Skewness Consequences of Seeking Alpha

Transcription

1 Skewness Consequences of Seeking Alpha Kerry Back Jones Graduate School of Business and Department of Economics Rice University, Houston, TX 77005, U.S.A. Alan Crane Jones Graduate School of Business Rice University, Houston, TX 77005, U.S.A. Kevin Crotty Jones Graduate School of Business Rice University, Houston, TX 77005, U.S.A. Abstract Residual coskewness is an important performance attribute. Theory predicts that alpha and residual coskewness relative to the market should be negatively correlated in the crosssection of returns. Empirically, higher alpha mutual funds indeed have worse (lower) residual coskewness. The trade-off exists for active funds even controlling for styles (which also exhibit the trade-off). Conditional on residual coskewness, active funds outperform (underperform) passive funds when residual coskewness is bad (good). Moments beyond skewness may also be important for performance evaluation, but we find no evidence of a trade-off between alpha and residual cokurtosis. addresses: Kerry.E.Back@rice.edu (Kerry Back), Alan.D.Crane@rice.edu (Alan Crane), Kevin.P.Crotty@rice.edu (Kevin Crotty) April 6, 2016

2 1. Introduction The usual measure of fund performance is alpha relative to a benchmark (Jensen, 1969). Alphas matter for testing asset pricing models, but their importance for investors is that mean-variance efficiency can be improved by a marginal investment in a positive alpha fund (Dybvig and Ross, 1985). This justifies alpha as a performance measure for quadratic utility investors. However, it is quite possible that non-quadratic-utility investors may not wish to invest in a positive alpha fund if it has undesirable higher moments. Furthermore, investors who measure performance by alpha but care about higher moments may be drawn into making bad investments, because fund managers may be able to create positive alphas by taking on undesirable higher moments. For example, Leland (1999) shows that negatively skewed strategies like selling covered calls produce positive alphas when the market return is lognormal. Investors who have decreasing absolute risk aversion prefer positive skewness (Arditti, 1967) and hence may dislike those positive alpha strategies. What matters about skewness to an investor considering an investment in an asset is the marginal effect on the investor s portfolio skewness. For the type of zero-beta investment analyzed by Dybvig and Ross (1985), this marginal effect is governed by residual coskewness, meaning the coskewness between the investor s portfolio (benchmark) and the residual in the projection of the asset return on the benchmark. If there is an asset with a positive alpha and positive residual coskewness, then an investment in the asset can improve the mean, variance, and skewness of the investor s portfolio (Section 2.1). If investors care exclusively about mean, variance, and skewness, then it should be impossible in equilibrium to make such improvements. This means that there should be a trade-off in equilibrium between alpha and residual coskewness positive alphas can be achieved only at the expense of negative residual coskewness and vice versa. We analyze the trade-off between alpha and residual coskewness in theory and in the cross-section of mutual fund returns. The theoretical trade-off between alpha and residual coskewness is based on two results. First, it is easy to see that the coskewness pricing model (Kraus and Litzenberger, 1

3 1976) implies a negative linear relationship between alpha and residual coskewness, when the benchmark is the market portfolio. 1 The key assumption of the coskewness pricing model is that there is a representative investor with decreasing absolute risk aversion whose utility is well approximated by a cubic function. This implies that the representative investor cares exclusively about mean, variance, and skewness, as discussed in the previous paragraph. We present a new theoretical result that does not depend on a cubic approximation of the utility function: 2 If there is a representative investor with constant relative risk aversion and the market return is lognormal, then alpha and residual coskewness have opposite signs for all returns that are affine functions of lognormal random variables. Like the coskewness pricing model, our result implies that alpha and residual coskewness relative to the market are negatively correlated. In the cross-section of mutual fund returns, we reject the linear relation implied by the coskewness pricing model, but we find strong support for the negative correlation of alpha and residual coskewness. The rejection of the coskewness pricing model is important, because it implies that the trade-off between alpha and residual coskewness is not constant in the cross-section of returns. Different strategies with the same residual coskewness can produce different alphas. Empirically, we find that active funds and passive funds with the same residual coskewness have different alphas, reflecting different strategies underlying the residual coskewness of active and passive funds (we discuss this further in the conclusion). Our main empirical results are as follows: (1) alpha and residual coskewness are negatively correlated in the cross-section of mutual funds, (2) the coskewness pricing model is rejected in the cross-section of mutual funds, (3) alpha and residual coskewness are neg- 1 The recent literature on fund performance focuses on Fama-French-Carhart alphas, which we discuss below. However, given the widely recognized failures of the Fama-French-Carhart model (see, for example, the new factor models proposed by Fama and French (2015) and Hou, Xue, and Zhang (2015)) and the practice in the industry to benchmark to passive indices, it is worthwhile to study alphas relative to the market index. Additionally, Berk and van Binsbergen (2016) and Barber, Huang, and Odean (2016) provide evidence that the market model best explains investor behavior. We adjust for size, value, and momentum styles in Section 4. 2 An objection to a cubic utility function is that a cubic function cannot be both monotone increasing and concave on the domain R +. 2

4 atively correlated for style (size, value, and momentum) benchmarks, (4) controlling for styles, alpha and residual coskewness remain negatively correlated for active mutual funds but not for passive funds, (5) whether we control for styles or not, active funds with bad (good) residual coskewness have higher (lower) alphas than passive funds. Thus, there are skewness consequences of the search for alpha by active funds: positive alphas coincide with undesirable residual coskewness. The results just described are for alpha and residual coskewness relative to the market. Rather than benchmarking to the market, many investors may benchmark instead to the market and value, size, and momentum factors. For such investors, alphas relative to the Fama-French-Carhart (hereafter FFC) factors are relevant. A representative investor cannot hold the FFC factors (other than the market, of course), so the theoretical results described above do not apply to alphas and residual coskewness relative to FFC factors. Consistent with this, we find less evidence empirically of a trade-off between alpha and residual coskewness for FFC benchmarks (taking the benchmark for each fund to be the projection of its return on the factors). The point estimates are even slightly positively correlated. Moments beyond mean, variance, and skewness may also be important for performance evaluation. For instance, decreasing absolute prudence (Kimball, 1990) implies a negative fourth derivative of the utility function and hence a preference for lower kurtosis (see also Haas, 2007). We extend our analysis to residual cokurtosis, which governs a fund s marginal contribution to a benchmark portfolio s kurtosis. Negative residual cokurtosis funds are attractive because adding them to a portfolio reduces portfolio kurtosis. Under the coskewness/cokurtosis pricing model, positive alpha funds should be undesirable on residual coskewness and/or residual cokurtosis. We reject the linear relation implied by the coskewness/cokurtosis pricing model, and we find no evidence of a trade-off between alpha and residual cokurtosis. The interest in our results depends on the extent to which investors care about skewness. In theory, preferences for skewness are common. For example, investors with constant 3

