Bayesian Alphas and Mutual Fund Persistence Jeffrey A. Busse Paul J. Irvine * February 00 Abstract Using daily returns, we find that Bayesian alphas predict future mutual fund Sharpe ratios significantly better than traditional measures. For investors that believe in managerial skill, Bayesian measures choose funds that subsequently outperform funds chosen by standard single- or four-factor alphas. For investors that are skeptical of managerial skill, Bayesian measures choose funds that subsequently outperform funds chosen by expenses. Over our entire sample period, we find that priors consistent with a moderate belief in managerial skill dominate the more extreme skeptical or diffuse prior beliefs. Since a model with diffuse prior beliefs in managerial skill best predicts actual future mutual fund cash flows, our results suggest that Bayesian alphas can help investors choose better performing funds. * Goizueta Business School, Emory University, Atlanta GA 303. We thank Jay Shanken and Robert Stambaugh for comments and Ron Harris for research assistance.
1. Introduction Measuring fund performance is important both because of the scale of the mutual fund industry ($6.97 trillion in assets, Investment Company Institute, 001) and also because of its implications for efficient market theory. This paper contributes to the literature on fund performance by bringing together two strains of research. The first examines the persistence of mutual fund performance. Hendricks, Patel, and Zeckhauser (1993), Goetzmann and Ibbotson (1994), Brown and Goetzmann (1995), and Gruber (1996) find persistence in fund returns and in some risk-adjusted performance measures over adjacent annual periods. Although Carhart (1997) attributes annual persistence to fund expenses and momentum security holdings, Bollen and Busse (00) find persistence beyond expenses or momentum across shorter, quarterly periods. The second literature began with Stambaugh (1997), who shows that the moments of short-lived security returns can be more accurately estimated by incorporating information on correlated assets with longer histories. Pastor and Stambaugh (PS, 001a) demonstrate that abnormal fund performance measures need not be restricted to information on fund and factor returns over the life of the mutual fund. They find that incorporating a long time-series of factor returns using Bayesian methods improves estimates of abnormal return. However, they provide no evidence that their estimates predict future performance better than conventional measures. 1 We use daily fund returns and the PS (001a) framework to construct Bayesian alphas for 30 mutual funds from 1985-1995. We use the Bayesian alphas to address two questions. First, should investors use Bayesian alphas to choose mutual funds? Second, do investors use Bayesian alphas (or correlated information) to choose mutual funds? We find that over an expansive range of prior beliefs about managerial skill, Bayesian alphas predict future fund performance better than single- (Jensen, 1968) or four-factor alphas (Carhart, 1997). The intuition behind using the Bayesian approach to predict future performance is as follows. Suppose that a small cap fund had a high singlefactor alpha last quarter because the returns of small cap stocks were greater than the returns of large cap stocks in the past quarter. The Bayesian approach applies 1 Pastor and Stambaugh (001b) use Bayesian techniques to form optimal portfolios of mutual funds. They find that different priors can lead to different ex-post performance of the optimal portfolio. 1
Stambaugh s (1997) insight that the small cap factor premium can be estimated more precisely by using historical information on the small cap premium that predates the recent measurement period. To predict the small cap fund s future performance, the Bayesian approach estimates a small stock premium over a long period relative to the recent quarter s premium. In this example, the Bayesian measure s ability to predict future performance better than the standard performance measure depends on what better reflects future small cap abnormal returns: more recent small cap abnormal returns or long-term mean small cap abnormal returns. In addition to using additional factor data, the Bayesian approach also explicitly incorporates fund expenses. We find that Bayesian alphas generally predict future Sharpe ratios significantly better than conventional measures. Although Bayesian alphas generally outperform conventional performance measures, an investor s belief about managerial skill greatly affects the ability of Bayesian alphas to predict future performance. Whether prior belief in managerial skill leads an investor to identify the best performing funds depends critically on whether managerial skill persists. In the first half of our sample period managerial skill persists, and a reasonable belief in managerial skill allows investors to use Bayesian measures to identify funds with subsequent Sharpe ratios that are considerably higher than sorting by the single- or four-factor alpha or by the expense ratio. In the second half of our sample period managerial skill does not persist, and Bayesian measures with priors that allow for managerial skill still dominate conventional performance measures, but underperform the skeptical investor s strategy of investing in funds with the lowest expenses. We also examine the extent to which Bayesian alphas predict investor cash flows. Several earlier papers study the relation between performance and cash flow. For example, Ippolito (1989), Gruber (1996), Chevalier and Ellison (1997), and Sirri and Tufano (1998) all find strong relations between performance measures, including returns and single- and multi-factor alphas, and subsequent cash flow. We find that Bayesian alphas predict cash flows about as well as single-factor alphas and slightly better than four-factor alphas. In addition, by varying the priors used to estimate the Bayesian alpha, we infer which set of prior beliefs best reflect investor behavior. Our evidence suggests
