# Double-Adjusted Mutual Fund Performance

Double-Adjusted Mutual Fund Performance
Busse, Jeffrey A; Jiang, Lei; Tang, Yuehua
2021-02-15 00:00:00
Abstract Mutual fund returns are significantly related to stock characteristics in the cross-section after controlling for risk via factor models. We develop a new double-adjusted approach that controls for both factor model betas and stock characteristics in one performance measure. The new measure substantially affects performance rankings, with a quarter of funds experiencing a change in their percentile ranking greater than 10. Double-adjusted performance produces strong evidence of persistence in relative performance. Inference based on the new measure often differs, sometimes dramatically, from that based on traditional performance estimates. Received November 22, 2019; editorial decision June 28, 2020; Editor: Jeffrey Pontiff. 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. The performance evaluation of mutual fund managers is an enduring topic within financial economics. At the core of any performance analysis is the model used to determine the fund’s benchmark. Among the alternative techniques utilized over the years, the factor model regression approach of Jensen (1968, 1969) and, more recently, Carhart (1997) and the characteristic-based benchmark approach of Daniel et al. (DGTW, 1997) stand out for their simplicity, intuitive interpretation, and widespread use.1 Both approaches are parsimonious, yet control for major influences identified in the empirical asset pricing literature as significantly affecting the cross-section of stock returns. For example, both Carhart (1997) and DGTW approaches control for fund exposure to varying degrees of stock market capitalization, book-to-market ratio, and momentum, either via factor model betas, as in Carhart (1997), or via benchmark portfolio returns, as in DGTW. Evaluating a fund by either approach provides insight into the types of stocks held by the fund through the regression factor loadings or specific characteristic benchmarks, while identifying a return hurdle for the fund commensurate with its stock portfolio. The parsimonious structure of the models, however, may limit their ability to fully control for the passive effects they target. For instance, the empirical asset pricing literature has examined the incremental effect stock characteristics have on the cross-section of stock returns beyond what is captured by factor model betas. Brennan, Chordia, and Subramanyam (1998) and Chordia, Goyal, and Shanken (2019) find that stock characteristics, such as market capitalization, book-to-market ratio, and momentum, are all statistically significantly related to average stock returns after controlling for factor model betas.2 That is, cross-sectionally, stock returns remain related to market capitalization, for example, even after controlling for market capitalization via Fama and French’s (1993) SMB factor. Thus, factor models provide incomplete descriptions of average stock returns that typically relate to stock characteristics. The systematic shortcomings of factor models can affect inferences drawn from them, such as in the evaluation of mutual fund performance. Similar to the evidence on stock returns, we find that although mutual fund factor loadings and holding characteristics are correlated, the correlations are modest in magnitude (e.g., the average of the absolute value is about 0.62), which indicates that factor loadings and characteristics do not convey identical information. We find that characteristics explain the cross-section of mutual fund returns after controlling for exposure to risk factors, consistent with the notion that controlling for factor loadings does not fully eliminate the effects of market capitalization, book-to-market ratio, and momentum in mutual fund returns. As a result, standard alpha measures from factor model regressions of mutual fund net returns are significantly related in the cross-section to the characteristics of mutual fund portfolio holdings.3 For example, over our sample period from 1980m4 to 2016m12, funds in the top quintile of stock momentum (i.e., those holding the highest momentum stocks) have an annualized Carhart four-factor alpha that is 2.31% (t-stat.=4.13) greater than funds in the bottom quintile. Thus, funds can show higher relative performance by passively loading on characteristics, even when the performance measure explicitly controls for those effects. Motivated by this evidence, we develop a new mutual fund performance measure that controls for both types of influences. We base our measure on two variations of a two-step procedure, where we sequentially control first for exposure to factors and then for the characteristics of a mutual fund’s stock holdings. We first estimate the standard Carhart (1997) four-factor alpha based on net shareholder returns for a sample of actively managed U.S. domestic equity funds. We then use either a regression or a portfolio sorting approach in a second-pass cross-sectional adjustment. In our first approach, we regress cross-sectionally the four-factor alphas on fund portfolio holding characteristics (i.e., fund portfolio holding value-weighted averages of market capitalization, book-to-market ratio, and momentum). Based on the cross-sectional regression estimates, we decompose the standard four-factor alpha into two components: (1) double-adjusted performance, which we define as the sum of the intercept and a fund’s residual from the cross-sectional regression, and (2) characteristics-driven performance, the component attributable to exposure to stock characteristics, estimated as the difference between the standard four-factor alpha and double-adjusted performance. As an alternative to the cross-sectional regression, our second approach subtracts the mean four-factor alpha of a portfolio of funds that invest in stocks with similar size, book-to-market, and momentum characteristics to produce the double-adjusted performance measure. In this alternative, the mean characteristic-matched fund alpha represents the characteristic component of performance. It is important to note that, with either approach, by design, our second-pass adjustment only affects fund relative performance rankings in the cross-section, leaving the global mean of the double-adjusted alpha equal to the mean of the standard four-factor alpha. It is worth mentioning that characteristics-driven performance could consist of two components. One component is attributable to the factor model’s inability to fully control for passive effects associated with the characteristic. It is this component that can muddy the waters when interpreting standard factor model alpha as representing genuine skill. The other component of characteristics performance is associated with the ability of the mutual fund industry to generate abnormal net return (i.e., after expenses and trading costs), on average, within the space of stocks associated with the characteristic. For instance, to the extent that the market is not perfectly efficient among small cap stocks, the mutual fund industry might generate abnormal net performance, on average, via its small cap stock investments. However, we find little cross-sectional correspondence between the characteristic component of performance and future abnormal fund performance, consistent with the interpretation that the bulk of characteristic performance does not capture genuine skill.4 As a result, we interpret double-adjusted performance as a clean measure of fund skill, that is, abnormal performance that exists after fully controlling for effects related to characteristic exposure. We find that between a quarter and a third of a typical fund’s standard four-factor alpha is attributable to stock characteristics. More importantly, we find that our second-pass adjustment affects the inference associated with relative fund performance, sometimes quite dramatically. For example, about 25% of funds experience a change in performance percentile of greater than 10% with the regression approach and 15% with the portfolio approach, and 10% of funds experience a change in performance percentile greater than 16% with the regression approach and 24% with the portfolio approach. To the extent that the characteristic component of performance reflects passive effects, rather than ability, these results suggest that standard factor model estimates of performance do a poor job ranking funds based on fund skill. Since a fund’s double-adjusted performance estimate sometimes differs substantially from its four-factor alpha, inference based on double-adjusted performance can differ from inference based on four-factor alpha. For example, studies of performance persistence examine consistency in relative fund rankings over time (e.g., Carhart 1997; Bollen and Busse 2005). Ranking funds based on standard four-factor performance, we find modest evidence of long-term performance persistence. By contrast, after controlling for both factor exposure and characteristics, we find that double-adjusted performance predicts four-factor alpha as far as 8 years following the initial ranking. Characteristic-driven performance shows no positive correspondence to future fund performance, consistent with the idea that this component of performance is not substantially associated with genuine skill. Furthermore, the appraisal ratio associated with the top-bottom portfolio of funds selected according to their double-adjusted performance measure is 0.61, whereas the corresponding appraisal ratio for funds selected according to the standard four-factor alpha is 0.39. Thus, after removing the portion of factor model alpha that is attributable to the characteristics of portfolio holdings, we document new evidence that mutual fund skill persists over long periods of time. We also find strong evidence of short-term persistence (i.e., over the next month) via our new measure, where past top performing funds generate statistically significant positive performance in the future. Finally, we find that double-adjusted performance predicts future performance better than other recent approaches proposed in the literature to estimate alpha more precisely, including those that augment factor models with nonbenchmark passive assets (Pástor and Stambaugh 2002) or active peer benchmarks (Hunter et al. 2014). Beyond performance persistence, studies that emphasize relative fund performance include numerous analyses that relate performance to a particular fund feature, such as return gap (Kacperczyk, Sialm, and Zheng 2008), active share (Cremers and Petajisto 2009), or the factor model R-squared (Amihud and Goyenko 2013). When we use standard four-factor alpha performance measures, we confirm the major findings of these earlier mutual fund studies. However, after we adjust for the characteristics of fund stock holdings in the second stage of our measurement procedure, we find important changes that affect the way we interpret the results. For instance, we find that the significant relation between a fund’s standard four-factor alpha and its active share or factor model R-squared disappears after further adjusting the standard alpha measure for fund portfolio characteristics. Only the return gap is significantly related to double-adjusted performance. Taken together, our results suggest that the performance a fund generates via its exposure to particular stock characteristics, much of which may be attributable to passive effects rather than active effects, drives many of the relations documented in the literature. Finally, double-adjusted performance also affects inference based on the value-added measure proposed by Berk and van Binsbergen (2015). Our evidence suggests that domestic equity funds as a whole have better managerial skill based on the value-added measure (i.e., higher cross-sectional mean) after accounting for portfolio holding characteristics. To the extent that our double adjustment produces a cleaner measure of fund skill, our results suggest that many prior findings may not be fully driven by fund skill. While it is debatable whether or not fund managers actively choosing to emphasize certain stock characteristics in their portfolios is a specific dimension of skill, it seems difficult to argue for an approach that only partially adjusts for a particular influence. Our results suggest that the most commonly used performance measures do just that. We should note that the goal of our paper is not to argue that mutual fund benchmark models should control for anomalies beyond market capitalization, book-to-market ratio, and momentum, for example, as in Carhart (1997). Our point is that, for whichever set of anomalies addressed in a model, adjusting for both the factor betas and stock characteristics more fully controls for those influences than utilizing only one type of approach, similar to the insight of Brennan, Chordia, and Subramanyam (1998), Chordia, Goyal, and Shanken (2019), and Raponi, Robotti, and Zaffaroni (2020). Our paper contributes to the literature on mutual fund performance that applies innovations from the broader empirical asset pricing literature. To this point, advancements have largely proceeded either by expanding the set of factors used in the regression model, as in the move from the one-factor model of Jensen (1968, 1969) to the multifactor models of Elton et al. (1993) and Carhart (1997), or by the more radical move to nonparametric benchmarks that control for stock holding characteristics, as in Daniel et al. (1997).5 Our paper is the first to incorporate both approaches in one measure to produce an estimate of performance that more comprehensively controls for influences that are not necessarily attributable to manager skill. We find that double-adjusted performance predicts future fund performance better than standard factor model alphas or other approaches proposed to estimate alpha more precisely (e.g., Pástor and Stambaugh 2002; Hunter et al. 2014). Our analysis also relates to previous findings that growth funds have higher four-factor alphas than value funds (Chan, Chen, and Lakonishok 2002) as well as studies that identify flaws with standard approaches, including Chan, Dimmock, and Lakonishok (2009) and Cremers, Petajisto, and Zitzewitz (2013). Our findings provide new insight into how traditional performance measures attribute performance, while at the same time raising questions regarding what constitutes fund skill. 1. Data and Variables 1.1 Data description We obtain data from several sources. We take fund names, monthly returns, total net assets (TNA), expense ratios, investment objectives, and other fund characteristics from the Center for Research in Security Prices (CRSP) Survivorship Bias Free Mutual Fund Database. We obtain mutual fund portfolio holdings from the Thomson Reuters Mutual Fund Holdings (formerly CDA/Spectrum S12) database. The database contains quarterly or semi-annual portfolio holdings for all U.S. equity mutual funds. To address the issue of missing funds in the Thomson Reuters Mutual Fund Holdings database,6 we follow Zhu (2020) and extend the MFLINKS tables available via WRDS (see Wermers 2000) using the “crsp_cl_grp” variable from the CRSP Mutual Fund database and the CRSP Holdings database to supplement the Thomson Reuters holdings data. Lastly, we merge the CRSP Mutual Fund database and the Thomson Reuters Mutual Fund Holdings database. We examine actively managed U.S. equity mutual funds from April 1980 to December 2016.7 Similar to prior studies (e.g., Kacperczyk, Sialm, and Zheng 2008), we base our selection criteria on objective codes and on disclosed asset compositions.8 We exclude funds that have the following Investment Objective Codes in the Thomson Reuters Mutual Fund Holdings database: International, Municipal Bonds, Bond and Preferred, Balanced, and Metals. We identify and exclude index funds using their names and the CRSP index fund identifier.9 To be included in the sample, a fund’s average percentage of stocks in the portfolio as reported by CRSP must be at least 70%. We exclude funds with fewer than 10 stocks to focus on diversified funds. Following Elton et al. (2001), Chen et al. (2004), and Yan (2008), we exclude funds with less than $15 million in TNA. We further follow Evans (2010) and use the date the fund ticker was created to address incubation bias.10 Our final sample consists of 3,395 unique actively managed U.S. equity mutual funds and 451,599 fund-month observations. 1.2 Variable construction 1.2.1 Fund characteristics To measure performance, we compute alphas using the Carhart (1997) four-factor model based on fund net returns over a 24-month estimation period as follows: ri,t-rf,t=αi+∑k=14βi,kFk,t+εi,t.(1) We require a minimum of 12 monthly observations in our estimation. The four-factor model includes the CRSP value-weighted excess market return (Mktrf), size (SMB), book-to-market (HML), and momentum (UMD) factors from Kenneth French’s website. We also compute the Daniel et al. (1997) CS benchmark-adjusted return. We form 125 portfolios in June of each year based on a three-way quintile sort along the size (using the NYSE size quintile), book-to-market ratio, and momentum dimensions. The abnormal performance of a stock position is its return in excess of its DGTW benchmark portfolio, and the DGTW-adjusted return for each fund aggregates over all the component stocks using the most recent portfolio dollar value weighting. Since the CRSP Mutual Fund database lists multiple share classes separately, we create a sample of unique funds by aggregating across each fund’s share classes and define fund variables as follows. Fund TNA is the sum of portfolio assets across all share classes of a fund. The variable Fund age is the age of the oldest share class in the fund. Family TNA is the log of aggregate total TNA of each fund (not restricted to domestic equity funds) in a fund family, excluding the fund itself. We use the “mgmt_name” variable in CRSP (representing the fund’s management company) to assign funds to fund families. Expense ratio is the average expense ratio value-weighted across all fund share classes. We define Fund cash flow as the average monthly net growth in fund assets beyond capital gains and dividends (as in Sirri and Tufano 1998). 1.2.2 Portfolio holding characteristics For each stock in a fund’s portfolio, we obtain stock-level characteristics from CRSP and COMPUSTAT, including market capitalization, book-to-market ratio, and momentum. We only keep stocks with CRSP share codes 10 or 11 (i.e., common stock) and NYSE, AMEX, or NASDAQ listing. For each fund in our sample, we use individual stock holdings to calculate the monthly fund-level market capitalization, book-to-market ratio, and momentum. To calculate the fund-level statistic, we weight each firm-level stock characteristic according to its dollar weight in the most recent fund portfolio. Since fund holdings are usually available at a quarterly frequency, we obtain monthly measures by keeping fund holdings constant between quarters. We calculate the book-to-market ratio of a firm as the book value of equity (assumed to be available 6 months after the fiscal year end) divided by month-end market capitalization. We take book value from COMPUSTAT supplemented by the book values from Kenneth French’s website. We winsorize the book-to-market ratios at the 0.5% and 99.5% levels to eliminate outliers, although our results are not sensitive to this winsorization. We define momentum as the 11-month cumulative stock return over the period from month t – 12 to t – 2, consistent with the UMD factor’s definition of momentum. 