Investment Management Assignment -

In this assignment you will look at Mean-variance optimization, statistical arbitrage and testing the CAPM.

Data Description - The Excel worksheet contains monthly closing prices from Dec 2009 to Dec-2017 for six Australian stocks and also for the All Ordinaries (AORD) index. It also has the cash rate target from the RBA for the same time period. Note that the cash rate is expressed as a percentage per annum (p.a.) with daily compounding (assume 30 days in each month).

Question 1 - Mean-variance optimization

You have been approached by a client who is interested in investing in the six stocks. After some questionnaires, you discover that you client's utility function can be approximated by a quadratic utility,

U(E[r_p], Var[r_p]) = E[r_p] - ½AVar[r_p], A > 0.

(a) Based on the closing prices provided, estimate the expected monthly (continuous) returns for CBA, MYR, BHP, TLS, FLT and WBC respectively. If the investor is risk-neutral (i.e. A = 0) and plans to invest 100% of wealth in one of the six stocks, what would be the optimal choice?

(b) Compute the sample standard deviations of monthly returns for each of the six stocks in part (a). If the investor is extremely risk averse (i.e. A → ∞), and plans to invest 100% of wealth in one of the six stocks, what would be the optimal choice?

(c) Compute the sample correlation with each pair of the six stocks in part (a). Which pair has the highest correlation and which pair has the lowest correlation?

(d) Assuming short-selling is not allowed, find the portfolio of the two most highly correlated stocks that will maximise your client's utility (Assume A = 4). Note that you can do this by using Solver in Excel or simply by trial and error (in a systematic way).

(e) Repeat part (d) with short-selling allowed.

(f) Repeat parts (d) and (e) for the two stocks that have the lowest correlation.

(g) Suppose now your client invests in the two stocks that have the highest correlation and also in the risk-free security, find the optimal portfolio that maximizes the expected utility of the client (assume A = 4 and short-selling is allowed). Hint: the best way is to first find the portfolio of the two stocks that maximizes the slope of the CAL and then find the best allocation between it and the risk-free security. Assume the risk-free interest rate is the average cash rate.

(h) Repeat part (g) if your client now decides to invest in the two stocks that have the lowest correlation and also in the risk-free security.

(i) Repeat part (g) if your client now decides to invest in AORD and also in the risk-free security.

(j) Overall, among all the portfolios from part (d) to part (i), which one is most optimal for the client? Provide a brief explanation.

Question 2 - Statistical Arbitrage

You are now discussing statistical arbitrage by investing in long-short portfolios with another client.

(a) Let R_i be the excess monthly return of stock i above the cash rate and R_M the excess monthly return of AORD. Based on the single index model (SIM),

R_i = α_i + β_iR_M + ε_i,
estimate the α_i and β_i i for each of the six stocks.

(b) Compute the expected excess returns, standard deviation of excess returns for each of the six stocks, and also the correlations between each pair of stocks under the assumptions of the SIM.

(c) Construct an arbitrage portfolio P by combining two stocks that are most highly correlated and the risk-free security. Make sure the portfolio P requires zero initial investment, has a zero beta coefficient (i.e. β_p = 0) and a positive expected return. Plot the accumulative percentage returns of portfolio P from Jan-2010 to Dec-2016.

(d) Repeat part (c) for an arbitrage portfolio, which consists of AORD (proxy for the market portfolio) and an equally-weighted portfolio of the six stocks. Plot the accumulative profits of the arbitrage portfolios in parts (c) and (d) on the same graph.

(e) Assume risk aversion A = 4, compute the expected utility of the arbitrage portfolios in parts (c) and (d). Which portfolio would you recommend to your client, if any? Provide a brief explanation.

Question 3 - Testing the CAPM

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 beta estimates and expected excess returns of the six stocks from Q2 parts (a) and (b), run the following regression

R^-_i = α + γβ^{^}_i + ε_i,

where R^-_i and β^{^}_i are the estimated expected excess return and beta coefficient of stock i. Report the estimates for α and γ and their p values.

(b) Based the estimates of α and γ, plot the best-fitted line and also the six stocks on a graph with betas on the horizonal axis and expected excess returns on the vertical axis. Which stocks are underpriced/overpriced according to the CAPM?

(c) Repeat part (a) now restricting α to be zero. Report the estimate for γ and its p value. Why are they different to those in part (a)?

(d) Repeat part (b) based on the regression results in part (c).

(e) Explain to your client why the market may not be perfectly efficient and therefore arbitrage opportunities may exist.

Attachment:- Assignment Files.rar