Introduction to Corporate Finance

PART 2: VALUATION, RISK AND RETURN

For instance, in Table 6.6 in the previous chapter (see page 207), we estimated the standard deviation

of returns for Archer Daniels Midland (ADM) to be 24.8%. The same table reported a standard deviation

for American Airlines (AMR) of 47.0%. Suppose we form a portfolio invested equally in ADM and AMR

shares. With portfolio weights of 0.50, you might guess that the standard deviation of this portfolio equals:

Portfolio standard deviation = (0.50)(24.8%) + (0.50)(47.0%) = 35.9%

As reasonable as that guess seems, it is wrong. It turns out that a portfolio invested in equal proportions

of ADM and AMR has a standard deviation of just 28%!

As a general rule, the standard deviation of a portfolio is almost always less than the weighted average of

the standard deviations of the shares in the portfolio. This is diversification at work. Combining securities

together eliminates some of their unsystematic risk, so the portfolio is less volatile than the average share

in the portfolio.^9

The standard deviation of a portfolio takes account of how the expected returns of the securities in

the portfolio co-vary with each other. This term is the product of the correlation coefficient of the expected

returns of the constituent securities with each other (ρ 12 ), and their respective standard deviations (σ 1 )

and (σ 2 ). For a portfolio with two securities (or more generally, two assets) the standard deviation of a

portfolio (σPortfolio) is given by:

σ=σ+ σ+ ρσσ

σ

σ

ρ

wwww

w

w

2

Where

=ProportionoftheportfolioinvestedinAsset1

=ProportionoftheportfolioinvestedinAsset2

=Asset1standard deviationofreturns

=Asset2standard deviationofreturns

=CorrelationcoefficientbetweenthereturnsofAsset1andAsset2

portfolio 1

2

1

2

2

2

2

2

12 1, 21 2

1

2

1

2

1, 2

In this example, the correlation coefficient of the two assets is 0.13382. Thus the portfolio standard

deviation can be calculated as follows:

Portfolio standard deviation = [(0.50)^2 (24.8%)^2 + (0.50)^2 (47.0%)^2 +

2(0.5)(0.5)(0.13382)(24.8%)(47.0%)]0.5 = 28%

We can write a more general expression to calculate portfolio standard deviation for a portfolio

containing n assets. As described previously, the expected returns on these assets are E(r 1 ), E(r 2 ),...., E(rn).

The portfolio weights are w 1 , w 2 , ....wn. The portfolio standard deviation (σPortfolio) is given by this equation:

Eq. 7.3 σ=∑ σ+∑∑ σσσ

=

prtfolio wwaa^2 aaw

a

n

aa aa

a

n

a

n

22

=1

,

1 =1

o 12

2

2212

1

wa 1 + wa 2 + ... + wan = 1

which can also be represented as

∑

a=

n

1

wa = 1

9 You can also see this effect of diversification in Table 6.6 on page 207. The average standard deviation of the 10 shares listed in that table

is 31.4%, but the standard deviation of a portfolio containing all 10 shares is just 19.1%.

correlation coefficient
A statistical measure of the
degree of interdependence
between two variables,
indicating how they vary
together. The correlation
coefficient can range from
–1 to +1. A value of +1
indicates the variables are
perfectly correlated, while
a value of –1 indicates they
are perfectly negatively
correlated. A value of zero
indicates that the variables
have zero interdependence.
In the context of portfolio
management, the risk of the
returns of a portfolio of two
assets is reduced when it
contains assets whose returns
are negatively correlated with
each other

Eq. 7.2