Principles of Corporate Finance_ 12th Edition

bre44380_ch07_162-191.indd 177 09/02/15 04:11 PM

Chapter 7 Introduction to Risk and Return 177

Calculating the expected portfolio return is easy. The hard part is to work out the risk of

your portfolio. In the past the standard deviation of returns was 13.2% for JNJ and 31.0% for

Ford. You believe that these figures are a good representation of the spread of possible future

outcomes. At first you may be inclined to assume that the standard deviation of the portfolio

is a weighted average of the standard deviations of the two stocks, that is, (.60 × 13.2) + (.40 

× 31.0) = 20.3%. That would be correct only if the prices of the two stocks moved in perfect

lockstep. In any other case, diversification reduces the risk below this figure.

The exact procedure for calculating the risk of a two-stock portfolio is given in Figure 7.12.

You need to fill in four boxes. To complete the top-left box, you weight the variance of the

returns on stock 1 ( σ 1

2

) by the square of the proportion invested in it ( x 1

2

). Similarly, to com-

plete the bottom-right box, you weight the variance of the returns on stock 2 ( σ 2

2

) by the

square of the proportion invested in stock 2 ( x 2

2

).

The entries in these diagonal boxes depend on the variances of stocks 1 and 2; the entries

in the other two boxes depend on their covariance. As you might guess, the covariance is a

measure of the degree to which the two stocks “covary.” The covariance can be expressed as

the product of the correlation coefficient ρ 12 and the two standard deviations:^29

Covariance between stocks 1 and 2 = σ 12 = ρ 12 σ 1 σ 2

For the most part stocks tend to move together. In this case the correlation coefficient ρ 12 is

positive, and therefore the covariance σ 12 is also positive. If the prospects of the stocks were

wholly unrelated, both the correlation coefficient and the covariance would be zero; and if the

stocks tended to move in opposite directions, the correlation coefficient and the covariance

would be negative. Just as you weighted the variances by the square of the proportion invested,

so you must weight the covariance by the product of the two proportionate holdings x 1 and x 2.

Once you have completed these four boxes, you simply add the entries to obtain the port-

folio variance:

Portfolio variance = x 1

2

σ 1

2

+ x 2

2

σ 2

2

+ 2(x 1 x 2 ρ 12 σ 1 σ 2 )

BEYOND THE PAGE

mhhe.com/brealey12e

How to calculate

correlation

coefficients

(^29) Another way to define the covariance is as follows:
Covariance between stocks 1 and 2 = σ 12 = expected value of ( ̃r 1 − r 1 ) × (r ̃ 2 − r 2 )
Note that any security’s covariance with itself is just its variance:
σ 11 = expected value of ( ̃r 1 − r 1 ) × (r ̃ 1 − r 1 )
= expected value of ( r̃ 1 − r 1 )^2 = variance of stock 1 = σ 12
◗ FIGURE 7.12
The variance of a
two-stock portfolio is
the sum of these four
boxes. x 1 , x 2  = pro-
portions invested in
stocks 1 and 2; σ 12 ,
σ 22 = variance of stock
returns; σ 12  = cova-
riance of returns
(ρ 12  σ 1  σ 2 ); ρ 12  = correla-
tion between returns on
stocks 1 and 2.
Stock 1
Stock 1
Stock 2
Stock 2
x 122 1
x 1 x 2 σ 12
= x 1 x 2 ρ 12 σ 1 σ 2
x 1 x 2 σ 12
= x 1 x 2 ρ 12 σ 1 σ 2
σ
x 222 σ 2