Residual Correlation and Predictability of Mutual Fund Performance

Transcription

1 Residual Correlation and Predictability of Mutual Fund Performance Wei Huang a, David Hunter a, Trang Phan b a Shidler College of Business, University of Hawaii at Manoa, Honolulu, HI 968, USA b Augustana College, Rock Island, IL 6101, USA Abstract We analyze the economics of R s return predictability. Correlations between idiosyncratic returns of portfolio assets, which we call Residual Correlation, explain the predictability of R much better than its remaining components, namely portfolio concentrations in, and average portfolio exposures to idiosyncratically volatile assets. Our results suggest that fund managers that show return predictability do so by holding many assets that share common exposures to either unmodeled common risk factors or unmodeled abnormal return events. Thus, abnormal return persistence in these funds could result from either skill or unmodeled risk. Regardless, we show that by attributing R s predictability into its component parts, residual correlation can identify abnormal return predictability in a broader variety of mutual fund objectives. Residual correlation outperforms R s prediction of abnormal returns in situations where peer funds share similar idiosyncratic volatility exposures, such as exists among closet index and sector funds. Keywords: G11, G3, mutual funds, performance measurement, idiosyncratic volatility, active management addresses: weih@hawaii.edu (Wei Huang), hunterd@hawaii.edu (David Hunter), trangphan@augustana.edu (Trang Phan)

2 1. Introduction Predictability of mutual fund abnormal returns, though pervasively persistent, is frequently modeled as an endowment that a few lucky managers possess while others do not. Surprisingly little is understood about the economic content of this predictability, meaning that existing studies reveal little about the variables portfolio managers might evaluate to produce this predictability. Our analysis begins with a powerful indicator of mutual fund return predictability, R s complement (1 R ), which Amihud and Goyenko (013) show has greater power to identify predictability than other previously identified measures of a mutual fund s return predictability. We decompose this measure into three component parts, namely, correlations between idiosyncratic returns of portfolio assets, which we call Residual Correlation (RC), portfolio concentrations in, and average portfolio exposures to idiosyncratically volatile assets (abbreviated as HI and AI, respectively). This decomposition also reveals economic content, because the same measures capture distinct portfolio actions. We show that RC strongly identifies a fund s abnormal return predictability. It captures R complement s predictability better than its two remaining components. Persistence of RC in some funds and a persistent absence of RC in others aligns with funds return predictability. Furthermore, a fund s persistently high RC can only result from two alternative scenarios: First, perhaps fund managers identify and then purchase assets that exploit outof-model risk factors. Such actions would give an illusion of predictable abnormal returns, but should actually represent risk exposure if a more precise model were possible. Second, perhaps fund managers anticipate and exploit future abnormal return events, so they purchase multiple assets to boost their returns from these events. The first example would indicate that managers exploit unobservable risk-seeking opportunities, while the second would indicate that managers are skilled. RC is distinct from other measures that identify abnormal return predictability because of the data it exploits. RC is constructed from periodic public disclosures of mutual fund holdings, which omit all unobservable intra-period trade effects. It significantly predicts fund

3 performance, even after controlling for fund characteristics and other active management measures. On average, funds with the highest RC have 0.14% higher return per month (1.68% annually) as measured by four factor alpha (alpha obtained from the Fama French three factor model augmented with the Carhart momentum factor), compared to funds in the lowest quintile portfolio. The other two components of R s complement, AI and HI, do not predict fund performance. These tests confirm that mutual funds strongest return predictability appears in funds with correlated residual returns in their reported asset holdings, while residual volatility exposures alone are insignificant. RC outperforms R s prediction of abnormal returns in situations where peer funds share similar idiosyncratic volatility exposures, such as exists among closet index and sector funds. We not only investigate the performance predictability of RC among broad ranges of funds, but we also examine this performance predictability among specific groups of funds. These groups include: index funds; closet indexes; and sector funds. Active management has not been investigated among these funds, because existing measures are unable to do so. Since RC does not depend on identification of arbitrary industry or benchmark measures, it is particularly well suited to identify active management among these more specialized groups of funds. RC consistently predicts performance among closet index funds, and among sector funds. For robustness, we confirm that RC cannot predict abnormal returns among index funds, providing supporting evidence that RC captures economically meaningful information about active management. This paper proceeds as follows. Section explains empirical methodology, Section 3 describes data and presents summary statistics, Section 4 provides empirical results, and Section 5 concludes.. Motivating Residual Correlation (RC) Existing research asserts that mutual fund returns are strongly predictable, yet it is unclear which portfolio decisions produce strong predictability. Consider a portfolio manager who knows that his success and compensation depend heavily upon if he outperforms his peers in strongly competitive mutual fund markets. Some predictability measures impose 3

