio P using the following strategy. First, invest either 100% or -100% in BHP, i.e., WBHP = 1 or WBHp = —1. Then, decide on the portfolio weights of ASX200 and the risk-free security (assume the risk-free rate is 2.3% p.a.), such that portfolio weights sum up to zero and the beta of the arbitrage portfolio is zero. What is the expected return, variance and utility of portfolio P (Assume risk aversion A = 4)? [1 mark]
(d) Replace BHP with each of the other 3 stocks and repeat the exercise in part (c). Which arbitrage portfolio should the investor choose and why? [1 mark]
(e) Briefly explain why a statistical arbitrage opportunity may not be desirable for a risk-averse investor even if the expected return is positive. [1 mark]
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Question 3 (Testing the CAPM - 5 marks)
Your client from Question 2 is quite skeptical about the profitability of the strategy that you've recommended. The client argues that if the market is efficient and the CAPM holds, then the only way to earn a higher risk premium is to take on more systematic risk (measured by beta).
(a) Based on the adjusted closing prices for BHP, CBA, TLS, WOW, FLT, MYR, COH, JBH, WES, QBE and ASX200, estimate the stock betas /3k and the annualized expected returns E[rk]. Plot the SML predicted by the CAPM, also indicate the positions of the 10 individual stocks and of ASX200 (you can treat ASX200 as a proxy for the market portfolio and assume the risk-free rate is 2.3% p.a.). Are the positions of the individual stock consistent with the CAPM prediction? Why or why not? [1.5 marks]
(b) Based on the stock beta estimates and expected returns of the 10 stocks from Part (a), run the following linear regression
E[rk] = a + 7i3k + Ek,
where E[rk] is the dependent variable and Ok is the independent variable. Essentially, you are trying to explain the variation in expected stock returns by variation in the stock betas. Report the estimates for a and 7 and their p values. Plot the best-fitted line together with the SML from Part (a), you should also indicate the positions of the 10 stocks. Finally, comment on the regression results in terms of the observed risk-return relationship. [2 marks]
(c) Plot the CML and indicate the positions of the 10 individual stocks in the (a, E[r]) space. Are the positions of the individual stocks consistent with CAPM, why or why not? Finally, explain to your client why the market may not be perfectly efficient and therefore arbitrage opportunities may exist. [1.5 marks]
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