Does Individual-Stock Skewness/Coskewness Determine Portfolio Skewness?

Transcription

1 Does Individual-Stock Skewness/Coskewness Determine Portfolio Skewness? Thomas Kim School of Business Administration University of California Riverside, CA 92521, U.S.A. Telephone: Fax: November 17, 2014 Abstract Historically, in the U.S., portfolios that contain a larger number of stocks have more negatively skewed return distributions. Individual stock characteristics that have been argued to be the determinants of portfolio skewness, such as co-skewness or idiosyncratic skewness, have a marginal effect on portfolio skewness when compared to the number of stocks. We find that investors with a smaller amount of wealth may prefer the positive skewness of under-diversified portfolios over the less standard deviation of fully diversified portfolios. This result provides a rational explanation for the wide-spread under-diversification of individual investors. JEL Code: G12 Keywords: Diversification; Portfolio Skewness; Co-skewness; Idiosyncratic Skewness; Individual Investors

2 Does Individual-Stock Skewness/Coskewness Determine Portfolio Skewness? Abstract Historically, in the U.S., portfolios that contain a larger number of stocks have more negatively skewed return distributions. Individual stock characteristics that have been argued to be the determinants of portfolio skewness, such as co-skewness or idiosyncratic skewness, have a marginal effect on portfolio skewness when compared to the number of stocks. We find that investors with a smaller amount of wealth may prefer the positive skewness of under-diversified portfolios over the less standard deviation of fully diversified portfolios. This result provides a rational explanation for the wide-spread under-diversification of individual investors. JEL Code: G12 Keywords: Diversification; Portfolio Skewness; Co-skewness; Idiosyncratic Skewness; Individual Investors 1

3 1. Introduction Market portfolios have negative skewness in general (Christie, 1982; French, Schwert, and Stambaugh, 1987; Hong and Stein, 2003) and investors demand compensation for this negative skewness (Kraus and Litzenberger, 1972; Harvey and Siddique, 1999, 2000; Mitton and Vorkink, 2007; Conrad, Dittmar, and Ghysels, 2013). Recently, Albuquerque (2012) finds an interesting contrast to this stylized result. Individual stock return skewness is, in general, positive. He argues that interactions between individual stock skewness generate negative skewness in market portfolios. To better understand the mechanism that converts positive stock skewness to negative market skewness, we examine what happens in the middle by investigating the skewness of smaller portfolios. We form portfolios of different sizes from actual stock returns and track how portfolio skewness changes as portfolio size grows. In addition, we explore individual stock characteristics that are associated with the skewness of various sized portfolios. While many asset pricing studies argue a stock characteristic is priced due to its contribution to portfolio skewness, the empirical relationship between a stock characteristic and portfolio skewness has not been thoroughly tested, especially for smaller size portfolios. Kraus and Litzenberger (1972) and Harvey and Siddique (1999, 2000) argue that a stock s idiosyncratic skewness does not affect prices, while Mitton and Vorkink (2007), Brunnermeier, Gollier, and Parker (2007), Barberis and Huang (2008) find that idiosyncratic skewness may influence prices. Other asset pricing factors, such as size or book-to-market, may be related to a stock s contribution to portfolio skewness as well (Chung, Johnson, and Schill, 2006). We empirically test the link 2

4 between a stock characteristic and portfolio skewness by constructing portfolios from stocks that share the same characteristics. For example, we examine whether a portfolio formed with only low co-skewness stocks actually exhibits low portfolio skewness. Since there are few theories that provide a priori knowledge regarding the skewness of smaller sized portfolios, we measure portfolio skewness using a bootstrap method. From the CRSP database, we randomly select stocks and form certain sized portfolios. The stock selection is repeated every year with replacements, generating a series of monthly returns for the entire data period from This process is repeated 100 times to create 100 series of monthly returns for each portfolio size. This bootstrap method can reveal the true distribution of a test statistic, portfolio skewness in this case, without imposing certain prerequisites (Efron, 1979). The empirically measured portfolio skewness exhibits the following. Historically, in the U.S., portfolios with a larger number of stocks have greater negative skewness. Setting a number of stocks on the x axis and portfolio skewness on the y axis, we acquire a function shaped similar to y = 1/x. The point of the first few stocks is where the number of stocks has the largest effect. The marginal effect of additional stocks on portfolio skewness decreases as portfolio size grows, and at the point where a portfolio contains more than 60 stocks, portfolio skewness is near its minimum. Another important feature we find is that the relationship between the number of stocks and portfolio skewness is robust to individual stock characteristics. Portfolio skewness decreases by adding any type of stocks to the portfolio, including high co-skewness stocks or high idiosyncratic stocks. The relationship between the degree of diversification and portfolio 3

5 skewness is robust even at a more micro level. When we construct portfolios from stocks that are more diversified at the firm level, portfolio skewness is initially at a lower level and becomes negative more quickly as the number of stocks increases. The literature regarding corporate diversification and stock performance documents discounts on diversified firm stocks (Denis, Denis, and Yost, 2002), and our results confirm that diversified firm stocks are also less desirable in terms of skewness. Apart from the negative relationship between diversification and portfolio skewness, individual stock characteristics affect the level of portfolio skewness. Portfolios formed with only low co-skewness stocks or only low idiosyncratic skewness stocks have lower levels of portfolio skewness when compared to other portfolios of the same size. This result is consistent with the studies on co-skewness (Kraus and Litzenberger, 1972; Harvey and Siddique, 1999, 2000), as well as the research on idiosyncratic skewness (Mitton and Vorkink, 2007; Boyer, Mitton, and Vorkink, 2010; Xing, Zhang, and Zhao, 2010). However, while the studies on coskewness argue that diversification eliminates the effect of idiosyncratic skewness, we find the effect of idiosyncratic skewness persists even in a portfolio with 100 stocks, indicating that idiosyncratic skewness is not easily diversified away. Fama-French (1992) asset pricing factors are not priced as compensation for portfolio skewness. Portfolios of small stocks or high book-to-market stocks have more positive skewness than otherwise. As such, small stocks or high book-to-market stocks outperform other stocks in terms of the first moment (average return) and the third moment (skewness). 4

