6.2The Principle of Diversification

Everyone familiar with the cliché, “Don’t put all your eggs in one basket,” knows that the

fraction of heads observed for a coin tossed 1,000 times is more likely to be closer to

one-half than a coin tossed 10 times. Yet, coin tossing is not a perfect analogy for invest-

ment diversification. Factor models help us break up the returns of securities into two

components: a component for which coin tossing as an analogy fails miserably (common

factors) and a component for which it works perfectly (the firm-specific components).

Insurance Analogies to Factor Risk and Firm-Specific Risk

To further one’s intuition about these two components of risk, think about two differ-

ent insurance contracts: fire insurance and health insurance. Fires are fairly indepen-

dent events across homes (or, at the very least, fires are independent events across

neighborhoods); thus, the fire-related claims on each company are reasonably pre-

dictable each year. As a consequence of the near-perfect diversifiability of these claims,

fire insurance companies tend to charge the expected claim for this diversifiable type

of risk (adding a charge for overhead and profit). By contrast, health insurance has a

mixture of diversifiable and nondiversifiable risk components. Diseases that require

costly use of the medical care system do not tend to afflict large portions of the pop-

ulation simultaneously. As the AIDS epidemic proves, however, health insurance com-

panies cannot completely eliminate some kinds of risk by having a large number of

policyholders. Should the HIVvirus mutate into a more easily transmittable disease,

many major health insurers would be forced into bankruptcy. As a result, insurers

should charge more than the expected loss (that is, a risk premium) for the financial

risk they bear from epidemics.

Factor risk is not diversifiable because the factors are common to many securities.

This means that the returns due to each factor’s realized values are perfectly correlated

across securities. In a one-factor market model, a portfolio with equal weights on a

thousand securities, each with the same market model beta, has the same market beta

(and thus, the same systematic risk) as each of the portfolio’s individual securities.¹⁰

This holds true in more general factor models, as the next section shows. Thus, even

the most extreme diversification strategy, such as placing an equal number of eggs in

all the baskets, does not reduce that portion of the return variance due to factor risk.

Quantifying the Diversification of Firm-Specific Risk

By contrast, it is relatively straightforward to demonstrate that the ˜risk of securities

is diversified away in large portfolios because the ˜s are uncorrelated across securi-

ties. Let us begin with two securities, denoted 1 and 2, each with uncorrelated ˜s that

¹⁰See

Chapter 5, Result 5.4.

Grinblatt382Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw382Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

182	Part IIValuing Financial Assets
EXHIBIT6.3	Firm-Specific Standard Deviation of a Portfolio

0.11

Firm-specific standard deviation

0.09

0.07

0.05

0.03

0.01

0	10	20	30 40 50	60	70	80
			Number of securities

have identical variances of, say, .1. By the now familiar portfolio variance formula

from Chapter 4, an equally weighted portfolio of the two securities, that is

x x .5, has the firm-specific variance

12

var (˜2˜2˜) .25(.1).25(.1) 2(.25)(.1) .05

) xvar ()xvar (

p1122

Thus, a portfolio of two securities halves the firm-specific variance of each of the two

securities.

An equally weighted portfolio of 10 securities, each with equal firm-specific vari-

ance, has the firm-specific variance

var (˜2˜2˜). . .2˜)

) xvar ()xvar (xvar (

p11221010

.01(.1).01(.1). . ..01(.1) 10(.01)(.1) .01

This is one-tenth the firm-specific variance of any of the individual securities.

Continuing this process for Nsecurities shows that the firm-specific variance of the

portfolio is 1/Ntimes the firm-specific variance of any individual security and that the

standard deviation is inversely proportional to the square root of N.Exhibit 6.3 summa-

rizes these results by plotting the standard deviation of the firm-specific ˜of a portfolio

against the number of securities in the portfolio. It becomes obvious that firm-specific risk

is rapidly diversified away as the number of securities in the portfolio increases.¹¹

Agood rule of thumb is that a portfolio with these kinds of weights will have a

firm-specific variance inversely proportional to the number of securities. This implies

the following result about standard deviations:

¹¹When firm-specific variances are unequal, the portfolio of the Nsecurities that minimizes var(˜)

p

has weights that are inversely proportional to the variances of the ˜s. These weights result in a

i

firm-specific variance for the portfolio that is equal to the product of the inverse of the number of

securities in the portfolio times the inverse of the average precision of a security in the portfolio, where

the precision of security i is 1/var(˜).As the number of securities in the portfolio increases, the firm-

i

specific variance rapidly gets smaller with large numbers of securities in this portfolio. Although the

inverse of the average precision is not the same as the average variance unless all the variances are

equal, the two will probably be reasonably close.

Grinblatt384Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw384Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

Chapter 6

Factor Models and the Arbitrage Pricing Theory

183

Result 6.1

If securities returns follow a factor model (with uncorrelated residuals), portfolios withapproximately equal weight on all securities have residuals with standard deviations that areapproximately inversely proportional to the square root of the number of securities.