4 specific portfolio constructs, but cannot explain the economics that motivate these constructions. For example, industry concentration (Kacperczyk et al., 005) might inspire a fund manager to concentrate his portfolio holdings within industries into particular assets where he anticipates his greatest returns. Alternatively, Active Share (Cremers and Petajisto, 009) might inspire an aspiring portfolio manager to hold benchmark-variant positions in any number of assets and proportionally scale his bets according to perceived informational potency. Both measures treat manager information (or perhaps skill) as an endowment, and thus cannot explain the economics that guide these positions. R, or rather its complement (1 R ), was shown by Amihud and Goyenko (013) to be an exceptionally powerful predictor of mutual fund abnormal returns. Its statistical power makes it stand out among competing measures. R s complement is also appealing because it parsimoniously makes no portfolio construction assumptions. Without deeper investigation, however, this parsimony also obscures the economics that truly support return predictability. For example, a portfolio manager that follows this research might presume that any increase to portfolio idiosyncratic volatility will improve his abnormal return predictability, which directly opposes the implications of Ang et al. (006). Using R s complement as a starting point, we decompose it into three distinct measures that each could predict mutual fund abnormal returns. These components are as follow: first, a square of a portfolio s average exposure to asset idiosyncratic risk (AI: i.e. squared portfolio weighted average idiosyncratic risk); second, a portfolio s concentrated exposure to portfolio idiosyncratic volatilities (HI, i.e. herfindahl index of portfolio idiosyncratic volatilities); and third, the average weighted pairwise correlations of a portfolio s asset residuals (RC: i.e. weighted average pairwise residual corrleations). We test which of these three measures best predict future mutual fund abnormal returns. Correlated residuals can occur in portfolio returns if either (1) portfolio assets share common exposures to out-of-model risk factors, or () portfolio assets share common exposures to unmodeled abnormal return events. The first of these, out-of-model risk factor exposures, could indicate that mutual fund predictability results from opportunistic risk shifting (Brown et al., 1996) of portfolio assets. For example, managers that generate return 4

5 predictability opportunistically identify and exploit unmodeled risk factors to elevate their abnormal returns above those of their competitors. The second of these would indicate that mutual fund abnormal return predictability results from informed fund managers who skillfully exploit their superior information..1. Decomposing R s Complement R s complement, (1 R ), strongly predicts mutual fund abnormal returns (Amihud and Goyenko, 013), so we first consider how to decompose R s complement into economically meaningful component parts. Without restriction on generality, let R estimates be generated from a factor model regression such as Fama and French (1993) s 3 factor model plus Carhart (1997) s fourth momentum factor. R s complement for a regression on portfolio p s time series returns is then simply the ratio of residual volatility (σɛ p ) to total return volatility (σp) for every vector of residuals ɛ in every portfolio p. (1 R p) = σ ɛ p (1) We show that Equation 1 s numerator decomposes into our aforementioned three key portfolio components: AI, HI, and RC. We also note how each component captures distinct portfolio considerations. The residual volatility of a portfolio p is the weighted sum of its N assets residual variances and covariances. For any a given portfolio p at time t, let w t,p be an N 1 column vector of asset weights (w i i 1... N), such that w t,p1 = 1, and Σ t be an N N matrix of asset residual variances/covariances. Define a matrix D t = diag(σ t ) to represent a diagonal matrix of asset residual standard deviations (σ ɛi ), let I represent a standard identity matrix, and let 1 denote a square N N matrix that has ones in every cell. For convenience, subscripts for time t and portfolio p are omitted from the equational notations below: 5

6 σ ɛ p = w Σw = w [DD] w + w [Σ DD] w () = Wt. Average Resid. Variances + Wt. Average Resid. Covariances The weighted average covariance portion of Equation () can be further split into the product of two components: σɛ p =w DDw + w D [ D 1 ΣD 1 I ] Dw (3) [ ] w =w D [D 1 ΣD 1 I] Dw DDw + (w D1Dw w DDw) (4) w D [1 I] Dw To understand the decomposition of Equation 3, note that the term D 1 ΣD 1 in Equation 3 is the matrix representation of asset correlations that has ones along the main diagonal, and N (N 1) pairwise asset correlations (ρ ɛi,ɛ j i, j 1... N, i j) in both the upper and lower triangular portions of the matrix. Subtraction of the identity matrix I from this correlation matrix simply eliminates all main diagonal perfect correlations. Thus, Equation 3 simply restates Equation in correlation matrix form. Rearranging Equation 4, we define HI, RC, and AI as follows: 6

7 σɛ p = AI (RC) + (1 RC) HI (5) where: RC = w D [D 1 ΣD 1 I] Dw (6) w D [1 I] Dw ( ) = Σ N i=1σ N w i w j σ ɛi σ ɛj j=1,j i Σ N m=1σ N n=1,n m w ρ ɛi,ɛ mw n σ ɛm σ j ɛn AI = w D1Dw (7) = ( ) Σ N i=1w i σ ɛi HI = w DDw (8) = Σ N i=1w i σ ɛ i Equation 6 shows that RC (for Residual Correlation ) is a simple weighted average of asset residual correlations. For any pairwise residual correlation of two assets, the weight is weight i,j = w i w j σ ɛi σ ɛj Σ N m=1 ΣN n=1,n m wmwnσɛmσɛn. The weight of each pairwise correlation of residual returns depends upon both asset weights, and the idiosyncratic risk of the two assets. Equation 7 shows that AI (for Squared Average Idiosyncratic Risk ) is a simple portfolio weighted average of a portfolio s asset idiosyncratic risk. This term is squared, which represents a simple monotonic transformation of the portfolio weighted average. Falkenstein (1996) shows that mutual funds have non-linear preferences for assets with high idiosyncratic risk. These high idiosyncratic risk stocks may have greater firm-specific information that an active manager may exploit. Ang et al. (006) show that stocks with high idiosyncratic risk have low average returns. In the absence of active management, a fund that spreads its holdings to a large number of high idiosyncratic risk assets will have a high AI, which should result in low performance due to the negative relationship identified by Ang et al. (006). Equation 8 shows that HI (for Herfindahl of Idiosyncratic Volatility ) represents a Herfindahl measure of a portfolio s concentration in idiosyncratic volatility. Research such 7

