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The Review of Financial Studies
, Volume Advance Article – Mar 15, 2018

42 pages

/lp/ou_press/skewness-consequences-of-seeking-alpha-zRO9440x7s

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.
- ISSN
- 0893-9454
- eISSN
- 1465-7368
- D.O.I.
- 10.1093/rfs/hhy029
- Publisher site
- See Article on Publisher Site

Abstract Mutual funds seek alpha, but coskewness is also an important performance attribute. Coskewness of fund returns is associated with market timing, liquidity management, and derivative use. Measures of active management associated with positive alphas are also associated with undesirable coskewness. When controlling for other characteristics, coskewness is positively associated with activity measures related to market timing and negatively associated with activity measures related to stock picking. In the cross-section of funds, the latter effect dominates, so funds generate undesirable coskewness in the pursuit of alpha. Money flows to funds with desirable coskewness. Received October 25, 2016; editorial decision January 29, 2018 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web Site next to the link to the final published paper online. Mutual fund performance is often measured by alpha relative to a benchmark (Jensen 1969). This makes sense for quadratic utility investors, because mean-variance efficiency can be improved by a marginal investment in a positive alpha fund (Dybvig and Ross 1985). However, there is evidence that investors care about moments beyond mean and variance, in particular that investors prefer positive skewness.1 For these investors, the marginal effect of an investment in a mutual fund on the investor’s portfolio skewness is also important. For an investor who holds the market, the marginal effect depends on coskewness with the market return. Absent investment skill, theory (e.g., Kraus and Litzenberger 1976) suggests that managers should not be able to produce both desirable alpha and coskewness. Consistent with this and prior empirical work by Moreno and Rodríguez (2009), we document a robust trade-off between CAPM alphas and coskewness in the cross-section of fund returns: funds with higher alphas tend to have lower (worse) coskewness and vice versa. We study the determinants of mutual fund coskewness. There are at least three possible sources of coskewness in fund returns that are distinct from the coskewness of stocks held by the funds. One is market timing. Funds that successfully time the market create return profiles that are convex in the market return (Treynor and Mazuy 1966)—effectively, they create protective puts at no cost—and hence have positive coskewness with the market. We account for market timing in two ways. First, we estimate how market exposure varies over time based on public information (Ferson and Schadt 1996) and flows (Edelen 1999) and we decompose coskewness into a part that is due to time-varying market exposures and a part that is due to other factors. Second, we infer the market timing return of each fund using the calculation of Kacperczyk, Van Nieuwerburgh, and Veldkamp (2014) and show that it is positively related to coskewness as expected. A second source of coskewness is cash management. If money flows into funds when expected returns are high and it takes some time for funds to invest new cash, then funds will be holding more cash when expected returns are high. This implies that their return profiles will be concave in the market return and hence have negative coskewness with the market.2 We show that coskewness is worse for funds that have more illiquid noncash holdings. This is consistent with such funds taking longer to invest new cash when expected returns are high. Furthermore, funds that hold more cash in general suffer less from illiquidity. A third source is that some funds use derivatives. For example, selling covered calls produces a return profile that is concave in the market return and hence has negative coskewness with the market.3 We find that funds’ use of options on individual stocks and their use of options on stock indices and stock index futures contribute significantly to coskewness. The evidence is consistent with funds writing covered calls on individual stocks (generating negative coskewness) and buying protective puts on the market (generating positive coskewness). Recent work on mutual fund performance has shown that various characteristics of funds predict performance, when performance is measured by alpha. We show that the same characteristics that predict positive alphas also predict undesirable coskewness with the market. We sort funds by Industry concentration (Kacperczyk, Sialm, and Zheng 2005), Return gap (Kacperczyk, Sialm, and Zheng 2008), Active share (Cremers and Petajisto 2009), $$1-R^2$$ (Amihud and Goyenko 2013), a time-varying Skill index (Kacperczyk, Van Nieuwerburgh, and Veldkamp 2014), and Active weight (Doshi, Elkamhi, and Simutin 2015). In each case, higher alpha portfolios have worse coskewness. To determine the relative importance of different sources of coskewness, we run a cross-sectional regression of coskewness on a variety of fund characteristics. Controlling for other activities, coskewness is positively related to Market timing (a component of the Skill index), Active weight, and Industry concentration. The positive coefficient on Market timing is expected, because successful market timing creates a convex return profile, as discussed above. The positive coefficients