Diversification Decisions of Individual Investorsand Asset Prices∗

William N. Goetzmann

Yale University

School of Management

Alok Kumar

University of Notre Dame

Mendoza College of Business

November 14, 2003

(Comments are welcome)

∗Please address all correspondence to William N. Goetzmann, Yale School of Management, 135 ProspectStreet, New Haven, CT 06511; Phone: 203-432-5950; email: william.goetzmann@yale.edu; OR AlokKumar, 239 Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556; Phone: 574-631-0354; email: akumar@nd.edu. We would like to thank an anonymous referee, John Campbell, PengChen, Simon Gervais, Terrance Odean, Vicente Pons, Jim Poterba, K. Geert Rouwenhorst, Paul Schultz,Mark Seasholes, Meir Statman, Ning Zhu, and seminar participants at the Conference on Household Portfolio-Choice and Financial Decision-Making at the University of Pennsylvania for helpful discussions and valuablecomments. In addition, we would like to thank Itamar Simonson for making the investor data available tous and Terrance Odean for answering numerous questions about the investor database. We are responsiblefor all remaining errors and omissions. An earlier version of the paper circulated under the title “EquityPortfolio Diversification”.

Diversification Decisions of Individual Investors

and Asset Prices

Abstract

In this paper, we examine if the diversification decisions of individual investors influence

asset prices. First, we show that a vast majority of individual investors in our sample are

under-diversified and the unexpectedly high idiosyncratic risk in their portfolios results in a

welfare loss – the least diversified group of investors earn 2.40% lower return annually than

the most diversified group of investors on a risk-adjusted basis. Next, we examine the deter-

minants of investors’ under-diversification and find that younger, low-income, and relatively

less sophisticated investors hold less diversified portfolios. In addition, investors who prefer

skewness, exhibit relatively stronger familiarity bias, and exhibit greater over-confidence are

less diversified. Finally, we show that the systematic under-diversification of individual in-

vestors influence asset prices. A zero-cost portfolio (DIV factor) that takes a long position

in stocks with the least diversified individual investor clientele and a short position in stocks

with the most diversified individual investor clientele earns an annual excess return of 7.44%

on a risk-adjusted basis. Furthermore, this factor has power to explain the cross-sectional

variation in returns for a considerable group of stocks.

U.S. equity risk has a large idiosyncratic component, much of which may be reduced through

portfolio diversification. Most rational models of investor choice suggest that investors hold

diversified portfolios to reduce or eliminate non-compensated risk and virtually all asset pric-

ing models posit that securities are priced by a diversified, marginal investor who demands

little or no compensation for holding idiosyncratic risk. However, if investors systematically

hold less than fully diversified portfolios and if they engage in narrow framing (Kahneman

and Lovallo 1993, Barberis, Huang, and Thaler 2003) where they do not adopt a unified view

of their financial portfolio, they are likely to demand compensation for the idiosyncratic risk

in their equity portfolios. Consequently, the manner in which investors construct their equity

portfolios is likely to influence asset prices.

In this paper, we focus on the portfolio decisions of a representative group of more than

40, 000 investors at a large discount brokerage house during a six year period (1991-96)

in recent U.S. capital market history. Using the historical performance for the equities in

these accounts, we examine the volatility and risk characteristics of chosen portfolios and

estimate the level of diversification in these portfolios. Our results indicate that a vast

majority of individual investors in our sample are under-diversified. Even accounting for the

likelihood we have selected a group of speculators, the magnitude of the idiosyncratic risk

taken by investors in our sample is surprising. We estimate the economic costs of under-

diversification, identify the determinants of this observed under-diversification, and examine

if portfolio decisions of individual investors generate pervasive forces that influence asset

prices.

Consistent with the results from previous studies (Blume and Friend 1975, Kelly 1995,

Odean 1999, Vissing-Jorgensen 1999, Barber and Odean 2000, Polkovnichenko 2003), we

find that more than 25% of investor portfolios in our sample contain only 1 stock, more

than 50% of them contain fewer than 3 stocks, and in any given month, only 5-10% of the

portfolios contain more than 10 stocks. As a consequence, investor portfolios have extremely

high volatility (more than 75% of investor portfolios have higher volatility than the market

portfolio) and they exhibit worse risk-return trade-off than randomly constructed portfolios.

The individual holding data allows us to do something previous authors studying indi-

vidual accounts have not been able to do – directly calculate the variance and covariance

of their equity holdings. This analysis allows us to decompose the level of diversification

of a household into two components: (i) the risk reduction due to holding more than one

security, and (ii) the risk reduction due to choosing imperfectly correlated stocks. We find

positive evidence of the former and negative evidence of the latter.

2

The unexpectedly high idiosyncratic risk in investor portfolios results in a welfare loss

as measured by their portfolio under-performance. We find that the least diversified (low-

est decile) group of investors earn 2.40% lower return annually than the most diversified

group (highest decile) of investors on a risk-adjusted basis. The economic costs of under-

diversification is higher for older investors and investors who trade infrequently – within

these two groups, the risk-adjusted performance differentials between the least diversified

and the most diversified investors are 3.60% and 3.12% respectively.

If investors pay a cost for improper diversification, a natural question arises: why do

they continue to hold under-diversified portfolios? In other words, if the observed under-

diversification is not intentional, why don’t investors learn to diversify? We find that the

degree of diversification varies considerably across investor households. Diversification level

increases with income as well as age which reflects an increasing degree of risk aversion

with age and income. We also find that investor sophistication, in particular, their financial

sophistication influences their portfolio choices. Relatively more sophisticated investors –

investors who hold mutual funds, trade in options and foreign equities, and engage in short-

selling – hold more diversified portfolios. In addition, investors with a preference for skewness

and those who exhibit widely documented behavioral biases such as over-confidence (Odean

1999) and familiarity (Grinblatt and Keloharju 2001, Huberman 2001, Zhu 2002) hold less

diversified portfolios.

The degree of diversification also varies across occupation categories in a manner that fur-

ther supports the view that investors’ diversification decisions depend upon their age, income

and their level of financial sophistication. Investors that belong to the non-professional job

category (blue-collar workers, clerical workers, sales and service workers, house-wives, and

students) hold the least diversified portfolios in our sample while investors who are retired

are on the other end of the diversification spectrum where they hold the most diversified

portfolios.

To gain further insights into the portfolio decisions of individual investors, we examine the

time-variation in the average diversification level of investor portfolios. We find that during

the 1991-96 sample period, the average number of stocks in investor portfolios increases

from 4 to 7. This results in a decrease in the average portfolio variance and it also has

a considerable impact on investors’ portfolio performance. The risk-adjusted performance

of investors portfolios, as measured by the 4-factor alpha, increases from −0.34% (t-stat

= −3.35) during the 1991-93 sub-period to −0.09% (t-stat = −0.81) during the 1994-96 sub-

period. This yields a monthly risk-adjusted performance improvement of 0.25% or 3.00% on

3

an annual basis.

The improved diversification over time does not necessarily imply that investors’ port-

folio composition skills have improved over time. We do not find any perceptible evidence

of diversification improvement by active means. Over time, there is no decrease in either

the excess average correlation (relative to benchmark portfolios) or the excess normalized

variance of investor portfolios. This suggests that a significant part of diversification im-

provement results from passive means where investors increase the number of stocks in their

portfolios without giving proper consideration to stock correlations. In addition, during the

1991-96 time-period the average correlation among stocks in the U.S. equity market declined

steadily (Campbell, Lettau, Malkiel, and Xu 2001) which leads to a further decrease in the

variance of investor portfolios.

Given the systematic under-diversification in individual investor portfolios, we examine

if their portfolio decisions generate pervasive forces that can influence asset prices. If indi-

vidual investors systematically hold less than fully diversified portfolios, they are likely to

demand compensation for the idiosyncratic risk in their equity portfolios. If this sensitivity

is widespread, their portfolio decisions are likely to generate pervasive forces that can influ-

ence returns. Consequently, ceteris paribus, stocks with a less diversified individual investor

clientele are likely to earn higher expected returns.

To set the stage, we examine the performance of a zero-cost portfolio that takes a long

position in stocks with the least diversified individual investor clientele and a short position in

stocks with the least diversified individual investor clientele. This trading strategy earns an

impressive annual excess return of 7.44% on a risk-adjusted basis. The performance declines

modestly when there is a delay between the portfolio formation date and the month in which

the returns to the zero-cost portfolio can be realized. For delays of 1, 2, and 3 months, the

annual risk-adjusted performance differentials are 5.76%, 4.32%, and 4.08% respectively.

We also examine the extent to which individual investors’ diversification decisions ex-

plain the cross-sectional variation in stock returns. To do so, we construct a diversification

factor (DIV) that represents the difference between the equal-weighted return of a portfolio

of stocks with the least diversified (lowest decile) individual investor clientele and the equal-

weighted return of a portfolio of stocks with the most diversified (highest decile) individual

investor clientele. We find that DIV is moderately correlated with the standard risk fac-

tors – the contemporaneous correlations with market, small-minus-big, high-minus-low, and

momentum factors are −0.147, 0.374, −0.044, and −0.067 respectively.

To examine the explanatory power of DIV for cross-sectional variation in stock returns, we

4

employ a five-factor time-series model which contains the three standard Fama-French factors

(Fama and French 1993), the momentum factor (Jegadeesh and Titman 1993, Carhart 1997),

and the diversification factor as explanatory variables. Our results indicate that DIV has

incremental explanatory power over the standard risk factors for small stocks, value stocks

and growth stocks. DIV has considerable explanatory power even when we consider a set of

random portfolios – depending on the portfolio size, the DIV factor loadings are significant

in 20-43% of the cases, and significantly positive in 76-100% of those cases. Taken together,

the results from our asset-pricing tests indicate that the diversification decisions of individual

investors get impounded into asset prices and have the power to explain the cross-sectional

variation in stock returns for a considerable group of stocks.

The rest of the paper is organized as follows: in the next section, we provide a brief

review of the literature on equity portfolio diversification. A brief description of the investor

database and the sample used in the study is provided in Section II. In Section III, we

provide evidence of under-diversification and examine the time variation in the average level

of diversification. In Section IV, we measure the economic costs of under-diversification and

in Section V, we examine the determinants of investors’ diversification decisions. We carry

out robustness tests in Section VI. In Section VII, we examine if the diversification decisions

of individual investors influence asset prices, and finally, we conclude in Section VIII with a

summary of our main results and a brief discussion.

I Background: Equity Portfolio Diversification and Asset Prices

There is a considerable empirical literature on the diversification decisions of households.

Blume and Friend (1975) use tax filing and survey data to investigate diversification in

household portfolios. They find that households are grossly under-diversified and the degree

of diversification (and hence the degree of risk aversion) increases with wealth. In contrast,

Lease, Lewellen, and Schlarbaum (1974) find that individual investors with accounts at a

retail brokerage house are quite diversified – only 23% of investors hold 5 or fewer stocks

and more than half of the investors hold 10 or more stocks.

More recently, Kelly (1995) and Polkovnichenko (2003) examine equity portfolio diversi-

fication among households in the U.S. Using data from the Surveys of Consumer Finances,

they document poor diversification among U.S. households. Kelly (1995) finds that in 1983

the median number of stocks in an investor portfolio is only two and less than one third of

the households hold more than ten stocks. Polkovnichenko (2003) documents an improved

5

level of diversification among individual investors but even in 1998, approximately 75% of

households hold 5 or fewer stocks.

A majority of these earlier studies use survey data which lacks information about the

composition of household portfolios or any information about their trading activities. In

contrast, our dataset provides monthly details of the composition of investor portfolios and

contains a direct account of investor trades during the six-year sample-period. The stock level

data allows us to measure the level of portfolio diversification and its economic costs more

accurately. Furthermore, details about the composition of investor portfolios and their trades

allows us to provide further insights into the reasons for sub-optimal levels of diversification.

Most importantly, the investor database allows us to assess the asset pricing implications

of individual investors’ diversification decisions. In spite of the fundamental role played

by portfolio choice in asset pricing, few prior studies have examined the direct relation

between investors’ portfolio diversification decisions and asset prices, perhaps due to data

limitations.1

Using the same investor dataset as the one used in our study, Barber and Odean (2000)

indicate that individual investors hold portfolios with fewer stocks and hence are inappro-

priately diversified.2 We take their observation much further – we compute the variance-

covariance matrices of investor portfolios, examine the cross-sectional differences in diversi-

fication, estimate the economic costs of under-diversification, and examine its asset pricing

implications.

While the number of stocks in a portfolio is a useful heuristic for identifying the degree of

diversification, this measure is not sufficient to determine the diversification characteristics

of a portfolio. Two individuals may both hold the same number of stocks in their portfo-

lios, but one may hold stocks with low correlations among them and the other may hold

strongly correlated stocks confined to a single industry – the volatility of these portfolios

will certainly differ. Furthermore, this diversification measure is likely to overstate the level

of diversification of a portfolio (Blume, Crockett, and Friend 1974, Vissing-Jorgensen 1999).

This measure is also unable to provide a firm basis for analyzing cross-sectional portfolio

risk differences conditional upon other factors.

A number of papers have examined the market participation decisions, asset allocation

decisions and diversification decisions of investors in broader contexts.3 In contrast, our focus

1There is one notable exception. Heaton and Lucas (2000b) argue that the changing diversification levelsof U.S. households was a key determinant of the stock price run-up during the internet bubble.

2Using an older version of this dataset, Odean (1999) also provides evidence of under-diversification.3A partial list includes Uhler and Cragg (1971), Guiso, Japelli, and Terlizze (1996), Bertaut (1998),

6

is on the level of diversification within an asset class and the implications (if any) of such

within-class diversification decisions on asset prices. According to the traditional economic

theory, investors adopt a unified view of their entire financial portfolio which includes their

equity portfolio, labor income, real-estate portfolio, etc. In this setting, hedging motives

influence investors’ broad asset allocation decisions and their diversification decisions. For

instance, investors may utilize their equity portfolio to hedge against background risks such

as their labor income risk, entrepreneurial risk, and real-estate risk.

An alternative psychological theory posits that investors engage in narrow framing (Kah-

neman and Lovallo 1993, Barberis, Huang, and Thaler 2003) where they do not adopt a

unified view of their financial portfolio. These investors are less likely to integrate the var-

ious risks they face in their aggregate financial portfolio but rather they evaluate those

risks individually. Consequently, these investors are likely to demand compensation for the

individual risks they face. For instance, investors who systematically hold less than fully

diversified equity portfolios are likely to demand compensation for the idiosyncratic risk in

their equity portfolios. The compensation they demand for holding idiosyncratic risk is likely

to depend upon the riskiness of their equity portfolio rather than on the riskiness of their

entire financial portfolio. If this sensitivity is widespread, investors’ equity portfolio decisions

are likely to generate pervasive forces which may influence returns.

Given this motivation, we focus on the level of diversification within an asset class and

examine the implications of such within-class diversification decisions on asset prices. Our

study joins a growing literature (Lee, Shleifer, and Thaler 1991, Kumar and Lee 2002, Barber,

Odean, and Zhu 2003, Goetzmann and Massa 2003, Kumar 2003, Jackson 2003, Graham

and Kumar 2003) in Behavioral Finance which examines the relation between systematic

individual investor behavior and stock prices.

Our results also provide a partial rationale for studies (Goyal and Santa-Clara 2003,

Malkiel and Xu 2002) which suggest that idiosyncratic risk is priced. It is quite clear that

the pricing impact of idiosyncratic risk depends upon the level of diversification of the rep-

resentative investor for each stock. If the idiosyncratic risk is high but the representative

investor is very well diversified, idiosyncratic risk is unlikely to influence prices, irrespective

of its magnitude. Similarly, if the idiosyncratic risk is low and the diversification level of

the representative investor is very low, idiosyncratic risk may still influence prices. Overall,

the level of diversification of the representative investor is likely to determine the sensitivity

Gentry and Hubbard (2000), Heaton and Lucas (2000a), Perraudin and Sorensen (2000), Benartzi andThaler (2001), Souleles (2001), and Moskowitz and Vissing-Jorgensen (2001).