6 Despite the varying effects of individual stock characteristics on portfolio skewness, a multivariate regression analysis demonstrates that the number of stocks is the single most important factor relating to portfolio skewness. While a stock s co-skewness or idiosyncratic skewness is argued to be the main determinant of portfolio skewness by the literature, these two characteristics have marginal explanatory powers on portfolio skewness when compared to the number of stocks. Further, co-skewness has an insignificant effect on portfolio skewness when the effect of idiosyncratic skewness is controlled, even in the cases of fairly large 100-stock portfolios. While our empirical analysis indicates a robust relationship between the degree of diversification and portfolio skewness, this finding would be not very important if investors hold large diversified portfolios. Several models indicate that investors may choose underdiversification to obtain benefits in terms of skewness (Simkowitz and Beedles, 1978; Mitton and Vorkink, 2007; Brunnermeier et al., 2007). However, Simkowitz and Beedles (1978) model assumes the change in standard deviation is minimal when compared to the change in skewness, while Mitton and Vorkink (2007) and the Brunnermeier et al. (2007) models assume a separate type of investor who prefers Lotto type returns. We solve a relatively parsimonious model with the assumption of a log utility function and find that the choice between less standard deviation and more positive skewness depends upon the wealth level of an investor. Investors with relatively smaller amounts of wealth, such as individual investors, would hold under diversified portfolios. This implication is consistent with the literature regarding individual investment behavior that documents severe under-diversification of individual investors (Barber and Odean, 5

7 2000; Benartzi and Thaler, 2001; Polkovnichenko, 2005; Mitton and Vorkink, 2007; Goetzmann and Kumar, 2008). From the empirically measured standard deviation and skewness, we compare the wealth levels of investors who choose to under-diversify. A calculation from the empirically measured standard deviation and skewness reveals that severely under-diversified investors holding four stocks have two-thirds the wealth level when compared to well-diversified investors holding over 100 stocks. As the difference in wealth level is not very large compared to the actual wealth distribution of U.S. households, there may exist more under-diversified investors than predicted 1 by existing theories regarding portfolio skewness.p0f Our results can be summarized as follows. Portfolio skewness largely depends upon the level of diversification. Individual stock characteristics that are argued to influence portfolio skewness have marginal effect. Investors may choose the degree of diversification according to their wealth level. In the next section, we present our data and methods. Section 3 reports the empirical portfolio skewness by the number of stocks. Section 4 discusses the effect of individual stock co-skewness and idiosyncratic skewness on portfolio skewness. Section 5 tests the relationship between the Fama-French (1992) asset pricing factors and portfolio skewness. Section 6 examines the determinants of portfolio skewness using a multivariate regression. Section 7 solves a model concerning the choice of diversification by rational investors, while Section 8 provides our conclusions. 1 For example, U.S. Census Data in 2011 indicates that the fourth quintile of U.S. household wealth is more than three times larger than the third quintile (median). 6

8 2. Data and Method We acquire stock returns data from the Center for Research in Stock Prices (CRSP) database. The sample period is from , as stock returns prior to the early 1960s are not very accurate according to Fama and French (1992). We merge previous year accounting data obtained from Compustat to the stock returns data. The results reported in this paper are based on monthly returns in excess of the risk free rate. Excess returns are obtained by subtracting the risk free rates from the CRSP returns, and the monthly risk free rates are from Ken French s Data Library. We also check to see whether daily returns show different patterns from the monthly returns and we find qualitatively similar results. We form portfolios with different numbers of stocks. We construct portfolios with the 2 following number of stocks: 4, 9, 16, 25, 36, 49, 64, 81, and 100.P1F P At the end of each calendar year, we randomly select stocks to form a portfolio and track the portfolio return for the next entire calendar year. The composition of a portfolio changes every year as a result. The yearly selection process is done every year in the sample, creating a series of monthly returns for the entire sample period ( ). This process is repeated 100 times, generating 100 time-series returns for each number of stocks. The process yields approximately (100 iterations x 9 different number of stocks x 46 years x 12 months = 496,800) monthly portfolio returns. 2 This design is to make it easy to form portfolios by selecting stocks from different ranks. For example, a portfolio immune to the size effect can be formed by picking the same number of stocks from each size quartile. We examine the validity of random selection by forming portfolios that are immune to size or book-to-market effects. We find the skewness pattern of the immunized portfolio is no different from that of the portfolios constructed by totally random draws. 7

