Mutual Fund Flowsand Optimal Manager Replacement∗

Thomas DanglDepartment of Managerial Economics

and Industrial OrganizationVienna University of Technology

Theresianumgasse 27, A-1040 ViennaPhone:0043-58801-33063

Thomas.dangl@tuwien.ac.at

Youchang WuDepartment of Finance, University of Vienna

Bruennerstrasse 72, 1210 ViennaPhone:0043-1-4277-38211youchang.wu@univie.ac.at

Josef ZechnerDepartment of Finance, University of Vienna

Bruennerstrasse 72, 1210 ViennaPhone:0043-1-4277-38072josef.zechner@univie.ac.at

May 2004

∗We thank Russ Wermers, Michael Brennan, the participants of the seminars at University of Tuebingen,University of Vienna for their suggestions and comments. Youchang Wu and Josef Zechner gratefully acknowl-edge the financial support by the Austrian Science Fund (FWF) under grant SFB 010 and by the GutmannCenter for Portfolio Management at the University of Vienna.

Abstract

This paper develops a continuous-time model in which a portfolio manager is hired

by a management company. Based on past portfolio returns, all agents, including the

portfolio manager, update their beliefs about the manager’s skills. In response, investors

can move capital into or out of the mutual fund, the portfolio manager can alter the risk

of the portfolio and the management company can replace the manager. The resulting

interplay between internal governance and product market discipline generates a rich set

of results in accordance with the empirical literature. In particular we show how fund

inflows are related to a fund’s performance, fund size, and the uncertainty about manage-

rial skills. We also derive optimal management replacement rules using the real options

approach and demonstrate how manager turnover depends on the model parameters.

1 Introduction

The portfolio management industry has undergone dramatic growth in the last few decades,

thereby also generating an increasing research interest among academics. A particular ques-

tion which has attracted much attention is the governance issue associated with delegated

portfolio management. Numerous authors have approached this question using the standard

principal-agent paradigm, trying to characterize the optimal contracts that alleviate the agency

problem between fund investors and managers,1 Some researchers have also empirically in-

vestigated the effectiveness of funds’ boards of directors in controlling agency costs2 and the

impact of incentive fees.3

This paper takes a broader view of the governance mechanisms in the portfolio manage-

ment industry by explicitly taking into account the role of the product market. A key char-

acteristic of the mutual fund industry is that fund investors are at the same time consumers.

The open-end structure of mutual funds allows individual investors to ”fire” the manager

whenever they feel dissatisfied with the management quality. This not only disciplines the

manager directly, as pointed out by Fama and Jensen (1983), but also gives the management

company strong incentives to fire underperforming managers in order to avoid losing market

share. The interplay between these external and internal governance is the key to understand

many phenomena observed in the mutual fund market. We formalize these two alternative

governance forces simultaneously in a fully-dynamic framework. More specifically, we an-

alyze how beliefs about managerial ability are updated over time and how investors allocate

their money in response to changing beliefs; we also derive the fund management company’s

optimal manager replacement rule and the equilibrium response of fund flows to an observed

manager replacement.

In our model, the portfolio manager may have some stock picking ability that generates

an expected rate of abnormal return by taking idiosyncratic risks. However, active portfo-

lio management exhibits a diseconomy of scale, as introduced by Berk and Green (2004).

1See, for example, Stoughton (1993), Heinkel and Stoughton (1994), Admati and Pfleiderer (1997),Das andSundaram (2002), Ou-Yang (2003).

2See, for example, Tufano and Sevick (1997) for the case of open-end funds and Guercio, Dann, and Partch(2003) for the case of closed-end funds.

3See, for example, Elton, Gruber, and Blake (2003).

1

Furthermore, the manager’s ability to manage a specific fund is not known either to the man-

agement company, to the fund investors or to the manager himself. All agents in the model

only know a common prior distribution of the ability and keep updating their beliefs using the

observed fund performance. Since investors keep moving money into or out of a fund based

on the updated beliefs about the managerial ability, there exists an equilibrium relation be-

tween beliefs and fund size. The management company’s problem is to determine a threshold

of the updated beliefs that triggers a manager replacement. Since the management company’s

right to replace the manager is essentially an American put option written on the stochastic

process of beliefs, the optimal threshold can be derived using the real options approach.

We consider two polar cases. In the first case, the manager is endowed with a given ability

which remains constant over time. In the second case the manager’s ability may change over

time. In this case we assume that the manager’s ability follows a driftless Wiener process, so

that the precision of the ability estimate remains constant.

