Increasing a Stock Position Financed by Reducing orSelling Short the Position in

the Risk-Free Asset.The marginal variance result of the last subsection generates

predictions about the impact of marginal changes in the weights of a portfolio. For

example, consider a $100,000 portfolio that has $60,000 invested in IBM, $30,000

invested in Compaq, and $10,000 invested in risk-free Treasury bills. If IBM’s return

positively covaries with the return of this $100,000 portfolio and Compaq’s return neg-

atively covaries with it, then the marginal variance result implies that:

1.Aportfolio with $60,001 invested in IBM, $30,000 in Compaq, and $9,999

invested in Treasury bills will have a higher return variance than the original

portfolio—$60,000 in IBM, $30,000 in Compaq, and $10,000 in T-bills—

because, at the margin, it has an additional dollar invested in the positively

covarying stock, financed by reducing investment in the risk-free asset by one

dollar.

2.Aportfolio with $60,000 invested in IBM, $30,001 in Compaq, and $9,999

invested in Treasury bills will have a lower return variance than the original

portfolio because, at the margin, it has an additional dollar invested in the

negatively covarying stock, financed by reducing investment in the risk-free

asset by one dollar.

3.Aportfolio with $59,999 invested in IBM, $30,000 in Compaq, and $10,001

invested in Treasury bills will have a lower return variance than the original

portfolio because at the margin, it has one less dollar invested in the

positively covarying stock and an additional dollar in the risk-free asset.¹²

Increasing a Stock Position Financed by Reducing orShorting a Position in a

Risky Asset.To generalize the interpretation of the covariance as a marginal vari-

ance, substitute the returns of some stock for the risk-free asset in equation (4.11). In

this case, the marginal variance is proportional to the difference in the covariances of

the returns of the two stocks with the portfolio. Adding a bit of the difference between

two stock returns is equivalent to slightly increasing the portfolio’s position in the first

stock and slightly decreasing its position by an offsetting amount in the second stock.

In the IBM, Compaq, Treasury bill example, suppose that IBM’s return has a

covariance of .03 with the return of the $100,000 portfolio, while Compaq’s return has

a covariance of .01. In this case:

¹²We

have confidence in these results because the change in the portfolio is small. Specifically, m

.00001 in cases (1) and (2) and m .00001 in case 3.

Grinblatt263Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw263Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

122Part IIValuing Financial Assets

1.Aportfolio with $60,001 invested in IBM, $29,999 in Compaq, and $10,000

invested in Treasury bills will have a higher return variance than the original

portfolio because, at the margin, it has an additional dollar invested in

positively covarying stock, financed by reducing investment in a stock with a

lower covariance by one dollar.

2.Aportfolio with $59,999 invested in IBM, $30,001 in Compaq, and $10,000

invested in Treasury bills will have a lower return variance than the original

portfolio because, at the margin, it has an additional dollar invested in the

negatively covarying stock, financed by reducing investment in a stock with a

higher covariance by one dollar.

Example 4.17:How to Use Covariances Alone to Reduce Portfolio Variance

IBM and AT&T stock have respective covariances of .001 and .002 with a portfolio.IBM and

AT&T are already in the portfolio.Now, change the portfolio’s composition slightly by hold-

ing a few more shares of IBM and reducing the holding of AT&T by an equivalent dollar

amount.Does this change increase or decrease the variance of the overall portfolio?

Answer:This problem takes an existing portfolio and adds some IBM stock to it, financed

by selling short an equal amount of AT&T (which is equivalent to reducing an existing AT&T

position).The new return is

˜m(˜r˜)

Rr

pIBMATT

The covariance of r˜r˜with the portfolio is .001.002, which is negative.If mis small

IBMATT

enough, the new portfolio will have a smaller variance than the old portfolio.More formally,

taking the derivative of the variance of this expression with respect to mand evaluating the

derivative at m 0 yields

2[cov(r˜,R˜)cov(r˜,˜R)]

IBMpATTp

which is negative.

Result 4.9 summarizes the conclusions of this section:

Result 4.9

If the difference between the covariances of the returns of stocks Aand B with the returnof a portfolio is positive, slightly increasing the portfolio’s holding in stock Aand reduc-ing the position in stock B by the same amount increases the portfolio return variance. Ifthe difference is negative, the change will decrease the portfolio return variance.

Why Stock Variances Have No Effect on the Marginal Variance.It is some-

what surprising that the variances of individual stocks play no role in the computa-

tion of what has been termed the marginal variance.However, bear in mind that

computations involving infinitesimal changes in a portfolio require the use of cal-

culus. In the IBM and Compaq examples, the portfolio changes, while extremely

small, are not infinitesimal, and the variance of IBM and Compaq affects the vari-

ance of the new portfolio return. However, the portfolio changes are so small that

any effect from the variances of IBM and Compaq are minuscule enough to be

swamped by the covariance effect.

The lesson about covariance as a marginal variance is important because it allows

us to understand the necessary conditions for identifying the precise portfolio weights

of portfolios that investors and corporate financial managers find useful. The first of

these portfolios is introduced in the next section.

Grinblatt265Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw265Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

123