that investors believe that some managers have the skill to earn abnormal returns, at least gross of expenses. Since the Bayesian estimates of abnormal performance predict future performance significantly better than traditional measures of alpha, our results provide compelling evidence that abnormal performance is estimated more precisely by using the information in the returns of non-benchmark passive assets over long time horizons, consistent with PS (001a). Since fund cash flows are as highly correlated with single-index alphas as with Bayesian alphas, our results suggest that investors would be better off by focusing more on Bayesian alphas and less on single-factor alphas when making their investment decisions. Although the Bayesian approach in our paper is most closely related to PS (001a), it is also related to other papers in the growing literature that use Bayesian methods to examine fund performance. The related work includes PS (001b), who use Bayesian methods to calculate the optimal portfolio of mutual funds, and Baks, Metrick and Wachter (001), who find that particular prior beliefs can justify investment in active mutual funds, even for investors that have little belief in either managerial skill or asset pricing models. The paper proceeds as follows. Section outlines the calculation of standard performance measures and the Bayesian alphas developed in PS (001a). Section 3 describes the data. Section 4 presents the empirical analysis. Section 5 concludes.. Performance measures.1. Standard performance measures Fund abnormal performance is often measured by alpha, defined as the intercept in a regression of the fund s excess returns on the returns of one or more passive indices. For the single-factor CAPM, the passive index is the excess return on the market portfolio, r A, t α A1 + βarb, t + εa, t =, (1) 3
where r A, t is the excess return for fund A at time t, r, is the excess return on the market portfolio at time t, and α A1 is the fund s single-factor alpha, widely used to denote managerial skill. The ability of the market portfolio to accurately price all assets has been called into question frequently over the last two decades. Passive portfolios composed of small stocks (Banz, 1981), high book-to-market stocks (Fama and French, 199), and stocks with high past returns (Jegadeesh and Titman, 1993) produce positive returns after controlling for market risk. Consequently, a positive single-factor alpha could represent holdings in these passive assets (see Elton, et al., 1993), rather than skill in selecting specific securities. To address this issue, multi-factor specifications add one or more additional passive indices to the market portfolio in equation (1). We follow Carhart (1997) and augment the market portfolio with the SMB and HML factors of Fama and French (1993), and a momentum factor. We typically interpret the intercept in this specification (the four-factor alpha) as managerial skill, which is skill after accounting for fund style. The single-factor alpha, in contrast, encompasses both managerial skill and skill associated with choosing a particular style. To distinguish between skill associated with stock selectivity and skill associated with fund style, we express the single-factor alpha as the sum of the four-factor alpha and the incremental abnormal returns attributable to fund exposure to the additional (nonmarket) passive factors. To differentiate the passive indices of the single- and four-factor specifications, we refer to the market portfolio as the benchmark factor and the SMB, HML, and momentum factors as non-benchmark factors. We first estimate non-benchmark alphas by regressing each of the non-benchmark factors on the benchmark portfolio, B t r, t α + βrb, t + ε, t =, () where r, are the non-benchmark factors and t r, are the excess returns of the B t benchmark portfolio. ext, we regress excess fund returns on a four-factor specification of equation (1), which we express in terms of the benchmark portfolio and three nonbenchmark portfolios, r A, t A + c A, r, t + ca, B rb, t + ua, t = δ. (3) 4
The intercept in equation (3), δ A, is fund A s four-factor alpha, and c, are the fund s A exposures to the non-benchmark factors. We then substitute equation () into equation (3). Solving for the single-factor alpha gives α A1 δa + c A, α =. (4) Equation (4) decomposes a fund s single-factor alpha into two components. The first component, δ A, is associated with managerial skill. The second component, which is a product of the non-benchmark alphas ( α ) and the fund s exposures to the nonbenchmark alphas ( c A, ), is associated with fund style. We estimate the standard single- and four-factor performance measures of equations (1)-(4) directly from the time series of returns for the fund and the benchmark and non-benchmark factors, all over the same time period... Bayesian performance measures The Bayesian alpha, introduced by PS (001a), is an alternative to the conventional measures in equations (1)-(4). The Bayesian measure uses factor returns in prior periods combined with a flexible set of prior beliefs about managerial skill and the validity of the CAPM to predict future fund performance. Stambaugh (1997) documents the advantages of using prior period factor returns. He finds that long-horizon returns provide more precise estimates of the moments of correlated short-horizon returns. In our context, the return history of mutual funds is relatively short. For example, according to Morningstar s Principia Pro, the median age of U.S. diversified equity funds is 4.1 years as of January 00. Conversely, we can calculate non-benchmark alphas, α, using data as far back as the early 1960s, the starting point for book values on Compustat. Using the long-period estimates of the non-benchmark α in equation (4) should produce more precise predictions of managerial performance associated with fund style. Incorporating past, non-benchmark asset returns is the next step in estimating Bayesian alphas. We use the techniques described in PS (001a) to calculate Bayesian alphas. For consistency, we use their notation whenever possible. To compare the Bayesian technique directly to the standard single- and four-factor alphas without giving the Bayesian alpha the advantage of additional passive indices, we use the same four passive 5