5 relative risk aversion care about skewness. There is a large literature focusing on investors preferences for skewness. Golec and Tamarkin (1998) and Garrett and Sobel (1999) explain horse track betting and lottery participation through skewness preferences. Kumar (2009) links skewness preferences to individual investors stock decisions, showing that those that play the lottery also choose stocks with high idiosyncratic skewness. Goetzmann and Kumar (2008) show that individuals who hold undiversified portfolios hold stocks with high levels of skewness, and Bailey, Kumar, and Ng (2011) show that individual investors also tend to choose mutual funds that hold stocks with high levels of skewness. Consistent with Goetzmann and Kumar (2008), Mitton and Vorkink (2007) show that the prevalence of undiversified portfolios can be explained by heterogeneous preferences for skewness. Kadan and Liu (2014) show that the riskiness measures of Aumann and Serrano (2008) and Foster and Hart (2009) depend on higher moments (as does expected utility when it is not quadratic). Kraus and Litzenberger (1976) document that returns are related to coskewness in the cross-section. Other empirical research on the coskewness (and cokurtosis) pricing model includes Harvey and Siddique (2000), Dittmar (2002), and Guidolin and Timmermann (2008). Harvey and Siddique (2000) test the coskewness model allowing for time-varying betas and coskewness. Dittmar (2002) shows that if there is a representative investor whose marginal utility function can be well approximated by a cubic function, then risk premia should be determined by betas, coskewness, and cokurtosis relative to the market portfolio. He shows that the coskewness-cokurtosis pricing model works well for industry portfolios, but the parameter estimates indicate that the representative investor s utility function is not concave over the entire relevant domain. Likewise, Post, van Vliet, and Levy (2008) provide evidence that the utility function is S-shaped. Guidolin and Timmermann (2008) estimate the coskewness-cokurtosis pricing model from global return data, assuming a regime-shifting model for the prices of risks. They show that coskewness and cokurtosis premia are comparable in size to the covariance premium. They also show that home bias for U.S. investors can be at least partly explained by the desirable coskewness and cokurtosis of the U.S. market 4

6 return relative to the global market return. Using earlier data, Arditti (1971) shows that on average mutual fund returns have total skewness that exceeds that of the market return. Duarte, Longstaff, and Yu (2007) examine alphas and skewness of fixed income arbitrage strategies commonly employed by hedge funds. They show that most of the strategies have positive total skewness. Polkovnichenko, Wei, and Zhao (2014) examine the performance of actively managed funds versus passive benchmarks in up and down markets, which is related to skewness. In contrast to previous studies, we study residual coskewness rather than skewness. We show that residual coskewness is the right measure of performance for an investor who cares about skewness and is considering a small investment in a fund. Moreno and Rodríguez (2009) create a coskewness factor by sorting stocks based on coskewness with the market (a Spanish stock index in their case) and following the procedure used by Fama and French (1993) to construct SMB and HML. They evaluate Spanish mutual fund performance using the Fama-French model augmented by the coskewness factor. The motivation for the coskewness factor is empirical, similar to SMB and HML. Covariance with the coskewness factor is obviously not the same as coskewness with the market, which is what we study. The remainder of the paper is organized as follows. Section 2 extends Dybvig and Ross (1985) to account for skewness and shows the theoretical trade-off between alpha and residual coskewness for the coskewness pricing model and in the presence of a CRRA representative investor. Section 3 documents the empirical trade-off between alpha and residual coskewness for actively managed mutual funds. Section 4 shows that the trade-off exists for active funds even controlling for styles and is larger than that found in passive funds. Section 5 addresses residual cokurtosis. Section 6 concludes. 2. Theory This section explains why residual coskewness is an important attribute of performance. We also show that alpha and residual coskewness are negatively linearly related when the 5

7 coskewness pricing model holds and have opposite signs when there is a representative investor with constant relative risk aversion The Importance of Residual Coskewness The importance of alpha for investors is that a small investment in a zero-beta version of a positive-alpha fund improves mean-variance efficiency, relative to holding the benchmark (Dybvig and Ross, 1985). A positive alpha does not imply that a fund mean-variance dominates the benchmark it only implies that a marginal investment creates a portfolio that mean-variance dominates the benchmark. In this section, we demonstrate that residual coskewness is an important attribute of performance. We show that mean-variance-skewness efficiency of a benchmark portfolio can be improved by a zero-beta investment in a fund with a positive alpha and positive residual coskewness. On the other hand, a zero-beta investment in a fund with negative residual coskewness has an undesirable effect on portfolio skewness. We consider an arbitrary benchmark return. The benchmark could be the market return. It could also be, for example, a combination of the Fama-French factors (see Section 3). Let R b denote the benchmark return. Given a return R, regress the excess return on the benchmark as 3 R Rf = α + β(rb Rf) + ε, (1) where R f denotes the risk-free return. Define residual coskewness as E[ε(R b R b ) 2 ] = E[εR 2 b ], where the overbar denotes expectation. For a constant λ 0, consider the return R λ = R b + λ[r R f β(r b R f )] = R b + λ[α + ε]. (2) This is the benchmark combined with an investment in a zero-beta version of R. The 3 This regression is redundant if the benchmark is formed by projecting on factors as described in Section 3 for the Fama-French factors, because in that case β in the regression equation (1) is equal to 1. 6

8 derivatives of the first three moments of R λ with respect to λ evaluated at λ = 0 are d R λ dλ = α, λ=0 d var(r λ ) dλ = 0, λ=0 1 3 de[(r λ R λ ) 3 ] dλ = E[ε(R b R b ) 2 ]. λ=0 (3a) (3b) (3c) The derivatives tell us the signs of the changes in the return moments produced by a small investment in the fund. From (3a), we see that an investment in the fund increases the mean return if the fund s alpha is positive. From (3b), we see that the investment has only a second-order effect on variance. 4 From (3b) and (3c), we see that an investment in the fund increases skewness if the fund has positive residual coskewness. On the other hand, at least part of the benefit of the alpha is lost if the fund has negative residual coskewness Alpha and Residual Coskewness The previous subsection shows that positive residual coskewness funds are desirable, as are positive alpha funds. This section investigates the theoretical trade-off between alpha and residual coskewness relative to the market. Let R m denote the market return. Given another return R, project its excess return on the market excess return as usual: R R f = α + β(r m R f ) + ε. With the market as the benchmark, residual coskewness is E[εR 2 m] = cov(ε, R 2 m). (4) The coskewness pricing model can be expressed in terms of residual coskewness. The 4 A slightly different investment in the fund than that in (2) has a first-order negative effect on variance. Such an investment is analyzed by Dybvig and Ross (1985). See Appendix A for a generalization of the Dybvig-Ross argument. 7