9 From the value-weighted or equal-weighted portfolio returns, portfolio statistics, such as standard deviation and skewness, are calculated. Portfolios betas are also estimated to examine the co-movements between a portfolio and the market. The beta of a portfolio is obtained by regressing the entire time-series return of the portfolio with the market return. Firm-level diversification is calculated from the Historical Segments Data in Compustat. The segments data begins in 1976, so relevant analyses are based on a shorter time series of stock returns from There are four categories of segments in the database: business, geographic, operation, or state. For each category, we calculate the Herfindahl index of sales in each segment. A smaller Herfindahl index indicates greater diversification and many of the firms have an index value of one, implying no diversification. Since there are various avenues of diversification, we define a firm s diversification level as the minimum of four Herfindahl indices from each category. Fama-French asset pricing factors, size and book-to-market, are calculated using the method in Fama and French (1992). Stocks are ranked into size or book-to-market quintiles every calendar year. 3. Number of Stocks and Portfolio Skewness Our first analysis is to form portfolios with different numbers of stocks without imposing restrictions. The mean, standard deviation, and skewness of the returns are calculated from the whole time-series returns of the formed portfolios. As there are 100 iterations of the time-series returns, the average of 100 portfolio statistics is reported in Table 1. The standard errors of the 8

10 averages are in parentheses. Columns 5 and 6 report the elasticity of the standard deviation or skewness to the number of stocks. The elasticity is calculated by dividing the percentage change of a portfolio statistic with the percentage change in the number of stocks. Column 7 provides the portfolio betas. Insert Table 1 about here. The first output that is immediately noticeable is that portfolio skewness becomes negative as the number of stocks increases. In other words, including more stocks in a portfolio makes the portfolio less desirable to investors in terms of skewness. Investors who value skewness face a tradeoff as additional diversification makes skewness worse, while standard deviation makes it better. Other than the standard deviation and skewness, the number of stocks has little effect on other portfolio properties. Column 2 in Table 1 indicates that portfolio mean returns are similar across different numbers of stocks. In Column 7, portfolio betas do not vary much either, 3 indicating little material difference in the co-movements with the market.p2f Insert Figure 1 about here. 3 In the brokerage account sample of Mitton and Vorkink (2007), more diversified portfolios have lower Sharpe Ratios. These portfolios are self-selected by the account holders, while our analysis uses random selections. 9

11 Figure 1 plots the standard deviation and skewness by the number of stocks. The figure demonstrates that the number of stocks does not have the same effect on the two moments. Both of the graphs have a shape similar to an inverse function, such as y = 1/x, but their slopes are significantly different. Most of the reduction in portfolio standard deviation occurs until the number of stocks reaches 30, and there is minimal change beyond that point. In contrast, the number of stocks has a greater prolonged effect on portfolio skewness. The largest reduction in skewness takes place in the first few stocks, but portfolio skewness continues to decrease until the number of stocks reaches around 60. Beyond 60, there is little change in skewness suggesting that portfolio skewness is near its minimum. To examine whether the relationship between the number of stocks and skewness is a time-specific phenomenon, we report the relationship between the number of stocks and portfolio skewness in different time periods. We cut our sample by decades and examine whether our previous results are time specific. From this point, we report only the statistics from the equally-weighted returns for visual convenience. Equally-weighted returns and value-weighted returns yield qualitatively similar results as seen in Table 1 and Figure 1. Insert Table 2 and Figure 2 about here. In Table 2 and Figure 2, we observe downward sloping curves in all of the decades. Thus, the negative relationship between the number of stocks and portfolio skewness is not a time specific phenomenon. 10

12 If diversification is such a powerful factor on portfolio skewness, similar results would be noticeable at a more micro level. A firm can be thought of as a portfolio holding multiple assets, and a firm that holds diversified assets may exhibit return skewness similar to that of a 4 diversified portfolio of stocks.p3f P To test this conjecture, we rank firm level diversification every year into quintiles (from Rank 1 to Rank 5) and form portfolios from the most diversified firms (Rank 5) and the least diversified firms (Rank 1). Insert Table 3 and Figure 3 about here. Table 3 and Figure 3 report portfolio skewness of the most diversified firms and the least diversified firms. Consistent with our logic, more diversified firms generate lower portfolio skewness. The portfolio of diversified firms becomes deeply negative as portfolio size increases, while the portfolio of non-diversified firms always reports positive skewness, albeit decreasing by the number of stocks. This result confirms that the degree of diversification is a main determinant of portfolio skewness. 4. Does Individual Stock Skewness Matter?: Co-skewness and Idiosyncratic Skewness Cross sectional differences in stock returns are often explained by a stock s contribution to portfolio skewness (Kraus and Litzenberger, 1972; Harvey and Siddique, 1999, 2000; Mitton and Vorkink, 2007). However, there is no consensus regarding which stock characteristic is 4 See Chung, Johnson, and Kim (2014) for further development of this argument. 11

13 linked to portfolio skewness. One strand of studies argues that only co-skewness of a stock with the market matters, as investors are ultimately holding market portfolios and idiosyncratic skewness is diversified away (Kraus and Litzenberger, 1972; Harvey and Siddique, 1999, 2000). Alternatively, investors valuing skewness may hold less diversified portfolios than market portfolios, and idiosyncratic skewness may affect these smaller portfolios (Mitton and Vorkink, 2007; Boyer et al., 2010). We empirically examine the effect of co-skewness and idiosyncratic skewness on portfolio skewness. Stocks are ranked into quintiles (Ranks 1-5) every year by their co-skewness with market returns and idiosyncratic skewness using the past five years of returns. Portfolios are formed from the lowest ranked stocks only or from the highest ranked stocks only. Table 4 and Figure 4 present the portfolio skewness of the highest vs. the lowest co-skewness stocks. Insert Table 4 and Figure 4 about here. Co-skewness affects the level of portfolio skewness, as a portfolio of high co-skewness stocks exhibits high portfolio skewness. However, the effect of diversification does not change with the stock characteristics. Additional diversification decreases portfolio skewness even if the portfolios are formed with only high co-skewness stocks. Also, co-skewness has a marginal effect on portfolio skewness when compared to diversification. The portfolio skewness of high co-skewness stocks can be equally achieved from low co-skewness stocks by simply having a 12

