John and Marsha on Portfolio Selection

JOHN AND MARSHA ON PORTFOLIO SELECTION

Minicase solution, Chapter 8

Principles of Corporate Finance, 10th 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.

Here is the 2×2 covariance matrix for the market and Pioneer.

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

It turns out that the Sharpe ratio is maximized by putting about 95% in the market and 5% in Pioneer.

|S&P 500 |Pioneer |Sharpe ratio |

|1.0 |0 |.4688 |

|.99 |.01 |.4694 |

|.98 |.02 |.4698 |

|.97 |.03 |.4701 |

|.96 |.04 |.4702 |

|.95 |.05 |.4702 |...