7

level of returns to idiosyncratic risk. Our evidence of systematic under-diversification among

individual investors suggests that the representative investor for a considerable number of

stocks is likely to be sub-optimally diversified, and thus, idiosyncratic risk may be priced

conditionally.

II Data and Sample Selection

The data for this study consists of trades and monthly portfolio positions of investors at a

major discount brokerage house in the U.S. for the period of 1991-96. This database has

been used in several studies including Odean (1998, 1999) and Barber and Odean (2000).

There are a total of 77, 995 households in the database of which 62, 387 have traded in stocks.

More than half of the households in our database have 2 or more accounts. Approximately

27% of the households have 2 accounts, 13% have 3 accounts, 6% have 4 accounts and 6%

have 5 or more accounts. All accounts for a given investor are combined to obtain a stock

portfolio at the household level.

The aggregate value of equity portfolios of investors in our database is $2.41 billion in a

typical month. This represents approximately 82% of investors’ aggregate portfolio at the

brokerage house – 62% in stocks and 20% in mutual funds. An average investor holds a

4-stock portfolio (median is 3) with an average size of $35,629 (median is $13,869). Less

than 10% of the investors hold portfolios over $100,000 and less than 5% of them hold more

than 10 stocks. The average portfolio turnover rate – the average of purchase and sales

turnover rates – is 7.59% (median is 2.53%) for our chosen sample. A typical investor makes

less than 10 trades per year and the average trade size is $8,779 (median is $5,239). The

average number of days an investor holds a stock is 187 trading days (median is 95). Further

details of the investor database is available in Barber and Odean (2000).

To gauge how representative our individual investor sample is of the overall population of

individual investors in the US, we compare the stock holdings of the investors in our sample

with those reported by the Census Bureau4 (Survey of Income and Program Participation

(SIPP), 1995) and the Federal Reserve5 (Survey of Consumer Finances (SCF), 1992, 1995).

According to the 1992 SCF, a typical household held $8,700 in stocks (median was $16,900).

The stock ownership declined marginally in the 1995 SCF where a typical household held

4Source: US Census Bureau Report, Asset Ownership of Households, 1995. The data are available athttp://www.census.gov/hhes/www/wealth/1995/wealth95.html.

5The report is available at http://www.federalreserve.gov/pubs/oss/oss2/95/scf95home.html.Also see Kennickell, Starr-McCluer, and Sunden (1997).

8

$8,000 in stocks (median was $15,300).

In the SIPP survey conducted by the Census Bureau, the real median value of stock

and mutual funds held by households increased from $7,331 in 1993 to $9,000 in 1995. The

median portfolio size of an investor in our sample is $13,869 and it matches quite well with

the average portfolio sizes reported in SCF 1992, SCF 1995, and SIPP 1995. In unreported

results, we find that the portfolio sizes are comparable when we examine the portfolio sizes

of investors in different age and income groups. Overall, these comparisons suggest that our

individual investor sample is likely to be a good representative of the households in the U.S.

Several other standard datasets are used in our study. For each stock in our sample, we

obtain monthly prices, returns, and market capitalization data from CRSP and quarterly

book value of common equity data from COMPUSTAT. We obtain the monthly time-series

of the 3 Fama-French factors and the momentum factor, monthly returns of various size and

B/M portfolios, and the NYSE size break-points and B/M break-points for each month from

Ken French’s data library.6

III Evidence of Under-Diversification

III.A Diversification Measures

To examine the extent of under-diversification among investor portfolios, we use three differ-

ent (but related) measures of diversification. The first measure (D1) is a normalized version

of portfolio variance. The expected portfolio variance of an equal weighted portfolio with N

stocks is given by:

σ2p =

1

Nσ2 +

(N − 1

N

)cov. (1)

where σ2 is the average variance of all stocks in the portfolio and cov is the average covariance

among all stocks in the portfolio. The normalized portfolio variance is obtained by dividing

the portfolio variance by the average variance of stocks in the portfolio:

D1 = NVEWP =σ2

p

σ2=

1

N+

(N − 1

N

) (cov

σ2

)=

1

N+

(N − 1

N

)corr. (2)

Here, corr is the average correlation among stocks in the portfolio. We measure the portfolio

variance in a normalized unit so that portfolios of different sizes can be aggregated.

The expression for normalized variance indicates clearly that the portfolio variance can be

reduced in two different ways. Firstly, it can be reduced by increasing the number of stocks

6Ken French’s data library is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.

9

in the portfolio (i.e., by increasing N) and secondly, it can be reduced by a proper selection

of stocks such that the average correlation (corr) among stocks in the portfolio is lower.

Variance reduction through proper stock selection reflects “skill” in portfolio composition

while addition of stocks in the portfolio without lowering the average portfolio correlation is

more likely to reflect a “passive” form of diversification.7

The degree of diversification of a portfolio can also be measured as its deviation from

the market portfolio (Blume and Friend 1975). The weight of each security in the market

portfolio is very small, so the diversification measure (D2) is approximately:

D2 =N∑

i=1

(wi − wm)2 =N∑

i=1

(wi − 1

Nm)2 ≈

N∑i=1

w2i (3)

where N is the number of securities held by the investor, Nm is the number of stocks in the

market portfolio, wi is the portfolio weight assigned to stock i in the investor portfolio and

wm is the weight assigned to a stock in the market portfolio (wm = 1/Nm). A lower value of

D2 reflects a higher level of diversification.

Finally, we also use the total number of stocks in the portfolio as a “crude” measure of

the level of diversification of a given portfolio:

D3 = N. (4)

Our first diversification measure is perhaps the most appropriate one since it recognizes

the importance of the covariance structure of a portfolio. More importantly, it allows us to

distinguish between “passive” (portfolio risk is reduced by increasing the number of stocks

in the portfolio) and “active” or skill-based (portfolio risk is reduced by choosing imperfectly

correlated stocks) portfolio diversification. The D2 measure provides a good approximation

to the level of diversification of a given portfolio but ignores the role of covariance. Our

third diversification measure is the most commonly used diversification measure but it is

likely to overstate the level of diversification of a portfolio (Blume, Crockett, and Friend

1974, Vissing-Jorgensen 1999).

III.B Diversification at an Aggregate Level

The observed degree of under-diversification among investor portfolios in our sample is quite

surprising. It is commonly believed that a well-diversified portfolio should consist of at least

7The idea of decomposing portfolio variance into two parts, one representing the effect of the number ofstocks (N) and the other representing the average correlation among stocks in the portfolio (corr) is proposedin Goetzmann, Li, and Rouwenhorst (2001). Also, see Elton and Gruber (1977).

10

10-15 stocks.8 In our sample, in any given monthly time-period, only 5-10% of the portfolios

consist of more than 10 stocks. More than 25% of investor portfolios contain only 1 stock,

more than 50% of them contain 1-3 stocks, and more than 70% of households hold 5 or

fewer stocks. This pattern of holding concentrated portfolios is observed throughout the

1991-96 sample period though, over time, there has been an increase in the average number

of stocks held by investors (see Table I, Panel A). For instance, the percent of investors

holding more than 5 stocks increased from 18% in 1991 to 29% in 1996. These aggregate

level diversification results are broadly consistent with the findings of Blume and Friend

(1975), Kelly (1995), and Polkovnichenko (2003).

To measure the level of diversification more formally, in each month, using past 5 years

of monthly stock returns data,9 we estimate the expected return vector and the covariance

matrix for the entire set of stocks traded by investors in our sample. These estimates are

then used to compute the normalized portfolio variance and the average portfolio correlation

for each investor portfolio.

Table I (Panel B) report the normalized portfolio variance statistics of investor portfolios.

As expected, the normalized variance decreases as the number of stocks in the portfolio (N)

increases. The normalized variance of concentrated portfolios is approximately 3-4 times

the normalized variance of well diversified portfolios. For example, in 1996, the normalized

variance of well-diversified portfolios with 11-15 stocks is 0.163 while concentrated portfolios

with only 2 stocks, on average, have a normalized variance of 0.407.

Over time, the normalized portfolio variance of investor portfolios has decreased but

to a large extent due to changes in the correlation structure of the U.S. equity market

(Campbell, Lettau, Malkiel, and Xu 2001). The reduction in variance in the group of

well-diversified portfolios is much larger than the variance improvement in the group of

concentrated portfolios. For example, the normalized variance of 2-stock portfolios has

improved from 0.508 in 1991 to 0.407 in 1996 – a 20% decline. However, during the same

period, the normalized variance of portfolios containing more than 15 stocks has decreased

from 0.291 to 0.130 – a 55% decline.

We also compute the average correlation among the stocks in investor portfolios (see

Table I, Panel C) and find that the average portfolio correlation decreases during our 1991-

96 sample period. The average portfolio correlation decreases for portfolios of all sizes

but the average correlation does not vary significantly across portfolios at a given instant

8This is a conservative estimate. Statman (1987) estimates this number to be 30 while a recent estimate(Statman 2002) suggests that a mean-variance optimal portfolio is likely to contain more than 100 stocks.

9Stocks with less than 2 years of monthly returns data are excluded from the analysis.

11

of time. The observed differences in average correlations are not statistically significant.

This suggests that the reduction in portfolio variance during the 1991-96 time-period occurs

primarily due to an increase in the number of stocks in the portfolio. There is no evidence

of an improvement in the stock selection ability of investors in our sample. In other words,

the portfolio composition skill of individual investors has not improved over time.

For robustness, we compute the average diversification measures for portfolios of different

sizes (see Table II) where the average portfolio value of a portfolio during the six-year sample-

period is used as a measure of portfolio size. As expected, we find that larger portfolios

contain a greater number of stocks and exhibit better diversification properties (Panel B).

However, improper diversification is not concentrated only among smaller portfolios. Even

among larger portfolios (quintile 5), more than one-third of the portfolios contain 5 or fewer

stocks (Panel A). Overall, these results suggest that a vast majority of investors in our sample

are under-diversified.

III.B.1 Investor Portfolios Relative to Benchmark Portfolios

To further examine the level of under-diversification among investor portfolios, we compare

the investor portfolios with two benchmark portfolios: (i) the market portfolio (S&P 500

index)10, and (ii) a large number of randomly constructed portfolios. The market portfolio

represents the risk-return trade-off the investors could have achieved with a passive trading

style just by investing in one of the many available index funds. The set of random portfo-

lios represent the risk-return trade-off a “naive” investor could have achieved by arbitrarily

picking stocks. Thus, these benchmarks by no means constitute a “desirable” set but rather

they represent a “minimum” level of risk-return trade-off an investor portfolio is expected

to exhibit.

Figure 1 shows the positions of investor portfolios relative to the market portfolio (and

the capital market line) in the mean-standard deviation (µ-σ) plane. Two monthly time-

periods are arbitrarily chosen in the first half of the sample period (February 1991 and June

1993) and two monthly time-periods are arbitrarily chosen in the second half of the sample

period (September 1995 and June 1996). The past 5 years of monthly returns data are

used to estimate the means and the standard deviations of the market portfolio and investor

portfolios. The riskfree rate corresponds to the 90-day T-Bill rate.

We find that only a very small fraction of investor portfolios are above the capital market

10We thank John Campbell for suggesting this benchmark.

12

line (CML). For instance, in a month chosen in the first year of the sample period (February

1991), only 9.53% of the portfolios are above the CML and in a month in the last year of our

sample (June 1996), 13.96% of the portfolios are above the CML. In other monthly time-

periods also, only a small fraction of investor portfolios exhibit a better risk-return trade-off

than the market portfolio. Consistent with our earlier evidence of improving diversification

characteristics, we find that investor portfolios are more “spread out” in the µ-σ plane

during the initial years but during the latter years a relatively larger proportion of investor

portfolios are closer to the CML. In spite of this improvement, only a small proportion of

investor portfolios are above the CML. Overall, the graphical evidence suggests that investor

portfolios have significantly higher volatility relative to the market portfolio.

To allow for a more accurate comparison between the volatilities of investor and market

portfolios, in Figure 2, we plot the rolling volatility (using a 12-month window) of the

market portfolio along with the rolling volatility statistics (25th percentile, median, and 75th

percentile) of investor portfolios. The plot shows quite clearly that in any given time-period,

more than 75% of investor portfolios have higher volatility than the market portfolio. The

magnitude of the risk taken by investors in our sample is quite surprising.

Comparing the variance of observed investor portfolios with the variance of randomly

constructed portfolios (our second benchmark), we again find that investor portfolios have

relatively higher risk exposures. First, we identify several sets of investor portfolios, each

set containing 2,000 k-stock portfolios, where k = 2, . . . , 15. Then, the average diversifica-

tion characteristics of these randomly chosen sets of portfolios are compared to the average

diversification characteristics of matching investor portfolios.

Figure 3 shows the average normalized variance of investor portfolios of different sizes

relative to the matching benchmark portfolios during the month of June 1996. We find

evidence of systematic under-diversification. The normalized variance of investor portfolios

is approximately 25% higher than the normalized variance of benchmark portfolios and this

difference increases with the average size of the investor portfolio. This suggests that investor

portfolios in our sample are worse in terms of their risk-return characteristics than even those

portfolios that in a sense provide a lower bound on the attainable risk-return trade-off.

III.C Diversification Over Time

During the 1991-96 sample period, the average number of stocks in investor portfolios has

increased almost monotonically from 4.28 in 1991 to 6.51 in 1996 – an increase of almost

13

48% (see Table III, Panel A). Furthermore, the normalized portfolio variance has steadily

decreased from 0.47 in 1991 to 0.31 in 1996 – a decrease of more than 34%. At a first glance,

these two observations seem to imply that the portfolio composition skills of investors have

improved over time. However, when we compare investor portfolios to a benchmark of

randomly constructed matching portfolios, we find that the average risk exposure of investor

portfolios is significantly higher than that of the benchmark portfolios. In fact, during our

sample period the excess normalized variance (relative to the benchmark portfolios) has

increased from 44.14% in 1991 to 67.80% in 1996. This suggests that the improvements in

the diversification characteristics of investor portfolios result to a large extent from changes

in the correlation structure of the U.S. equity market and not necessarily from an improved

ability to form better diversified portfolios.11

We also compare the average correlation of investor portfolios with a set of randomly

chosen benchmark portfolios. Each month 2,000 portfolios containing upto 10 stocks are

formed by selecting stocks randomly from the set of stocks in our sample. Using the his-

torical monthly returns data, the portfolio correlation matrix is estimated and the average

correlation among the stocks in a chosen portfolio is computed. Finally, the average correla-

tion for a month is obtained by averaging the average correlations of these 2,000 randomly

chosen portfolios. As expected, we find that the average correlations for both sets of port-

folios decrease during the 1991-96 time period but the average correlation among stocks in

actual investor portfolios is significantly higher than the average correlation among stocks

in randomly constructed portfolios. For instance, the excess average correlation is 87.24% in

1991 and 61.69% in 1996. Again, these results suggest that investors’ portfolio composition

skills have not improved.

In the analysis above we have combined portfolios of different sizes and find that at an

aggregate level reduction in portfolio variance over time is driven primarily by the changing

market correlation structure. However, potential improvements in portfolio variance cross-

sectionally are not revealed by this analysis. In Figure 4, we show the cross-sectional variation

in average correlation across investor portfolios with different number of stocks for two

monthly time-periods. The two monthly periods are chosen in the first and the last years

of our sample period. For comparison, we also plot the average correlations of matched

random portfolios. The average correlations of investor portfolios containing k-stocks and

11Malkiel and Xu (1997) report a similar finding by tracking the variation in correlations among industryportfolios during the 1970-95 time-period. They find that the mean correlation among portfolios decreasesover time thereby suggesting that the risk reduction benefits of holding a diversified portfolio has increasedover time. Also, see Campbell, Lettau, Malkiel, and Xu (2001).

