4.8Interpreting the Covariance as a Marginal Variance

It is useful to interpret the covariance between the return of a stock and the return of

a portfolio as the stock’s marginal variance, which is the change in the variance of a

portfolio for a small increase in the portfolio weight on the stock.

Result 4.8

The covariance of a stock return with the return of a portfolio is proportional to the vari-ance added to the portfolio return when the stock’s portfolio weight is increased by a smallamount, keeping the weighting of other stocks fixed by financing the additional holdingsof the stock with another investment that has zero covariance with the portfolio.

AProof Using Derivatives from Calculus

To show that the covariance is a marginal variance, add $m,per dollar invested in the

portfolio, of stock k to the portfolio and finance this purchase by borrowing $mof a

risk-free security with return rwhich, being risk free, has zero variance and a zero

f

covariance with the portfolio. If, prior to the addition of stock k, the portfolio had return

˜,the new portfolio return, R˜, is

R

p

˜˜m(˜r)

R Rr

pkf

where ris the risk-free return. Because the risk-free return is constant and has no effect

f

on the variance, we know from the portfolio variance formula, equation (4.9a), that the

²

variance of this portfolio, , is

2 ˜m(˜222
var[Rrr)] m2m	(4.11)
pkfpkpk

where

2˜	2˜),	and	cov(˜,r˜).
var(R),	var(r		R
pp	kk		pkpk

The derivative of the variance in equation (4.11) with respect to mis

²

d

2

2(m)

dm^kpk

Evaluate this derivative at m 0 to determine how the portfolio variance is affected

by a marginal addition of stock k.At m 0, the derivative has the value

¹⁰Although

not pictured in Exhibit 4.5, smaller correlations increase the standard deviation when short

sales of one of the investments occurs.

¹¹While short positions are not displayed in Exhibit 4.5, they can easily be added to an analogous

diagram. See exercise 4.20.

Grinblatt261Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw261Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

121

2
d
2	(4.12)
dmpk

Equation (4.12) implies that adding stock k to the portfolio increases the portfolio vari-

ance if the return on the stock covaries positively with the portfolio return. The addi-

tion of stock k decreases the portfolio variance if its return covaries negatively with the

portfolio return. This result applies only for small m(that is, for a sufficiently small

amount of stock k added to the portfolio), and it assumes that this amount is appropri-

ately financed with an opposite, and similarly small, short position in the risk-free asset,

so that the new portfolio weights still sum to 1 after the stock is added to the portfolio.

Numerical Interpretations of the Marginal Variance Result