how much diversification is...

How much diversification is enough?

by

Meir Statman Glenn Klimek Professor of Finance

Santa Clara University Leavey School of Business

Santa Clara, CA 95053 [email protected]

September 2002

I thank Roger Clarke, Ramie Fernandez, William Goetzmann, Mark Kutzman and Jonathan Scheid and acknowledge financial support from The Dean Witter Foundation.


Abstract

Levels of diversification in the portfolios of investors present a puzzle. The benefits of diversification, measured by the rules of mean-variance portfolio theory, have increased in recent years, yet levels of diversification did not increase, remaining much below their optimal levels. We find that today’s optimal level of diversification, measured by the rules of mean-variance portfolio theory, exceeds 120 stocks, and argue that the diversification puzzle is solved within Shefrin and Statman’s (2000) behavioral portfolio theory.

Investors in behavioral portfolio theory construct their portfolios as layered pyramids where bottom layers are designed for downside protection while top layers are designed for upside potential. Risk-aversion gives way to risk-seeking at the uppermost layers as they desire to avoid poverty give way to the desire for riches. Some investors fill the uppermost layers with the few stocks of an undiversified portfolio while others fill them with lottery tickets. Neither lottery buying nor undiversified portfolios are consistent with mean-variance portfolio theory but both are consistent with behavioral portfolio theory.

Behavioral portfolios, such as those reflected in the rules of “core and satellite,” are sensible ways to allocate portfolio assets between the upside potential and downside protection layers. A well-diversified core forms is the downside protection layer of the portfolio and a less diversified satellite forms the upside potential one.

The rules of diversification in behavioral portfolio theory are not as precise as the rules in mean-variance portfolio theory, but they are clear enough. Investors, financial advisors, and companies sponsoring 401(k) plans must be careful to draw the line between upside potential and downside protection such that dreams of riches do not plunge investors into poverty.


Levels of diversification in the portfolios of investors present a puzzle. The benefits of

diversification, measured by the rules of mean-variance portfolio theory, have increased in recent

years, yet levels of diversification did not increase, remaining much below their optimal levels.

We find that today’s optimal level of diversification, measured by the rules of mean-variance

portfolio theory, exceeds 120 stocks and argue that the diversification puzzle is solved within

Shefrin and Statman’s (2000) behavioral portfolio theory.

Campbell, Lettau, Malkiel and Xu (2000) studied U.S. stocks and found “a clear

tendency for correlations among individual stocks to decline over time. Correlations based on

five years of monthly data decline from 0.28 in the early 1960s to 0.08 in 1997…” (p. 23). They

concluded that “[d]eclining correlations among stocks imply that the benefits of portfolio

diversification have increased over time.” (p. 25). Campbell et al. cited a conventional rule of

thumb, supported by the results of Bloomfield, Leftwhich and Long (1977) that a portfolio of 20

stocks “attains a large fraction of the total benefits of diversification” (p. 25), while Statman

(1987) showed that an optimally diversified portfolio must include at least 30 stocks. Yet actual

levels of diversification were much lower than 20 or 30 in 1977 and 1987 and they remain much

lower than these figures more recently. Goetzmann and Kumar (2001) who studied more than

40,000 stock accounts at a brokerage firm found that the mean number of stocks in a portfolio in

the 1991-1996 period was 4 and that the median number was 3, little changed from the 3.41

average reported in 1967 by the Federal Reserve Board Survey of Financial Characteristics of

Consumers (1967).

We argue that investors fail to diversify their stock portfolios because they consider

individual stocks in their portfolios as the equivalent of individual lottery tickets and do not

1

diversify among stocks for the same reason that they do not diversify among lottery tickets. A

few stocks, like a few lottery tickets, provide a chance for great riches but a well-diversified

portfolio of stocks, like a well-diversified portfolio of lottery tickets, guarantees mediocrity.

Neither lottery buying nor undiversified portfolios are consistent with mean-variance portfolio

theory but both are consistent with behavioral portfolio theory.