factors introduced in section.1. We again classify the market portfolio as the benchmark factor and the SMB, HML, and momentum factors as non-benchmark factors..3. Bayesian alpha estimation We use the PS (001a,b) framework to estimate the posterior means of the elements of equation (4). We first specify a conditional prior estimate of α, the nonbenchmark a ε, t, we specify the prior for We use α as: 1 α Σ ~ 0, σα Σ. (5) s σ α, the marginal prior variance of each element in α, to adjust the prior belief in the ability of the benchmark, in this case the market portfolio, to explain the returns on the non-benchmark assets. A prior of σ = 0 is equivalent to setting α = 0, corresponding to perfect confidence in the market portfolio s ability to price the returns α of the non-benchmark assets, SMB, HML, and momentum. Alternatively, σ α = corresponds to a diffuse prior for α. Prior beliefs between these two extremes signify that the prior belief in the market portfolio s ability to price the non-benchmark assets is centered at α = 0, but allows for some belief in model mispricing. We set s in equation (5) equal to the average of the diagonal elements of Ó from OLS estimation of equation () for each non-benchmark factor. variance of We construct the priors for the estimation of equation (3) conditional on normal distributions: u A, t. Conditional on σ u, we specify the priors for σ u, the δ A and c A as independent The classification of factors as benchmark or non-benchmark portfolios is somewhat arbitrary. For example, PS (001a) use a three-factor benchmark model that incorporates the market, SMB, and HML factors. As non-benchmark portfolios, PS (001a) also consider the characteristic-based portfolios of Daniel and Titman (1997), the industry-based portfolios suggested by Moskowitz and Grinblatt (1999), and momentum portfolios. However, a four-benchmark model could include momentum as a benchmark portfolio. 6
and δ A ( ) σ u ~ δ 0, σ (6) E σu σu δ Following PS (001b), we set ( ) σ u σ Φ u ~ c0, c. (7) E σu c A σ δ, the marginal prior variance of δ A, to finite values and δ 0 equal to 1 multiplied by the fund s expense ratio divided by the number of trading days in the year (since we use daily returns). Intuitively, the resulting priors are centered at a belief that managers possess no skill, and investing in mutual funds will underperform the benchmark and non-benchmark assets by the fund s expenses. However, a finite prior variance allows varying degrees of belief in the possibility of managerial skill. We set ( ) E σ equal to the cross-sectional mean of u σ ˆ u from OLS regressions of equation (3) for each fund type, sorted by investment objective. We set the parameters c 0 and Φ c equal to the OLS estimate of the sample cross-sectional moments of ĉ A, again estimated separately for each fund type. We combine the priors specified in equations (5), (6), and (7) with data on nonbenchmark and fund returns to produce estimates of the posterior means of equation (4), where we designate the posteriors by tilde: α ~ ~ A δa c ~ A α ~ 1 = +,. (8) We use the posterior estimates in equation (8) as the Bayesian estimates of alpha, α ~ A1, managerial skill, δ ~ A, and fund style, c ~ A α ~,. The Bayesian skill estimate, A δ ~, is closer to the standard four-factor alpha skill estimate as the priors on managerial skill are more diffuse (i.e., as σ δ grows large). The Bayesian alpha, A1 α ~, is closest to the standard single-factor alpha as the priors become more diffuse, corresponding to large σ α and σ δ. The Bayesian alpha estimate differs somewhat from the standard single-factor alpha because the c 0 are not fund specific and because we estimate the that precedes the single-factor alpha measurement period. α ~ using information 7
3. Data The mutual fund sample, taken from Busse (1999), consists of daily returns on 30 open end equity funds. Included are non-specialized domestic funds, the names of which are taken from the December 1984 edition of Weisenberger s Mutual Funds Panorama. To be included in the sample, funds must have at least $15 million in total net assets (as of December 1984) and daily net asset values and dividends available from Interactive Data Corporation. The daily AVs and dividends are combined to form daily total returns (see Busse, 1999). The sample is divided into three investment objectives: maximum capital gains (68 funds), growth (107 funds), and growth and income (55 funds). Panel A of Table 1 reports summary statistics of the mutual fund sample. The prior for δ A in equation (6) incorporates fund expenses. We take annual fund expenses from the 1985-1995 editions of Weisenberger. We use quarterly total net assets from Standard & Poor s Micropal to estimate quarterly normalized net cash flow for each fund. We define normalized cash flow as TAt TA CFt = TA t ( + R ) 1 t 1 t, (9) where TA t are the fund s total net assets at the end of quarter t, and R t is the fund s total return during quarter t. Our model includes one benchmark factor and three non-benchmark factors. The Center for Research in Security Prices (CRSP) YSE/AMEX/asdaq value-weighted market return series is our benchmark return. Our non-benchmark factors include a market capitalization factor, a book-to-market factor, and a momentum factor. For the market capitalization factor, we use a daily frequency, value-weighted version of the Fama and French (1993) small market capitalization minus big market capitalization (SMB) factor. For the book-to-market factor, we use a daily frequency, value-weighted version of the Fama French high book-to-market minus low book-to-market (HML) factor. For the momentum factor, we use a daily frequency, value-weighted version of the one-year high return minus low return (up minus down, UMD) factor. Carhart (1997) uses a variation of this momentum factor at a monthly frequency. We construct the SMB and HML factors as in Fama and French (1993), except at a daily frequency. We construct the UMD factor as described on Ken French s website 8
