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. Email 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.
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
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
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
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
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. Our performance measure is in the spirit of this literature, but differs from it in
two respects. First, we exploit the information embedded in the performance of a fund’s
family. 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
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 b
θ of a fund. One part of b
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 b
θ, and a fund’s realized return is a
signal of this unknown quantity. We calculate a conditional distribution of b
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
dGt
= (rt 1n + ησ m + αt − ft ) dt + σ m dWmt + σ t BdWt ,
Gt
(1)
where Gt is the n × 1 vector of net asset values per share with dividends reinvested; rt is the
risk-free rate process; 1n 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
The n × 1 vector αt captures the contribution of the composite skill; an element αit is the
abnormal expected rate of return of fund i generated by active management of the fund. We
refer to αit simply as the alpha of fund i. The n × 1 vector ft 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.
σ m dWmt , where Wmt is a scalar Brownian motion.
One is the systematic component
The second is the idiosyncratic com-
ponent σ t BdWt , where Wt is a vector of standard Brownian motions that are pairwise
independent, and are independent of Wmt . The n × n diagonal matrix σ t has elements σ it
along the main diagonal, representing the volatility of idiosyncratic returns. Matrix BB0
is symmetric and nonsingular, with ones along the main diagonal and off-diagonal elements
ρij , 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 ρij = 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
bt − γσ t σ t At ,
αt = σ t θ
(2)
where
def
bt def
θ
= bθ t + βθF ; θ t =
θ1t ... θnt
0
def
; β =
β 1 ... β n
and b is a diagonal matrix with vector 1n − β along the diagonal.
4
0
;
(3)
bt has
The vector σ t θ
Matrix B is the Cholesky decomposition of the correlation matrix.
While we focus our discussion in the paper on the empirically more relevant case with ρij ≥ 0, our model
is valid for all ρij ∈ (−1, 1), i 6= j. When ρij = −1 or 1, BB0 is singular. When ρij < 0, the common-noise
effect that we discuss below is of the opposite sign.
5
7
elements σ it [(1 − β i ) θit + β i θF t ] , and it represents the effect of active management on the
expected returns.
Here, θit is the skill of the fund manager i; θF t is the quality of the
common resources of the family; β i ∈ [0, 1] is the degree to which a manager uses the
common resources; and b
θit = (1 − β i ) θit + β i θF t is the composite skill of fund i. For a
manager with no individual skill, θit = 0. A fund with a pool of excellent analysts has large
θF 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 θit and θF t .
Fund assets Ait are elements of vector At . 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 2it Ait . 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:
b
b
dθ t = k θ − θ t dt + Ωdwt ,
(4)
where the constant k governs the speed at which bθ t reverts to the long-run mean θ, and wt
is a vector of n + 1 pairwise independent standard Brownian motions, each independent of
Wmt and Wt . 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 db
θi in (4) is (1 − β i )ω i dwit + β i ω F dwn+1,t .
8
Denote the instantaneous volatility of the composite skill of fund i by ωbi , and we have
q
ωbi = (1 − β i )2 ω 2i + β 2i ω 2F .
(6)
Equations (4)-(6) nest three important cases. First, when k = ω
b i = 0 for all i, funds
bt = θ. Second, when k = 0 and ωbi > 0 for all i, funds’ composite
have constant skills, and θ
skills follow random walks. Finally, when k > 0 and ωbi > 0 for all i, funds’ skills are
mean-reverting.
When ω
b i > 0 for all i, the instantaneous covariance matrix ΩΩ0 is positive definite, and
the instantaneous correlation of the true composite skills for a pair of funds i and j is
def
λij =
β i β j ω 2F
.
ω
biω
bj
(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 λij
increases with β i and β j , and decreases with the ratios of ω i /ω F and ω j /ω F . A value λij = 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 λij = 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 = βj = 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
2.1
Investor Evaluation of Fund Performance
Information is symmetric but incomplete.
We assume all variables in equation (1) are
bt and the idiosyncratic shocks dWt . Summarizing
observable, except the composite skills θ
all the observable terms by dξ t , we can rewrite equation (1) as
def
dξ t =
σ −1
t
dGt
− (rt 1n − γσ t σ t At + ησ m − ft ) dt − σ m dWmt
Gt
(8)
bt dt + BdWt .
=θ
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 BdWt is the noise correlated across funds.
At any time t, information is the history of fund returns represented by the filtration
def
Ft = σ {ξ s }ts=0 . Given a multivariate normal prior distribution with mean vector m0 and
covariance matrix V0 , the conditional distribution of the composite skills is also multivariate
normal.6
def
bt |F ) and the conditional covariance
Proposition 1. The conditional mean vector mt = E(θ
t
def
bt |F ) for t ≥ 0 follow the processes:
matrix Vt = V ar(θ
t
dmt = k θ − mt dt + St dWtF ,
(9)
dVt
−1
= ΩΩ0 − 2kVt − Vt (BB0 ) Vt ,
dt
(10)
where
def
−1
St = Vt (BB0 )
def
,
dWtF = (dξ t − mt dt) .
6
(11)
(12)
bt , which has the same dimension as the observation
Investor beliefs are conditional distributions for θ
equation. A conditional distribution can be calculated numerically for individual components θi and θF .
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
When k = 0, equation (9) has the analytic solution:
where
Vt = BPt B0 + V∗ ,
(13)

 0
if ω
b i = 0 f or all i,
n×n ,
∗
V =
 BDΠ1/2 D0 B0 , if ω
b i > 0 f or all i,
(14)
is the constant covariance matrix in the steady state, and where the matrices D, Π, and Pt
are as defined in Appendix A.1.
Proof. See Appendix A.1.
The conditional mean mt follows a multi-variate Orsten-Uhlenbeck process in equation
(9), with long-run mean θ.