8 as Kacperczyk et al. (005) argue that skilled managers that concentrate their portfolio exposures into select (expectedly outperforming) assets significantly outperform their peers. Such funds should show a significantly greater HI estimate than their peers. Funds may concentrate their portfolios in high idiosyncratic risk assets, i.e. they might prefer big bets on idiosyncratic risk of a single (or a few) individual assets. For example, they might have superior skill and therefore seek to exploit their information. As their superior information pays off, it manifests as high abnormal returns and consequentially increased idiosyncratic volatility. In this case, we would expect a positive relationship between HI and fund performance. Alternatively, suppose the managers have no skill, but instead manipulate their risk exposures to exploit their opportunity to earn high returns. When lucky, such funds might enjoy a correlation between HI and abnormal returns, but this should not persist long term. RC reflects how strongly idiosyncratic returns correlate. A fund manager may either skillfully anticipate future abnormal returns, or he may exploit out-of-model risk factor exposures. Both actions would generate correlated residuals. We devise one test to distinguish between skill or risk taking by following the risk-shifting arguments of Brown et al. (1996). If correlated residuals result from manager skill, then managers with the highest past performance will be those with the highest future RC. Conversely, if correlated residuals result from risk shifting, then managers with the lowest past performance will be those with the highest future RC. We construct this simple test to distinguish manager skill or risk shifting within RC predictability. Equation 5 shows that the numerator of R s complement (portfolio idiosyncratic volatility) is a convex combination in RC of average exposures to asset idiosyncratic volatilities (AI), and concentrated exposures to idiosyncratic volatilities (HI). R complement s decompositional form is thus as follows: σ ɛ p = AI (RC) + (1 RC) HI (9) Equation 9 shows that when asset residual correlations are low, then R s complement is greatest when portfolios concentrate their exposures in high idiosyncratic volatility stocks. 8

9 When residual correlations are high, the R s complement is greatest when average asset residual volatilties are high. Our tests in this paper focus upon the three measures, AI, HI, σp σp and RC, because they directly correspond to R complement s decomposition. In unreported results, we have repeated these tests with the denominator term omitted with no significant changes to our results. 3. Data and summary statistics Our mutual fund data come from two sources. The first is the CRSP Survivorship Bias Free Mutual Fund Database includes information funds characteristics such as fund returns, total net assets, investment objective, turnover ratio, expense ratios and other types of fees. CRSP Mutual Fund Database records different share classes of the same fund as distinct funds. The second source is CDA/Spectrum S1 mutual fund holding database. The CDA database is collected from mutual funds reports filed with SEC and from funds voluntary reports. The two databases are merged using MFLINKS file of Wharton Research Data Services (WRDS). We aggregate different share classes of the same fund in CRSP as follows. For age and qualitative characteristics (name, investment objective), we keep the value of the oldest share class. For TNA of the fund, we add up TNAs of its share classes. For other quantitative characteristics, we take the weighted average of other quantitative characteristics (return, expense ratio, management fee, turnover ratio), using the most recent TNA of each share class as weights. We focus on actively managed domestic equity mutual funds, eliminating balanced, bond, money market, and international funds. Following Huang et al. (011), we select funds with the following Lipper objectives: CA, CG, CS, EI, FS, G, GI, H, ID, LCCE, LCGE, LCVE, MC, MCCE, MCGE, MCVE, MLCE, MLGE, MLVE, MR, NR, S, SCCE, SCGE, SCVE, SG, SP, TK, TL, UT. When the Lipper code is missing, we select fund with the following Strategic Insights objectives: AGG, ENV, FIN, GMC, GRI, GRO, HLT, ING, NTR, SCG, SEC, TEC, UTI, GLD, RLE. When both Lipper and SI codes are missing, we select funds with the following Wiesenberger objectives: G, G-I, G-S, GCI, IEQ, ENR, FIN, GRI, HLT, LTG, MCG, SCG, TCH, 9

