Mutual Fund Families and Performance Evaluation

Transcription

1 Mutual Fund Families and Performance Evaluation David P. Brown Youchang Wu June 2012 Abstract We develop a model of evaluating mutual fund skills based on a fund s own performance and the performance of its family. Our model highlights two competing effects of family performance on the estimated skill of a member fund: a positive commonskill effect due to the reliance on common resources, and a negative common-noise effect due to the correlation of unobservable shocks to fund returns. Our analysis pins down a few key variables that determine the sensitivities of investor beliefs to fund and family performance. Consistent with the predictions of our model, family performance has a stronger impact on money flow to a member fund in larger families, and families with a larger fraction of team-managed funds, while the sensitivity of flow to a fund s own performance decreases with family size and increases with the correlation of idiosyncratic returns within families. Both authors are in the Department of Finance, Investment and Banking, School of Business, University of Wisconsin-Madison. addresses: dbrown@bus.wisc.edu, and ywu@bus.wisc.edu. We thank Michael Brennan, Pierre Collin-Dufresne, Thomas Dangl, Zhiguo He, Bryan Lim, Lubos Pastor, Matthew Spiegel, Hong Yan, Tong Yao, Josef Zechner, and seminar participants at the Utah Winter Finance Conference in 2012, American Finance Association meetings in 2012, Western Finance Association meetings in 2011, Financial Intermediation Research Society meetings in 2011, China International Conference in Finance in 2011, University of Wisconsin-Madison, University of Illinois at Urbana-Champaign, University of Illinois at Chicago, University of Technology Sydney for helpful discussions.

2 Investment outcomes are driven by skill and by luck. A fundamental issue in delegated portfolio management is performance evaluation; that is, to distinguish skill from luck. This distinction is crucial for appropriate selection of funds and compensation of fund managers. Most methods of performance evaluation focus on the records of individual funds in isolation, apart from any relevant information contained in other funds in the same family. Our objectives are to provide a theoretical framework of performance evaluation for mutual funds within families, and to examine empirically how investors incorporate both fund and family performance information when they allocate money across funds. Most mutual funds belong to a family where individual managers share common resources. One example of a shared resource is an information system that provides managers tools for portfolio analysis, risk management, and performance measurement. Another is a legal department that counsels portfolio managers. Legal expertise is particularly important for understanding patents; for evaluation of companies facing litigation or engaged in bankruptcy or corporate transactions; and for understanding contractual terms of bond indentures. Fund managers may also have access to the same set of outside experts, such as reports from particular sell-side firms. Finally, family pools of security analysts and traders provide investment ideas and transaction services to portfolio managers. As a result of family membership, a fund s risk-adjusted performance (its alpha) is determined by the quality of common resources in addition to the expertise of the manager. This is supported by empirical evidence. Baks (2003), for example, attributes the majority of funds abnormal returns to family membership rather than to individual managers. With this in mind, we develop a model in which the risk-adjusted performance of one fund is driven by its composite skill, which is a summary measure of the quality of family resources and its manager s expertise. Our model recognizes that returns of other funds in the family contain information about the family component of the composite skill. Therefore, the conditional estimate of a fund s composite skill is based on both its own performance and the performance of other funds in the family. Our model also recognizes that because of reliance on common family resources, the returns of funds in a family contains correlated noise. 1 As a consequence, family performance 1 Elton, Gruber, and Green (2007) find that mutual fund returns are more closely correlated within families 1

3 helps to filter out such noise in a member fund s returns. A positive shock leads to good performance by many funds in a family. By comparing one fund s performance with that of the rest of the family, we estimate the fund s composite skill more precisely. This provides another reason to incorporate family performance information into the evaluation of a member fund. In our model, the key distinction between the skill and the noise is that the skill determines the mean of the risk adjusted returns (i.e., the alpha) while the noise, represented by unobservable idiosyncratic shocks to fund returns, leads to temporary fluctuations around the mean. In general, we expect both alphas and noise to be more closely correlated within a family than across families. Family resources induce member funds to tilt their portfolios in similar directions, meaning that funds tend to over- or underweight the same securities relative to their benchmarks. For example, a single idea from the analyst pool can lead several managers to simultaneously increase or decrease positions in a security. As a result, both alphas and short-term fluctuations in returns are highly correlated within the family. However, this is not necessarily always the case. Some family resources affect the alphas of all member funds, but have little impact on the correlations of short-term fluctuations in their returns. Examples are the trading desks retained to execute trades, the risk management process, and the screening mechanisms that the family uses to hire analysts. By contrast, one can also think of situations in which alphas are uncorrelated across funds in the family, but short-term fluctuations are positively correlated. For example, a family may have a focus on certain geographic areas or industries, or securities with certain characteristics, but within these categories individual managers are responsible for selecting stocks independently. We model learning and investors optimal responses to fund performance and family performance. Investors observe the performance of funds in a family. Each fund manager s skill is an unknown latent variable. The quality of the common resources of the family (the family skill) is also unknown. A fund s alpha increases with its composite skill, and decreases with fund size. Fund returns are subject to correlated idiosyncratic shocks. Investors estimate funds composite skills, conditional on returns, and allocate wealth across the funds, generating flow into and out of funds. than across families. 2