More than 50 years ago, Friedman and Savage (1948) noted that risk-aversion and risk-

seeking share roles in our behavior; people who buy insurance policies often buy lottery tickets

as well. Four years later, Markowitz (1952a, 1952b) wrote two papers. In one he extended

Friedman and Savage’s insurance-lottery framework while in the other he created the mean-

variance framework. People in the mean-variance framework, unlike people in the insurance-

lottery framework, never buy lottery tickets; they are always risk-averse, never risk seeking.

Risk-averse people can be expected to buy insurance policies while risk-seeking people

can be expected to buy lottery tickets. But why would people buy both? Friedman and Savage

(1948) answered the question by noting that people buy lottery tickets because they aspire to

reach the riches of higher social classes while they buy insurance as protection against falls into

the poverty of lower social classes.

Markowitz (1952a) clarified the Friedman-Savage framework by noting that people

aspire to move up from their current social class. So people with $10,000 might accept lottery-

like odds in the hope of winning $1 million, while people with $1 million might accept lottery-

like odds in the hope of winning $100 million. Kahneman and Tversky (1979) extended the

work of Friedman and Savage (1948) and Markowitz (1952a) into prospect theory. Prospect

theory describes people who accept lottery-like odds when they are below their levels of

aspirations but reject such odds when they are above their levels of aspirations.

2

The framework of Friedman-Savage (1948), Markowitz (1952a) and Kahneman and

Tversky (1979) is a keystone in Shefrin and Statman’s (2000) behavioral portfolio theory. People

in behavioral portfolio theory act as if they contain many “doers” each with a different goal and

attitude towards risk. People in the simple version of the theory have two doers, a “downside

protection” doer whose goal is to avoid poverty and an “upside potential” doer whose goal is a

shot at riches. Lottery tickets are best for upside potential doers with high aspiration levels and

little money. However, upside potential doers with lower aspiration levels can meet their needs

through call options and those with even lower aspiration level can buy stocks.

Diversification in mean-variance portfolio theory

The optimal level of diversification is determined by marginal analysis; diversification

should be increased as long as its marginal benefits exceed its marginal costs. The benefits of

diversification, in mean-variance portfolio theory, are in the reduction of risk while the costs are

transaction and holding costs. Risk is measured in the mean-variance framework by the standard

deviation of portfolio returns.

Declining correlations increase the marginal benefits of diversification; Campbell et al

(2001) estimated that 50 stocks were required in the 1986-1997 period to reduce the excess

standard deviation of portfolios to a levels achieved by 20 stocks in the 1963-1985 period. But

was a 20-stock portfolio the optimal portfolio in the early periods? And is a 50-stock portfolio

optimal today?

Statman (1987) compared the marginal benefits of diversification to its costs using data

available in the mid-1980s and concluded that at least 30 stocks were required for an optimally

diversified portfolio. He noted that investors could have diversified into 500 stocks by holding a

3

mutual fund, such as the Vanguard 500 index fund, at an annual cost (at the time) of 0.49%. He

calculated the marginal benefits of diversification by comparing the expected return of a

portfolio of say, 30 stocks, to the expected return of a 500-stock portfolio, levered so that its

expected standard deviation is equal to the expected standard deviation of a 30-stock portfolio.

For example, Statman estimated at 0.52% the benefit of increasing diversification from 30 stocks

to 500 stocks. An increase of diversification from 30 stocks to 500 stocks is worthwhile since the

0.52% benefit exceeds the 0.49% cost of the Vanguard Index 500 fund. The advantage of a

levered 500-stock portfolio over a 30-stock portfolio is even greater once we consider the costs

of buying and holding a portfolio of individual stocks. For example, more than 100 stocks were

required to exceed the risk reduction benefits of a levered 500-stock portfolio if the annualized

cost of buying and holding a portfolio of individual stocks is 0.35%.

The expected standard deviation declines as portfolios become increasingly diversified.