10 UTL, GPM. If a fund has none of these objective codes but it has a CS policy or has the percentage of common shares in the porfolio between 80%-105%, then the fund will be included. The percentage of common shares is calculated as the time-series average for each fund. We exclude all funds that contain the following words in the fund name: INTERNA- TIONAL, GLOBAL, BOND, BALANCED, MONEY MARKET. We exclude funds with TNA less than $5 million because inclusion of smaller funds may cause a survivorship bias problem. We follow Cremers and Petajisto (009) to include only funds that have more than 67% of value of stock holdings over the fund total net asset. We also require that funds hold ten stocks or more and that fund characteristics (TNA; Age; Turnover; Expenses) are non-missing. After all exclusions, our final sample includes 391 actively managed equity funds Our sample period is from 1980 until 01. To decompose R s complement into AI and HI, we follow Equations 7 and 8. This requires estimates of portfolio weights w and asset idiosyncratic risk σ ɛi for each asset i in i = 1... N assets. Asset weights are from CRSP s mutual fund holdings database and the CDA Spectrum holdings in earlier years, and we calculate monthly idiosyncratic volatility for all stocks held by mutual funds that have more than fifteen daily return observations in a month. RC may be calculated without data-intensive estimation of pairwise asset correlations by recognizing that total portfolio idiosyncratic volatility (σɛ p ) can be estimated directly from daily portfolio returns, and Equation 5 can be rewritten as: RC = σ ɛ p HI AI HI (10) RC is specified to represent existing portfolio positions, to avoid possible intra-period trade effects. Unobservable trades between reporting periods may create an illusion of residual correlation where none really exists, and RC should not account for such actions. For this reason we intentionally discard the actual reported returns of mutual funds when we calculate σɛ p, and instead exploit the daily returns of a passive portfolio that is constructed 10

11 from each fund s most recently reported portfolio holdings. 1 This passive portfolio construction is not unlike the passive portfolio that was used to construct Kacperczyk et al. (008) s return gap. Thus, HI, AI, and RC are each estimated from every fund s most recently reported portfolio holdings, and thus are not directly influenced by unobservable portfolio actions that might occur after portfolio holdings are reported. Our estimation of idiosyncratic volatilities, both for individual assets and for hypothetical portfolios of reported asset holdings follow the techniques of Ang et al. (006) who calculate monthly idiosyncratic risk of stocks based on the standard deviation of daily residual returns. Specifically, we estimate residual returns using the four factor risk model (Fama French three factor model, augmented with the Carhart momentum factor) for every stock or hypothetical portfolio that has more than 15 daily return observations in a given month. A stock s idiosyncratic risk (σ ɛ ) is the standard deviation of those daily residual returns multiplied by the square root of the number of return observations during the month. Since R, AI, HI, and RC are estimated monthly from daily asset and portfolio returns, σp each estimate is highly variable from month-to-month. For stability, we average these values over the six months prior to each month t in our analyses. Table 1 presents summary statistics of the mutual funds in our sample. Our sample size is slightly larger than those of existing mutual fund predictability studies because we don t eliminate sector funds and index funds from our sample. For robustness, we also tested our result with a smaller sample in which we eliminate sector and index funds and confirmed the results to be similar. Otherwise, our sample data show similar characteristics as prior studies. Unlike prior studies, we also present the three underlying components in R s complement. RC ranges from to 0.78, with a mean of 0.0 and a median of This suggests that the RC is highly skewed, and most funds have a residual correlation that is close to zero. Despite this, a significant subset of funds generate significantly strong RC. In unreported results, we find that over 10% of sample funds have an average RC estimate that is 1 We calculated daily portfolio return from asset daily return, assuming buy-and-hold weights. However, the results are not different when we use constant asset weights (under the daily re-balancing assumption). 11

12 more than double the sample mean. These funds, though a small subset of all funds, remain economically significant if they produce predictable future abnormal returns. Although 0.0 may seem to be a very small average RC, note that residuals are typically expected to be uncorrelated, and this sample average is significantly larger than a market weighted average RC of (not shown in the table), which we estimated in the same manner over the same sample. AI averages.1, which indicates that funds squared average asset idiosyncratic risks (AI) are expectedly more than double their total portfolio volatility (). This simply confirms that the average mutual fund constructs portfolios that diversify away asset idiosyncratic risk exposures. HI averages , which indicates that funds concentrated exposure to idiosyncratically volatile assets is also quite small. Each measure captures sizable subsets of funds that possess either substantially larger investment in idiosyncratically volatile assets or significantly greater concentration in idiosyncratically volatile assets. 4. Empirical Results In this section, we illustrate RC s critical role in determining R complement s prediction of mutual fund abnormal returns. We demonstrate that the three distinct components of R complement do not strongly correlate with each other. We then show that mutual funds persist most strongly at either very high RC exposures or very low RC exposures. We show that mutual fund abnormal return predictability is most strongly correlated with RC, above not only over the alternative components of R s complement, but also above other existing measures of abnormal return persistence. We present this evidence as from both regression and n-tile portfolio sorts. We then show that RC captures abnormal return persistence among funds that other measures have previously failed. Finally, we test if changes in RC can be explained by manager skill or manager risk shifting behavior. AI RC,, and HI σp σp each capture distinct dynamics within R s complement. Table presents time series Spearman correlations between R s complement and its three com- AI ponent parts. RC,, and HI σp σp are estimated monthly, using every fund s most recently reported portfolio holdings and asset weights (no more than 6 months old), daily asset 1