14

2,000 random portfolios with k-stocks are compared for k = 2, . . . , 15. The procedure for

constructing random portfolios is similar to the one described earlier.

Three immediate observations can be made from the figure. First, the average correlations

for both investor portfolios and random portfolios are lower in 9601 in comparison to 9101.

This is consistent with our earlier finding that the average portfolio variance decreases over

time. Secondly, in both monthly time-periods, the average correlations of investor portfolios

are higher than those of randomly chosen portfolios for all values of k12 where the correlation

differences are statistically significant (p-value < 0.05). Finally, we find that the average

correlation decreases with k for the set of random portfolios but for investor portfolios,

the average correlation increases as k increases. These observations suggest that investor

portfolios of all sizes have relatively poorer diversification characteristics than the benchmark

of randomly constructed portfolios and this result holds throughout our six-year sample

period.

III.D Following 1991 Investors Over Time

The above documented improvements in the mean diversification measures do not reveal

if the improved portfolio diversification is a result of “learning” by “original” investors or

whether “new” investors start with better diversified portfolios. To investigate if there is

any evidence of learning by “original” investors, we track the diversification characteristics

of investor portfolios who are present at the beginning of our sample period (January 1991).

Table III (Panel B) reports the results. Even for the group of investors who are present

at the beginning of our sample period, we find a considerable improvement in their diversifi-

cation characteristics. For instance, during the 1991-96 sample period, the average number

of stocks in their portfolios has increased almost monotonically from 3.82 in 1991 to 5.53

in 1996 – an increase of almost 45% (see Table III, Panel A). Furthermore, the normalized

portfolio variance has steadily decreased from 0.48 in 1991 to 0.33 in 1996 – a decrease of

more than 31%. Overall, the improvements in the diversification characteristics of 1991 in-

vestors (Panel B) mirror the diversification improvements at an aggregate level (Panel A)

though these improvements are marginally lower.

12There is an exception. In 9601, for k = 2, the average correlation of random portfolios is higher thanthat of investor portfolios.

15

IV Economic Costs of Under-Diversification

Are the economic costs of under-diversification significant or are investors rationally choos-

ing under-diversified portfolios given that a variety of frictions such as transaction costs,

information gathering costs, etc. are likely to influence their portfolio choices? To estimate

the welfare loss from improper diversification, we examine both the ex ante and the realized

performance of investor portfolios.

IV.A Ex Ante Portfolio Performance

The ex ante performance measures provide a snapshot of the diversification characteristics

of investor portfolios at the time investors make their portfolio choices. To examine the ex

ante performance, we measure the proportion of investor portfolios that are above the capital

market line (CML) when they choose their portfolios. The portfolios above the CML exhibit

better risk-return tradeoffs relative to the market portfolio.

Figure 5 shows the positions of concentrated portfolios (portfolios with 1-3 stocks) and

relatively more diversified portfolios (portfolios with 7 or more stocks) relative to the market

portfolio and the CML in the µ-σ plane. About 28% of portfolios that have 7 or more stocks

are above the CML while only 17% of concentrated portfolios are above the CML. The results

are shown for one time period (September 1995) but qualitatively similar results are obtained

for other time periods. These results suggest that a greater proportion of less diversified

portfolios are inefficient. As a consequence, the economic costs of under-diversification are

likely to be higher for the less diversified group of investors.

IV.B Economic Costs of Under-Diversification at an Aggregate Level

To examine the relation between portfolio diversification and realized performance, we use

two performance measures: (i) the mean monthly excess (relative to the market) portfolio re-

turn and (ii) the Sharpe ratio. First, using the average sample-period diversification measure

(D1) for each investor, we rank investors and divide them into ten groups (deciles). Then, we

compute the two performance measures for each investor in each of these ten diversification

groups.

Table IV reports the performance statistics for each of the ten investor groups. In Panel

A, we report the cross-sectional statistics of the mean monthly portfolio excess return mea-

sure and in Panel B, we report the cross-sectional statistics of the Sharpe Ratio measure. The

16

mean values of the performance measures show that as the level of diversification increases,

both performance measures increase. For instance, the mean monthly excess portfolio re-

turn for decile 1 investors is −0.12% but for the decile 10 investor group, it is 0.05%. On

an annual basis, the most diversified investor group earns 2.04% higher return than the

least diversified investor group. Furthermore, the mean Sharpe Ratio differential between

the extreme diversification groups is 0.08. As we show later (Section V.C), relatively less

diversified investors trade more frequently and thus the net returns they earn is likely to be

even lower and consequently, the performance differentials between the least diversified and

the most diversified groups of investors are likely to be even higher.

Given the large idiosyncratic risk exposures of the less diversified investor group, not

surprisingly, we find a greater number of extreme performers in this group. The standard

deviation of the performance measures are greater in low diversification groups. For instance,

the standard deviation of the mean monthly excess portfolio return measure is 1.74 for

investors with the least diversified portfolios (decile 1) but only 0.80 for investors with the

most diversified portfolios (decile 10). The 10th and the 90th percentile measures provide

additional evidence of extreme performance in the lower diversification investor groups. In

addition, we find that in deciles 1-5, more than a quarter of investors have negative Sharpe

Ratios. These investors earn lower return than even the riskfree rate of return while taking

considerable risks.

IV.C Cross-Sectional Variation in Economic Costs of Under-Diversification

How do the economic costs of under-diversification vary cross-sectionally across different

investor groups? Are there certain types of investors who pay higher economic costs for

not diversifying appropriately? To answer these questions, we define investor groups on

the basis of their age, income, occupation, and trading frequency and examine the cross-

sectional performance differentials between the most diversified and the least diversified set

of investors within those investor groups.

Table V reports the results. In Panel A, we report the raw monthly portfolio returns and

the Sharpe Ratio while in Panel B we report the CAPM and the 4-Factor alphas. Consistent

with our previous results, we find that for the aggregate investor group, all four performance

differentials between the most diversified and the least diversified investor categories are

positive and statistically significant. For instance, the 4-Factor alpha differential is 0.20%

monthly which is equivalent to an annual performance differential of 2.40%.

17

We also find that the performance differentials between the extreme diversification cat-

egories are significantly positive within almost all investor groups. For instance, within the

older investor group (see Panel B), the mean 4-factor alpha is −0.74% for the least diversified

investors and −0.44% for the most diversified set of investors. The monthly risk-adjusted

performance differential of 0.30% translates into an annual performance differential of 3.60%.

The performance differentials are also greater than average for retired investors and investors

who trade less frequently. The 4-Factor alpha indicates annual performance differentials of

3.96% and 3.12% respectively.

Overall, our cross-sectional performance results reveal that the economic costs of under-

diversification are likely to be significant for a majority of investors in our sample.

IV.D Time Variation in Economic Costs of Under-Diversification

Earlier we documented that the average portfolio diversification of individual investors im-

proved over time during the 1991-96 sample-period (see Section III.C). Does improved port-

folio diversification result in better portfolio performance? To address this issue, we carry

out a split-sample test and examine the performance of the aggregate investor portfolio dur-

ing the 1991-93 and 1994-96 sub-periods. The aggregate investor portfolio is constructed by

combining the portfolios of all investors in the sample.

We find that there is a decrease in the level of under-performance in the aggregate investor

portfolio as the overall level of diversification improves. The aggregate investor portfolio

earns a mean return of 1.14% with a standard deviation of 3.49% during the 1991-93 sub-

period but it earned a higher mean return (1.27%) with a slightly lower standard deviation

(3.29%) during the 1994-96 sub-period. There is also an improvement in the aggregate

portfolio’s risk-adjusted performance – the 4-Factor alpha is −0.34% (t-stat = −3.35) during

the first half of the sample-period but it is considerably lower (−0.09%) and statistically

insignificant (t-stat = −0.81) during the second half of the sample-period. This yields a

monthly risk-adjusted performance improvement of 0.25% or 3.00% on an annual basis.

Overall, consistent with previous studies (Brennan and Torous 1999, Meulbroek 2002),

our results suggest that better portfolio diversification yields better risk-adjusted perfor-

mance. However, a majority of investors in our sample could have achieved these levels of

risk-adjusted performance by simply investing in one of the many available index funds.

18

V Why Don’t Investors Diversify?

If investors pay a cost for improper diversification, why don’t they diversify? Why do they

continue to hold only a handful of stocks and why is the average correlation among stocks in

their portfolios so high? Are investors aware of the benefits of diversification but choose to

hold under-diversified portfolios or is the observed under-diversification a result of investors’

inability to diversify appropriately?

To gain insights into the diversification decisions of individual investors, we consider three

broad set of determinants of portfolio diversification: (i) investors’ personal characteristics

and their levels of financial sophistication, (ii) their behavioral biases, and (iii) their prefer-

ence for holding stocks with certain characteristics. We carry out a series of non-parametric

and parametric tests to estimate the importance and relative strengths of these potential

determinants of portfolio diversification.

V.A Investor Demographics and Sophistication

Investors’ attitude towards risk is likely to influence their diversification decisions. An in-

vestor with a high (low) tolerance for risk may hold a less (more) diversified portfolio.

Previous studies (e.g., Blume and Friend (1975)) document that risk aversion increases with

age and wealth. As a consequence, portfolio diversification is likely to increase with age and

income. Alternatively, portfolio diversification may increase with age because with experi-

ence, investors acquire more information about the market (King and Leape 1987). Taken

together, these results suggest that portfolio diversification is likely to increase with age and

income.

Investors’ sophistication level, in particular, their financial sophistication, is likely to

be an important determinant of their diversification decisions. For instance, the average

correlation among stocks in investor portfolios may be high because investors do not fully

understand why diversification reduces portfolio risk. They may incorrectly believe that

any multiple-stock portfolio, irrespective of its covariance structure, is well-diversified. As a

result, they may hold portfolios that are inappropriately diversified.

To examine this possibility, we employ multiple proxies for investor sophistication. We

assume that investors who engage in short-selling, trade in options and foreign equities

(ADRs, foreign stocks, and closed-end country funds) or hold mutual funds are relatively

more sophisticated than the average investor. Furthermore, we assume that the amount of

19

available resources to an investor and her education level are likely to determine her level of

financial sophistication. Wealthier investors and those who hold professional jobs are likely

to have access to more resources and they are also likely to be better educated. With these

assumptions, we use occupation and income as additional proxies for investor sophistication.

To examine the relation between investors’ personal characteristics, their levels of finan-

cial sophistication, and their diversification decisions, we define investor groups on the basis

of (i) age, (ii) income, (iii) occupation, and (iv) financial sophistication, and measure the

average diversification characteristics of these groups of investors. The results are reported

in Table VI. We find that all three diversification measures vary in a predictable manner

with age – older investors are more diversified than younger investors. Similarly, we find

that high-income investors are marginally better diversified than low-income investors. Us-

ing the Kolmogorov-Smirnov (KS) test (Press, Teukolsky, Vetterling, and Flannery 1992),

we find that the diversification differences between the extreme age and income groups are

significant at the 5% level.

In order to understand better why diversification increases with age, we investigate the

relation between age and the frequency of trading. Younger investors may be less diversi-

fied because of their higher degree of over-confidence (Odean 1999). We find that trading

frequency, a proxy for investor over-confidence, decreases with age. The portfolio turnover

rate is 6.82% for the bottom age decile (age between 26-36) and 5.02% for the top age decile

(age between 70-82). The difference between the turnover distributions of the two groups

is statistically significant (p-value < 0.05). This suggests that young, active investors are

over-focused and hold concentrated and under-diversified portfolios.

In Table VI, we also report the average diversification measures for three broad occupation

categories: (i) professional category, consisting of investors that hold technical or managerial

positions, (ii) non-professional category, consisting of investors who are blue-collar workers,

sales and service workers, clerical workers, house-wives or students, and (iii) the retired

category. We find that the non-professional category holds the least diversified portfolios

while investors in the retired category fall on the other end of the diversification spectrum.

The average diversification levels of investor portfolios in the professional category falls in

between the average diversification levels of non-professional and retired categories.

The diversification characteristics of investor groups formed on the basis of their financial

sophistication provide stronger evidence of a positive relation between investor sophistication

and portfolio diversification. For instance, investors who trade in foreign equities hold an

average of 7 stocks (median = 5) and exhibit relatively better diversification characteristics

20

according to other diversification measures. Other investor groups formed on the basis of

the financial sophistication measures also exhibit better than average diversification charac-

teristics. Overall, these results suggest that the level of diversification is positively related

with investors’ age, income, and their financial sophistication.

V.B Preference for Mutual Funds

Do individual investors compensate for their lack of diversification by holding mutual funds?

To answer this question, we first compare the diversification levels of investors who hold

mutual funds with the diversification levels of investors who do not invest in mutual funds.

We find that investors who hold mutual funds also hold better diversified portfolios. For

instance, investors that hold large portfolios (highest quintile) and invest in mutual funds

hold an average of 11 stocks in their portfolios while those that do not invest in mutual

funds hold 7-stock portfolios. The normalized variance differential between the two groups

is −0.061 which is statistically significant at the 1% level. The diversification level differences

are significant for portfolios of all sizes though they are more pronounced for larger portfolios.

Overall, the evidence suggests that investors who hold mutual funds are aware of the

benefits of diversification and thus they hold better diversified stock portfolios. Our results

do not suggest that individual investors compensate for their lack of diversification by holding

mutual funds.

V.C Investor Over-Confidence

Lack of diversification may also result from various psychological factors, and in particular,

due to an “illusion of control” (Langer 1975). In experimental settings it has been observed

that when factors such as involvement, choice and familiarity are introduced into chance

situations, people begin to believe that they can control the outcome of those chance events.

Some investors may develop an illusory sense of control because they are directly involved

in the investment process where they make their own choices instead of relying on others (as

in the case of mutual funds) for their investment decisions.

An illusion of control may create an inappropriate level of over-confidence. Over-confident

investors may mistakenly believe that they can earn superior performance by active trading

and consequently they may choose not to diversify. A sense of over-confidence can also

emerge among investors simply because they may believe that their stock-picking abilities

21

are superior to that of the market (Kelly 1995). Furthermore, over-confident investors may

develop a false perception that they can manage their portfolio risks better by a thorough

understanding of a small number of firms rather than diversifying. Using survey data from

a set of large and experienced investors, DeBondt (1998) finds that such a belief is quite

common among investors.

To examine the relation between investor over-confidence and their diversification deci-

sions, we perform a double sort on portfolio size and portfolio turnover (a proxy for investor

over-confidence) variables. We identify 25 investor groups and for each of these groups,

we compute equal-weighted average diversification measures. The results are reported in

Table VII. We find that investors with higher monthly portfolio turnover rates, i.e., active

investors, hold fewer stocks and their portfolios have higher normalized portfolio variance.

For instance, investors who hold large portfolios (top quintile) and trade infrequently hold

an average of more than 10 stocks in their portfolios while those who trade frequently hold

7-stock portfolios. In addition, the normalized variance differential (see Panel B) between

the two groups is −0.051 which is statistically significant at the 1% level.

These results suggest that over-confident investors hold relatively less diversified port-

folios. Given that diversification level is positively related to performance (see Tables IV

and V), our results are consistent with the findings of Odean (1999) who documents that

over-confident investors trade more actively and thus earn a lower net return. The lower

level of diversification among active investors is probably another manifestation of their

over-confidence.

V.D Preference for Local Stocks and Familiarity Bias

Familiarity with a certain set of stocks may further exacerbate the illusion of control where

investors may fail to realize that more knowledge about the chosen set of stocks does not nec-

essarily imply control over the outcome (i.e., returns earned by the portfolio). Furthermore,

as Merton (1987) suggests, investors may limit the number of stocks in their portfolios due to

search and monitoring costs.13 To minimize these costs, investors may invest in stocks they

already know about. Several studies, including Huberman (2001), Zhu (2002) and Ivkovic

and Weisbenner (2003) find that individual investors exhibit a preference for local stocks,

i.e., the stocks that they are familiar with.