(http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/), except at a daily frequency. To compute daily excess returns on the funds and on the CRSP market return, we use the CRSP monthly T30RET 30-day t-bill return divided equally over the trading days in the month as the risk free rate. Panel B of Table 1 reports summary statistics for the factors. 4. Empirical Analysis 4.1. Methodology For each fund, we estimate Bayesian alphas each quarter during the 1985-1995 sample period. The quarterly forecast window provides two advantages. First, a quarterly window is the time period over which Bollen and Busse (00) find performance persistence unrelated to fund fees. Second, by updating fund information quarterly, the procedure incorporates the PS (001a) suggestion to incorporate time-varying fund betas. As we update our estimates of the fund factor exposures each quarter, we capture time variation in fund exposures to the factors. We estimate the Bayesian alphas using ten different skill priors ranging from 10-1 to 10-3 and seven model mispricing priors ranging from 10-9 to 10-3. We find that these priors span the set of priors that give different posterior estimates of Bayesian alphas. We also vary the time series length of historical data used to compute α ~, the posterior estimate of the non-benchmark factor alphas. We use six different historical time periods: the past 1,, 5, 10, 0, and all years, where the factor data begin in July 1963. The different priors and historical factor time series result in 7 10 6 = 40 different quarterly Bayesian alphas for each fund. Each quarter, for each combination of skill prior, model mispricing prior, and historical factor time series, we sort funds into deciles based on their estimated Bayesian alpha. We call this the ranking period. ext, for each decile, we compute performance and cash flows during the following post-ranking period quarter. The post-ranking period performance of the deciles indicates the extent to which the Bayesian alphas predict future performance. Cash flows indicate the extent to which investors use the same type of information incorporated in the Bayesian alphas to determine where to invest. For the post-ranking period performance measure, we use the Sharpe ratio. By sorting on Bayesian alpha and examining the subsequent Sharpe ratio rather than the 9
Bayesian alpha, we avoid induced correlation between the ranking and post ranking period measures arising from using some of the same historical factor return data in both measures. We choose the Sharpe ratio over standard single- or four-factor alphas because the standard alphas serve as benchmarks against which we compare how well the Bayesian alpha predicts future performance. We evaluate the relation between the ranking period Bayesian alpha and postranking period Sharpe ratio or cash flow two different ways. First, to examine whether the relation is statistically significant, we compute Spearman rank correlation coefficients between the ranking period decile ranking and the post-ranking period Sharpe or cash flow decile ranking. Second, to examine the economic relation, we compute the difference in post-ranking period Sharpe ratio or cash flow between the top and bottom deciles. The Bayesian version of equation (4) decomposes the Bayesian alpha into two components, one attributable to managerial skill and one associated with fund style. In addition to examining the relations between the total Bayesian alpha and the subsequent Sharpe ratio and cash flow, we also examine the relation between each component of the Bayesian alpha and the subsequent Sharpe ratio and cash flow. 4.. Contribution of managerial skill and fund style Before examining the relations between the Bayesian measures and subsequent performance or cash flow, we quantify the contributions of managerial skill and fund style to the Bayesian alpha for each combination of model mispricing prior, skill prior, and historical time period. The contributions indicate which component of the Bayesian alpha (i.e., managerial skill or fund style) dominates for a given set of priors. We define the contribution of managerial skill as frac ~ δ =. (10) A δ ~ δa + c~ A α ~, Fund style accounts for the remaining fraction of the Bayesian alpha, 1 frac. δ Figure 1 shows the fraction of the Bayesian alpha attributable to managerial skill for each combination of skill prior and model mispricing prior. The reported fractions 10
represent averages across the six historical time period lengths (1,, 5, 10, 0, and all years), although the contribution pattern is very similar for all six historical time period lengths. The table shows that the skill contribution increases as the skill prior variance increases and as the model mispricing prior variance decreases. On average, the skill contribution is greater than the style contribution, and ranges from a low of 40.8% for a low skill prior variance and a high model mispricing prior variance to 93.5% for a high skill prior variance and a low model mispricing prior variance. These results reflect that, as the skill prior variance increases, the model allows for managerial skill. As the model mispricing prior variance increases, the model allows for non-zero non-benchmark alphas, which, ceteris paribus, reduces the skill contribution. 