14 fewer number of stocks in a portfolio. In a later section, we will compare the relative importance of these factors on portfolio skewness. Table 5 and Figure 5 contrast the cases of the highest idiosyncratic skewness stocks and the lowest. Insert Table 5 and Figure 5 about here. We find that idiosyncratic skewness is also associated with the level of portfolio skewness. The relationship between diversification and portfolio skewness is robust to idiosyncratic skewness as well. Note that the effect of idiosyncratic skewness is not completely diversified away even when a portfolio contains 100 stocks. A 100-stock portfolio with high idiosyncratic skewness stocks has significantly higher portfolio skewness than the other 100- stock portfolio. If the effect of idiosyncratic skewness is to be diversified, as argued by the papers emphasizing co-skewness, a 100-stock portfolio of high idiosyncratic skewness stocks should have a similar skewness to a low idiosyncratic skewness portfolio. One may think that a 100-stock portfolio is not large enough to obtain the benefit of diversification, but recall that in our random sampling analysis in Table 1, portfolio skewness does not decrease much when a portfolio has more than 60 stocks. 13

15 5. Asset Pricing Factors as Compensation for Portfolio Skewness Chung et al. (2006) attempts to explain the Fama-French (1992) asset pricing factors as compensation for the risk of higher moments. In addition, a factor, like size, may be mechanically related to skewness as small firms may experience large negative shocks more frequently. In this section, we examine whether Fama-French (1992) asset pricing factors are associated with portfolio skewness. We rank firms in quintiles by their previous year s market value or book-to-market value. Table 6 and Figure 6 compare the skewness of the smallest stock portfolios with the skewness of the largest stock portfolios. Insert Table 6 and Figure 6 about here. We find a downward sloping pattern in both of the portfolios indicating that the effect of diversification persists. Meanwhile, the level of portfolio skewness is higher for a small stock portfolio suggesting that small stock premiums cannot be explained as compensation for the possibility of extremely low returns. A portfolio of small stocks is less likely to experience extremely low returns as the return is positively skewed. This result may seem puzzling to the notion that small firms should be experiencing more negative shocks (negative skewness), but it seems that small firms have more positive shocks (positive skewness) that offset negative shocks (negative skewness). 14

16 A similar pattern can be found regarding the book-to-market factor. Table 7 and Figure 7 report the skewness of portfolios created from the highest and the lowest book-to-market quintiles. Insert Table 7 and Figure 7 about here. Again, the book-to-market factor does not change the trend of downward sloping skewness. High book-to-market stocks have more positive portfolio skewness, in general, indicating that book-to-market premiums are not compensation for negative skewness as well. Stocks with size premiums or book-to-market premiums are better in terms of the first moment (average return) and the third moment (skewness of return). 6. A Test on the Determinant of Portfolio Skewness Thus far, we find that portfolio skewness is associated with various stock characteristics. In this section, we investigate which characteristic has the most significant effect on portfolio skewness using a multivariate regression approach. We re-examine the returns of the randomly formed portfolios used in Table 1 for this purpose. Portfolio skewness is calculated every calendar year from the monthly returns of the portfolio, and the corresponding stock characteristics are the averages from the stocks included in the portfolio. Our estimation equation is: 15

17 Portfolio Skewness ii,tt = aa + bb1 NNNNNNNNNNNN oooo SSSSSSSSSSSS ıııı aa PPPPPPPPPPPPPPPPPP ıı,tt +bb2 CCCCCCCCCCCCCCCCCCCC ıı,tt + bb3 IIIIIIIIIIIIIIIIIIIIIIIIII ssssssssssssssss ıı,tt +bb4 MMMMMMMMMMMM VVVVVVVVVV ıı,tt + bb5 BBBBBBBB tttt MMMMMMMMMMMM VVVVVVVVVV ıı,tt + εε ii,tt (2) where the portfolio skewness of a portfolio i in year t is regressed by the number of stocks and the average of co-skewness, idiosyncratic skewness, market value, and the book-to-market value acquired from the stocks in a portfolio. The estimation method is a Generalized Method of Moments (GMM) that corrects for heteroscedasticity and autocorrelation in error terms. Insert Table 8 about here. Table 8 reports the results. Given that we have over 31,000 yearly observations of portfolio skewness, it is safe to assume that a t-stat less than three indicates insignificant 5 explanatory power.p4f P The number of stocks has the most significant impact on portfolio skewness (t-stat: ), trailed by idiosyncratic skewness (t-stat: 8.51) and firm size (t-stat: 7.23). Despite the significance of the t-statistics, the signs of idiosyncratic skewness and firm size are the opposite of the univariate results in Tables 5 and 6. The flipping signs indicate that the effect of these characteristics is conditional upon other factors or that the effect is marginal. Surprisingly, co-skewness has an insignificant coefficient, even when controlling for only idiosyncratic 5 See Kim and Stoll (2014) for a discussion regarding the relationship between the number of observations and t- statistics. 16

