4.6Variances of Portfolios and Covariances between Portfolios

Covariances of stock returns are critical inputs for computing the variance of a portfo-

lio of stocks.

Variances for Two-Stock Portfolios

Consider an investor who holds an airline stock and is considering adding a second

stock to his or her portfolio. You would expect the new portfolio to have more vari-

ability if the second stock were another airline stock as opposed to, say, a pharmaceu-

tical stock. After all, the prices of two airline stocks are more likely to move together

than the prices of an airline stock and a pharmaceutical stock. In other words, if the

returns of two stocks covary positively, a portfolio that has positive weights on both

stocks will have a higher return variance than if they covary negatively.

Computing Portfolio Variances with Given Inputs.To verify this relation between

portfolio variance and the covariance between a pair of stocks, expand the formula for

the variance of a portfolio

var(xr˜xr˜) E{[xr˜xr˜E(xrxr)]²}

112211221122

It is possible to rewrite this equation as

var(x˜xr˜) E{[xr˜xr˜(xrxr)]²}

r

112211221122

E{[x(˜r)x(˜2

rrr)]}

111222

Squaring the bracketed term and expanding yields

(xr˜xr˜2˜r)²2˜r)²r˜r)(˜r)]

var) E[x(rx(r2xx(r

1122111222121122

2˜r)²2˜r)²]2xxE[(˜r)(˜r)]

xE[(r]xE[(rrr

111222121122

2˜2˜)2xxcov(r˜, r˜)

xvar(r)xvar(r

11221212

2222

xx2xx(4.9a)

11221212

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Markets and Corporate			Companies, 2002
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Chapter 4

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111

Equation (4.9a) indicates that the variance of a portfolio of two investments is the sum

of the products of the squared portfolio weights and the variances of the investment

returns, plus a third term, which is twice the product of the two portfolio weights and

the covariance between the investment returns. Thus, consistent with the intuition pro-

vided above, when the portfolio has positive weight on both stocks, the larger the

covariance, the larger is the portfolio variance.

Example 4.12 illustrates how to apply equation (4.9a).

Example 4.12:Computing the Standard Deviation of Portfolios of Two Stocks

Stock A has a standard deviation () of 30 percent per year.Stock B has a standard devi-

A

ation () of 10 percent per year.The annualized covariance between the returns of the two

B

stocks () is .0002.Compute the standard deviation of portfolios with the following sets of

AB

portfolio weights:

1.	x .75	x .25	3.x 1.5x.5
	A	B	AB
2.	x .25	x .75	4.x xx 1 x
	A	B	AB

Answer:The stocks’variances are the squares of their standard deviations.The variance

of stock A’s return is therefore .09, stock B’s is .01.Applying equation (4.9a) gives the vari-

ances for the four cases.

1.	22(.01) .75(.09) .252(.75)(.25)(.0002) .051325
	22(.01)
2.	.25(.09) .752(.75)(.25)(.0002) .011325 22(.01)
3.	1.5(.09) (.5)2(1.5)(.5)(.0002) .2047
4.	.09x2.01(1 x)2 2(.0002)x(1 x)

Because these are variances, the standard deviations are the square roots of these results.

They are approximately

1.	22.65%	3.45.24%
2.	10.64%	4..09x22
		.01(1x)2(.0002)x(1x)

Using Historical Data to Derive the Inputs: The Backward-Looking Approach.

Example 4.12 calculates the standard deviations of portfolios of two stocks, assuming

that the variances (or standard deviations) and covariances of the stocks are given to

us. As noted earlier, these variances and covariances must be estimated. Acommon

approach is to look backward, using variances and covariances computed from histor-

ical returns as estimates of future variances and covariances. Example 4.13 shows how

to use this approach to predict the variance following a major merger.

Example 4.13:Predicting the Return Variance of ChevronTexaco Following the

Mergerof Chevron and Texaco

In October 2000, Chevron and Texaco announced plans to merge, with Chevron, at about

twice the market capitalization of Texaco, being the acquiring firm.Exhibit 4.2 presents the

12 monthly returns of Texaco and Chevron in the year prior to the announcement.Predict

the variance of the merged entity, ChevronTexaco, using this data.Assume that Chevron is

2/3 of the merged entity for your answer.