We first characterize the inference of managerial ability using the nonlinear filtering the-

ory developed by Lipster and Shiryayev (1978). We then analyze two closely-related ques-

tions under the equilibrium condition that the expected abnormal return to fund investors is

zero:

(i). What are the determinants of fund flows? We find that (1) the expected rate of net

fund inflows is negatively correlated with fund size and management fee, but positively

correlated with uncertainty about managerial ability; (2) the net fund inflows are pos-

itively correlated with the fund’s idiosyncratic returns but negatively correlated with

the market return; (3) the sensitivity of net fund inflows to fund performance increases

with uncertainty about managerial ability; and (4) the volatility of net fund inflows de-

creases with fund size and increases with uncertainty about managerial ability. We also

find that fund size is positively related to historical performance but uncorrelated with

future performance.

(ii). Given the equilibrium relation between fund size and beliefs about managerial ability,

what is the optimal manager replacement rule that maximizes the value of the manage-

ment company? Our analysis shows that (1) in the case of constantly changing ability,

2

manager changes are preceded by poor risk adjusted performance and a decrease in

net fund inflows; (2) in the case of constant ability, the optimal manager replacement

threshold increases over the manager’s tenure; it can reach a level significantly above

the expected ability of a new manager; (3) manager tenure is positively correlated with

fund size; (4) the conditional probability of manager departure is non-monotonic in

manager tenure, it increases at the early stage of manager tenure and then keeps de-

creasing.

Our results help to understand a large number of findings documented in the empirical liter-

ature on mutual fund flows and portfolio manager turnover.

Since the early 1990s, a considerable amount of effort has been devoted to research on the

behavior of mutual fund flows. A well-documented finding is that there is a strong positive

relation between a fund’s money inflows and its past performance, indicating that investors

are chasing good past performers.4 Similar to Berk and Green (2004), our model generates

results where rational investors behave in a way fully consistent with this empirical obser-

vation. Another stylized fact that have drawn much attention is that the flow-performance

relation is not only positive but also convex, meaning that out-performers receive dispro-

portionately more fund inflows, while under-performers are not penalized in an offsetting

manner.5 This feature is captured by the positive drift of the flow dynamics we derive. In ad-

dition, it has also been documented that fund inflows, as well as their volatility and sensitivity

to past performance, are negatively correlated with fund size and fund age.6 Finally, Warther

(1995) reports a negative relation between the market return and subsequent aggregate fund

inflows. Our model is consistent with all these results.

There is also a strand of literature on the turnover of portfolio managers. Khorana (1996)

documents an inverse relation between the probability of manager replacement and past fund

performance, measured by asset growth rate and portfolio returns. Chevalier and Ellison

(1999a) find a similar result. They also find that managerial turnover is more performance-

sensitive for younger managers. Chevalier and Ellison (1999b) find that managers with longer

4See, for example, Ippolito (1992), Gruber (1996), Chevalier and Ellison (1997), Sirri and Tufano (1998),Bergstresser and Poterba (2002),and Boudoukh, Richardson, Stanton, and Whitelaw (2003).

5See Chevalier and Ellison (1997) and Sirri and Tufano (1998).6See, for example, Chevalier and Ellison (1997), Sirri and Tufano (1998), Bergstresser and Poterba (2002),

Boudoukh, Richardson, Stanton, and Whitelaw (2003).

3

tenure tend to manage larger funds and have a higher probability to retain their positions.7

Hu, Hall, and Harvey (2000) classify the managerial departure into promotion and demotion,

and find that the probability of a manager being fired or demoted is negatively correlated

with the fund’s current and past performance whereas the promotion probability is positively

related to the fund’s current and past performance. Khorana (2001) documents significant

performance improvements after the replacement of an underperforming manager and per-

formance deterioration after the replacement of outperforming managers. Ding and Wermers

(2002) find that fund managers underperform their counterparts during the year of their re-

placement, and the underperformance vanishes after the manager is replaced. These findings

are in accordance with the predictions from our model.