on Active weight and Industry concentration suggest that these characteristics are also related to market timing.4 Together, these three variables account for 15% of the explained variation in coskewness. Variables related to cash management (illiquidity of holdings and abnormal cash holdings) explain an additional 7%. Derivatives usage and other disclosed activities account for 5% of the explained variation in coskewness. Stock selection, either dynamic or passive, explains the bulk of the remaining explained variation. Dynamic selection activities like turnover and Active share contribute negatively to coskewness, accounting for 15% of the explained variation in coskewness. Finally, the chosen fund style is an important determinant of coskewness. About half of the explained variation is due to the underlying styles of the funds. There is a clear trade-off between CAPM alphas and coskewness in the cross-section of fund returns: funds with higher alphas tend to have lower (worse) coskewness and vice versa. This trade-off cannot be generated by the first two sources of coskewness described above. Successful market timers create positive alphas and positive coskewness. Funds that have a greater need to hold cash when the market risk premium is high suffer both in average returns and in coskewness. So, each of the first two sources of coskewness would produce a positive relation between alphas and coskewness. The third source would create a negative relation under the assumptions of Jagannathan and Korajczyk (1986) and Leland (1999); however, only 24% of the funds in our sample report using options on equities, equity indices, or equity index futures. The fact that the relation between fund alphas and coskewness is negative suggests that it is predominantly driven by stock picking (other unobserved activities of mutual funds could also contribute to the relation; for example, funds could create synthetic options by dynamic trading). The presence of a trade-off between alpha and coskewness among stocks is implied by the coskewness pricing model (Kraus and Litzenberger 1976) and has been documented by Kraus and Litzenberger (1976), Harvey and Siddique (2000), Dittmar (2002), Guidolin and Timmermann (2008), and Christoffersen et al. (2016). In fact, portfolios formed from sorts on stock picking characteristics, such as Active share and $$1\!-\!R^2$$, exhibit larger estimated trade-offs than what we observe for stock portfolios on average, indicating that funds that are active in stock picking tend to pick stocks whose high (conditional) alphas carry especially high coskewness costs.5 We control for styles throughout. Harvey and Siddique (2000) show that styles associated with CAPM alphas are related to coskewness. Furthermore, standard style factors (SMB, HML, and UMD) exhibit a trade-off between alpha and coskewness: each has a positive alpha and negative coskewness. Styles account for some of the trade-off between alpha and coskewness in mutual fund returns, but the trade-off is significant even controlling for styles. If retail investors do not monitor coskewness (even though their utilities ultimately depend on it because they are not quadratic), then there is scope for active funds to exploit the trade-off the market presents, producing alpha at a coskewness cost. We show that fund flows do respond to coskewness in addition to responding to alphas.6 However, the flow response to coskewness is weaker among funds that serve fewer institutional investors, so some retail investors are apparently less aware of or at least less concerned about coskewness. This is consistent with funds having scope to exploit the trade-off between alpha and coskewness available in the market. We assess the importance of mutual fund coskewness both empirically and theoretically. Empirically, the difference in alpha between the bottom and top coskewness quartiles of funds is 4.8% per year, which is clearly economically meaningful. Furthermore, we use standard sets of stock portfolios to estimate the price of coskewness and conclude that a one-standard-deviation change in coskewness (based on the standard deviation across funds) is equivalent to a change in alpha of around 2.5% per year. For the theoretical assessment, we follow Dahlquist, Farago, and Tédongap (2017) in modeling return skewness via exponential-lognormal distributions and in assuming investors have the generalized disappointment aversion preferences of Routledge and Zin (2010). We find that our estimated price of coskewness is consistent with this model for reasonable parameter values. For the fund characteristics associated with abnormal performance, we adjust CAPM alphas of the high-minus-low portfolios by penalizing negative coskewness and rewarding positive coskewness based on the price of coskewness. The adjusted alpha is significant only for Active weight (Doshi, Elkamhi, and Simutin 2015). We conclude that Active weight is a characteristic that signals outperformance even when coskewness costs are considered. Moreno and Rodríguez (2009) also incorporate coskewness in mutual fund performance evaluation. They create a coskewness factor by sorting stocks based on coskewness with the market and following a procedure similar to that used by Fama and French (1993) to construct SMB and HML.7 One important difference between their paper and ours is that they study exposure to their coskewness factor whereas we study coskewness characteristics of mutual funds. We focus on the coskewness characteristic because we assume