13Goetzmann, Massa, and Simonov (2003) show that portfolio diversification is inversely related to thelevel of professional specialization because with increased specialization, investors are able to allocate lesstime to gathering useful financial information.

22

To examine if investors’ preference for local stocks induces them to hold less diversified

portfolios, we compute a local bias (or familiarity bias) measure for each investor. The

familiarity bias (FB) measure is defined as, FB = Dact − Dmkt, where Dact is the distance

between an investor’s location and her stock portfolio and Dmkt is the distance between an

investor’s location and the market portfolio.14

We perform a double sort on portfolio size and familiarity bias variables and identify 25

investor groups. For each of these groups, we compute equal-weighted average diversification

measures. The results are presented in Table VIII. We find that across portfolios of all sizes,

investors with higher FB own less diversified portfolios. There is a mild non-linearity where

FB quintile 2 portfolio rather than quintile 1 portfolio has the highest level of diversification.

Nevertheless, the mean diversification difference between low FB and high FB investors is

significant, especially for larger portfolios.

Overall, these results suggest that investors with stronger familiarity bias tend to hold

relatively less diversified portfolios and the familiarity bias appears to be an important

determinant of investors’ diversification decisions.

V.E Preference for Skewness and Other Stock Characteristics

Do investors choose to hold under-diversified portfolios even though they are aware of its

benefits because they prefer skewness? It is known that portfolio variance and portfolio

skewness are highly correlated – as a portfolio becomes more diversified, both its variance

and skewness declines (e.g., Simkowitz and Beedles (1978), Golec and Tamarkin (1998)).

Investors with a preference for skewness may choose to hold portfolios with high variance

(i.e., an under-diversified portfolio) and thus may rationally choose to hold under-diversified

portfolios.

To examine if a preference for positive skewness influences investors’ diversification deci-

sions, we compare the skewness tilts of the aggregate portfolios of investor groups formed by

sorting on the diversification measure. First, at the end of each month, we rank investors on

the basis of the diversification level of their end-of-month portfolios and form diversification

quintiles. Next, at the end of each month, we also rank all CRSP stocks on the basis of

their three moments (mean, standard deviation, and skewness) and form moment quintiles.

Finally, we construct an aggregate portfolio for each of the diversification quintiles by com-

bining the portfolios of all investors within a group. The weights in an aggregate group

14See Coval and Moskowitz (2001) and Zhu (2002) for details of this measure.

23

portfolio assigned to stocks in the three moment quintiles reveal the group’s preference for

mean, standard deviation, and skewness. The monthly weights from our sample-period are

averaged and reported in Table IX.

We find that investors in each diversification quintile prefer stocks with higher mean (see

Panel A). In addition, comparing investors in the top and the bottom diversification quintiles,

we find that less diversified investors have a greater preference (relative to highly diversified

investors) for stocks with higher mean returns. Examining investors’ preference for higher

moments (see Panels B and C), we again find that all five investor groups exhibit a greater

preference for stocks with lower standard deviation and lower skewness. However, comparing

investors in top and bottom diversification quintiles, we find that less diversified investors

have a greater preference (relative to highly diversified investors) for stocks with higher

standard deviation and higher skewness. Relative to more diversified investors (highest

quintile), less diversified investors (lowest quintile) allocate 4.7% more weight to stocks with

high standard deviation (highest quintile) and 3.7% more weight to stocks with high skewness

(highest quintile). Overall, we find moderate evidence that skewness preference influences

investors’ portfolio diversification decisions.

In addition to skewness, do investors exhibit a strong preference for other stock charac-

teristics (e.g., firm size) and thus hold relatively less diversified portfolios? To examine this

possibility, we follow the procedure described above and examine the weights assigned to

stocks categorized in two different ways on the basis of their size and B/M measures. The

first set consists of small-cap (size deciles 1-3), mid-cap (size deciles 4-7), and large-cap (size

deciles 8-10) categories and the second set consists of growth (B/M deciles 1-3), blend (B/M

deciles 4-7), and value (B/M deciles 8-10) categories.

The results are reported in Table X. We find that investors in the top decile exhibit a

weak preference for small-cap and value stocks – relative to less diversified investors (lowest

decile), more diversified investors (highest decile) allocate 1.1% more weight to small-cap

stocks and 1.4% more weight to value stocks. Other than this weak pattern, we do not find

any evidence of style variation even when we perform a finer partition (deciles instead of

quintiles) of investors using their diversification levels. Overall, these results suggest that

the style preference of investors is not a key determinant of their portfolio diversification

decisions.

24

V.F Summing Up: Regression Results

To further explore the determinants of investors’ diversification decisions and to estimate

the relative explanatory power of different determinants of diversification, we estimate three

cross-sectional regressions. The three diversificationmeasures, −D1, −D2, and D3, computed

for the entire sample-period, are used as dependent variables and a set of household and

portfolio characteristics are used as independent variables.

In the regression specification, Income is the total household income and Age is the age

of the head of the household. The Professional and Retired dummy variables represent

the occupation categories where the professional job category includes investors who hold

technical and managerial positions. The remaining investors belong to the non-professional

category which consists of blue-collar workers, sales and service workers, clerical workers,

house-wives, and students. Portfolio Turnover is the average of monthly buy and sell turnover

rates, Portfolio Size is the average size of the household portfolio during the sample-period,

and Portfolio Performance is the risk-adjusted sample-period performance (Sharpe Ratio)

of the household.

Four additional dummy variables are employed as independent variables: (i) MFund

Dummy which is set to one if an investor held mutual fund in at least one month during

the sample-period, (ii) Foreign Dummy which is set to one if an investor made at least

one trade in a foreign asset (ADR, foreign stock or a closed-end country fund) during the

sample-period, (iii) Short-Sell Dummy which is set to one if an investor executed at least

one short-sell during the sample-period, and (iv) Option Dummy which is set to one if an

investor made at least one trade in an option during the sample-period. Finally, Familiarity

Bias is the difference between the weighted distance of an actual investor portfolio from her

location (∑N

i=1 wiDi, where N is the number of stocks in the portfolio, wi is the weight of

stock i in the portfolio, and Di is the distance between an investor’s zipcode and the zipcode

of a firm’s headquarters) and the weighted distance of the market portfolio from her location.

The estimation results are presented in Table XI where both independent and dependent

variables have been standardized so that the coefficient estimates can be compared directly

within a regression specification and also across the three specifications. There is a positive

relation between Age and diversification (coefficient estimate = 0.065, t-stat = 6.16) as well

as Income and diversification (coefficient estimate = 0.012, t-stat = 2.32). These estimates

suggest that older and high-income investors are more diversified.

We also find that investor sophistication dummies are positive and significant – the coeffi-

cient estimates for MFund, Foreign, Short-Sell, and Option dummies are 0.030, 0.208, 0.068,

25

and 0.020. This suggests that more sophisticated investors hold relatively more diversified

portfolios. The positive sign on the Professional Dummy further support the view that more

sophisticated investors and those with higher income hold more diversified portfolios. In

addition, the negative signs on Portf Turnover and Familiarity Bias variables suggest that

both behavioral biases, over-confidence and familiarity, influence investors’ diversification

decisions. Finally, the Portf Size, and Portf Performance variables have expected signs –

larger portfolios are more diversified and more diversified portfolios earn higher returns.

Comparing the magnitudes of the coefficient estimates, we find that Portfolio Turnover

and Foreign Dummy variables have significantly larger coefficients (in absolute terms). This

suggests that investor over-confidence and their awareness of the benefits of diversification

as reflected by their willingness to hold foreign equities are the two strongest determinants

of investors’ diversification decisions. In contrast, the Income and Professional Dummy

variables have the smallest coefficients and thus they appear to be the weakest determinants

of investors’ diversification decisions.

Overall, our regression estimates support the results from non-parametric tests presented

earlier. On the basis of our results from our non-parametric and parametric tests, we can

draw the following conclusions about the determinants of portfolio diversification: (i) in-

vestors’ over-confidence and their level of sophistication are the strongest determinants of

portfolio diversification, (ii) age, income, preference for skewness, and preference for local

stocks (familiarity bias) are other significant determinants of portfolio diversification, and

(iii) investors’ style preferences do not explain their diversification choices.

VI Robustness Tests

VI.A Small Portfolio Size and High Transaction Costs

It is conceivable that investors do not diversify appropriately due to the small size of their

portfolios. The inability of investors to buy in round lots, transaction costs (Brennan 1975)

and overall higher stock prices may prevent investors from diversifying, especially those who

hold smaller portfolios.

Consistent with these predictions, we find that smaller portfolios are relatively less di-

versified than larger portfolios. However, about 8% of investors with very small portfolios

(portfolio size < $5,830) and approximately 14% of investors with moderate size portfolios

($5,830 ≤ portfolio size < $10,560) hold more than 5 stocks in their portfolios (see Table

26

II, Panel A). Furthermore, in our sample, relatively less diversified investors trade more fre-

quently and pay significant transaction costs. The average annual trading cost for investors

in our sample is 1.46% of their total income. In addition, for the investor database used

in this study, Barber and Odean (2001) estimate that the trading cost of active investors

is 3.90% of their annual income. These results suggest that investors in our sample pay

significant transaction costs but still choose to remain under-diversified.

Taken together, this evidence suggests that neither small size of the portfolio nor higher

transaction costs act as deterrents to diversification. Even investors holding smaller portfolios

would have been able to diversify if they desired to do so.

VI.B Play Money Accounts

One might argue that investor portfolios are under-diversified because these portfolios repre-

sent their “play money” accounts meant primarily for gambling and entertainment purposes

while the bulk of their actual investment including retirement money is elsewhere. This

seems quite unlikely. Approximately 42% of accounts15 in our sample are retirement ac-

counts (IRA or Keogh). Furthermore, at any given instant of time, the aggregate value of

investor equity portfolios is more than $2 billion. We also find that the average ratio of ac-

count size to annual income level is 0.79 where the average portfolio value during the six-year

sample-period is used as a measure of portfolio size. The portfolio size to income ratio is

much higher for lower income groups. For example, this ratio is 3.62 for investors that earn

less than $15,000 per year and 1.79 for investors with annual income between $20,000 and

$30,000.

This evidence suggests that the money in the investment accounts we examine does not

represent an insignificant fraction of a household’s financial portfolio. It is very unlikely

that individual investors’ portfolios are under-diversified because they represent their “play

money” accounts.

VII Diversification Decisions and Asset Prices

According to the traditional economic theory, investors adopt a unified view of their entire

financial portfolio which includes their equity portfolio, labor income, real-estate portfolio,

15There are 158,031 accounts in our sample which includes 64,416 IRA and 1,299 Keogh accounts. Atypical household holds multiple accounts – out of 77,995 households in our sample, 43,706 hold at least oneretirement account.

27

etc. In this setting, hedging motives influence investors’ broad asset allocation decisions and

their diversification decisions. For instance, investors may utilize their equity portfolio to

hedge against background risks such as their labor income risk, entrepreneurial risk, and

real-estate risk.

An alternative psychological theory posits that investors engage in narrow framing (Kah-

neman and Lovallo 1993, Barberis, Huang, and Thaler 2003) where they do not adopt a

unified view of their financial portfolio. These investors are less likely to integrate the vari-

ous risks they face in their aggregate financial portfolio but rather they evaluate those risks

individually.16 Consequently, these investors are likely to demand compensation for the

individual risks they face.

More specifically, investors who systematically hold less than fully diversified equity port-

folios are likely to demand compensation for the idiosyncratic risk in their equity portfolios.

The compensation they demand for holding idiosyncratic risk is likely to depend upon the

riskiness of their equity portfolio rather than on the riskiness of their entire financial portfo-

lio. If this sensitivity is widespread, investors’ equity portfolio decisions are likely to generate

pervasive forces which may influence returns. As a result, ceteris paribus, stocks with a less

diversified individual investor clientele are likely to yield higher expected returns.

VII.A Stock Level Diversification Measure

To examine the relation between individual investor diversification decisions and stock re-

turns, we estimate the average diversification level of each stock’s individual investor clientele

(ADIV) at the end of each month using the following relation:

ADIVimt =Nit∑k=1

wiktDmkt. (5)

Here, ADIVimt is the average diversification level (mth diversification measure) of the in-

dividual investor clientele (ADIV) of stock i at the end of month t, Nit is the number of

investors who hold stock i at the end of month t, wikt is the ownership weight of investor k

in stock i at the end of month t, and Dmkt is the mth diversification measure of investor k

at the end of month t. The total ownership weight is given by:

wikt =Sikt∑Nit

k=1 Sikt

(6)

16There is some empirical evidence that supports this hypothesis. Massa and Simonov (2002) find thatinvestors’ equity portfolio decisions are not influenced by hedging motives – the correlations between equityportfolio returns and labor income as well as entrepreneurial income are ignored.

28

where Sikt is the total number of shares of stock i owned by investor k at the end of month

t. In our analysis we use the first diversification measure (normalized variance), i.e., m = 1,

but our results are quite similar with the other two diversification measures.

Are there certain types of stocks that attract a less diversified individual investor clien-

tele? To answer this question, at the end of each month, we rank all stocks in our sample on

the basis of its end-of-month ADIV measure and form ADIV decile portfolios. Furthermore,

at the end of each month, we obtain the size, B/M, price, and institutional ownership (IO)

decile rankings of all stocks in our sample using the NYSE breakpoints. Using the prior

month stock characteristic rankings, we compute a size, B/M, price, and IO decile score for

each ADIV portfolio. This decile score is an equal-weighted average of the characteristic

decile ranks of stocks that belong to the ADIV portfolio.

The average decile scores and the systematic risk exposures of the ten ADIV portfolios

are reported in Table XIII . We find that stocks with a less diversified individual investor

clientele are smaller in size and have lower market beta. The size decile score is 3.53 for

ADIV decile 1 portfolio compared with 4.47 for ADIV decile 10 portfolio. Consistent with

this evidence, we find that the SMB exposure of ADIV decile 1 portfolio is higher (1.29) than

the SMB exposure (1.00) of ADIV decile 10 portfolio. Other than this weak pattern, stocks

with different characteristics seem to be evenly distributed across the ten ADIV groups.

VII.B Performance of a Diversification Based Trading Strategy

If individual investor diversification decisions influence stock returns, stocks with a less di-

versified individual investor clientele are likely to earn higher returns than stocks with a more

diversified individual investor clientele. To test this possibility, we compute the performance

of a zero-cost portfolio that takes a long position in stocks with the least diversified indi-

vidual investor clientele and a short position in stocks with the most diversified individual

investor clientele.

At the end of each month, we sort all stocks in our sample using their month-end ADIV

measures and form ten decile portfolios. Portfolio 1 consists of stocks with the lowest ADIV

measure (i.e., stocks that have the least diversified individual investor clientele) while port-

folio 10 consists of stocks with the largest ADIV measure (i.e., stocks that have the most

diversified individual investor clientele). For each of the ADIV decile portfolios constructed

at the end of month t, we compute its return in month (t + k) as an equal weighted av-

erage of returns of stocks in that portfolio. k represents the number of months between

29

the portfolio formation date and the month in which portfolio returns can be realized. For

robustness, we choose four different values of k (k = 1, 2, 3, 4). Finally, a monthly portfolio

return time-series is obtained for each of the ten ADIV decile portfolios.

Table XIII (Panel A) reports the means and the standard deviations of ADIV portfolio

returns time-series. Portfolio 1 which contains stocks with the least diversified individual

investor clientele has the highest mean monthly return (2.19%). The mean monthly return

decreases as one moves to higher ADIV decile portfolios. The decline in monthly returns is

not monotonic, but nonetheless, portfolio 10 which contains stocks with the least diversified

individual investor clientele has a lower mean return (1.63%).