2. Double-Adjusted Mutual Fund Performance Measures 2.1 Relation between standard measures and characteristics or factor loadings To provide initial evidence that standard mutual fund performance measures imperfectly control for the effects of the characteristics of the stocks held in fund portfolios, we examine the four-factor alpha of funds sorted into quintiles based on their holding value-weighted average market capitalization, book-to-market ratio, or momentum (lagged 1 month). Beginning with the 24th month during our 1980m4–2016m12 sample period, we estimate each month the standard four-factor alpha over the past 24 months ending that month. We then sort funds into quintiles based on their portfolio holding characteristics averaged over the past 24-month period, lagged 1 month (i.e., the 24-month average holding characteristics align with the 24-month regression time frame), and we compute the mean fund four-factor alpha for each characteristic quintile. Table 1 presents the results of this analysis, with panel A reporting summary statistics of the fund characteristics. Panel B reports the mean four-factor alpha (computed from 24 monthly returns) of funds in each characteristic quintile.11 The results indicate that for sorts associated with all three characteristics, the difference between the top quintile (which includes funds that hold stocks of the greatest market capitalization, book-to-market ratio, or momentum) and the bottom quintile (which includes funds that hold stocks with the smallest market capitalization, book-to-market ratio, or momentum) is statistically significant at the 10% level or lower. The magnitude of these differences is economically large. For instance, funds in the bottom quintile of stock holding book-to-market ratio have an annualized four-factor alpha that is 1.38% (t-stat. = 2.46) higher than funds in the top quintile. Funds in the top quintile of stock holding momentum have an annualized four-factor alpha that is 2.31% (t-stat. = 4.13) higher than funds in the top quintile. Thus, funds show higher four-factor performance by loading on characteristics, even when those characteristics are explicitly addressed in the four-factor model. Table 1 Fund holding characteristics versus four-factor alpha A. Fund holding characteristics statistics . Characteristic . Mean . SD . 1st percentile . Median . 99th percentile . Market cap ($ million) 37,957 40,207 336 19,375 147,944 Book-to-market 0.44 0.20 0.12 0.41 1.10 Return12m1 (%) 23.82 33.79 −31.35 19.09 150.79 A. Fund holding characteristics statistics . Characteristic . Mean . SD . 1st percentile . Median . 99th percentile . Market cap ($ million) 37,957 40,207 336 19,375 147,944 Book-to-market 0.44 0.20 0.12 0.41 1.10 Return12m1 (%) 23.82 33.79 −31.35 19.09 150.79 B. Four-factor alpha performance of holding characteristics sorts . Quintile . Market cap . Book-to-market ratio . Return12m1 . Bottom −0.10 0.69 −1.55 2 0.28 −0.60 −0.82 3 −0.51 −0.77 −0.29 4 −0.76 −0.58 −0.06 Top −0.85 −0.69 0.76 Top-bottom −0.75 −1.38** 2.31*** t-statistic (−1.62) (−2.46) (4.13) B. Four-factor alpha performance of holding characteristics sorts . Quintile . Market cap . Book-to-market ratio . Return12m1 . Bottom −0.10 0.69 −1.55 2 0.28 −0.60 −0.82 3 −0.51 −0.77 −0.29 4 −0.76 −0.58 −0.06 Top −0.85 −0.69 0.76 Top-bottom −0.75 −1.38** 2.31*** t-statistic (−1.62) (−2.46) (4.13) C. Four-factor alpha versus holding characteristics: Cross-sectional regressions . Market cap −0.270** −0.124 (−2.28) (−1.06) Book-to-market −1.052** −0.076 (−2.27) (−0.17) Return12m1 0.045*** 0.034*** (4.12) (2.96) Constant −0.408 −0.408 −0.408 −0.408 (−1.40) (−1.40) (−1.40) (−1.40) Adj. R-squared .021 .036 .043 .078 No. of months 441 441 441 441 C. Four-factor alpha versus holding characteristics: Cross-sectional regressions . Market cap −0.270** −0.124 (−2.28) (−1.06) Book-to-market −1.052** −0.076 (−2.27) (−0.17) Return12m1 0.045*** 0.034*** (4.12) (2.96) Constant −0.408 −0.408 −0.408 −0.408 (−1.40) (−1.40) (−1.40) (−1.40) Adj. R-squared .021 .036 .043 .078 No. of months 441 441 441 441 Panel A reports statistics for fund portfolio holding stock characteristics, including market capitalization, book-to-market ratio, and momentum (total return from t – 12 to t – 2). We weight each firm-level stock characteristic according to its dollar weight in the most recent fund portfolio holdings to calculate the monthly fund-level statistic. Panel B reports mean Carhart (1997) net four-factor alphas (each estimated over a 24-month period) for funds sorted into quintiles based on average portfolio characteristics during a 24-month ranking period (lagged 1 month with respect to the 24-month alpha estimation window). We compute t-statistics of the differences between the top and bottom quintiles with Newey-West (1987) correction for time-series correlation with 23 lags. Panel C reports average coefficients from monthly cross-sectional regressions, αi,t=a+∑m=1MZi,m,t-1cm+ηi,t,(2) where αi,t represents fund i’s four-factor alpha during month t, Zi,m,t-1 represents lagged 24-month average fund holding characteristics, including portfolio value-weighted measures of market capitalization, book-to-market ratio, or momentum (total return from t – 12 to t – 2). We standardize each of the holding characteristics by subtracting its monthly cross-sectional mean before including them in the regressions. We estimate the t-statistics in parentheses as in Fama and MacBeth (1973) with Newey-West (1987) correction for time-series correlation with 23 lags. The results reflect 441 individual monthly observations over a 1980m4–2016m12 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab Table 1 Fund holding characteristics versus four-factor alpha A. Fund holding characteristics statistics . Characteristic . Mean . SD . 1st percentile . Median . 99th percentile . Market cap ($ million) 37,957 40,207 336 19,375 147,944 Book-to-market 0.44 0.20 0.12 0.41 1.10 Return12m1 (%) 23.82 33.79 −31.35 19.09 150.79 A. Fund holding characteristics statistics . Characteristic . Mean . SD . 1st percentile . Median . 99th percentile . Market cap ($ million) 37,957 40,207 336 19,375 147,944 Book-to-market 0.44 0.20 0.12 0.41 1.10 Return12m1 (%) 23.82 33.79 −31.35 19.09 150.79 B. Four-factor alpha performance of holding characteristics sorts . Quintile . Market cap . Book-to-market ratio . Return12m1 . Bottom −0.10 0.69 −1.55 2 0.28 −0.60 −0.82 3 −0.51 −0.77 −0.29 4 −0.76 −0.58 −0.06 Top −0.85 −0.69 0.76 Top-bottom −0.75 −1.38** 2.31*** t-statistic (−1.62) (−2.46) (4.13) B. Four-factor alpha performance of holding characteristics sorts . Quintile . Market cap . Book-to-market ratio . Return12m1 . Bottom −0.10 0.69 −1.55 2 0.28 −0.60 −0.82 3 −0.51 −0.77 −0.29 4 −0.76 −0.58 −0.06 Top −0.85 −0.69 0.76 Top-bottom −0.75 −1.38** 2.31*** t-statistic (−1.62) (−2.46) (4.13) C. Four-factor alpha versus holding characteristics: Cross-sectional regressions . Market cap −0.270** −0.124 (−2.28) (−1.06) Book-to-market −1.052** −0.076 (−2.27) (−0.17) Return12m1 0.045*** 0.034*** (4.12) (2.96) Constant −0.408 −0.408 −0.408 −0.408 (−1.40) (−1.40) (−1.40) (−1.40) Adj. R-squared .021 .036 .043 .078 No. of months 441 441 441 441 C. Four-factor alpha versus holding characteristics: Cross-sectional regressions . Market cap −0.270** −0.124 (−2.28) (−1.06) Book-to-market −1.052** −0.076 (−2.27) (−0.17) Return12m1 0.045*** 0.034*** (4.12) (2.96) Constant −0.408 −0.408 −0.408 −0.408 (−1.40) (−1.40) (−1.40) (−1.40) Adj. R-squared .021 .036 .043 .078 No. of months 441 441 441 441 Panel A reports statistics for fund portfolio holding stock characteristics, including market capitalization, book-to-market ratio, and momentum (total return from t – 12 to t – 2). We weight each firm-level stock characteristic according to its dollar weight in the most recent fund portfolio holdings to calculate the monthly fund-level statistic. Panel B reports mean Carhart (1997) net four-factor alphas (each estimated over a 24-month period) for funds sorted into quintiles based on average portfolio characteristics during a 24-month ranking period (lagged 1 month with respect to the 24-month alpha estimation window). We compute t-statistics of the differences between the top and bottom quintiles with Newey-West (1987) correction for time-series correlation with 23 lags. Panel C reports average coefficients from monthly cross-sectional regressions, αi,t=a+∑m=1MZi,m,t-1cm+ηi,t,(2) where αi,t represents fund i’s four-factor alpha during month t, Zi,m,t-1 represents lagged 24-month average fund holding characteristics, including portfolio value-weighted measures of market capitalization, book-to-market ratio, or momentum (total return from t – 12 to t – 2). We standardize each of the holding characteristics by subtracting its monthly cross-sectional mean before including them in the regressions. We estimate the t-statistics in parentheses as in Fama and MacBeth (1973) with Newey-West (1987) correction for time-series correlation with 23 lags. The results reflect 441 individual monthly observations over a 1980m4–2016m12 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab An important caveat exists in the way we interpret the results of Table 1, panel B. Although the results show a significant correlation between characteristic exposure and four-factor alpha, the existence of the correspondence, by itself, is not sufficient to conclude that the four-factor model inadequately controls for passive effects related to size, book-to-market, or momentum. For instance, the significant correspondence between fund characteristic exposure and performance could be driven, in part, by the fund industry delivering alpha, on average, by investing in stocks of a particular characteristic. To the extent that the market is not perfectly efficient among small cap stocks, for example, the mutual fund industry might generate abnormal performance via small cap stock investments. We later examine the extent to which the component of performance associated with characteristic exposure predicts future fund performance, with an expectation that representations of genuine fund skill should correlate with future fund performance. For now, we can only say that four-factor mutual fund performance remains cross-sectionally correlated to characteristics that closely relate to SMB, HML, and UMD factor loadings. The evidence in panel B is consistent with recent developments in the asset pricing literature. Brennan, Chordia, and Subramanyam (1998) find that, after adjusting for risk factors, stock characteristics, such as market capitalization, book-to-market ratio, and momentum, capture additional aspects of the cross-section of stock returns. Similarly, Chordia, Goyal, and Shanken (2019) find that both factor loadings and stock characteristics explain cross-sectional variation of stock returns. To provide an additional indication of the relation between standard factor model alphas and the characteristics of the funds’ stock holdings, we regress cross-sectionally the Carhart (1997) four-factor fund alphas, αi , on their 24-month average fund holding characteristics. That is, αi,t=a+∑m=1MZi,m,t-1cm+ηi,t,(2) where Zi,m,t-1 represents lagged fund holding characteristics, including portfolio value-weighted measures of market capitalization, book-to-market ratio, and momentum. We standardize each of the holding characteristics by subtracting its monthly cross-sectional mean before including them in the regressions. In regression (2), we examine specifications that control for each characteristic by itself (i.e., M = 1) as well as a specification that controls for all independent variables jointly (i.e., M = 3). Table 1, panel C, shows the results, where we compute the mean regression coefficients across all sample months. To address time series correlation due to the overlap in estimation windows, we calculate Fama and MacBeth (1973) t-statistics with Newey-West (1987) correction for time-series correlation with 23 lags. Similar to the inference associated with the results in panel B, the results in panel C again show that, on average, standard fund performance measures are sensitive to the characteristics of the stocks held in fund portfolios. All three univariate regression results show a statistically significant relation at the 5% or 1% level between the performance measure and the value-weighted mean stock characteristic. Evidence again suggests that, on average over the full sample period, four-factor alphas remain significantly related to size, value, and momentum effects.12 Although the results shown in Table 1, panels B and C, reflect average effects over the full 1980m4–2016m12 sample period, we find that the extent to which the four-factor model adjusts for the market capitalization, book-to-market, and momentum effects varies depending on the sample period. In particular, during periods of time when an especially strong cross-sectional relation exists between the characteristic and stock returns, the four-factor model under-adjusts for the effect. For example, during months when the momentum effect is strong, funds with high momentum holding characteristics show positive performance via the four-factor model compared to funds with low momentum holdings. Consequently, the inference associated with a particular set of four-factor model performance estimates could reflect performance estimates that do not accurately adjust for market capitalization, book-to-market, and momentum effects.13 Finally, we calculate the fraction of the cross-sectional variance of fund returns explained by factor loadings and characteristics following the procedure in Chordia, Goyal, and Shanken (2019). Each month t, we estimate the cross-sectional relation between fund returns and fund factor loadings and/or characteristics, ri,t-rf,t=γ0,t+∑k=14γ1,k,tβi,k,t-1+∑m=13γ2,m,tZi,m,t-1+ei,t,(3) where ri,t represents fund i’s return during month t, βi,k,t-1 represents fund i’s factor loadings on Mktrf, SMB, HML, and UMD, estimated with monthly fund returns over the prior 24 months, and Zi,m,t-1 represent fund i’s holding characteristics in month t –1, including portfolio value-weighted measures of market capitalization, book-to-market ratio, and momentum. Although no consensus in the asset pricing literature exists on the underlying reasons both factor loadings and characteristics relate to stock returns,14 our view is that as long as Equation (3) more fully captures the passive influences of size, book-to-market ratio, and momentum on stock returns compared to using only one type of control, we should use it to identify performance attributable to fund manager skill. Similar to Carhart (1997) and DGTW (1997), we assume that the passive influences of size, book-to-market ratio, and momentum are not considered performance. We first compute the cross-sectional variance of the fitted value of regression (3) each month based on both factor loadings and characteristics. We then calculate the two components of the fitted value—one driven by factor loadings and the other driven by characteristics—and calculate the cross-sectional variances of the two components. Lastly, we compute the ratio of these latter two variances to that of the full model each month and compute the average ratio across all sample months. We find that both factor loadings and characteristics contribute an economically meaningful fraction of the cross-sectional variance of fund returns, with factor loadings accounting for 57.4% and characteristics accounting for 49.2%.15 We also compute the time-series average of the cross-sectional correlations between fund factor loadings and holding characteristics and find that while factor loadings are correlated with fund holding characteristics (as expected), the correlations are not particularly high. The correlations between the SMB loading and holding size, between the HML loading and holding book-to-market ratio, and between the UMD loading and holding stock momentum are −0.79, 0.68, and 0.38, respectively. This evidence further suggests that factor loadings and characteristics do not convey identical information.16 2.2 Definition of double-adjusted performance measures Section 2.1 shows strong and robust evidence that mutual fund returns are significantly related to stock characteristics in the cross-section after controlling for risk via factor models, and vice versa. Thus, controlling only for factor loadings, as in Carhart (1997), or only for characteristics, as in DGTW, may overlook the other effect, and in so doing materially affect the performance measures. To control for both types of return influences, based on Equation (2), we formally define mutual fund double-adjusted performance as αi*=αi-∑m=1MZi,mcm.(4) We define characteristic-driven performance as αichar=∑m=1MZi,mcm.(5) We utilize two alternative approaches to calculate our double-adjusted performance measure, both based on a two-step procedure. In both alternatives, we first compute in-sample alphas via the Carhart (1997) four-factor model as in Equation (1) over a 24-month estimation period, rolling this window a month at a time.17 In our first approach, for each month in our sample period, we regress cross-sectionally the four-factor alphas on fund portfolio holding characteristics averaged over the past 24 months, lagged 1 month, using all sample funds in that month.18 We standardize each of the holding characteristics by subtracting its monthly cross-sectional mean before including them in the regressions. The demeaning procedure ensures that the intercept of each monthly regression equals the cross-sectional mean four-factor alpha, so that our second-stage adjustment only affects relative performance rankings. In this approach, we define double-adjusted performance as the sum of the intercept and the residual of a fund from the cross-sectional regression. Characteristics-driven performance, the component attributable to exposure to stock characteristics, is the difference between the standard four-factor alpha and double-adjusted performance. Controlling for the factor model first and characteristics second follows Brennan, Chordia, and Subramanyam (1998). In our second alternative approach, for each month in our sample period, we assign each fund to a cell based on either a three-way sequential tercile or a quartile sort (i.e., 3 × 3×3 or 4 × 4×4) on fund portfolio stock holding characteristics (size, book-to-market ratio, and momentum, in that order) averaged over the past 24 months, lagged 1 month.19 We calculate the mean alpha in each cell and subtract the global mean alpha of all sample funds from it. In this approach, the characteristic-matched demeaned alpha represents the fund’s characteristics-driven performance. The difference between the fund’s standard four-factor alpha and its characteristic-matched demeaned alpha is the fund’s double-adjusted alpha. Note that subtracting the global mean alpha from the mean characteristic alphas again ensures that our procedure only affects relative performance rankings, leaving the global mean double-adjusted alpha equal to the mean standard four-factor alpha. For both alternative approaches, the sum of the two performance components, the double-adjusted performance and the characteristics-driven performance, always equals the standard four-factor alpha, as in Equations (4) and (5).20 We measure performance based on net shareholder returns. The advantage of using net shareholder returns is that prior research indicates that fund actions that occur between quarterly portfolio holding snapshots materially affect shareholder returns (see Kacperczyk, Sialm, and Zheng 2008; Puckett and Yan 2011; Busse et al. 2020). Moreover, the relative fund performance literature that our paper builds on emphasizes performance measures based on net returns, likely because net returns reflect actual fund performance to investors.21 We note that net shareholder returns incorporate fund transaction costs and expenses, which both may be related, on average, to fund characteristic exposure. When based on net shareholder returns, the mean levels of expenses and transaction costs associated with exposure to a characteristic are thus removed from the double-adjusted measure during the second stage. Since we rely on quarterly fund portfolio holdings to measure fund characteristics, our second-stage control does not account for intraquarterly characteristics that differ from quarter-end characteristics. Consequently, for a subset of our sample, we use fund transaction data from Abel Noser Solutions, a leading provider of trade execution analysis for institutional investors, to infer fund holding characteristics between the quarter-end holding snapshots. We first merge mutual fund holding data from Thomson Reuters with trade data to get daily holding data for a sample of 556 actively managed U.S. equity mutual funds from Abel Noser from January 1999 to September 2011.22 We combine daily portfolio weights with stock characteristics to get daily portfolio holding value-weighted averages of size, book to market, and momentum (506,806 fund-day observations). When we compare these daily fund holding characteristics to fund holding characteristics at month-end based on Thomson Reuters holdings data, we find little difference.23 Therefore, the portfolio snapshots that we use in the paper capture the vast majority of a fund’s characteristic exposure across time. 2.3 Effects of double adjustment The results in Section 2.1 demonstrate that standard multifactor abnormal performance estimates are cross-sectionally correlated with fund characteristics. Our double adjustment procedure removes the characteristics-related component from the factor model performance estimate. In this section, we examine the extent to which the second adjustment in our two-stage procedure affects performance. We begin by estimating the fraction of standard alphas that is driven by exposure to characteristics. Later, we estimate the difference in fund percentile performance rankings before and after the second adjustment. That is, we examine whether the difference between standard performance measures and our new performance measure is economically meaningful. Based on Equations (4) and (5), for a given fund, we can estimate the fraction of its standard performance measure that is attributable to characteristics, that is, the ratio of the absolute value of the characteristic-driven component to the sum of the absolute values of the double-adjusted and characteristics components: fracichar=abs(αichar)absαi*+absαichar.