9 usual statement of the pricing model is: For some λ 1 > 0 and λ 2 > 0 and for all returns R, R R f = λ 1 cov(r, R m ) λ 2 cov ( R, (R m R m ) 2). (5) Substitute R = R f + α + β(r m R f ) + ε to calculate the right-hand side of (5) as λ 1 cov(βr m + ε, R m ) λ 2 cov ( βr m + ε, (R m R m ) 2) [ = β λ 1 var(r m ) λ 2 cov ( R m, (R m R m ) 2) ] λ 2 cov ( ε, (R m R m ) 2). (6) Substitute this into (5) and apply (5) for the return R = R m (with β = 1 and ε = 0) to see that the expression in square braces in (6) is the market risk premium. Thus, for any return R, R R f = β( R m R f ) λ 2 cov ( ε, (R m R m ) 2). (7) This expresses the coskewness pricing model in terms of residual coskewness. An equivalent statement of the coskewness pricing model (7) is the following negative linear relation between alphas and residual coskewness: α = λ 2 cov ( ε, (R m R m ) 2). (8) The coskewness pricing model is derived from a cubic approximation to the representative investor s utility function. It is natural to wonder whether the approximation is adequate, especially since cubic utility is not an attractive assumption (note 2). The following shows that CRRA utility is a sufficient condition to imply a negative correlation between alpha and residual coskewness for lognormal returns (though it does not imply a linear relation). The proof is in Appendix B. Proposition. Suppose the representative investor has constant relative risk aversion and the market return R m is lognormal. Then, α cov ( ε, (R m R m ) 2) 0 (9) 8

10 for every return R = R f + α + β(r m R f ) + ε that is an affine function of a lognormal random variable. Furthermore, if the market is complete, then the coskewness pricing model (8) holds for the returns on the mean-variance frontier. The first statement of the proposition is illustrated in Figure 1. The last statement of the proposition follows from the first statement, because the mean-variance frontier is spanned by the risk-free return and a lognormal return when the market is complete, under the other assumptions of the proposition. Letting θ denote the risk aversion of the representative investor, the unique stochastic discount factor M is proportional to the representative investor s marginal utility: M = λr θ m for λ > 0. Define the return R = M E[M 2 ] = The mean-variance frontier is the set of returns R θ m λe[rm 2θ ]. (1 + δ)r f δr for δ R; furthermore, δ > 0 for the efficient returns on the frontier (Chamberlain and Rothschild, 1983; Hansen and Richard, 1987). The alpha and residual coskewness of the frontier returns both scale with δ, and the ratio of alpha to residual coskewness that is, λ 2 in (8) for the frontier returns is the alpha of R divided by its residual coskewness. As the proposition shows, that ratio is negative, so λ 2 in (8) is positive, consistent with the representative investor s preference for positive skewness. 3. Alpha and Residual Coskewness of Active Mutual Funds This section describes our data and methodology and the joint distributions of alpha and residual coskewness estimates. We use daily net returns data from the CRSP Survivor- Bias-Free U.S. Mutual Fund Database. We merge the daily fund data to holdings data from Thomson Reuters using the WRDS MF Links file. For a given fund, we consider average returns across share classes, weighting by total net assets in each class. We exclude index 9

11 and sector funds as well as international and emerging market funds. We include funds with at least sixty days of returns. The sample runs from September 1, 1998 through June 30, 2014 and includes 4,262 funds. We obtain benchmark market excess returns and risk-free returns from Kenneth French s website. 5 We calculate alphas and residual coskewness using the market as the benchmark and also using benchmarks formed from the FFC factors. First, consider the market as the benchmark. Let R e m denote the market excess return. For each fund, we estimate α and β by OLS from the usual regression equation: R R f = α + βr e m + ε. (10) Let γ denote residual coskewness. Given a sample of size T, we estimate residual coskewness as ˆγ = 1 T T ε t (Rmt) e 2, (11) t=1 where the ε t are the fitted residuals from the regression (10). This estimation of the parameter vector (α, β, γ) is a special case of the generalized method of moments (GMM). To estimate the performance dimensions using the FFC factors to construct benchmarks, we estimate the α and factor loadings of each fund from the regression R R f = α + β 1 R e m + β 2 R SMB + β 3 R HML + β 4 R UMD + ε. (12) The excess return R e b def = β 1 R e m + β 2 SMB + β 3 HML + β 4 R UMD (13) is the excess return in the span of the market excess return and the FFC factors that is most highly correlated with R R f. The natural benchmark for R for investors who can invest in the market and the FFC factors is R f + Rb e. Given a sample of size T, we estimate residual 5 library.html. 10

12 coskewness as ˆγ = 1 T T ε t (Rbt) e 2, t=1 where the ε t are the fitted residuals from the regression (12). Again, this is a special case of GMM. Table 1 reports summary statistics for the distributions of alpha and residual coskewness point estimates, using the market as the benchmark and also for FFC benchmarks. Figure 2 displays the histograms of the distributions. The median alpha is positive for the market benchmark but negative for FFC benchmarks. Most active funds (69%) have negative residual coskewness under the market benchmark. The residual coskewness distribution is much tighter around zero and more symmetric when the FFC factors are used to form benchmarks than when the market is used as the benchmark Correlation of Alpha and Residual Coskewness Using the market as the benchmark, the correlation between alpha and residual coskewness estimates in the cross-section of active funds is 26%. 6 The correlation between alpha and residual coskewness estimates for FFC benchmarks is 2%. These facts are consistent with the theoretical result in Section 2.2, which establishes a negative relation for the market benchmark but which does not apply to FFC benchmarks (because the representative investor does not hold the FFC factors). Figure 3 presents the joint distributions of the alpha and residual coskewness point estimates. We can see the negative correlation for the market benchmark in Panel A. On the other hand, Panel B shows the lack of correlation for FFC benchmarks. Table 2 provides further information about the joint distribution of the alpha and residual coskewness estimates. Panel A presents the joint distribution of the point estimates relative to the market benchmark. The estimates have the same sign for 35% of the funds and opposite signs for 65% of the funds. The most common combination is a positive alpha estimate 6 When assessing correlations, we winsorize the distributions of alpha and residual coskewness point estimates at the 1/99% levels to limit effects of outliers. 11