4.3. Main Results 4.3.1. Performance predictability Figure, Panel A shows the difference in average post-ranking period Sharpe ratios for funds in the highest Bayesian alpha decile and funds in the lowest Bayesian alpha decile for each combination of model mispricing and skill prior. Here, we use all years of historical information to estimate the non-benchmark alphas. The x-axis represents different priors for belief in managerial skill, σ δ, ranging from 10-1, which almost precludes the possibility that managerial skill exists, to the more diffuse prior of 10-3. The y-axis corresponds to the model mispricing prior, which reflects uncertainty over the CAPM s ability to price the non-benchmark factors. The model mispricing uncertainty, σ α, varies from a high level of confidence in the CAPM (10-9 ) to a nearly diffuse prior (10-3 ). The z-axis represents the difference in average post-ranking period Sharpe ratios between funds in the highest and lowest ranking period Bayesian alpha deciles. The figure suggests that the Bayesian alpha estimates are useful predictors of future fund performance. For all combinations of skill prior and model mispricing prior, the difference in future performance between the highest Bayesian alpha decile and lowest Bayesian alpha decile is positive, averaging 0.0138. Further, sorting by Bayesian alphas predicts future fund performance significantly better than sorting by single- or four-factor alpha. In particular, the difference in average future Sharpe ratios between 11
funds in the highest and lowest single-factor alpha decile is 0.0074. The corresponding difference using the four-factor alpha is 0.0091. Thus, the average difference is considerably higher for the Bayesian alpha decile sorts than for the single-or four-factor alpha decile sorts. Panels B and C of Figure examine the relations between the skill and style components of the Bayesian alpha and subsequent performance. Panel B indicates a strong relation between Bayesian skill and subsequent performance. Skill predictability peaks with a skill prior that allows for some managerial skill but does not fully disregard fund expenses. The predictability of Bayesian style in Panel C is also sensitive to the skill prior variance. The panel suggests that Bayesian style predicts future performance best for precise priors against the existence of managerial skill. The skill prior affects style predictability because the factor loadings in the Bayesian estimation of equation (3) are sensitive to the skill estimate. In this sample, the fund posterior factor loading estimates, c ~ A,, are closer to the factor loading priors, c 0, when the skill prior is precise. As the skill prior becomes more diffuse, the posterior factor loading estimates move toward the actual ranking period factor loadings. We find that the posterior factor loadings associated with precise skill priors predict the funds actual next-period factor loadings slightly better than the loading estimates associated with diffuse skill priors. For example, in a pooled cross-sectional regression of the actual future factor loading on the Bayesian posterior factor loading with a skill prior variance of 10-1 (10-3 ), c a + b~ c, (11) A,, t = A,, t 1 the r-squares are 0.560, 0.350, and 0.303 (0.55, 0.345, and 0.94) for the SMB, HML, and UMD factors, respectively. Thus, the style posteriors predict future Sharpe ratios better when the skill prior is precise. Panels B and C also indicate that the model mispricing prior variance does not affect the predictability of the skill or style component of the Bayesian alpha. The model prior variance does not affect skill because it does not enter, directly or indirectly, into the skill estimate in equation (6). For this methodology, which is based on rankings, the model prior does not affect the predictability of style because the prior changes the style 1
components uniformly across funds (shrinking them toward 0 as the prior variance decreases) without affecting relative rankings. Table reports summary statistics for each value of the three parameters that we vary (skill prior, model mispricing prior, or historical factor length) averaged over all other parameter values. The table shows that the Spearman rank correlation between the ranking period Bayesian alpha decile and the subsequent Sharpe ratio decile ranking is statistically significant in most instances, indicating a strong relation between Bayesian alpha and subsequent risk-adjusted performance. The difference in average post-ranking period Sharpe ratios for funds in the highest Bayesian alpha decile and funds in the lowest Bayesian alpha decile gives similar inference. The Bayesian measures have more predictive power when we incorporate either very short or very long time series of historical factor returns. When we incorporate only five years of historical factor returns, the difference in average future Sharpe ratios between funds in the highest and lowest Bayesian alpha decile is less than the corresponding single-factor alpha sort difference. This result suggests time variation in non-benchmark abnormal returns, which we examine more closely later. The table also indicates that, on average, the Bayesian alphas are better predictors of future Sharpe ratios for more diffuse skill priors. Recall that as the skill prior becomes more diffuse, the Bayesian measure of managerial skill is closer to the standard fourfactor alpha. As the skill prior variance decreases, the Bayesian measure of managerial skill is closer to the prior mean of 1 multiplied by the fund s expense ratio. The better predictability associated with more diffuse skill priors suggests that some amount of managerial skill exists among mutual fund managers and that it persists across quarters, consistent with Bollen and Busse (00). The implication is that investing in funds with higher estimates of managerial skill under a diffuse prior produces higher Sharpe ratios than investing in funds with the lowest expense ratios (i.e., investing based on the precise prior that managers have no skill). As the table indicates, it is only in the case with precise priors that managers have no skill that the average post-ranking period top decilebottom decile Sharpe ratio difference is greater for the four-factor alpha sort than for the Bayesian sorts. 13