(6) The remaining fraction, 1-fracichar , is attributable to double-adjusted performance. Table 2, panel A, reports statistics for the Equation (6) ratio. The median ratio across our sample is 0.23 and 0.32 for the regression and portfolio approaches, respectively. If we focus on the subsample of fund observations where the two components have the same sign (about half of the full sample), the results are similar, as the median ratio is 0.22 and 0.31 for the regression and portfolio approaches, respectively, as shown in Table IA.2 in the Internet Appendix. The statistics indicate that characteristics account for between roughly one quarter and one third of traditional four-factor abnormal performance estimates for a typical fund. Table 2 Double-adjusted performance effects . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . A. Performance attribution Double-adjusted Regression .703 .204 .338 .570 .765 .893 .957 .978 Portfolio .637 .145 .255 .471 .683 .842 .934 .966 Characteristics Regression .297 .022 .043 .107 .235 .430 .662 .796 Portfolio .363 .034 .066 .158 .317 .529 .745 .855 B. Change in performance rank Rank (%) Regression 0.00 −17.25 −11.43 −4.11 0.33 5.06 10.77 15.00 Portfolio 0.00 −25.03 −17.12 −6.69 0.23 7.42 16.71 23.02 Abs. rank (%) Regression 6.90 0.22 0.51 1.72 4.63 9.50 16.06 21.40 Portfolio 10.27 0.36 0.84 2.68 7.06 14.55 23.99 30.89 . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . A. Performance attribution Double-adjusted Regression .703 .204 .338 .570 .765 .893 .957 .978 Portfolio .637 .145 .255 .471 .683 .842 .934 .966 Characteristics Regression .297 .022 .043 .107 .235 .430 .662 .796 Portfolio .363 .034 .066 .158 .317 .529 .745 .855 B. Change in performance rank Rank (%) Regression 0.00 −17.25 −11.43 −4.11 0.33 5.06 10.77 15.00 Portfolio 0.00 −25.03 −17.12 −6.69 0.23 7.42 16.71 23.02 Abs. rank (%) Regression 6.90 0.22 0.51 1.72 4.63 9.50 16.06 21.40 Portfolio 10.27 0.36 0.84 2.68 7.06 14.55 23.99 30.89 This table shows the decomposition of standard net four-factor alpha into the double-adjusted performance and characteristic-related components and the effects of the double-adjusted procedure on relative performance ranking. Panel A reports statistics associated with the fraction of standard four-factor alpha attributable to characteristics, fracichar=abs(αichar)absαi*+absαichar.(6) and the fraction of double-adjusted performance, 1-fracichar . Panel B reports statistics that describe the change in the performance percentile from standard four-factor alpha to the double-adjusted measure using the regression or portfolio approach. The results reflect 451,599 fund-month observations over a 1980m4–2016m12 sample period. Open in new tab Table 2 Double-adjusted performance effects . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . A. Performance attribution Double-adjusted Regression .703 .204 .338 .570 .765 .893 .957 .978 Portfolio .637 .145 .255 .471 .683 .842 .934 .966 Characteristics Regression .297 .022 .043 .107 .235 .430 .662 .796 Portfolio .363 .034 .066 .158 .317 .529 .745 .855 B. Change in performance rank Rank (%) Regression 0.00 −17.25 −11.43 −4.11 0.33 5.06 10.77 15.00 Portfolio 0.00 −25.03 −17.12 −6.69 0.23 7.42 16.71 23.02 Abs. rank (%) Regression 6.90 0.22 0.51 1.72 4.63 9.50 16.06 21.40 Portfolio 10.27 0.36 0.84 2.68 7.06 14.55 23.99 30.89 . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . A. Performance attribution Double-adjusted Regression .703 .204 .338 .570 .765 .893 .957 .978 Portfolio .637 .145 .255 .471 .683 .842 .934 .966 Characteristics Regression .297 .022 .043 .107 .235 .430 .662 .796 Portfolio .363 .034 .066 .158 .317 .529 .745 .855 B. Change in performance rank Rank (%) Regression 0.00 −17.25 −11.43 −4.11 0.33 5.06 10.77 15.00 Portfolio 0.00 −25.03 −17.12 −6.69 0.23 7.42 16.71 23.02 Abs. rank (%) Regression 6.90 0.22 0.51 1.72 4.63 9.50 16.06 21.40 Portfolio 10.27 0.36 0.84 2.68 7.06 14.55 23.99 30.89 This table shows the decomposition of standard net four-factor alpha into the double-adjusted performance and characteristic-related components and the effects of the double-adjusted procedure on relative performance ranking. Panel A reports statistics associated with the fraction of standard four-factor alpha attributable to characteristics, fracichar=abs(αichar)absαi*+absαichar.(6) and the fraction of double-adjusted performance, 1-fracichar . Panel B reports statistics that describe the change in the performance percentile from standard four-factor alpha to the double-adjusted measure using the regression or portfolio approach. The results reflect 451,599 fund-month observations over a 1980m4–2016m12 sample period. Open in new tab One might anticipate that removing the characteristics component of performance from the standard measure of abnormal performance could materially affect fund performance rankings. When we compare percentile performance rankings of standard four-factor performance estimates to our double-adjusted performance estimate, the mean (median) absolute change in percentile performance estimate is 6.9% (4.6%) based on the regression double-adjustment approach and 10.3% (7.1%) based on the portfolio approach. As a point of comparison, the median change in performance from a Fama-French three-factor performance estimate to the Carhart four-factor performance estimate is 3%. Furthermore, many funds experience dramatic changes in performance: about 25% of funds experience a change in performance percentile greater than 10% with the regression approach and 15% with the portfolio approach, and 10% of funds experience a change in performance percentile greater than 16% with the regression approach and 24% with the portfolio approach.24 3. Impact on Analysis of Mutual Fund Performance Persistence Both the fraction of standard alpha attributable to characteristics and the degree to which the new double-adjusted measure affects fund performance suggest that the new performance measure could affect the inference of studies that analyze relative performance rankings. Central to the empirical mutual fund literature, studies that focus on relative performance rankings include analyses on performance persistence. Long-term persistence studies, such as Carhart (1997), analyze the tendency for relative performance rankings to persist for at least 1 year beyond the ranking period. Short-term persistence studies, such as Bollen and Busse (2005), analyze persistence in relative performance rankings over shorter time periods, up to one quarter, for example.25 In this section, we explore how the double-adjusted performance measure affects inference in the analysis of performance persistence, over both long and short post-ranking periods. Specifically, we examine persistence in standard alpha performance measures as well as the two components of performance defined in Equations (4) and (5), that is, our double-adjusted measure and the component attributable to characteristics. To the extent that our double-adjusted measure of performance represents a cleaner estimate of fund skill, analyzing both components of performance will indicate whether evidence of persistence is attributable to fund manager skill or to effects related to characteristics. 3.1 Short-term persistence We begin with short-term persistence, where we examine whether fund performance during a ranking period persists to the following month (i.e., the 1-month post-ranking period). Each month, we sort funds into deciles based on performance measures estimated over the 24-month time period ending that month. The post-ranking month is the month immediately following the 24-month ranking period. We sort based on four different performance measures: standard four-factor alpha, the two components of standard performance, and, for comparison purposes, the 24-month average DGTW CS measure. We move the ranking and post-ranking periods forward 1 month at a time. For post-ranking performance, we concatenate the returns of all post-ranking months for each decile and estimate four-factor alphas across the concatenated time series of monthly returns (similar to Carhart 1997). We examine post-ranking four-factor performance, rather than the characteristic-based DGTW measure, because four-factor performance utilizes actual shareholder returns, rather than a proxy for returns gleaned from fund portfolio holdings. Table 3 shows the short-term persistence results. The table reports the 1-month post-ranking performance measures estimated over the time series of concatenated post-ranking months. The results show strong evidence of persistence in the standard four-factor alpha. The 4.58% annualized difference in post-ranking top-bottom performance is both statistically and economically significant. We also find strong evidence that the double-adjusted performance measure predicts future four-factor performance. The annualized top-bottom post-ranking abnormal net return difference based on the regression (portfolio) double-adjusted approach is 4.37 (4.21) with t-statistic of 7.43 (7.62). By contrast, the returns associated with characteristics only weakly correspond to future four-factor performance, with the difference in the post-ranking performance of the top and bottom deciles showing t-statistics of 1.67 (1.65) based on the regression (portfolio) double-adjusted approach. Table 3 Short-term persistence sorts . Model . . . Double-adjusted . Characteristics . . Decile . Four-factor . Regression . Portfolio . Regression . Portfolio . DGTW CS . Bottom −3.20 −2.90 −2.93 −2.05 −1.69 −0.87 2 −1.74 −2.01 −1.85 −1.52 −1.83 −0.81 3 −1.54 −1.40 −1.45 −1.37 −1.39 −0.89 4 −1.02 −0.82 −1.01 −1.28 −0.82 −1.09 5 −0.86 −0.93 −0.77 −1.02 −1.19 −1.13 6 −0.89 −0.90 −0.75 −0.90 −1.27 −0.79 7 −0.66 −0.43 −0.50 −0.72 −0.60 −1.07 8 −0.21 −0.48 −0.35 −0.37 −0.62 −0.78 9 −0.15 −0.47 −0.56 −0.27 0.27 −0.76 Top 1.39 1.47 1.28 0.62 0.41 −0.36 Top-bottom 4.58*** 4.37*** 4.21*** 2.66* 2.10* 0.51 t-statistic (5.13) (7.43) (7.62) (1.67) (1.65) (0.49) . Model . . . Double-adjusted . Characteristics . . Decile . Four-factor . Regression . Portfolio . Regression . Portfolio . DGTW CS . Bottom −3.20 −2.90 −2.93 −2.05 −1.69 −0.87 2 −1.74 −2.01 −1.85 −1.52 −1.83 −0.81 3 −1.54 −1.40 −1.45 −1.37 −1.39 −0.89 4 −1.02 −0.82 −1.01 −1.28 −0.82 −1.09 5 −0.86 −0.93 −0.77 −1.02 −1.19 −1.13 6 −0.89 −0.90 −0.75 −0.90 −1.27 −0.79 7 −0.66 −0.43 −0.50 −0.72 −0.60 −1.07 8 −0.21 −0.48 −0.35 −0.37 −0.62 −0.78 9 −0.15 −0.47 −0.56 −0.27 0.27 −0.76 Top 1.39 1.47 1.28 0.62 0.41 −0.36 Top-bottom 4.58*** 4.37*** 4.21*** 2.66* 2.10* 0.51 t-statistic (5.13) (7.43) (7.62) (1.67) (1.65) (0.49) The table reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on different performance measures during a 24-month ranking period: (a) standard four-factor alpha, (b) the two components of four-factor alpha (based on both the regression and portfolio approaches), and (c) the 24-month average DGTW characteristic selectivity (CS) measure. The four-factor alpha in the post-ranking month is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. The results reflect 440 individual monthly observations over a 1980m5–2016m12 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab Table 3 Short-term persistence sorts . Model . . . Double-adjusted . Characteristics . . Decile . Four-factor . Regression . Portfolio . Regression . Portfolio . DGTW CS . Bottom −3.20 −2.90 −2.93 −2.05 −1.69 −0.87 2 −1.74 −2.01 −1.85 −1.52 −1.83 −0.81 3 −1.54 −1.40 −1.45 −1.37 −1.39 −0.89 4 −1.02 −0.82 −1.01 −1.28 −0.82 −1.09 5 −0.86 −0.93 −0.77 −1.02 −1.19 −1.13 6 −0.89 −0.90 −0.75 −0.90 −1.27 −0.79 7 −0.66 −0.43 −0.50 −0.72 −0.60 −1.07 8 −0.21 −0.48 −0.35 −0.37 −0.62 −0.78 9 −0.15 −0.47 −0.56 −0.27 0.27 −0.76 Top 1.39 1.47 1.28 0.62 0.41 −0.36 Top-bottom 4.58*** 4.37*** 4.21*** 2.66* 2.10* 0.51 t-statistic (5.13) (7.43) (7.62) (1.67) (1.65) (0.49) . Model . . . Double-adjusted . Characteristics . . Decile . Four-factor . Regression . Portfolio . Regression . Portfolio . DGTW CS . Bottom −3.20 −2.90 −2.93 −2.05 −1.69 −0.87 2 −1.74 −2.01 −1.85 −1.52 −1.83 −0.81 3 −1.54 −1.40 −1.45 −1.37 −1.39 −0.89 4 −1.02 −0.82 −1.01 −1.28 −0.82 −1.09 5 −0.86 −0.93 −0.77 −1.02 −1.19 −1.13 6 −0.89 −0.90 −0.75 −0.90 −1.27 −0.79 7 −0.66 −0.43 −0.50 −0.72 −0.60 −1.07 8 −0.21 −0.48 −0.35 −0.37 −0.62 −0.78 9 −0.15 −0.47 −0.56 −0.27 0.27 −0.76 Top 1.39 1.47 1.28 0.62 0.41 −0.36 Top-bottom 4.58*** 4.37*** 4.21*** 2.66* 2.10* 0.51 t-statistic (5.13) (7.43) (7.62) (1.67) (1.65) (0.49) The table reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on different performance measures during a 24-month ranking period: (a) standard four-factor alpha, (b) the two components of four-factor alpha (based on both the regression and portfolio approaches), and (c) the 24-month average DGTW characteristic selectivity (CS) measure. The four-factor alpha in the post-ranking month is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. The results reflect 440 individual monthly observations over a 1980m5–2016m12 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab Thus, our interpretation is that the double-adjusted performance measure captures fund skill more precisely than standard alpha (as reflected in the higher Table 3 t-statistics) because the double-adjustment process extracts the characteristic component that only weakly relates to future performance. One explanation for the weak predictability of the characteristics-related component of alpha is that characteristic premiums in stock returns do not persist over time. We test this conjecture and indeed find that, after removing the premiums associated with factor loadings, the characteristic premiums of size, value, and momentum in the stock market do not persist from 1 month to the next (see Table IA.3 in the Internet Appendix).26 We also find statistically significant positive four-factor performance in the top post-ranking decile sorted by the double-adjusted measure. That is, funds that performed well in the past (i.e., the top decile) produce a statistically significant positive abnormal performance of approximately 1.28% to 1.47% annualized (with t-statistics equal to 2.49 and 2.22 for the regression and portfolio approach double-adjusted alphas, respectively) over the subsequent month. This result suggests that the evidence of short-term persistence based on the double-adjusted measure is not solely attributable to persistence in the poorly performing funds. Lastly, we find that the 24-month average DGTW CS measure does not predict future four-factor fund performance, with a statistically insignificant 0.51% difference between the top and bottom post-ranking deciles. Together with the other persistence results, this evidence suggests that controlling for both risk factors and characteristics provides a cleaner picture of fund manager skill, insofar as such controls produce a performance measure that more closely aligns with future performance. 3.2 Long-term persistence Next, we turn to long-term persistence. We use the same set of performance estimates that we use in the short-term persistence analysis. We aggregate the ranking period alphas in each calendar year (i.e., we average monthly alphas over the 12 months in a calendar year, with each monthly alpha estimated over a 24-month window ending that month) and move the ranking period forward 1 year at a time. We sort funds into deciles based on the ranking period alphas and compute mean returns each month for each decile in post-ranking periods ranging from 1 to 10 years. We then estimate four-factor alphas for each decile over each post-ranking year using concatenated time series of post-ranking returns.27 Table 4 shows the long-term persistence results based on net fund returns. Table IA.4 in the Internet Appendix provides analogous results based on gross fund returns (i.e., where we add one-twelfth of the annual expense ratio back to the shareholder monthly return). Panel A sorts based on standard four-factor alpha; panels B and C sort based on double-adjusted alpha; and panels D and E sort based on characteristic-driven alpha. We utilize both the regression-based approach for computing the double-adjusted performance (panels B and D) and the portfolio approach (panels C and E). The results for each post-ranking year reflect noncumulative post-ranking periods, so that the year 10 results reflect standard four-factor performance only during the tenth year after the initial ranking, rather than the performance across all 10 post-ranking years. Table 4 Long-term persistence sorts . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . A. Four-factor alpha 1 −1.74 −2.13 −2.35 −0.80 −0.70 −1.46 −1.04 −1.09 −1.02 −0.75 2 −1.39 −0.92 −1.45 −1.04 −0.88 −1.17 −0.60 −1.08 −1.36 −0.73 3 −1.14 −1.27 −1.23 −1.61 −0.55 −0.88 −0.65 −1.08 −0.88 −0.53 4 −1.19 −1.14 −1.24 −1.13 −0.84 −0.64 −0.95 −1.37 −1.15 −0.83 5 −0.77 −0.79 −1.18 −1.53 −0.83 −0.77 −0.89 −1.61 −0.67 −0.35 6 −0.97 −0.73 −0.92 −0.71 −0.67 −0.57 −1.23 −0.93 −0.84 −0.50 7 −1.03 −0.80 −0.88 −0.95 −1.08 −0.70 −0.39 −0.91 −0.90 −0.73 8 −0.78 −0.64 −0.50 −0.60 −0.74 −0.59 −0.55 −0.27 −0.49 −1.01 9 −0.37 −0.64 −0.67 −0.53 −0.79 −0.41 −0.58 −0.80 −0.29 −0.71 10 −0.02 −0.21 0.15 −0.14 −0.51 0.02 −0.11 −0.26 −0.09 −1.19 10-1 1.72* 1.93** 2.50*** 0.66 0.20 1.47** 0.93 0.83 0.93 −0.44 t-stat (1.96) (2.49) (3.18) (1.00) (0.28) (1.98) (1.21) (1.15) (1.21) (−0.50) B. Double-adjusted alpha, regression approach 1 −1.74 −2.15 −2.15 −1.00 −0.66 −1.75 −1.28 −1.47 −1.40 −1.20 2 −1.50 −1.01 −1.46 −1.14 −1.33 −1.10 −0.96 −1.37 −0.75 −1.19 3 −1.08 −0.97 −1.35 −1.23 −0.75 −0.86 −0.85 −0.71 −1.03 −0.38 4 −0.99 −1.01 −1.18 −1.32 −0.93 −1.03 −0.80 −1.47 −1.25 −0.59 5 −1.18 −1.37 −1.32 −1.01 −1.03 −0.84 −1.06 −1.38 −0.99 −0.52 6 −1.04 −1.03 −0.77 −1.03 −1.09 −0.42 −0.73 −0.68 −0.96 −0.84 7 −1.05 −0.79 −0.79 −0.79 −0.77 −0.42 −0.34 −0.88 −0.80 −0.66 8 −0.34 −0.35 −0.64 −0.64 −0.53 −0.48 −0.41 −0.87 −0.22 −0.96 9 −0.62 −0.61 −0.48 −0.33 −0.45 −0.17 −0.19 −0.65 −0.10 −0.35 10 0.15 −0.03 −0.10 −0.47 −0.10 −0.17 −0.37 0.10 −0.16 −0.85 10-1 1.88*** 2.12*** 2.06*** 0.52 0.56 1.58*** 0.92 1.56** 1.24* 0.36 t-stat (3.07) (4.04) (3.97) (1.02) (0.96) (2.88) (1.44) (2.54) (1.86) (0.58) C. Double-adjusted alpha, portfolio approach 1 −1.67 −2.25 −2.06 −0.90 −1.00 −1.72 −1.26 −1.20 −1.28 −1.34 2 −1.57 −1.24 −1.62 −1.38 −0.97 −1.18 −0.94 −1.08 −0.90 −0.82 3 −0.95 −0.80 −1.58 −1.30 −0.61 −0.68 −1.18 −1.39 −0.97 −0.20 4 −1.09 −1.05 −1.07 −1.19 −1.03 −0.96 −0.70 −1.28 −0.81 −1.03 5 −1.45 −1.27 −1.09 −1.21 −0.83 −0.76 −0.92 −1.23 −0.94 −0.44 6 −0.92 −0.84 −0.91 −0.78 −0.94 −0.52 −0.63 −1.19 −0.96 −0.54 7 −0.70 −0.49 −0.64 −0.67 −0.77 −0.56 −0.52 −0.49 −1.00 −0.81 8 −0.76 −0.51 −0.65 −0.68 −0.53 −0.26 −0.63 −0.80 −0.27 −0.78 9 −0.53 −0.89 −0.45 −0.66 −0.80 −0.51 −0.06 −0.85 −0.31 −0.68 10 0.23 0.01 −0.20 −0.27 −0.12 −0.07 −0.21 0.09 −0.08 −0.89 10-1 1.90*** 2.26*** 1.86*** 0.63 0.88 1.65*** 1.05* 1.29** 1.19* 0.45 t-stat (3.31) (4.32) (3.65) (1.25) (1.60) (3.12) (1.74) (2.16) (1.93) (0.72) D. Characteristic-related performance, regression approach 1 −1.72 −0.12 −1.30 −0.49 −0.33 −0.28 −0.11 −1.11 −0.67 0.95 2 −0.30 −0.83 −0.64 −1.00 −0.25 −0.51 −0.36 −1.51 −1.12 0.47 3 −0.80 −0.64 −1.28 −1.07 −0.55 −0.48 −0.59 −1.11 −0.68 −0.52 4 −0.73 −0.88 −1.23 −1.15 −0.87 −0.31 −0.38 −0.92 −0.18 −1.35 5 −0.98 −0.65 −1.31 −1.33 −0.87 −1.46 −0.62 −0.56 −0.75 −0.59 6 −0.59 −0.98 −1.19 −0.87 −0.73 −0.59 −1.55 −1.13 −0.52 −0.89 7 −0.78 −1.14 −1.06 −0.59 −0.65 −1.09 −0.60 −0.24 −0.62 −1.15 8 −0.93 −0.76 −0.99 −1.07 −0.51 −0.81 −1.19 −0.94 −0.22 −1.34 9 −1.11 −1.42 −0.84 −1.07 −0.88 −0.73 −0.51 −0.69 −1.00 −1.01 10 −1.53 −1.74 −0.10 −0.27 −1.63 −0.80 −0.81 −1.42 −1.38 −1.73 10-1 0.19 −1.62 1.21 0.22 −1.30 −0.51 −0.70 −0.30 −0.71 −2.68* t-stat (0.12) (−1.02) (0.69) (0.14) (−0.89) (−0.37) (−0.53) (−0.27) (−0.61) (−1.80) E. Characteristic-related performance, portfolio approach 1 −0.69 −1.32 −1.35 −1.10 −0.54 −1.56 −1.26 −0.49 −0.94 −0.29 2 −1.04 −0.66 −1.65 −0.89 −0.80 −0.50 −0.34 −0.92 −0.64 0.27 3 −1.06 −0.96 −1.27 −1.02 −0.81 −0.35 0.42 −1.47 −0.73 −0.19 4 −0.89 −1.41 −0.83 −1.14 −0.39 −0.49 −0.53 −1.66 −0.90 −0.83 5 −0.97 −0.41 −0.72 −0.87 −0.61 −0.21 −0.45 −0.92 −0.88 −1.29 6 −0.90 −0.94 −0.53 −0.73 −0.65 −0.31 −1.24 −1.14 −0.92 −0.98 7 −0.93 −0.36 −1.15 −1.42 −0.10 −0.66 −0.88 −1.25 −0.39 −0.92 8 −0.85 −0.52 −1.25 −0.81 −0.52 −0.89 −0.68 −0.05 −0.46 −0.75 9 −1.08 −1.24 −0.72 −0.29 −1.28 −1.06 −0.61 −0.39 −0.61 −0.56 10 −0.89 −1.43 −0.55 −0.68 −1.75 −1.00 −1.06 −1.25 −0.75 −1.64 10-1 −0.20 −0.11 0.80 0.42 −1.21 0.56 0.20 −0.76 0.19 −1.35 t-stat (−0.15) (−0.08) (0.65) (0.41) (−1.20) (0.52) (0.19) (−0.81) (0.19) (−1.06) . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . A. Four-factor alpha 1 −1.74 −2.13 −2.35 −0.80 −0.70 −1.46 −1.04 −1.09 −1.02 −0.75 2 −1.39 −0.92 −1.45 −1.04 −0.88 −1.17 −0.60 −1.08 −1.36 −0.73 3 −1.14 −1.27 −1.23 −1.61 −0.55 −0.88 −0.65 −1.08 −0.88 −0.53 4 −1.19 −1.14 −1.24 −1.13 −0.84 −0.64 −0.95 −1.37 −1.15 −0.83 5 −0.77 −0.79 −1.18 −1.53 −0.83 −0.77 −0.89 −1.61 −0.67 −0.35 6 −0.97 −0.73 −0.92 −0.71 −0.67 −0.57 −1.23 −0.93 −0.84 −0.50 7 −1.03 −0.80 −0.88 −0.95 −1.08 −0.70 −0.39 −0.91 −0.90 −0.73 8 −0.78 −0.64 −0.50 −0.60 −0.74 −0.59 −0.55 −0.27 −0.49 −1.01 9 −0.37 −0.64 −0.67 −0.53 −0.79 −0.41 −0.58 −0.80 −0.29 −0.71 10 −0.02 −0.21 0.15 −0.14 −0.51 0.02 −0.11 −0.26 −0.09 −1.19 10-1 1.72* 1.93** 2.50*** 0.66 0.20 1.47** 0.93 0.83 0.93 −0.44 t-stat (1.96) (2.49) (3.18) (1.00) (0.28) (1.98) (1.21) (1.15) (1.21) (−0.50) B. Double-adjusted alpha, regression approach 1 −1.74 −2.15 −2.15 −1.00 −0.66 −1.75 −1.28 −1.47 −1.40 −1.20 2 −1.50 −1.01 −1.46 −1.14 −1.33 −1.10 −0.96 −1.37 −0.75 −1.19 3 −1.08 −0.97 −1.35 −1.23 −0.75 −0.86 −0.85 −0.71 −1.03 −0.38 4 −0.99 −1.01 −1.18 −1.32 −0.93 −1.03 −0.80 −1.47 −1.25 −0.59 5 −1.18 −1.37 −1.32 −1.01 −1.03 −0.84 −1.06 −1.38 −0.99 −0.52 6 −1.04 −1.03 −0.77 −1.03 −1.09 −0.42 −0.73 −0.68 −0.96 −0.84 7 −1.05 −0.79 −0.79 −0.79 −0.77 −0.42 −0.34 −0.88 −0.80 −0.66 8 −0.34 −0.35 −0.64 −0.64 −0.53 −0.48 −0.41 −0.87 −0.22 −0.96 9 −0.62 −0.61 −0.48 −0.33 −0.45 −0.17 −0.19 −0.65 −0.10 −0.35 10 0.15 −0.03 −0.10 −0.47 −0.10 −0.17 −0.37 0.10 −0.16 −0.85 10-1 1.88*** 2.12*** 2.06*** 0.52 0.56 1.58*** 0.92 1.56** 1.24* 0.36 t-stat (3.07) (4.04) (3.97) (1.02) (0.96) (2.88) (1.44) (2.54) (1.86) (0.58) C. Double-adjusted alpha, portfolio approach 1 −1.67 −2.25 −2.06 −0.90 −1.00 −1.72 −1.26 −1.20 −1.28 −1.34 2 −1.57 −1.24 −1.62 −1.38 −0.97 −1.18 −0.94 −1.08 −0.90 −0.82 3 −0.95 −0.80 −1.58 −1.30 −0.61 −0.68 −1.18 −1.39 −0.97 −0.20 4 −1.09 −1.05 −1.07 −1.19 −1.03 −0.96 −0.70 −1.28 −0.81 −1.03 5 −1.45 −1.27 −1.09 −1.21 −0.83 −0.76 −0.92 −1.23 −0.94 −0.44 6 −0.92 −0.84 −0.91 −0.78 −0.94 −0.52 −0.63 −1.19 −0.96 −0.54 7 −0.70 −0.49 −0.64 −0.67 −0.77 −0.56 −0.52 −0.49 −1.00 −0.81 8 −0.76 −0.51 −0.65 −0.68 −0.53 −0.26 −0.63 −0.80 −0.27 −0.78 9 −0.53 −0.89 −0.45 −0.66 −0.80 −0.51 −0.06 −0.85 −0.31 −0.68 10 0.23 0.01 −0.20 −0.27 −0.12 −0.07 −0.21 0.09 −0.08 −0.89 10-1 1.90*** 2.26*** 1.86*** 0.63 0.88 1.65*** 1.05* 1.29** 1.19* 0.45 t-stat (3.31) (4.32) (3.65) (1.25) (1.60) (3.12) (1.74) (2.16) (1.93) (0.72) D. Characteristic-related performance, regression approach 1 −1.72 −0.12 −1.30 −0.49 −0.33 −0.28 −0.11 −1.11 −0.67 0.95 2 −0.30 −0.83 −0.64 −1.00 −0.25 −0.51 −0.36 −1.51 −1.12 0.47 3 −0.80 −0.64 −1.28 −1.07 −0.55 −0.48 −0.59 −1.11 −0.68 −0.52 4 −0.73 −0.88 −1.23 −1.15 −0.87 −0.31 −0.38 −0.92 −0.18 −1.35 5 −0.98 −0.65 −1.31 −1.33 −0.87 −1.46 −0.62 −0.56 −0.75 −0.59 6 −0.59 −0.98 −1.19 −0.87 −0.73 −0.59 −1.55 −1.13 −0.52 −0.89 7 −0.78 −1.14 −1.06 −0.59 −0.65 −1.09 −0.60 −0.24 −0.62 −1.15 8 −0.93 −0.76 −0.99 −1.07 −0.51 −0.81 −1.19 −0.94 −0.22 −1.34 9 −1.11 −1.42 −0.84 −1.07 −0.88 −0.73 −0.51 −0.69 −1.00 −1.01 10 −1.53 −1.74 −0.10 −0.27 −1.63 −0.80 −0.81 −1.42 −1.38 −1.73 10-1 0.19 −1.62 1.21 0.22 −1.30 −0.51 −0.70 −0.30 −0.71 −2.68* t-stat (0.12) (−1.02) (0.69) (0.14) (−0.89) (−0.37) (−0.53) (−0.27) (−0.61) (−1.80) E. Characteristic-related performance, portfolio approach 1 −0.69 −1.32 −1.35 −1.10 −0.54 −1.56 −1.26 −0.49 −0.94 −0.29 2 −1.04 −0.66 −1.65 −0.89 −0.80 −0.50 −0.34 −0.92 −0.64 0.27 3 −1.06 −0.96 −1.27 −1.02 −0.81 −0.35 0.42 −1.47 −0.73 −0.19 4 −0.89 −1.41 −0.83 −1.14 −0.39 −0.49 −0.53 −1.66 −0.90 −0.83 5 −0.97 −0.41 −0.72 −0.87 −0.61 −0.21 −0.45 −0.92 −0.88 −1.29 6 −0.90 −0.94 −0.53 −0.73 −0.65 −0.31 −1.24 −1.14 −0.92 −0.98 7 −0.93 −0.36 −1.15 −1.42 −0.10 −0.66 −0.88 −1.25 −0.39 −0.92 8 −0.85 −0.52 −1.25 −0.81 −0.52 −0.89 −0.68 −0.05 −0.46 −0.75 9 −1.08 −1.24 −0.72 −0.29 −1.28 −1.06 −0.61 −0.39 −0.61 −0.56 10 −0.89 −1.43 −0.55 −0.68 −1.75 −1.00 −1.06 −1.25 −0.75 −1.64 10-1 −0.20 −0.11 0.80 0.42 −1.21 0.56 0.20 −0.76 0.19 −1.35 t-stat (−0.15) (−0.08) (0.65) (0.41) (−1.20) (0.52) (0.19) (−0.81) (0.19) (−1.06) The table reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on four-factor alpha (panel A), double-adjusted performance (panels B and C), or characteristic-related performance (panels D and E). Panels B and D utilize the regression double-adjustment approach, and panels C and E utilize the portfolio double-adjustment approach. The post-ranking performance measure, four-factor alpha, for each decile over each post-ranking year is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. The results for the first (tenth) post-ranking year comprise 432 (324) individual post-ranking monthly observations over a 1981–2016 (1990–2016) sample period. For instance, we base 1-year post-ranking period performance on 36 annual ranking periods (each year from 1980 to 2015) and a concatenated set of 1-year post-ranking periods (each year from 1981 to 2016), where each post-ranking period immediately follows its ranking period. We base the 10-year post-ranking performance on the concatenated set of 27 post-ranking periods (from 1990 to 2016) that begin the tenth year after the ranking period. *p < .1; **p < .05; ***p < .01. Open in new tab Table 4 Long-term persistence sorts . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . A. Four-factor alpha 1 −1.74 −2.13 −2.35 −0.80 −0.70 −1.46 −1.04 −1.09 −1.02 −0.75 2 −1.39 −0.92 −1.45 −1.04 −0.88 −1.17 −0.60 −1.08 −1.36 −0.73 3 −1.14 −1.27 −1.23 −1.61 −0.55 −0.88 −0.65 −1.08 −0.88 −0.53 4 −1.19 −1.14 −1.24 −1.13 −0.84 −0.64 −0.95 −1.37 −1.15 −0.83 5 −0.77 −0.79 −1.18 −1.53 −0.83 −0.77 −0.89 −1.61 −0.67 −0.35 6 −0.97 −0.73 −0.92 −0.71 −0.67 −0.57 −1.23 −0.93 −0.84 −0.50 7 −1.03 −0.80 −0.88 −0.95 −1.08 −0.70 −0.39 −0.91 −0.90 −0.73 8 −0.78 −0.64 −0.50 −0.60 −0.74 −0.59 −0.55 −0.27 −0.49 −1.01 9 −0.37 −0.64 −0.67 −0.53 −0.79 −0.41 −0.58 −0.80 −0.29 −0.71 10 −0.02 −0.21 0.15 −0.14 −0.51 0.02 −0.11 −0.26 −0.09 −1.19 10-1 1.72* 1.93** 2.50*** 0.66 0.20 1.47** 0.93 0.83 0.93 −0.44 t-stat (1.96) (2.49) (3.18) (1.00) (0.28) (1.98) (1.21) (1.15) (1.21) (−0.50) B. Double-adjusted alpha, regression approach 1 −1.74 −2.15 −2.15 −1.00 −0.66 −1.75 −1.28 −1.47 −1.40 −1.20 2 −1.50 −1.01 −1.46 −1.14 −1.33 −1.10 −0.96 −1.37 −0.75 −1.19 3 −1.08 −0.97 −1.35 −1.23 −0.75 −0.86 −0.85 −0.71 −1.03 −0.38 4 −0.99 −1.01 −1.18 −1.32 −0.93 −1.03 −0.80 −1.47 −1.25 −0.59 5 −1.18 −1.37 −1.32 −1.01 −1.03 −0.84 −1.06 −1.38 −0.99 −0.52 6 −1.04 −1.03 −0.77 −1.03 −1.09 −0.42 −0.73 −0.68 −0.96 −0.84 7 −1.05 −0.79 −0.79 −0.79 −0.77 −0.42 −0.34 −0.88 −0.80 −0.66 8 −0.34 −0.35 −0.64 −0.64 −0.53 −0.48 −0.41 −0.87 −0.22 −0.96 9 −0.62 −0.61 −0.48 −0.33 −0.45 −0.17 −0.19 −0.65 −0.10 −0.35 10 0.15 −0.03 −0.10 −0.47 −0.10 −0.17 −0.37 0.10 −0.16 −0.85 10-1 1.88*** 2.12*** 2.06*** 0.52 0.56 1.58*** 0.92 1.56** 1.24* 0.36 t-stat (3.07) (4.04) (3.97) (1.02) (0.96) (2.88) (1.44) (2.54) (1.86) (0.58) C. Double-adjusted alpha, portfolio approach 1 −1.67 −2.25 −2.06 −0.90 −1.00 −1.72 −1.26 −1.20 −1.28 −1.34 2 −1.57 −1.24 −1.62 −1.38 −0.97 −1.18 −0.94 −1.08 −0.90 −0.82 3 −0.95 −0.80 −1.58 −1.30 −0.61 −0.68 −1.18 −1.39 −0.97 −0.20 4 −1.09 −1.05 −1.07 −1.19 −1.03 −0.96 −0.70 −1.28 −0.81 −1.03 5 −1.45 −1.27 −1.09 −1.21 −0.83 −0.76 −0.92 −1.23 −0.94 −0.44 6 −0.92 −0.84 −0.91 −0.78 −0.94 −0.52 −0.63 −1.19 −0.96 −0.54 7 −0.70 −0.49 −0.64 −0.67 −0.77 −0.56 −0.52 −0.49 −1.00 −0.81 8 −0.76 −0.51 −0.65 −0.68 −0.53 −0.26 −0.63 −0.80 −0.27 −0.78 9 −0.53 −0.89 −0.45 −0.66 −0.80 −0.51 −0.06 −0.85 −0.31 −0.68 10 0.23 0.01 −0.20 −0.27 −0.12 −0.07 −0.21 0.09 −0.08 −0.89 10-1 1.90*** 2.26*** 1.86*** 0.63 0.88 1.65*** 1.05* 1.29** 1.19* 0.45 t-stat (3.31) (4.32) (3.65) (1.25) (1.60) (3.12) (1.74) (2.16) (1.93) (0.72) D. Characteristic-related performance, regression approach 1 −1.72 −0.12 −1.30 −0.49 −0.33 −0.28 −0.11 −1.11 −0.67 0.95 2 −0.30 −0.83 −0.64 −1.00 −0.25 −0.51 −0.36 −1.51 −1.12 0.47 3 −0.80 −0.64 −1.28 −1.07 −0.55 −0.48 −0.59 −1.11 −0.68 −0.52 4 −0.73 −0.88 −1.23 −1.15 −0.87 −0.31 −0.38 −0.92 −0.18 −1.35 5 −0.98 −0.65 −1.31 −1.33 −0.87 −1.46 −0.62 −0.56 −0.75 −0.59 6 −0.59 −0.98 −1.19 −0.87 −0.73 −0.59 −1.55 −1.13 −0.52 −0.89 7 −0.78 −1.14 −1.06 −0.59 −0.65 −1.09 −0.60 −0.24 −0.62 −1.15 8 −0.93 −0.76 −0.99 −1.07 −0.51 −0.81 −1.19 −0.94 −0.22 −1.34 9 −1.11 −1.42 −0.84 −1.07 −0.88 −0.73 −0.51 −0.69 −1.00 −1.01 10 −1.53 −1.74 −0.10 −0.27 −1.63 −0.80 −0.81 −1.42 −1.38 −1.73 10-1 0.19 −1.62 1.21 0.22 −1.30 −0.51 −0.70 −0.30 −0.71 −2.68* t-stat (0.12) (−1.02) (0.69) (0.14) (−0.89) (−0.37) (−0.53) (−0.27) (−0.61) (−1.80) E. Characteristic-related performance, portfolio approach 1 −0.69 −1.32 −1.35 −1.10 −0.54 −1.56 −1.26 −0.49 −0.94 −0.29 2 −1.04 −0.66 −1.65 −0.89 −0.80 −0.50 −0.34 −0.92 −0.64 0.27 3 −1.06 −0.96 −1.27 −1.02 −0.81 −0.35 0.42 −1.47 −0.73 −0.19 4 −0.89 −1.41 −0.83 −1.14 −0.39 −0.49 −0.53 −1.66 −0.90 −0.83 5 −0.97 −0.41 −0.72 −0.87 −0.61 −0.21 −0.45 −0.92 −0.88 −1.29 6 −0.90 −0.94 −0.53 −0.73 −0.65 −0.31 −1.24 −1.14 −0.92 −0.98 7 −0.93 −0.36 −1.15 −1.42 −0.10 −0.66 −0.88 −1.25 −0.39 −0.92 8 −0.85 −0.52 −1.25 −0.81 −0.52 −0.89 −0.68 −0.05 −0.46 −0.75 9 −1.08 −1.24 −0.72 −0.29 −1.28 −1.06 −0.61 −0.39 −0.61 −0.56 10 −0.89 −1.43 −0.55 −0.68 −1.75 −1.00 −1.06 −1.25 −0.75 −1.64 10-1 −0.20 −0.11 0.80 0.42 −1.21 0.56 0.20 −0.76 0.19 −1.35 t-stat (−0.15) (−0.08) (0.65) (0.41) (−1.20) (0.52) (0.19) (−0.81) (0.19) (−1.06) . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . A. Four-factor alpha 1 −1.74 −2.13 −2.35 −0.80 −0.70 −1.46 −1.04 −1.09 −1.02 −0.75 2 −1.39 −0.92 −1.45 −1.04 −0.88 −1.17 −0.60 −1.08 −1.36 −0.73 3 −1.14 −1.27 −1.23 −1.61 −0.55 −0.88 −0.65 −1.08 −0.88 −0.53 4 −1.19 −1.14 −1.24 −1.13 −0.84 −0.64 −0.95 −1.37 −1.15 −0.83 5 −0.77 −0.79 −1.18 −1.53 −0.83 −0.77 −0.89 −1.61 −0.67 −0.35 6 −0.97 −0.73 −0.92 −0.71 −0.67 −0.57 −1.23 −0.93 −0.84 −0.50 7 −1.03 −0.80 −0.88 −0.95 −1.08 −0.70 −0.39 −0.91 −0.90 −0.73 8 −0.78 −0.64 −0.50 −0.60 −0.74 −0.59 −0.55 −0.27 −0.49 −1.01 9 −0.37 −0.64 −0.67 −0.53 −0.79 −0.41 −0.58 −0.80 −0.29 −0.71 10 −0.02 −0.21 0.15 −0.14 −0.51 0.02 −0.11 −0.26 −0.09 −1.19 10-1 1.72* 1.93** 2.50*** 0.66 0.20 1.47** 0.93 0.83 0.93 −0.44 t-stat (1.96) (2.49) (3.18) (1.00) (0.28) (1.98) (1.21) (1.15) (1.21) (−0.50) B. Double-adjusted alpha, regression approach 1 −1.74 −2.15 −2.15 −1.00 −0.66 −1.75 −1.28 −1.47 −1.40 −1.20 2 −1.50 −1.01 −1.46 −1.14 −1.33 −1.10 −0.96 −1.37 −0.75 −1.19 3 −1.08 −0.97 −1.35 −1.23 −0.75 −0.86 −0.85 −0.71 −1.03 −0.38 4 −0.99 −1.01 −1.18 −1.32 −0.93 −1.03 −0.80 −1.47 −1.25 −0.59 5 −1.18 −1.37 −1.32 −1.01 −1.03 −0.84 −1.06 −1.38 −0.99 −0.52 6 −1.04 −1.03 −0.77 −1.03 −1.09 −0.42 −0.73 −0.68 −0.96 −0.84 7 −1.05 −0.79 −0.79 −0.79 −0.77 −0.42 −0.34 −0.88 −0.80 −0.66 8 −0.34 −0.35 −0.64 −0.64 −0.53 −0.48 −0.41 −0.87 −0.22 −0.96 9 −0.62 −0.61 −0.48 −0.33 −0.45 −0.17 −0.19 −0.65 −0.10 −0.35 10 0.15 −0.03 −0.10 −0.47 −0.10 −0.17 −0.37 0.10 −0.16 −0.85 10-1 1.88*** 2.12*** 2.06*** 0.52 0.56 1.58*** 0.92 1.56** 1.24* 0.36 t-stat (3.07) (4.04) (3.97) (1.02) (0.96) (2.88) (1.44) (2.54) (1.86) (0.58) C. Double-adjusted alpha, portfolio approach 1 −1.67 −2.25 −2.06 −0.90 −1.00 −1.72 −1.26 −1.20 −1.28 −1.34 2 −1.57 −1.24 −1.62 −1.38 −0.97 −1.18 −0.94 −1.08 −0.90 −0.82 3 −0.95 −0.80 −1.58 −1.30 −0.61 −0.68 −1.18 −1.39 −0.97 −0.20 4 −1.09 −1.05 −1.07 −1.19 −1.03 −0.96 −0.70 −1.28 −0.81 −1.03 5 −1.45 −1.27 −1.09 −1.21 −0.83 −0.76 −0.92 −1.23 −0.94 −0.44 6 −0.92 −0.84 −0.91 −0.78 −0.94 −0.52 −0.63 −1.19 −0.96 −0.54 7 −0.70 −0.49 −0.64 −0.67 −0.77 −0.56 −0.52 −0.49 −1.00 −0.81 8 −0.76 −0.51 −0.65 −0.68 −0.53 −0.26 −0.63 −0.80 −0.27 −0.78 9 −0.53 −0.89 −0.45 −0.66 −0.80 −0.51 −0.06 −0.85 −0.31 −0.68 10 0.23 0.01 −0.20 −0.27 −0.12 −0.07 −0.21 0.09 −0.08 −0.89 10-1 1.90*** 2.26*** 1.86*** 0.63 0.88 1.65*** 1.05* 1.29** 1.19* 0.45 t-stat (3.31) (4.32) (3.65) (1.25) (1.60) (3.12) (1.74) (2.16) (1.93) (0.72) D. Characteristic-related performance, regression approach 1 −1.72 −0.12 −1.30 −0.49 −0.33 −0.28 −0.11 −1.11 −0.67 0.95 2 −0.30 −0.83 −0.64 −1.00 −0.25 −0.51 −0.36 −1.51 −1.12 0.47 3 −0.80 −0.64 −1.28 −1.07 −0.55 −0.48 −0.59 −1.11 −0.68 −0.52 4 −0.73 −0.88 −1.23 −1.15 −0.87 −0.31 −0.38 −0.92 −0.18 −1.35 5 −0.98 −0.65 −1.31 −1.33 −0.87 −1.46 −0.62 −0.56 −0.75 −0.59 6 −0.59 −0.98 −1.19 −0.87 −0.73 −0.59 −1.55 −1.13 −0.52 −0.89 7 −0.78 −1.14 −1.06 −0.59 −0.65 −1.09 −0.60 −0.24 −0.62 −1.15 8 −0.93 −0.76 −0.99 −1.07 −0.51 −0.81 −1.19 −0.94 −0.22 −1.34 9 −1.11 −1.42 −0.84 −1.07 −0.88 −0.73 −0.51 −0.69 −1.00 −1.01 10 −1.53 −1.74 −0.10 −0.27 −1.63 −0.80 −0.81 −1.42 −1.38 −1.73 10-1 0.19 −1.62 1.21 0.22 −1.30 −0.51 −0.70 −0.30 −0.71 −2.68* t-stat (0.12) (−1.02) (0.69) (0.14) (−0.89) (−0.37) (−0.53) (−0.27) (−0.61) (−1.80) E. Characteristic-related performance, portfolio approach 1 −0.69 −1.32 −1.35 −1.10 −0.54 −1.56 −1.26 −0.49 −0.94 −0.29 2 −1.04 −0.66 −1.65 −0.89 −0.80 −0.50 −0.34 −0.92 −0.64 0.27 3 −1.06 −0.96 −1.27 −1.02 −0.81 −0.35 0.42 −1.47 −0.73 −0.19 4 −0.89 −1.41 −0.83 −1.14 −0.39 −0.49 −0.53 −1.66 −0.90 −0.83 5 −0.97 −0.41 −0.72 −0.87 −0.61 −0.21 −0.45 −0.92 −0.88 −1.29 6 −0.90 −0.94 −0.53 −0.73 −0.65 −0.31 −1.24 −1.14 −0.92 −0.98 7 −0.93 −0.36 −1.15 −1.42 −0.10 −0.66 −0.88 −1.25 −0.39 −0.92 8 −0.85 −0.52 −1.25 −0.81 −0.52 −0.89 −0.68 −0.05 −0.46 −0.75 9 −1.08 −1.24 −0.72 −0.29 −1.28 −1.06 −0.61 −0.39 −0.61 −0.56 10 −0.89 −1.43 −0.55 −0.68 −1.75 −1.00 −1.06 −1.25 −0.75 −1.64 10-1 −0.20 −0.11 0.80 0.42 −1.21 0.56 0.20 −0.76 0.19 −1.35 t-stat (−0.15) (−0.08) (0.65) (0.41) (−1.20) (0.52) (0.19) (−0.81) (0.19) (−1.06) The table reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on four-factor alpha (panel A), double-adjusted performance (panels B and C), or characteristic-related performance (panels D and E). Panels B and D utilize the regression double-adjustment approach, and panels C and E utilize the portfolio double-adjustment approach. The post-ranking performance measure, four-factor alpha, for each decile over each post-ranking year is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. The results for the first (tenth) post-ranking year comprise 432 (324) individual post-ranking monthly observations over a 1981–2016 (1990–2016) sample period. For instance, we base 1-year post-ranking period performance on 36 annual ranking periods (each year from 1980 to 2015) and a concatenated set of 1-year post-ranking periods (each year from 1981 to 2016), where each post-ranking period immediately follows its ranking period. We base the 10-year post-ranking performance on the concatenated set of 27 post-ranking periods (from 1990 to 2016) that begin the tenth year after the ranking period. *p < .1; **p < .05; ***p < .01. Open in new tab Compared to the short-term persistence results, we see weaker persistence in the long term, as one might expect given results previously documented in the literature. The results in panel A show mixed evidence of long-term persistence in standard four-factor alpha, largely consistent with Carhart (1997). Although four post-ranking years (years 1, 2, 3, and 6) are statistically significantly consistent with past top performers outperforming past bottom performers, the remaining 6 post-ranking years show a statistically insignificant difference (at the 10% significance level) between past top and bottom performers. By contrast to the standard alpha results in panel A, the two iterations of the double-adjusted results in panels B and C (varying by double-adjusted approach) show a statistically significant four-factor performance difference between past top and bottom double-adjusted performers for 6 and 7 of the 10 post-ranking years.28 Thus, sorting on performance that excludes the portion attributable to the characteristics of portfolio holdings predicts future four-factor performance better than sorting on total four-factor performance. To the extent that the double-adjusted measure provides a more precise estimate of genuine fund skill, we document new evidence that mutual fund skill persists over long periods of time. Using a four-factor model, Carhart (1997) found little evidence of persistence in mutual fund performance in the 5 years after ranking by four-factor alpha. By contrast, our new measure shows marginal evidence of skill predictability even in the eighth post-ranking year. Note, however, that in contrast to the short-term persistence results, the evidence of predictability in net returns is driven more by the bottom-ranked funds, as the top decile in panels B and C fail to produce statistically significant positive four-factor abnormal net returns during any post-ranking year (t-statistics not shown).29 Regardless of the post-ranking year, the results in panels D and E show no evidence that the portion of standard alpha attributable to characteristics positively predicts future four-factor performance. These results help to explain why we see stronger evidence of predictability based on the double-adjusted measure than on the standard four-factor alpha. In particular, the standard alpha includes performance attributable to characteristics, which does not predict future performance. By removing a noisy component, the double-adjusted performance measure is associated with higher measurement precision (i.e., lower standard error) and thus has greater power to detect skill. The combination of genuine skill that does help forecast future performance (as in panels B and C) plus characteristic-driven performance that does not (as in panels D and E) produces the weaker evidence of persistence that we see in panel A. The interplay between the standard four-factor performance measure and the double-adjusted performance measure becomes clear when we cumulate the post-ranking four-factor performance across the post-ranking years, rather than examining the performance of each post-ranking year in isolation as in Table 4. Figure 1 shows the cumulative abnormal post-ranking four-factor performance difference between top and bottom decile funds, where the three plot lines represent alternative sorts based on the standard four-factor performance, double-adjusted performance, and the characteristics component of performance. Panel A reflects the regression approach for determining double-adjusted performance, and panel B reflects the portfolio approach. In both panels, the upward-trending plots associated with the standard and double-adjusted sorts reflect the evidence of long-run persistence in relative performance shown in Table 4. That is, funds originally ranked in the top decile (based on standard four-factor alpha or double-adjusted performance) outperform funds originally ranked in the bottom decile, and the outperformance persists over many years. By contrast, the cumulative four-factor performance for funds sorted by their characteristic-driven component of performance indicates that the characteristic component of performance does not forecast subsequent four-factor performance over any post-ranking time frame. The plots suggest that whatever evidence exists that standard four-factor performance or fund manager skill persists is largely driven by the double-adjusted component of performance, and the characteristic-driven portion of performance is not associated with an enduring component of four-factor performance. Figure 1 Open in new tabDownload slide Cumulative abnormal net returns for long-term persistence sorts The figure shows cumulative post-ranking four-factor alpha from net fund returns for top-bottom portfolios of funds sorted by four-factor alpha, double-adjusted performance, or the characteristic component of performance during the initial ranking period. Panel A reflects the regression approach for determining double-adjusted performance, and panel B reflects the portfolio approach. The horizontal axes represent the post-ranking month number. Figure 1 Open in new tabDownload slide Cumulative abnormal net returns for long-term persistence sorts The figure shows cumulative post-ranking four-factor alpha from net fund returns for top-bottom portfolios of funds sorted by four-factor alpha, double-adjusted performance, or the characteristic component of performance during the initial ranking period. Panel A reflects the regression approach for determining double-adjusted performance, and panel B reflects the portfolio approach. The horizontal axes represent the post-ranking month number. This last point could justify, perhaps to a great extent, the second-stage adjustment that is central to our double-adjusted measure. In examining the relations evident in Table 1, panel B, one could argue that skillful fund managers are especially adept at uncovering profitable investment opportunities among certain types of stocks. For example, it seems reasonable to expect the stock market to be less efficient among smaller cap stocks, perhaps because some large funds are reluctant to trade less liquid, smaller cap stocks. If that were the case, then the second-stage adjustment would unfairly punish those managers, by removing premiums that extends beyond that which is passively associated with the characteristic. However, the results in Table 4 and the evidence in Figure 1 all indicate that the component of performance that we remove in the second stage of our procedure, that is, the characteristics-driven component, does not forecast future performance. To the extent that we unfairly remove fund skill in our second stage, then we would expect to see some evidence that the extracted component significantly relates to future performance. 