13 residual returns, and daily residual returns of a hypothetical portfolio formed from each fund s most recently reported holdings. Daily residual returns of both individual assets and the hypothetical holdings portfolio were estimated from 4-factor model regressions that included the 3 factors of Fama and French (1993), and the momentum factor of Carhart (1997). Equations 10, 7, and 8, guide the construction of each estimate. Pairwise Spearman correlations between RC, AI, and HI and R s complement are calculated cross-sectionally σp σp at every month t in the sample, then these correlations are averaged across time to produce the estimates presented in Table. R complement s sensitivities in Equation 5 is visible from table. Observe that R s complement is most strongly and positively correlated to RC and HI. This implies that R σp complement is greatest when either RC is high or HI is high. Table also reveals that RC, AI, and HI have very low correlations with each other, meaning that each capture distinct dynamics of R s complement. Of the three components, HI and AI share the greatest correlation of 0.4. RC is only slightly correlated with HI correlation of 0.3, and RC has almost no correlation with AI. Thus, most of the variation σp in these three components is unrelated to the variation of another. RC tends to be large and highly persistent in some mutual fund portfolios, and persistently absent from others. We show this by constructing a transition matrix that shows by ranked RC quintile at time t, the average percentage of funds that are in the each of the 5 ranked RC quintiles in the next year t + 1. Every year in our sample period, we sort funds into 5 quintiles based on RC. Among the funds that have non-missing data for two continuous years, we calculate the percentage of funds in each quintile that switch to each of the five quintiles in the next year. On average, about 15% of the funds drop from the sample in the next year. RC persistence is bi-modal. It is strongest among the highest RC funds (quintile 5) with 71% of funds in quintile 5 remaining in the same high quintile in the following year. The next strongest persistence arises among the lowest estimated RC (quintile 1), with 6% of funds in quintile 1 remaining there in the next year. Table 3 also reveals that RC is strongly persistent, regardless of which ranked RC quintile 13 at a

14 a fund begins. Notice that throughout the table, the highest transition percentages fall upon the main diagonal of the matrix. The significance of RC persistence is particularly salient when one considers that holdings turnover in sample mutual funds is very close to 100% year over year (shown in Table 1). This means that the holdings that produce persistent RC (recall that RC is calculated using past reported fund holdings) in high RC funds are frequently changing, yet they persistently retain high levels of RC. In contrast, funds with persistently low RC also frequently change their holdings without any consequential increase to RC. Table 3 suggests that RC is more characteristic of a fund or its manager than it is of its asset holdings. By distinguishing the return predictability of each of R complement s three components, one can discern the economic content of mutual fund return predictability. Predictability in RC can only result when multiple portfolio assets returns simultaneously respond to out-ofmodel return effects. Only two explanations can account for a fund persistently incurring such return effects: (1) the fund has exposed multiple portfolio assets to some unmodeled risk factor and earns a higher average risk premia as a result; or () the fund has obtained multiple portfolio assets that are similarly mispriced, and the mispricing simultaneously corrects in the future. The first of these relates to unobservable risk seeking by managers, while the second relates to informational superiority or skill. Predictability in HI would result if managers selectively concentrate their portfolios in fewer assets that they expect will earn higher abnormal returns. As with RC, this expectation could arise from a manager s exposure to risk premia or endowment of skill. Importantly, predictability of HI would require distinct portfolio actions than would predictability in RC. A manager that seeks predictability from HI would bear significant idiosyncratic risk in his portfolio from reduced diversification, but one that does the same using RC would not. Likewise, predictability in AI would also indicate that managers should pursue distinct portfolio actions from the other two measures. Predictability in AI would result if managers σp earn higher abnormal returns by holding assets with higher average idiosyncratic volatilities than other assets in the market. Falkenstein (1996) noted that mutual funds do, indeed, hold assets with greater idiosyncratic volatilities than the market. Such predictability would 14

15 reward asset managers without any need for forecasting, risk shifting, or skill. We investigate the performance predictability of R s complement and its three component parts. To do this, we regress future abnormal returns on past observations of R s complement, RC, HI, and AI and control variables, including lagged abnormal returns, expense ratio, turnover, the natural log of fund age (in months), the natural log of each σp σp fund s total net assets (TNA), and the squared natural log of each fund s total net assets. To permit comparisons between regression coefficients, all independent variables in the regression were standardized except the lagged abnormal return. The dependent variable and the lagged dependent variable were not standardized to set the estimated coefficients to units of monthly abnormal returns. The abnormal return dependent variable was estimated by regressing each fund s past 36 month performance on the three factor risk model of Fama and French (1993) plus the momentum factor of Carhart (1997). This produced at each month t estimates of past abnormal returns and risk factor exposures. Future abnormal returns were obtained by calculating the difference between each fund s time t + 1 gross total return (reported net return plus 1 month expense ratio) and the expected return from the time t 36 month estimated coefficients and the time t + 1 realized risk factor premia. In additional to using future abnormal returns as dependent variable, we also run the same regressions with time t + 1 estimates of characteristic selectivity (CS) and characteristic timing (CT) of Daniel et al. (1997) as dependent variables. These regressions were run as stacked panels, and t-statistics were adjusted to control for correlated errors across both time and funds. Table 4 presents our results. The first four columns of the table present results using time t + 1 abnormal returns where t + 1 represents the next quarter (3 months). First notice that the first column of the table confirms Amihud and Goyenko (013) s result that R s complement significantly predicts mutual fund performance. The next three columns then test the predictive power of each of R complement s three component parts. They show that only RC significantly identifies R complement s predictability, and its coefficient increases in magnitude and nearly matches R complement s statistical significance. We repeated each of these regressions letting t + 1 represent one month instead of one quarter and obtained consistent estimates and statistical significance. RC explains R com- 15