Several theoretical models have been advanced to explain subsets of the phenomena men-

tioned above. Berk and Green (2004) present an interesting discrete-time learning model to

explain one seemingly puzzling phenomenon mentioned above: Given that the evidence on

the predictability of fund performance is rather weak, why do investors always chase past

winners? There are three key elements in their answer to this question: First, there is com-

petitive provision of capital by investors to mutual funds. Second, there is decreasing return

to scale in active portfolio management. Finally, past returns contain information about man-

agerial ability. The combination of these three elements explains both the lack of performance

persistence and the flow-performance relation: Good performance leads to higher assessment

of managerial ability, which in turn leads to fund inflows, however this will hurt future fund

performance due to the decreasing return to scale. Since the capital provision is competitive,

the fund size will be adjusted to a level where no predictability in fund performance exists.

Our model on mutual fund flows incorporates all the three key elements of Berk and

Green (2004). However, in contrast to Berk and Green (2004), we explicitly distinguish

between fund management companies and fund managers. While Berk and Green (2004)

assume an exogenous shut-down threshold, we allow the fund management company to fire

the manager if it is optimal to do so. This not only enables us to explain both fund flows

and managerial turnover in a unified framework, but also shed some new light on the flow-

performance relation. In particular, while the Berk and Green (2004) model predicts that

7The positive tenure-fund size relation is also documented by Fortin, Michelson, and Jordan-Wagner (1999)and Boyson (2003).

4

when the fund performs very badly, there will be a100%cash outflow due to the shut-down

of the fund, our model predicts that there will be a cash inflow due to the replacement of

the underperforming manager. Therefore, in contrast to Berk and Green (2004), our model

implies a globally convex flow-performance relation.

Our model also differs from the Berk and Green (2004) model in other important aspects.

Instead of the discrete-time setup in Berk and Green (2004), we derive a continuous-time

formulation. Instead of assuming a convex cost function, we model directly the increasing

difficulty of generating abnormal return when the fund becomes larger. Our formulation not

only makes the analysis much easier to handle, but also yields new results. For example, while

Berk and Green (2004) predict that the proportional fund inflows, after controlling for fund

age, are independent of fund size, our model shows that the proportional fund inflows depend

negatively on fund size, even after controlling for fund age, as documented by the empirical

investigations mentioned above. Our setup also predicts a negative relation between fund

inflows and past market return, as documented by Warther (1995).

Lynch and Musto (2003) develop a two-period model to explain the convexity of the

flow-performance relation. Their key insight is that funds that underperform will change

their strategies while those outperforming will not. Therefore bad past performance con-

tains less information about future performance than good past performance does. Taking

this effect into account, rational investors will be less sensitive to past performance when

it is poor. Our model differs from the Lynch and Musto (2003) model in several aspects.

First, Lynch and Musto (2003) do not allow for diseconomies of scale. As a result, active

funds will on average earn above normal returns and good performance should exhibit strong

persistence. Both of these implications are in contradiction with the majority of empirical

findings. Second, whereas Lynch and Musto (2003) analyze the decision to change strate-

gies in a two-period model, we model the manager replacement decision in a fully dynamic

framework, taking into account flow responses, replacement costs, and the option value of

postponing the replacement. Third, although firing bad managers increases the convexity of

the flow-performance relation, such a convexity exists in our model even without this consid-

eration.

Heinkel and Stoughton (1994) consider a two-period contracting problem between a port-

5

folio manager and a client. Their model contains both adverse selection and moral hazard

components. Our model differs in several important dimensions. First, we develop a fully

dynamic model in continuous time. Second, we do not model a direct contracting relation-

ship between the investor and the portfolio manager. Instead, the principal-agent relationship

we consider is between portfolio managers and management companies. Third, our model

formalizes simultaneously the external governance of the product market and the internal

governance mechanism.

Another closely-related paper is Jovanovic (1979), which considers a situation where a

worker’s productivity in a particular job is not known ex ante and becomes known more

precisely as his job tenure increases. While we model learning in a way similar to Jovanovic

(1979), the latter paper does not address the specific issues arising in the mutual fund market.

In fact, it abstracts completely from product market considerations.

The rest of the paper is structured as follows: Section 2 and Section 3 analyze the dynam-

ics of equilibrium fund size and optimal manager replacement rules under the assumption

that the true managerial ability evolves randomly over time. Section 4 reexamines these two

issues under the assumption that the true managerial ability is constant over time.