3.3 Sharpe and appraisal ratio of persistence sort One implication of the persistence results is that we may be able to identify funds with similar four-factor alpha but lower levels of risk. That is, it seems likely that selecting funds based on the double-adjusted measure of performance, rather than standard four-factor alpha, would result in a portfolio of funds less influenced by the characteristic component of performance, which, going forward, would be expected to increase a fund’s risk, but not its abnormal net return. To examine this possibility, we compute the Sharpe and appraisal ratios, respectively defined as the ratio of the mean to the standard deviation of the 120 post-ranking monthly excess and abnormal net returns for the top-bottom portfolio of funds associated with the double-adjusted performance sorts in Table 4, panels B and C, and the standard four-factor alpha sorts in Table 4, panel A. Although mutual funds are not shortable, such that the top-bottom difference is not a tradable strategy, we examine the Sharpe and appraisal ratio of the top-bottom difference simply to differentiate which sorting measures can identify skilled and unskilled fund managers with the highest precision. Table 5 reports these Sharpe and appraisal ratios. Table 5 Sharpe and appraisal ratios of persistence sorts Sort variable . Short-term sort . Long-term sort . A. Sharpe ratio Four-factor alpha 0.209 0.263 Double-adj. alpha – regression 0.336 0.307 Chara.-related alpha – regression 0.078 0.018 Double-adj. alpha – portfolio 0.353 0.384 Chara.-related alpha – portfolio 0.078 0.058 DGTW CS 0.072 −0.061 B. Appraisal ratio Four-factor alpha 0.257 0.387 Double-adj. alpha – regression 0.371 0.607 Chara.-related alpha – regression 0.083 −0.131 Double-adj. alpha – portfolio 0.381 0.644 Chara.-related alpha – portfolio 0.082 −0.040 DGTW CS 0.025 0.012 Sort variable . Short-term sort . Long-term sort . A. Sharpe ratio Four-factor alpha 0.209 0.263 Double-adj. alpha – regression 0.336 0.307 Chara.-related alpha – regression 0.078 0.018 Double-adj. alpha – portfolio 0.353 0.384 Chara.-related alpha – portfolio 0.078 0.058 DGTW CS 0.072 −0.061 B. Appraisal ratio Four-factor alpha 0.257 0.387 Double-adj. alpha – regression 0.371 0.607 Chara.-related alpha – regression 0.083 −0.131 Double-adj. alpha – portfolio 0.381 0.644 Chara.-related alpha – portfolio 0.082 −0.040 DGTW CS 0.025 0.012 This table reports the Sharpe ratio (panel A) and the appraisal ratio (panel B) of the top-minus-bottom decile strategies in the short-term and long-term persistence sorts. For the short-term persistence sorts as in Table 3, we compute the Sharpe and appraisal ratios as the mean to the standard deviation of the 1-month post-ranking monthly excess and abnormal net returns for the top-bottom portfolio of funds. For the long-term persistence sorts as in Table 4, we calculate the Sharpe and appraisal ratios, respectively defined as the ratio of the mean to the standard deviation of the 120 post-ranking monthly excess and abnormal net returns for the top-bottom portfolio of funds. In both short- and long-term persistence analysis, we sort funds into deciles based on the following performance measures: (a) the standard four-factor alpha, (b) the double-adjusted performance (based on regression or portfolio approach), (c) the characteristics-related alpha (based on regression or portfolio approach), and (d) the DGTW CS measure. Open in new tab Table 5 Sharpe and appraisal ratios of persistence sorts Sort variable . Short-term sort . Long-term sort . A. Sharpe ratio Four-factor alpha 0.209 0.263 Double-adj. alpha – regression 0.336 0.307 Chara.-related alpha – regression 0.078 0.018 Double-adj. alpha – portfolio 0.353 0.384 Chara.-related alpha – portfolio 0.078 0.058 DGTW CS 0.072 −0.061 B. Appraisal ratio Four-factor alpha 0.257 0.387 Double-adj. alpha – regression 0.371 0.607 Chara.-related alpha – regression 0.083 −0.131 Double-adj. alpha – portfolio 0.381 0.644 Chara.-related alpha – portfolio 0.082 −0.040 DGTW CS 0.025 0.012 Sort variable . Short-term sort . Long-term sort . A. Sharpe ratio Four-factor alpha 0.209 0.263 Double-adj. alpha – regression 0.336 0.307 Chara.-related alpha – regression 0.078 0.018 Double-adj. alpha – portfolio 0.353 0.384 Chara.-related alpha – portfolio 0.078 0.058 DGTW CS 0.072 −0.061 B. Appraisal ratio Four-factor alpha 0.257 0.387 Double-adj. alpha – regression 0.371 0.607 Chara.-related alpha – regression 0.083 −0.131 Double-adj. alpha – portfolio 0.381 0.644 Chara.-related alpha – portfolio 0.082 −0.040 DGTW CS 0.025 0.012 This table reports the Sharpe ratio (panel A) and the appraisal ratio (panel B) of the top-minus-bottom decile strategies in the short-term and long-term persistence sorts. For the short-term persistence sorts as in Table 3, we compute the Sharpe and appraisal ratios as the mean to the standard deviation of the 1-month post-ranking monthly excess and abnormal net returns for the top-bottom portfolio of funds. For the long-term persistence sorts as in Table 4, we calculate the Sharpe and appraisal ratios, respectively defined as the ratio of the mean to the standard deviation of the 120 post-ranking monthly excess and abnormal net returns for the top-bottom portfolio of funds. In both short- and long-term persistence analysis, we sort funds into deciles based on the following performance measures: (a) the standard four-factor alpha, (b) the double-adjusted performance (based on regression or portfolio approach), (c) the characteristics-related alpha (based on regression or portfolio approach), and (d) the DGTW CS measure. Open in new tab As expected, selecting funds based on double-adjusted performance produces greater average Sharpe and appraisal ratios compared to selecting funds based on the standard four-factor alpha. The Sharpe ratio associated with the top-bottom portfolio of funds selected according to their double-adjusted performance measure is 0.307 (0.384) based on the regression (portfolio) approach, whereas the corresponding Sharpe ratio for funds selected according to their standard four-factor alpha is 0.263. Similarly, the appraisal ratio associated with the top-bottom portfolio of funds selected according to their double-adjusted performance measure is 0.607 (0.644) based on the regression (portfolio) approach, whereas the corresponding appraisal ratio for funds selected according to their standard four-factor alpha is 0.387. To put it differently, although the plot lines corresponding to the standard four-factor and double-adjusted performance sorts in panels A and B of Figure 1 are close to each other, the confidence intervals of the former are substantially larger than the latter. The Sharpe ratio (appraisal ratio) associated with the top-bottom portfolio of funds selected according to their characteristic-driven alpha is 0.018 and 0.058 (-0.131 and -0.040), respectively, based on the regression and portfolio approach. Moreover, we also find similar patterns in the Sharpe and appraisal ratios associated with the top-bottom portfolio of funds in the short-term persistence sorts examined in Table 3. 3.4 Comparison to nonbenchmark passive assets and active peer benchmark approaches Similar in some respects to our motivation, Pástor and Stambaugh (2002) and Hunter et al. (2014) advocate new empirical methodologies for increasing precision in mutual fund risk-adjusted performance via factor models. Both papers add new factors that explain some of the residuals from a time series regression of fund returns on the standard model. Pástor and Stambaugh (2002) call their additional factors nonbenchmark passive assets. In our analysis, we choose nonbenchmark passive assets from a list of nine Russell indexes that span market cap and growth/value dimensions.30 Hunter et al.’s (2014) active peer benchmarks are factors that comprise the residuals from a four-factor model regression of the equal-weighted gross return (the net shareholder return plus the expense ratio) of funds with the same benchmark to the analyzed fund. To compute, we follow the procedure used by Hunter et al. (2014). We compare the extent to which double-adjusted performance forecasts future fund performance to these two alternative approaches using the short- and long-term persistence analyses examined in subsections 3.1 and 3.2. We follow the same methodology as before, except the decile sorts are based on the alphas from the specifications that incorporate the nonbenchmark passive asset or the active peer benchmark. We estimate standard four-factor alpha, the Sharpe ratio, and the appraisal ratio of the post-ranking period returns based on concatenating each decile’s post-ranking periods, similar to Carhart (1997). In the short-term persistence tests, we re-sort each month and use a 1-month post-ranking period. In the long-term persistence tests, we re-sort each year and track decile performance across the subsequent 10 years. Moreover, to make an “apples to apples” comparison, we compute all of the alternative alpha measures over a common sample limited by the availability of the closest matching passive benchmarks required to estimate performance in Pástor and Stambaugh (2002) and Hunter et al. (2014). Table 6 reports the results, with the short- and long-term persistence four-factor alpha results in panels A and B, respectively, and the Sharpe and appraisal ratio analyses in panel C. One clear advantage for the double-adjusted measures in the short-term analysis is the relatively high t-statistic on the difference in post-ranking performance between the top and bottom deciles. This finding is consistent with the results in Tables 3–5 that suggest that the double-adjusted performance measures benefit by removing the relatively noisy component of performance associated with passive characteristics. Among all of the alternative measures, performance based on nonbenchmark passive assets forecasts future alpha the weakest across both short and long horizons. In the long-term persistence sorts, the double-adjusted performance results appear to be marginally better than performance based on active peer benchmarks, with the magnitude of the decile 10 – decile 1 difference in post-ranking four-factor alpha insignificantly greater than those based on the active peer benchmark alpha sorts during 8 of the 10 post-ranking years. These performance differences cumulate across time into the visually noticeable advantage for the double-adjusted measure reflected in Figure 2. Figure 2 Open in new tabDownload slide Long-term persistence sorts: Double-adjusted alpha versus other enhanced alpha measures The figure shows cumulative post-ranking four-factor alpha from net fund returns for top-bottom portfolios of funds sorted by double-adjusted performance (based on regression or portfolio approach), alpha based on the active peer benchmark model of Hunter et al. (HKKW, 2014), or alpha based on the model with nonbenchmark passive assets of Pástor and Stambaugh (PS, 2002). The horizontal axes show the post-ranking month number. All performance measures are based on a common sample. Figure 2 Open in new tabDownload slide Long-term persistence sorts: Double-adjusted alpha versus other enhanced alpha measures The figure shows cumulative post-ranking four-factor alpha from net fund returns for top-bottom portfolios of funds sorted by double-adjusted performance (based on regression or portfolio approach), alpha based on the active peer benchmark model of Hunter et al. (HKKW, 2014), or alpha based on the model with nonbenchmark passive assets of Pástor and Stambaugh (PS, 2002). The horizontal axes show the post-ranking month number. All performance measures are based on a common sample. Table 6 Comparison to performance measures based on nonbenchmark passive assets and active peer benchmarks A. Short-term persistence . . Model . . Double-adjusted . Nonbenchmark passive assets . Active peer benchmark . . Regression . Portfolio . Bottom −2.78 −2.78 −1.94 −2.38 2 −1.77 −1.90 −1.54 −1.47 3 −1.34 −1.23 −1.16 −1.65 4 −0.82 −0.84 −1.40 −0.96 5 −1.03 −0.81 −0.97 −1.17 6 −1.00 −0.99 −0.90 −0.72 7 −0.39 −0.53 −0.87 −0.47 8 −0.75 −0.44 −0.52 −0.71 9 −0.44 −0.53 0.11 −0.59 Top 1.34 1.07 0.23 1.13 Top-bottom 4.12*** 3.85*** 2.17*** 3.51*** t-statistic (7.07) (7.32) (2.89) (5.01) A. Short-term persistence . . Model . . Double-adjusted . Nonbenchmark passive assets . Active peer benchmark . . Regression . Portfolio . Bottom −2.78 −2.78 −1.94 −2.38 2 −1.77 −1.90 −1.54 −1.47 3 −1.34 −1.23 −1.16 −1.65 4 −0.82 −0.84 −1.40 −0.96 5 −1.03 −0.81 −0.97 −1.17 6 −1.00 −0.99 −0.90 −0.72 7 −0.39 −0.53 −0.87 −0.47 8 −0.75 −0.44 −0.52 −0.71 9 −0.44 −0.53 0.11 −0.59 Top 1.34 1.07 0.23 1.13 Top-bottom 4.12*** 3.85*** 2.17*** 3.51*** t-statistic (7.07) (7.32) (2.89) (5.01) B. Long-term persistence . . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . B1. Double-adjusted alpha, regression approach 1 −1.92 −2.19 −2.09 −1.03 −0.67 −2.20 −1.40 −1.38 −1.45 −1.00 2 −1.38 −0.96 −1.28 −1.18 −1.48 −1.01 −0.99 −1.34 −0.88 −1.34 3 −1.10 −0.97 −1.26 −1.30 −0.90 −0.79 −0.82 −0.73 −0.98 −0.60 4 −0.95 −1.00 −1.28 −1.27 −1.04 −1.21 −0.96 −1.56 −1.36 −0.36 5 −1.20 −1.51 −1.36 −1.12 −1.11 −0.69 −0.97 −1.32 −0.86 −0.61 6 −1.11 −0.93 −0.77 −0.90 −1.09 −0.59 −0.64 −0.74 −1.01 −0.82 7 −0.96 −0.92 −0.53 −0.79 −0.71 −0.67 −0.65 −0.72 −0.67 −0.70 8 −0.38 −0.50 −0.68 −0.94 −0.49 −0.39 −0.21 −0.95 −0.41 −0.91 9 −0.64 −0.55 −0.53 −0.33 −0.62 −0.19 −0.22 −0.83 −0.19 −0.41 10 0.11 0.00 0.08 −0.33 −0.12 −0.13 −0.35 0.02 −0.31 −1.00 10-1 2.02*** 2.19*** 2.17*** 0.71 0.54 2.07*** 1.04* 1.41** 1.15* 0.00 t-stat (3.34) (3.98) (3.92) (1.41) (0.99) (3.73) (1.76) (2.37) (1.88) (0.00) B2. Double-adjusted alpha, portfolio approach 1 −1.83 −2.30 −1.82 −0.83 −1.10 −2.16 −1.22 −1.28 −1.37 −1.31 2 −1.57 −1.30 −1.71 −1.40 −1.11 −1.11 −1.04 −1.05 −0.92 −0.73 3 −0.84 −0.63 −1.50 −1.35 −0.63 −0.78 −1.13 −1.51 −0.93 −0.32 4 −1.08 −1.22 −1.00 −1.25 −1.08 −0.88 −0.91 −1.02 −0.81 −0.93 5 −1.40 −1.17 −1.02 −1.28 −0.86 −0.78 −0.76 −1.40 −0.81 −0.46 6 −0.95 −0.95 −0.92 −0.79 −0.96 −0.48 −0.62 −1.19 −1.17 −0.55 7 −0.76 −0.55 −0.70 −0.63 −0.74 −0.69 −0.57 −0.53 −1.13 −0.84 8 −0.82 −0.54 −0.40 −0.87 −0.57 −0.53 −0.70 −0.72 −0.26 −0.71 9 −0.38 −0.86 −0.44 −0.55 −0.92 −0.36 −0.06 −0.82 −0.39 −0.87 10 0.09 0.03 −0.22 −0.29 −0.22 0.05 −0.30 −0.11 −0.16 −1.00 10-1 1.91*** 2.33*** 1.60*** 0.54 0.88* 2.20*** 0.93 1.17** 1.22** 0.31 t-stat (3.41) (4.48) (3.06) (1.12) (1.71) (4.18) (1.63) (2.10) (2.22) (0.55) B3. Nonbenchmark passive assets 1 −0.93 −1.28 −1.21 −1.05 −0.97 −1.10 −0.67 −1.47 −1.54 −0.98 2 −1.27 −1.31 −1.45 −1.00 −0.71 −1.23 −1.00 −1.27 −1.05 −1.24 3 −0.86 −1.01 −0.95 −0.76 −0.50 −0.74 −0.80 −1.23 −0.90 −0.48 4 −0.97 −1.07 −1.02 −1.10 −0.84 −0.69 −0.46 −1.06 −0.92 −0.08 5 −1.43 −0.60 −1.27 −1.02 −0.79 −0.64 −0.60 −0.87 −0.69 −0.70 6 −1.02 −0.65 −0.97 −0.96 −0.50 −0.50 −0.87 −0.74 −0.78 −0.97 7 −0.93 −0.87 −0.78 −0.75 −0.91 −0.63 −0.25 −1.03 −0.99 −0.93 8 −0.78 −0.80 −0.54 −1.04 −1.13 −0.77 −0.28 −1.03 −0.52 −1.50 9 −0.62 −1.00 −0.60 −0.49 −0.68 −0.28 −1.13 −0.25 −0.35 −0.54 10 −0.84 −0.91 −0.44 −0.66 −0.85 −0.76 −0.46 −0.14 −0.04 −0.70 10-1 0.09 0.37 0.78 0.39 0.12 0.34 0.22 1.33** 1.51*** 0.28 t-stat (0.13) (0.58) (1.31) (0.71) (0.20) (0.56) (0.40) (2.28) (2.61) (0.49) B4. Active peer benchmark 1 −1.74 −1.66 −1.79 −1.15 −0.73 −1.69 −1.14 −1.74 −1.53 −0.34 2 −1.41 −1.10 −1.01 −0.96 −1.16 −0.79 −0.87 −1.36 −1.03 −1.00 3 −1.19 −1.66 −1.23 −1.13 −0.55 −1.28 −0.75 −1.03 −0.92 −0.55 4 −1.52 −0.87 −1.59 −0.99 −1.04 −0.32 −1.38 −0.91 −1.26 −0.74 5 −1.04 −1.03 −1.21 −1.19 −1.10 −0.57 −0.81 −1.39 −0.80 −0.54 6 −0.93 −0.71 −0.83 −1.13 −0.77 −0.74 −0.64 −0.96 −0.44 −0.82 7 −0.69 −1.00 −0.94 −0.87 −1.01 −1.21 −0.77 −0.75 −0.59 −0.56 8 −0.54 −0.76 −0.82 −0.98 −0.71 −0.58 −0.28 −1.06 −0.47 −1.04 9 −0.60 −0.29 −0.36 −0.44 −0.89 −0.44 −0.16 −0.52 −0.77 −0.55 10 0.18 −0.38 0.05 −0.33 −0.33 −0.10 −0.36 0.10 −0.42 −1.42 10-1 1.91*** 1.28** 1.84*** 0.82 0.41 1.59*** 0.78 1.84*** 1.11* −1.09** t-stat (3.22) (2.35) (3.40) (1.55) (0.81) (2.75) (1.25) (3.38) (1.95) (−2.10) B. Long-term persistence . . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . B1. Double-adjusted alpha, regression approach 1 −1.92 −2.19 −2.09 −1.03 −0.67 −2.20 −1.40 −1.38 −1.45 −1.00 2 −1.38 −0.96 −1.28 −1.18 −1.48 −1.01 −0.99 −1.34 −0.88 −1.34 3 −1.10 −0.97 −1.26 −1.30 −0.90 −0.79 −0.82 −0.73 −0.98 −0.60 4 −0.95 −1.00 −1.28 −1.27 −1.04 −1.21 −0.96 −1.56 −1.36 −0.36 5 −1.20 −1.51 −1.36 −1.12 −1.11 −0.69 −0.97 −1.32 −0.86 −0.61 6 −1.11 −0.93 −0.77 −0.90 −1.09 −0.59 −0.64 −0.74 −1.01 −0.82 7 −0.96 −0.92 −0.53 −0.79 −0.71 −0.67 −0.65 −0.72 −0.67 −0.70 8 −0.38 −0.50 −0.68 −0.94 −0.49 −0.39 −0.21 −0.95 −0.41 −0.91 9 −0.64 −0.55 −0.53 −0.33 −0.62 −0.19 −0.22 −0.83 −0.19 −0.41 10 0.11 0.00 0.08 −0.33 −0.12 −0.13 −0.35 0.02 −0.31 −1.00 10-1 2.02*** 2.19*** 2.17*** 0.71 0.54 2.07*** 1.04* 1.41** 1.15* 0.00 t-stat (3.34) (3.98) (3.92) (1.41) (0.99) (3.73) (1.76) (2.37) (1.88) (0.00) B2. Double-adjusted alpha, portfolio approach 1 −1.83 −2.30 −1.82 −0.83 −1.10 −2.16 −1.22 −1.28 −1.37 −1.31 2 −1.57 −1.30 −1.71 −1.40 −1.11 −1.11 −1.04 −1.05 −0.92 −0.73 3 −0.84 −0.63 −1.50 −1.35 −0.63 −0.78 −1.13 −1.51 −0.93 −0.32 4 −1.08 −1.22 −1.00 −1.25 −1.08 −0.88 −0.91 −1.02 −0.81 −0.93 5 −1.40 −1.17 −1.02 −1.28 −0.86 −0.78 −0.76 −1.40 −0.81 −0.46 6 −0.95 −0.95 −0.92 −0.79 −0.96 −0.48 −0.62 −1.19 −1.17 −0.55 7 −0.76 −0.55 −0.70 −0.63 −0.74 −0.69 −0.57 −0.53 −1.13 −0.84 8 −0.82 −0.54 −0.40 −0.87 −0.57 −0.53 −0.70 −0.72 −0.26 −0.71 9 −0.38 −0.86 −0.44 −0.55 −0.92 −0.36 −0.06 −0.82 −0.39 −0.87 10 0.09 0.03 −0.22 −0.29 −0.22 0.05 −0.30 −0.11 −0.16 −1.00 10-1 1.91*** 2.33*** 1.60*** 0.54 0.88* 2.20*** 0.93 1.17** 1.22** 0.31 t-stat (3.41) (4.48) (3.06) (1.12) (1.71) (4.18) (1.63) (2.10) (2.22) (0.55) B3. Nonbenchmark passive assets 1 −0.93 −1.28 −1.21 −1.05 −0.97 −1.10 −0.67 −1.47 −1.54 −0.98 2 −1.27 −1.31 −1.45 −1.00 −0.71 −1.23 −1.00 −1.27 −1.05 −1.24 3 −0.86 −1.01 −0.95 −0.76 −0.50 −0.74 −0.80 −1.23 −0.90 −0.48 4 −0.97 −1.07 −1.02 −1.10 −0.84 −0.69 −0.46 −1.06 −0.92 −0.08 5 −1.43 −0.60 −1.27 −1.02 −0.79 −0.64 −0.60 −0.87 −0.69 −0.70 6 −1.02 −0.65 −0.97 −0.96 −0.50 −0.50 −0.87 −0.74 −0.78 −0.97 7 −0.93 −0.87 −0.78 −0.75 −0.91 −0.63 −0.25 −1.03 −0.99 −0.93 8 −0.78 −0.80 −0.54 −1.04 −1.13 −0.77 −0.28 −1.03 −0.52 −1.50 9 −0.62 −1.00 −0.60 −0.49 −0.68 −0.28 −1.13 −0.25 −0.35 −0.54 10 −0.84 −0.91 −0.44 −0.66 −0.85 −0.76 −0.46 −0.14 −0.04 −0.70 10-1 0.09 0.37 0.78 0.39 0.12 0.34 0.22 1.33** 1.51*** 0.28 t-stat (0.13) (0.58) (1.31) (0.71) (0.20) (0.56) (0.40) (2.28) (2.61) (0.49) B4. Active peer benchmark 1 −1.74 −1.66 −1.79 −1.15 −0.73 −1.69 −1.14 −1.74 −1.53 −0.34 2 −1.41 −1.10 −1.01 −0.96 −1.16 −0.79 −0.87 −1.36 −1.03 −1.00 3 −1.19 −1.66 −1.23 −1.13 −0.55 −1.28 −0.75 −1.03 −0.92 −0.55 4 −1.52 −0.87 −1.59 −0.99 −1.04 −0.32 −1.38 −0.91 −1.26 −0.74 5 −1.04 −1.03 −1.21 −1.19 −1.10 −0.57 −0.81 −1.39 −0.80 −0.54 6 −0.93 −0.71 −0.83 −1.13 −0.77 −0.74 −0.64 −0.96 −0.44 −0.82 7 −0.69 −1.00 −0.94 −0.87 −1.01 −1.21 −0.77 −0.75 −0.59 −0.56 8 −0.54 −0.76 −0.82 −0.98 −0.71 −0.58 −0.28 −1.06 −0.47 −1.04 9 −0.60 −0.29 −0.36 −0.44 −0.89 −0.44 −0.16 −0.52 −0.77 −0.55 10 0.18 −0.38 0.05 −0.33 −0.33 −0.10 −0.36 0.10 −0.42 −1.42 10-1 1.91*** 1.28** 1.84*** 0.82 0.41 1.59*** 0.78 1.84*** 1.11* −1.09** t-stat (3.22) (2.35) (3.40) (1.55) (0.81) (2.75) (1.25) (3.38) (1.95) (−2.10) C. Sharpe and appraisal ratios of persistence sorts . Sort variable . Short-term sort . Long-term sort . Sharpe ratio Double-adj. alpha – regression 0.307 0.333 Double-adj. alpha – portfolio 0.322 0.392 Nonbenchmark passive assets 0.142 0.174 Active peer benchmark 0.170 0.322 Appraisal ratio Double-adj. alpha – regression 0.356 0.640 Double-adj. alpha – portfolio 0.369 0.661 Nonbenchmark passive assets 0.146 0.269 Active peer benchmark 0.253 0.510 C. Sharpe and appraisal ratios of persistence sorts . Sort variable . Short-term sort . Long-term sort . Sharpe ratio Double-adj. alpha – regression 0.307 0.333 Double-adj. alpha – portfolio 0.322 0.392 Nonbenchmark passive assets 0.142 0.174 Active peer benchmark 0.170 0.322 Appraisal ratio Double-adj. alpha – regression 0.356 0.640 Double-adj. alpha – portfolio 0.369 0.661 Nonbenchmark passive assets 0.146 0.269 Active peer benchmark 0.253 0.510 The table reports the results of persistence analysis for two alternative precision-enhanced alpha measures proposed by Pástor and Stambaugh (2002) and Hunter et al. (2014). For comparison purposes, we also report results from double-adjusted performance (based on both the regression and portfolio approaches) using the same sample. Panel A analyzes short term performance persistence (analogous to the analysis in Table 3). It reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on those performance measures during a 24-month ranking period. The four-factor alpha in the post-ranking month is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. Panel B analyzes long-term performance persistence (analogous to the analysis in Table 4). It reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on alpha from a model that includes double-adjusted alpha based on the regression approach (panel B1), double-adjusted alpha based on the portfolio approach (panel B2), nonbenchmark passive assets (panel B3), or active peer benchmarks (panel B4). The post-ranking performance measure, four-factor alpha, for each decile over each post-ranking year is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. Panel C reports the Sharpe ratio and the appraisal ratio of the top-minus-bottom decile strategies in the short-term and long-term persistence sorts. *p < .1; **p < .05; ***p < .01. Open in new tab Table 6 Comparison to performance measures based on nonbenchmark passive assets and active peer benchmarks A. Short-term persistence . . Model . . Double-adjusted . Nonbenchmark passive assets . Active peer benchmark . . Regression . Portfolio . Bottom −2.78 −2.78 −1.94 −2.38 2 −1.77 −1.90 −1.54 −1.47 3 −1.34 −1.23 −1.16 −1.65 4 −0.82 −0.84 −1.40 −0.96 5 −1.03 −0.81 −0.97 −1.17 6 −1.00 −0.99 −0.90 −0.72 7 −0.39 −0.53 −0.87 −0.47 8 −0.75 −0.44 −0.52 −0.71 9 −0.44 −0.53 0.11 −0.59 Top 1.34 1.07 0.23 1.13 Top-bottom 4.12*** 3.85*** 2.17*** 3.51*** t-statistic (7.07) (7.32) (2.89) (5.01) A. Short-term persistence . . Model . . Double-adjusted . Nonbenchmark passive assets . Active peer benchmark . . Regression . Portfolio . Bottom −2.78 −2.78 −1.94 −2.38 2 −1.77 −1.90 −1.54 −1.47 3 −1.34 −1.23 −1.16 −1.65 4 −0.82 −0.84 −1.40 −0.96 5 −1.03 −0.81 −0.97 −1.17 6 −1.00 −0.99 −0.90 −0.72 7 −0.39 −0.53 −0.87 −0.47 8 −0.75 −0.44 −0.52 −0.71 9 −0.44 −0.53 0.11 −0.59 Top 1.34 1.07 0.23 1.13 Top-bottom 4.12*** 3.85*** 2.17*** 3.51*** t-statistic (7.07) (7.32) (2.89) (5.01) B. Long-term persistence . . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . B1. Double-adjusted alpha, regression approach 1 −1.92 −2.19 −2.09 −1.03 −0.67 −2.20 −1.40 −1.38 −1.45 −1.00 2 −1.38 −0.96 −1.28 −1.18 −1.48 −1.01 −0.99 −1.34 −0.88 −1.34 3 −1.10 −0.97 −1.26 −1.30 −0.90 −0.79 −0.82 −0.73 −0.98 −0.60 4 −0.95 −1.00 −1.28 −1.27 −1.04 −1.21 −0.96 −1.56 −1.36 −0.36 5 −1.20 −1.51 −1.36 −1.12 −1.11 −0.69 −0.97 −1.32 −0.86 −0.61 6 −1.11 −0.93 −0.77 −0.90 −1.09 −0.59 −0.64 −0.74 −1.01 −0.82 7 −0.96 −0.92 −0.53 −0.79 −0.71 −0.67 −0.65 −0.72 −0.67 −0.70 8 −0.38 −0.50 −0.68 −0.94 −0.49 −0.39 −0.21 −0.95 −0.41 −0.91 9 −0.64 −0.55 −0.53 −0.33 −0.62 −0.19 −0.22 −0.83 −0.19 −0.41 10 0.11 0.00 0.08 −0.33 −0.12 −0.13 −0.35 0.02 −0.31 −1.00 10-1 2.02*** 2.19*** 2.17*** 0.71 0.54 2.07*** 1.04* 1.41** 1.15* 0.00 t-stat (3.34) (3.98) (3.92) (1.41) (0.99) (3.73) (1.76) (2.37) (1.88) (0.00) B2. Double-adjusted alpha, portfolio approach 1 −1.83 −2.30 −1.82 −0.83 −1.10 −2.16 −1.22 −1.28 −1.37 −1.31 2 −1.57 −1.30 −1.71 −1.40 −1.11 −1.11 −1.04 −1.05 −0.92 −0.73 3 −0.84 −0.63 −1.50 −1.35 −0.63 −0.78 −1.13 −1.51 −0.93 −0.32 4 −1.08 −1.22 −1.00 −1.25 −1.08 −0.88 −0.91 −1.02 −0.81 −0.93 5 −1.40 −1.17 −1.02 −1.28 −0.86 −0.78 −0.76 −1.40 −0.81 −0.46 6 −0.95 −0.95 −0.92 −0.79 −0.96 −0.48 −0.62 −1.19 −1.17 −0.55 7 −0.76 −0.55 −0.70 −0.63 −0.74 −0.69 −0.57 −0.53 −1.13 −0.84 8 −0.82 −0.54 −0.40 −0.87 −0.57 −0.53 −0.70 −0.72 −0.26 −0.71 9 −0.38 −0.86 −0.44 −0.55 −0.92 −0.36 −0.06 −0.82 −0.39 −0.87 10 0.09 0.03 −0.22 −0.29 −0.22 0.05 −0.30 −0.11 −0.16 −1.00 10-1 1.91*** 2.33*** 1.60*** 0.54 0.88* 2.20*** 0.93 1.17** 1.22** 0.31 t-stat (3.41) (4.48) (3.06) (1.12) (1.71) (4.18) (1.63) (2.10) (2.22) (0.55) B3. Nonbenchmark passive assets 1 −0.93 −1.28 −1.21 −1.05 −0.97 −1.10 −0.67 −1.47 −1.54 −0.98 2 −1.27 −1.31 −1.45 −1.00 −0.71 −1.23 −1.00 −1.27 −1.05 −1.24 3 −0.86 −1.01 −0.95 −0.76 −0.50 −0.74 −0.80 −1.23 −0.90 −0.48 4 −0.97 −1.07 −1.02 −1.10 −0.84 −0.69 −0.46 −1.06 −0.92 −0.08 5 −1.43 −0.60 −1.27 −1.02 −0.79 −0.64 −0.60 −0.87 −0.69 −0.70 6 −1.02 −0.65 −0.97 −0.96 −0.50 −0.50 −0.87 −0.74 −0.78 −0.97 7 −0.93 −0.87 −0.78 −0.75 −0.91 −0.63 −0.25 −1.03 −0.99 −0.93 8 −0.78 −0.80 −0.54 −1.04 −1.13 −0.77 −0.28 −1.03 −0.52 −1.50 9 −0.62 −1.00 −0.60 −0.49 −0.68 −0.28 −1.13 −0.25 −0.35 −0.54 10 −0.84 −0.91 −0.44 −0.66 −0.85 −0.76 −0.46 −0.14 −0.04 −0.70 10-1 0.09 0.37 0.78 0.39 0.12 0.34 0.22 1.33** 1.51*** 0.28 t-stat (0.13) (0.58) (1.31) (0.71) (0.20) (0.56) (0.40) (2.28) (2.61) (0.49) B4. Active peer benchmark 1 −1.74 −1.66 −1.79 −1.15 −0.73 −1.69 −1.14 −1.74 −1.53 −0.34 2 −1.41 −1.10 −1.01 −0.96 −1.16 −0.79 −0.87 −1.36 −1.03 −1.00 3 −1.19 −1.66 −1.23 −1.13 −0.55 −1.28 −0.75 −1.03 −0.92 −0.55 4 −1.52 −0.87 −1.59 −0.99 −1.04 −0.32 −1.38 −0.91 −1.26 −0.74 5 −1.04 −1.03 −1.21 −1.19 −1.10 −0.57 −0.81 −1.39 −0.80 −0.54 6 −0.93 −0.71 −0.83 −1.13 −0.77 −0.74 −0.64 −0.96 −0.44 −0.82 7 −0.69 −1.00 −0.94 −0.87 −1.01 −1.21 −0.77 −0.75 −0.59 −0.56 8 −0.54 −0.76 −0.82 −0.98 −0.71 −0.58 −0.28 −1.06 −0.47 −1.04 9 −0.60 −0.29 −0.36 −0.44 −0.89 −0.44 −0.16 −0.52 −0.77 −0.55 10 0.18 −0.38 0.05 −0.33 −0.33 −0.10 −0.36 0.10 −0.42 −1.42 10-1 1.91*** 1.28** 1.84*** 0.82 0.41 1.59*** 0.78 1.84*** 1.11* −1.09** t-stat (3.22) (2.35) (3.40) (1.55) (0.81) (2.75) (1.25) (3.38) (1.95) (−2.10) B. Long-term persistence . . Post-ranking year . Decile . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . B1. Double-adjusted alpha, regression approach 1 −1.92 −2.19 −2.09 −1.03 −0.67 −2.20 −1.40 −1.38 −1.45 −1.00 2 −1.38 −0.96 −1.28 −1.18 −1.48 −1.01 −0.99 −1.34 −0.88 −1.34 3 −1.10 −0.97 −1.26 −1.30 −0.90 −0.79 −0.82 −0.73 −0.98 −0.60 4 −0.95 −1.00 −1.28 −1.27 −1.04 −1.21 −0.96 −1.56 −1.36 −0.36 5 −1.20 −1.51 −1.36 −1.12 −1.11 −0.69 −0.97 −1.32 −0.86 −0.61 6 −1.11 −0.93 −0.77 −0.90 −1.09 −0.59 −0.64 −0.74 −1.01 −0.82 7 −0.96 −0.92 −0.53 −0.79 −0.71 −0.67 −0.65 −0.72 −0.67 −0.70 8 −0.38 −0.50 −0.68 −0.94 −0.49 −0.39 −0.21 −0.95 −0.41 −0.91 9 −0.64 −0.55 −0.53 −0.33 −0.62 −0.19 −0.22 −0.83 −0.19 −0.41 10 0.11 0.00 0.08 −0.33 −0.12 −0.13 −0.35 0.02 −0.31 −1.00 10-1 2.02*** 2.19*** 2.17*** 0.71 0.54 2.07*** 1.04* 1.41** 1.15* 0.00 t-stat (3.34) (3.98) (3.92) (1.41) (0.99) (3.73) (1.76) (2.37) (1.88) (0.00) B2. Double-adjusted alpha, portfolio approach 1 −1.83 −2.30 −1.82 −0.83 −1.10 −2.16 −1.22 −1.28 −1.37 −1.31 2 −1.57 −1.30 −1.71 −1.40 −1.11 −1.11 −1.04 −1.05 −0.92 −0.73 3 −0.84 −0.63 −1.50 −1.35 −0.63 −0.78 −1.13 −1.51 −0.93 −0.32 4 −1.08 −1.22 −1.00 −1.25 −1.08 −0.88 −0.91 −1.02 −0.81 −0.93 5 −1.40 −1.17 −1.02 −1.28 −0.86 −0.78 −0.76 −1.40 −0.81 −0.46 6 −0.95 −0.95 −0.92 −0.79 −0.96 −0.48 −0.62 −1.19 −1.17 −0.55 7 −0.76 −0.55 −0.70 −0.63 −0.74 −0.69 −0.57 −0.53 −1.13 −0.84 8 −0.82 −0.54 −0.40 −0.87 −0.57 −0.53 −0.70 −0.72 −0.26 −0.71 9 −0.38 −0.86 −0.44 −0.55 −0.92 −0.36 −0.06 −0.82 −0.39 −0.87 10 0.09 0.03 −0.22 −0.29 −0.22 0.05 −0.30 −0.11 −0.16 −1.00 10-1 1.91*** 2.33*** 1.60*** 0.54 0.88* 2.20*** 0.93 1.17** 1.22** 0.31 t-stat (3.41) (4.48) (3.06) (1.12) (1.71) (4.18) (1.63) (2.10) (2.22) (0.55) B3. Nonbenchmark passive assets 1 −0.93 −1.28 −1.21 −1.05 −0.97 −1.10 −0.67 −1.47 −1.54 −0.98 2 −1.27 −1.31 −1.45 −1.00 −0.71 −1.23 −1.00 −1.27 −1.05 −1.24 3 −0.86 −1.01 −0.95 −0.76 −0.50 −0.74 −0.80 −1.23 −0.90 −0.48 4 −0.97 −1.07 −1.02 −1.10 −0.84 −0.69 −0.46 −1.06 −0.92 −0.08 5 −1.43 −0.60 −1.27 −1.02 −0.79 −0.64 −0.60 −0.87 −0.69 −0.70 6 −1.02 −0.65 −0.97 −0.96 −0.50 −0.50 −0.87 −0.74 −0.78 −0.97 7 −0.93 −0.87 −0.78 −0.75 −0.91 −0.63 −0.25 −1.03 −0.99 −0.93 8 −0.78 −0.80 −0.54 −1.04 −1.13 −0.77 −0.28 −1.03 −0.52 −1.50 9 −0.62 −1.00 −0.60 −0.49 −0.68 −0.28 −1.13 −0.25 −0.35 −0.54 10 −0.84 −0.91 −0.44 −0.66 −0.85 −0.76 −0.46 −0.14 −0.04 −0.70 10-1 0.09 0.37 0.78 0.39 0.12 0.34 0.22 1.33** 1.51*** 0.28 t-stat (0.13) (0.58) (1.31) (0.71) (0.20) (0.56) (0.40) (2.28) (2.61) (0.49) B4. Active peer benchmark 1 −1.74 −1.66 −1.79 −1.15 −0.73 −1.69 −1.14 −1.74 −1.53 −0.34 2 −1.41 −1.10 −1.01 −0.96 −1.16 −0.79 −0.87 −1.36 −1.03 −1.00 3 −1.19 −1.66 −1.23 −1.13 −0.55 −1.28 −0.75 −1.03 −0.92 −0.55 4 −1.52 −0.87 −1.59 −0.99 −1.04 −0.32 −1.38 −0.91 −1.26 −0.74 5 −1.04 −1.03 −1.21 −1.19 −1.10 −0.57 −0.81 −1.39 −0.80 −0.54 6 −0.93 −0.71 −0.83 −1.13 −0.77 −0.74 −0.64 −0.96 −0.44 −0.82 7 −0.69 −1.00 −0.94 −0.87 −1.01 −1.21 −0.77 −0.75 −0.59 −0.56 8 −0.54 −0.76 −0.82 −0.98 −0.71 −0.58 −0.28 −1.06 −0.47 −1.04 9 −0.60 −0.29 −0.36 −0.44 −0.89 −0.44 −0.16 −0.52 −0.77 −0.55 10 0.18 −0.38 0.05 −0.33 −0.33 −0.10 −0.36 0.10 −0.42 −1.42 10-1 1.91*** 1.28** 1.84*** 0.82 0.41 1.59*** 0.78 1.84*** 1.11* −1.09** t-stat (3.22) (2.35) (3.40) (1.55) (0.81) (2.75) (1.25) (3.38) (1.95) (−2.10) C. Sharpe and appraisal ratios of persistence sorts . Sort variable . Short-term sort . Long-term sort . Sharpe ratio Double-adj. alpha – regression 0.307 0.333 Double-adj. alpha – portfolio 0.322 0.392 Nonbenchmark passive assets 0.142 0.174 Active peer benchmark 0.170 0.322 Appraisal ratio Double-adj. alpha – regression 0.356 0.640 Double-adj. alpha – portfolio 0.369 0.661 Nonbenchmark passive assets 0.146 0.269 Active peer benchmark 0.253 0.510 C. Sharpe and appraisal ratios of persistence sorts . Sort variable . Short-term sort . Long-term sort . Sharpe ratio Double-adj. alpha – regression 0.307 0.333 Double-adj. alpha – portfolio 0.322 0.392 Nonbenchmark passive assets 0.142 0.174 Active peer benchmark 0.170 0.322 Appraisal ratio Double-adj. alpha – regression 0.356 0.640 Double-adj. alpha – portfolio 0.369 0.661 Nonbenchmark passive assets 0.146 0.269 Active peer benchmark 0.253 0.510 The table reports the results of persistence analysis for two alternative precision-enhanced alpha measures proposed by Pástor and Stambaugh (2002) and Hunter et al. (2014). For comparison purposes, we also report results from double-adjusted performance (based on both the regression and portfolio approaches) using the same sample. Panel A analyzes short term performance persistence (analogous to the analysis in Table 3). It reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on those performance measures during a 24-month ranking period. The four-factor alpha in the post-ranking month is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. Panel B analyzes long-term performance persistence (analogous to the analysis in Table 4). It reports annualized post-ranking percentage four-factor alphas from net fund returns for funds sorted into deciles based on alpha from a model that includes double-adjusted alpha based on the regression approach (panel B1), double-adjusted alpha based on the portfolio approach (panel B2), nonbenchmark passive assets (panel B3), or active peer benchmarks (panel B4). The post-ranking performance measure, four-factor alpha, for each decile over each post-ranking year is the intercept of the regression of the concatenated time series over the entire sample period of post-ranking monthly fund returns on Mktrf, SMB, HML, and UMD factors. Panel C reports the Sharpe ratio and the appraisal ratio of the top-minus-bottom decile strategies in the short-term and long-term persistence sorts. *p < .1; **p < .05; ***p < .01. Open in new tab Panel C of Table 6 reports the Sharpe and appraisal ratios of the top-bottom portfolio of funds for the four sorts in panels A and B. For both the short- and long-term persistence analyses and based on using both the Sharpe and appraisal ratios to measure performance, the top-bottom portfolio of funds selected based on the double-adjusted approach appears to dominate that selected based on the alphas of Pástor and Stambaugh (2002) and Hunter et al. (2014). Overall, the Table 6 results suggest that double-adjusted performance forecasts future performance at least as precisely as other relatively new measures of fund performance. 4. Impact on Analysis of Return Gap, Active Share, R-squared, and Value-Added Measure Beyond studies of performance persistence, many other analyses examined in the recent mutual fund literature emphasize relative performance, especially relating it to a specific fund feature (rather than stock characteristic). In this section, we examine whether the inference one takes away from these analyses can be sensitive to more fully controlling for fund holding characteristics. Given the prevalence of this type of analysis in the mutual fund literature, numerous suitable candidates for examination exist. We focus on the following three relatively recent studies: Kacperczyk, Sialm, and Zheng (2008) on the return gap, Cremers and Petajisto (2009) on active share, and Amihud and Goyenko (2013) on factor model R-squared. Our main goal in this analysis is to examine whether the original interpretations that relate these measures to genuine fund skill are robust to our double-adjusted decomposition. For example, if a particular fund feature (such as active share) correlates with the characteristic component of alpha rather than the double-adjusted component of alpha, then that would suggest that high active share captures the tendency for a fund to load on characteristics (e.g., momentum stocks) rather than to show genuine stock selection skill (i.e., skill after controlling for stock characteristics). Thus, based on the results in this section, when the original findings appear to be an artifact of the factor model’s inability to fully control for passive effects related to stock characteristics, we offer an alternative interpretation. To study the implications of double-adjusted performance on the analysis of these fund features, we first replicate some of the main analyses by examining the relation between each of the measures and fund four-factor alpha. We then relate the various fund features to the two components of four-factor alpha, our double-adjusted measure and the portion of performance associated with characteristic exposure. Relating the fund features to the two components of performance will help disentangle which of the two components drives the main findings. We sort funds into quintiles based on each feature each month and then examine the subsequent performance of the quintiles. For performance during the post-ranking month, we use the four-factor alpha calculated as the difference between the realized net fund return and the sum of the product of the factor betas estimated over the previous 24 months and the factor returns during the month. Furthermore, each month we perform a second-pass regression or portfolio cross-sectional adjustment on these monthly out-of-sample four-factor alphas using the lagged 1-month fund holding characteristics and decompose them into characteristic-related and double-adjusted components. 4.1 Return gap The return gap measure (Kacperczyk, Sialm, and Zheng 2008) is the difference between fund gross returns and holdings-based returns. We compute gross fund returns by adding one-twelfth of the year-end expense ratio to the monthly net fund returns during the year. We calculate the holdings-based gross portfolio return each month as the return of the disclosed portfolio by assuming constant fund portfolio holdings from the fund’s most recent disclosure. For our analysis of the return gap, we sort based on the average return gap over the prior 12 months, consistent with the original study, and then examine performance over the following month. The results in Table 7, panel B1, indicate that the return gap is positively related to subsequent double-adjusted fund performance, with a statistically significant difference between the top and bottom post-ranking performance deciles. The results also indicate that the return gap is not related to the characteristic-driven component of fund performance. These results are consistent with the interpretation that the return gap proxies for an unobserved action of the fund manager that affects performance not attributable to exposure to stock characteristics. That performance could relate to transaction costs and interim trading activity (e.g., stock picking, timing the entry or exit of positions, or unusual trading ability), but cannot be attributed to the size, book-to-market ratio, or price momentum of fund holdings. Our findings, therefore, are consistent with the authors’ original interpretation of their results. Table 7 Fund characteristic sorts A. Fund characteristic statistics . . . . Percentile . Characteristic . Mean . SD . 1st . 50th . 99th . Return gap −0.02 0.38 −1.12 −0.02 1.12 Active share 0.82 0.15 0.33 0.87 0.99 R-squared 0.90 0.10 0.47 0.93 0.99 A. Fund characteristic statistics . . . . Percentile . Characteristic . Mean . SD . 1st . 50th . 99th . Return gap −0.02 0.38 −1.12 −0.02 1.12 Active share 0.82 0.15 0.33 0.87 0.99 R-squared 0.90 0.10 0.47 0.93 0.99 B. Performance of fund characteristic sorts . . Model . . Four-factor alpha . Double-adjusted alpha . Chara.-related alpha . Quintile . Regression . Portfolio . Regression . Portfolio . B1. Return gap Bottom −1.58 −1.70 −1.59 0.12 0.00 2 −1.02 −0.98 −0.88 −0.04 −0.14 3 −0.79 −0.73 −0.78 −0.07 −0.01 4 −0.62 −0.57 −0.67 −0.05 0.04 Top −0.29 −0.33 −0.37 0.03 0.09 Top-bottom 1.29*** 1.37*** 1.22 −0.08 0.08 t-statistic (4.59) (5.90) (5.85) (−0.58) (0.53) B2. Active share Bottom −1.21 −0.73 −0.65 −0.47 −0.55 2 −1.10 −0.83 −0.78 −0.27 −0.33 3 −0.74 −0.71 −0.77 −0.03 0.02 4 −0.25 −0.57 −0.66 0.32 0.41 Top −0.29 −0.73 −0.73 0.44 0.45 Top-bottom 0.92* 0.00 −0.07 0.92** 1.00** t-statistic (1.77) (0.00) (−0.53) (2.11) (2.14) B3. R-squared Bottom −0.28 −0.54 −0.66 0.25 0.37 2 −0.87 −1.06 −1.07 0.18 0.20 3 −1.01 −1.06 −1.01 0.05 0.00 4 −1.30 −1.13 −1.11 −0.17 −0.19 Top −1.23 −0.89 −0.85 −0.33 −0.38 Top-bottom −0.94** −0.35 −0.19 −0.59*** −0.75*** t-statistic (−2.35) (−1.09) (−0.71) (−2.79) (−2.97) B. Performance of fund characteristic sorts . . Model . . Four-factor alpha . Double-adjusted alpha . Chara.