The vector dWtF is the difference of dξ t and the conditional
means mt dt, and it is a Brownian motion under filtration Ft . This vector has zero mean, unit
variance, and correlation matrix BB0 , and it represents normalized unexpected idiosyncratic
fund returns or, simply, unexpected returns.
The matrix St in equation (9) characterizes responses of investor beliefs to mutual fund
performance. Elements of St are sensitivities of conditional means to unexpected returns.
Defined in equation (11), they increase with uncertainty about composite skills, which is in
matrix Vt . Elements on and off the main diagonal of Vt are conditional variances and covariances, respectively. Generally, if skills are estimated precisely, St 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 St 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 off-
diagonal 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 Vt in equation (13) obtains when k = 0, i.e., when the skills
11
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 Vt may either decrease
or increase with time, depending on the level of initial uncertainty V0 . In practice, however,
we expect Vt decreases early in the life of a family, because investors initially know little
about the family’s skills and V0 is large.
The steady-state covariance matrix, V∗ , is the solution to equation (10) with
dVt
dt
= 0n×n .
In comparison to Vt , 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 Vt after the passage
of enough time, i.e., in old families.
For these reasons, we first study the steady-state
−1
covariance matrix V∗ and sensitivity matrix S∗ = V∗ (BB0 )
in Section 3, and then the
time-dependent case in Section 4.
2.2
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
Ft . In equilibrium, the size of fund i satisfies the condition E(αit |F t ) =fit or, specifically:
1
Ait =
γ
mit
fit
− 2
σ it
σ it
.
(15)
A mutual fund family maximizes total fee income f 0 A by optimizing the fee ratios and
idiosyncratic volatilities. An optimal fee satisfies
fit
1
= mit .
σ it
2
(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
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 = (fi ) . Because σ it ≥ 0, equation (15) implies that
a fund is viable, i.e., it has Ait > 0 and earns a positive fee, only if the expected composite
skill mt is positive. Otherwise, the fund is either reorganized or closed.
Equations (15) and (16) determine the equilibrium size of a fund. For mit > 0, size is
m2it
Ait =
,
4γfi
and, using Ito’s lemma and equation (9), the instantaneous growth rate of assets is
dmit (dmit )2
dAit
= 2
+
Ait
mit
m2it
1
1
1
= 2k
θi − mit dt + 2 Sit BB0 S0it dt + 2
Sit dWtF .
mit
mit
mit
where Sit is the i–th row of St .7 By writing Sit dWtF =
P
j
(17)
sij dWjtF , we see that one fund’s
asset growth rate responses to the performance of all funds in the family.
If sij > 0,
unexpectedly good performance by fund j increases the size of fund i, while if sij < 0, the
relation is negative. We now show how the coefficients sij 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 λi = λ ∈ [0, 1] is constant across all funds. Similarly, we set θi = 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
ρij = ρ ∈ [0, 1) for all 1 ≤ i, j ≤ n, i 6= j. As a consequence, matrix V∗ has homogeneous
elements on the main diagonal, say, vn , and homogeneous elements off the diagonal, say, v n .
Similarly, S∗ is homogeneous, with elements sn and sn 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:
dmt = sn dWitF + sn (n − 1) dXitF ,
(18)
def
where dWitF is the unexpected idiosyncratic return of fund i, and dXitF =
1
n−1
P
j6=i
dWjtF is
the average of unexpected idiosyncratic returns of the other funds in the family. We refer
to dXitF 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 dXitF .
Our primary interest is the coefficients sn and sn (n − 1), which are the sensitivities of
investor beliefs to fund and family performances, respectively.
Section 3.1 describes the
elements vn and v n of the covariance matrix, while Section 3.2 describes sn and sn (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
K1
≤ ω
b,
vn = ω
b q
(Kρ − Kλ )2 (n − 1) + K12
(19)
and the conditional covariance of each pair of funds is
K2 − Kρ Kλ
vn = ω
b q
,
(Kρ − Kλ )2 (n − 1) + K12
(20)
and K1 , K2 , Kρ , andKλ are functions of ρ and λ given in Appendix A.2. When ρ = λ, vn = ω
b;
14
otherwise, vn < ω
b.
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
v2
v2
p
√
√
1 p
=
ω
b
1+ρ 1+λ+ 1−ρ 1−λ ,
2
p
p
√
√
1
ω
b
=
1+ρ 1+λ− 1−ρ 1−λ ,
2
(21)
(22)
and the limiting values of the variance and covariance as the family size grows are
def
v =
(23)
lim v n
(24)
n→∞
def
v =
p
p
√
ρλ + 1 − ρ 1 − λ ,
p
= ω
b ρλ.
lim vn = ω
b
n→∞
Proof. See Appendix A.3.
Equations (21) and (23) show clearly that the conditional variance vn 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
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, vn 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 φn = v n /vn for any n ≥ 2. It is φn = 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
beliefs to own performance is
sn = vn
Kλ
1− 1−
Kρ
1
n−1
ρ
+λ+ρ−1
!
,
(25)
and the sensitivity to the average performance of other funds is
sn (n − 1) =
1
(vn − sn ) .
ρ
(26)
If ρ = λ, then sn = vn , and sn = 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
s2
s2
√
√
1
1+λ
1−λ
=
ω
b √
+√
,
2
1+ρ
1−ρ
√
√
1
1+λ
1−λ
=
−√
,
ω
b √
2
1+ρ
1−ρ
(27)
(28)
and the limiting values of the sensitivity coefficients as the family size grows are
√
1−λ
s = lim sn = ω
b√
,
n→∞
1−ρ


0
if ρ = λ = 0
def
√
s = lim sn (n − 1) =
.
√
n→∞
 ω
b √λρ − √1−λ
,
otherwise.
1−ρ
def
(29)
(30)
Proof. See Appendix A.5.