18 skewness in Model 3. The results indicate that the number of stocks in a portfolio is the most important factor on portfolio skewness and other stock characteristics have secondary effects at most. 7. Investor Utility and Choice between Standard Deviation and Skewness We find that the most powerful method to acquire better portfolio skewness is to underdiversify. This under-diversification leads to poorer standard deviation. Thus, portfolio skewness matters when an investor is willing to exchange a better skewness for a poorer standard deviation. In this section, we investigate whether a rational investor would make such an exchange. Arrow (1971) argues that a utility function of a rational investor should have: (a) positive marginal utility for wealth, (b) decreasing marginal utility for wealth, and (c) non-increasing absolute risk aversion. Property (b) is equivalent to a preference for less standard deviation, 6 while Property (c) is equivalent to a preference for more positive skewness.p5f P A form of utility function that satisfies these properties is the power utility function. A typical power utility function has the following form: U(w) = ww1 γγ 1 1 γγ (3) function. where γ is a constant. If γ approaches one, the power utility function becomes a log utility 6 See Arditti (1967), Tsiang (1972), or Kraus and Litzenberger (1972). 17

19 wr1r over if is as is Now, investors choose one portfolio over the other if: EE[UU(ww 1 )] > EE[UU(ww 2 )] (4) where wr1r the series of wealth amounts resulting from Portfolio 1, and wr2r of wealth amounts resulting from Portfolio 2. We can think of wr1r the series the outcome of a less diversified portfolio and w2 as the outcome of a more diversified portfolio. Investors will choose wr2r their expected utility is greater. Tsiang (1972) finds that expected utility from a random variable w can be expanded by a Taylor series expansion and be written as: E[U(w)] = UU(ww ) + UU (ww ) mm UU (ww ) mm 3 + (5) 3! where mr2r, mr3r,, are the second, third, and higher central moments of wealth distribution. 7 The omitted portion is the higher polynomials that have very little influence on expected utility.p6f Then Equation (5) can be converted into a comparison between the moments of two distributions as: EE[UU(ww 1 )] > EE[UU(ww 2 )] if, 7 The Taylor series expansion indicates that the denominator of n th moment is an n factorial (n!). Thus, higher moments have larger denominators that discount their effects on overall utility. 18

20 UU(ww 1 ) + UU (ww 1 ) mm 1,2 + 2 UU (ww 1 ) mm 1,3 > UU(ww 3! 2 ) + UU (ww 2 ) mm 2,2 + 2 UU (ww 2 ) mm 2,3 3! (6) We can safely assume that the two distributions have the same mean from the intuition that the degree of diversification would have little effect on the portfolio return and from our empirical results in Table 1. Thus, only the risk from the higher moments determines the investors choice. UU (ww 1 ) mm 1,2 + 2 UU (ww 1 ) mm 1,3 > UU (ww 3! 2 ) mm 2,2 + 2 UU (ww 2 ) mm 2,3 3! (7) We simplify the equation by plugging in the log utility function, which is a form of a power utility function. 1 mm 1,2 + 2 mm 1,3 > 1 mm 2,2 + 2 mm 2,3 (ww 1 ) 2 2 (ww 1 ) 3 3! (ww 2 ) 2 2 (ww 2 ) 3 3! (8) Equation (8) can be rearranged as: mm 2,2 mm 1,2 > 2 3ww (mm 2,3 mm 1,3 ) (9) 19

21 from from Thus, the wealth series wr1r a less diversified portfolio is preferred to wr2r a more diversified portfolio if the difference in their second moments (m 2,2 m 1,2 ) is larger than twothirds of the difference in their third moments (m 2,3 m 1,3 ) divided by average wealth level. Equation (9) demonstrates that the difference in portfolio skewness may dominate the difference in portfolio standard deviation. In other words, rational investors may choose more positive skewness over less standard deviation. As we see empirically, increasing the number of stocks in portfolios creates a tradeoff between standard deviation and skewness, and investors may stop increasing the number of stocks at some point where the marginal gain in terms of standard deviation is not as desirable as marginal loss in terms of skewness. The selection between standard deviation and skewness depends upon the average level of wealth ww. The difference in skewness (m 2,3 m 1,3 ) becomes much less important compared to the difference in standard deviation (m 2,2 m 1,2 ) as wealth level ww gets larger. Investors who have a greater amount of wealth, such as institutional investors, may put less weight on portfolio skewness and hold more diversified portfolios. In contrast, individual investors, who typically have a smaller amount of wealth, may not hold too many stocks in their portfolios as portfolio skewness is more important to them. Intuitively, investors with a smaller amount of wealth would be more averse to large negative shocks (negative skewed returns), as small wealth investors may be ruined by such negative shocks. In a similar vein, Wagner (2011) argues that investors may choose under-diversification when there is a greater risk of liquidation. 20

22 Equation (9) can be modified to calculate the wealth level associated with the degree of portfolio diversification. First, we set the utility level from the two portfolios the same in Equation (10): mm 2,2 mm 1,2 = 2 3ww (mm 2,3 mm 1,3 ) (10) Then, the empirical standard deviation we acquired in Table 1 is plugged into Equation (10). We acquire a wealth level that makes an investor indifferent between a less diversified portfolio and a next-level more diversified portfolio. This wealth level is a threshold where a wealthier investor prefers more diversified portfolios and a less wealthy investor prefers underdiversified portfolios. The absolute wealth level does not have much meaning in this analysis, 8 but we can compare the relative size of each threshold.p7f Insert Table 9 about here. Table 9 reports the threshold. The threshold is not monotonically increasing, but less diversified portfolios have thresholds approximately two-thirds of those for more diversified portfolios. The difference in threshold is not very large, especially when compared to the overall wealth distribution of U.S. households. For example, U.S. Census Data in 2011 indicates that the 8 The absolute level is difficult to interpret as one cannot find the dollar value of wealth corresponding to a certain degree in portfolio statistics. 21