When we assume that the returns to portfolios formed at the end of month t can be

realized in month t +1 (i.e., k = 1), the average return differential between the two extreme

ADIV portfolios (portfolios 1 and 10) is 0.56% per month or 6.72% per annum and the

difference is statistically significant (t-value = 3.34). The performance declines considerably

when there is a delay between the portfolio formation date and the month in which portfolio

returns can be realized. For delays of 1, 2, and 3 months, the annual performance differentials

are 5.04%, 3.96%, and 3.96% respectively.

When we control for risk (see Panel B), the performance differential between the two

extreme ADIV portfolios (portfolios 1 and 10) is even more impressive – 0.62% per month

or 7.44% per annum. As before, the performance declines considerably when there is a

delay between the portfolio formation date and the month in which portfolio returns can be

realized. For delays of 1, 2, and 3 months, the annual risk-adjusted performance differentials

are 5.76%, 4.32%, and 4.08% respectively.

At first, the higher returns of the low ADIV portfolio which consists of stocks with less

diversified individual investor clientele may appear inconsistent with our previous evidence

that less sophisticated investors tend to be less diversified and consequently earn lower risk-

adjusted returns. However, we find that less diversified investors do not concentrate their

holdings among stocks that have less diversified individual investor clienteles. Their portfo-

lios also contain a significant number of stocks that have well-diversified individual investor

clienteles and these stocks do not necessarily earn higher returns.17

Overall, these results indicate that a trading strategy based on individual investors’

diversification levels earns economically significant excess returns. This provides preliminary

evidence on the influence of individual investors’ diversification decisions on stock returns.

17We thank Paul Schultz for directing our attention to this apparent puzzle.

30

VII.C The Diversification (DIV) Factor

To examine the extent to which individual investors’ diversification decisions explain the

cross-sectional variation in stock returns, we construct a diversification factor (DIV) that

represents the difference between the equal-weighted return of a portfolio of stocks with the

least diversified (lowest decile) individual investor clientele and the equal-weighted return of

a portfolio of stocks with the most diversified (highest decile) individual investor clientele.

The DIV factor represents the returns to a diversification based trading strategy where there

is no delay between the portfolio formation date and the month in which portfolio returns

can be realized

Table XIV reports the basic statistics of the DIV factor and four other risk factors

(market or RMRF, small-minus-big or SMB, high-minus-low or HML, and momentum or

UMD) and shows the correlation matrix. We find that the DIV factor has a mean monthly

return of 0.56% which is greater than the mean monthly returns of the SMB (0.17%) and

the HML (0.45%) factors but lower than the mean monthly returns of the RMRF (1.00%)

and the UMD (0.88%) factors. In addition, we find that the DIV factor has lower volatility

than the other four factors. Examining the correlations among the DIV factor and other

risk factors, we find that DIV is moderately correlated with the standard risk factors – the

contemporaneous correlations with RMRF, SMB, HML, and UMD factors are −0.147, 0.374,

−0.044, and −0.067 respectively.

VII.D Multi-factor Model Estimation

To examine the explanatory power of DIV for cross-sectional variation in stock returns, we

employ a five-factor time-series model which contains the three standard Fama-French factors

(Fama and French 1993), the momentum factor (Jegadeesh and Titman 1993, Carhart 1997),

and the diversification factor as explanatory variables. The following time-series factor model

is estimated:

Rpt−Rft = αp+β1pRMRFt+β2pSMBt+β3pHMLt+β4pUMDt+β5pDIVt+εpt t = 1, 2, . . . , T.

Here, Rpt is the rate of return on quintile portfolio p, Rft is the riskfree rate of return,

RMRFt is the market return in excess of the riskfree rate, SMBt is the difference between

the value-weighted return of a portfolio of small stocks and the value-weighted return of

a portfolio of large stocks, HMLt is the difference between the value-weighted return of a

portfolio of high B/M stocks and the value-weighted return of a portfolio of low B/M stocks,

31

UMDt is the difference between the value-weighted return of a portfolio of stocks with high

returns during months t− 12 to t− 2 and the value-weighted return of a portfolio of stocks

with low returns during months t−12 to t−2, DIVt is the return of the DIV factor in month

t, and εpt is the residual return on the portfolio.

To set the stage, we consider portfolios obtained from a uni-dimensional sort along the

size and book-to-market (B/M) dimensions. At the end of each year, we sort the entire

universe of stocks for which returns data is available from CRSP according to their market

capitalizations at the end of November. Using the NYSE break-points, we group stocks

into size quintiles where portfolio membership is not modified during the course of the year.

Portfolio 1 consists of stocks with the lowest market capitalization while portfolio 5 contains

stocks with the largest market capitalization. For each portfolio, we compute the monthly

portfolio return as an equal-weighted average of all stocks in the portfolio and construct a

monthly portfolio return time-series. A similar procedure is carried out to obtain the returns

time-series for the five B/M quintile portfolios.

Table XV presents the time-series factor model estimation results for each of the five size-

and B/M-quintile portfolios. For size quintile portfolio 1, the loading on the DIV factor is

positive (0.221) and statistically significant (t-value = 2.929). For the remaining 4 size port-

folios, DIV loadings are positive but statistically insignificant. Examining the DIV loadings

for B/M quintile portfolios, we find that the loading is positive and statistically significant

for both “growth” (B/M portfolio 1) and “value” (B/M portfolio 5) portfolios. Again, for the

remaining 3 B/M portfolios, DIV loadings are positive but statistically insignificant. These

results indicate that DIV has incremental explanatory power over the standard risk factors

for small stocks, value stocks and growth stocks.

To further explore the incremental power of the DIV factor in explaining cross-sectional

variation in stock returns, we perform double sorts using firm size and B/M variables and

form 25 size-B/M sorted portfolios. The factor model estimation results for these portfolios

are presented in Table XVI. For brevity, we only report the loadings on the DIV factor.

Panel A presents the loadings for equal-weighted size-B/M portfolios while Panel B presents

the loadings for value-weighted size-B/M portfolios. We find that the loadings on the DIV

factor are positive and significant for small-cap and value portfolios. The results hold for

both equal-weighted and value-weighted portfolios though the results are weaker for value-

weighted portfolios.

For robustness, we consider a large number of randomly formed portfolios of different

sizes and carry out the five-factor model estimation for each of these random portfolios. The

32

results are presented in Table XVII and Figure 6. We find that the DIV factor loading is

positive and statistically significant for a considerable number of portfolios. For instance,

when we consider 5,000 random portfolios with 100 stocks each, we find that the DIV loading

is significant in almost 25% of those portfolios and it is significantly positive in 94% of those

cases. In addition, the explanatory power of DIV is greater for larger portfolios. Depending

on the portfolio size, the DIV loading is significant in 20-43% of the cases, and significantly

positive in 76-100% of those cases.

Overall, the results from our asset-pricing tests suggest that the diversification decisions of

individual investors get impounded into asset prices and have the power to explain the cross-

sectional variation in stock returns for a considerable group of stocks. As discussed earlier,

these results provide a partial rationale for studies (Goyal and Santa-Clara 2003, Malkiel

and Xu 2002) that provide evidence that idiosyncratic risk is priced. Furthermore, given the

relation between diversification and investor characteristics, our results are also consistent

with theories that posit that the demographic characteristics of the investor population is

likely to influence asset prices (e.g., Bakshi and Chen (1994), Goyal (2003)).

VIII Summary and Conclusion

In this paper, using the end-of-month portfolio positions and trades of a representative

group of more than 40, 000 investors at a large discount brokerage during a six year period

(1991-96) in recent U.S. capital market history, we examine if the diversification decisions of

individual investors influence asset prices. First, we show that a vast majority of individual

investors in our sample are under-diversified. Over time, the degree of diversification among

investor portfolios has improved but these improvements result primarily from changes in

the correlation structure of the U.S. equity market.

The unexpectedly high idiosyncratic risk in investor portfolios results in a welfare loss

– the least diversified (lowest decile) group of investors earn 2.40% lower return annually

than the most diversified (highest decile) group of investors on a risk-adjusted basis. The

economic costs of under-diversification is higher for older investors and investors who trade

infrequently – within these two groups, the risk-adjusted performance differentials between

the least diversified and the most diversified investors are 3.60% and 3.12% respectively.

We find that investors’ over-confidence and their degree of sophistication are the strongest

determinants of their diversification decisions. Furthermore, we find that investors’ age,

income, preference for skewness, and preference for local stocks (familiarity bias) are other

33

key determinants of their diversification decisions. In contrast, investors’ style preferences,

the small size of their portfolios, and higher transaction costs cannot explain their systematic

levels of under-diversification.

Our asset pricing tests reveal that the systematic under-diversification of individual in-

vestors influence asset prices. A zero-cost portfolio (DIV factor) that takes a long position

in stocks with the least diversified individual investor clientele and a short position in stocks

with the most diversified individual investor clientele earns an annual excess return of 7.44%

on a risk-adjusted basis. Furthermore, this factor has power to explain the cross-sectional

variation in returns for a considerable group of stocks.

The DIV factor has incremental explanatory power over the standard risk factors for

small stocks, value stocks and growth stocks. We find that the DIV factor has considerable

explanatory power even when we consider a set of random portfolios – depending on the

portfolio size, the DIV factor loadings are significant in 20-43% of the cases, and significantly

positive in 76-100% of those cases. Taken together, the results from our asset-pricing tests

indicate that the diversification decisions of individual investors get impounded into asset

prices and have the power to explain the cross-sectional variation in stock returns for a

considerable group of stocks.

Our findings raise a number of interesting questions. Do institutional investors hold

under-diversified portfolios and if they do, do their diversification decisions influence stock

returns too? Given that we find a positive relation between the level of sophistication and

portfolio diversification, on average, we expect institutional investors to be better diversified

than individual investors. Furthermore, due to their greater financial sophistication, insti-

tutional investors may adopt a unified view of their financial portfolio. They may integrate

the various risks they face and consequently, they may not demand compensation for the

idiosyncratic risk in their equity portfolios. However, if there is considerable heterogeneity in

the diversification levels and the sophistication levels of institutional investors across stocks,

like individual investors, institutional investors may demand a compensation for the idiosyn-

cratic risk in their equity portfolios. In sum, the pricing impact of idiosyncratic risk may

be estimated more accurately by examining the combined diversification levels of individual

and institutional investors. We leave these questions for future research.

34

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Table IAggregate Level Diversification Measures: Summary Statistics

This table reports the aggregate level diversification statistics of investor portfolios for each of the six yearsin our sample-period. Panel A reports the percent of investor portfolios holding a certain number of stocks,Panel B reports the mean normalized variance for portfolios with different number of stocks, and Panel Creports the mean average correlation for portfolios with different number of stocks. The normalized variance(D1) of a portfolio is defined as, D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given portfolio

and corr is the average correlation among the stocks in the portfolio. The average correlation of investorportfolios are estimated using past 5 years of monthly returns data. The individual investor holdings dataare from a large discount brokerage house in the U.S. for the 1991-96 time-period.

Panel A: Percent of Portfolios

N(stocks) 1991 1992 1993 1994 1995 1996

1 33.02 29.71 27.88 27.06 26.75 25.50

2 20.55 19.60 18.65 17.91 17.99 17.37

3 13.51 13.59 13.14 13.03 12.50 12.01

4 8.86 9.20 9.50 9.46 9.36 9.30

5 6.11 6.55 6.87 6.87 6.70 6.59

6-10 12.36 14.49 15.56 16.26 16.81 17.40

11-15 3.28 3.93 4.80 5.18 5.30 6.13

Over 15 2.31 2.93 3.59 4.23 4.59 5.70

Panel B: Normalized Portfolio Variance

2 0.645 0.612 0.601 0.589 0.570 0.563

3 0.508 0.470 0.459 0.443 0.417 0.407

4 0.441 0.397 0.385 0.366 0.337 0.329

5 0.396 0.347 0.338 0.322 0.293 0.278

6-10 0.355 0.300 0.291 0.267 0.234 0.218

11-15 0.309 0.246 0.239 0.217 0.182 0.163

Over 15 0.291 0.224 0.220 0.192 0.151 0.130

Panel C: Average Correlation among Stocks in the Portfolio

2 0.323 0.251 0.228 0.203 0.160 0.146

3 0.312 0.250 0.231 0.203 0.157 0.143

4 0.314 0.251 0.233 0.202 0.154 0.143

5 0.314 0.246 0.231 0.202 0.158 0.139

6-10 0.325 0.259 0.245 0.210 0.161 0.139

11-15 0.329 0.260 0.249 0.214 0.165 0.140

Over 15 0.341 0.271 0.264 0.224 0.168 0.143

40

Table IIPortfolio Size and Diversification Characteristics: Summary Statistics

This table reports the diversification statistics of portfolios of different sizes. Portfolio size quintiles aredefined using the average portfolio size of investor portfolios during the entire 6-year sample period. Thereare 8, 207 investors in each portfolio size quintile. In Panel A, for each portfolio size quintile, we report thepercent of investors holding a certain number of stocks. In Panel B, for each portfolio size quintile, we reportthe means and the medians of three diversification measures, namely, number of stocks in the portfolio (D3),sum of squared portfolio weights (D2), and normalized portfolio variance (D1). D3 measure is the numberof stocks in the portfolio, D2 is the sum of squared portfolio weights, and D1 measure is the normalizedportfolio variance which is defined as, D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given

portfolio and corr is the average correlation among the stocks in the portfolio. The average correlation ofinvestor portfolios are estimated using past 5 years of monthly returns data. The individual investor holdingsdata are from a large discount brokerage house in the U.S. for the 1991-96 time-period.

Panel A: Percent of Investors

Portfolio Size Quintiles

N(stocks) All Small Q2 Q3 Q4 Large

1 6.72 12.88 7.70 5.53 4.46 3.02

1-3 34.78 49.34 44.23 35.76 27.93 16.66

3-5 19.77 14.88 21.04 24.88 22.99 15.08

5-7 9.84 4.36 7.69 10.83 14.13 12.17

7-10 7.49 1.77 3.94 7.19 10.60 13.97

10-15 4.65 0.65 1.68 3.06 5.70 12.14

Over 15 3.34 0.13 0.48 0.96 2.27 12.86

Panel B: Diversification Measures

Portfolio Size Quintiles

All Small Q2 Q3 Q4 Large

Portf Size (Thousands) < 5.83 5.83-10.56 10.56-18.25 18.25-37.04 > 37.04

Mean

Num of Stocks (D1) 4.71 2.54 3.27 4.01 4.97 8.74

Sq Portf Weights (D2) 0.54 0.70 0.60 0.54 0.49 0.40

Norm Var (D3) 0.43 0.48 0.46 0.44 0.42 0.37

Median

Num of Stocks (D1) 3.17 2.00 2.57 3.20 3.98 6.33

Sq Portf Weights (D2) 0.52 0.72 0.59 0.51 0.44 0.32

Norm Var (D3) 0.43 0.49 0.46 0.43 0.40 0.35

41

Table IIITime Variation in Portfolio Diversification

This table reports the observed and the expected aggregate level diversification measures for each of the sixyears in our sample-period. Three diversification measures are reported: (i) number of stocks in the portfolio(D3), (ii) sum of squared portfolio weights (D2), and (iii) normalized portfolio variance (D1). D3 measureis the number of stocks in the portfolio, D2 is the sum of squared portfolio weights, and D1 measure is thenormalized portfolio variance which is defined as, D1 = 1

N+

(N−1

N

)corr, where N is the number of stocks in a

given portfolio and corr is the average correlation among the stocks in the portfolio. The average correlationof investor portfolios are estimated using past 5 years of monthly returns data. The expected values of D1and average portfolio correlation are computed using a set of random portfolios – each month, 2, 500 investorportfolios are randomly chosen and each stock in each of these portfolios is replaced by a randomly chosenstock. The portfolio weights are kept fixed. Panel A reports the means for all investors while Panel B reportsthe results for only those investors who were present at the beginning of our sample period (January 1991).The individual investor holdings data are from a large discount brokerage house in the U.S. for the 1991-96time-period. The Kolmogorov-Smirnov test is used to examine the statistical significance of the differencein diversification measures. * and ** denote significance at the 5% and 1% levels respectively.