Consider first the simple cases of the corollary. Provided that composite skills are stochastic, i.e., ω
b > 0, s2 and s, are positive, which means that unexpectedly good performance
by a fund raises beliefs about its composite skill. Furthermore, s2 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
skills.
The cross-coefficients, s2 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, s2 = 12 ω
b
1 + λ − 1 − λ , which measures the
pure common-skill effect, and is positive as long as λ > 0.
Similarly, when λ = 0, we
1
have s2 = 12 ω
−
b ( √1+ρ
√ 1 ),
1−ρ
as long as ρ > 0.
Between these two extreme cases, s2 represents a mixture of both
which measures the pure common-noise effect, and is negative
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, s2 = 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 each family size n, and
for any given fund in the family, the ratio
s̄n (n−1)
sn
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 This ratio
increases with λ, and decreases with ρ, and is zero when λ = ρ.
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 6= 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
F
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 sn (n − 1) dXi,t as n → ∞ is finite,
even though s̄ is not.
10
The parameters β, ω, and ω F have no direct impact on the ratio s2 /s2 , other than their impact through
λ.
18
The figure provides important insights about family size. The positive impact of λ and
the negative impact of ρ on the ratio
s̄n (n−1)
sn
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 vnt 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, snt and snt (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 V0 , 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 vn0 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 vn0 , 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 Vt and St 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
variances vn,t and covariances v n,t , and in Panels (c) and (d) are the sensitivity coefficients
sn,t and sn,t (n − 1), respectively. The parameter values are as follows. We set β = 1/3, so
that fund skill θi is twice as important as family skill θF . The initial variance of composite
skills is v0 = 0.12, and the initial correlation of composite skills is φn,0 = v n,0 /vn,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 = 0.15. 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 sn,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 vn,t and v n,t would converge to zero. In
our case, however, skills are stochastic. Therefore, investors remain uncertain about skills,
and vn,t and v n,t remain positive throughout the lives of the funds.
The sensitivities of beliefs to performance, sn,t and sn,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 vn,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 φn,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
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 φn,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 s5,t and 4s5,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 φn,0 . Interestingly, these are similar to those in Panels (a) and (b), but in reverse order. As φn,0 increases,
s5,t falls and 4s5,t rises. Since φn,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
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, snt and snt .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. Our
sample covers the period from January 1999 through December 2009.11 The data are at the
share class level.12 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 1999. The database uses different classification systems for years prior to 1999. Furthermore, for most funds, the management company code, a data
item we use to identify the fund family, begins in 1999.
12
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
(5) Mid-Cap Funds (MC). We use the MFLINKS database to link the share classes to the
fund it belongs to, and aggregate the data to the fund level to avoid double counting. A
fund’s total net asset value (TNA) is the sum of all its share classes. Its return and expense
ratio are asset-weighted average of each share class. The fund age is defined as the age of
its oldest share class. An important variable for our purpose is a fund’s family affiliation.
This is identified by a management company code (together with company name). We also
collect manage names of each fund. This variable contains one or more manager names, or
the phrase “team managed”. We exclude the time period before a fund’s total net asset value
reaches five million dollars, and funds with fewer than 36 monthly return observations. The
final sample consists of 2256 funds, affiliated with 638 fund families.13 Summary statistics
of various variables of interest to our study are in Table 1.
We calculate the monthly fund flow for each fund by subtracting its monthly return
from its monthly asset (TNA) growth rate. To mitigate the effects of extreme observations
or potential data errors, we winsorize the monthly flow at the 1st and 99th percentiles,
although our results are almost identical with or without the winsorization.
According to equation (8), investor learning in our model is based on a fund’s alpha
normalized by the standard deviation of its idiosyncratic returns. This corresponds to the
Treynor and Black (1973) information ratio widely used in portfolio performance evaluation.
It accounts for the noisiness of a fund’s excess return relative to its benchmark. Following
this model structure, we use information ratio as our measure of fund performance, and
estimate it using a rolling window of 36 months. For each fund with at least 30 return
observations in a given rolling period, we first estimate its alpha and residual returns using
the standard Fama-French three-factor (market, size and book-to-market) model and the
Carhart (1997) four-factor (Fama-French factors plus the momentum factor) model. We
then divide the alpha by the standard deviation of the residual terms over the same period
to get the information ratio.14
Our estimates of fund performance using the three-factor model and four-factor model
13
703 funds in our sample change family affiliation at least once. For example, nine Merrill Lynch funds
switch to their family affiliation to Blackrock in the first quarter of 2007.
14
All the factor returns, as well as the Treasury rates, are obtained from the professor Ken French website
at Dartmouth College.
23
are very similar. Funds in our sample on average earn a negative alpha of around ten basis
points per month, net of expenses. The average monthly volatility of residual returns is
around 1.4%, which is around 5% per annum. The average monthly information ratio is
around -0.10, which corresponds to an annualized information ratio of -0.35.
Family performance is calculated as the average information ratio of all funds in the
same family except the fund under consideration. For each fund family, we also compute
the average pairwise correlation of idiosyncratic returns between member funds. These
correlations are also estimated with a rolling window of 36 months. We require a fund to
be affiliated with a family in an entire estimation window for it to enter the calculation of
family performance and correlations.
Figure 4 shows the average correlation of residuals within families. For comparison, we
identify in our sample 239 standalone equity funds, i.e., those in a single-fund family, and
plot the average pairwise correlation between them. The difference between the within-family
and across-family correlations is quite substantial. While the average correlation between
standalone funds is below 5% for most of the sample period, the average within-family
correlation is almost always above 15%. This significant gap of more than 10% throughout
the sample period is evidence of a strong family effect in fund returns. Its magnitude is more
striking than what Elton, Gruber, and Green (2007) find in their study. This highlights the
importance of accounting for the common noise effect when evaluating funds using family
performance.
