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
0
.4688
.99
.01
.4694
.98
.02
.4698
.97
.03
.4701
.96
.04
.4702
.95
.05
.4702
.94
.06
.4699
We can follow the same procedures for Global Mining. Global’s covariance is .03123 and its correlation with the market is extremely high at .976. (Perhaps John has underestimated Global’s standard deviation and thus overestimated its correlation with the market.) The 2×2 covariance matrix is:
Global’s return is not attractive: with a beta of , it should offer an expected rate of return of .05 + ×.075 = .1415, over 14%. But John’s estimate is only %. Therefore he should sell Global. In fact he should eliminate it from his portfolio.
Suppose that John’s starting portfolio matches the market and includes % in Global. Then he should sell all the Global shares and put the proceeds back into the overall market. The resulting portfolio weights are % in the market and % in Global. That is, the portfolio should “underweight” Global by % in order to reduce holdings of Global to zero. The underweight increases the Sharpe ratio from .4688 to .4693:
S&P 500
Global
Sharpe ratio
.99
.01
.4680
0
.4688
.4691
.4693
The Sharpe ratio gets still better if the portfolio weight for Global is reduced below %. A weight below % means selling short. In order to sell short, John would have to borrow Global shares, sell them (with an obligation to repurchase and return the shares later) and invest the sale proceeds in the market. But we doubt that John is allowed to sell short from the portfolio he manages. We discuss short selling in Chapter 21.