4 This model of Bayesian learning is based on a well-established theory of continuous-time filtering, and it is an extension of the work of Dangl, Wu, and Zechner (2008). The enviroment mimics that of Berk and Green (2004). There is perfect capital mobility, decreasing returns to scale, and competitive capital provision. In this setting, mutual fund flow directly reflects innovations in investors beliefs about a fund s composite skills. We characterize the optimal updating of beliefs about funds composite skills. Not surprisingly, the estimate of a funds composite skill is positively related to its own unexpected risk-adjusted return. Good performance indicates either a skilled fund manager or high quality of family resources. The more interesting question concerns the effect of family performance (measured by the average performance of other funds in the family) on the estimated skill of a member fund. Our model highlights two competing effects: a positive common-skill effect and a negative common-noise effect. The positive effect arises because family performance partially reveals the quality of family resources, while the negative effect arises because family performance also partially reveals the common shocks to the returns of all funds in the family. In the steady state, in which the uncertainty about the composite skills is constant, the overall effect is either positive or negative, depending on two correlations. The estimate of a fund s composite skill increases with family performance when the changes of unobservable composite skills in the family are highly correlated, but the unobservable shocks are relatively independent of each other. Alternatively, the estimate decreases with family performance when unobservable shocks are highly correlated, but the instantaneous correlations of composite skills are low. By varying the number of funds in the family, we further find that this pattern is stronger in bigger families. While the sign of the cross-sensitivity in the steady state is fully determined by the relative magnitude of the two correlations mentioned above, this is not the case in the nonsteady state. When the fund family is young, in addition to the correlation structure of noise and the dynamics of the true skill processes, learning is also heavily influenced by the initial beliefs. The cross-sensitivity is positive, as long as the degree of reliance on family resources and the uncertainty about the quality of family resources are relatively high. Due to the availability of a larger number signals, uncertainty about composite skills declines faster in larger families, and investor beliefs are less sensitive to a fund s own performance and more 3

5 sensitive to family performance. Our model generates a number of predictions about the sensitivities of mutual fund flow to fund and family performance. We empirically test these predictions and find supportive evidence. For a median fund, we find a positive spillover effect. That is, a fund receives higher flow when other funds in the same family perform well. This suggests that the common-skill effect dominates the common-noise effect. More important, flow to a member fund is more sensitive to family performance in larger families, and in families with a larger fraction of team-managed funds. Furthermore, the sensitivity of flow to a fund s own performance declines with fund age, fund size, and family size, and increases with correlations of idiosyncratic fund returns. These patterns support the predictions of our model. Our work contributes to the literature in several ways. First, from a theoretical point of view, we study a dynamic model of multivariate learning. Our main results are relevant for performance evaluation in general, beyond the application to a family of mutual funds. The idea that peer performance can be used to filter out common shocks to multiple agents is well-recognized in the literature on relative performance evaluation (see, for example, Holmstrom (1982) and Gibbons and Murphy (1990)). This consideration generally leads to a negative relation between the optimal compensation of an agent and the performance of his peers. Our model allows for common components in both shocks and skills. As a result, peer performance can have a positive impact on beliefs about an agent s skills, provided that the common-skill effect dominates the common-noise effect. Second, we present new empirical evidence on the determinant of mutual fund flow. We show that mutual fund flow responds to fund performance and family performance in a manner that is largely consistent with optimal learning about funds skills. Third, from a practical point of view, our results suggests that an accurate evaluation of a fund s composite skill should incorporate the performance of all funds within a family. For a fund family whose managers rely little on family resources and whose funds returns exhibit a high correlation in temporary fluctuations, an estimate of a fund s skill should negatively weight the family performance. By contrast, for a family in which the common resources are the main driver of fund alphas, and there is little correlation in the temporary fluctuations in fund returns, the estimate should positively weight family performance. 4

6 There is a large body of literature on mutual fund performance evaluation. Aragon and Ferson (2006) provide an extensive review. Most methods of evaluation rely solely on a fund s own return or portfolio holding information. Several recent papers propose methods incorporating additional information. For example, Pastor and Stambaugh (2002) estimate the alpha of an actively managed fund using the returns on seemingly unrelated nonbenchmark passive assets. Cohen, Coval, and Pastor (2005) judge a fund manager s skill by the extent to which his or her investment decisions resemble those of managers with distinguished track records. Jones and Shanken (2005) measure performance using the distribution of other funds alphas in additon to the information in a fund s own return history. two respects. family. Our performance measure is in the spirit of this literature, but differs from it in First, we exploit the information embedded in the performance of a fund s Second, we derive our measure from a model of optimal learning. Our work is closely related to studies of mutual fund flow. Many authors find that mutual funds with good past performance attract more fund flow (see, for example, Sirri and Tufano (1998). More relevant to our paper, Nanda, Wang, and Zheng (2004) find that the stellar performance of one fund has a positive spillover onto the inflow to other funds in the same family, and Sialm and Tham (2011) find that the prior stock price performance of the management company affects the money flow of the affiliated funds. Berk and Green (2004) develop a learning model that can explain the positive response of fund flow to past performance, even though performance is not persistent. Dangl, Wu, and Zechner (2008) model simultaneously mutual fund flow and termination of fund managers in response to past performance. Other learning-based models for the flow-performance relation include those by Lynch and Musto (2003) and Huang, Wei, and Yan (2007). All these models are silent about spillovers within fund families. The current paper extends the continuous-time model of Dangl, Wu, and Zechner (2008) to account for multiple funds within a family. We derive the optimal response of mutual fund flow to fund and family performances in an economy with rational investors, and find strong empirical patterns of spillovers within families that are consistent with our model. 2 2 The recent literature on mutual funds has shown a growing interest in fund families. See for example, Mamaysky and Spiegel (2002), Massa (2003), Gervais, Lynch, and Musto (2005), Massa, Gaspar, and Matos (2006), Ruenzi and Kempf (2008), Pomorski (2009), Bhattacharya, Lee, and Pool (2010), Warner and Wu 5