For example, assuming that the correlation between stocks is 0.08, the standard deviation of a

20-stock portfolio is only 35 percent of the standard deviation of a 1-stock portfolio. (See Figure

1). However, a 20-stock portfolio is not necessarily optimal even if it attains a large fraction of

the total benefits of diversification.

The optimal level of diversification depends on expected correlations among individual

stocks, the cost of buying and holding stocks and mutual funds and the expected equity premium.

They have all changed. The expected correlation used by Statman, based on data in Elton and

Gruber (1977), was 0.15. The more recent figure, according to Campbell et al (2001), was 0.08.

The current expense ratio of the Vanguard Total Stock Market Index fund, a fund that did not

exist in the mid-1980s, is 0.20%, lower than the mid-1980s expense ratio of the Vanguard Index

4

500 fund1. Yet the Vanguard Total Stock Market Index fund contained 3,444 stocks in March

2002, many more than the 500 stocks of the Vanguard Index 500 fund. The equity premium in

the mid-1980s was estimated at 8.2%, based on realized returns during 1926-1984. The equity

premium, based on realized returns during 1926-2001, is 8.79%, but today there is little

agreement that it is a fair estimate of the expected equity premium. Fama and French (2001)

estimated the expected equity premium based on P/E ratios and dividend yield. The average of

the two is 3.44%.

Assume that all stocks have an identical expected return, R, an identical expected

standard deviations, σ, and that each pair of stocks has an identical expected correlation, ρ.

Consider a portfolio of n randomly chosen and equally weighted stocks. The expected return of

the portfolio is equal to R, the expected return of a single stock. The expected standard

deviation of a n-stock portfolio is :

1 n-1 σn = σ

n +

n ρ

(1)

The expected standard deviation of the portfolio declines when the number of stocks in

the portfolio increases.

Compare a portfolio of n stocks to a portfolio with a larger number of stocks, m. We set

m to be 3,444, the number of stocks in the Vanguard Total Stock Market Index fund. If investors

can borrow and lend at a common rate of Rf, they can lever a portfolio of m stocks such that the

1 The actual mean annual cost of the Vanguard Total Stock Market Index fund was lower than 0.20% during the 1997-2001 period. Indeed, the mean annual return of the Vanguard fund was higher by 0.06% than the mean annual return of the Wilshire 500 Index, which it tracks. However, good fortune is not guaranteed to continue. We assume that the annual cost of the Vanguard fund is 0.20%.

5

expected standard deviation of the levered m-stock portfolio is equal to σn, the expected standard

deviation of an n-stock portfolio. The expected return of the levered m-stock portfolio is:

σn Rnm = Rf +σm

EP

Where σm is the expected standard deviation of an m-stock portfolio, and EP, the

expected equity premium, is the difference between R and Rf.

The difference between the expected return of an n-stock portfolio, R, and the expected

return of its corresponding levered m-stock portfolio, Rnm, is the benefit of increased

diversification from n to m stocks, expressed in units of expected returns.

Bnm = Rnm - R σn = [Rf + σm EP] - [Rf + EP]

1 n-1 n

+n

ρ

1 m-1= ( m

+m

ρ- 1 ) EP

σn = ( σm

- 1 ) EP

Consider the case where the expected correlation between any pair of stocks is 0.08,

equal to the Campbell et al (2001) estimate for 1997, and where the expected equity premium is

8.79%, the mean realized equity premium during 1926-2001. The expected annual benefit of

increasing diversification from 20 stocks to the 3,444 stocks of the Total Market fund is 2.22%

while the expected annual benefit when diversification increases from 50 stocks to 3,444 stocks

is 0.94%. (See Table 1). The expected gross annual cost of an increase in diversification from

6

20 or 50 stocks to 3,444 is 0.20%, the expense ratio of the Vanguard Total Market Stock fund,

but the net cost is smaller since a portfolio of individual stocks involves transaction and holding

costs. While the cost of buying individual stocks might be incurred only once and stocks can be

held for decades, additional costs are likely since portfolios must be revamped when some

companies merge and other companies go bankrupt. Morever, costs are associated with keeping

track of individual stocks. Consider 0.05% as a conservative estimate of the expected annual

costs of buying and holding portfolios of individual stocks. If so, the net cost of increasing

diversification from 20, 50, or 100 stocks to 3,444 is 0.15%, the difference between the 0.20%

cost of the Vanguard Total Market Stock fund and the 0.05% cost of buying and holding a

portfolio of individual stocks.