23 fourth quintile of U.S. household wealth is more than three times larger than the third quintile (median). Table 9 demonstrates that an investor with a wealth level around 30 will hold a large portfolio containing 100 stocks, while an investor with a 33% lower wealth level (20) will hold a four-stock portfolio. The wealth level does not have to be fractions of others to make an investor under-diversified, indicating that under-diversification can be a quite popular phenomenon. Our analysis is consistent with the empirical findings regarding individual investor behavior. From data provided by a large U.S. discount brokerage house, Barber and Odean (2000), Mitton and Vorkink (2007), and Goetzmann and Kumar (2008) find typical individual investors hold approximately four stocks in their portfolios. In addition, Polkovnichenko (2005) provides evidence of under-diversification among U.S. households. Benartzi and Thaler (2001) note under-diversification in retirement and pension accounts. While many of these studies attempt to explain this under-diversification in the context of behavioral bias, our analysis suggests that the under-diversification of many individual investors is a rational choice, and the degree of diversification can vary by relatively small differences in the average wealth level. 8. Conclusion We analyze the return skewness of smaller portfolios. We find that historically, in the U.S., portfolios with a larger number of stocks have more negative skewness. The negative relationship between the number of stocks and portfolio skewness is robust to stock characteristics such as co-skewness, idiosyncratic skewness, size, or book-to-market. The variables that are traditionally rendered as a main determinant of portfolio skewness, such as co- 22

24 skewness, have marginal or no correlation with the empirical portfolio skewness as compared to the number of stocks in a portfolio. This phenomenon demonstrates that diversification creates a tradeoff instead of a onesided benefit. More diversified portfolios are better in terms of standard deviation, but in worse in terms of skewness. A marginal change in standard deviation by additional diversification may be small compared to a marginal change in skewness, indicating that the benefits of further diversification may be offset by the worsening in skewness. Using a log utility function, we demonstrate that some investors, who have a relatively smaller amount of wealth, may choose the more positive skewness of under-diversified portfolios rather than the less standard deviation of more diversified portfolios. Our finding is consistent with the empirically observed under-diversification of individual investors. The relationship between diversification and portfolio risk profiles warrants further studies on this topic. Other than Albuquerque (2012), there are few theories that explain the contrast between positive individual stock skewness and negative portfolio skewness. Our findings indicate that the empirical mechanism behind portfolio skewness is more complex than assumed by many of the existing studies. Further, the effect of higher moments can sometimes be more important than standard deviation, which is a central variable in asset pricing, and full diversification may be avoided by a rational investor more often than expected. 23

25 References Albuquerque, R., 2012, Skewness in stock returns: Reconciling the evidence on firm versus aggregate returns, Review of Financial Studies 25, Arditti, F., 1967, Risk and the required return on equity, Journal of Finance 22, Arrow, K., 1971, Essays in the theory of risk-bearing, Chicago, IL: Markham Publishing Co. Barber, B.M. and Odean, T., 2000, Trading is hazardous to your wealth: The common stock investment performance of individual investors, Journal of Finance 55, Barberis, N. and Huang, M., 2008, Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review 98, Benartzi, S. and Thaler, R.H., 2001, Naive diversification strategies in retirement saving plans, American Economic Review 91, Boyer, B., Mitton, T., and Vorkink, K., 2010, Expected idiosyncratic skewness, Review of Financial Studies 23, Brunnermeier, M.K., Gollier, G., and Parker, J., 2007, Optimal beliefs, asset prices and the preferences for skewed returns, American Economic Review 97, Christie, A.A., 1982, The stochastic behavior of common stock variances: Value, leverage, and interest rate effects, Journal of Financial Economics 10,

26 Chung, P.Y., Johnson, H., Kim, T., 2014, Asset returns, asymmetric correlation, skewness, and suppressor variables, University of California Working Paper. Chung, P.Y., Johnson, H., Schill, M.J., 2006, Asset pricing when returns are nonnormal: Fama- French factors versus higher-order systematic comoments, Journal of Business 79, Conrad, J., R.F. Dittmar, and E. Ghysels, 2013, Ex ante skewness and expected stock returns, Journal of Finance 68, Denis, D.J., D.K. Denis, and K. Yost, 2002, Global diversification, industrial diversification, and firm value, Journal of Finance 57, Efron, B., 1979, Bootstrap methods: Another look at the jackknife, The Annals of Statistics 7, Fama, E. and French, K., 1992, The cross-section of expected stock returns, Journal of Finance 47, French, K.R., Schwert, G.W., and Stambaugh, R.F., 1987, Expected stock returns and volatility, Journal of Financial Economics 19, Goetzmann W.N. and Kumar A., 2008, Equity portfolio diversification, Review of Finance 12, Harvey, C.R. and Siddique, A., 1999, Autoregressive conditional skewness, Journal of Financial and Quantitative Analysis 34, Harvey C.R. and Siddique, A., 2000, Conditional skewness in asset pricing tests, Journal of Finance 55,