13 coupled with negative residual coskewness (43% of funds). These funds look more attractive under mean-variance preferences than under mean-variance-skewness preferences. Among the funds with positive alpha estimates, 83% have negative residual coskewness estimates. The same figure for funds with negative alpha estimates is only 54%. Panel C presents similar information for statistically significant estimates. The largest sets of significant estimates are for positive alphas and negative residual coskewness. Among funds with significant positive alphas, more than 1/3 have significant negative residual coskewness, but less than 1% have significant positive residual coskewness. Thus, for both the point estimates and in terms of statistical significance, the joint distribution of alpha and residual coskewness estimates is generally consistent with the theoretical result in Section 2.2, which predicts that alpha and residual coskewness relative to the market should have opposite signs. The theory in Section 2.2 applies only when the market is used as the benchmark. Consistent with this, Panel B of Table 2 shows that the joint distribution of alpha and residual coskewness estimates is much more balanced for FFC benchmarks than for the market benchmark: 46% have the same sign and 54% have opposite signs. There is some negative correlation of statistically significant estimates for FFC benchmarks. Panel D of Table 2 shows that, among funds with significant positive FFC alphas, 13.5% have significant negative residual coskewness and only 1.6% have significant positive residual coskewness. Because the theory pertains only to the market benchmark, most of our remaining empirical analysis utilizes the market benchmark only Rejection of the Coskewness Pricing Model The coskewness pricing model (8) asserts that there is a linear relation (without an intercept) between alpha and residual coskewness. We test the model by running cross-sectional regressions of the (winsorized) alpha estimates on powers of the (winsorized) residual coskewness estimates. Table 3 shows that the intercept and powers greater than one of residual coskewness are statistically significant. Furthermore, the adjusted R 2 increases substantially when the squared coskewness term is introduced. Thus, we reject the coskewness pricing 12

14 model. In the affine specification in the first column of Table 3, we estimate that a one standard deviation increase in residual coskewness produces a 0.28 standard deviation reduction in alpha. In the other columns, the coefficient on residual coskewness is the marginal effect of a change in residual coskewness starting from zero residual coskewness. The marginal effects are consistently negative in the non-affine specifications and are larger in absolute value in the non-affine specifications than in the affine specification. In all cases, the effect on alpha of varying residual coskewness seems economically significant. 4. Active Funds, Passive Funds, and Styles This section addresses the performance of active funds relative to passive style benchmarks and relative to passive funds in general. We show that there is a negative correlation between alpha and residual coskewness (relative to the market) among investment styles. However, there remains a negative correlation between alpha and residual coskewness among active funds even controlling for styles. In contrast, there is no negative relation between alpha and residual coskewness for passive funds after controlling for styles. Conditional on residual coskewness, active funds have higher alphas than passive funds when residual coskewness is low (bad) and lower alphas than passive funds when residual coskewness is high (good) Investment Styles To investigate the effects of styles on residual coskewness, we first estimate the residual coskewness of the FFC factors relative to the market benchmark. Table 4 reports that SMB, HML, and UMD all have desirable alphas and undesirable residual coskewness during our sample period. Again, this is consistent with theory. Table 4 also reports estimates of alpha and residual coskewness relative to the market for a number of Russell and S&P benchmark indices. The indices exhibit size effects and value effects for residual coskewness that are the opposite of the alpha effects: Smaller capitalization indices have worse (more negative) residual coskewness than do larger cap indices, and value indices have worse residual coskewness 13

15 than do growth indices. 7 Among the 17 index and factor returns, the correlation between alpha and residual coskewness is 73%. To investigate the effects of investment styles on the alphas and residual coskewness of active mutual fund returns, we classify funds into style categories using a holdings-based approach following Daniel et al. (1997) and Wermers (2003). Specifically, we classify each holding into one of five characteristic quintiles using the stock assignment file from Russ Wermer s website. 8 The characteristics we consider are size, book-to-market ratio, and momentum. To aggregate the holding-specific classification to the fund level, we first valueweight the quintile assignments for each reporting date. We then average across reporting dates to arrive at fund-specific average size, book-to-market, and momentum quintiles. We sort funds into terciles based on these averages. Table 5 reports average values of alpha and residual coskewness for funds sorted as described above. We intersect separate sorts on two characteristics at a time. Like the indices reported in Table 4, the mutual fund returns exhibit a size effect for residual coskewness that is the opposite of the effect for alpha. Funds that hold larger stocks have smaller alphas but higher residual coskewness. The size effects for residual coskewness and alpha exist in every book-to-market tercile (Panel A) and every past return tercile (Panel B). The mutual fund returns also exhibit a value effect for residual coskewness that is the opposite of the value effect for alphas. This is again like the indices. Funds that hold more value stocks have higher alphas and lower residual coskewness in every size tercile (Panel A) and every past return tercile (Panel C). There is not a clear momentum effect in our sample (which includes the momentum crash in the spring of 2009). In the six terciles (three for size and three for book-to-market), funds holding past losers beat funds holding past winners in three, and the reverse is true in the other three. However, there is still a negative correlation between 7 The S&P 500, 400, and 600 indices provide benchmarks for large, mid-cap, and small-cap equities, respectively. The Russell 1000 measures large-cap performance; the Russell Midcap consists of the smallest 800 or so firms in the Russell The Russell 2000 index measures small-cap performance

16 alpha and residual coskewness in the sense that residual coskewness moves in the opposite direction of alpha in five of the six terciles. Tables 4 and 5 document that the negative relation between alpha and residual coskewness is partly due to styles: Styles that do better on alpha do worse on residual coskewness. However, Table 5 also shows that the negative relation is not entirely due to styles. The table reports the the cross-sectional correlation between alpha and residual coskewness in each of the 27 bins. The correlation is negative in 24 of the 27 bins. Thus, even controlling for styles, there is a negative relation between alpha and residual coskewness in the cross-section of active fund returns. As a complement to the holdings analysis, we control for styles by adjusting the active fund returns for their FFC factor exposures. Using the estimates of the FFC model for each fund, we form style-adjusted returns R Rb e, where Re b is the benchmark excess return in (13). We estimate the market-model alpha and residual coskewness of each style-adjusted return. In the cross-section of active funds, the correlation between alpha and residual coskewness for style-adjusted returns is This estimate is statistically significant at the one percent level Active versus Passive Funds To determine whether active funds outperform or underperform passive funds in producing alpha conditional on residual coskewness, we augment the active fund sample with passive funds. These passive funds include traditional index funds as well as ETFs. We regress estimated alphas on estimated residual coskewness for the resulting full sample of active and passive funds with a dummy variable for passive funds. We also include dummy variables for the 27 (3 3 3) styles discussed in the previous subsection. As shown previously, there is a negative relation between alpha and residual coskewness for active funds even when we condition on style. This is confirmed in Table 6 by the significant negative coefficient on ˆγ in Columns 2 through 5. However, this trade-off between alpha and residual coskewness conditional on style does not exist for passive funds. The point estimate of the 15