Panel A: Diversification Measures of All Investors

Div Measure 1991 1992 1993 1994 1995 1996 1996−1991 (%)

Num of Stocks (D1) 4.28 4.79 5.25 5.54 5.63 6.32 47.64∗∗

Sq Portf Weights (D2) 0.558 0.519 0.495 0.485 0.476 0.468 −16.19∗∗

Norm Var (D3) 0.470 0.411 0.390 0.363 0.325 0.309 −34.24∗∗

Avg Correl 0.319 0.251 0.234 0.201 0.151 0.137 −57.03∗∗

Expected D3 0.327 0.262 0.257 0.233 0.197 0.184 −43.78∗∗

Expected Avg Correl 0.170 0.141 0.139 0.118 0.092 0.085 −50.15∗∗

Excess D3 (%) 44.14∗∗ 56.75∗∗ 51.77∗∗ 56.21∗∗ 64.90∗∗ 67.80∗∗

Excess Avg Correl (%) 87.24∗∗ 77.60∗∗ 68.08∗∗ 70.75∗∗ 63.39∗∗ 61.69∗∗

Panel B: Diversification Measures of January 1991 Investors Only

Div Measure 1991 1992 1993 1994 1995 1996 1996−1991 (%)

Num of Stocks (D1) 3.82 4.30 4.67 4.93 4.92 5.53 44.80∗∗

Sq Portf Weights (D2) 0.601 0.568 0.550 0.538 0.523 0.524 −12.86∗∗

Norm Var (D3) 0.480 0.421 0.403 0.379 0.338 0.329 −31.41∗∗

Avg Correl 0.318 0.250 0.234 0.206 0.155 0.142 −55.23∗∗

Expected D3 0.327 0.262 0.257 0.233 0.197 0.184 −43.78∗∗

Expected Avg Correl 0.170 0.141 0.139 0.118 0.092 0.085 −50.15∗∗

Excess D3 (%) 47.32∗∗ 60.62∗∗ 57.04∗∗ 63.27∗∗ 71.38∗∗ 78.90∗∗

Excess Avg Correl (%) 86.33∗∗ 76.85∗∗ 68.58∗∗ 74.91∗∗ 67.35∗∗ 67.62∗∗

42

Table IVEconomic Costs of Under-Diversification: Summary Statistics

This table reports the performance statistics for investor groups (deciles) formed by sorting on the portfoliodiversification measure (normalized variance). The normalized variance (D1) of a portfolio is defined as,D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given portfolio and corr is the average correlation

among the stocks in the portfolio. The average correlation of investor portfolios are estimated using past 5years of monthly returns data. The mean diversification measure for the 1991-96 sample period is used todefine the investor groups. For each investor group, the following performance measures are reported: (i) themean monthly excess (relative to the market) portfolio return (Panel A) and the mean Sharpe Ratio (PanelB). For each performance measure, the following statistics are computed: mean, median, cross-sectionalstandard deviation, 25th percentile, 75th percentile, 10th percentile, and 90th percentile. The individualinvestor holdings data are from a large discount brokerage house in the U.S. for the 1991-96 time-period.

Panel A: Monthly Excess (Relative to the Market) Return Statistics

Div Decile Mean Median StdDev 25th Pctl 75th Pctl 10th Pctl 90th Pctl

Low Div −0.12 −0.18 1.74 −1.19 0.93 −2.34 2.17

D2 −0.10 −0.14 1.60 −1.09 0.90 −2.08 1.92

D3 −0.12 −0.19 1.55 −1.10 0.78 −2.03 1.90

D4 −0.10 −0.14 1.46 −1.02 0.76 −1.87 1.78

D5 −0.09 −0.17 1.34 −0.94 0.71 −1.74 1.64

D6 −0.04 −0.12 1.25 −0.87 0.69 −1.56 1.63

D7 −0.10 −0.17 1.16 −0.84 0.57 −1.52 1.48

D8 −0.01 −0.10 1.08 −0.72 0.63 −1.30 1.40

D9 0.00 −0.11 0.96 −0.63 0.54 −1.08 1.29

High Div 0.05 −0.07 0.80 −0.50 0.47 −0.83 1.13

Panel B: Sharpe Ratio Statistics

Div Decile Mean Median StdDev 25th Pctl 75th Pctl 10th Pctl 90th Pctl

Low Div 0.08 0.09 0.19 −0.03 0.21 −0.16 0.33

D2 0.09 0.09 0.17 −0.02 0.20 −0.12 0.31

D3 0.08 0.08 0.16 −0.03 0.19 −0.12 0.29

D4 0.09 0.09 0.15 −0.02 0.19 −0.11 0.28

D5 0.09 0.09 0.15 −0.01 0.19 −0.10 0.28

D6 0.10 0.10 0.14 0.00 0.20 −0.08 0.29

D7 0.10 0.10 0.14 0.01 0.19 −0.08 0.28

D8 0.11 0.12 0.13 0.02 0.21 −0.06 0.28

D9 0.13 0.12 0.13 0.04 0.21 −0.03 0.29

High Div 0.16 0.16 0.11 0.08 0.23 0.01 0.31

43

Table VEconomic Costs of Under-Diversification: Cross-Sectional Variation

This table reports the cross-sectional variation in the differential performance of highly diversified portfoliosrelative to under-diversified portfolios. Investor groups are formed by performing single sorts on age, tradingfrequency, and portfolio size variables. The portfolio turnover measure, which is the mean of purchase andsales turnover, is used as a measure of trading frequency. In addition, income and occupation codes areused to define income and occupation groups. Diversification quintile portfolios are defined using the meandiversification measures (normalized variance) of investor portfolios during the 6-year sample period. Thenormalized variance (D1) of a portfolio is defined as, D1 = 1

N +(

N−1N

)corr, where N is the number of

stocks in a given portfolio and corr is the average correlation among the stocks in the portfolio. The averagecorrelation of investor portfolios are estimated using past 5 years of monthly returns data. Four performancemeasures are reported: (i) raw realized monthly portfolio return, (ii) Sharpe Ratio, (iii) CAPM Alpha, and(iv) 4-Factor Alpha. Panel A reports the mean monthly return and the Sharpe Ratio measures while PanelB reports the CAPM and 4-Factor Alphas. The individual investor holdings data are from a large discountbrokerage house in the U.S. for the 1991-96 time-period. The Kolmogorov-Smirnov test is used to examinethe statistical significance of the difference in performance measures. * and ** denote significance at the 5%and 1% levels respectively.

Panel A: Raw Monthly Portfolio Return and Sharpe Ratio

Raw Monthly Return Sharpe Ratio

Investor Group Low Div High Div High−Low Low Div High Div High−Low

All Investors

1.18 1.36 0.18∗∗ 0.08 0.16 0.08∗∗

Age Groups

Below 45 (Younger) 1.21 1.30 0.09∗∗ 0.09 0.14 0.05∗∗

45-65 1.17 1.43 0.26∗∗ 0.08 0.16 0.08∗∗

Above 65 (Older) 1.15 1.31 0.16∗∗ 0.08 0.17 0.09∗∗

Income Groups

Below 40K (Low) 1.11 1.34 0.23∗∗ 0.07 0.15 0.08∗∗

40-75K 1.19 1.34 0.15∗∗ 0.09 0.15 0.06∗∗

Above 75K (High) 1.27 1.40 0.13∗∗ 0.09 0.16 0.07∗∗

Occupation Groups

Non-Professional 1.29 1.40 0.11∗∗ 0.09 0.16 0.07∗∗

Professional 1.18 1.40 0.22∗∗ 0.09 0.15 0.06∗∗

Retired 1.07 1.30 0.23∗∗ 0.08 0.17 0.09∗∗

Trading Frequency

Infrequent Traders 0.96 1.29 0.33∗∗ 0.07 0.17 0.10∗∗

Moderate 1.06 1.43 0.37∗∗ 0.08 0.16 0.08∗∗

Active Traders 1.60 1.45 −0.15∗∗ 0.11 0.14 0.03

44

Table V(Continued)

Economic Costs of Under-Diversification: Cross-Sectional Variation

Panel B: CAPM and 4-Factor Alphas

CAPM Alpha 4-Factor Alpha

Investor Group Low Div High Div High−Low Low Div High Div High−Low

All Investors

−0.18 −0.01 0.17∗∗ −0.67 −0.47 0.20∗∗

Age Groups

Below 45 (Younger) −0.20 −0.09 0.11∗∗ −0.53 −0.49 0.04

45-65 −0.19 0.04 0.23∗∗ −0.63 −0.50 0.13∗∗

Above 65 (Older) −0.17 −0.02 0.15∗∗ −0.74 −0.44 0.30∗∗

Income Groups

Below 40K (Low) −0.20 0.04 0.24∗∗ −0.60 −0.47 0.13∗∗

40-75K −0.21 −0.03 0.17∗∗ −0.56 −0.40 0.16∗∗

Above 75K (High) −0.07 0.01 0.08∗∗ −0.53 −0.39 0.14∗∗

Occupation Groups

Non-Professional −0.13 −0.00 0.13∗∗ −0.55 −0.46 0.09∗∗

Professional −0.19 0.07 0.26∗∗ −0.59 −0.44 0.15∗∗

Retired −0.22 −0.00 0.22∗∗ −0.75 −0.41 0.33∗∗

Trading Frequency

Infrequent Traders −0.33 −0.02 0.31∗∗ −0.65 −0.39 0.26∗∗

Moderate −0.28 0.03 0.31∗∗ −0.61 −0.48 0.13∗∗

Active Traders 0.13 −0.06 −0.19∗∗ −0.43 −0.57 −0.14∗∗

45

Table VIInvestor Demographics, Sophistication, and Portfolio Diversification

This table reports the mean and the median diversification measures of investor groups formed on the basisof age, income, occupation, and financial sophistication. Three diversification measures are reported: (i)number of stocks in the portfolio (D3), (ii) sum of squared portfolio weights (D2), and (iii) normalizedportfolio variance (D1). The D1 measure is defined as, D1 = 1

N +(

N−1N

)corr, where N is the number of

stocks in a given portfolio and corr is the average correlation among the stocks in the portfolio. The threeoccupation categories are defined as: (i) professional category, consisting of investors that hold technicalor managerial positions, (ii) non-professional category, consisting of investors who are blue-collar workers,sales and service workers, clerical workers, house-wives or students, and (iii) the retired category. In thesophistication groups, Short Sellers are investors who engaged in at least one short-sell during the sample-period, Option Traders are investors who executed at least one option trade during the sample-period, HoldsForeign investors are those who executed at least one foreign equity trade during the sample-period, andHolds MFund are investors who held mutual funds during the sample-period. Age is the age of the headof the household and income is the total annual income of the household. The individual investor holdingsdata are from a large discount brokerage house in the U.S. for the 1991-96 time-period.

Mean Median

Group D3 D2 D1 Income Age D3 D2 D1 Income Age

All Investors

4.71 0.544 0.431 89.36 50.34 3.17 0.518 0.426 87.50 48.00

Age

Below 45 3.80 0.588 0.450 94.18 38.21 2.74 0.572 0.449 87.50 38.00

45-65 4.67 0.539 0.426 95.09 53.29 3.25 0.509 0.419 87.50 52.00

Above 65 5.69 0.490 0.408 68.33 72.90 3.79 0.441 0.399 62.50 72.00

Income

Below 40K 4.02 0.535 0.383 25.91 54.19 3.10 0.518 0.378 25.00 54.00

40-75K 4.84 0.513 0.379 70.02 50.01 4.00 0.493 0.363 62.50 48.00

Above 75K 5.04 0.505 0.378 170.00 48.61 4.12 0.488 0.372 112.50 48.00

Occupation

Non-Professional 4.06 0.557 0.416 103.46 48.24 3.08 0.527 0.410 87.50 48.00

Professional 4.86 0.513 0.383 82.48 48.71 4.00 0.549 0.373 62.50 48.00

Retired 6.15 0.447 0.358 62.46 67.08 4.71 0.417 0.350 45.00 70.00

Sophistication

Short Sellers 5.86 0.509 0.404 90.10 50.52 3.78 0.481 0.396 87.50 48.00

Option Traders 5.52 0.522 0.405 90.63 48.93 3.56 0.500 0.403 87.50 46.00

Holds Foreign 6.69 0.458 0.380 89.76 51.82 4.94 0.415 0.367 62.50 50.00

Holds MFund 5.77 0.486 0.399 90.47 51.01 3.91 0.445 0.388 87.50 48.00

46

Table VIITrading Frequency and Portfolio Diversification

This table reports mean diversification measures, normalized portfolio variance or D1 (Panel B) and numberof stocks in the portfolio or D3 (Panel A), for investor groups (quintiles) defined by performing a doublesort on trading frequency and portfolio size variables. The normalized variance (D1) of a portfolio is definedas, D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given portfolio and corr is the average

correlation among the stocks in the portfolio. The average correlation of investor portfolios are estimatedusing past 5 years of monthly returns data. The portfolio turnover measure, which is the mean of purchaseand sales turnover, is used as a measure of trading frequency. The portfolio size is the average size of aninvestor portfolio during the 6-year sample period. The individual investor holdings data are from a largediscount brokerage house in the U.S. for the 1991-96 time-period. The Kolmogorov-Smirnov test is used toexamine the statistical significance of the difference in diversification measures. * and ** denote significanceat the 5% and 1% levels respectively.

Panel A: Number of Stocks (D3 Measure)

Turnover Quintiles

Portf Size Infrequent Q2 Q3 Q4 Active Active−Infrequent

Small 2.86 3.19 2.61 2.26 2.13 −0.73

Q2 3.88 3.83 3.17 2.87 2.69 −1.19∗

Q3 4.90 4.56 3.98 3.58 3.30 −1.60∗∗

Q4 5.81 5.48 4.86 4.59 4.01 −1.80∗∗

Large 10.54 9.77 8.53 7.55 6.76 −3.78∗∗

Panel B: Normalized Variance (D1 Measure)

Turnover Quintiles

Portf Size Infrequent Q2 Q3 Q4 Active Active−Infrequent

Small 0.464 0.452 0.480 0.500 0.506 0.043∗∗

Q2 0.436 0.440 0.460 0.474 0.488 0.052∗∗

Q3 0.411 0.418 0.441 0.451 0.466 0.055∗∗

Q4 0.399 0.399 0.415 0.425 0.448 0.049∗∗

Large 0.348 0.353 0.367 0.380 0.400 0.051∗∗

47

Table VIIIFamiliarity Bias and Portfolio Diversification

This table reports mean diversification measures, normalized portfolio variance or D1 (Panel B) and numberof stocks in the portfolio or D3 (Panel A), for investor groups (quintiles) defined by performing a doublesort on familiarity bias and portfolio size variables. The normalized variance (D1) of a portfolio is definedas, D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given portfolio and corr is the average

correlation among the stocks in the portfolio. The average correlation of investor portfolios are estimatedusing past 5 years of monthly returns data. The familiarity bias is the difference between the weighteddistance (

∑Ni=1 wiDi, where N is the number of stocks in the portfolio, wi is the weight of stock i in the

portfolio, and Di is the distance between an investor’s zipcode and the zipcode of a firm’s headquarters)of an actual investor portfolio from her location and the weighted distance of the market portfolio fromher location. The portfolio size is the average size of an investor portfolio during the 6-year sample period.The individual investor holdings data are from a large discount brokerage house in the U.S. for the 1991-96time-period. The Kolmogorov-Smirnov test is used to examine the statistical significance of the differencein diversification measures. * and ** denote significance at the 5% and 1% levels respectively.