The table also indicates that the Bayesian alphas are better predictors of future Sharpe ratios for precise model variance priors, i.e., stronger priors that the nonbenchmark assets do not produce abnormal returns in a CAPM context. This result is not because the non-benchmark assets do not produce abnormal single-factor returns over our sample period. In fact, during our sample period, the mean abnormal return contribution of the SMB, HML, and UMD non-benchmark factors to our fund returns is positive (0.00693% per day or about 1.8% annualized), where we estimate the mean contribution as α A, = ca, α. (1) The reason that a more precise model mispricing prior leads to better performance predictability is because of two relations. First, as the model mispricing prior decreases, a higher fraction of the Bayesian alpha is attributable to skill (as seen in Figure 1). Second, the skill component of the Bayesian alpha predicts future performance better than the style component (see Figure, Panels B and C). Finally, for all model mispricing priors, the average post-ranking period top decile-bottom decile Sharpe ratio difference is greater for the Bayesian alpha sorts than for the single- or four-factor alpha sorts. 4.3.. Performance and cash flow If Bayesian alphas predict future performance, do investors use the information reflected in Bayesian alphas to invest in mutual funds? To answer this question, Figure 3 plots the difference in quarterly post-ranking period net cash flows between the top and bottom deciles of funds sorted according to their Bayesian alpha during a quarterly ranking period. The figure reports the results using all years of past factor returns. The figure shows that investors behave as though they have diffuse managerial skill priors. In other words, investors believe managers can add value, and they invest in mutual funds with the highest estimates of managerial skill. Furthermore, the plot suggests that cash flows are largely insensitive to the model mispricing prior variance. This result suggests that investors respond to the Bayesian alpha similarly regardless of whether it is driven by the skill component or the style component. Table 3 reports summary statistics for each value of the three parameters that vary (skill prior, model mispricing prior, or historical factor length) averaged over all other 14
parameter values. The table shows that the Spearman rank correlations between the ranking period Bayesian alpha decile and the subsequent net cash flow decile are strongly statistically significant on average across all parameters. This result is strong evidence that Bayesian alphas capture a performance dynamic that investors respond to. Similar to Figure 3, the table suggests that investors do not preclude the possibility of managerial skill, since the Spearman rank correlation coefficient between performance and cash flow increases almost monotonically as the skill prior variance increases. Furthermore, the greatest Spearman rank correlation is associated the most diffuse skill prior. The table also weakly suggests that investors do not have precise priors that the CAPM correctly prices the non-benchmark factors. Since prior studies also document a strong relation between various performance measures and subsequent cash flow, it is interesting to compare the Bayesian performance-flow relation to that of standard performance measures. The Spearman rank correlation coefficient between standard single- (four-) factor alpha deciles and subsequent net cash flow deciles in our sample is 0.964 (0.97). The difference in net cash flow between the top and bottom deciles formed according to the prior standard single- (four-) factor alpha is 0.054 (0.040). The single-factor results are consistent with the strongest Bayesian cash flow results in Table 3, which are associated with a nearly diffuse skill prior. Since the single-factor alpha is closest to the Bayesian alpha under diffuse priors, this result suggests that investors do not focus on anything specific to the Bayesian alpha that is not already reflected in the single-factor alpha. Recall from Table that over this sample period the Bayesian alphas predict future performance best for moderate skill priors and precise model mispricing priors. Since the cash flow evidence in Table 3 suggests that investors are more inclined to follow more diffuse skill and model mispricing priors, this result suggests that investors would have been better off by being more skeptical about managerial skill. However, it is not clear if this implication holds over time, a point which we address later. 4.3.3. Sub-period analysis To examine the robustness of our predictability results over time, we repeat the analysis separately for our sample period divided into two subperiods, 1985 through the 15
first half of 1990 and the second half of 1990 through 1995. Figure 4 and Table 4 present results similar to those reported in Figure and Table by subperiod. The results for the two subperiods are strikingly different from one another. In the first subperiod, the Bayesian alpha, estimated with a belief in the possibility of managerial skill, produces differences in average future Sharpe ratios between top and bottom decile funds as high as 0.05 in Figure 4. The Spearman rank correlation coefficients between Bayesian decile and subsequent Sharpe ratio decile are statistically significant at the 10% level for all parameter averages and at the 5% level for all except those with strong priors against managerial skill. Traditional methods also predict future performance well in this subperiod. The single-factor alpha produces differences in future Sharpe ratios of 0.0157, and the four-factor alpha produces differences of 