17 slope coefficient for passive funds is positive (Columns 4 and 5 of Table 6). For the full specification (Column 5 of Table 6), the alphas of funds with zero residual coskewness (the intercepts) are significantly higher for passive funds than for active funds, and the slope coefficients are also significantly different. The regressions reported in Table 6 use residual coskewness estimates that are normalized to have a unit cross-sectional standard deviation. Using the point estimates for the full specification, passive funds outperform active funds conditional on residual coskewness for all values γ of normalized residual coskewness satisfying γ 0 γ The cut-off value 0.43 for normalized residual coskewness corresponds to a raw residual coskewness value of 0.087, which is the 44th percentile of the distribution of residual coskewness estimates for both active and passive funds (see Table 1 for information about the distribution of estimates for active funds). The regressions lines for active and passive funds in Column 5 of Table 6 are plotted in Panel A of Figure 4. The vertical line is at the cut-off value To the left of that value, active funds outperform passive funds conditional on residual coskewness; to the right, passive funds outperform. Panels B and C of Figure 4 present the densities of the residual coskewness estimates of active and passive funds. Both densities have long left tails. Thus, the regression indicates that passive funds outperform active funds conditional on residual coskewness in a slight majority of cases (as stated in the previous paragraph, 56% of the mass of both densities lie to the right of the cut-off value). However, the long left tails of the densities indicate more extreme outperformance by active funds. Importantly, the region of outperformance by active funds is the region of bad residual coskewness. To complement the regression analysis, we also sort funds by residual coskewness. We pool active and passive funds and sort into quintiles. Within each quintile, we calculate the average alphas of the active and passive funds. We also calculate average style-adjusted alphas of active and passive funds within each quintle. Panel A of Table 7 presents the raw 16

18 alphas. Conditional on low (undesirable) residual coskewness, active funds have significantly higher alphas than do passive funds, but the reverse is true for high (desirable) residual coskewness. Panel B of Table 7 shows that the same pattern holds after adjusting for styles (to adjust for styles, we subtract the mean alpha of the style bin in which a fund resides, based on the sort described earlier). The differences in alphas are economically as well as statistically significant, with active funds beating passive funds by 0.53 (0.81) basis points per day in raw (style-adjusted) alphas when residual coskweness is low and passive funds beating active funds by 0.57 (0.55) basis points per day in raw (style-adjusted) alphas when residual coskewness is high. 5. Incorporating Kurtosis into Performance Evaluation Investors may dislike kurtosis in a similar spirit to aversion to variance. The marginal effect of an investment on portfolio kurtosis is determined by residual cokurtosis, defined as E[ε(R b R b ) 3 ], where, as in Section 2.1, R b is the benchmark return, ε is the residual in the projection of a return R on the benchmark and a constant, and the overbar denotes expectation. Recall the return (2), which is the benchmark combined with an investment in a zero-beta version of R, with constant λ 0: R λ = R b + λ[r R f β(r b R f )] = R b + λ(α + ε). The derivative of the fourth moment of R λ with respect to λ evaluated at λ = 0 is 1 4 de[(r λ R λ ) 4 ] dλ = E[ε(R b R b ) 3 ]. (14) λ=0 The derivative tells us the sign of the change in kurtosis produced by a small investment in the fund with return R. 9 From (14), we see that an investment will reduce kurtosis if the fund has negative residual cokurtosis. Thus, residual cokurtosis should be important for investors who care about kurtosis in addition to expected returns, variance, and skewness. 9 As with our treatment of skewness, we define kurtosis to be the fourth central moment (i.e., not normalized by the fourth power of standard deviation). 17

19 The coskewness pricing model assumes kurtosis is unpriced, but we empirically reject that model in Section 3.2. The more general coskewness/cokurtosis pricing model allows kurtosis to be priced. The model is: For some λ 1 > 0, λ 2 > 0, and λ 3 > 0 and for all returns R, R R f = λ 1 cov(r, R m ) λ 2 cov ( R, (R m R m ) 2) + λ 3 cov ( R, (R m R m ) 3). (15) A similar argument to that in Section 2.2 shows that the coskewness/cokurtosis pricing model (15) is equivalent to alpha being linearly related to both residual coskewness and residual cokurtosis: α = λ 2 cov ( ε, (R m R m ) 2) + λ 3 cov ( ε, (R m R m ) 3). (16) Thus, positive alpha funds are undesirable on at least one of the other characteristics if the model holds. We test the linear relation (16) empirically. Following the notation in Section 3, we calculate residual cokurtosis using the market as the benchmark and also using benchmarks formed from the FFC factors. Let κ denote residual cokurtosis. Given a sample of size T, we estimate residual cokurtosis relative to the market as ˆκ = 1 T T ε t (Rmt e R m) e 3 (17) t=1 where the ε t are the fitted residuals from the regression (10). Similarly, we estimate residual cokurtosis relative to a fund-specific benchmark return R b constructed using the FFC factors as where the R e bt ˆκ = 1 T T ε t (Rbt e R b) e 3 t=1 are the fitted values of the benchmark excess return (13) with sample mean R b e, and the ε t are the fitted residuals from the regression (12). Histograms of the estimated residual cokurtosis are displayed in Figure 5. Approximately half of the funds are estimated to have desirable (negative) residual cokurtosis. As with residual coskewness, the residual cokurtosis distributions are tighter around zero when the 18