2 Learning and mutual fund flows

2.1 The dynamics of NAV

The net asset value per share, denoted by NAV, of an open-end fund is assumed to evolve as

follows:

dNAVt

NAVt= [r +λσm+αt −δt −bt ]dt+σmdWmt +σit dWit (1)

wherer is risk free rate,λ is the market price of risk,αt is the expected rate of abnormal return

generated by the manager due to his stock-picking ability,Wmt andWit are two uncorrelated

standard Wiener processes driving the market return and the idiosyncratic component of the

fund’s return respectively. The subscriptt denotes the incumbent manager’s tenure,σm is

the fund’s constant exposure to market risk whileσit is the fund’s time-varying exposure to

6

idiosyncratic risk,δt is the instantaneous dividend yield andbt is the management fee ratio.

Our specification of theNAV dynamics explicitly accounts for active portfolio manage-

ment. The termr + λσm is the fair return given the fund’s exposure to systematic risk. The

expected rate of abnormal return,αt , can be interpreted as Jensen’sα. The dividend yield,δt ,

and the management fee ratio,bt , are subtracted from theNAV return because they represent

cash paid out to the investors and the management company respectively. The exposure to

market risk,σm, is assumed to be constant since we do not intend to model the manager’s

market-timing ability.8 In contrast to the constant exposure to the market risk, the fund’s

exposure to the idiosyncratic risk,σit , can be changed by the manager at any time, under the

constraint that it must be positive.

The expected abnormal rate of return is assumed to have the following functional form:

αt = θσit − γAtσ2it (2)

whereθt is the incumbent manager’s stock-picking ability,At is the market value of assets

under management andγ is a positive constant characterizing the decreasing return to scale

in active portfolio management.

The manager’s abilityθt is assumed to be fund specific, which implies that the manager’s

track record in other funds does not matter. Thus, we assume that the abnormal return is a

joint product of the manager and the management company. A good manager in one company

is not necessarily a good manager in another company because every company has its own

organizational structures, research networks and business culture. This assumption implies

that our model is essentially a match model in the spirit of Jovanovic (1979). It allows us

to abstract from the observable heterogeneity among the manager candidates available to

replace the incumbent manager.

Our specification of the expected abnormal return has several desirable features that we

would like to capture. More specifically, the specification has the following properties.

(i). ∂αt∂θt

= σit > 0. For any given level of idiosyncratic risk and fund size, managers with

8There is little evidence showing that mutual fund managers have market-timing ability. However, recentstudies using portfolio holdings data do provide support for the notion that some managers have superior stock-picking ability. See, for example, Grinblatt and Titman (1993), Daniel, Grinblatt, Titman, and Wermers (1997),Wermers (2000).

7

higher ability can generate higher expected rate of abnormal return.

(ii). ∂αt∂At

=−γσ2it < 0. For any given level of managerial ability and idiosyncratic risk, larger

funds will have a lower expected rate of abnormal return. This is the diseconomy of

scale effect in active portfolio management that we want to capture. An important rea-

son of this effect is the price impact of large portfolio transactions. Consider a manager

who is able to identify a small number of undervalued stocks. If he is managing a small

fund, he can invest all his funds in these stocks and earn a high abnormal rate of return.

However, if he is managing a large fund, doing so would move the purchase prices of

those stocks up and erode his performance.9 10

(iii). ∂2αt∂σ2

it=−2γAt < 0. The marginal return of taking idiosyncratic risk is decreasing, and it

decreases faster for funds with bigger size. The marginal return of taking idiosyncratic

risk must be decreasing in order to rule out unlimited expected profit opportunities.

Consider a manager who can identify a small number of undervalued securities, the

idiosyncratic risk of his fund will increase as he puts more and more money in those

stocks.11 This will increase the fund’s expected rate of abnormal return. However, the

marginal return must be decreasing because the more money he invests in those stocks,

the more likely he will move the price to his disadvantage. This effect is much stronger

when the fund is big. Therefore, the marginal return decreases faster when the fund has

a larger size.

(iv). ∂2αt∂At∂σit

=−2σit < 0. This has two implications: (1) The larger the fund size, the smaller

the marginal return of taking idiosyncratic risk. This should be the case due to the

larger price impact associated with larger fund size. (2) the more idiosyncratic risk

a fund takes, the larger the diseconomy of scale effect it has to face. A fund with

high idiosyncratic risk may either concentrate on a small number of securities, or hold

illiquid stocks. Such funds would suffer the most from being large. However, a well-

9Many authors have investigated this issue empirically. See for example, Perold and Salomon (1991), Indro,Jiang, Hu, and Lee (1999), Chen, Hong, Huang, and Kubik (2003).

10Note that this condition does not imply that for a given level of managerial ability, a larger fund will alwayshave a lower expected rate of abnormal return than a smaller fund. This is true only if both funds take the samelevel of idiosyncratic risk.