7 The paper is organized as follows. Section 1 describes the structure of our model of a mutual fund family. Section 2 derives investors responses to fund performance in families, and the dynamics of fund size in equilibrium. In section 3 we examine the sensitivities of investor beliefs about a fund s composite skill to its own performance and the performance of other funds in the family in the steady state, under the assumption that composite skills follow a random walk. Section 4 analyzes how these sensitivities change over time. We consider both the case with constant skills and the case with mean-reverting skills. Section 5 presents the empirical evidence for key predictions of our model. Section 6 concludes. The proofs of all propositions are in the Appendix. 1 A Family of Mutual Funds We model n actively managed mutual funds within a family. The quality of management is an unobservable factor governing the success or failure of a fund. Quality varies through time, and is a linear combination of two components, which together form the composite skill θ of a fund. One part of θ is the skill of the fund manager. The second part is the quality of the common resources available to fund managers within the family. A fund s alpha and its expected return are increasing functions of θ, and a fund s realized return is a signal of this unknown quantity. We calculate a conditional distribution of θ for all funds in the family using fund returns as a continuous signal. Generalizing Dangl, Wu, and Zechner (2008), we assume that the funds rates of return, net of fees, are given by dg t G t = (r t 1 n + ησ m + α t f t ) dt + σ m dw mt + σ t BdW t, (1) where G t is the n 1 vector of net asset values per share with dividends reinvested; r t is the risk-free rate process; 1 n is an n 1 vector of ones; η is the market price of risk; and σ m is an n 1 vector of exposures to the market risk factor of the funds portfolios. 3 Together, these determine the expected rates of returns in the absence of portfolio management skills. (2011), and Khorana and Servaes (2011). 3 Our model can be easily generalized to allow for multiple systematic risk factors. 6

8 The n 1 vector α t captures the contribution of the composite skill; an element α it is the abnormal expected rate of return of fund i generated by active management of the fund. We refer to α it simply as the alpha of fund i. The n 1 vector f t is the instantaneous rate of management fees. In total, the drift in equation (1) is the vector of expected rates of return net of fees. Innovations in fund returns have two components. One is the systematic component σ m dw mt, where W mt is a scalar Brownian motion. The second is the idiosyncratic component σ t BdW t, where W t is a vector of standard Brownian motions that are pairwise independent, and are independent of W mt. The n n diagonal matrix σ t has elements σ it along the main diagonal, representing the volatility of idiosyncratic returns. Matrix BB is symmetric and nonsingular, with ones along the main diagonal and off-diagonal elements ρ ij, which are the correlations of idiosyncratic shocks. 4 A fund s idiosyncratic risk σ it is governed by the scale of the manager s portfolio tilt, which is the difference between the fund s weights in individual securities and the weights of a benchmark portfolio with only systematic risks and zero alpha. A fund with no tilt has σ it = 0. As the manager increases the scale of a tilt, with the expectation of increasing fund alpha, σ it increases. If managers of two funds i and j follow independent strategies and have orthogonal tilts, the idiosyncratic shocks are uncorrelated, and ρ ij = 0. For various reasons noted above, however, we expect fund managers within a family to follow positively correlated strategies. 5 Fund alphas follow the process α t = σ t θt γσ t σ t A t, (2) where ( def def θ t = bθ t + βθ F ; θ t = θ 1t... θ nt ) ; β def = ( β 1... β n ) ; (3) and b is a diagonal matrix with vector 1 n β along the diagonal. The vector σ t θt has 4 Matrix B is the Cholesky decomposition of the correlation matrix. 5 While we focus our discussion in the paper on the empirically more relevant case with ρ ij 0, our model is valid for all ρ ij ( 1, 1), i j. When ρ ij = 1 or 1, BB is singular. When ρ ij < 0, the common-noise effect that we discuss below is of the opposite sign. 7