20 FFC factors are used to form benchmarks than when the market is used as the benchmark. The second panel of Figure 5 shows the joint distribution of alpha and residual cokurtosis estimates. There appears to be little dependence between alpha and residual cokurtosis. In fact, the distribution of funds across the four combinations of positive/negative with alpha/residual cokurtosis is almost uniform (28% positive alpha/negative cokurtosis, 24% positive alpha/positive cokurtosis, 25% negative alpha/negative cokurtosis, and 23% negative alpha/positive cokurtosis). The absence of a trade-off between alpha and residual cokurtosis is shown more rigorously in Table 8, which presents regressions of estimated alphas on estimated residual coskewness and residual cokurtosis in both linear and quadratic specifications. The significance of the intercept and of the quadratic terms rejects the coskewness/cokurtosis pricing model. Furthermore, the linear and squared terms in residual cokurtosis are insignificant whenever the product ˆγ ˆκ is included in the regression. We conclude that there is no direct tradeoff between alpha and residual cokurtosis. 10 The fact that the product is significant does mean that the trade-off between alpha and residual coskewness is correlated with residual cokurtosis. The cause of that relationship may deserve further study. 6. Conclusion Residual coskewness is an important attribute of performance. In theory, a search for alpha relative to the market will create undesirable (negative) residual coskewness. Empirically, alpha and residual coskewness relative to the market are indeed negatively correlated in the cross-section of actively managed mutual funds. Thus, at least part of the value of the alpha of an actively managed mutual fund is typically offset by an undesirable effect on portfolio skewness. Our empirical results are consistent with the existence of a CRRA representative investor, though we reject the linear relation between alpha and residual coskewness implied by the coskewness pricing model. 10 Table 8 presents regressions for active funds only. The results are very similar for passive funds. 19

21 Like alpha, residual coskewness is related to investment styles like size, value, and momentum. Styles that do better on alphas do worse on residual coskewness. However, the trade-off between alpha and residual coskewness holds for active funds even accounting for funds styles. In contrast, there is no trade-off for passive funds after accounting for styles. Whether we control for styles or not, active funds with bad (good) residual coskewness have higher (lower) alphas than passive funds. The results suggest that funds that successfully search for alpha do so at the expense of residual coskewness. Active funds employ many different strategies, so it is probably impossible to say in general why they produce a different trade-off between alpha and residual coskewness than do passive funds. However, as an illustrative example, consider momentum. Table 4 documents that UMD has both a higher alpha and a smaller (in absolute value) residual coskewness than either SMB or HML; thus, the ratio of alpha to absolute residual coskewness is higher for UMD than for SMB or HML. There may be some active funds that play momentum and also some active funds that play contrarian strategies (shorting momentum see Edelen et al. (2016) for evidence that institutions are often on the wrong side of anomaly strategies). If active funds play both sides of momentum and momentum has a high ratio of alpha to absolute residual coskewness, then active funds with bad (good) residual coskewness will have higher (lower) alphas than passive funds, as we find in the data. This is just an example and does not explain all of our results, because the difference between active and passive funds holds even after controlling for size, value, and momentum styles. We leave further study of the strategies producing the different trade-offs to future research. 20

22 Appendix A. An Alternative to the Zero-Beta Investment in a Fund For λ 0 and k > 0, consider the return R kλ = R b + λ[r R f β(r b R f )] kλ(r b R f ). (A.1) This is the benchmark combined with an investment in a zero-beta version of R and adjusted by some position in the excess return R b R f. Dybvig and Ross (1985) analyze the return R kλ with k = α/( R b R f ). The derivatives of the first three moments of R kλ with respect to λ evaluated at λ = 0 are d R kλ dλ = α k( R b R f ), λ=0 1 2 d var(r kλ) dλ = k var(r b ), λ=0 1 3 de[(r kλ R kλ ) 3 ] dλ = E[ε(R b R b ) 2 ] ke[(r b R b ) 3 ]. λ=0 (A.2a) (A.2b) (A.2c) For k > 0 and small λ > 0, we see from (A.2b) that the return R kλ has a lower variance than does the benchmark. Furthermore, if k is sufficiently small and λ is sufficiently small, given k, then (A.2a) and (A.2c) still imply that R kλ has a higher mean return than the benchmark if alpha is positive and higher skewness than the benchmark if residual coskewness is positive. Thus, a small investment in the fund can produce a return that dominates the benchmark with respect to mean, variance, and skewness if the fund has a positive alpha and positive residual coskewness. On the other hand, if residual coskewness is negative and k is small, then a marginal investment will reduce portfolio skewness. 21

23 Appendix B. Proof of the Proposition Suppose R = A + Be x and R m = e y, where A and B are constants and x n(µ x, σ x ) and y n(µ y, σ y ) are joint normal with correlation ρ. Let θ denote the relative risk aversion of the representative investor. The SDF M is proportional to R θ m and E[MR f ] = 1, so M = Rm θ R f E[Rm θ ]. The equation E[MR] = 1 implies Therefore Likewise, Also, 1 = E[RR θ m ] R f E[Rm θ ] = E[(A + Bex )e θy ] R f E[e θy ] E[R] A = (R f A)e θρσxσy. E[R m ] = R f e θσ2 y. = A R f + E[R] A R f e θρσxσy. cov(r, R m ) = (E[R] A)E[R m ] (e ρσxσy 1) ( ) var(r m ) = E[R m ] 2 e σ2 y 1. Thus, the beta in the regression of R on R m is β = E[R] A E[R m ] ( e ρσ xσ y ) 1 e σ2 y 1 = (R f A)e θρσxσy E[R m ] ( e ρσ xσ y ) 1. (B.1) e σ2 y 1 22

24 The alpha of R is E[R] R f β(e[r m ] R f ) = (R f A) ( e θρσxσy 1 ) ( ) βr f e θσ2 y 1 [ = (R f A) e θρσxσy 1 R ( fe θρσxσy e ρσ xσ y 1 E[R m ] e σ2 y 1 [ ( e ρσ xσ y 1 = (R f A) e θρσxσy 1 e θρσxσy e σ2 y 1 = (R f A)e θρσxσy e σ2 y 1 = (R f A)e θρσxσy e σ2 y 1 = (R f A)e θρσxσy e σ2 y 1 ) ( ) ] e θσ2 y 1 ) ( 1 e θσ2 y [ ( ) (1 e σ2 y 1 e θρσ xσ y ) (e ρσ xσ y 1) ] [b ba θ + a θ a + ab θ b θ ) ] ( ) ] 1 e θσ2 y [ ] (a θ 1)(1 b) (b θ 1)(1 a), (B.2) where for the last two lines we define a = e ρσxσy and b = e σ2 y. The residual coskewness is E[εRm] 2 = cov(ε, Rm) 2 = cov(r βr m, Rm) 2 = cov(r, Rm) 2 β cov(r m, Rm) 2 = (E[R] A)E[Rm](e 2 2ρσxσy 1) βe[r m ]E[Rm](e 2 2σ2 y 1) [ ( e = (E[R] A)E[Rm] 2 ρσ xσ y ) ] 1 e 2ρσxσy 1 (e 2σ2y 1) e σ2 y 1 = (R ] f A)e θρσxσy E[Rm] [(e 2 2ρσxσy 1)(e σ2y 1) (e ρσ xσ y 1)(e 2σ2y 1) e σ2 y 1 ] = (R f A)e θρσxσy E[Rm] [e 2 2ρσxσy + e σ2y e σ 2y+ρσ xσ y e ρσxσy = (R f A)e θρσxσy E[Rm](e 2 ρσxσy 1)(e ρσxσy e σ2 y ) = (R f A)e θρσxσy E[Rm](a 2 1)(a b). (B.3) where a = e ρσxσy and b = e σ2 y as before. From (B.2) and (B.3), we see that the product of alpha with residual coskewness is a positive multiple of [ ] (a θ 1)(1 b) (b θ 1)(1 a) (a 1)(a b). (B.4) 23