11The systematic risk can be kept constant by shorting an index.

8

diversified fund, such as an index fund will not be hurt as much by its large size.

Our formulation of the abnormal return can be interpreted in another way.12 Suppose the

manager divides his assets under management into two parts: one inactive part with system-

atic riskσm and zero idiosyncratic risk, and one actively managed part with systematic risk

σm and idiosyncratic riskεi . The weights of these two components are1−wt andwt respec-

tively. The inactively managed part delivers no abnormal return and does not suffer from

any diseconomy of scale. The actively managed part produces an expected rate of abnormal

return due to the managerial abilityθ. But the abnormal return per unit of idiosyncratic risk

decreases with the size of the actively managed part. It decreases faster when the idiosyn-

cratic risk is higher, since the portfolio with higher idiosyncratic risk is less liquid. Therefore,

the abnormal return of the fund can be written as

αt = wtεi(θt −wtεiγAt). (3)

This specification is equivalent to Equation (2) since the idiosyncratic risk of the whole port-

folio is σit = wtεi .

2.2 Inference about managerial ability

We assume that neither the fund management company, nor the fund investors, nor the man-

ager himself, can observe the managerial abilityθt directly. Therefore, theWit process is not

observable either. In other words, when agents in our model observe a high (low) market

adjusted NAV return, they cannot be sure whether this is due to good (bad) luck or due to

the manager’s ability. All the other terms in Equation (1) are assumed to be observable. The

agents have a common normally distributed prior belief aboutθ0, with a meana0 and vari-

ancev0, and they all use the Bayesian rule to update their beliefs as they observe the realized

NAV process.

For the true abilityθt , we consider two polar cases. In one case, which is considered in

Section 4, the manager has a constant ability. In another case, which is considered in this and

the next section, the true abilityθt is assumed to be changing randomly over time and can be

12Berk and Green (2004) motivate their model in a similar way.

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described by a driftless Wiener process:

dθt = v0dWθt (4)

wheret denotes the manager tenure andWθt denotes a standard Wiener process drivingθt ,

which is assumed to be uncorrelated with the Wiener processes driving the market return

and idiosyncratic return, i.e.,Wmt andWit . We think both scenarios capture some features of

reality. The constantly-changing ability assumption can be motivated as follows. A manager

may improve his investment management skill by learning from his experience, this might

lead to an upward trend inθt . However, it is well-recognized that the business environment is

changing rapidly over time, implying that old strategies and procedures can be outdated very

fast. Sometimes past experience might even be an obstacle to future success. When these

two effects offset each other on average, we end up with a driftless process for managerial

ability. From a technical point of view, we will see shortly that the assumption of a randomly

evolving true ability simplifies our analysis significantly.

To characterize the learning process, we substitute Equation (2) into Equation (1), and

move all the directly observable terms to the left-hand side and denote the resulting expression

by dπt :

dπt :=dNAVt

NAVt− [(r +λσm− γAtσ2

it −δt −bt)dt+σmdWmt]. (5)

Note that the realizeddWmt can be directly observedex postas long as the agents know both

the true expected and realized market return, which we assume they do.

Using the new notation, we can rewrite Equation (1) as

dπt = θtσit dt+σit dWit . (6)

The learning in our model consists of updating the beliefs about a manager’s time-varyingθt

from the observed history ofdπ since the start of his tenure.

Under our model specifications, it follows directly from nonlinear filtering theory devel-

oped by Lipster and Shiryayev (1978) that the conditional distribution ofθt is normal at every

10

point of time, with a constant conditional variancev0. The conditional mean ofθt , denoted

by at , evolves according to the following differential equation13:

dat =v0

σit[dπt −atσit dt]. (7)

The fact that the conditional variance is constant over the manager tenure is due to our

specification of theθt process. Since the true managerial ability keeps changing over time,

the precision of the agents’ estimate of managerial ability does not increase over manager

tenure, although the mean estimate is continuously updated based on the realizeddπt process.

Under more general specifications of theθt process, the conditional variance will usually be

a deterministic function of the manager tenure.

Define the innovation process,Zt as the normalized deviation ofdπt from its conditional

mean,atσit dt:

dZt := [dπt −atσit dt]/σit , Z0 = 0.