9 elements σ it [(1 β i ) θ it + β i θ F t ], and it represents the effect of active management on the expected returns. common resources of the family; β i Here, θ it is the skill of the fund manager i; θ F t is the quality of the [0, 1] is the degree to which a manager uses the common resources; and θ it = (1 β i ) θ it + β i θ F t is the composite skill of fund i. For a manager with no individual skill, θ it = 0. A fund with a pool of excellent analysts has large θ F t. For a manager working independently of the analyst pool, β i = 0. We expect managers to rely on the pool for investment ideas, i.e., β i > 0. In this case, the fund s alpha increases directly with both θ it and θ F t. Fund assets A it are elements of vector A t. The parameter γ > 0 captures the decreasing returns to scale in active portfolio management. The i-th element of the final term in equation (2) is γσ 2 ita it. Thus, the alpha of fund i decreases with its own size, and at a higher rate when a fund is not well-diversified, i.e., when σ it is high. Funds with concentrated stock positions suffer most from the price impact of large portfolio transactions. Equation (3) also implies that the marginal return from taking idiosyncratic risk decreases, especially for large funds. This deters funds from taking unlimited idiosyncratic risk. Mutual funds operate in a rapidly changing business environment. Past success or experience is no guarantee of future performance. To capture this characteristic of the industry, the unobservable composite skills follow a stochastic process: d θ t = k (θ θ ) t dt + Ωdw t, (4) where the constant k governs the speed at which θ t reverts to the long-run mean θ, and w t is a vector of n + 1 pairwise independent standard Brownian motions, each independent of W mt and W t. Volatility coefficients are in the n (n + 1) matrix Ω = [bω. βω F ], (5) where ω is a diagonal matrix with coefficients ω i along the main diagonal. The instantaneous volatility of the skill of manager i is ω i 0, while that of the family resources is ω F 0. Thus, the stochastic component of an element d θ i in (4) is (1 β i )ω i dw it + β i ω F dw n+1,t. 8

10 Denote the instantaneous volatility of the composite skill of fund i by ω i, and we have ω i = (1 β i ) 2 ω 2 i + β2 i ω 2 F. (6) Equations (4)-(6) nest three important cases. First, when k = ω i = 0 for all i, funds have constant skills, and θ t = θ. Second, when k = 0 and ω i > 0 for all i, funds composite skills follow random walks. Finally, when k > 0 and ω i > 0 for all i, funds skills are mean-reverting. When ω i > 0 for all i, the instantaneous covariance matrix ΩΩ is positive definite, and the instantaneous correlation of the true composite skills for a pair of funds i and j is λ ij def = β iβ j ω 2 F ω i ω j. (7) This measures the variation in the quality of common resources of the family as a driver of composite skills, relative to the variation in managers skills. It is easy to see that λ ij increases with β i and β j, and decreases with the ratios of ω i /ω F and ω j /ω F. A value λ ij = 0 indicates either that the quality of the common resources is constant (ω F = 0), or that one or both of the managers works independently of those resources (β i = 0). A value λ ij = 1 indicates instead that the skills of the individual managers are fixed (ω i = ω j = 0), or that the managers act in concert and rely entirely on the common resources for alpha generation ( βi = β j = 1 ). 2 Family Performance and fund flow in Equilibrium Investors form beliefs about the conditional distribution of the unobservable composite skills of mutual funds, using the returns of all funds in a family as a continuous signal. They then allocate their money, pursuing those funds that offer a positive expected alpha. Section 2.1 shows how the conditional distribution is calculated, while Section 2.2 describes the dynamic equilibrium. 9

11 2.1 Investor Evaluation of Fund Performance Information is symmetric but incomplete. We assume all variables in equation (1) are observable, except the composite skills θ t and the idiosyncratic shocks dw t. Summarizing all the observable terms by dξ t, we can rewrite equation (1) as ( ) def dξ t = σ 1 dgt t (r t 1 n γσ t σ t A t + ησ m f t ) dt σ m dw mt G t = θ t dt + BdW t. (8) The equation demonstrates that the difference between the vector of fund returns and the observable components of the return, normalized by idiosyncratic volatilities, is a signal of composite skills, where BdW t is the noise correlated across funds. At any time t, information is the history of fund returns represented by the filtration F t def = σ {ξ s } t s=0. Given a multivariate normal prior distribution with mean vector m 0 and covariance matrix V 0, the conditional distribution of the composite skills is also multivariate normal. 6 Proposition 1. The conditional mean vector m t def = E( θ t F t ) and the conditional covariance matrix V t def = V ar( θ t F t ) for t 0 follow the processes: dm t = k ( θ m t ) dt + St dw F t, (9) dv t dt = ΩΩ 2kV t V t (BB ) 1 V t, (10) where S t def = V t (BB ) 1, (11) dw F t def = (dξ t m t dt). (12) 6 Investor beliefs are conditional distributions for θ t, which has the same dimension as the observation equation. A conditional distribution can be calculated numerically for individual components θ i and θ F. Learning these components separately is important for the hiring and firing decisions in fund families, and investor responses to these decisions, but it is not necessary for investors to form an expectation of fund alphas in our model. 10

