How much diversification is enough?

Transcription

1 How much diversification is enough? by Meir Statman Glenn Klimek Professor of Finance Santa Clara University Leavey School of Business Santa Clara, CA September 2002 I thank Roger Clarke, Ramie Fernandez, William Goetzmann, Mark Kutzman and Jonathan Scheid and acknowledge financial support from The Dean Witter Foundation.

2 How much diversification is enough? 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.

3 How much diversification is enough? 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 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

4 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 riskseeking 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 meanvariance framework. People in the mean-variance framework, unlike people in the insurancelottery 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 lotterylike odds in the hope of winning $1 million, while people with $1 million might accept lotterylike 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

5 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 period to reduce the excess standard deviation of portfolios to a levels achieved by 20 stocks in the 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

6 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 The more recent figure, according to Campbell et al (2001), was 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

7 500 fund 1. 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 The equity premium, based on realized returns during , 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) σ n = σ 1 + n-1 n n ρ 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 R f, 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 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

8 expected standard deviation of the levered m-stock portfolio is equal to σ n, the expected standard deviation of an n-stock portfolio. The expected return of the levered m-stock portfolio is: σ n R nm = R f + σm EP Where σ m is the expected standard deviation of an m-stock portfolio, and EP, the expected equity premium, is the difference between R and R f. The difference between the expected return of an n-stock portfolio, R, and the expected return of its corresponding levered m-stock portfolio, R nm, is the benefit of increased diversification from n to m stocks, expressed in units of expected returns. B nm = R nm - R σ n = [R f + σ m EP] - [R f + EP] = ( σ n σ m - 1 ) EP 1 n-1 ( + n n = 1 m-1 m + m ρ ρ - 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 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

9 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 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 The 0.28 figure is equal to Campbell et al s (2001) estimate of the realized correlation in the early 1960s 2. 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

10 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

11 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

12 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 bluecollar 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 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

13 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 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 ) 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 ) 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 nongamblers 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

14 since portfolios of a few stocks can bring extraordinary riches but diversified portfolios can bring only ordinary riches 3. 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

15 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

16 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 welldiversified 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

17 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

18 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

19 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: Charles Schwab and Company (2002). Core & Explore Details 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: Waring, Barton, Duane Whitney, John Pirone and Charles Castille (2000). Optimizing Manager Structure and Budgeting Manager Risk, The Journal of Portfolio Management, Spring: 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,

20 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 Excess of the benefit of diversification over the 0.15% net cost of the Vanguard Total Stock Market fund. Benefit of diversification when the equity premium is 8.79% and the correlation between any two stocks is Excess of the benefit of diversification over the 0.15% net cost of the Vanguard Total Stock Market fund. Benefit of diversification when the equity premium is 3.44% and the correlation between any two stocks is Excess of the benefit of diversification over the 0.15% net cost of the Vanguard Total Stock Market fund. Benefit of diversification when the equity premium is 3.44% and the correlation between any two stocks is Excess of the benefit of diversification over the 0.15% net cost of the Vanguard Total Stock Market fund % 22.09% 7.82% 7.67% 8.70% 8.55% 3.06% 2.91% % 8.33% 2.47% 2.32% 3.32% 3.17% 0.97% 0.82% % 3.93% 1.06% 0.91% 1.60% 1.45% 0.42% 0.27% % 2.07% 0.54% 0.39% 0.87% 0.72% 0.21% 0.06% % 1.66% 0.44% 0.29% 0.71% 0.56% 0.17% 0.02% % 1.38% 0.37% 0.22% 0.60% 0.45% 0.14% -0.01% % 1.17% 0.31% 0.16% 0.52% 0.37% 0.12% -0.03% % 0.79% 0.22% 0.07% 0.37% 0.22% 0.09% -0.06% % 0.53% 0.16% 0.01% 0.27% 0.12% 0.06% -0.09% % 0.33% 0.11% -0.04% 0.19% 0.04% 0.04% -0.11% % 0.25% 0.09% -0.06% 0.16% 0.01% 0.04% -0.11% % 0.11% 0.06% -0.09% 0.10% -0.05% 0.02% -0.13% % 0.00% 0.03% -0.12% 0.06% -0.09% 0.01% -0.14% % -0.05% 0.02% -0.13% 0.04% -0.11% 0.01% -0.14% % -0.06% 0.02% -0.13% 0.03% -0.12% 0.01% -0.14% % -0.11% 0.01% -0.14% 0.01% -0.14% 0.00% -0.15% % -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 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.

21 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) Standard Deviation of the Portfolio Number of Stocks in the Portfolio

22 Figure 2: The optimal level of diversification by the rules of mean-variance portfolio theory Cost and benefit of diversification 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Number of stocks in the portfolio Cost of diversification (0.20% Cost of Vanguard Total Stock Market Index fund less 0.05% cost of buying and holding individual stocks) Benefit of diversification when the correlation between any two stocks is 0.08 and the equity premium is 3.44%. The break-even portfolio contains more than 120 stocks. Benefit of diversification when the correlation between any two stocks is 0.08 and the equity premium is 8.79%. The break-even portfolio contains more than 300 stocks.