investors care about skewness and we are interested in the value that funds do or do not create for investors. Moreno and Rodríguez include the coskewness factor with the market and also with the Fama-French-Carhart (FFC) factors and compute mutual fund alphas. The price of their factor would be the price of coskewness if their factor were the portfolio return most highly correlated with the squared market return. Because the factor has essentially an ad hoc construction, the price of the factor may be less than the true price of coskewness (our estimate of the price of coskewness may also be biased downward due to an errors-in-variables attenuation bias). Moreno and Rodríguez show a trade-off between exposure to the factor and alphas. They also provide some analyses relating fund coskewness to fund size, expenses, turnover, and coarse investment categories. However, they do not study the fund characteristics recently shown to predict alphas, they do not study how flows respond to coskewness, and they do not study market timing, the effect of cash management on coskewness, or the use of derivatives by funds. 1. Coskewness: Definition and Estimation 1.1 Definition of coskewness Coskewness is the covariance of a return or excess return with a squared benchmark return. We use the market as the benchmark. Let $$R_m$$ denote the market return, and let $$R_f$$ denote the risk-free return. Given a return $$R$$, project its excess return on the market excess return as usual: \begin{equation} R - R_f = \alpha + \beta (R_m-R_f) + \varepsilon\,. \end{equation} (1) The random variable $$\alpha + \varepsilon$$ is the payoff of a zero-cost portfolio, which is an excess return. We define coskewness based on it as \begin{equation} {\text{cov}}(\alpha+\varepsilon,(R_m\!-\!R_f)^2) = \mathsf{E}[\varepsilon (R_m\!-\!R_f)^2]\,. \end{equation} (2) To see why coskewness should matter to an investor, consider a return $$R$$ and a constant $$\nu\geq 0$$ and construct the return \begin{equation} R_\nu \;=\; R_m + \nu[R-R_f-\beta(R_m-R_f)] \;=\; R_m + \nu [\alpha + \varepsilon]\,. \end{equation} (3) This is the benchmark combined with an investment in a zero-beta version of $$R$$. The derivatives of the first three moments of $$R_\nu$$ with respect to $$\nu$$ evaluated at $$\nu=0$$ are \begin{align} \left.\frac{\mathrm{d}\,\bar{R}_\nu}{\mathrm{d}\,\nu}\right\rvert_{\nu=0}&= \alpha \,,\\ \end{align} (4a) \begin{align} \left.\frac{\mathrm{d}\,{\text{var}}(R_\nu)}{\mathrm{d}\,\nu}\right\rvert_{\nu=0}&= 0\,, \\ \end{align} (4b) \begin{align} \frac{1}{3}\cdot\left.\frac{\mathrm{d}\,\mathsf{E}[(R_\nu-\bar{R}_\nu)^3]}{\mathrm{d}\,\nu}\right\rvert_{\nu=0}&= \mathsf{E}[\varepsilon (R_m\!-\!R_f)^2] \,. \end{align} (4c) The derivatives tell us the signs of the changes in the first three return moments produced by a small investment in the return, relative to holding the market. From (4a), we see that the investment increases the mean return if the alpha is positive. From (4b), we see that the investment has only a second-order effect on variance. Thus, a marginal investment in a return with a positive alpha can improve mean-variance efficiency (Dybvig and Ross 1985). From (4c), we see that a marginal investment in the return increases skewness if the return has positive coskewness. Therefore, a marginal investment in a return with a positive alpha and positive coskewness can improve mean-variance-skewness efficiency. If investors care only about the first three moments of returns, then there should be an exact negative linear relation between the CAPM alpha and coskewness. This is the coskewness pricing model (Kraus and Litzenberger 1976). See Appendix A for a derivation of this trade-off from the coskewness pricing model. More generally, if investors care about more than the first two moments of returns, then there should generally be a negative (and possibly nonlinear) relation between alphas and coskewness, as long as investors have decreasing absolute risk aversion and hence like both the first and third moments. We interpret the trade-off between alpha and residual coskewness as a cost of seeking alpha. This trade-off also could be viewed as an alpha cost to seeking positive coskewness. Since it is well established that fund managers have the incentive to “seek alpha,” we believe the former interpretation is most natural. The majority of tests, with the exception of those in Table IA.2, which regresses average returns on coskewness and other factor loadings, reflect this view by estimating the coskewness costs of seeking alpha. 1.2 Relation of coskewness to market timing measures A closely related measure of coskewness is obtained from the quadratic regression \begin{equation} R-R_{f} = a + b_{1} (R_{m} - R_{f}) + b_{2} (R_{m} - R_{f})^2 + \epsilon \,. \end{equation} (5) Note that $$a$$ and $$b_1$$ from (5) are not equivalent to $$\alpha$$ and $$\beta$$ from (1). It follows from the Frisch-Waugh Theorem that the multivariate regression coefficient $$b_{2}$$ equals the coskewness defined in (2) multiplied by a positive factor that depends on moments of the market return. See Appendix A for further detail on the relation between $$b_{2}$$ and coskewness from (2). The quadratic regression (5) is the regression of Treynor and Mazuy (1966), who interpret $$b_2>0$$ as an indication of successful market timing. Because the funds in our sample operate over different time periods, the market moments relating $$b_2$$ to coskewness (2) will be different for different funds; hence, the two parameters will