0.0189. Since past performance is an indicator of future performance during this subperiod, fund performance persists during this time frame, a period when hot money does well. During this subperiod, a Bayesian framework that permits some belief in managerial skill performs very well. However, when priors tilt away from a belief in managerial skill, the Bayesian estimates do not predict future performance as well. In these instances, investors, because of their strong priors against the existence of skill, essentially choose funds with the lowest expenses. In the second subperiod, the Bayesian alphas do not predict future performance well, especially for those that allow for the possibility of managerial skill. However, conventional measures predict future performance even worse than the Bayesian measures on average, and actually produce a slightly perverse ranking. When ranking on single- (four-) factor alpha, the subsequent Sharpe ratio of top decile funds is actually 0.0005 (0.000) less than the Sharpe ratio of bottom decile funds. On average, the Bayesian alphas predict future performance better than the standard measures across all parameters except when we use five years of historical factor data. During this subperiod, managerial skill, as measured by conventional alphas, does not persist, and the best strategy is to invest in funds with low expenses. This is precisely the strategy that investors with strong priors against managerial skill undertake, and it is this group of investors that outperforms during this subperiod. 16
4.4. Explaining the Results The main differences in estimating Bayesian alphas and standard single- or fourfactor alphas are that the Bayesian estimates use historical factor data that begins before the fund returns and the Bayesian estimates explicitly account for fund expenses. In this section, we show how these refinements enhance the predictability of the Bayesian alphas compared to the standard measures and examine the sensitivity of the Bayesian estimates to the length of historical factor data that we use. 4.4.1. Historical factor data The range of historical factor data that we use to estimate the Bayesian alphas affects how well the Bayesian measures predict future performance. Stambaugh (1997) indicates that more data is preferred because it permits a better estimate of the nonbenchmark factor abnormal returns. The results in Table confirm that, overall, the Bayesian alphas predict future performance better when we estimate them using more years of historical factor data. For example, the average Spearman correlation between the Bayesian alpha decile ranking and subsequent Sharpe ratio decile ranking is 0.865 when using all historical factor return data and 0.8 when using one year of historical data. The length of historical factor data impacts the Bayesian alpha only through the style component (via the non-benchmark abnormal returns in equation (5)). Consequently, the extent to which the abnormal returns of the non-benchmark factors are predictable over various spans of time determines how well various estimates of the Bayesian alpha predict future performance. The overlap in the time series of non-benchmark factor returns used to estimate the historical α is longer, in many iterations, than the 43 post-ranking period quarters. For example, when using 0 years of past factor data, adjacent historical factor returns have 79 quarters of returns in common, which induces severe autocorrelation in the historical abnormal return estimates. Consequently, it is not feasible to estimate the relation between the historical α and the future α, t+ 1 in a regression framework, even with GMM techniques. Instead, we examine predictability by computing the mean squared error between the α estimates using 1,, 5, 10, 0, or all years of historical 17
data and the future α, t+ 1. Table 5 shows the results. The results indicate that, for all three non-benchmark factors, the mean squared errors are smaller when using all available historical return data than when using only one year of historical data. That is, future nonbenchmark abnormal returns are better predicted using long- rather than short-horizon means. Table indicates that the decline in performance predictability is not monotonic as the amount of historical data decreases. The effectiveness of the Bayesian procedure declines dramatically as the horizon decreases to ten and then five years of historical data. However, the ability of Bayesian alphas to predict fund performance improves again for shorter estimation periods of one and two years. This pattern is more pronounced in Panel A of Table 4, where, during the 1990-1995 subperiod, there is an insignificantly negative relation between the five-year Bayesian alpha and subsequent Sharpe ratio. The poor results at the five-year horizon are attributable to poor non-benchmark abnormal return predictability at that horizon. In particular, during this subperiod, the mean abnormal returns for the SMB, HML, and UMD factors estimated using five years of historical factor returns are 0.0051%, 0.0107%, and 0.0306% per day, respectively. Funds with higher Bayesian style estimates have large UMD loadings at the expense of small SMB loadings and negative HML loadings. However, although the future UMD abnormal return is large (0.0487% per day), the future HML abnormal return is more than double (0.035% per day) its five-year historical level. Consequently, the funds that have high style rankings (with negative exposure to the HML factor) subsequently perform worse than expected, and this leads to the poor performance predictability during this time period for Bayesian measures based on five years of historical factor returns. 4.4.. Fund fees and predictability In addition to using a longer time series of factor returns, Bayesian alphas also explicitly account for fund expenses. To see why this enhances the Bayesian measure s ability to predict future performance, we first examine the simple relation between expenses and subsequent performance. We sort funds into deciles based on the expense ratio during a quarterly ranking period. We use the annual expense ratio since funds deduct their annually reported expenses from their total net assets proportionately each day. ext, we examine the average Sharpe ratio for each decile during the following 18