16 plement s prediction of mutual fund abnormal returns, while the other two components do not. For brevity, Table 4 only presents Column 5 s predicton of one month results, where RC was controlled. Columns 6 and 7 of Table 3 present the coefficient estimates when t + 1 values of CS and CT are predicted. These regressions show that RC predicts the abnormal returns that are captured by a fund s characteristic selectivity, and not its characteristic timing. CS measures abnormal return attributable to a fund s stock selection, thus RC appears to correlate more strongly with a manager s skill than his timing of risk factor exposures. Its important to note here that RC only captures residual correlation. One may perhaps think that if RC can predict mutual fund abnormal returns, then perhaps a direct estimate of total return asset correlations (instead of residual correlations) could also replicate Table 4 s results. We repeated Table 4 s tests using total return correlations. This can be done by simply replacing residual measures with total return measures in Equations 10, 7, and 8. In fact, none of these total return measures could identify any abnormal return predictability, so the results are not presented here. If RC can predicts fund performance, a natural question is whether a long-short portfolio in RC sorted funds can generate abnormal returns. We thus examine the relative performance of funds in quintiles of holdings portfolio sorted by RC. Every month, we sort all mutual funds into five quintile portfolios based on RC. For each quintile portfolio, we compute the equally weighted average next month gross return (reported net return plus expense ratio). We further compute the difference of equally weighted average returns of the top and the bottom quintile portfolios. Table 5 reports the time-series average of these portfolio returns, the return difference of top and bottom quintile, the CAPM alpha, Fama French three factor alpha and the four factor alpha. The robust Newey-West (1987) t-statistics are reported. Panel A of table 5 exhibits that top quintile portfolio of funds sorted by RC significantly outperforms the bottom quintile by various performance measures (gross raw return and risk adjusted returns that include CAPM alpha, Fama French three factor alpha, and Carhart four factor alpha). On average, a long position in the portfolio with the highest RC and 16

17 a short position in the portfolio with the lowest RC generate a gross raw return of 0.18% and a four factor alpha of 0.14% per month. This translates to an annual outperformance of.16% (gross raw return) and 1.68% (four factor alpha). To see if the long-short strategy may work for portfolios sorted by the two other components of R complement, we sort all mutual funds into five quintile portfolios based on the ratio of HI to σp and the ratio of AI to σp. The difference in performance of top and bottom quintile portfolios sorted by the ratio of AI to σp is not significant and slightly negative. Likewise, there is no significant difference in performance of top and bottom quintile portfolios sorted by the ratio of HI to σp. Consider that the expense ratio affects investors net profit, we further conduct tests with the reported net returns (return after expenses). Panel B of table 5 demonstrates that on an after-expenses-return basis, top quintile portfolio of funds sorted by weighted average RC significantly outperform bottom quintile portfolio in the month following portfolio formation, both for net returns and for three or four factor alphas. It s been known that the holdings of mutual funds are usually not immediately publicly available at the report date. For this reason, we examine if investors can still profit from the performance predictability of RC if the information of fund holdings are available a quarter after the report date. Panel C shows that the performance predictability of RC remains valid, even when the performance in the fourth month after portfolio formation is used. For example, an investor that only observes delayed holdings can realize significantly positive predictable returns (before expenses) at a magnitude of 0.17% per month (.04% per year). Panel D suggests that after fees, return predictability remains, however alpha predictability is not significant at conventional levels. There are two dimensions of active management, stock selection which involves picking individual stocks that are expected to outperform their peers; and factor timing which involves time-varying bets on systematic risk factors. Many recent measures of active management, such as the Industry Concentration Index 3 of Kacperczyk et al (005), Active The results are not reported for brevity but are available upon request. 3 Kacpercyk et al (005) define ICI as follow: Each stock is assigned to one of the 10 industries (as classified in Kenneth French Web page). Every month, ICI is calculated for each fund: ICI t = Σ 10 j=1 (w j,t w j,t ) where w j,t is the weight the fund loads on industry j, and w j,t is the weight of industry j in the stock market. 17