25 This is 0 at a = 1 and at a = b, so (9) holds for a = 1 and a = b. Assume now that a 1 and a b. We claim that (B.4) is negative in this situation. Obviously, (B.4) is 0 when θ = 0. We will show that the derivative of (B.4) with respect to θ is negative; hence (B.4) is negative for θ > 0. The derivative of (B.4) with respect to θ is [ ] (1 a)b θ log b (1 b)a θ log a (a 1)(a b) [ (1 a)a = a θ b θ θ (log a)(log b) log a ] (1 b)bθ (a 1)(a b). (B.5) log b The factor is positive. So, (B.5) is negative if and only if Condition (B.6) holds because [ (1 a)a θ a θ b θ (log a)(log b)(a 1) log a ] (1 b)bθ (a b) < 0. (B.6) log b x 1 log x xθ (B.7) is an increasing function of x. Clearly, the factor x θ is an increasing function of x. Also, (x 1)/ log x is an increasing function of x. To see this, note that Also, d dx ( ) x 1 = log x log x (x 1)/x (log x) 2. (B.8) log x + 1 (y x) > log y, x for all y, because the left-hand side as a function of y is the equation of the tangent line at (x, log x) to the graph of the logarithm function, and the tangent line lies above the graph, due to concavity. Setting y = 1 gives log x > x 1 x so (B.8) is positive. Hence, (B.7) is an increasing function of x, and (B.6) holds. 24

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29 Table 1: Distributions of Point Estimates of Alpha and Residual Coskewness Alpha and residual coskewness (γ) are estimated for active mutual funds over the time period September 1, 1998 through June 30, 2014 using daily returns. ˆα is measured in basis points (bps) and residual coskewness ˆγ is measured in % 3. The benchmark return is either the market or a portfolio of the Fama-French-Carhart factors. Market Benchmark Fama-French-Carhart Benchmark ˆα ˆγ ˆα ˆγ 10th th Median th th IQRange Observations

30 Table 2: Joint Distribution of Alpha and Residual Coskewness Alpha and residual coskewness (γ) are estimated for active mutual funds over the time period September 1, 1998 through June 30, 2014 using daily returns. Each cell of Panels A and B reports the percentage of funds satisfying the row and column condition. Each cell of Panels C and D reports the percentage of funds for which the null hypotheses α = 0 and γ = 0 are rejected for both the parameter appearing in the row label and the parameter appearing in the column label and for which the point estimates satisfy the conditions in the row and column labels. The tests are two-sided and conducted at a significance level of 5%. Panel A: Market Benchmark (Point Estimates) ˆγ > 0 ˆγ < 0 Total ˆα > ˆα < Total Panel B: Fama-French-Carhart Benchmarks (Point Estimates) ˆγ > 0 ˆγ < 0 Total ˆα > ˆα < Total Panel C: Market Benchmark (Hypothesis Tests) ˆα > 0 ˆα < 0 ˆγ > 0 ˆγ < 0 ˆα > ˆα < ˆγ > ˆγ < Panel D: Fama-French-Carhart Benchmarks (Hypothesis Tests) ˆα > 0 ˆα < 0 ˆγ > 0 ˆγ < 0 ˆα > ˆα < ˆγ > ˆγ <

31 Table 3: Cross-sectional Relation between Alpha and Residual Coskewness Alpha and residual coskewness (γ) are estimated for active mutual funds over the time period September 1, 1998 through June 30, 2014 using daily returns. Point estimates (winsorized at 1%/99%) are divided by their cross-sectional standard deviations and regressions are run using the standardized estimates. All columns report OLS estimates with t statistics reported in parentheses. Statistical significance is represented by * p < 0.10, ** p < 0.05, and *** p < ˆα ˆα ˆα ˆα ˆγ (-9.03) (-14.23) (-15.38) (-17.82) ˆγ (-6.45) (-1.97) (-3.18) ˆγ ˆγ 4 (1.88) (5.67) 0.02 (4.97) Constant (-7.75) (-6.12) (-7.95) (-9.24) Observations Adjusted R

32 Table 4: Alpha and Residual Coskewness for FFC Factors and Indices Alpha is measured in basis points (bps) and residual coskewness (γ) is measured in % 3. Standard errors of the parameter estimates are reported to the right of the estimates. The benchmark is the market portfolio. SMB, HML, and UMD are the FFC factors. The indices are the Russell 1000/Midcap/2000 total, growth, and value indices and the S&P 500 (total, growth, value), 400, and 600 indices. The S&P 500, 400, and 600 indices provide benchmarks for large, mid-cap, and small-cap equities, respectively. The Russell 1000 measures large-cap performance; the Russell Midcap consists of the smallest 800 or so firms in the Russell The Russell 2000 index measures small-cap performance. The sample uses daily returns from September 1, 1998 through June 30, ˆα SE(ˆα) ˆγ SE(ˆγ) (bps) (bps) (% 3 ) (% 3 ) FFC Factors SMB HML UMD Size-based Benchmark Indices S&P S&P S&P Russell Russell Midcap Russell Value/Growth Benchmark Indices S&P 500 Growth S&P 500 Value Russell 2000 Growth Russell 2000 Value Russell Midcap Growth Russell Midcap Value Russell 1000 Growth Russell 1000 Value

33 Table 5: Alpha, Residual Coskewness, and Investment Styles Alpha and residual coskewness (γ) are estimated for active mutual funds over the time period September 1, 1998 through June 30, 2014 using daily returns and winsorized at the 1/99% level. Style classifications are based on funds holdings. At each reporting date, we assign each stock to size, book-to-market, and momentum quintiles based on Russ Wermer s stock assignment file. For each characteristic and each fund, we value weight the rankings and average across reporting dates. We sort funds into terciles on each characteristic and intersect the sorts. The table presents the average values for each cell and the cross-sectional correlation within each cell. Panel A: Size and Value Small Medium Large Growth ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Neutral ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Value ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Panel B: Size and Momentum Small Medium Large Loser ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Average ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Winner ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ)