Answer:The variances of Chevron and Texaco’s monthly returns are 0.00850 and

0.00768, respectively, and the covariance between them is 0.00689.Thus, the monthly return

variance of the combined entity is

22

var(ChevronTexaco) (2/3)(0.00850) (1/3)(0.00768) 2(2/3)(1/3)(0.00689) 0.00770

The relatively high covariance between Chevron and Texaco’s returns generates a variance

of the combined entity that is not substantially less than that of the two separate entities.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

112	Part IIValuing Financial Assets

EXHIBIT4.2

Monthly Returns forChevron and Texaco from October1999through September2000

Chevron

Texaco

Oct-99	2.89%	2.77%
Nov-99	2.33%	0.09%
Dec-99	2.19%	10.76%
Jan-00	3.46%	2.65%
Feb-00	9.93%	9.50%
Mar-00	23.77%	13.31%
Apr-00	7.91%	8.60%
May-00	9.36%	17.98%
Jun-00	8.25%	7.29%
Jul-00	6.87%	7.19%
Aug-00	7.84%	5.19%
Sep-00	0.87%	1.91%

Correlations, Diversification, and Portfolio Variances

Substituting equation (4.8)—the formula for computing covariances from correlations

and standard deviations—into equation (4.9a), the portfolio variance formula, generates

a formula for the variance of a portfolio given the correlation instead of the covariance.

var(x˜xr˜2222
r) xx2xx	(4.9b)
11221122121212

Equation (4.9b) illustrates the principle of diversification. For example, if both

stock variances are .04, a portfolio that is half invested in each stock has a variance of

.02.02 .25(.04).25(.04)2(.5)(.5) (.2)(.2)

12

This variance is lower than the .04 variance of each of the two stocks as long as is

less than 1. Since is generally less than 1, more stocks almost always imply lower

variance.

Equation (4.9b) also yields the following result:

Result 4.2

Given positive portfolio weights on two stocks, the lower the correlation, the lower the vari-ance of the portfolio.

Portfolio Variances When One of the Two Investments in the Portfolio Is Riskless.

Aspecial case of equation (4.9b) occurs when one of the investments in the portfolio

is riskless. For example, if investment 1 is riskless, and are both zero. In this

112

case, the portfolio variance is

22

x

22

The standard deviation is either

x, if xis positive

222

or

x, if xis negative

222

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

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Hence, when is zero, the formula for the standard deviation of a portfolio of two

1

investments is the absolute value of the portfolio-weighted average of the two standard

deviations, 0 and . Note also that since x (1 x), a positive weight on a risk-

221

free asset and risky asset return, which necessarily implies that both weights are less

than 1,reduces variance relative to a 100 percent investment in the risky asset. More-

over, a negative weight on a risk-free asset (implying x

1) results in a larger vari-

2

ance than a 100 percent investment in the risky asset. Hence, leverage increases risk

(see part bof exercise 4.18).

Portfolio Variances When the Two Investments in the Portfolio Have Perfectly

Correlated Returns.Equation (4.9b) also implies that when two securities are per-

fectly positively or negatively correlated, it is possible to create a riskless portfolio

from them, that is, one with a variance and standard deviation of zero, as Example 4.14

illustrates.⁸

Example 4.14:Forming a Riskless Portfolio from Two Perfectly Correlated

Securities

The Acquiring Corporation has just announced plans to purchase the Target Corporation,

which started trading for $50 per share just after the announcement.In one month, share-

holders of the Target Corporation will exchange each share of Target Corporation stock they

own for 2 shares of Acquiring Corporation stock, which currently sells for $20 per share, plus

$10 in cash.As a consequence of this announcement, shares of the Target Corporation

began to move in lockstep with shares of the Acquiring Corporation.Although returns on the

two stocks became perfectly positively correlated once the announcement occurred, the cash

portion of the exchange offer made the shares of the Target Corporation less volatile than

shares of the Acquiring Corporation.Specifically, immediately after the announcement, the

standard deviation of the returns of the Acquiring Corporation’s shares is 50% per annum

while those of the Target Corporation have a standard deviation of 40% per annum.What

portfolio weights on the Target and Acquiring Corporations generate a riskless investment

during the month prior to the actual merger?

Answer:Using equation (4.9b), the portfolio weights xand 1 xon the Acquiring and

Target Corporation’s shares, respectively, make the variance of the portfolio’s return

²x^{2 222}

.5 .4(1 x) 2x(1 x)(.5)(.4) (.5x .4(1 x))

which equals zero when x 4.Thus, the weight on Acquiring is 4, the weight on Target

is 5.