27 Hong, H. and Stein, J.C., 2003, Differences of opinion, short-sales constraints, and market crashes, Review of Financial Studies 16, Kim, T. and Stoll, H.R., 2014, Are trading imbalances indicative of private information? Journal of Financial Markets 20, Kraus, A. and Litzenberger R.H., 1972, Skewness preference and the valuation of risk assets, Journal of Finance 31, Mitton, T. and Vorkink K., 2007, Equilibrium underdiversification and the preference for skewness, Review of Financial Studies 20, Polkovnichenko, V., 2005, Household portfolio diversification: A case for rank-dependent preferences, Review of Financial Studies 18, Simkowitz, M.A. and Beedles, W.A., 1978, Diversification in a three moment world, Journal of Financial and Quantitative Analysis 13, Tsiang, S., 1972, The rationale of the mean-standard deviation analysis, skewness preference, and the demand for money, American Economic Review 62, Wagner, W., 2011, Systemic liquidation risk and the diversity-diversification trade-off, Journal of Finance 66, Xing, Y., Zhang, X. and Zhao, R., 2010, What does the individual option volatility smirk tell us about future equity returns? Journal of Financial and Quantitative Analysis 45,

28 Table 1. Portfolio Skewness by Number of Stocks We construct portfolios from the CRSP database by randomly selecting stocks at the end of each calendar year. The return of the formed portfolio is tracked for the next calendar year, creating monthly equal-weighted or value-weighted portfolio returns. The yearly selection process is done for the entire sample period from The whole process is repeated 100 times with replacements, yielding 100 time-series observations for each number of stocks category. Accordingly, the reported statistics in this table are the averages from the 100 time-series observations, except for the elasticity, which is calculated from the averages. The standard errors of the averages are in parentheses. Portfolio mean, standard deviation, and skewness are calculated from the whole time-series returns of each portfolio and are reported in Columns 2, 3, and 4, respectively. Columns 5 and 6 provide the elasticity of standard deviation or skewness to the number of stocks. The elasticity is calculated by dividing the percentage change of a portfolio statistic with the percentage change in number of stocks. Column 7 reports portfolio betas, calculated by regressing a portfolio s whole time-series return with the market return. Equal-weighted returns Number of Stocks Average Return 0.95% (0.03%) 0.97% (0.02%) 0.95% (0.02%) 0.98% (0.01%) 0.98% (0.01%) 0.96% (0.01%) 0.97% (0.01%) 0.97% (0.01%) 0.97% (0.01%) Std. Dev (0.0009) 0.08 (0.0004) 0.07 (0.0003) 0.07 (0.0002) 0.07 (0.0002) 0.06 (0.0001) 0.06 (0.0002) 0.06 (0.0001) 0.06 (0.0001) Skewness 1.26 (0.14) 0.55 (0.05) Elasticity: Std. Dev. Elasticity: Skewness Beta 1.13 (0.009) 1.13 (0.006) 1.12 (0.004) 1.13 (0.004) 1.13 (0.003) 1.12 (0.003) 1.13 (0.002) 1.13 (0.002) 1.13 (0.002) 27

29 Value-weighted Returns Number of Stocks Average Return Std. Dev. Skewness Elasticity: Std. Dev. Elasticity: Skewness β % (0.02%) 0.83% (0.02%) 0.80% (0.02%) 0.82% (0.01%) 0.83% (0.01%) 0.82% (0.01%) 0.82% (0.01%) 0.82% (0.01%) 0.82% (0.01%) 0.09 (0.0007) 0.07 (0.0003) 0.07 (0.0002) 0.06 (0.0002) 0.06 (0.0001) 0.06 (0.0001) 0.06 (0.0001) 0.06 (0.0001) 0.06 (0.0001) 0.69 (0.06) 0.16 (0.08) (0.04) (0.010) (0.008) 1.13 (0.005) 1.13 (0.004) 1.13 (0.003) 1.13 (0.003) 1.13 (0.003) 1.13 (0.002) 1.13 (0.002) 1.13 (0.002) 28

30 Table 2. Portfolio Skewness by Number of Stocks: By Decade We construct portfolios from the CRSP database by randomly selecting stocks at the end of each calendar year. The return of the formed portfolio is tracked for the next calendar year, creating monthly equal-weighted or value-weighted portfolio returns. The yearly selection process is completed for the entire sample period from The process is repeated 100 times with replacements, yielding 100 time-series observations for each number of stocks category. Portfolio standard deviation and skewness are calculated by decade, 1960s, 1970s, 1980s, 1990s, and 2000s, from equal-weighted portfolio returns. The average of the portfolio statistics is reported by decade and number of stocks in a portfolio, and the standard errors of the averages are in parentheses. Number of Stocks Portfolio Skewness 1960s 1970s 1980s 1990s 2000s 0.31 (0.05) 0.20 (0.04) (0.06) 0.56 (0.04) 0.63 (0.04) (0.11) (0.05) (0.05) (0.04) (0.13) 0.85 (0.13) 0.43 (0.10) 0.19 (0.06) (0.04) (0.05) (0.11) 0.55 (0.08) 0.39 (0.08) 0.22 (0.04) 0.17 (0.04)