It turns out that the optimal level of diversification is greater than 300 stocks when the

equity premium is 8.79% and the correlation is 0.08. The benefit of increasing diversification

from 300 to 3,444 is 0.15%, equal to the 0.15% net cost of replacing a 300-stock portfolio with

the Index fund. The optimal level increases from 300 to 430 stocks if the net cost of the Index

fund is 0.10%. (See Table 1 and Figure 2).

The benefits of diversification are smaller when the equity premium is smaller. The

optimal level of diversification declines to 120 stocks when the correlation remains at 0.08 but

the expected equity premium declines from 8.79% to 3.44%. Similarly, the benefits of

diversification are smaller when the correlation is higher. While the optimal level of

diversification is 300 stocks when the equity premium is 8.79% and the correlation is 0.08, the

optimal level is only 70 stocks when the correlation is 0.28. The 0.28 figure is equal to

Campbell et al’s (2001) estimate of the realized correlation in the early 1960s2.

2 Good stock selection skills might overcome the disadvantages of limited diversification. For example, the gross benefit from increasing diversification from 20 stocks to 3,444 stocks is 0.87% if the correlation is 0.08 and the

7

The conservative estimate of the current optimal level for diversification is 120 stocks,

based on the 3.44% Fama and French estimate of the equity premium, the 0.08 Campbell et al

estimate of the recent correlation among U.S. stocks, and a 0.05% annual expense of holding

individual stocks. This estimate is much higher than the rule of thumb reported by Campbell et

al (2001) where 20 stocks make a diversified portfolio, or the Rule of Five, holding no fewer

than five stocks, advocated by the National Association of Investment Clubs. (Wasik, 1995) In

turn, the numbers of stocks advocated in diversification rules of thumb are higher than the

average number of stocks held in actual portfolios. Why do investors fail to diversify to levels

consistent with the mean-variance portfolio theory? We argue, consistent with behavioral

portfolio theory, that investors fail to diversify because undiversified portfolios give them a

chance, however small, to reach their aspired riches.

Diversification in behavioral portfolio theory

Mangalindan (2002) told the story of David Callisch, a man with an undiversified

portfolio. When Callisch joined Altheon WebSystems, Inc. in 1997 he asked his wife “to give

him four years and they would score big,” and his “bet seemed to pay off when Altheon went

public.” By 2000, Callisch’s Altheon shares were worth $10 million. “He remembers making

plans to retire, to go back to school, to spend time with his threes sons. His relatives, his

colleagues, his broker all told him to diversify his holdings. He didn’t.” Now, in 2002, his

shares are worth a small fraction of their 2000 value.

Callisch’s aspirations are common, shared by the many who gamble on individual stocks

and lottery tickets. Most lose, but some win. Brenner (1990) quoted a lottery winner, a clerk in

equity premium is 3.44%. The net benefit of increasing diversification, once we subtract the 0.15% net cost of the Index fund, is 0.72%. Investors who can beat the market by more than 0.72% per year overcome the disadvantage

8

the New York subway system. “I was able to retire from my job after 31 years. My wife was

able to quit her job and stay home to raise our daughter. We are able to travel whenever we want

to. We were able to buy a co-op, which before we could not afford.” (p. 43).

People who hold undiversified portfolios, like people who buy lottery tickets, behave as

gamblers since they accept higher risk without compensation in the form of higher expected

returns. While gambling behavior is usually recognized as inconsistent with mean-variance

portfolio theory, it is often dismissed as no more than a minor irritant to that theory, consisting of

minor amounts of “play money” that people gamble for “entertainment.” But gambling behavior

is a major puzzle to mean-variance portfolio theory since it consumes major amounts.