quarter. Table 6 shows the results. Consistent with Gruber (1996) and Carhart (1997), among others, the table shows a significant relation between expenses and subsequent performance. This result indicates that, on average, funds do not fully recoup their expenses with skill. The table indicates that the relation is statistically significant at the 5% level in both the first half and the second half of the sample period. Since funds report returns net of expenses, performance measures based on returns, such as standard single- or four-factor alphas, indirectly account for fund expenses. If the Bayesian alphas predict future performance better than standard single- or four-factor alphas because the Bayesian measures explicitly account for expenses, then the expense ratio should have incremental explanatory power over the standard alpha in explaining future performance. We examine the incremental explanatory power of the expense ratio with cross-sectional regressions, Sharpe = + t 0αt 1 γ1expt 1 γ, (13) where α t 1 is the quarter t-1 standard single- or four-factor alpha and Exp t-1 is the quarter t-1 expense ratio. When we run regression equation (13) with the single- (four-) factor alpha as a regressor, γ 1 has a t-statistic of 3.713 ( 3.174), which is statistically significant at the 1% (1%) level. The results indicate significant incremental predictive power for the expense ratio beyond standard return-based performance measures. We find similar but slightly weaker results when we run regression equation (13) over subperiods. From 1985-1990 the t-statistics for γ 1 are.65 and 1.810 for the single- and four-factor model, respectively; from 1990-1995 the t-statistics for γ 1 are 3.063 and.76 for the single- and four-factor model, respectively. 5. Conclusion Magazines such as Business Week and Money as well as mutual fund information providers such as Morningstar regularly report mutual fund performance measures. Many of the reported measures are similar in spirit to the standard risk-adjusted single- or multifactor alphas widely examined in the academic literature. For a given fund, these performance measures are invariably based on fund and factor returns that span the same 19
time period. Although such measures may accurately reflect past performance, they don t always accurately predict future performance (see Carhart, 1997). In this paper, we use the techniques of Pastor and Stambaugh (001a,b) to contrast Bayesian estimates of mutual fund abnormal performance with the traditional fund measures. Our results suggest that using historical data and the Bayesian procedure produces a powerful method for predicting future performance that is superior to conventional performance measures. For example, from its August 1996 inception through the end of 1999, the Munder etet Fund returned an average of 93.44% per year, earning Morningstar s highest five star rating in January 000. Although the fund s performance was stellar by almost any standard measure, it was not necessarily reflective of future performance. By using factor data that precedes the fund s inception date, and balancing this with the fund s high 1.59% annual expense ratio, a Bayesian measure would be much less sanguine about the etet Fund s future prospects. Overall, sorts based on Bayesian alphas computed with all years of historical data significantly outperform standard single- or four-factor alpha sorts in predicting future performance. Furthermore, for an investor who believes in managerial skill, the Bayesian alphas predict future fund performance better than traditional alpha-based sorts regardless of whether mutual fund skill persists. For investors that are skeptical of managerial skill, the Bayesian approach dominates a strategy of investing in low-expense funds when skill persists. It is only during periods of time when skill does not persist and for investors that are skeptical of managerial skill that the Bayesian strategy performs similarly to a strategy of investing in low-expense funds. 0
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Table 1. Summary statistics of mutual fund and factor sample The table reports average quarterly statistics for the mutual fund sample (Panel A) and the benchmark and non-benchmark factors (Panel B). The single -factor model is r A, t α A1 + βarb, t + εa, t =, (1) where r A is the excess fund return, and r B is the excess return on the CRSP YSE/AMEX/asdaq value-weighted market portfolio. The four-factor model is r A, t A + ca, r, t + c A, B rb, t + u A, t = δ, (3) where r are the SMB, HML, and UMD factors. CRSPVWex is the excess return on the CRSP YSE/AMEX/asdaq value-weighted market portfolio. The sample consists of 30 mutual funds. The mutual fund sample period is from January, 1985 to December 9, 1995. The factor sample period is from July 1, 1963 to December 9, 1995. Panel A. Mutual funds Single -factor Four-factor Period R σ α β R δ c c c c B SMB HML UMD R 1985-1995 0.0567% 0.786% -0.00345% 1.017 0.793-0.00649% 0.985 0.188-0.137 0.094 0.856 1985-1990 0.067 0.837-0.0038 0.950 0.807-0.0017 0.950 0.191-0.179 0.057 0.860 1990-1995 0.0500 0.731-0.00464 1.091 0.779-0.01 1.03 0.184-0.091 0.134 0.85 Panel B. Factors Factor R σ α R σ α B1. 6307-841 B. 8501-951 CRSPVWex 0.0097% 0.775% - 0.0384% 0.846% - SMB 0.068 0.414 0.06% 0.008 0.546 0.006% HML 0.059 0.406 0.08 0.010 0.351 0.019 UMD 0.0416 0.496 0.041 0.0393 0.415 0.03 B3. 8501-9006 B4. 9007-951 CRSPVWex 0.0415 1.003-0.0353 0.654 - SMB -0.0145 0.64-0.011 0.00 0.430 0.0 HML 0.0034 0.34 0.013 0.0171 0.361 0.05 UMD 0.0303 0.43 0.01 0.0483 0.407 0.043 3