JOHN AND MARSHA ON PORTFOLIO SELECTION Minicase solution, Chapter 8 Principles of Corporate Finance, 12th Edition R. A. Brealey, S. C. Myers and F. Allen John neglected to mention the standard deviation of the S&P 500. We will assume 16%. Recall that stock i’s beta is just the ratio of its covariance with the market (σim) to the market variance σm2, where σm2 = .162 = .0256. For Pioneer Gypsum, β = .65 = σim/.0256, which gives a covariance of σim = .01664. The covariance also equals the correlation coefficient ρ times the product of the stock’s and market’s standard deviations σi and σm. For Pioneer, σim = ρσiσm = .01664 = ρ×.32×.16, which implies ρ = .325. Now calculate the portfolio return rP, portfolio standard deviation σP and the Sharpe ratio for different fractions invested in the market and Pioneer. For example, suppose that the market gets 99% of investment and Pioneer 1%. rP = .99×.125 + .01×.11 = .12485 σP2 = .992×.0256 + 2×.99×.01×.01664 + .012×.1024 = .0254 σP = √.0254 = .1595 Sharpe ratio = (rP – rf)/σP = (.12485 - .05)/.1595 = .4694 Pioneer .01664 .1024 S&P 500 Here is the 2×2 covariance matrix for the market and Pioneer. S&P 500 .0256 .01664 Pioneer Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. It turns out that the Sharpe ratio is maximized by putting about 95% in the market and 5% in Pioneer. S&P 500 1.0 .99 .98 .97 .96 .95 .94 Global .03123 .0576 S&P 500 .0256 .03123 We can follow the same procedures for Global Mining. Global’s covariance is .03123 and its correlation with the market is .8125. The 2×2 covariance matrix is: S&P 500 Global Pioneer 0 .01 .02 .03 .04 .05 .06 Sharpe ratio .4688 .4694 .4698 .4701 .4702 .4702 .4699 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 本文来源：https://www.wddqw.com/doc/58ce096032126edb6f1aff00bed5b9f3f80f7249.html