For the portfolio in Example 4.14, the standard deviation, the square root of the

variance, is the absolute value of .5x.4(1 x). This is the portfolio-weighted aver-

age of the standard deviations. In sum, we have:

Result 4.3

The standard deviation of either (1) a portfolio of two investments where one of the invest-ments is riskless or (2) a portfolio of two investments that are perfectly positively corre-lated is the absolute value of the portfolio-weighted average of the standard deviations ofthe two investments.

⁸Perfect correlations often arise with derivative securities. Aderivativeis a security whose value

depends on the value of another security. For example, the value of a call option written on a stock—the

right to buy the stock for a given amount—described in Chapter 3, or a put option—the right to sell the

stock for a given amount—described in Chapter 2, depend entirely on the value of the stock. Derivative

securities are discussed further in Chapters 7 and 8.

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Markets and Corporate			Companies, 2002
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114Part IIValuing Financial Assets

For special cases (1) and (2), Result 4.3 suggests that when the weight on the riskier

of the two assets is positive, the equations for the standard deviation and of the mean

of the portfolio are, respectively, the portfolio-weighted averages of the individual asset

return standard deviations and the individual asset return means.

Portfolios of Many Stocks

To compute the variance of a portfolio of stocks, one needs to know the following:

•	The variances of the returns of each stock in the portfolio.
•	The covariances between the returns of each pair of stocks in the portfolio.
•	The portfolio weights.

APortfolio Variance Formula Based on Covariances.Given the information

described above, it is possible to compute the variance of the return of a portfolio of

an arbitrary number of stocks with the formula given in the result below:

Result 4.4The formula for the variance of a portfolio return is given by:

NN
2xx	(4.9c)
pijij
i 1j 1

where covariance between the returns of stocks i and j.

ij

Example 4.15 illustrates how to apply equation (4.9c).

Example 4.15:Computing the Variance of a Portfolio of Three Stocks

Consider three stocks, AT&T, SBC, and Verizon, denoted respectively as stocks 1, 2, 3.

Assume that the covariance between the returns of AT&Tand SBC (stocks 1 and 2) is .002;

between SBC and Verizon (2 and 3), .005;and between AT&T and Verizon (1 and 3), .001.

The variances of the three stocks are respectively .04, .06, and .09, and the weights on

AT&T, SBC, and Verizon are respectively x 1/3;x 1/6;and x 1/2.Calculate the vari-

123

ance of the portfolio.

Answer:From equation (4.9c), the variance of the portfolio is:

xxxxxxxxxxxxxxxxx

x

111112121313212122222323313132323333

.04.002.001.002.06.005.001.005.09

.03

91861836126124

In equation (4.9c), note that is the variance of the ith stock return when j i

ij

and that N²terms are summed. Therefore, the expression for the variance of a portfo-

lio contains Nvariance terms—one for each stock—but N²Ncovariance terms. For

a portfolio of 100 stocks, the expression would thus have 100 variance terms and 9,900

covariance terms. Clearly, the covariance terms are more important determinants of the

portfolio variance for large portfolios than the individual variances.

Note also that a number of the terms in the expression for a portfolio’s variance

repeat; for example, see equation (4.9c) and the answer to Example 4.15. Therefore,

you sometimes will see equation (4.9c) written as

N

222xx

x2

pjjijij

j 1i<j

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

115

For two stocks, this reads as

22222

xx2xx

p11221212

APortfolio Variance Formula Based on Correlations.Covariances also can be

computed from the corresponding correlations and either the variances or standard devi-

ations of the stock returns. Substituting equation (4.8), , into equation (4.9c)

ijijij

yields an equation for the variance of a portfolio of many stocks in terms of correla-

tions and standard deviations

NN

2xx

pijijij

i 1j 1

Covariances between Portfolio Returns and Stock Returns

Some of the portfolios that we will be interested in (for example, the minimum vari-

ance portfolio) can be found by solving for the weights that generate portfolios that

have prespecified covariances with individual stocks. Here, the following result is often

used:

Result 4.5

For any stock, indexed by k,the covariance of the return of a portfolio with the return ofstockk is the portfolio-weighted average of the covariances of the returns of the invest-ments in the portfolio with stock k’s return, that is

N

x

pkiik

i 1