Our model predicts that the degree of reliance on common resources is an important
determinant of sensitivity of fund flow to family performance. Unfortunately this variable is
not easy to measure directly. We use the fraction of funds in a family that are designated
as “team-managed” as a proxy for the importance of common resources shared by member
funds. Fund management is inevitably a teamwork process, however, a fund that is explicitly
designated as team-managed is likely to have a larger management team, draw inputs from
more sources, and rely less on a single individual. Furthermore, such anonymous teams are
more likely to be engaged in the management of multiple funds. Compared to a family
in which each fund is associated with one or two manager names, a family in which most
funds are team-managed is more likely to embrace a culture of make decision collectively.
24
Therefore, funds in such families are likely to share more common resources. About 25% of
the observations in our sample are from funds managed by an anonymous team.
The last row of Table 1 reports the average size of a fund’s affiliating family. It shows
that on average the funds in our sample operate in a family with 11.4 funds (within the
categories we select for this study).
5.3
Empirical results
We now test the predictions P1-P4 of our theory. We use the Fama-MacBeth procedure
to explore the cross-sectional variations in the flow sensitivities. We regress monthly flow
on our measures of fund and family performance, Perf and FamPerf, respectively, and their
interactions with various fund and family characteristics, including fund size (Log(T N A)),
fund age (Log(AGE)), the number of funds in the family (Log(N )), the fraction of teammanaged funds in the family (T eam), and the correlation of residual returns in a family
(Rho). It is the signs of the interaction terms that allow us to test the predictions. To
control for the direct impact of those characteristics on fund flow, we include them directly
in the regressions. We also include the square of fund performance as a regressor to account
for the convexity of flow-performance relation. Since fund age and fund assets are highly
correlated, we investigate their relations with fund flow in separate models. To simplify
the interpretation the regression results, all the fund and family characteristics (other than
performance) are adjusted by their median values of sample.
There are altogether 96 monthly cross-sectional regressions for each model. Table 2
reports the time series averages of the coefficients, and the t-statistics with Newey-Westcorrection for autocorrelation with three lags. The first two columns report the results
when performance and correlations are estimated from the three-factor model, while last
two columns correspond to the four-factor model. The two sets of results are very similar,
suggesting the robustness of our results with respect to the asset pricing models used for
performance evaluation.
The first two rows in the table confirm the well-documented convex relation between flow
and fund performance. The coefficients of both fund performance and its square are positive
and highly significant. The coefficient of family performance, F amP erf , is also positive,
25
and significant at the 1% level. This suggests that for a median fund fund, good family
performance is a positive signal about the its composite skill. This is consistent with the
spillover effect found in Nanda, Wang, and Zheng (2004). It suggests that for the median
fund, the common-skill effect dominates the common-noise effect.
Both fund size and fund age are negatively related to fund flow. Furthermore, their
interactions with fund performance have negative coefficients that are highly significant,
suggesting that the flow is less sensitive to fund performance for larger and older funds. This
is what our model predicts (P1). However, fund size and age do not seem to affect the
sensitivity to family performance significantly.
Prediction P2 hypothesizes that as the number of funds in a family increases, fund flow
becomes less sensitive to fund performance and more sensitive to family performance. This
is well-supported by the data. The coefficients of Log(N )∗Perf and Log(N )∗FamPerf are
of the predicted signs in each of the four columns, and all are statistically significant.
Prediction P3 states that the fund flow is more sensitive to family performance when
funds share more common resources. Our proxy for the importance of common resources,
i.e., the fraction of team-managed funds in a family (T eam), indeed shows a strong positive
impact on this sensitivity. The interaction term T eam∗F amP erf are positive and significant
at the 1% level across all columns. The economic magnitude of the impact is also very
strong. Take the coefficients in Column 1 as an example, while the sensitivity to family
performance is 0.879 for a median family with about 25% of funds team-managed, this
sensitivity is more than doubled for a family in which all member funds are team-manged
(1.976=0.879+0.75*1.462). On the other hand, team-management does not seem to have a
significant impact on the sensitivity of flow to fund performance.
Prediction P4 states that correlation of residual returns within a family increases the
sensitivity of flow to fund performance but decreases the sensitivity to family performance.
This again has received some support from the data. Higher correlation of residual returns,
Rho, does imply a higher flow sensitivity to fund performance, as shown by the positive
coefficient of the interaction term Rho ∗ P erf . This effect is significant at the 1% level
across all columns. The effect of Rho on the sensitivity to family performance is negative, as
predicted by our model, but it is not statistically significant. The reason for the weakness of
26
the second effect may be that Rho not only represents the correlation of noise in fund returns,
but also reflects to some degree the importance of family resources for fund performance.
This confounds the relation between Rho and the flow sensitivity.
Overall, the empirical evidence suggests that investors do look beyond the performance of
an individual fund when they allocate money across funds. All the four predictions coming
out of model receive some empirical support, although only prediction P2 is supported both
by the sensitivity to fund performance and the sensitivity to family performance. This
shows both the usefulness and the limitation of our stylized learning model in explaining the
patterns of mutual fund flow.
We conduct various tests to check the robustness our results. For example, we try different
combinations of the regressors, we test our model predictions using fund asset growth rate
instead of fund flow, we measure the importance of common resource using the fund-level
team-management dummies instead of the fraction of the team-managed funds at the family
level, and measure performance using multi-factor alpha instead of information ratio, our
results remain largely unchanged in all those tests.
6
Conclusion
The risk-adjusted performance of a mutual fund depends not only on the expertise of the
acting fund manager, but also on the resources of the family to which it belongs. Reliance
on family resources produces a common component of the unobservable alpha-generating
skill across funds in the same family. It also introduces correlations in the noise in returns
of funds in the family. Both of these suggest that it can be helpful to take into account the
performance of all funds within a family when we evaluate the composite skill of a single
fund. We build on this idea, and develop a model that characterizes the optimal evaluation
of a fund’s skill based on its own performance and its family performance.