12 When k = 0, equation (9) has the analytic solution: V t = BP t B + V, (13) where 0 V n n, if ω i = 0 for all i, = BDΠ 1/2 D B, if ω i > 0 for all i, (14) is the constant covariance matrix in the steady state, and where the matrices D, Π, and P t are as defined in Appendix A.1. Proof. See Appendix A.1. The conditional mean m t follows a multi-variate Orsten-Uhlenbeck process in equation (9), with long-run mean θ. The vector dwt F is the difference of dξ t and the conditional means m t dt, and it is a Brownian motion under filtration F t. This vector has zero mean, unit variance, and correlation matrix BB, and it represents normalized unexpected idiosyncratic fund returns or, simply, unexpected returns. The matrix S t in equation (9) characterizes responses of investor beliefs to mutual fund performance. Elements of S t are sensitivities of conditional means to unexpected returns. Defined in equation (11), they increase with uncertainty about composite skills, which is in matrix V t. Elements on and off the main diagonal of V t are conditional variances and covariances, respectively. Generally, if skills are estimated precisely, S t is small and beliefs are insensitive to unexpected returns. If instead little is known about skills, unexpected returns are important signals of skill. Therefore, investors respond strongly to past performance when they are least confident in their knowledge about either the skills of managers or the quality of common resources, or both. An element on the main diagonal of S t is the sensitivity of a fund s conditional mean to its own unexpected return. We expect that this coefficient is positive, because good performance increases the estimate of the composite skill of a fund s manager. An offdiagonal element is the sensitivity of the mean to the unexpected return of a second fund. This cross-coefficient can be either positive or negative for reasons that we discuss later. The analytic solution for V t in equation (13) obtains when k = 0, i.e., when the skills 11

13 are constant or follow a random walk. In the case of mean reversion, i.e, k > 0, numerical solutions to equation (10) are easily calculated. In theory, elements of V t may either decrease or increase with time, depending on the level of initial uncertainty V 0. In practice, however, we expect V t decreases early in the life of a family, because investors initially know little about the family s skills and V 0 is large. The steady-state covariance matrix, V, is the solution to equation (10) with dvt dt = 0 n n. In comparison to V t, V is relatively simple because it is time-independent. It is also independent of prior beliefs, and has the simple form given in equation (14) when k = 0. Furthermore, V as the limiting value is a good approximation to V t after the passage of enough time, i.e., in old families. For these reasons, we first study the steady-state covariance matrix V and sensitivity matrix S = V (BB ) 1 in Section 3, and then the time-dependent case in Section Equilibrium Fund Flow As in Berk and Green (2004), our investors provide capital to mutual funds competitively and without transaction costs. Active management may generate alpha, but the rents are captured by the mutual fund company due to the competition among investors. Investors direct assets toward funds with positive expected alpha, net of fees, and pull assets from funds with negative expected net alpha, and their evaluations are based on the information F t. In equilibrium, the size of fund i satisfies the condition E(α it F t ) =f it or, specifically: A it = 1 γ ( mit f ) it. (15) σ it σ 2 it A mutual fund family maximizes total fee income f A by optimizing the fee ratios and idiosyncratic volatilities. An optimal fee satisfies f it σ it = 1 2 m it. (16) The ratio on the left-hand side is determined for each fund i in equilibrium, but neither the fee nor the idiosyncratic risk is unique. A fund may set a high fee, attract a low level of 12

14 assets, and take large positions in mispriced assets. Or, it may set a low fee, attract a high level of investment, and stick closely to a benchmark portfolio. Provided that the fund s fee and idiosyncratic risk satisfy equation (16), the total fee income is the same in either case. For this reason, without loss of generality, we follow Dangl, Wu, and Zechner (2008) and set fees equal to the constant vector f = (f i ). Because σ it 0, equation (15) implies that a fund is viable, i.e., it has A it > 0 and earns a positive fee, only if the expected composite skill m t is positive. Otherwise, the fund is either reorganized or closed. Equations (15) and (16) determine the equilibrium size of a fund. For m it > 0, size is A it = m2 it 4γf i, and, using Ito s lemma and equation (9), the instantaneous growth rate of assets is da it A it = 2 dm it m it + (dm it) 2 m 2 it = 2k 1 ( ) 1 θi m it dt + S m it m 2 it BB S itdt S it dwt F. it m it (17) where S it is the i th row of S t. 7 By writing S it dwt F = j s ijdwjt F, we see that one fund s asset growth rate responses to the performance of all funds in the family. If s ij > 0, unexpectedly good performance by fund j increases the size of fund i, while if s ij < 0, the relation is negative. We now show how the coefficients s ij are determined. 3 Sensitivity of Investor Beliefs: The Steady State In this section we characterize learning of composite skills in the steady state. We focus on the case with k = 0, i.e., when composite skills follow a random walk. This case has the advantage of having nondegenerate analytical solutions, and is a good approximation for the case in which the mean reversion rate of skills is low. Also, we assume our mutual fund family to be homogeneous with β i = β [0, 1] and ω i = ω 0, for all funds. This implies that λ i = λ [0, 1] is constant across all funds. Similarly, we set θ i = θ, for all funds, and 7 The second term in each line is a positive drift in the size of the fund that is due to the convex relation between the assets and investors mean beliefs about the composite skill of the fund. 13

