Chapter 6: Efficient Diversification

Problem 6.16 Assume expected returns and standard deviations for all securities, as well as the riskfree rate for lending and borrowing, are known. Will investors arrive at the same optimal risky portfolio? Explain

Problem 6.16 Solution • If the lending and borrowing rates are equal and there are no other constraints on portfolio choice, then the optimal risky portfolios of all investors will be identical. • However, if the borrowing and lending rates are not equal, then borrowers (who are relatively risk averse) and lenders (who are relatively risk tolerant) will have different optimal risky portfolios.

Problem 6.18 What is the relationship of the portfolio standard deviation to the weighted average of the standard deviations of the component assets?

Problem 6.18 Solution • In the special case that all assets are perfectly positively correlated, the portfolio standard deviation is equal to the weighted average of the component-asset standard deviations. • Otherwise, as the formula for portfolio variance (Equation 6.6) shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. • The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights.

Problem 6.20 Investors expect the market rate of return this year to be 10%. The expected rate of return on a stock with a beta of 1.2 is currently 12%. If the market return this year turns out to be 8%, how would you revise your expectation of the rate of return on the stock?

Problem 6.20 Solution • The expected rate of return on the stock will change by beta times the unanticipated change in the market return: 1.2 ( .08 - .10) = -2.4% • Therefore, the expected rate of return on the stock should be revised to: .12 - .024 = 9.6%

Problem 6.1 Forming a portfolio of two risky assets, what must be true of the correlation coefficient between their returns if there are to be gained from diversification? Explain.

So long as the correlation coefficient is below 1.0, the portfolio will benefit from diversification because returns on component securities will not move in perfect lockstep. The portfolio standard deviation will be less than a weighted average of the standard deviations of the component securities.

Problem 6.2 When adding a risky asset to a portfolio of many risky assets, which property of the asset is more important, its standard deviation or its covariance with the other assets? Explain.

The covariance with the other assets is more important. Diversification is accomplished via correlation with other assets. Covariance helps determine that number.