Marsha:
Actually only about 500,000, dear. The co variances above the diagonal are the same as
the co variances below. But you are right, most of the estimates would be out-of-date or just
garbage.
John:
To say nothing about the expected returns: Garbage in, garbage out.
3

Marsha:
But John, you don’t need to solve for 1,000 portfolio weights. You only need a handful.
Here’s the trick: Take your benchmark, the S&P 500, as security 1. That’s what you would end up
with as an indexer. Then consider a few securities you really know something about. Pioneer could
be security 2, for example. Global, security 3. And so on. Then you could put your wonderful
financial mind to work.
John:
I get it. Active management means selling off some of the benchmark portfolio and
investing the proceeds in specific stocks like Pioneer. But how do I decide whether Pioneer really
improves the portfolio? Even if it does, how mush should I buy?
Marsha:
Just maximize the Sharpe ratio, dear.
John:
I’ve got it! The answer is yes!
Marsha:
What’s the question?
John:
You asked me to marry you. The answer is yes. Where should we go on our honeymoon?
Marsha:
How about Australia? I’d love to visit the Melbourne Stock Exchange.
The following table reproduces John’s notes on Pioneer Gypsum and Global Mining.
Calculate the expected return, risk premium, and standard deviation of a portfolio invested
partly in the market and partly in Pioneer. (You can calculate the necessary inputs from the
beats and standard deviations given in the table). Does adding Pioneer to the market
benchmark improve the Sharpe ratio? How mush should Jon invest in Pioneer and how
much in the market?
Pioneer Gypsum
Global Mining
Expected return
11.0%
12.9%
Standard deviation
32%
20%
Beta
0.65
1.22
Stock price
$87.50
$105.00
Repeat the analysis for Global Mining. What should John do in this case? Assume that
Global accounts for .75 of the S&P index.
Solution
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 (σ
im
) to the market variance
σ
m
2
, where σ
m
2
= .16
2
= .0256. For Pioneer Gypsum, β = .65 = σ
im
/.0256, which gives a
covariance of σ
im
= .01664. The covariance also equals the
correlation coefficient ρ times the product of the stock’s and market’s standard deviations σ
i
and
σ
m
. For Pioneer,
σ
im
= ρσ
i
σ
m
= .01664 = ρ×.32×.16, which implies ρ = .325.