15 ρ ij = ρ [0, 1) for all 1 i, j n, i j. As a consequence, matrix V has homogeneous elements on the main diagonal, say, v n, and homogeneous elements off the diagonal, say, v n. Similarly, S is homogeneous, with elements s n and s n on and off the diagonal, respectively. The dynamics of the conditional mean in equation (9) are simple in the homogeneous family. This estimate of composite skills follows the process: where dw F it dm t = s n dw F it + s n (n 1) dx F it, (18) is the unexpected idiosyncratic return of fund i, and dx F it def = 1 n 1 j i dw jt F the average of unexpected idiosyncratic returns of the other funds in the family. We refer to dx F it as the family performance. Equation (18) has the obvious advantage over equation (9) in that the performance of the other funds is summarized in the single statistic dx F it. Our primary interest is the coefficients s n and s n (n 1), which are the sensitivities of investor beliefs to fund and family performances, respectively. is Section 3.1 describes the elements v n and v n of the covariance matrix, while Section 3.2 describes s n and s n (n 1). 3.1 The Conditional Variances of Composite Skills Proposition 2 below characterizes the conditional covariance matrix V of composite skills in the steady state. Proposition 2. When composite skills follow a random walk, the conditional covariance matrix V in the steady state is homogeneous for a homogeneous n-fund family. The conditional variance of each fund is v n = ω K 1 (K ρ K λ ) 2 (n 1) + K 2 1 ω, (19) and the conditional covariance of each pair of funds is v n = ω K 2 K ρ K λ, (20) (K ρ K λ ) 2 (n 1) + K1 2 and K 1, K 2, K ρ, andk λ are functions of ρ and λ given in Appendix A.2. When ρ = λ, v n = ω; 14

16 otherwise, v n < ω. Proof. See Appendix A.2 The nature of the conditional moments is particularly simple in families of two funds, and in the limit as the number of funds increases. Corollary 1. When composite skills follow a random walk, in the steady state of a homogeneous two-fund family, we have v 2 = 1 ( 2 ω ) 1 + ρ 1 + λ + 1 ρ 1 λ, (21) v 2 = 1 ( 2 ω ) 1 + ρ 1 + λ 1 ρ 1 λ, (22) and the limiting values of the variance and covariance as the family size grows are ( v def ) = lim v n = ω ρλ + 1 ρ 1 λ, (23) n v def = lim n v n = ω ρλ. (24) Proof. See Appendix A.3. Equations (21) and (23) show clearly that the conditional variance v n is highest when the correlation of noise ρ is equal to the instantaneous correlation of true skills λ for the two special cases of n = 2 and n. This is also true for the general case of n. Therefore, investors are most uncertain when ρ = λ. As we see in Section 3.2, learning about a fund s composite skill is entirely based on its own performance, and the uncertainty is highest because of the lack of other sources of information. Equations (21) and (23) further show that the uncertainty about skills declines as ρ and λ deviate from each other in the steady state. The precision of investors beliefs about composite skills is greatest when either (i) noise in fund returns are uncorrelated (ρ = 0), and funds fully rely on family resources (λ = 1); or (ii) noise in fund returns is almost perfectly correlated (rho 1), and fund alphas depend solely on the skills of individual managers (λ = 0). In the first case, noise in returns of funds in the family tends to be averaged out, allowing the quality of the family resources to be estimated with high precision. In the 15

17 second case, there is no common component in skills across funds, one fund s return can be used to reduce the noise of another fund. As the number of funds goes to infinity, v n goes to zero in the two cases above, i.e., skills are perfectly revealed. The intuition is as follows. In the first case, family resources are the only component of composite skills. Since the noise is uncorrelated, the law of large numbers guarantees a single unobservable variable to be fully revealed as the number of signals goes to infinity. 8 In the second case, manager skills are the only component of the composite skills. Since variations in manager skills are assumed to be independent, by the law of large numbers the average skill of all managers converges to the population mean, which is a known constant. As a result, the average return of funds in the family fully reveals the perfectly correlated shocks. The difference between a fund s return and the average return then perfectly reveals the skills of it s manager. Since conditional matrix V is homogeneous, the conditional correlation of composite skills of a pair of funds is simply the ratio φ n = v n /v n for any n 2. It is φ n = ρ when λ = ρ, and approaches 1 as either ρ or λ approaches 1. When λ 1.0, composite skills consist mainly of common family resources. Therefore investors estimates of those skills are highly correlated. Alternatively when ρ 1.0, the funds idiosyncratic returns are highly correlated, and the returns of one fund contain nearly the same error as the returns of the other funds. Again, estimates of composite skills are highly correlated. 3.2 The Sensitivity of Beliefs to Performance Proposition 3 below characterizes the sensitivity of investor beliefs to past performance in the steady state. Proposition 3. When composite skills follow a random walk, the matrix S in the steady state is homogeneous for a homogeneous n-fund family. For each fund, the sensitivity of 8 Note that v does not go to zero if λ < 1, because idiosyncratic shocks to each fund s returns prevents perfect learning about individual manager s skills. 16

18 beliefs to own performance is s n = v n (1 ( 1 K ) λ K ρ ρ 1 + λ + ρ 1 n 1 and the sensitivity to the average performance of other funds is ), (25) s n (n 1) = 1 ρ (v n s n ). (26) If ρ = λ, then s n = v n, and s n = 0. Proof. See Appendix A.4. Again, the nature of these results is particularly easy to understand in the two-fund family and in the limit as the family size increases. Corollary 2. When composite skills follow a random walk, in the steady state of a two-fund family, we have s 2 = 1 ( ) 1 + λ 1 λ 2 ω +, (27) 1 + ρ 1 ρ s 2 = 1 ( ) 1 + λ 1 λ 2 ω, (28) 1 + ρ 1 ρ and the limiting values of the sensitivity coefficients as the family size grows are s def = lim s def = lim n s n (n 1) = 1 λ s n = ω, (29) n 1 ρ 0 if ρ = λ = 0 ( λ ω ρ ) 1 λ 1 ρ, otherwise.. (30) Proof. See Appendix A.5. Consider first the simple cases of the corollary. Provided that composite skills are stochastic, i.e., ω > 0, s 2 and s, are positive, which means that unexpectedly good performance by a fund raises beliefs about its composite skill. Furthermore, s 2 and s decrease with λ and increase with ρ, so that investors put more weight on a fund s own performance when idiosyncratic returns are highly correlated within a family, and there is low correlation of 17

