Diversification Decisions of Individual Investors and Asset Prices

Transcription

1 Diversification Decisions of Individual Investors and 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 Prospect Street, New Haven, CT 06511; Phone: ; ORAlok Kumar, 239 Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556; Phone: ; We would like to thank an anonymous referee, John Campbell, Peng Chen, 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 valuable comments. In addition, we would like to thank Itamar Simonson for making the investor data available to us and Terrance Odean for answering numerous questions about the investor database. We are responsible for all remaining errors and omissions. An earlier version of the paper circulated under the title Equity Portfolio Diversification.

2 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 determinants 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 investors 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.

3 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 pricing 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 ( ) 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 underdiversification, 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 individual 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

4 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 (lowest 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 underdiversification 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 underdiversification 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 shortselling 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 further 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 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 sub-period to 0.09% (t-stat = 0.81)during the subperiod. This yields a monthly risk-adjusted performance improvement of 0.25% or 3.00% on 3

5 an annual basis. The improved diversification over time does not necessarily imply that investors portfolio 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 improvement 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 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 individual 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 influence 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 explain 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 equalweighted 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 factors the contemporaneous correlations with market, small-minus-big, high-minus-low, and momentum factors are 0.147, 0.374, 0.044, and respectively. To examine the explanatory power of DIV for cross-sectional variation in stock returns, we 4

6 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 % 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 diversification 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

7 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 inappropriately diversified. 2 We take their observation much further we compute the variancecovariance matrices of investor portfolios, examine the cross-sectional differences in diversification, 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 portfolios, 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 1 There is one notable exception. Heaton and Lucas (2000b) argue that the changing diversification levels of U.S. households was a key determinant of the stock price run-up during the internet bubble. 2 Using an older version of this dataset, Odean (1999) also provides evidence of under-diversification. 3 A partial list includes Uhler and Cragg (1971), Guiso, Japelli, and Terlizze (1996), Bertaut (1998), 6

8 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 (Kahneman 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 various 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 representative 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 and Thaler (2001), Souleles (2001), and Moskowitz and Vissing-Jorgensen (2001). 7

9 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 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 Bureau 4 (Survey of Income and Program Participation (SIPP), 1995) and the Federal Reserve 5 (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 4 Source: US Census Bureau Report, Asset Ownership of Households, The data are available at 5 The report is available at Also see Kennickell, Starr-McCluer, and Sunden (1997). 8

10 $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 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 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 different (but related)measures of diversification. The first measure (D 1 )is a normalized version of portfolio variance. The expected portfolio variance of an equal weighted portfolio with N stocks is given by: σp 2 = 1 ( ) N 1 N σ2 + cov. (1) N where σ 2 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: D 1 = NV EWP = σ2 p σ = 1 ( )( ) N 1 cov 2 N + = 1 ( ) N 1 N σ 2 N + corr. (2) N 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 6 Ken French s data library is available at 9

11 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 (D 2 )is approximately: N N D 2 = (w i w m ) 2 = (w i 1 N ) 2 wi 2 (3) i=1 i=1 N m i=1 where N is the number of securities held by the investor, N m is the number of stocks in the market portfolio, w i is the portfolio weight assigned to stock i in the investor portfolio and w m is the weight assigned to a stock in the market portfolio (w m =1/N m ). A lower value of D 2 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: D 3 = 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 D 2 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 7 The idea of decomposing portfolio variance into two parts, one representing the effect of the number of stocks (N) and the other representing the average correlation among stocks in the portfolio (corr) is proposed in Goetzmann, Li, and Rouwenhorst (2001). Also, see Elton and Gruber (1977). 10

12 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 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 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 stocks is while concentrated portfolios with only 2 stocks, on average, have a normalized variance of 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 in 1991 to in 1996 a 20% decline. However, during the same period, the normalized variance of portfolios containing more than 15 stocks has decreased from to 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 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 8 This 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. 9 Stocks with less than 2 years of monthly returns data are excluded from the analysis. 11

13 of time. The observed differences in average correlations are not statistically significant. This suggests that the reduction in portfolio variance during the 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 sampleperiod 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 portfolios 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 timeperiods 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 10 We thank John Campbell for suggesting this benchmark. 12

14 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 timeperiods 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 (25 th percentile, median, and 75 th 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 diversification 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 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 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

15 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 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 historical 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 correlation 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 portfolios decrease during the 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 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 crosssectionally 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 11 Malkiel and Xu (1997) report a similar finding by tracking the variation in correlations among industry portfolios during the time-period. They find that the mean correlation among portfolios decreases over time thereby suggesting that the risk reduction benefits of holding a diversified portfolio has increased over time. Also, see Campbell, Lettau, Malkiel, and Xu (2001). 14

16 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 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 k 12 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 diversification characteristics. For instance, during the 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 investors (Panel B)mirror the diversification improvements at an aggregate level (Panel A) though these improvements are marginally lower. 12 There is an exception. In 9601, for k = 2, the average correlation of random portfolios is higher than that of investor portfolios. 15

17 IV Economic Costs of Under-Diversification Are the economic costs of under-diversification significant or are investors rationally choosing 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 portfolioandthecmlintheµ-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 return and (ii)the Sharpe ratio. First, using the average sample-period diversification measure (D 1 )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 measure and in Panel B, we report the cross-sectional statistics of the Sharpe Ratio measure. The 16

18 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 return 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 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 10 th and the 90 th 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 crosssectional 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

19 We also find that the performance differentials between the extreme diversification categories 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 underdiversification 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 improved over time during the sample-period (see Section III.C). Does improved portfolio 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 during the and 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 subperiod but it earned a higher mean return (1.27%)with a slightly lower standard deviation (3.29%)during the 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 performance. 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

20 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 preference 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 investor 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 experience, 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