Our theoretical model highlights two potential impacts of one fund’s performance on the
assessed skill of another fund in the family. When one fund performs well, it indicates the
quality of the common resource is high.
This is good news about the composite skill of
another fund. We call this positive effect the “common-skill effect.” When a fund is doing
27
well, this also suggests that it may have had some good luck. Due to the correlation of the
noise in returns, we may also attribute a greater portion of another fund’s performance to
good luck. This is bad news about the composite skill of another fund. We call this negative
effect the “common-noise effect.” The overall effect depends then on the relative strength
of these two opposite effects. Our theoretical analysis pins down a few key variables that
determine the sensitivities of beliefs about fund skills to both fund and family performance,
including the reliance of fund performance on common resources, correlation of noise in fund
returns, fund age, and the parameters governing the dynamics of the true skills.
Our empirical analysis shows patterns in the mutual fund flow-performance relation that
are consistent with our model predictions. Family performance on average has a positive
effect on fund flow to a member fund. Its has a stronger impact 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.
We have focused on a Berk and Green (2004) type of equilibrium without any frictions. As
a result, there is no predictability in mutual fund returns. Family performance information
incorporated in the updating of beliefs is immediately reflected in fund flow. In practice, it is
very likely that transaction costs in reallocating money across funds may lead to temporary
deviations from such an equilibrium. In this case, family performance information is not
only useful in predicting fund flow, but it also has some predictive power for fund returns.
This is a fruitful avenue for future research.
A
Appendix
A.1
Proof of Proposition 1
Equations (9)-(12) follow from Theorem 12.7 of Lipster and Shiryaev (2001), using (4) and
(8) as the state and observation equations, respectively.
For the analytic solution to the Ricatti equation (10) given in equation (13) for k = 0, we
consider two cases: (i) the unknown composite skills are constant, so that ΩΩ0 = 0n×n , and
28
(ii) the unknown composite skills follow a random walk, so that ΩΩ0 is a positive definite
matrix. In either case, in the steady state, the covariance matrix of skills V∗ is defined by
the equation:
−1
∗
0n×n = ΩΩ0 − V (BB0 )
V∗ .
(A.1)
The solution to this equation can be written as
V∗ =


0n×n ,
case (i),
 BDΠ1/2 D0 B0 , case (ii),
(A.2)
where D and Π are, respectively, the n × n matrix of orthonormal eigenvectors and the
diagonal matrix of eigenvalues of the symmetric, positive-definite matrix
0−1
DΠD0 = B−1 ΩΩ0 B
.
(A.3)
We write a general solution to equation (10) as
Vt = BPt B0 +V∗ ,
where Pt is to be found.
(A.4)
Substitution of (A.4) into equation (10) gives a homogeneous
equation:


−Pt Pt ,
case (i),
dPt
=
 −P DΠ1/2 D0 −DΠ1/2 D0 P −P P , case (ii).
dt
t
t
t t
(A.5)
For case (i), it is easy to show that15
Pt = Q−1
t ,
(A.6)
where the matrix Qt satisfies dQt = In×n , and has the elements

 Q (0) + t, i = j,
ij
Qij (t) =
 Q (0) ,
i=
6 j.
ij
15
Note that for any invertible Qt ,
dQ−1
t
dt
dQt −1
= −Q−1
t
dt Qt .
29
(A.7)
For case (ii), we define the matrix Qt by
0
Pt = DQ−1
t D.
(A.8)
Thus, equation (A.5) implies that Qt solves the linear equation
dQt
= Π1/2 Qt + Qt Π1/2 + In×n .
dt
(A.9)
A solution to this equation has the individual elements
 √
 e2 πi t Q (0) + √1 e2√πi t − 1 , i = j,
ij
2 πi
Qij (t) =
√
√
 e( πi + πj )t Q (0) ,
i=
6 j.
ij
(A.10)
In each case, the initial values Qij (0) are calculated as the solution to equation (A.4)
for a given V0 . Diagonal elements Qii (t) −→ ∞, so Pij (t) −→ 0 for all i and j. Thus,
t→∞
t→∞
Vt −→ V∗ .
t→∞
Given a solution Qt , equations (14), (A.4), (A.6), and (A.8) together specify the solution
for Vt in equation (10) in the text.
A.2
Proposition 2
−1
To prove Proposition 2, we first derive the matrix (BB0 )
.
Lemma 1. For all integers n ≥ 2, the inverse of the correlation matrix of idiosyncratic
−1
returns in the family of n funds, (BB0 ) , is homogeneous with elements bn on the main
diagonal and off-diagonal elements bn , where
bn =
1 + (n − 2) ρ
,
(1 − ρ) (1 + (n − 1) ρ)
(A.11)
and
bn = −
ρ
.
(1 − ρ) (1 + (n − 1) ρ)
(A.12)
Proof of Lemma 1. In the homogeneous n−fund family, each pair of funds has correlation of
30
idisyncratic returns equal to ρ. The correlation matrix is written
BB0 = (1 − ρ) I+ρ110 .
where I is the n × n identity matrix and 1 is a n × 1 vector with elements 1. The proof is
complete when we show that BB0 P = I, where
P = bn − bn I+bn 110 ,
with bn and bn given by equations (A.11) and (A.12), respectively. We calculate
BB0 P = [(1 − ρ) I+ρ110 ] bn − bn I+bn 110
= (1 − ρ) bn − bn I+ρ bn − bn 110 + (1 − ρ) bn 110 + ρ110 bn 110
= I+ ρ bn − bn + (1 − ρ) bn + nρbn 110 .
A bit of algebra shows that the second term of the final line is zero.
Q.E.D.
