1. Consider the following estimates on the US, UK and Japanese stock market:
US UK Japan
Means 0.12 0.15 0.14
St. Dev. 0.15 0.24 0.22
Correlation matrix:
US UK Japan
US 1.000 0.500 0.266
UK 0.500 1.000 0.358
Japan 0.266 0.358 1.000
These are estimated from historical data on asset returns in these countries. Assume that the risk-free rate is 5%. Suppose you are a US investor contemplating investing in the Japanese and UK stock markets. You currently have a 100% U.S. equity portfolio.
(a) Given the data above, what are the Sharpe ratios on your US portfolio and UK portfolio? Given the correlation between US and UK above, do you believe you should add UK portfolio to your US only portfolio?
(b) To see how robust your conclusion is on the issue of adding UK stocks to your portfolio, you can compute a "hurdle rate." That is, given the correlations, volatilities and US Sharpe ratio, you attempt to determine what expected return on UK stocks you should at least have to improve your Sharpe ratio. What is the hurdle rate? Show your reasoning and computations.
(c) Compute a similar hurdle rate for the Japanese equity market. Are the differences between your answers to c) and b) surprising? Why or why not?
2. The expected return on the Market Portfolio M is E(RM)=15%, the standard deviation is sM=25% and the risk-free rate is Rf=5%. The CAPM is assumed to hold.
(a) Draw on a diagram the Capital Market Line derived from the above data. Make sure to clarify the intercept and the slope.
(b) Compute the expected return of two well-diversified portfolios (i.e. portfolios on the CML), one with standard deviation of 15%, and the other with standard deviation of 25%.
(c) Suppose that a portfolio with standard deviation of 10% has an expected return of 11%. Is this compatible with the CAPM? Explain.
3. Suppose that the US market and the Japanese market are segmented, but the CAPM holds within each market. A US company (ZYX) wishes to raise additional capital and its CEO figures that the beta of ZYX with respect to the US market is 1.25. The beta of ZYX with respect to the Japanese market is 0.75. Suppose that the risk-free rate for both countries is 4%, the market expected return for the US is 12%, and the market expected return for Japan is 15%.
(a) Where would it be easier for ZYX to raise capital? That is, which investors (US or Japan) would require lower rate of return for investing in ZYX?
(b) If the two markets became integrated, so that Japanese investors could invest in US (and vice versa), the new market portfolio consists 60% of the US stocks and 40% of the Japanese stocks. What is the expected return of this new index? Suppose that the beta of company ZYX in this integrated market is 0.9, would the company be better off to raise the capital in this aggregated market? Explain.
4. Two portfolio managers, Mr. P and Mr. Q, claim that they are both good at picking under-valued stocks. Over the years, the average return on the portfolio managed by Mr. P has been 14.1%, with standard deviation 15.5%, while the average return of Mr. Q's portfolio has been 15%, with standard deviation 17%. Over the same period, the average return on the market portfolio has been 12%, with standard deviation 12%. You estimate that the covariance between Mr. P's portfolio and the market has been sPM=0.018, while the covariance between Mr. Q's portfolio and the market has been sQM=0.0216. Finally, you estimate that the average return on money market funds has been 4% (risk-free rate).
(a) Compute the expected returns on Mr. P's and Mr. Q's portfolios that would be consistent with CAPM.
(b) Given the CAPM as the benchmark, is either of the two managers over-performing the market? What if you use the Sharpe Ratio as benchmark? Explain your answer carefully.
5. There are three stocks. Stock A had a beta of 0.6 and investors expect it to return 8.8%. Stock B has a beta of 1.2 and investors expect it to return 13.6%. If the CAPM holds in this market, what should be stock C's expected rate of return if its beta is 1.05?