-related alpha . Quintile . Regression . Portfolio . Regression . Portfolio . B1. Return gap Bottom −1.58 −1.70 −1.59 0.12 0.00 2 −1.02 −0.98 −0.88 −0.04 −0.14 3 −0.79 −0.73 −0.78 −0.07 −0.01 4 −0.62 −0.57 −0.67 −0.05 0.04 Top −0.29 −0.33 −0.37 0.03 0.09 Top-bottom 1.29*** 1.37*** 1.22 −0.08 0.08 t-statistic (4.59) (5.90) (5.85) (−0.58) (0.53) B2. Active share Bottom −1.21 −0.73 −0.65 −0.47 −0.55 2 −1.10 −0.83 −0.78 −0.27 −0.33 3 −0.74 −0.71 −0.77 −0.03 0.02 4 −0.25 −0.57 −0.66 0.32 0.41 Top −0.29 −0.73 −0.73 0.44 0.45 Top-bottom 0.92* 0.00 −0.07 0.92** 1.00** t-statistic (1.77) (0.00) (−0.53) (2.11) (2.14) B3. R-squared Bottom −0.28 −0.54 −0.66 0.25 0.37 2 −0.87 −1.06 −1.07 0.18 0.20 3 −1.01 −1.06 −1.01 0.05 0.00 4 −1.30 −1.13 −1.11 −0.17 −0.19 Top −1.23 −0.89 −0.85 −0.33 −0.38 Top-bottom −0.94** −0.35 −0.19 −0.59*** −0.75*** t-statistic (−2.35) (−1.09) (−0.71) (−2.79) (−2.97) Panel A reports fund characteristic summary statistics. Panel B reports mean annualized post-ranking percentage four-factor alphas, double-adjusted alpha, and characteristics related alpha based on either the regression approach or the portfolio approach for funds sorted into quintiles based on the return gap (panel B1), active share (panel B2), or R-squared (panel B3). The four-factor alpha in the post-ranking month is calculated as the difference between the realized fund return and the sum of the product of the factor betas estimated over the previous 24-month and the factor returns during the month. We compute t-statistics of the differences between the top and bottom quintiles with Newey-West (1987) correction for time-series correlation with three lags. The results reflect between 385 and 441 individual monthly observations over the 1980–2016 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab Table 7 Fund characteristic sorts A. Fund characteristic statistics . . . . Percentile . Characteristic . Mean . SD . 1st . 50th . 99th . Return gap −0.02 0.38 −1.12 −0.02 1.12 Active share 0.82 0.15 0.33 0.87 0.99 R-squared 0.90 0.10 0.47 0.93 0.99 A. Fund characteristic statistics . . . . Percentile . Characteristic . Mean . SD . 1st . 50th . 99th . Return gap −0.02 0.38 −1.12 −0.02 1.12 Active share 0.82 0.15 0.33 0.87 0.99 R-squared 0.90 0.10 0.47 0.93 0.99 B. Performance of fund characteristic sorts . . Model . . Four-factor alpha . Double-adjusted alpha . Chara.-related alpha . Quintile . Regression . Portfolio . Regression . Portfolio . B1. Return gap Bottom −1.58 −1.70 −1.59 0.12 0.00 2 −1.02 −0.98 −0.88 −0.04 −0.14 3 −0.79 −0.73 −0.78 −0.07 −0.01 4 −0.62 −0.57 −0.67 −0.05 0.04 Top −0.29 −0.33 −0.37 0.03 0.09 Top-bottom 1.29*** 1.37*** 1.22 −0.08 0.08 t-statistic (4.59) (5.90) (5.85) (−0.58) (0.53) B2. Active share Bottom −1.21 −0.73 −0.65 −0.47 −0.55 2 −1.10 −0.83 −0.78 −0.27 −0.33 3 −0.74 −0.71 −0.77 −0.03 0.02 4 −0.25 −0.57 −0.66 0.32 0.41 Top −0.29 −0.73 −0.73 0.44 0.45 Top-bottom 0.92* 0.00 −0.07 0.92** 1.00** t-statistic (1.77) (0.00) (−0.53) (2.11) (2.14) B3. R-squared Bottom −0.28 −0.54 −0.66 0.25 0.37 2 −0.87 −1.06 −1.07 0.18 0.20 3 −1.01 −1.06 −1.01 0.05 0.00 4 −1.30 −1.13 −1.11 −0.17 −0.19 Top −1.23 −0.89 −0.85 −0.33 −0.38 Top-bottom −0.94** −0.35 −0.19 −0.59*** −0.75*** t-statistic (−2.35) (−1.09) (−0.71) (−2.79) (−2.97) B. Performance of fund characteristic sorts . . Model . . Four-factor alpha . Double-adjusted alpha . Chara.-related alpha . Quintile . Regression . Portfolio . Regression . Portfolio . B1. Return gap Bottom −1.58 −1.70 −1.59 0.12 0.00 2 −1.02 −0.98 −0.88 −0.04 −0.14 3 −0.79 −0.73 −0.78 −0.07 −0.01 4 −0.62 −0.57 −0.67 −0.05 0.04 Top −0.29 −0.33 −0.37 0.03 0.09 Top-bottom 1.29*** 1.37*** 1.22 −0.08 0.08 t-statistic (4.59) (5.90) (5.85) (−0.58) (0.53) B2. Active share Bottom −1.21 −0.73 −0.65 −0.47 −0.55 2 −1.10 −0.83 −0.78 −0.27 −0.33 3 −0.74 −0.71 −0.77 −0.03 0.02 4 −0.25 −0.57 −0.66 0.32 0.41 Top −0.29 −0.73 −0.73 0.44 0.45 Top-bottom 0.92* 0.00 −0.07 0.92** 1.00** t-statistic (1.77) (0.00) (−0.53) (2.11) (2.14) B3. R-squared Bottom −0.28 −0.54 −0.66 0.25 0.37 2 −0.87 −1.06 −1.07 0.18 0.20 3 −1.01 −1.06 −1.01 0.05 0.00 4 −1.30 −1.13 −1.11 −0.17 −0.19 Top −1.23 −0.89 −0.85 −0.33 −0.38 Top-bottom −0.94** −0.35 −0.19 −0.59*** −0.75*** t-statistic (−2.35) (−1.09) (−0.71) (−2.79) (−2.97) Panel A reports fund characteristic summary statistics. Panel B reports mean annualized post-ranking percentage four-factor alphas, double-adjusted alpha, and characteristics related alpha based on either the regression approach or the portfolio approach for funds sorted into quintiles based on the return gap (panel B1), active share (panel B2), or R-squared (panel B3). The four-factor alpha in the post-ranking month is calculated as the difference between the realized fund return and the sum of the product of the factor betas estimated over the previous 24-month and the factor returns during the month. We compute t-statistics of the differences between the top and bottom quintiles with Newey-West (1987) correction for time-series correlation with three lags. The results reflect between 385 and 441 individual monthly observations over the 1980–2016 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab 4.2 Active share Next, we examine the relation between fund active share (Cremers and Petajisto 2009) and performance. Active share captures the percentage of a manager’s portfolio that differs from its benchmark index. It is calculated by aggregating the absolute differences between the weight of a portfolio’s actual holdings and the weight of its closest matching index. Here we sort into active share quintiles each month and examine performance of the quintiles during the following month. The results in panel B2 of Table 7 indicate a statistically significant relation between active share and the performance driven by the characteristics of the fund stock holdings, with the results significant at the 5% level for both the regression and portfolio approaches. By contrast, we find no statistically significant correspondence between active share and double-adjusted fund performance for either double adjustment approach. Thus, the significant relation between active share and standard four-factor alpha is driven by the characteristic-related component of performance, rather than performance unrelated to characteristics. Greater deviations from one’s benchmark produces performance that our results tie back to stock characteristics, but that is not necessarily associated with stock-picking skill. In untabulated results, we find that the relation between active share and four-factor alpha is driven by high active share funds holding smaller cap stocks, consistent with Frazzini, Friedman, and Pomorski (2016). 4.3 R-squared Finally, we examine the relation between R-squared (Amihud and Goyenko 2013) and performance. We obtain a fund’s R-squared by regressing its excess returns on the Carhart four-factor model over a 24-month estimation period. Each month, we sort our sample funds into R-squared quintiles and examine performance of the quintiles over the following month. Panel B3 of Table 7 shows the results. Similar to the active share results, the R-squared results show a significant relation (here the relation is an inverse one) between R-squared and the characteristic component of performance, rather than double-adjusted performance. A low R-squared indicates fund returns are not well explained by the four factors of the regression model, which the original study interprets as high stock selectivity. One could hypothesize that characteristics help explain fund returns in instances where factors do not well explain fund returns, which could lead to the strong inverse relation we find between R-squared and the characteristic component of performance. In untabulated results, we find that the reason funds with low R-squared are associated with higher four-factor alpha is that they hold smaller cap stocks and growth stocks. 4.4 Fund features robustness test As a robustness test, we use Fama-MacBeth regressions to examine the same relations between the various fund features and performance that we examined via the portfolio sorting approach. We regress future monthly performance on each of the three fund features, perfi,t=a+b×fundchari,t-1+γXi,t-1+ηi,t,(7) where perfi,t refers to fund i’s standard four-factor alpha, double-adjusted performance measure, or characteristic component of performance for month t, and fundchari,t-1 represents fund i’s lagged return gap, active share, or log transformed R-squared.31 We examine alternative specifications that exclude and include fund-level control variables, denoted by Xi in Equation (7). Table 8 reports the cross-sectional regression coefficients averaged across time along with Fama-MacBeth t-statistics with Newey-West (1987) correction for time-series correlation with three lags. To a large extent, the inference that we take away from the cross-sectional results matches the quintile analysis interpretations associated with Table 7. With and without fund-level controls, active share is statistically significantly related to the characteristic component of performance, but not to double-adjusted performance. R-squared also shows a similar pattern. Any significant relation between these measures and standard performance, therefore, appears to be driven by the portion of standard performance attributable to stock holding characteristics. By contrast, the return gap significantly relates to double-adjusted performance. Table 8 Fund characteristic regressions . . Double-adjusted alpha . Chara.-related alpha . Variable . Four-factor alpha . Regression . Portfolio . Regression . Portfolio . A. Return gap Return gap 0.139*** 0.115*** 0.141*** 0.117*** 0.125*** 0.106*** −0.002 -0.002 0.012 0.008 (5.52) (4.65) (6.52) (5.45) (6.42) (5.53) (−0.22) (−0.19) (0.95) (0.61) Constant -0.818*** -0.045 −0.809*** 0.040 −0.802*** −0.048 −0.012 -0.064 -0.022 0.026 (−2.96) (−0.07) (−2.91) (0.07) (−2.87) (−0.10) (−0.65) (−0.24) (−0.79) (0.08) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .008 .040 .007 .034 .006 .031 .016 .073 .009 .044 No. of months 429 429 429 429 429 429 429 429 429 429 B. Active share Active share 2.355* 2.870** 0.043 0.654 −0.178 0.430 2.276** 2.179** 2.424** 2.309** (1.89) (2.22) (0.07) (0.91) (−0.55) (1.04) (2.19) (2.18) (2.10) (2.03) Constant -2.672** −2.537** -0.749 −0.488 −0.561 −0.369 −1.897** −1.978** -2.020** -2.031* (−2.84) (−2.00) (−1.63) (−0.58) (−1.49) (−0.57) (−2.13) (−2.28) (−2.04) (−1.88) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .020 .052 .004 .033 .001 .028 .135 .169 .075 .101 No. of months 385 385 385 385 385 385 385 385 385 385 C. R-squared log TR-sq −0.603** −0.647** −0.297 −0.417* −0.234 −0.351* −0.271*** −0.195* −0.342** −0.265** (−2.28) (−2.40) (−1.28) (−1.75) (−1.24) (−1.79) (−2.67) (−1.96) (−2.54) (−2.04) Constant 0.719 1.501 −0.096 1.149 −0.278 0.805 0.713*** 0.288 0.914** 0.628 (0.84) (1.54) (−0.13) (1.27) (−0.45) (1.04) (2.65) (0.76) (2.47) (1.38) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .017 .048 .015 .041 .011 .035 .025 .082 .021 .055 No. of months 441 441 441 441 441 441 441 441 441 441 . . Double-adjusted alpha . Chara.-related alpha . Variable . Four-factor alpha . Regression . Portfolio . Regression . Portfolio . A. Return gap Return gap 0.139*** 0.115*** 0.141*** 0.117*** 0.125*** 0.106*** −0.002 -0.002 0.012 0.008 (5.52) (4.65) (6.52) (5.45) (6.42) (5.53) (−0.22) (−0.19) (0.95) (0.61) Constant -0.818*** -0.045 −0.809*** 0.040 −0.802*** −0.048 −0.012 -0.064 -0.022 0.026 (−2.96) (−0.07) (−2.91) (0.07) (−2.87) (−0.10) (−0.65) (−0.24) (−0.79) (0.08) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .008 .040 .007 .034 .006 .031 .016 .073 .009 .044 No. of months 429 429 429 429 429 429 429 429 429 429 B. Active share Active share 2.355* 2.870** 0.043 0.654 −0.178 0.430 2.276** 2.179** 2.424** 2.309** (1.89) (2.22) (0.07) (0.91) (−0.55) (1.04) (2.19) (2.18) (2.10) (2.03) Constant -2.672** −2.537** -0.749 −0.488 −0.561 −0.369 −1.897** −1.978** -2.020** -2.031* (−2.84) (−2.00) (−1.63) (−0.58) (−1.49) (−0.57) (−2.13) (−2.28) (−2.04) (−1.88) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .020 .052 .004 .033 .001 .028 .135 .169 .075 .101 No. of months 385 385 385 385 385 385 385 385 385 385 C. R-squared log TR-sq −0.603** −0.647** −0.297 −0.417* −0.234 −0.351* −0.271*** −0.195* −0.342** −0.265** (−2.28) (−2.40) (−1.28) (−1.75) (−1.24) (−1.79) (−2.67) (−1.96) (−2.54) (−2.04) Constant 0.719 1.501 −0.096 1.149 −0.278 0.805 0.713*** 0.288 0.914** 0.628 (0.84) (1.54) (−0.13) (1.27) (−0.45) (1.04) (2.65) (0.76) (2.47) (1.38) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .017 .048 .015 .041 .011 .035 .025 .082 .021 .055 No. of months 441 441 441 441 441 441 441 441 441 441 The table reports mean coefficients from monthly cross-sectional regressions of fund performance on past fund characteristics, perfi,t=a+b×fundchari,t-1+γXi,t-1+ηi,t,(7) where perfi represents fund i’s four-factor alpha, double-adjusted alpha, or characteristic-related alpha (based on the regression or portfolio approach), and fundchari represents fund i’s return gap (panel A), active share (panel B), or log transformed R-squared (log TR-sq, panel C). We estimate the regressions with and without fund-level control variables. We estimate the t-statistics in parentheses as in Fama and MacBeth (1973) with Newey-West (1987) correction for time-series correlation with three lags. The results reflect between 385 and 441 individual monthly regressions over the 1980–2016 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab Table 8 Fund characteristic regressions . . Double-adjusted alpha . Chara.-related alpha . Variable . Four-factor alpha . Regression . Portfolio . Regression . Portfolio . A. Return gap Return gap 0.139*** 0.115*** 0.141*** 0.117*** 0.125*** 0.106*** −0.002 -0.002 0.012 0.008 (5.52) (4.65) (6.52) (5.45) (6.42) (5.53) (−0.22) (−0.19) (0.95) (0.61) Constant -0.818*** -0.045 −0.809*** 0.040 −0.802*** −0.048 −0.012 -0.064 -0.022 0.026 (−2.96) (−0.07) (−2.91) (0.07) (−2.87) (−0.10) (−0.65) (−0.24) (−0.79) (0.08) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .008 .040 .007 .034 .006 .031 .016 .073 .009 .044 No. of months 429 429 429 429 429 429 429 429 429 429 B. Active share Active share 2.355* 2.870** 0.043 0.654 −0.178 0.430 2.276** 2.179** 2.424** 2.309** (1.89) (2.22) (0.07) (0.91) (−0.55) (1.04) (2.19) (2.18) (2.10) (2.03) Constant -2.672** −2.537** -0.749 −0.488 −0.561 −0.369 −1.897** −1.978** -2.020** -2.031* (−2.84) (−2.00) (−1.63) (−0.58) (−1.49) (−0.57) (−2.13) (−2.28) (−2.04) (−1.88) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .020 .052 .004 .033 .001 .028 .135 .169 .075 .101 No. of months 385 385 385 385 385 385 385 385 385 385 C. R-squared log TR-sq −0.603** −0.647** −0.297 −0.417* −0.234 −0.351* −0.271*** −0.195* −0.342** −0.265** (−2.28) (−2.40) (−1.28) (−1.75) (−1.24) (−1.79) (−2.67) (−1.96) (−2.54) (−2.04) Constant 0.719 1.501 −0.096 1.149 −0.278 0.805 0.713*** 0.288 0.914** 0.628 (0.84) (1.54) (−0.13) (1.27) (−0.45) (1.04) (2.65) (0.76) (2.47) (1.38) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .017 .048 .015 .041 .011 .035 .025 .082 .021 .055 No. of months 441 441 441 441 441 441 441 441 441 441 . . Double-adjusted alpha . Chara.-related alpha . Variable . Four-factor alpha . Regression . Portfolio . Regression . Portfolio . A. Return gap Return gap 0.139*** 0.115*** 0.141*** 0.117*** 0.125*** 0.106*** −0.002 -0.002 0.012 0.008 (5.52) (4.65) (6.52) (5.45) (6.42) (5.53) (−0.22) (−0.19) (0.95) (0.61) Constant -0.818*** -0.045 −0.809*** 0.040 −0.802*** −0.048 −0.012 -0.064 -0.022 0.026 (−2.96) (−0.07) (−2.91) (0.07) (−2.87) (−0.10) (−0.65) (−0.24) (−0.79) (0.08) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .008 .040 .007 .034 .006 .031 .016 .073 .009 .044 No. of months 429 429 429 429 429 429 429 429 429 429 B. Active share Active share 2.355* 2.870** 0.043 0.654 −0.178 0.430 2.276** 2.179** 2.424** 2.309** (1.89) (2.22) (0.07) (0.91) (−0.55) (1.04) (2.19) (2.18) (2.10) (2.03) Constant -2.672** −2.537** -0.749 −0.488 −0.561 −0.369 −1.897** −1.978** -2.020** -2.031* (−2.84) (−2.00) (−1.63) (−0.58) (−1.49) (−0.57) (−2.13) (−2.28) (−2.04) (−1.88) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .020 .052 .004 .033 .001 .028 .135 .169 .075 .101 No. of months 385 385 385 385 385 385 385 385 385 385 C. R-squared log TR-sq −0.603** −0.647** −0.297 −0.417* −0.234 −0.351* −0.271*** −0.195* −0.342** −0.265** (−2.28) (−2.40) (−1.28) (−1.75) (−1.24) (−1.79) (−2.67) (−1.96) (−2.54) (−2.04) Constant 0.719 1.501 −0.096 1.149 −0.278 0.805 0.713*** 0.288 0.914** 0.628 (0.84) (1.54) (−0.13) (1.27) (−0.45) (1.04) (2.65) (0.76) (2.47) (1.38) Controls No Yes No Yes No Yes No Yes No Yes Adj. R-squared .017 .048 .015 .041 .011 .035 .025 .082 .021 .055 No. of months 441 441 441 441 441 441 441 441 441 441 The table reports mean coefficients from monthly cross-sectional regressions of fund performance on past fund characteristics, perfi,t=a+b×fundchari,t-1+γXi,t-1+ηi,t,(7) where perfi represents fund i’s four-factor alpha, double-adjusted alpha, or characteristic-related alpha (based on the regression or portfolio approach), and fundchari represents fund i’s return gap (panel A), active share (panel B), or log transformed R-squared (log TR-sq, panel C). We estimate the regressions with and without fund-level control variables. We estimate the t-statistics in parentheses as in Fama and MacBeth (1973) with Newey-West (1987) correction for time-series correlation with three lags. The results reflect between 385 and 441 individual monthly regressions over the 1980–2016 sample period. *p < .1; **p < .05; ***p < .01. Open in new tab 4.5 Value-added measure Berk and van Binsbergen (2015) propose measuring mutual fund skill as the aggregate dollar value that a fund extracts from the capital markets, computed as the product of the standard monthly gross alpha and fund TNA as of previous month-end. As an example, based on this approach, a large fund could rank highly by combining a small positive gross alpha with its large TNA. By contrast, for a similar ranking, a small fund would need a large alpha to offset its smaller base of assets. We examine whether double-adjusted performance affects inference associated with this relatively new approach for assessing mutual fund skill. We compute Berk and van Binsbergen’s (2015) value-added measure three ways: based on standard four-factor gross alpha (computed from net returns plus the expense ratio), similar to Berk and van Binsbergen (2015), and also based on the two components of gross alpha, double-adjusted alpha and the characteristics-driven component of performance. We report double-adjusted value-added performance statistics in Table 9, with panel A reporting statistics that characterize the distribution of the measures and panel B reporting the mean levels for fund TNA terciles. Table 9 Double-adjusted value-added performance A. Value-added measures of the full sample . . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . Four-factor alpha −0.165 −27.147 −11.552 −2.241 −0.004 2.215 11.482 26.991 Double-adjusted Regression 0.032 −25.406 −10.820 −2.097 0.006 2.144 10.993 25.841 Portfolio 0.067 −24.192 −10.420 −2.037 0.008 2.081 10.610 24.711 Characteristics Regression −0.196 −9.651 −3.985 −0.737 −0.007 0.683 3.746 8.950 Portfolio −0.232 −13.843 −5.809 −1.072 0.002 1.042 5.547 13.279 A. Value-added measures of the full sample . . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . Four-factor alpha −0.165 −27.147 −11.552 −2.241 −0.004 2.215 11.482 26.991 Double-adjusted Regression 0.032 −25.406 −10.820 −2.097 0.006 2.144 10.993 25.841 Portfolio 0.067 −24.192 −10.420 −2.037 0.008 2.081 10.610 24.711 Characteristics Regression −0.196 −9.651 −3.985 −0.737 −0.007 0.683 3.746 8.950 Portfolio −0.232 −13.843 −5.809 −1.072 0.002 1.042 5.547 13.279 B. Value-added measures of funds sorted into terciles by TNA . . 1 (small) . 2 . 3 (large) . Four-factor alpha 0.015 0.008 −0.517 Double-adjusted Regression 0.012 −0.001 0.084 Portfolio 0.010 0.010 0.182 Characteristics Regression 0.003 0.009 −0.601 Portfolio 0.005 −0.002 −0.699 B. Value-added measures of funds sorted into terciles by TNA . . 1 (small) . 2 . 