Goetzmann and Kumar (2001) found that, on average, the value of investors’ undiversified

portfolios was 79% of their annual income.

While gambling behavior is a puzzle to mean-variance portfolio theory, it is a main

feature of behavioral portfolio theory. Investors in the simple version of Shefrin and Statman’s

(2000) behavioral portfolio theory divide their money into two layers of a portfolio pyramid, a

downside protection layer designed to protect them from poverty and an upside potential layer

designed to make them rich. Investors in the complete version of the theory divide their money

into many layers corresponding to many goals and levels of aspiration. Investors such as Mr.

Kallisch and lottery buyers such as the New York subway clerk aspire to retire, buy houses,

travel, and spend time with their children. They buy bonds in the hope of protection from

poverty, stock mutual funds in the hope of moderate riches and individual stocks and lottery

tickets in the hope of great riches.

Investors who place great importance on their upside potential layers of their portfolios

do not necessarily neglect the downside protection ones. Indeed, investors form their portfolios

of a limited 20-stock diversification

9

as if they fill the downside protection layers of their portfolios before they move on to fill the

upside potential ones. Gambling in America (1976) reported that gamblers have more

substantial downside protection layers than non-gamblers. The proportions of both stock owners

and bond owners among gamblers is higher than their proportions among non-gamblers.

Moreover, Gambling in America reported that “gamblers were more likely to have their future

secured by social security and pension plans than non-gamblers and hold 60 percent more

assets…” (p. 66)

The demographics of gamblers are similar to those of undiversified investors.

Goetzmann and Kumar (2001) found that the proportion of investors with undiversified

portfolios investors is higher among members of the non-professional category, such as blue-

collar and clerical workers, than among members of the professional category. Lottery gamblers

are similar to undiversified investors if education proxies for occupation. Clotfelter and Cook

(1989, p. 96) found that the proportion of lottery buyers is higher among those with low levels of

education than among people with high levels. While 49% of those with less than high school

education bought lottery tickets during the week of the survey, only 30% of college graduates

did.

Goetzmann and Kumar (2001) found that the degree of diversification is higher for old

investors than for young ones. This is the case for gambling as well. The authors of Gambling

in America (1976) wrote: “Gambling is a young person’s pursuit.” (p. 7). They reported that

73% of 18-24 year olds gambled but only 23% of 65-year olds or older did (p. 2-3). The age

pattern of participation in lotteries is somewhat different from the age pattern of gambling in

general. Clotfelter and Cook (1989, p. 96) found that the proportion lottery buyers among those

10

who are 65-year old or older is indeed lower than the proportion among younger people but

participation in lotteries increases with age up to age 65 and peaks in the 45-65 age group.

Goetzmann and Kumar (2001) found no relationship between income and diversification

in one period but in another period they found that those with higher incomes held more

diversified portfolios than those with low incomes. While Clotfelter and Cook (1989, p. 99-100)

found no systematic relationship between income and the absolute amount spent on lotteries,

they found a strong relationship between income and the relative amount. In particular, people

with low income spent higher proportions of their income on lotteries than people with high

income. Similarly, Gambling in America (1976 pp. 103-4) reported that people with low

incomes spent higher proportions of their income on gambling than people with high income.

Goetzmann and Kumar (2001) found that investors who trade most heavily are likely to

have the lowest levels of diversification and suggested that the overconfidence underlies that

behavior. But the need to get ahead in life and the need for excitement might underlie that

behavior as well. Gamblers reported higher needs than non-gamblers for money, chances to get

ahead, excitement and challenges. Gambling in America (1976, p. 82) asked gamblers and non-

gamblers to rate their needs on a scale from a low of 1 to a high of 8. The mean score of the

need for chances to get ahead among gamblers was 5.35, higher than the 4.69 mean score among

non-gamblers. The mean score for the need for excitement among gamblers was 4.24, much

higher than the 2.89 mean score among non-gamblers. Heavy trading and low diversification

found by Goetzmann and Kumar seem to reflect the preferences of gamblers. Heavy trading

satisfies the need for excitement while low diversification satisfies the need to get ahead in life,

11

since portfolios of a few stocks can bring extraordinary riches but diversified portfolios can bring

only ordinary riches3.

