VK Asset Management



Asset Management

Instructor: Dr. Youchang Wu

Exercises on portfolio optimization &CAPM

The excel sheet AM2007_EX1.xls contains monthly returns of five industry portfolios, which are downloaded from professor French’s website: .

1. Calculate the historical average monthly returns for each portfolio. Annualize the returns.

2. Compute the variance-covariance matrix. Annualize.

3. Calculate the annual standard deviations of each portfolio and correlations among the portfolios.

4. Identify the portfolio weights for the minimum variance portfolio. What are the expected return and standard deviation of this portfolio?

5. Use the historical averages of annual returns as expected returns. Assume the risk-free rate to be 4%. Compute the portfolio weight of the tangency portfolio.

6. Plot the portfolio frontier of the risky assets. Also plot a minimum of at least 100 “random” portfolios consisting of these assets – check that none has a better return-risk profile than the frontier portfolios.

7. Mr. Wu is willing to accept a portfolio standard deviation of 30%. Compute his optimal portfolio composition and portfolio characteristics, if he holds (or shorts)

a. Only the risky portfolios,

b. Risky portfolios plus the risk-free asset.

8. What is the beta of the minimum variance portfolio with respect to the tangency portfolio?

9. What would be the market portfolio if CAPM is true?

10. Check whether the expected return of minimum variance portfolio is consistent with the CAPM formula.

11. Now suppose that the risky free asset does not exist and that the market portfolio is a 50:50 combination of the minimum variance portfolio and tangency portfolio you just identify in Exercise 4 and 5.

a. Identify the zero beta portfolio (zero-beta with respect to the market portfolio)

b. Calculate the expected return of the zero-beta portfolio based on portfolio weights and check whether it is consistent with the zero-beta CAPM formula.

c. Check whether the zero-beta CAPM formula holds for each industry portfolio.

12. The following information is given: E(rm(=0.16, (m=0.2, E(rA(=0.17, (A=0.2, E(rB(=0.18, (B=0.3, [pic]=0.08, where m stands for the market portfolio, A and B for individual stocks and z for the zero beta portfolio.

a. Does CAPM hold in this case?

b. Now suppose you make a mistake when computing the market portfolio: you assume that the market portfolio consists of 50% of stock A and 50% of stock B. We denote this “wrong” market portfolio as m´. The returns rA and rB are not correlated. Would in this case the “CAPM” hold for portfolios p consisting of combinations of stock A and stock B:

[pic]?

rz’ and rm’ stand for the returns of the zero-beta portfolio and the “wrong” market portfolio respectively. [pic]denotes the beta of the portfolio with respect to the “market portfolio” m’.

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