19 skills. The cross-coefficients, s 2 are s are either positive, negative, or zero, depending on the relative sizes of ρ and λ. Each coefficient decreases with ρ, and increases with λ. In the two-fund family with ρ = 0, s 2 = 1 2 ω ( 1 + λ 1 λ ), which measures the pure common-skill effect, and is positive as long as λ > 0. Similarly, when λ = 0, we have s 2 = 1 2 ω( 1 1+ρ 1 1 ρ ), which measures the pure common-noise effect, and is negative as long as ρ > 0. Between these two extreme cases, s 2 represents a mixture of both effects. It is positive when λ > ρ, suggesting that in the face of high correlation of skills and low correlation of noise, the optimal estimate of the composite skill of one fund puts a positive weight on the performance of the second fund. This means that the common-skill effect dominates the common-noise effect. Alternatively, when λ < ρ, the common-noise effect is dominant. Finally, when λ = ρ, the two effects offset each other, s 2 = 0, and the evaluation of a fund s skill is entirely based on its own performance. Equation (30) shows that the common-skill and common-noise effects work in the same way in large families. The common-skill effect dominates the common noise effect if and only if λ > ρ > 0. 9 Consider now the general case described by Proposition 3. for any given fund in the family, the ratio sn(n 1) s n For each family size n, and represents the sensitivity to the average performance of all other funds relative to the sensitivity to the performance of the fund in question. The ratio depends only on the correlation of true composite skills λ and the correlation of idiosyncratic shocks ρ, and is independent of other parameters. 10 increases with λ, and decreases with ρ, and is zero when λ = ρ. This ratio We plot the ratio as a function of ρ (Panel A) and λ (Panel B) for alternative values of n in Figure 1. The solid line represents the case of a two-fund family. The other lines, each with increasingly steeper slopes, represent fund families with n = 5, n = 20, and the limiting case described in Corollary 2, respectively. 9 Equation (30) also shows that s explodes as ρ 0, provided λ 0. This does not imply, however, that investors beliefs have unbounded volatility. Because noise in fund returns is uncorrelated in this case, and variations in managers skills are independent by assumption, they both are averaged out as the number of funds goes to infinity. This allows perfect learning of the quality of family resources, and implies that dxi,t F converges to zero in the limiting case, and that the limiting variance of s n (n 1) dxi,t F as n is finite, even though s is not. 10 The parameters β, ω, and ω F have no direct impact on the ratio s 2 /s 2, other than their impact through λ. 18

20 The figure provides important insights about family size. The positive impact of λ and the negative impact of ρ on the ratio sn(n 1) s n are progressively stronger as the size of the family grows, indicating that investors put more weight in bigger families on the unexpected family performance when they evaluate any single fund. For example, in a family with more than two funds, while the ratio of cross-sensitivity to direct sensitivity is never lower than -1, it can be substantially bigger than 1 when either λ is high or ρ is low, indicating that investors react to the family performance more strongly than to a fund s own performance. This is because a low ρ allows the average performance of a large family to reveal the quality of the family resources more accurately, while a high λ implies that family resources play an important role in the alpha generation. 4 Sensitivity of Investor Beliefs: The Non-Steady State We now analyze investor learning in the non-steady state. We assume again, as in Section 3, that the family is homogeneous. The Riccati equation (10) in this family is fully characterized by a pair of differential equations, one for the diagonal v nt and the other for the off-diagonal elements v nt. These are solved numerically. Also investors mean beliefs about skills follow equation (18), although now the coefficients, s nt and s nt (n 1), change over time. While correlation of noise ρ and instantaneous correlation of true skills λ determine the learning in the steady state, the initial covariance matrix V 0, together with ρ and the dynamics of the true skill processes, drives the learning in the non-steady state. At the time when funds are created, investors are uncertain about both funds skills and the quality of family resources. Equation (3) suggests that the initial variance v n0 and covariance v n0 can be calculated as simple functions of β and initial variances and covariance of fund and family skills. While modeling how investors form initial beliefs about fund and family skills is beyond the scope of this paper, it is reasonable to argue that the ratio of initial covariances v n0 to variance v n0, i.e., the correlation of composite skills under investors information set at time 0, is an increasing function of the importance of family resources β. Figure 2 plots the elements of V t and S t as functions of fund age during the first 5 years of life for three families with n = 2, 5, and 20 funds. In Panels (a) and (b) are the 19