3 (large) . Four-factor alpha 0.015 0.008 −0.517 Double-adjusted Regression 0.012 −0.001 0.084 Portfolio 0.010 0.010 0.182 Characteristics Regression 0.003 0.009 −0.601 Portfolio 0.005 −0.002 −0.699 The table reports value-added performance statistics based on the full sample of funds (panel A) and for terciles of funds sorted by fund TNA (panel B). To calculate the value-added measure of Berk and van Binsbergen (2015), we first compute monthly out-of-sample alpha from gross fund returns (net fund returns plus one-twelfth of the annual expense ratio) and decompose it into two components: double-adjusted alpha and the characteristics-related alpha. We then compute the value-added measure as the product of the monthly alpha and fund TNA as of previous month-end using each of the gross alpha measures. Panel A reports the mean value-added performance and value-added performance by percentile of the full sample over the period 1980m4–2016m12, and panel B reports the mean value-added performance for each TNA tercile. The statistics in both panels reflect monthly value-added in millions of dollars (inflation adjusted to January 2000). Open in new tab Table 9 Double-adjusted value-added performance A. Value-added measures of the full sample . . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . Four-factor alpha −0.165 −27.147 −11.552 −2.241 −0.004 2.215 11.482 26.991 Double-adjusted Regression 0.032 −25.406 −10.820 −2.097 0.006 2.144 10.993 25.841 Portfolio 0.067 −24.192 −10.420 −2.037 0.008 2.081 10.610 24.711 Characteristics Regression −0.196 −9.651 −3.985 −0.737 −0.007 0.683 3.746 8.950 Portfolio −0.232 −13.843 −5.809 −1.072 0.002 1.042 5.547 13.279 A. Value-added measures of the full sample . . . Percentile . . Mean . 5th . 10th . 25th . 50th . 75th . 90th . 95th . Four-factor alpha −0.165 −27.147 −11.552 −2.241 −0.004 2.215 11.482 26.991 Double-adjusted Regression 0.032 −25.406 −10.820 −2.097 0.006 2.144 10.993 25.841 Portfolio 0.067 −24.192 −10.420 −2.037 0.008 2.081 10.610 24.711 Characteristics Regression −0.196 −9.651 −3.985 −0.737 −0.007 0.683 3.746 8.950 Portfolio −0.232 −13.843 −5.809 −1.072 0.002 1.042 5.547 13.279 B. Value-added measures of funds sorted into terciles by TNA . . 1 (small) . 2 . 3 (large) . Four-factor alpha 0.015 0.008 −0.517 Double-adjusted Regression 0.012 −0.001 0.084 Portfolio 0.010 0.010 0.182 Characteristics Regression 0.003 0.009 −0.601 Portfolio 0.005 −0.002 −0.699 B. Value-added measures of funds sorted into terciles by TNA . . 1 (small) . 2 . 3 (large) . Four-factor alpha 0.015 0.008 −0.517 Double-adjusted Regression 0.012 −0.001 0.084 Portfolio 0.010 0.010 0.182 Characteristics Regression 0.003 0.009 −0.601 Portfolio 0.005 −0.002 −0.699 The table reports value-added performance statistics based on the full sample of funds (panel A) and for terciles of funds sorted by fund TNA (panel B). To calculate the value-added measure of Berk and van Binsbergen (2015), we first compute monthly out-of-sample alpha from gross fund returns (net fund returns plus one-twelfth of the annual expense ratio) and decompose it into two components: double-adjusted alpha and the characteristics-related alpha. We then compute the value-added measure as the product of the monthly alpha and fund TNA as of previous month-end using each of the gross alpha measures. Panel A reports the mean value-added performance and value-added performance by percentile of the full sample over the period 1980m4–2016m12, and panel B reports the mean value-added performance for each TNA tercile. The statistics in both panels reflect monthly value-added in millions of dollars (inflation adjusted to January 2000). Open in new tab Based on our sample of domestic equity funds over the period from 1980m4 to 2016m12, panel A indicates a negative cross-sectional mean value added of -$165,000 per month based on the standard four-factor alpha, with dollar amounts inflation adjusted to January 2000 as in Berk and van Binsbergen (2015). Based on a 1977–2011 sample period and a sample that includes funds that hold U.S. equity and funds that hold non-U.S. equity, Berk and van Binsbergen (2015) report a positive cross-sectional mean value added of $140,000 per month for their four-factor measure.32 More interestingly, when we decompose four-factor alpha into its two components, we find that the double-adjusted portion shows a positive cross-sectional mean for our sample funds, amounting to $32,000 ($67,000) per month based on the regression (portfolio) approach, whereas the characteristic component of performance shows a negative cross-sectional mean of -$196,000 (-$232,000) per month. The negative characteristic component is driven by relatively large funds, as reflected in panel B. Given the influence of TNA in the value added measure, large funds disproportionately affect aggregate measures of value added. For instance, larger TNA funds are more likely to show negative characteristic-driven performance (see Table IA.6 of the Internet Appendix) because they tend to have greater portfolio weights in large cap stocks (e.g., Yan 2008; Busse et al. 2020), and the four-factor model under adjusts for the premium associated with holding small cap stocks (as Table 1 suggests). Our evidence thus suggests that the sample of domestic equity mutual funds as a whole has better managerial skill based on the value-added measure after accounting for portfolio holding characteristics. 5. Conclusion Many mutual fund studies incorporate either factor model regressions or characteristic benchmarks in their performance analyses. But by estimating the alternative measures separately, rather than in a unified framework, each performance estimate only partially controls for passive influences on fund returns. Motivated by recent developments in the empirical asset pricing literature, we find that both factor loadings and portfolio characteristics explain a significant portion of the cross-section of fund returns. Thus, we advocate adjusting for both factor exposure and stock characteristics simultaneously in one measure. We find that stock characteristics drive up to about a third of a typical fund’s four-factor alpha, an amount that, when taken away, can dramatically affect the inference drawn from a sample of performance estimates. When we reexamine recent mutual fund analyses that emphasize relations between specific fund features and relative performance, we find that, quite often, the feature correlates with performance attributable to the stock characteristics of the fund’s portfolio holdings, rather than the performance that remains after controlling for those effects. Moreover, more fully controlling for the impact of characteristics alters how one interprets the results of performance persistence. We find that this new proxy for mutual fund skill forecasts future fund performance much longer than standard measures do, up to 8 years in our analysis. Further, our double-adjusted performance also predicts future fund performance better than other precision-enhanced alpha measures proposed in the literature (e.g., Pástor and Stambaugh 2002; Hunter et al. 2014). Our interpretation is that double-adjusted performance provides a cleaner signal of future fund performance that may be beneficial to investors. We appreciate the comments of an anonymous referee and Jeff Pontiff (the editor). We thank Vikas Agarwal, Farid Aitsahlia, Darwin Choi, Tarun Chordia, Kent Daniel, Stephen Dimmock, Fangjian Fu, Dashan Huang, Jennifer Huang, Ravi Jagannathan, Narasimhan Jegadeesh, Raymond Kan, Nitish Kumar, Roger Loh, Spencer Martin, Andy Naranjo, Breno Schmidt, Jay Shanken, Clemens Sialm, Mikhail Simutin, Avanidhar Subrahmanyam, Sterling Yan, and Baozhong Yang and seminar participants at the 2016 American Finance Association Meetings, 2015 Citigroup Global Quant Conference, 2017 China International Conference in Finance, 2015 European Finance Association Meetings, 2015 SMU Finance Summer Camp, Cheung Kong Graduate School of Business, Emory University, Georgia State University, Louisiana State University, Peking University, PBC School of Finance at Tsinghua University, Singapore Management University, University of Georgia, University of International Business and Economics, University of Melbourne, University of New South Wales, and University of Puerto Rico for helpful comments. We are grateful to Russ Wermers for providing the characteristic selectivity measure, Clemens Sialm for providing the industry concentration and return gap measures, Richard Evans for providing data on fund ticker creation date, and Martijn Cremers and Antti Petajisto for providing the active share measure. We thank Jing Ding for excellent research assistance. Lei Jiang acknowledges financial support from Tsinghua University Initiative Scientific Research Program, the National Science Foundation of China (71572091), and Tsinghua National Laboratory for Information Science and Technology. Yuehua Tang acknowledges the D. S. Lee Foundation Fellowship of Singapore Management University. Footnotes 1 For instance, for the period of 2000 and 2016, we find that 137 empirical papers related to mutual fund performance published in the Journal of Finance, the Journal of Financial Economics, or the Review of Financial Studies utilized the Carhart (1997) four-factor model, the Fama and French (1993) three-factor model, or the DGTW characteristic-based benchmark model. 2 Using a different methodology with individual stocks as test assets, Raponi, Robotti, and Zaffaroni (2020) find that firm characteristics (including firm size, book-to-market, and momentum) empirically explain a larger fraction of variation in expected return than do Fama-French factor betas. 3 Corroborative of our evidence, Daniel et al. (1997) show that their fund-characteristic-adjusted selectivity measure (i.e., characteristics selectivity) significantly loads on the Carhart (1997) four factors. 4 We find that one important reason for the weak predictability of the characteristics-related component of alpha is that characteristic premiums in stock returns on size, value, and momentum (after removing the premiums associated with factor loadings) do not persist over time (e.g., from 1 month to the next). 5 Additional advancements include conditional models that allow for time-varying factor loadings (Ferson and Schadt 1996) or time-varying alphas (Christopherson, Ferson, and Glassman 1998) and, more recently, a model that simultaneously accommodates security selection, market timing, and volatility timing (Ferson and Mo 2016). 6 See “Update: Data Quality Problems in Thomson Reuters Ownership Data - Mutual Fund (S12) and Institutional (S34) Holdings” on the WRDS website: https://wrds-web.wharton.upenn.edu/wrds/news/index.cfm?display=read& news_id=616. 7 Our sample period begins in April 1980 because portfolio holdings data from Thomson Reuters begin at the end of the first quarter in 1980. 8 First, we select funds with the following Lipper classification codes: EIEI, G, LCCE, LCGE, LCVE, MCCE, MCGE, MCVE, MLCE, MLGE, MLVE, SCCE, SCGE, or SCVE. If a fund does not have a Lipper classification code, we select funds with Strategic Insight objectives (AGG, GMC, GRI, GRO, ING, or SCG). If neither the Strategic Insight nor the Lipper objective is available, we use the Wiesenberger Fund Type Code and select funds with objectives G, G-I, AGG, GCI, GRI, GRO, LTG, MCG, or SCG. If none of these objectives is available, we keep a fund if it has a CS policy (i.e., the fund holds mainly common stocks). 9 Similar to Busse and Tong (2012) and Ferson and Lin (2014), we exclude from our sample funds whose names contain any of the following text strings: Index, Ind, Idx, Indx, Mkt, Market, Composite, S&P, SP, Russell, Nasdaq, DJ, Dow, Jones, Wilshire, NYSE, iShares, SPDR, HOLDRs, ETF, Exchange-Traded Fund, PowerShares, StreetTRACKS, 100, 400, 500, 600, 1000, 1500, 2000, 3000, and/or 5000. We also remove funds with a CRSP index fund flag equal to “D” (pure index fund) or “E” (enhanced index fund). 10 We address incubation bias as follows. As in Evans (2010), we use the fund ticker creation date to identify funds that are incubated. If a fund is classified as incubated, we eliminate all data before the ticker creation date. The ticker creation date data cover all funds in existence at any point in time between January 1999 and January 2008. For a small set of funds that are not covered in the ticker creation date data (i.e., those terminated before January 1999 or those that first appear after January 2008), we remove the first 3 years of return history as suggested by Evans (2010). 11 Since there is a 23-month overlap in the estimation periods of two consecutive monthly performance measures, we compute t-statistics of the differences between the top and bottom quintiles with Newey-West (1987) correction for time-series correlation with 23 lags. 12 In untabulated results, when we run regressions similar to Equation (2), except with the DGTW CS performance measure as the regressand and fund four-factor model loadings as regressors, we find a statistically significant relation between CS and fund factor loadings, which is consistent with the evidence in table 3 of Daniel et al. (1997). Thus, characteristic-based benchmarks also inadequately control for fund characteristic exposure. 13 Each month, we sort fund four-factor alphas into quintiles (Q1 to Q5) based on lagged 1-month fund holding characteristics (market capitalization, book-to-market, and momentum). We obtain a monthly time series of the difference in four-factor alphas for Q5-Q1 for each of the three characteristics. We then sort the sample months into quintiles based on the extent to which each effect exists within a universe of stocks by estimating the cross-sectional relation (i.e., gamma) between each stock’s 1-month lag characteristic and monthly return. Table IA.1 in the Internet Appendix reports the difference in four-factor alpha between funds with top or bottom quintile exposure to a given characteristic during periods of time when the relation between stock returns and market capitalization, book-to-market ratio, or momentum ranges from strongly positive to strongly negative. See the table notes to Table IA.1 in the Internet Appendix for further details. 14 One potential reason proposed in the literature is that both factor loadings and characteristics are related to true unobservable betas and therefore related to expected stock returns (e.g., Gomes, Kogan, and Zhang 2003; Lin and Zhang 2013). 15 As mentioned by Chordia, Goyal, and Shanken (2019), the sum of the contributions of the two components need not equal 100% because correlations between factor loadings and characteristics lead to correlations between the component attributable to factor loadings and the component attributable to characteristics. 16 Our results are robust if we utilize alternative approaches to better capture time variation in the factor loadings. First, we use daily fund returns from Morningstar Direct to estimate factor loadings over the prior 3 months. Second, we estimate factor loadings conditional on fund characteristics. Using these alternative factor loadings, we continue to find that factor loadings and characteristics each explain a significant fraction of the cross-sectional variance of fund returns. We also find that factor loadings are not highly cross-sectionally correlated with characteristics. 17 It is worth noting that we are careful in choosing characteristic and fund performance measurement periods. When we decompose the in-sample alpha for a 24-month period, we use 1-month-lagged 2-year average characteristics in the decomposition (i.e., 24-month average holding characteristics that align with the 24-month regression time frame for alpha). In our later analysis in Section 4, we also decompose out-of-sample 1-month alpha, calculated as the difference between the realized fund return and the sum of the product of the factor betas estimated over the previous 24 months and the factor returns during the month. We use the 1-month-lagged characteristics when decomposing out-of-sample alpha. Note that all of our results are qualitatively similar if we use a 36-month period to estimate the factor model. 18 As an alternative to using mean fund portfolio characteristics measured over the prior 24 months in the second-stage adjustment, we use holding characteristics as of the last month within the 24-month period. Double-adjusted performance estimates based on the two alternative measurements show high cross-sectional correlation with one another: 0.98 for the regression approach and 0.92 for the portfolio approach. 19 To ensure well-populated cells, we utilize tercile sorts (i.e., 3 × 3×3) during the first portion of our sample period (1980–1994) when fewer funds exist. We sort funds into quartiles (i.e., 4 × 4×4) beginning in 1995 when the number of mutual funds in our sample increases dramatically (i.e., above 640 funds with at least 10 funds in each cell). 20 The correlation between the two alternative double-adjusted performance measures in our sample is 0.94. 21 This literature includes, among others, Carhart (1997), Pástor and Stambaugh (2002), Bollen and Busse (2005), Kacperczyk, Sialm, and Zheng (2005, 2008), Cremers and Petajisto (2009), Amihud and Goyenko (2013), and Hunter et al. (2014). 22 See Busse et al. (2020) for more details on matching Thomson Reuters Mutual Fund Holdings and Abel Noser trading data. 23 In untabulated results, we find that the differences of portfolio holding log(Size), log(B/M), and momentum (i.e., past 11-month cumulative return, lagged 1 month) between the month-end and the average of all trading days in the month are small on average and statistically insignificant at the 10% level. 24 Our evidence in Table 2 is not driven by fund investment style. We find similar evidence of performance ranking changes across subsamples of funds classified based on the Investment Objective Code in the Thomson Reuters Mutual Fund Holdings database (i.e., Aggressive Growth, Growth, Growth and Income, and Others). 25 Additional persistence studies include Grinblatt and Titman (1992), Hendricks, Patel, and Zeckhauser (1993), Goetzmann and Ibbotson (1994), Brown and Goetzmann (1995), Malkiel (1995), Elton, Gruber, and Blake (1996), Busse and Irvine (2006), and Busse and Tong (2012), among many others. 26 We find qualitatively similar results if we examine short-term performance persistence with a 1-quarter, rather than a 1-month, post-ranking period. 27 We keep all funds in our post-ranking periods until the month it disappears from our sample so that mergers and liquidations do not bias the results. Across our sample, approximately 4% of funds disappear annually. About 69% (84%) of the sample funds are still alive 10 (5) years after the initial ranking year. 28 The evidence of significant differences when utilizing gross fund returns in Table IA.4 in the Internet Appendix suggests that the double-adjusted performance measure is not simply isolating differences in expense ratios. As an alternative, we also examine the extent to which the various performance measures forecast double-adjusted performance. The results, which we provide in Table IA.5 in the Internet Appendix, are qualitatively similar to the results presented in Table 4. 29 Our evidence of long-term persistence is consistent with the fact that there exist institutional impediments or market frictions that prevent investors from penalizing poorly performing funds, as discussed by Gruber (1996). For instance, poorly performing mutual funds cannot be sold short, and negative alpha funds can therefore persist, as documented in the prior literature (e.g., Carhart 1997). 30 Specifically, there are 19 closest matching passive indexes for our sample funds (i.e., with lowest active share as defined in Cremers and Petajisto 2009 and Petajisto 2013). We narrow the 19 indexes to nine Russell indexes based on highest return correlation as described in appendix A of Hunter et al. (2014). The nine indexes are Russell 1000, Russell 1000 Growth, Russell 1000 Value, Russell Midcap, Russell Midcap Growth, Russell Midcap Value, Russell 2000, Russell 2000 Growth, and Russell 2000 Value. When the active share-based benchmark is not available, we choose the best-fit benchmark by maximizing the return correlations with funds. 31 Following Amihud and Goyenko (2013), we use the logistic transformation of R-squared in our regressions since the distribution of R-squared is skewed toward 1.0. 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