18 Share 4 and Tracking Error 5 of Cremers and Petajisto (009) are similar in that their measures successfully define stock selection of actively managed mutual funds and show that stock selection can earn abnormal returns. 6 In this section, we examine the performance predictability of RC, controlling for other active management measures. Panel A of Table 6 presents the coefficients of panel data regression of funds four factor alphas (in the next quarter) on various measures of active management, with other fund characteristics controlled for. We confirm that Industry Concentration Index significantly positively predict fund performance as demonstrated in Kacpercyk et al (005). Active Share marginally positively predict performance. This result is consistent with Cremers and Petajisto (009) who shows that while Active Share has significant predictive power for benchmark adjusted fund performance, its performance predictability is not significant for the four factor alpha. More important, we find that RC significantly positively predicts fund performance even after other active management measures are controlled for. Moreover, the coefficients of other active management measures lose significance when RC is added to the models. RC is a more powerful predictor than other active management measures, which is visible in coefficient magnitudes (when active management measures are regressed indepedently against subsequent four factor alpha)and in both coefficient magnitudes and statistical significance (when active management measures are regressed jointly against subsequent four factor alpha). To better understand why RC is a more powerful predictor of fund performance, we examine the correlation of R complement and its three components with active shares and industry concentration index. Panel B of Table 6 shows the correlations among the Industry Concentration Index, Active Share, R complement and the three components, namely, RC, 4 Active Share is defined as the percentage of a fund s portfolio holdings that differ from the fund s benchmark index. Specifically, AS = 1 ΣN i=1 w fund,i w index,i where w fund,i and w index,i are the weights of asset i in the fund portfolio and in the index, and the sum is taken over the universe of all assets. See Cremers and Petajisto (009) for the intuitive economic interpretation of Active Share. 5 Tracking error is defined as the time-series standard deviation of the difference between a fund return R fund,t and its benchmark index return R index,t : Tracking error = σ (Rfund,t R index,t ). 6 On the other hand, as suggested by Cremers and Petajisto (009), broad systematic factors may either be too efficiently priced or too difficult for fund managers to predict, so funds that focus on factor bets generally do not outperform. 18

19 the ratio of HI to σp and the ratio of AI to σp. Not surprisingly, the Industry Concentration Index and Active Share are highly correlated with R complement, with correlation coefficients of 0.64 and 0.77, respectively. This suggests that all three measures are similar indicators that capture stock selection of mutual funds. In contrast, the correlation coefficients of Industry Concentration Index and Active Share with RC are lower at 0.54 and 0.4, respectively. Furthermore, the Industry Concentration Index is moderately correlated with the ratio of HI to σp (0.47) while Active Share is highly correlated with the ratio of HI to σp (0.70). Active share is also relatively highly correlated with the ratio of AI to σp (0.50). The results suggest that both Industry Concentration Index and active share are associated with not only RC but also with other components of R complement that turn out to have little predictive power of fund performance. In particular, Industry Concentration Index is associated to both RC and the ratio of HI to σp. Active share, on the other hand, seems to be highly associated to both the ratio of HI to σp and the ratio of AI to σp. Thus, the correlation structure suggests that, compared to RC, other active management measures are noisier indicators that contain non-predictive components of R complement. We have shown that RC can identify active management and predict performance among the broad categories of equity mutual funds. We further apply our measure to the three specific groups of funds such as sectors fund, closet indexers and index funds. These funds have generally been omitted from other studies of active management. Unlike other existing active management measures, RC does not depend on identification in industry or benchmark measures, which makes it particularly well suited to identifying active management among these more specialized groups of funds with narrower industry or benchmark focus. Panel A of Table 7 tests active management among sector funds, defined in various ways. First, we form the group of sector funds by objective codes. Second, we define sector funds as the funds that hold only one industry in the funds portfolios. Third, sector funds are defined as the funds that hold one or two industries in the funds portfolios. Next, sector funds are defined as the funds that are in the highest decile of Industry Concentration Index. Finally, sector funds are defined as the funds that are in the highest decile of Herfindal Industry Concentration. Regardless of how we define sector funds, RC consistently predicts active 19

20 management performance while none of the alternative measures of active management consistently do so. Panel B of Table 7 tests active management among closet indexers, which are defined as in Petajisto (013). At the end of each month, all funds are sorted into quintiles by Active Share and tracking error. Funds in the lowest Active Share quintile (except those in the highest tracking error quintile) are defined as closet indexers. We used measures of Active Share and tracking error as used in Cremers and Petajisto (009) for the short period of 1990 to 006. In a different test, we used the measure of Active Share as used in Petajisto (013) for the extended period of 1980 to 010, with tracking error calculated relative to S&P 500 as in Wermers (003). In both samples, RC significantly predicts active management performance while none of the other measures does so. Panel C of Table 7 tests performance predictability of active management measures among index funds. We form broad group of all index funds using both funds objective codes and names. Regression results suggest that RC along with Industry Concentration Index and active share do not have significant return predictability among index funds. Having shown strong performance predictability of RC, we examine the determinants of the active management measure. Table 8 shows coefficients of a panel data regression on RC. The Residual Correlation shown here is the 1 month average of monthly RC. All independent variables are measured at the end of the previous year (so that it s nonoverlapping with the estimation period of RC). We use the lagged performance as measured by the Fama-French-Carhart four factor alpha, expenses, turnover, fund size, fund age as explanatory variables. Because RC and many of the independent variables are persistent over time, we cluster standard errors by funds. We find that the lagged fund performance is positively correlated with subsequent RC. In addition, funds with higher past turnover exhibit higher RC. Fund size is positively related to RC. The expense ratio is not statistically significant. Overall, funds with larger size, higher turnover, and better past performance tend to have higher RC measure. 0