Proof of Proposition 2. The matrix equation (A.1) in a homogeneous family is described by
two scalar equations. Each element of the main diagonal is the equation
ω
b 2 = vn2 bn + 2vn v n bn (n − 1) + v 2n (n − 1) bn + bn (n − 2) ,
(A.13)
where bn and bn are given by Lemma 1. Similarly, each off-diagonal element is the equation:
ω
b 2 λ = vn2 bn + 2vn v n bn + bn (n − 2) + v 2n bn (n − 2) + bn n − 1 + (n − 2)2 .
(A.14)
The equations identify vn and v 2n . To solve them, first substitute for bn and bn , using
equations (A.11) and (A.12), respectively. Then substitute for v n , using v n = φn vn , where
φn is the correlation of two funds composite skills.
Set equal the ratios of the left- and
right-hand sides. After some algebra a quadratic equation emerges:
0 = K1 φ2n − 2K2 φn + K3 ,
31
(A.15)
with the coefficients
K1 = 1 + (n − 1) (λ + ρ − 1) ,
(A.16)
K2 = (n − 1) ρλ + 1 = (n − 1) (1 − ρ) (1 − λ) + K1 > 0,
(A.17)
K3 = ρ + λ + λρ (n − 2) > 0.
(A.18)
Equation (A.15) identifies φn . The discriminant, K22 − K1 K3 , is positive and the equation
has two real roots. One root has the following equivalent solutions:
φn = ρ + Kρ
Kρ − Kλ
Kλ − Kρ
= λ + Kλ
,
K1
K1
(A.19)
where
Ki =
p
(1 − i) ((n − 1) i + 1), i = ρ, λ.
(A.20)
Numerical evaluation of the second root gives values |φn | > 1, which is inconsistent with the
definition of a correlation.
The equivalence of the solutions in (A.19), together with the
definitions of K1 , Kρ , and Kλ , demonstrates that φn is a symmetric function of ρ and λ.
Substitute again for bn and bn , using equations (A.11) and (A.12), and for v n using
v n = φn vn in equation (A.13). After algebra, the equation is then written in either of two
forms:
vn = ω
bq
Ki2
, i = ρ, λ.
(A.21)
(φn − i)2 (n − 1) + Ki2
Together, equations (A.19) and (A.21) give the solution for vn in equation (19). It is then
evident that vn is a symmetric function of ρ and λ. The equation (20) follows from these
results using v n = φn vn . Because vn and φn are symmetric, so too is v n . Setting ρ = λ in
equation (A.19) gives φn = ρ = λ. Equation (A.21) gives vn = ω
b when ρ = λ, and vn ≤ ω
b
ω when ρ = λ.
generally. Finally, v n = ρb
32
A.3
Proof of Corollary 1
First set n = 2, and substitute for bn and bn , using (A.11) and (A.12). Equations (A.13)
and (A.14) are then
ω
b 2 (1 − ρ) (1 + ρ) = vn2 − 2v2 v 2 ρ + v 22
(A.22)
ω
b 2 λ (1 − ρ) (1 + ρ) = −ρvn2 + 2vn v n − ρv 2n .
(A.23)
and
Substitute for v 2 , using v 2 = φ2 v2 , take the ratios of these equation, and then solve for φ2 .
With some algebra we get
p
p
(1 + ρ) (1 + λ) − (1 − ρ) (1 − λ)
p
.
φ2 = p
(1 + ρ) (1 + λ) + (1 − ρ) (1 − λ)
(A.24)
Substitute again for v 2 , using v 2 = φ2 v2 , in equation (A.22), and solve the equation for vn ,
using (A.24). The result is (21). Equation (22) follows from v 2 = φ2 v2 .
For the limiting results in (23) and (24), we consider two cases: ρ > 0 and ρ = 0. When
ρ > 0, the equations (A.22) and (A.23) in the limit as n → ∞ are
ω
b 2 (1 − ρ) ρ = v 2 ρ − 2ρvv + v 2 ,
ω
b 2 λρ = v 2 .
The solutions to these equations are in equations (23) and (24).
When ρ = 0, equations (A.22) and (A.23) are equivalent to
ω
b 2 = vn2 + v 2n (n − 1) ,
ω
b 2 λ = 2vn v n + (n − 2) v 2n .
When λ = 0, vn = ω
b and v n = 0 for all n ≥ 2, and therefore v = ω
b and v = 0. When
λ > 0 , a limiting solution to (A.22) and (A.23) as n → ∞ is v n → 0, nv 2n → ω
b 2 λ, and v 2 =
ω
b 2 (1 − λ) . These are the solutions given in equations (23) and (24).
33
Q.E.D
A.4
Proof of Proposition 3
The matrix S∗ in the homogeneous family has on- and off-diagonal elements, sn and sn ,
−1
respectively. Using S∗ = V∗ (BB0 )
−1
and the solution for (BB0 )
in Lemma 1, these are
(n − 1) ρ
1
sn = vn + (vn − v n )
,
(1 − ρ) 1 + (n − 1) ρ
sn =
(A.25)
1
v n − ρvn
.
1 − ρ 1 + (n − 1) ρ
(A.26)
Substitute for v n in these equations using v n = φn vn , and use equations (A.19), and (A.20)
to obtain:
Kλ (n − 1) ρ
,
sn = vn 1 − 1 −
Kρ
K1
1
Kλ
sn = vn
1−
.
K1
Kρ
Finally, substitute for K1 using equation (A.16) to obtain equations (25) and (26), respectively. Q.E.D.
A.5
Proof of Corollary 2
First set n = 2. Substitute in equation (A.25) for v2 and v 2 , using equations (21) and (22),
respectively. After some algebra, we obtain equation (27). With these same substitutions,
equation (A.26) becomes equation (28).
Now consider the limiting case as n → ∞. The limit of equation (A.25) is
lim sn = (v − v)
n→∞
1
,
1−ρ
where v and v are the limiting values of vn and v n , respectively. We substitute for v and v,
using equations (23) and (24), respectively, to obtain equation (29).