21 variances v n,t and covariances v n,t, and in Panels (c) and (d) are the sensitivity coefficients s n,t and s n,t (n 1), respectively. The parameter values are as follows. We set β = 1/3, so that fund skill θ i is twice as important as family skill θ F. The initial variance of composite skills is v 0 = 0.12, and the initial correlation of composite skills is φ n,0 = v n,0 /v n,0 = 1/3. The mean-reversion rate is k = 0.05, and the volatilities of fund and family skills are equal, ω i = ω F = With β = 1/3, this gives λ = 0.2. We also set ρ = 0.2. Because λ = ρ, according to Propositions 2 and 3, in the limit as t, the cross-sensitivity s n,t 0,. Investors observe fund performance and develop more precise estimates of composite skills as time passes, so the variances and covariances in Panels (a) and (b) decline as the funds grow older. If skills were constant, the values of v n,t and v n,t would converge to zero. In our case, however, skills are stochastic. Therefore, investors remain uncertain about skills, and v n,t and v n,t remain positive throughout the lives of the funds. The sensitivities of beliefs to performance, s n,t and s n,t (n 1) in Panels (c) and (d), respectively, also decline with fund age. This is because the influence of new information on beliefs declines as prior estimates of skills become more precise, and because the conditional moments v n,t and v n,t decline with age. Given equation (17), we expect fund age to influence fund flow. That is, flow should become less sensitive to performance as funds grow older. Investors learn about skills more rapidly in large families than in small families. The conditional variances and covariances in Figure 2 decrease with age at a faster pace in large families. Family size has two effects: (i) each fund s performance is a signal about the quality of common resources in the family, so the composite skill is estimated more precisely as the number of signals increases, and (ii) each fund s performance is a signal about the noise in other funds returns, and noise is estimated more precisely as n increases. These are, of course, the common-skill and noise effects described above for the steady state. These cross-fund learning effects speed up the learning process in the large families, as long as one is not completely offset by the other. Under our parameterization, although there is no cross-fund learning in the steady state since λ = ρ, such learning is present early in the life of a fund family, because of the difference between φ n,0 and ρ. The sensitivity of beliefs to fund performance decreases with family size n in Panel (c), while the sensitivity to family performance increases with n in Panel (d). This first effect 20

22 is easy to understand. Due to the larger number of signals in larger families, uncertainty about composite skills decline faster, investor beliefs should therefore be less sensitive to fund performance. The second effect is sensible as well. The performance of a larger family is an average of a larger number of funds, so it is a more precise signal of skills than that of a small family. While the sensitivity of beliefs is generally low when the uncertainty is low, a signal can still generate a strong response its precision is relatively high. Given equation (17), we expect that this family-size effect should show up in fund flow. The sensitivity of flow to fund performance should decline and the sensitivity to family performance should increase with the size of the fund family. We investigate further the influence of the correlations ρ and φ n,0 on the sensitivity of investor beliefs to performance in Figure 3. Here, the family size is n = 5, and as in Figure 2, five years of results are shown. Each of the panels shows sensitivity coefficients; in the left- and right-hand side panels are s 5,t and 4s 5,t, respectively. Panels (a) and (b) show three cases that vary ρ, the correlation of noise in fund returns. We see that the sensitivities to fund and family performance increase and decrease, respectively, with ρ. This suggests that investor beliefs should be more responsive to own fund performance, and less responsive to family performance in families in which idiosyncratic returns are highly correlated. Finally, Panels (c) and (d) show cases that vary with the initial correlation φ n,0. Interestingly, these are similar to those in Panels (a) and (b), but in reverse order. As φ n,0 increases, s 5,t falls and 4s 5,t rises. Since φ n,0 increases with the importance of common resources in fund performance, we conclude that investor beliefs should respond strongly to own fund performance in families in which fund mangers act relatively independently, and they should respond strongly to family performance when managers rely on a common analyst pool and other family resources. 5 Empirical Analysis Our model generates a number of testable hypotheses. We now summarize and empirically test a few key predictions of the model. 21

23 5.1 Hypotheses Equation (17) describes the evolution of fund size in response to unexpected fund and family performance, which is governed by two sensitivity coefficients, s nt and s nt.this is closely related to fund flow extensively studied in the mutual fund literature, since a major determinant of a fund s asset growth rate is money flow into and out the fund. Furthermore, since the ex ante abnormal performance of a fund is zero in equilibrium, we can measure the unexpected performance directly by the ex post abnormal performance. Our analysis in Sections 3 and 4 thus leads to the following predictions about mutual fund flow: P1: The sensitivities of flow to both fund performance and family performance decrease with fund age and fund size; P2: The sensitivity of flow to fund performance decreases, while the sensitivity of flow to family performance increases, with the number of funds in the family; P3: The sensitivity of flow to fund performance decreases, while the sensitivity of flow to family performance increases, with the importance of common resources in generating fund performance; P4: The sensitivity of flow to fund performance increases, while the sensitivity flow to family performance decreases, with the correlation of noise in fund returns. 5.2 Data and Summary Statistics We use the CRSP survivor-bias-free mutual fund database for our empirical tests. sample covers the period from January 1999 through December The data are at the share class level. 12 Our We select all equity fund share classes that fall into the following major fund categories according to Lipper investment objective code: (1) Growth Funds (G); (2) Growth and Income Funds (GI); (3) Equity Income Funds (EI); (4) Small-Cap Fund (SG); 11 The Lipper fund classification information begins in the year The database uses different classification systems for years prior to Furthermore, for most funds, the management company code, a data item we use to identify the fund family, begins in To attract investors with different preferences, a mutual fund typically has multiple share classes, all tied to the same underlying portfolio but differing in fee structure. 22