not be proportional to each other empirically in the cross-section of funds. Because of the importance of coskewness (2) shown in (4c), we calculate it instead of $$b_2$$. However, qualitative cross-sectional results for one parameter also will be generally true for the other. When we reference work from the market timing literature, we generally refer to estimates of $$b_2$$ as coskewness. A persistent finding in the market timing literature is that $$b_2<0$$ for many funds, indicating “perverse” market timing ability by fund managers. Many papers argue that $$b_2$$ is a poor measure of market timing for various reasons. For instance, $$b_2$$ can be affected by time-varying betas (Edelen 1999; Ferson and Schadt 1996; Ferson and Warther 1996), it can be affected by the use of options or stocks exhibiting option-like features (Jagannathan and Korajczyk 1986), and it is affected by dynamic trading between return measurement dates (Pfleiderer and Bhattacharya 1983). In general, the market timing literature has not considered the implications of the coskewness pricing model in its relation to $$b_2$$. We are aware of only one exception, the study of market timing in bond funds by Chen, Ferson, and Peters (2010), which briefly mentions that the quadratic coefficient also has the interpretation as coskewness. More recent papers in the market timing literature make use of holdings-based measures (Jiang, Yao, and Yu 2007; Kacperczyk, Van Nieuwerburgh, and Veldkamp 2014). As discussed in the introduction, successful market timing should result in a positive correlation between alpha and coskewness, rather than the negative correlation documented in Section 2. In Section 3, we show that the holdings-based market-timing measure of Kacperczyk, Van Nieuwerburgh, and Veldkamp (2014) is positively related to coskewness when we control for other fund activities. 1.3 Data 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. The sample contains active domestic equity mutual funds.8 We exclude index and target date funds. To account for the incubation bias documented by Evans (2010), we only include fund returns for dates later than the fund’s reported first offer date and for dates for which the fund name is not missing in the CRSP dataset. We also exclude any funds whose total net assets are below $15 million or that hold fewer than ten stocks in each reporting period. Our analysis includes holdings-based style categories following Daniel et al. (1997) and Wermers (2003) (described in Section 2.3) and holdings-based attribution analyses (described in Section 2.4). The sample excludes any funds for which we cannot determine these values. We exclude funds with fewer than 60 days of returns. We also exclude any fund for which less than 90% of the daily observations in CRSP contain reported returns, which eliminates funds that report returns weekly rather than daily. The sample runs from September 1, 1998 through June 30, 2014, and includes 3,001 funds. The sample starts in 1998 as this is the initiation of daily return reporting for mutual funds by CRSP. We also study stock portfolios formed from sorts on characteristics that have been shown to spread returns. We use the same sample horizon as the mutual fund sample. We obtain benchmark market excess returns and risk-free returns from Kenneth French’s Web site.9 We use several portfolios from French’s Web site, specifically the 100 Fama-French portfolios formed by bivariate sorts on size and book-to-market, investment, or profitability as well as the 25 size and momentum portfolios. We use CRSP daily and monthly stock returns for various holdings-based measures described throughout the text. 1.4 Estimation of coskewness For each asset $$i$$, we estimate $$\alpha_i$$ and $$\beta_i$$ by OLS from the usual regression equation (1). Let $$\gamma_i$$ denote coskewness. Given a sample of size $$T_i$$, we estimate coskewness as \begin{equation} \qquad \widehat{\gamma}_i = \frac{1}{T_i} \sum_{t=1}^{T_i} \varepsilon_{it} (R_{mt} - R_{ft})^2\,, \end{equation} (6) where the $$\varepsilon_{it}$$ are the fitted residuals from the regression (1). For most of our results, we estimate the regression (1) and coskewness (6) once for each asset, using the full time series available for the asset. However, we also allow for time variation in parameters by estimating rolling regressions using 5 years of daily data in each regression. 2. The Trade-off between Alpha and Coskewness In this section, we show that alpha and coskewness are negatively related in the cross-section of mutual fund returns. We decompose alphas into an alpha due to average holdings and the residual, which is due to trading. Both parts are negatively related to coskewness. When we sort stocks by activity measures shown previously to predict alphas, we find that higher alpha portfolios have worse coskewness. These results are robust to controlling for styles. 2.1 Coskewness costs of funds First, we report the distribution of alpha and coskewness estimates for our sample. Table 1 reports summary statistics for funds over the full time-series (panel A) and for funds over 5-year windows rolling forward daily (panel B). Coskewness is negative for 68% of the funds, consistent with prior evidence by Moreno and Rodríguez (2009) that the median fund has negative coskewness. It is also consistent with the persistent finding in the market timing literature that $$b_2<0$$ for most funds. The coskewness pricing model implies that coskewness should be negatively related to alpha. Empirically, a