To derive equation (30), we consider two cases, ρ > 0 and ρ = 0.
equation (A.26) gives
lim sn (n − 1) =
n→∞
34
v − ρv 1
.
1−ρ ρ
When ρ > 0, the
Substitute for v and v, using equations (23) and (24), respectively. The result is the second
line in equation (29).
When ρ = 0, the equation (A.26) gives
sn (n − 1) = v n (n − 1) .
(A.27)
When λ = 0, the proof of Proposition 4 gives v n = 0 for all n ≥ 2. Therefore,
lim sn (n − 1) = 0.
n→∞
b 2 λ as n → ∞. When λ > 0 the
The proof of Propostion 4 also gives v n → 0 and nv 2n → ω
limit of equation (A.27) is
b 2 λ/v n = ∞.
lim sn (n − 1) = lim ω
n→∞
These are the results in equation (30).
n→∞
Q.E.D.
35
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38
2
(a) Sensitivity ratio as a function of r
n=5
n=¥
-1
0
1
n=2
n=20
0
.2
.4
.6
Correlation of nonsystematic shocks (r)
.8
1
(a) Sensitivity ratio as a function of ρ
4
(b) Sensitivity ratio as a function of l
n=5
n=¥
-1
0
1
2
3
n=2
n=20
0
.2
.4
.6
Correlation of skills (l)
.8
1
(b) Sensitivity ratio as a function of λ
Figure 1: Ratio of cross-sensitivity to direct sensitivity. Plotted here is the ratio
s̄n (n−1)
, where sn and s̄n (n − 1) are the sensitivities of the conditional mean belief to a funds
sn
own unexpected return and the average unexpected return of other funds in the same family,
respectively. n is the number of funds in the family. The ratio is plotted as a function of ρ,
which is the correlation of non-systematic shocks in Panel (a), and λ, which is the correlation
of increments in composite skills in Panel (b). The solid line represents a family with two
funds, the dashed line represents a family of five funds, the dot-dashed line represents a
family of 20 funds, while the dotted line represents the limiting case when the number of
funds in a family goes to infinity. λ = 0.2 in Panel (a), ρ = 0.2 is Panel (b).
39
(a) Variance of composite skill
(b) Covariance of composite skills
0.12
0.05
n=2
5
20
0.115
n=2
5
20
0.045
0.11
0.04
0.105
0.1
0.035
0.095
0.03
0.09
0.025
0.085
0.08
0
1
2
3
Fund age (years)
4
0.02
0
5
(c) Sensitivity to fund performance
1
2
3
Fund age (years)
4
5
(d) Sensitivity to family performance
0.115
0.14
n=2
5
20
0.11
n=2
5
20
0.12
0.105
0.1
0.1
0.08
0.095
0.06
0.09
0.04
0.085
0.02
0.08
0.075
0
1
2
3
Fund age (years)
4
0
0
5
1
2
3
Fund age (years)
4
5
Figure 2: Uncertainty and sensitivities in the non-steady state: role of family size
Panels (a) and (b) show conditional variances and covariances, vn,t and v n,t , respectively, and
Panels (c) and (d) show sensitivities of the conditional mean to fund and family performance,
sn,t and sn,t (n−1), respectively. Each panel shows families with n = 2, 5 and 20 funds. Each
family is homogeneous and is described by the following parameters: Correlation of noise
in returns, ρ = 0.2; initial variance and covariance, vn,0 = 0.12 and v n,0 = 0.12φ5,0 = 0.04;
mean reversion rate of skills, k = 0.05; volatilities of fund and family skills, ω = ω F = 0.15;
and reliance on family resources, β = 1/3.
40
(a) Sensitivity to fund performance
(b) Sensitivity to family performance
0.13
0.04
ρ=0.1
0.2
0.3
0.125
0.12
ρ=0.1
0.2
0.3
0.03
0.02
0.115
0.01
0.11
0.105
0
0.1
−0.01
0.095
−0.02
0.09
−0.03
0.085
0.08
0
1
2
3
Family age (years)
4
−0.04
0
5
(c) Sensitivity to fund performance
1
2
3
Family age (years)
4
5
(d) Sensitivity to family performance
0.13
0.04
φ
φ
=0.1
n,0
0.125
0.12
=0.1
n,0
0.03
0.2
0.3
0.2
0.3
0.02
0.115
0.01
0.11
0.105
0
0.1
−0.01
0.095
−0.02
0.09
−0.03
0.085
0.08
0
1
2
3
Family age (years)
4
−0.04
0
5
1
2
3
Family age (years)
4
5
Figure 3: Sensitivities of beliefs in the non-steady state: role of ρ and φn,0
The left and right panels show sensitivities of the conditional mean to fund and family
peformance, s5,t and 4s5,t , respectively. Each family is homogeneous and has 5 funds. In
Panels (a) and (b) the correlations of noise in returns, ρ, varies and the initial correlation
of composite skills is φ5,0 = 0.20. In Panels (c) and (d), φ5,0 varies and ρ = 0.2. The
other parameters are: initial variance and covariance, v5,0 = 0.12 and v 5,0 = 0.12φ5,0 ; mean
reversion rate of skills, k = 0.05; volatilities of fund and family skills, ω = ω F = 0.15; and
reliance on family resources, β = 1/3.
41
Average correlation of residual returns
Rho_Standalone
0
.05
.1
.15
.2
.25
Rho_Within_Family
2001m7
2003m7
2005m7
Month
2007m7
2009m7
Figure 4: Correlation of residual returns: within families vs. across families
Rho W ithin F amily represents the average correlation of residual returns within families. It is calculated by first taking the average of pairwise correlation within a family,
and then averaging across families, using the number of funds as the weight for each family. Rho Standalone is average pairwise correlation of residual returns between standalone
funds. The residual returns are calculated using the Carhart four-factor model. Both residual
returns and correlations are estimated with a rolling window of 36 months.