21 5. Conclusion Mutual funds asset return correlations that are independent of systematic risk factors identify funds that earn predictable abnormal returns. These residual correlations (RC) also provide an economic interpretation of R complement s predictability of mutual fund performance. That is, that select funds persistently identify assets that share similar exposures to future out-of-model abnormal returns. This could represent either manager skill or risk-seeking investment activities. We calculate mutual fund portfolio residual correlations using each fund s periodically reported asset holdings. We derive its relationship to other variables within R s complement, and confirm that RC most powerfully identifies abnormal return predictability, even after controlling for fund characteristics and other return predictability measures. On average, funds in highest quintile portfolio as sorted by RC have an annual outperformance of about 1.68% per year on a risk adjusted basis, or.16% on a total return basis compared to funds in the lowest quintile portfolio. In contrast, the other two components, namely, the concentration of idiosyncratic volatility and average exposure to idiosyncratic volatility do not significantly predict fund performance. Using RC, we also find evidence of funds with predictable abnormal returns among groups of funds that are typically discarded from analyses of predictability, namely, sector funds and closet index funds. 1

22 References Amihud, Y. and Goyenko, R. (013). Mutual fund s R as predictor of performance. Review of Financial Studies, 6(3): Ang, A., Hodrick, R. J., Xing, Y., and Zhang, X. (006). The cross-section of volatility and expected returns. Journal of Finance, 61(1): Brown, K. C., Harlow, W. V., and Starks, L. T. (1996). Of tournaments and temptations: An analysis of managerial incentives in the mutual fund industry. Journal of Finance, 51(1): Carhart, M. (1997). On persistence in mutual fund performance. Journal of Finance, 5:57 8. Cremers, M. and Petajisto, A. (009). How active is your fund manager? a new measure that predicts performance. Review of Financial Studies, : Daniel, K., Grinblatt, M., Titman, S., and Wermers, R. (1997). Measuring mutual fund performance with characteristic-based benchmarks. Journal of Finance, 5: Falkenstein, E. G. (1996). Preferences for stock characteristics as revealed by mutual fund portfolio holdings. Journal of Finance, 51(1): Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33:3 56. Huang, J., Sialm, C., and Zhang, H. (011). Risk shifting and mutual fund performance. Review of Financial Studies, 4(8): Kacperczyk, M., Sialm, C., and Zheng, L. (005). On the industry concentration of actively managed equity mutual funds. Journal of Finance, 60(4): Kacperczyk, M., Sialm, C., and Zheng, L. (008). Unobserved actions of mutual funds. Review of Financial Studies, 1: White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4):

23 Table 1: This table reports summary statistics of sample mutual funds. Fund characteristics are reported by CRSP. Gross returns are calculated as CRSP reported net return plus expenses (expense ratio). R is defined as in Amihud and Goyenko (013), obtained from regressions of fund excess monthly gross return (reported net return plus expense ratio) over the one-month T-bill rate on the Fama and French (1993) and Carhart (1997) (FFC) four factor model over a period of the previous 4 months. RC; HI and AI three components of R s Complement (1 R ). RC is defined as the weighted average of pairwise correlation of daily residual returns, where the weight of each pairwise correlation depends upon both asset allocation weights, and the idiosyncratic risk of the two assets. It is calculated using RC = σ ɛp HI AI HI. AI σ is p the ratio of squared weighted average standard deviations of daily asset residual returns divided by the total return volatility of a hypothetical portfolio constructed from a funds most recently reported (no older HI than 6 months) portfolio holdings. σ is the ratio of a Herfindahl measure of asset daily residual p volatilities divided by the total hypothetical portfolio volatilities that were just described. Asset daily residual returns are calculated using the FFC four factor model within each monthly period. For stability, we average RC, AI σ, AI p σ, and R s complement over the prior six months (month t-5 to month t) to have p the variables of month t. Sample period is Summary Statistics Total Number of Funds: 3,91 Variable Mean Median Minimum Maximum Number Of Assets TNA in millions Age (Years) Expenses (%) Turnover (%) Gross return (%) R Residual Correlation (RC) Herfindahl Idiosyncratic Vol ( HI Average Idiosyncratic Risk ( AI ) ) are 3

24 Table : This table shows the time-series average of cross-sectional Spearman Correlations between funds R complement and its three component parts: RC, HI σ and AI p σ. R s complement is defined as (1R ) and p is calculated as a ratio with a funds residual return variance in the numerator and its total return variance in the denominator. Both variances are calculated using daily returns within a single month t. Each funds residual returns are constructed by creating a hypothetical portfolio of its most recently reported asset holdings (not older than 6 months). Residual returns are estimated from regressions of excess holding portfolio daily returns over the daily T-Bill rate on the FFC four factor model. RC, HI σ and AI p σ are p computed as before. Sample period is 1980 to 01. Correlations of R Complement and its components R HI Complement RC σp RC 0.6 HI σp AI σp Table 3: This table shows the persistence of mutual funds ranked RC estimates. RC is computed as before. Every year in the sample period ( ), funds are sorted into quintiles of 1-month-average RC. The proportion of funds switching to each quintile in the following years 1-month-average of Residual Correlation is reported. Residual Correlation Persistence Next Year RC Quintile RC Quintile % 0.59% 9.48% 6.33%.04% 1.15% 39.18% 3.94% 1.44% 3.9% % 4.79% 35.80% 3.61% 6.68% 4 5.7% 1.31% 4.46% 41.51% 15.99% 5.46% 3.70% 6.33% 16.54% 70.97% 4