High allocations to the upside potential layers of portfolios in the form of individual

stocks or lottery tickets are related to demographics but demographics do not explain it all.

People who gamble on stocks or lottery tickets are those who find themselves below their

aspirations levels for riches. Such people might be predominantly young, poor and low income

levels of education and occupation. But this is not always the case. Mr. Kallisch gambled on an

undiversified portfolio even though his level of education was high and so were his income and

occupational status. Mr. Kallisch gambled because he was below his aspiration level for riches.

How much diversification is enough? Investors who follow the rules of mean-variance

portfolio theory, like investors who follow the rules of behavioral portfolio theory increase

diversification as long as the marginal benefits of diversification exceed their costs. But the

benefits and costs of diversification by the rules of mean-variance portfolio theory are different

then those by the rules of behavioral portfolio theory.

Reduction of risk is always a benefit in mean-variance portfolio theory. The optimal

number of stocks in a portfolio exceeds 120 since the benefits of diversification at lower levels of

diversification exceed their costs. But reduction of risk is not always a benefit in behavioral

portfolio theory. While investors in behavioral portfolio theory, like investors in mean-variance

portfolio theory, prefer low risk over high risk in the downside protection layers of their

portfolios, they prefer high risk over low risk in the upside potential layers. So investors in

behavioral portfolio theory hold money market accounts, bonds and diversified stock mutual

funds in their downside protection layers, but they hold a handful of stocks, like a handful of

3 The elements of play and excitement of lottery playing and stock trading are discussed in Statman (2002) “Lottery players/stock traders.”

12

lottery tickets, in their upside potential layers. The optimal number of individual stocks by the

rules of behavioral portfolio is the number that balances the chance for an uplift into riches with

the chance of a descent into poverty. But what is the right balance?

The desire of investors for the riches of lotteries and individual stocks is strong.

Centuries of education and preaching have not uprooted lotteries and they are not likely to uproot

undiversified portfolios. Moreover, there is no good reason to uproot the desire for upside

potential, manifested in undiversified portfolios, once the need for downside protection is

satisfied. The rules of optimal diversification in behavioral portfolio theory are similar to the

rules of suitability that govern brokers and financial advisors.

Suitability regulations require brokers to make sure that investors desire for upside

potential does not breach their need for downside protection. Roach (1978) quoted from a

Securities Exchange Commission decision where a broker was found liable for recommending a

particular stock to investors. “Whether or not customers Z and E considered a purchase of the

stock… a suitable investment is not the test for determining the propriety of applicants’ conduct.

The test is whether [the broker] fulfilled the obligation he assumed when he undertook to counsel

the customers of making only such recommendation as would be consistent with the customer’s

financial situation and needs.” Roach noted: “Both the NASD and the Commission here

suggests that suitability is an objective concept which the broker is obliged to observe regardless

of a customer’s wishes… The NASD’s statement that the customer’s ‘own greed’ may well have

been their motivation reinforces the idea that the customer is not sovereign for suitability

purposes.” (p. 1126).

Behavioral portfolios, such as those reflected in the rules of “core and satellite” and “risk

budget” are sensible ways to allocate portfolio assets between the upside potential and downside

13

protection layers. Pietranico and Riepe (2002) describe Core and Explore, Schwab’s version of

core and satellite, as comprised of a well-diversified “core,” serving as the “foundation” layer of

the portfolio and a less diversified layer of “explore,” seeking “returns that are higher than the

overall market, which entails greater risk.” Similarly, Waring et al (2000) describe portfolios

where the risk budget is allocated to active funds in the hope of upside potential, while the safe

budget is allocated to index funds for downside protection.