42
Table 1: Summary statistics
Our equity mutual fund sample consists of 2256 funds belonging to 638 fund families, starting
from January 1999 to December 2009. 703 funds changed family affiliations at least once.
Total net asset (TNA), return, and expense ratio are aggregated across all share classes
of the same fund. Fund age is measured by the age of the oldest share class. Fund flow
is calculated as asset growth rate minus fund return. Alpha, volatility of residuals, and
information ratio (=alpha divided by volatility of residuals) are estimated by the FamaFrench three-factor model and the Carhart four factor model using a rolling window of 36
months. Correlation of residuals is the average pairwise correlation of factor model residuals
within a fund’s affiliating family, estimated also using a rolling window of 36 months. Number
of funds within family is the number of funds that a fund’s affiliating family is simultaneously
managing. Team-managed dummy is an indicator equal to one if a fund is designated as
“team-managed” and zero otherwise.
Variable
Mean
Std. Dev.
N
1275.647
5155.121
225778
Fund age (year)
12.499
13.204
226546
Monthly return (%)
0.351
5.627
223756
Monthly fund flow (%)
0.354
4.813
222009
Annual expense ratio (%)
1.248
0.456
221929
Alpha (3-factor, %)
-0.097
0.389
131308
Alpha (4-factor, %)
-0.098
0.374
131308
Volatility of residuals (3-factor, %)
1.474
0.91
131308
Volatility of residuals (4-factor, %)
1.389
0.837
131308
Information Ratio (3-factor)
-0.094
0.24
131308
Information Ratio (4-factor)
-0.097
0.247
131308
Correlation of residuals (3-factor)
0.179
0.192
140333
Correlation of residuals (4-factor)
0.177
0.182
140333
Team-managed dummy
0.25
0.433
226546
11.402
11.273
226546
Monthly TNA (million dollar)
Number of funds within family
43
Table 2: Responses of fund flows to fund and family performances
This table shows the results of Fama-MacBeth regressions of monthly fund flows (in percentage) on
various explanatory variables. P erf is a fund’s monthly alpha estimated over the 36-month period
ending at the end of the previous month, divided by the fund’s monthly idiosyncratic volatility
estimated over the same period; P erf 2 is the square of P erf ; F amP erf is the average P erf of
all other funds in the same family; Log(T N A), Log(Age), and Log(N ) are the natural logarithms
of lagged total net asset value, fund age and number of funds in the family, respectively; expense
is the lagged expense ratio (in percentage); T eam is a dummy variable equal to one if the fund is
team-managed. Rho is the average pairwise correlation of residual returns estimated for each fund
family over the 36-month period ending at the end of the previous month. The model also includes
ten interaction terms of fund and family performance with various fund and family characteristics,
including Log(AGE), Log(T N A), Log(N ), T eam, and Rho. All the regressors, except for the
dummy variable T eam, the three performance measures P erf , P erf 2 , and F amP erf , and the
interaction terms, are adjusted by the median value of the sample in each month. The first two
columns report the results when performance and correlations are estimated from the three-factor
model, while last two columns report results estimated using the four-factor model. The t-statistics
are Newey-Wested corrected (with three lags) for autocorrelation in the estimated coefficients.
44
P erf
P erf 2
F amP erf
Expense
Log(Age)
Log(Age) ∗ P erf
Log(Age) ∗ F amP erf
(1)
4.827***
(13.36)
3.553***
(8.29)
0.879***
(3.14)
-0.447***
(-8.12)
-0.575***
(-12.10)
-0.795***
(-6.34)
0.156
(0.63)
Log(T N A)
Log(T N A) ∗ P erf
Log(T N A) ∗ F amP erf
Log(N )
Log(N ) ∗ P erf
Log(N ) ∗ F amP erf
T eam
T eam ∗ P erf
T eam ∗ F amP erf
Rho
Rho ∗ P erf
Rho ∗ F amP erf
Constant
Observations
R2
0.016
(0.50)
-0.434***
(-4.53)
0.623**
(2.24)
0.330**
(2.19)
0.315
(0.85)
1.463***
(3.50)
0.418***
(3.57)
2.186***
(5.90)
-0.944
(-1.16)
0.311***
(3.31)
116365
0.064
* p < 0.1, ** p < 0.05, *** p < 0.01
45
(2)
4.839***
(13.86)
3.747***
(8.16)
0.949***
(3.14)
-0.402***
(-6.03)
-0.084***
(-6.39)
-0.171***
(-2.90)
-0.106
(-1.24)
0.053
(1.64)
-0.357***
(-3.86)
0.653**
(2.01)
0.404***
(2.71)
0.353
(1.00)
1.419***
(3.34)
0.400***
(3.14)
2.266***
(6.03)
-1.139
(-1.38)
0.194**
(2.07)
116365
0.056
(3)
4.618***
(13.84)
3.140***
(8.44)
0.925***
(3.75)
-0.411***
(-8.23)
-0.582***
(-12.60)
-0.799***
(-6.80)
0.083
(0.37)
0.017
(0.53)
-0.405***
(-4.36)
0.544**
(2.06)
0.290*
(1.95)
0.288
(0.79)
1.391***
(3.41)
0.401***
(3.37)
2.153***
(5.60)
-0.803
(-1.03)
0.355***
(3.85)
116365
0.062
(4)
4.612***
(14.08)
3.261***
(8.40)
0.954***
(3.32)
-0.368***
(-6.15)
-0.083***
(-5.58)
-0.152***
(-2.92)
-0.102
(-1.19)
0.054
(1.62)
-0.337***
(-3.99)
0.569*
(1.81)
0.367**
(2.49)
0.318
(0.89)
1.406***
(3.28)
0.409***
(3.09)
2.260***
(6.07)
-0.931
(-1.18)
0.234**
(2.54)
116365
0.053