Panel A: Number of Stocks (D3 Measure)

Familiarity Bias (FB) Quintiles

Portf Size Low FB Q2 Q3 Q4 High FB High−Low High−Q2

Small 2.25 2.94 2.91 2.82 1.94 −0.31 −1.00∗∗

Q2 2.85 3.95 3.81 3.38 2.25 −0.60 −1.70∗∗

Q3 3.51 4.87 4.46 4.17 2.63 −0.88∗ −2.24∗∗

Q4 4.21 6.30 5.59 4.82 3.12 −1.09∗∗ −3.18∗∗

Large 6.65 11.75 9.25 8.36 3.89 −2.76∗∗ −7.86∗∗

Panel B: Normalized Variance (D1 Measure)

Familiarity Bias (FB) Quintiles

Portf Size Low FB Q2 Q3 Q4 High FB High−Low High−Q2

Small 0.490 0.466 0.465 0.472 0.516 0.027 0.050∗∗

Q2 0.472 0.427 0.436 0.459 0.510 0.038∗ 0.083∗∗

Q3 0.449 0.404 0.422 0.434 0.500 0.051∗∗ 0.096∗∗

Q4 0.434 0.377 0.388 0.425 0.480 0.046∗ 0.104∗∗

Large 0.390 0.334 0.353 0.371 0.453 0.063∗∗ 0.120∗∗

48

Table IXSkewness Preference and Portfolio Diversification

This table reports the portfolio weights of different categories of stocks in the aggregate investor groupportfolio. Stock categories are formed on the basis of their return moments (mean, standard deviation andskewness) and investor groups are defined on the basis of their average diversification measure (normalizedvariance) during the sample-period. The normalized variance (D1) of a portfolio is defined as, D1 = 1

N+(

N−1N

)corr, where N is the number of stocks in a given portfolio and corr is the average correlation among

the stocks in the portfolio. The average correlation of investor portfolios and the return moments of stocksare estimated using past 5 years of monthly returns data. The aggregate portfolio of an investor group isconstructed at the end of each month by combining the portfolios of all investors that belong to the group.Panel A reports the portfolio weights for stock categories formed on the basis of first return moment orthe mean, Panel B reports the portfolio weights for stock categories formed on the basis of second returnmoment or the standard deviation, and Panel C reports the portfolio weights for stock categories formedon the basis of third return moment or skewness. The individual investor holdings data are from a largediscount brokerage house in the U.S. for the 1991-96 time-period. The Kolmogorov-Smirnov test is used toexamine the statistical significance of the difference in portfolio weights. * and ** denote significance at the5% and 1% levels respectively.

Panel A: Preference for MeanMean Quintile

Div Quintile Low Q2 Q3 Q4 HighLow Div 0.031 0.155 0.251 0.291 0.270Q2 0.033 0.154 0.266 0.277 0.263Q3 0.033 0.142 0.285 0.296 0.242Q4 0.025 0.141 0.307 0.318 0.207High Div 0.017 0.129 0.358 0.327 0.167

High−Low −0.014 −0.026∗ 0.107∗∗ 0.036∗ −0.103∗∗

Panel B: Preference for Standard DeviationStd Dev Quintile

Div Quintile Low Q2 Q3 Q4 HighLow Div 0.324 0.257 0.223 0.125 0.069Q2 0.307 0.249 0.238 0.152 0.046Q3 0.343 0.270 0.197 0.140 0.048Q4 0.402 0.267 0.182 0.112 0.036High Div 0.464 0.278 0.153 0.082 0.022

High−Low 0.140∗∗ 0.021∗ −0.070∗∗ −0.043∗∗ −0.047∗∗

Panel C: Preference for SkewnessSkewness Quintile

Div Quintile Low Q2 Q3 Q4 HighLow Div 0.255 0.352 0.202 0.110 0.077Q2 0.277 0.340 0.220 0.092 0.061Q3 0.279 0.349 0.213 0.098 0.057Q4 0.285 0.356 0.212 0.096 0.049High Div 0.299 0.362 0.212 0.084 0.040

High−Low 0.044∗∗ 0.011 0.010 −0.026∗ −0.037∗

49

Table XEquity Style Preference and Portfolio Diversification

This table reports the portfolio weights of different categories of stocks (or style categories) in the aggregateinvestor group portfolio. Style categories are formed at the end of each month on the basis of previous month’smarket capitalization and book-to-market (B/M) measures and investor groups are defined on the basisof their average diversification measure (normalized variance) during the sample-period. The normalizedvariance (D1) of a portfolio is defined as, D1 = 1

N +(

N−1N

)corr, where N is the number of stocks in a given

portfolio and corr is the average correlation among the stocks in the portfolio. The average correlation ofinvestor portfolios are estimated using past 5 years of monthly returns data. The aggregate portfolio ofan investor group is constructed at the end of each month by combining the portfolios of all investors thatbelong to the group. Small-cap (growth) style consists of stocks in size (B/M) deciles 1-3, mid-cap (blend)style consists of stocks in size (B/M) deciles 4-7, and large-cap (value) style consists of stocks in size (B/M)deciles 7-10. The individual investor holdings data are from a large discount brokerage house in the U.S. forthe 1991-96 time-period. The Kolmogorov-Smirnov test is used to examine the statistical significance of thedifference in portfolio weights. * and ** denote significance at the 5% and 1% levels respectively.

Size Book-to-Market

Div Decile Small-Cap Mid-Cap Large-Cap Growth Blend Value

Low Div 0.208 0.204 0.588 0.473 0.322 0.205

D2 0.241 0.223 0.536 0.501 0.289 0.209

D3 0.245 0.210 0.545 0.489 0.293 0.218

D4 0.246 0.218 0.536 0.506 0.284 0.211

D5 0.243 0.202 0.555 0.486 0.295 0.219

D6 0.236 0.207 0.557 0.485 0.295 0.219

D7 0.236 0.201 0.563 0.490 0.293 0.216

D8 0.232 0.205 0.563 0.480 0.299 0.221

D9 0.238 0.201 0.562 0.471 0.305 0.224

High Div 0.219 0.197 0.584 0.469 0.312 0.219

High−Low 0.011∗∗ −0.007∗∗ −0.004 −0.004 −0.010∗∗ 0.014∗∗

50

Table XIRegression Estimates: Determinants of Portfolio Diversification

This table reports the estimates of cross-sectional regressions where the average diversification measure (−D1,−D2, and D3) of a household are the dependent variables and a set of household characteristics are used asindependent variables. Income is the total household income and Age is the age of the head of the household.The Professional and Retired dummy variables represent the occupation categories where the professionaljob category includes investors who hold technical and managerial positions. The remaining investors belongto the non-professional category which consists of blue-collar workers, sales and service workers, and clericalworkers. Portfolio Turnover is the average of monthly buy and sell turnovers, Portfolio Size is the averagesize of the household portfolio, and Portfolio Performance is the risk-adjusted performance (Sharpe Ratio) ofthe household. Four additional dummy variables are employed as independent variables: (i) MFund Dummywhich is set to one if an investor held mutual fund in at least one month during the sample-period, (ii)Foreign Dummy which is set to one if an investor made at least one trade in a foreign asset (ADR, foreignstock or a closed-end country fund) during the sample-period, (iii) Short-Sell Dummy which is set to one ifan investor executed at least one short-sell during the sample-period, and (iv) Option Dummy which is setto one if an investor made at least one trade in an option during the sample-period. Finally, FamiliarityBias is the difference between the weighted distance (

∑Ni=1 wiDi, where N is the number of stocks in the

portfolio, wi is the weight of stock i in the portfolio, and Di is the distance between an investor’s zipcodeand the zipcode of a firm’s headquarters) of an actual investor portfolio from her location and the weighteddistance of the market portfolio from her location. Both independent and dependent variables have beenstandardized so that the coefficient estimates can be compared directly within a regression specification andalso across the three specifications. The individual investor holdings data are from a large discount brokeragehouse in the U.S. for the 1991-96 time-period.

Dependent Variable: Portfolio Diversification (−D1, −D2, D3)

D3: Nstocks −D2: −SqPortfWts −D1: −NormVar

Variable Estimate t-stat Estimate t-stat Estimate t-stat VIF

Intercept 0.019 2.19 0.040 3.74 0.021 1.91

Income 0.013 1.99 0.009 1.75 0.012 2.32 1.02

Age 0.074 6.97 0.083 8.19 0.065 6.16 1.06

Professional Dummy 0.016 2.40 0.037 2.43 0.019 2.16 1.47

Retired Dummy 0.018 1.30 0.022 1.28 0.011 0.62 1.83

Portf. Turnover −0.378 −14.14 −0.546 −14.72 −0.433 −12.55 1.10

Portf. Size 0.275 4.18 0.127 4.04 0.100 4.16 1.07

Portf. Performance 0.095 10.66 0.100 10.57 0.103 10.19 1.01

MFund Dummy 0.097 11.07 0.122 12.39 0.030 2.79 1.02

Foreign Dummy 0.184 22.01 0.225 23.62 0.208 20.39 1.08

Short-Sell Dummy 0.069 6.29 0.053 5.17 0.068 6.30 1.33

Option Dummy 0.013 2.12 0.009 1.86 0.020 2.62 1.31

Familiarity Bias −0.044 −6.31 −0.072 −6.86 −0.094 −8.07 1.01

Num. of investors 9, 246 9, 246 9, 246

Adj. R2 0.205 0.164 0.121

51

Table XIIDiversification Level of Individual Investor Clientele and Stock Characteristics

This table reports the decile scores and the factor exposures of ADIV portfolios. The average diversificationlevel (ADIV) of each stock is estimated at the end of each month using the end-of-month portfolio positionsand diversification measures (normalized variances) of investor portfolios. The normalized variance (D1) ofa portfolio is defined as, D1 = 1

N+

(N−1

N

)corr, where N is the number of stocks in a given portfolio and corr

is the average correlation among the stocks in the portfolio. The average correlation of investor portfoliosare estimated using past 5 years of monthly returns data. Using the end-of-month ADIV measures, eachmonth we rank all stocks in our sample and form ten ADIV decile portfolios. Portfolio 1 consists of stockswith the lowest ADIV measure (i.e., stocks that have the least diversified individual investor clientele) whileportfolio 10 consists of stocks with the largest ADIV measure (i.e., stocks that have the most diversifiedindividual investor clientele). At the end of each month, we also rank all CRSP stocks according to theirsize, B/M, price, and institutional ownership characteristics, and using the NYSE breakpoints, we form size,B/M, price, and IO decile portfolios. Using the prior month stock characteristic rankings, we compute asize, B/M, price, and IO decile score for each ADIV portfolio where the decile score is an equal-weightedaverage of the characteristic decile ranks of stocks that belong to the ADIV portfolio. For each of the ADIVdecile portfolios, we also compute its monthly return as an equal weighted average of returns of the stocksin that portfolio. The factor exposures of ADIV portfolios are estimated by fitting a 4-Factor model tothe monthly ADIV returns time-series for our sample-period. RMRF is the market return in excess of theriskfree rate, SMB is the difference between the value-weighted return of a portfolio of small stocks and thevalue-weighted return of a portfolio of large stocks, HML is the difference between the value-weighted returnof a portfolio of high B/M stocks and the value-weighted return of a portfolio of low B/M stocks, and UMDis the difference between the value-weighted return of a portfolio of stocks with high returns during monthst − 12 to t − 2 and the value-weighted return of a portfolio of stocks with low returns during months t − 12to t− 2. The individual investor holdings data are from a large discount brokerage house in the U.S. for the1991-96 time-period.

Decile Score Factor Exposure

ADIV Portfolio Size B/M Price InstiOwn RMRF SMB HML UMD

Low Avg Div 3.53 4.73 5.57 4.70 1.01 1.29 0.25 −0.16

D2 3.92 4.57 5.55 4.82 1.20 1.15 0.23 −0.24

D3 4.05 4.60 5.55 4.77 1.18 1.35 0.31 −0.21

D4 4.43 4.56 5.76 4.82 1.25 1.18 0.26 −0.25

D5 4.65 4.64 5.89 4.83 1.17 1.10 0.16 −0.25

D6 4.75 4.64 6.04 4.91 1.21 1.09 0.36 −0.28

D7 4.70 4.70 6.14 5.11 1.18 1.32 0.48 −0.28

D8 4.45 4.67 6.09 4.96 1.25 1.08 0.37 −0.27

D9 4.44 4.88 5.88 4.95 1.26 1.16 0.45 −0.30

High Avg Div 4.47 5.07 5.87 5.14 1.12 1.00 0.24 −0.15

52

Table XIIIPerformance of Diversification Based Trading Strategies

This table reports raw and risk-adjusted performance measures of trading strategies based on the averagediversification levels (ADIV) of stocks’ individual investor clienteles. For each stock, ADIV is estimated atthe end of each month using the end-of-month portfolio positions and diversification measures (normalizedvariance) of investor portfolios. The normalized variance (D1) of a portfolio is defined as, D1 = 1

N +(N−1

N

)corr, where N is the number of stocks in a given portfolio and corr is the average correlation among

the stocks in the portfolio. The average correlation of investor portfolios are estimated using past 5 yearsof monthly returns data. Each month we rank all stocks in our sample using the ADIV measure and formten decile portfolios. Portfolio 1 consists of stocks with the lowest ADIV measure (i.e., stocks that havethe least diversified individual investor clientele) while portfolio 10 consists of stocks with the largest ADIVmeasure (i.e., stocks that have the most diversified individual investor clientele). For each of the decileportfolios constructed at the end of month t, we compute its monthly return in month (t + k) as an equalweighted average of returns of the stocks in that portfolio. k represents the delay (in number of months)between the portfolio formation date and the return computation month (k = 1, 2, 3, 4). A monthly portfolioreturn time-series is obtained for each of the ten ADIV decile portfolios which is used to compute theraw and the risk-adjusted portfolio performance measures. Panel A reports the mean monthly return andstandard deviation of ADIV decile portfolios while Panel B reports the 4-factor alphas of these portfolios.The individual investor holdings data are from a large discount brokerage house in the U.S. for the 1991-96time-period. * and ** denote significance at the 5% and 1% levels respectively.

Panel A: Raw Performance (Monthly Return)t + 1 t + 2 t + 3 t + 4

ADIV Portfolio Mean StdDev Mean StdDev Mean StdDev Mean StdDevLow Avg Div 2.19 4.49 1.94 4.13 1.72 4.06 1.58 4.19D2 1.71 4.63 1.78 4.46 1.56 4.49 1.37 4.32D3 1.71 4.89 1.43 4.51 1.61 4.36 1.51 4.66D4 1.59 4.70 1.47 4.58 1.24 4.22 1.36 4.45D5 1.55 4.42 1.48 4.16 1.42 4.25 1.04 4.16D6 1.66 4.35 1.52 4.14 1.26 4.13 1.30 4.16D7 1.50 4.76 1.37 4.10 1.24 4.01 1.13 4.09D8 1.44 4.47 1.40 4.62 1.37 4.91 1.26 4.86D9 1.53 4.55 1.44 4.17 1.17 3.93 1.20 4.21High Avg Div 1.64 4.14 1.52 3.85 1.39 3.65 1.25 3.76

Low−High 0.56∗∗ 0.42∗∗ 0.33∗ 0.33∗

Panel B: Risk-Adjusted Performance (4-Factor Alpha)t + 1 t + 2 t + 3 t + 4

ADIV Portfolio Alpha t-stat Alpha t-stat Alpha t-stat Alpha t-statLow Avg Div 0.64 2.70 0.47 2.06 0.37 1.77 0.33 1.59D2 0.07 0.29 0.28 1.17 0.21 0.84 0.10 0.47D3 −0.01 −0.04 −0.03 −0.12 0.20 0.84 0.30 1.16D4 −0.11 −0.51 −0.15 −0.62 −0.15 −0.72 −0.05 −0.25D5 −0.01 −0.06 0.03 0.17 0.14 0.63 −0.14 −0.75D6 −0.01 −0.08 0.08 0.42 −0.19 −0.90 −0.04 −0.18D7 −0.23 −0.91 −0.17 −0.92 −0.16 −0.83 −0.14 −0.71D8 −0.28 −1.32 −0.19 −0.67 −0.05 −0.17 −0.20 −0.69D9 −0.22 −1.08 −0.09 −0.45 −0.32 −1.85 −0.13 −0.64High Avg Div 0.02 0.09 −0.01 −0.02 0.01 0.03 −0.01 −0.04

Low−High 0.62∗∗ 0.48∗∗ 0.36∗∗ 0.34∗∗

53

Table XIVCorrelations Among the Standard Risk Factors and the Diversification Factor

This table reports the correlations among the standard risk factors and the diversification factor during the1991-96 sample period. RMRF is the market return in excess of the riskfree rate, SMB is the differencebetween the value-weighted return of a portfolio of small stocks and the value-weighted return of a portfolioof large stocks, HML is the difference between the value-weighted return of a portfolio of high B/M stocksand the value-weighted return of a portfolio of low B/M stocks, and UMD is the difference between the value-weighted return of a portfolio of stocks with high returns during months t−12 to t−2 and the value-weightedreturn of a portfolio of stocks with low returns during months t−12 to t−2. The diversification factor DIV isthe difference between the equal-weighted return of a portfolio of stocks with least diversified (lowest decile)individual investors and the equal-weighted return of a portfolio of stocks with most diversified (highestdecile) individual investors. The individual investor holdings data are from a large discount brokerage housein the U.S. for the 1991-96 time-period.