Conclusion

The optimal number of individual stocks in a portfolio by the rules of mean-variance

portfolio theory is greater than 120, but the average number of stocks in actual portfolios is much

lower than that. Goetzmann and Kumar (2001) found that the mean number of stocks in more

than 40,000 stock portfolios was 4 and the median was 3, much lower than 120 and not much

different from the 3.41 average number of stocks in portfolios in 1967, as reported in a Federal

Reserve Board survey. Lack of diversification is costly. Investors who hold only 4 stocks in

their portfolios forego the equivalent of a 3.3% annual return relative to investors who hold the

3,444 stocks of the Vanguard Total Index Stock Market Index fund. Why do investors forego

the benefits of diversification?

Goetzmann and Kumar (2001) argued that investors forego the benefits of diversification

because diversified portfolios are difficult to implement. They wrote that “investors realize the

benefits of diversification but face a daunting task of implementing and maintaining a well-

diversified portfolio.” (p. 20). But this cannot be true. Index funds, such as the Vanguard Total

Stock Market Index fund, provide easy ways to implement and maintain well-diversified

portfolios. Such funds have been advocated for many years in newspaper and magazine articles

14

directed at individual investors. The minimum amount required for a Vanguard Total Stock

Market Index fund account is $3,000, much lower than the $13,869 median value of the accounts

studied by Goetzmann and Kumar (2001).

The persistence of undiversified portfolios, like the persistence of lotteries, tells us that

mean-variance portfolio theory fails to describe the behavior of investors. The behavior of

investors is described better in Shefrin and Statman’s (2000) behavioral portfolio theory.

Investors in behavioral portfolio theory construct their portfolios as layered pyramids where

bottom layers are designed for downside protection while top layers are designed for upside

potential. Risk-aversion gives way to risk-seeking at the uppermost layers as they desire to avoid

poverty give way to the desire for riches. Some investors fill the uppermost layers with the few

stocks of an undiversified portfolio while others fill them with lottery tickets.

Zernike (2002) wrote about the havoc that the early 2000s stock slide is playing with

older Americans’ dreams. She described the undiversified portfolio of Gena and John Lovett,

people in their late 50s. “Our retirement is one-half of what it was a year ago,” said Gena.

“And because John works for G.E. we have mostly G.E. stock. I suppose we should have

diversified, but G.E. stock was supposed to be wonderful. John’s simply not looking at

retirement. We simply told our kids that we’re spending their inheritance.” (p. A1)

Postponing retirement beyond the late 50s and spending the kids inheritance are sad but

not disastrous breaches of the downside protection layer. Gena and John Lovett are no longer

rich, but neither are they poor. But sad consequences of undiversified portfolios can easily turn

into disastrous ones if G.E. is replaced by Enron or WorldCom and if no downside protection

layer underlies the upside potential one.

15

The rules of diversification in behavioral portfolio theory are not as precise as the rules in

mean-variance portfolio theory, but they are clear enough. Investors, financial advisors, and

companies sponsoring 401(k) plans must be careful to draw the line between upside potential and

downside protection such that dreams of riches do not plunge investors into poverty.

16

Reference:

Brenner, Reuven and Gabrielle Brenner (1990). Gambling and Speculation: A Theory, a History and a Future of Some Human Decisions. New York: Cambridge University Press.

Bloomfield, Ted, Richard Leftwich and John Long (1977). “Portfolio strategies and

performance,” Journal of Financial Economics 5: 201-218. Charles Schwab and Company (2002). “Core & Explore – Details” www.schwab.com Campbell, Felicia (1976). “Gambling: A Positive View,” in William R. Eadington (ed.),

Gambling and Society, Springfield, Ill: Thomas. Goetzmann, William and Alok Kumar (2001), “Equity Portfolio Diversification,” National

Bureau of Economic Research working paper series. Mangalindan, Mylene (2002), “Hoping is Hard in Silicon Valley,” The Wall Street Journal, July

15: C1, C14. Roach, E. Arvid (1978). “The Suitability Obligations of Brokers: Present Law and the Proposed

Federal Securities Code.” The Hastings Law Journal 29: 1069-1159. Waring, Barton, Duane Whitney, John Pirone and Charles Castille (2000). “Optimizing

Manager Structure and Budgeting Manager Risk,” The Journal of Portfolio Management, Spring: 90-104.