34 Panel C: Value and Momentum Growth Neutral Value Loser ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Average ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ) Winner ˆα (bps) ˆγ (% 3 ) Corr(ˆα, ˆγ)

35 Table 6: The Alpha Residual Coskewness Trade-offs for Active and Passive Funds Alpha and residual coskewness (γ) are estimated for active and passive funds over the time period September 1, 1998 through June 30, 2014 using daily returns. Point estimates (winsorized at 1%/99%) are divided by their cross-sectional standard deviations (pooling active and passive funds), and regressions are performed on the standardized estimates. The first column reports the OLS regression of residual coskewness on alpha for all funds including an indicator for index funds, Passive Fund. Columns two through five include all funds and controls for mutual fund style. Style classifications are based on funds holdings. At each reporting date, we assign each stock to size, book-to-market, and momentum quintiles based on Russ Wermer s stock assignment file. For each characteristic and each fund, we value weight the rankings and average across reporting dates. We sort funds into terciles on each characteristic and intersect the sorts. We include dummy variables in the regression for each of the 27 classifications. t statistics are in parentheses, and statistical significance is represented by * p < 0.10, ** p < 0.05, and *** p < ˆα ˆα ˆα ˆα ˆα ˆγ (-8.86) (-4.11) (-4.10) (-4.44) (-4.49) Passive Fund (1.48) (0.09) (2.23) ˆγ*Passive Constant (-7.35) (2.22) (2.43) Observations Adjusted R Style Controls No Yes Yes Yes Yes 34

36 Table 7: Alphas of Active and Passive Funds Conditional on Residual Coskewness Alpha and residual coskewness (γ) are estimated for active and passive funds over the time period September 1, 1998 through June 30, 2014 using daily returns. Point estimates are winsorized at 1%/99%. We partition the sample into quintiles of residual coskewness (pooling active and passive funds) and calculate the mean alphas of active and passive funds within each quintile. The style return for each fund is the mean alpha of the bin in which the fund resides, based on the sort described in Table 6. Panel A: Raw Alphas (bps) Active Obs Passive Obs Dif. t(dif.) Lo Q Q Q Hi Panel B: Alphas in Excess of Style Returns (bps) Active Obs Passive Obs Dif. t(dif.) Lo Q Q Q Hi

37 Table 8: Cross-sectional Relation between Alpha, Residual Coskewness, and Residual Cokurtosis Alpha, residual coskewness (γ), and residual cokurtosis (κ) are estimated for active mutual funds over the time period September 1, 1998 through June 30, 2014 using daily returns. Point estimates (winsorized at 1%/99%) are divided by their cross-sectional standard deviations and regressions are run using the standardized estimates. All columns report OLS estimates with t statistics reported in parentheses. Statistical significance is represented by * p < 0.10, ** p < 0.05, and *** p < ˆα ˆα ˆα ˆα ˆγ (-9.66) (-10.15) (-13.87) (-14.39) ˆκ (4.69) (-0.69) (0.68) (-0.77) ˆγ ˆκ ˆγ 2 ˆκ 2 (-5.54) (-2.68) (-4.96) (-4.14) (-3.74) (-1.41) Constant (-7.55) (-7.95) (-4.86) (-5.38) Observations Adjusted R

38 Figure 1: Theoretical Trade-Off Between Alpha and Residual Coskewness This depicts the alpha and residual coskewness relative to the market of returns A + Be x, assuming x is normally distributed, there is a representative investor with constant relative risk aversion, and the market return is lognormal. The parameters of a return A + Be x that determine its alpha and residual coskewness are A and the log beta, meaning the beta of x with respect to the log market return see (B.2) and (B.3). By varying A and the log beta, we trace out the locus of alpha and residual coskewness pairs. In this example, the risk aversion of the representative investor is 3, the expected market return is 0.08, and the standard deviation of the market return is The figure plots alpha and residual coskewness pairs for two A values and log betas ranging from 4 to 4. Positive (negative) values of the log beta are indicated by ρ > 0 (ρ < 0) A=0, ρ<0 A=0, ρ>0 A=2, ρ<0 A=2, ρ>0 Residual Coskewness Alpha 37

39 Figure 2: Univariate Distributions of Alpha and Residual Coskewness Estimates Alpha and residual coskewness are estimated for active mutual funds using daily returns over the time period September 1, 1998 through June 30, 2014 relative to the market benchmark and relative to Fama-French-Carhart benchmarks. Panel A: Market Benchmark (a) Alpha (b) Residual Coskewness Panel B: Fama-French-Carhart Benchmarks (a) Alpha (b) Residual Coskewness 38

40 Figure 3: Joint Distributions of Alpha and Residual Coskewness Estimates Alpha and residual coskewness are estimated for active mutual funds using daily returns over the time period September 1, 1998 through June 30, 2014 relative to the market benchmark and relative to Fama-French-Carhart benchmarks. The (a) and (b) panels present different views of the same data. Panel A: Market Benchmark (a) Alpha and Residual Coskewness (b) Alpha and Residual Coskewness Panel B: Fama-French-Carhart Benchmarks (a) Alpha and Residual Coskewness (b) Alpha and Residual Coskewness 39

41 Figure 4: Alphas of Active and Passive Funds Conditional on Residual Coskewness Alpha and residual coskewness are estimated for active and passive mutual funds using daily returns over the time period September 1, 1998 through June 30, Panel (a) presents the regression lines for active and passive funds from Column 5 of Table 6. The alphas are alphas in excess of the average alpha of the fund s style. Both alphas and residual coskewness are normalized to have unit cross-sectional standard deviations. Panels (b) and (c) present the densities of the normalized residual coskewness estimates. The vertical lines represent the value of residual coskewness for which the predicted alpha is equal for active and passive funds. (a) Trade-off (b) Active (c) Passive 40

42 Figure 5: Residual Cokurtosis Estimates Alpha and residual cokurtosis are estimated for active mutual funds using daily returns over the time period September 1, 1998 through June 30, 2014 relative to the market benchmark and relative to Fama-French-Carhart benchmarks. Panel A: Point Estimates (a) Market Benchmark (b) FFC Benchmark Panel B: Joint Distribution of Alpha and Residual Cokurtosis (a) Market Benchmark (b) FFC Benchmark 41