Factor Mean (%) StdDev (%) RMRF SMB HML UMD

Market (RMRF) 1.00 2.86

Small-Minus-Big (SMB) 0.17 2.46 0.070

High-Minus-Low (HML) 0.45 2.42 −0.374 −0.312

Momentum (UMD) 0.88 2.32 0.206 −0.047 −0.075

Diversification (DIV) 0.56 1.84 −0.147 0.374 −0.044 −0.067

54

Table XVPricing of Size and B/M Portfolios

This table reports the factor model estimates for the 5 equal-weighted size-quintile and the 5 equal-weightedB/M-quintile portfolios. The quintile portfolios are formed at the end of each year in December using NYSEsize break-points and then held fixed throughout the following year. The following time-series factor modelis estimated:

Rpt −Rft = αp + β1pRMRFt + β2pSMBt + β3pHMLt + β4pUMDt + β5pDIVt + εpt t = 1, 2, . . . , T.

Here, Rpt is the rate of return on quintile portfolio p, Rft is the riskfree rate of return, RMRFt is the marketreturn in excess of the riskfree rate, SMBt is the difference between the value-weighted return of a portfolioof small stocks and the value-weighted return of a portfolio of large stocks, HMLt is the difference betweenthe value-weighted return of a portfolio of high B/M stocks and the value-weighted return of a portfolio oflow B/M stocks, UMDt is the difference between the value-weighted return of a portfolio of stocks with highreturns during months t− 12 to t− 2 and the value-weighted return of a portfolio of stocks with low returnsduring months t − 12 to t − 2, DIVt is the difference between the equal-weighted return of a portfolio ofstocks with least diversified (lowest decile) individual investors and the equal-weighted return of a portfolioof stocks with most diversified (highest decile) individual investors in month t, and εpt is the residual returnon the portfolio. The individual investor holdings data are from a large discount brokerage house in the U.S.for the 1991-96 time-period. The Newey-West adjusted t-statistics of the coefficient estimates are reportedin the parentheses.

Panel A: Size Quintile Portfolios

Size Portf Intercept RMRF SMB HML UMD DIV Adj. R2

Small-Cap 0.333 0.977 1.413 0.664 −0.308 0.221(1.272) (10.921) (8.406) (5.488) (−2.905) (2.929) 0.800

Q2 −0.069 1.081 1.010 0.078 −0.082 0.015(−0.778) (40.239) (36.851) (4.004) (−3.411) (0.409) 0.983

Q3 0.012 1.067 0.645 0.030 0.025 0.045(0.113) (30.417) (15.289) (0.727) (0.840) (0.916) 0.944

Q4 −0.035 1.069 0.265 0.065 −0.011 0.022(−0.389) (40.727) (7.936) (2.600) (−0.267) (0.433) 0.951

Large-Cap −0.014 1.067 −0.113 0.048 −0.053 0.026(−0.326) (46.754) (−8.112) (2.620) (−2.557) (0.984) 0.983

Panel B: B/M Quintile Portfolios

B/M Portf Intercept RMRF SMB HML UMD DIV Adj. R2

Growth −0.483 1.146 1.236 0.033 −0.246 0.239(−1.656) (13.937) (10.507) (0.380) (−3.101) (2.310) 0.878

Q2 0.157 1.003 0.988 0.258 −0.158 0.080(1.202) (21.040) (13.859) (5.265) (−3.038) (1.420) 0.907

Q3 0.416 0.913 0.850 0.333 −0.142 0.011(2.871) (16.199) (17.197) (4.857) (−2.572) (0.179) 0.908

Q4 0.383 0.881 0.875 0.523 −0.157 0.092(3.593) (23.206) (9.371) (10.316) (−3.448) (1.460) 0.905

Value 0.670 1.012 1.169 0.859 −0.251 0.259(4.094) (19.681) (6.818) (6.875) (−2.493) (3.061) 0.831

55

Table XVIDIV Factor Loadings in Size-B/M Portfolios

This table reports the DIV factor loadings for portfolios formed by performing double sorts on firm size andbook-to-market (B/M) measures. The portfolios are formed at the end of each year in December and thenheld fixed throughout the following year. The following time-series factor model is estimated:

Rpt −Rft = αp + β1pRMRFt + β2pSMBt + β3pHMLt + β4pUMDt + β5pDIVt + εpt t = 1, 2, . . . , T.

Here, Rpt is the rate of return on quintile portfolio p, Rft is the riskfree rate of return, RMRFt is the marketreturn in excess of the riskfree rate, SMBt is the difference between the value-weighted return of a portfolioof small stocks and the value-weighted return of a portfolio of large stocks, HMLt is the difference betweenthe value-weighted return of a portfolio of high B/M stocks and the value-weighted return of a portfolio oflow B/M stocks, UMDt is the difference between the value-weighted return of a portfolio of stocks with highreturns during months t− 12 to t− 2 and the value-weighted return of a portfolio of stocks with low returnsduring months t − 12 to t − 2, DIVt is the difference between the equal-weighted return of a portfolio ofstocks with least diversified (lowest decile) individual investors and the equal-weighted return of a portfolioof stocks with most diversified (highest decile) individual investors in month t, and εpt is the residual returnon the portfolio. Panel A reports the DIV factor loadings for equal-weighted size-B/M portfolios while PanelB reports the loadings for value-weighted size-B/M portfolios. The individual investor holdings data arefrom a large discount brokerage house in the U.S. for the 1991-96 time-period. The Newey-West adjustedt-statistic of the coefficient estimate is used to determine the statistical significance of the estimate. * and** denote significance at the 5% and 1% levels respectively.

Panel A: Equal-Weighted Size-B/M Portfolios

Size Quintile

B/M Quintile Small-Cap Q2 Q3 Q4 Large-Cap

Growth 0.387∗∗ 0.108 0.062 0.071 −0.004

Q2 0.197∗ −0.040 0.007 0.003 −0.006

Q3 0.072 −0.018 −0.115 −0.034 0.005

Q4 0.170∗ −0.147 0.153∗∗ −0.016 0.088∗

Value 0.287∗∗ 0.117∗∗ 0.290∗∗ 0.191∗∗ 0.286∗∗

Panel B: Value-Weighted Size-B/M Portfolios

Size Quintile

B/M Quintile Small-Cap Q2 Q3 Q4 Large-Cap

Growth 0.296∗∗ 0.047 0.038 0.078 0.055

Q2 0.121∗ −0.027 −0.004 −0.033 −0.039

Q3 0.004 −0.059 −0.158 −0.057 −0.052

Q4 0.070 −0.222 0.143∗∗ −0.023 0.055

Value 0.101∗ 0.100∗ 0.195∗∗ 0.234∗∗ 0.201∗∗

56

Table XVIIDIV Factor Loadings in Random Portfolios: Summary Statistics

This table reports the summary statistics of DIV factor loading in random portfolios. For a given portfoliosize k (k = 25, 50, 75, 100, 125, 250, and 500), 5, 000 portfolios are constructed by randomly choosing k stocksfrom the sample of stocks that traded during the 1991-96 period. An equal-weighted monthly return time-series for the 1991-96 time-period is obtained for each of these portfolios and the following time-series factormodel is estimated:

Rpt −Rft = αp + β1pRMRFt + β2pSMBt + β3pHMLt + β4pUMDt + β5pDIVt + εpt t = 1, 2, . . . , T.

Here, Rpt is the rate of return on random portfolio p, Rft is the riskfree rate of return, RMRFt is the marketreturn in excess of the riskfree rate, SMBt is the difference between the value-weighted return of a portfolioof small stocks and the value-weighted return of a portfolio of large stocks, HMLt is the difference betweenthe value-weighted return of a portfolio of high B/M stocks and the value-weighted return of a portfolio oflow B/M stocks, UMDt is the difference between the value-weighted return of a portfolio of stocks with highreturns during months t− 12 to t− 2 and the value-weighted return of a portfolio of stocks with low returnsduring months t − 12 to t − 2, DIVt is the difference between the equal-weighted return of a portfolio ofstocks with least diversified (lowest decile) individual investors and the equal-weighted return of a portfolioof stocks with most diversified (highest decile) individual investors in month t, and εpt is the residual returnon the portfolio. In the table below, column 2 reports the percent of DIV loadings that are statisticallysignificant at the 5% level, column 3 reports the percent of DIV loadings that are significantly positive,column 4 reports the percent of DIV loadings that are significantly negative, column 5 reports the mean ofthe significantly positive coefficients, and column 6 reports the mean of the significantly negative coefficients.Standard errors are reported in parentheses.

Portf Size Significant (%) Sig Pos (%) Sig Neg (%) Mean Pos Coeff Mean Neg Coeff

25 20.22 75.77 24.23 0.627 −0.523

(0.009) (0.011)

50 23.22 89.23 10.77 0.466 −0.377

(0.006) (0.009)

75 24.18 93.15 6.85 0.396 −0.333

(0.004) (0.009)

100 24.94 94.39 5.61 0.350 −0.291

(0.003) (0.009)

125 28.70 96.38 3.62 0.325 −0.258

(0.003) (0.008)

250 33.84 99.05 0.95 0.257 −0.158

(0.002) (0.010)

500 43.84 100.00 0.210

(0.001)

57

0 5 10 15 20−5

0

5

Mkt Portf

Rf

Capital Mkt Line

Exp

Mon

thly

Ret

(%

)

Time Period: 9102Investors above CML = 9.53%

0 5 10 15 20−5

0

5

Time Period: 9306Investors above CML = 10.55%

0 5 10 15 20−5

0

5

Monthly Std Dev (%)

Exp

Mon

thly

Ret

(%

)

Time Period: 9509Investors above CML = 20.15%

0 5 10 15 20−5

0

5

Monthly Std Dev (%)

Time Period: 9606Investors above CML = 13.96%

Figure 1. Investor portfolios relative to the market portfolio. This figure shows the positionsof investor portfolios relative to the market portfolio (and the Capital Market Line). Two monthlytime-periods are chosen in the first half of the sample period (February 1991 and June 1993) and twomonthly time-periods are arbitrarily chosen in the second half of the sample period (September 1995 andJune 1996). The past 5 years of monthly returns data are used to estimate the means and the standarddeviations of the market portfolio and investor portfolios. The riskfree rate corresponds to the 90-day T-Billrate. The individual investor holdings data are from a large discount brokerage house in the U.S. for the1991-96 time-period.

58

Dec 1992 Dec 1993 Dec 1994 Dec 1995 Dec 19960

5

10

15

Time (Jan 1992 − Dec 1996)

Mon

thly

Sta

ndar

d D

evia

tion

(Per

cent

)

Market (S&P 500)

Investor Portfolio: 25th Pctl

Investor Portfolio: Median

Investor Portfolio: 75th Pctl

Figure 2. Volatility of investor portfolios relative to the volatility of the market (S&P 500)portfolio. The figure shows selected volatility statistics of investor portfolios and the volatility of themarket portfolio using a 12-month rolling window. The individual investor holdings data are from a largediscount brokerage house in the U.S. for the 1991-96 time-period.

59

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Average Number of Stocks in the Portfolio

Nor

mal

ized

Var

ianc

e

Time Period: 9606

Actual Investor Portfolios

Randomly Chosen Portfolios

Figure 3. Normalized variance of investor portfolios relative to a set of randomly chosenportfolios. This figure shows the normalized variance of actual investor portfolios and 2,000 randomlyconstructed portfolios (the benchmark portfolios) during the month of June 1996. Similar results areobtained for other months during our 1991-96 sample period. The individual investor holdings data arefrom a large discount brokerage house in the U.S. for the 1991-96 time-period.

60

0 2 4 6 8 10 12 14 160.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Actual Investor Portfolios

Randomly Chosen Portfolios

Average Number of Stocks in the Portfolio

Avg

Cor

rela

tion

am

ong

Stoc

ks in

the

Por

tfol

io

Time Periods: 9101, 9601

9101

9601

Randomly Chosen Portfolios

Actual Investor Portfolios

Figure 4. Cross-sectional variation in average correlation. The figure shows the variation in averagecorrelation across portfolios with different number of stocks. The average correlations of investor portfolioscontaining k-stocks and 2,000 random portfolios with k-stocks are compared for k = 2, . . . , 15. Two monthlytime-periods are chosen, one in the first year of the sample period (January 1991) and the other in the lastyear of the sample period (January 1996). The individual investor holdings data are from a large discountbrokerage house in the U.S. for the 1991-96 time-period.

61

0 5 10 15 20−5

0

5

Mkt Portf

Rf

Monthly Std Dev (%)

Exp

Mon

thly

Ret

(%

)

Time Period: 9509Investors above CML = 16.95%

0 5 10 15 20−5

0

5Capital Mkt Line

Monthly Std Dev (%)

Exp

Mon

thly

Ret

(%

)

Time Period: 9509Investors above CML = 28.32%

Figure 5. Diversification and the position of investor portfolios relative to the market. Thisfigure shows the positions of two types of investor portfolios relative to the market portfolio (and theCapital Market Line): (a) portfolios with 1-3 stocks, (b) portfolios with 7 or more stocks. The results areshown for September 1995 but similar results are observed during other monthly time-periods. The past 5years of monthly returns data are used to estimate the means and the standard deviations of the marketportfolio and investor portfolios. The riskfree rate corresponds to the 90-day T-Bill rate. The individualinvestor holdings data are from a large discount brokerage house in the U.S. for the 1991-96 time-period.

62

0.1 0.2 0.3 0.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14Mean = 0.21, Median = 0.20, StdDev = 0.05, N = 5000

DIV Coefficient

Prop

ortio

n of

Por

tfolio

s

−0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

DIV Coefficient

Mean = 0.35, Median = 0.40, StdDev = 0.33, N = 5000

Portfolio Size = 500 Portfolio Size = 50

Figure 6. Distributions of DIV factor loadings. This figure shows the DIV factor loading distributionsfor two portfolio sizes: 50 and 500. The DIV loading is estimated for 5, 000 randomly chosen portfoliosusing a multi-factor time-series model. The factor model contains the standard risk factors (market,small-minus-big, high-minus-low, momentum) and an additional diversification factor. Only the statisticallysignificant (at the 5% level) loadings are used to obtain the DIV loading distributions. The individualinvestor holdings data are from a large discount brokerage house in the U.S. for the 1991-96 time-period.

63