Wasik, John (1995). The Investment Club Book. New York: Warner Books. Zernike, Kate (2002). “Stocks’ Slide is Playing Havoc with Older Americans’ Dreams,” The

New York Times, July 14: A1, 16.

17

Table 1: The optimal level of diversification by the rules of mean-variance portfolio theory

Number of Stocks in the Portfolio

Benefit of diversification

when the equity premium is

8.79% and the correlation

between any two stocks is 0.08.

Excess of the benefit of

diversification over the 0.15% net cost of the Vanguard Total Stock Market

fund.







fund.







fund.







fund.1 22.24% 22.09% 7.82% 7.67% 8.70% 8.55% 3.06% 2.91%4 8.48% 8.33% 2.47% 2.32% 3.32% 3.17% 0.97% 0.82%

10 4.08% 3.93% 1.06% 0.91% 1.60% 1.45% 0.42% 0.27%20 2.22% 2.07% 0.54% 0.39% 0.87% 0.72% 0.21% 0.06%25 1.81% 1.66% 0.44% 0.29% 0.71% 0.56% 0.17% 0.02%30 1.53% 1.38% 0.37% 0.22% 0.60% 0.45% 0.14% -0.01%40 1.32% 1.17% 0.31% 0.16% 0.52% 0.37% 0.12% -0.03%50 0.94% 0.79% 0.22% 0.07% 0.37% 0.22% 0.09% -0.06%70 0.68% 0.53% 0.16% 0.01% 0.27% 0.12% 0.06% -0.09%

100 0.48% 0.33% 0.11% -0.04% 0.19% 0.04% 0.04% -0.11%120 0.40% 0.25% 0.09% -0.06% 0.16% 0.01% 0.04% -0.11%180 0.26% 0.11% 0.06% -0.09% 0.10% -0.05% 0.02% -0.13%300 0.15% 0.00% 0.03% -0.12% 0.06% -0.09% 0.01% -0.14%430 0.10% -0.05% 0.02% -0.13% 0.04% -0.11% 0.01% -0.14%500 0.09% -0.06% 0.02% -0.13% 0.03% -0.12% 0.01% -0.14%

1000 0.04% -0.11% 0.01% -0.14% 0.01% -0.14% 0.00% -0.15%3444 0.00% -0.15% 0.00% -0.15% 0.00% -0.15% 0.00% -0.15%

The annual benefit of an increase in diversification from 120 stocks to 3,444 stocks (as in the Vanguard Total Stock Market Index fund) is 0.16% when the equity premium is 3.44% and the correlation is 0.08. The net annual cost of such increase in diversification is 0.15%, composed of the 0.20% annual cost of the Vanguard fund less an assumed 0.05% annual cost of buying and holding 120 individual stocks. So the optimal level of diversification exceeds 120 stocks.

Figure 1: The decline in the standard deviation of portoflios as diversification increases. (The correlation between the returns of any two

stocks is 0.08)

0.000

0.200

0.400

0.600

0.800

1.000

0 50 100 150 200Number of Stocks in the Portfolio

Stan

dard

Dev

iatio

n of

the

Port

folio

Figure 2: The optimal level of diversification by the rules of mean-variance portfolio theory

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

0 100 200 300 400

Number of stocks in the portfolio

Cos

t and

ben

efit

of d

iver

sific

atio

n

Cost of diversification(0.20% Cost of VanguardTotal Stock Market Indexfund less 0.05% cost ofbuying and holdingindividual stocks)

Benefit of diversificationwhen the correlationbetween any two stocks is0.08 and the equitypremium is 3.44%. Thebreak-even portfoliocontains more than 120stocks.

Benefit of diversificationwhen the correlationbetween any two stocks is0.08 and the equitypremium is 8.79%. Thebreak-even portfoliocontains more than 300stocks.

how much diversification is...

Documents