31 Table 3. Firm-level Diversification and Portfolio Skewness The Historical Segments Data in Compustat reports four categories of segments (business, operating, geographic, and state) and sales in each segment. We calculate the Herfindahl index of sales by segment in each category, and use the minimum of the Herfindahl indices as the measure of firm-level diversification. Stocks are ranked into quintiles every year by their firmlevel diversification. We construct portfolios from the CRSP database by randomly selecting stocks from the most diversified rank (Rank 5) only or from the least diversified rank (Rank 1) only. Other than this restriction to the sample, the portfolio formation process is identical to the method used in Table 1. We report the average of portfolio skewness by firm-level diversification and number of stocks in a portfolio. Standard errors of the averages are reported in parentheses. Column 4 reports the difference in skewness and the standard error of the difference. Number of Stocks 4 Skewness of the Most Diversified Stocks Portfolio 1.06 (0.17) Skewness of the Least Diversified Stocks Portfolio 1.84 (0.19) Difference in Skewness (Standard Errors) (0.25) (0.08) 1.66 (0.18) (0.20) (0.14) (0.14) (0.12) (0.12) (0.11) (0.11) (0.10) (0.10) (0.07) (0.07) (0.06) (0.06) (0.05) (0.06) 30

32 Table 4. Stock Co-skewness and Portfolio Skewness We calculate the co-skewness of a stock with the monthly market return in Ken French s Data Library using the past five years of monthly returns. Stocks are ranked into quintiles every year by the measured co-skewness. We construct portfolios from the CRSP database by randomly selecting stocks from the highest co-skewness rank (Rank 5) only or from the lowest coskewness rank (Rank 1) only. Other than this restriction to the sample, the portfolio formation process is identical to the method used in Table 1. We report the average of portfolio skewness by co-skewness and number of stocks in a portfolio. Standard errors of the averages are reported in parentheses. Column 4 reports the difference in skewness and the standard error of the difference. Number of Stocks 4 Skewness of the Highest Co-skewness Stocks Portfolio 2.30 (0.23) Skewness of the Lowest Co-skewness Stocks Portfolio 1.69 (0.13) Difference in Portfolio Skewness (Standard Errors) 0.61 (0.26) (0.17) 1.33 (0.16) 0.53 (0.23) (0.15) 0.93 (0.09) 0.56 (0.17) (0.18) 0.81 (0.09) 0.51 (0.20) (0.12) 0.60 (0.10) 0.29 (0.16) (0.07) 0.42 (0.10) 0.39 (0.12) (0.08) 0.25 (0.10) 0.47 (0.13) (0.11) 0.50 (0.11) (0.04) (0.09) 0.60 (0.10) 31

33 Table 5. Stock Idiosyncratic Skewness and Portfolio Skewness We calculate the idiosyncratic skewness of a stock following the method of Mitton and Vorkink (2007). Stocks are ranked into quintiles every year by the measured idiosyncratic skewness. We construct portfolios from the CRSP database by randomly selecting stocks from the highest idiosyncratic skewness rank (Rank 5) only or from the lowest idiosyncratic skewness rank (Rank 1) only. Other than this restriction to the sample, the portfolio formation process is identical to the method used in Table 1. We report the average of portfolio skewness by idiosyncratic skewness and number of stocks in a portfolio. Standard errors of the averages are reported in parentheses. Column 4 reports the difference in skewness and the standard error of the difference. Number of Stocks 4 Skewness of the Highest Idiosyncratic Skewness Stocks Portfolio 2.17 (0.20) Skewness of the Lowest Idiosyncratic Skewness Stocks Portfolio 1.72 (0.23) Difference in Portfolio Skewness (Standard Errors) 0.45 (0.30) (0.08) 0.93 (0.12) 0.29 (0.14) (0.04) 0.45 (0.07) 0.42 (0.08) (0.04) 0.37 (0.07) 0.27 (0.08) (0.04) 0.40 (0.04) (0.04)

34 Table 6. Size and Portfolio Skewness We calculate the market value of a stock following the method of Fama and French (1992). Stocks are ranked into quintiles every year by the market value in June. We construct portfolios from the CRSP database by randomly selecting stocks from the largest market value rank (Rank 5) only or from the smallest market value rank (Rank 1) only. Other than this restriction to the sample, the portfolio formation process is identical to the method used in Table 1. We report the average of portfolio skewness by firm market value and number of stocks in a portfolio. Standard errors of the averages are reported in parentheses. Column 4 reports the difference in skewness and the standard error of the difference. Number of Stocks 4 Skewness of the Largest Stocks Portfolio 0.07 Skewness of the Smallest Stocks Portfolio 2.42 (0.22) Difference in Portfolio Skewness (Standard Errors) (0.22) (0.11) (0.11) (0.06) (0.06)

35 Table 7. Book-to-Market Value and Portfolio Skewness We calculate the market value of a stock following the method of Fama and French (1992). Stocks are ranked into quintiles every year by their book-to-market value. The book-to-market value is calculated from previous year s accounting value and market value. We construct portfolios from the CRSP database by randomly selecting stocks from the highest book-tomarket value rank (Rank 5) only or from the lowest book-to-market value rank (Rank 1) only. Other than this restriction to the sample, the portfolio formation process is identical to the method used in Table 1. We report the average of portfolio skewness by book-to-market and number of stocks in a portfolio. Standard errors of the averages are reported in parentheses. Column 4 reports the difference in skewness and the standard error of the difference. Number of Stocks 4 Skewness of the Highest Book-to-Market Stocks Portfolio 1.93 (0.15) Skewness of the Lowest Book-to-Market Stocks Portfolio 1.22 (0.17) Difference in Portfolio Skewness (Standard Errors) 0.71 (0.23) (0.13) 0.52 (0.06) 1.13 (0.14) (0.08) 0.29 (0.04) 0.86 (0.09) (0.04) (0.04) (0.04) (0.04) (0.04)