It follows thatZt is a standard Wiener process conditional on the common information set of

all agents in the model. Unlike the unobservableWit process, theZt process is derived from

an observable process and is thus observable. Rewriting the dynamics ofNAVt andat in terms

of dZt , we have:

dNAVt

NAVt= [r +λσm+(atσit − γAtσ2

it )−δt −bt ]dt+σmdWmt +σit dZt (8)

dat = v0dZt . (9)

2.3 Equilibrium fund size

We assume that after paying a one-time setup cost, the management company charges a pro-

portional feebt for its services. The operating cost of managing the fund, including the

compensation to the fund manager, is assumed to be a fixed fractions of the total fee in-

13See theorem12.1 of Lipster and Shiryayev (1978). For an intuitive explanation of the non-linear filter-ing theory and its application in finance, see Gennotte (1986). Recent financial research using this techniqueincludes Brennan (1998) and Xia (2001).

11

come. Therefore, the instantaneous net profit of the management company isbt(1− s)At .

This specification can be motivated as a linear sharing rule between the fund manager and the

management company.

In the open-end fund market, investors will allocate their money to funds whose expected

rate of abnormal return is higher than the management fee and withdraw money from funds

whose expected rate of abnormal return is lower than the management fee. For simplicity, we

assume that such fund flows are free of charge, i.e., there is perfect capital mobility. Since

there is no asymmetric information in our model, the investors will update their beliefs about

the managerial ability exactly as the manager and the management company do, and they

will also watch the size and idiosyncratic risk of the fund to judge whether it is worthwhile

to invest in it. Due to the diseconomy of scale, the performance of funds attracting more

money tends to decrease and the performance of funds experiencing money outflows tends

to improve. In equilibrium, the size of every fund will be adjusted to such a level that the

expected abnormal return is equal to the management fee:14

atσit − γAtσ2it = bt , (10)

Given an updated belief about managerial ability, the management company and the manager

will choose an optimal feebt and an optimal level of idiosyncratic riskσit to maximize the

value of the management company.15 We assume that bothbt andσit can be changed without

any cost. Since learning of managerial ability is independent of thebt and σit processes,

at any point of time,bt andσit will be set at the level such that the instantaneous net fee

income is maximized, under the investor’s free movement constraint (Equation 10) and the

constraints thatbt , σit andAt must all be positive. Formally, the manager (or the management

14Ippolito (1992), Edelen (1999) and Wermers (2000) show that this condition holds approximately in reality.15Since there is no asymmetric information, it makes no difference whether the management company or

the manager sets the fee ratio and the level of idiosyncratic risk. Therefore from now on we will adopt theconvention that the fee ratio is set by the management company while the level of risk is determined by themanager.

12

company) solves the following maximization problem,

maxbt ,σit bt(1−s)At

s.t. atσit − γAtσ2it = bt

bt > 0,σit > 0,At > 0.

Solving for At from the free movement constraint and substituting it into the objective

function, i.e., the net fee income, we see immediately that the net fee income depends only

on the ratiobtσit

, but not onbt or σit per se. Since the net fee income is a quadratic function

of btσit

, one can easily see that it is maximized atbtσit

= at2 . Whenat ≤ 0, the problem has

no solution, since the constraints cannot be met jointly. This means that managers whose

mean estimated ability is non-positive will be driven out of business: they get no assets to

manage even if they are not fired by the management company. However, whenat > 0, the

management company has the flexibility of choosing any combination ofbt andσit such that

the ratio equalsat2 . Therefore, our model does not determine a unique optimal value ofbt or

σit , but it does predict a negative one-to-one correspondence betweenbt andσit for a given

at . This prediction is consistent with empirical findings (see for example, (Golec 1996)).

Since the profits of the management company are only determined by the ratiobtσit

, we

assume for the moment thatbt is constant over time and denote it byb. This allows us to

focus on the influence of beliefs about managerial ability.16

Clearly, whenbt is fixed atb, σit will be adjusted as the beliefs about managerial ability

are updated over time, so does the fund sizeAt . More specifically, we have the following

proposition:

Proposition 1. Under the conditions of perfect capital mobility and constant fee ratiob, the

equilibrium sizeA∗t of an open-end fund and the optimal level of idiosyncratic riskσ∗it are

16In reality, management fees are usually very stable. They serve as a contract between investors and fundmanagement company. It makes no sense to change them very frequently. So our assumption is not veryunrealistic.

13

given respectively by

A∗t =

a2t

4bγ if at > 0

/0 if at ≤ 0(11)

σ∗it =

2bat

if at > 0

/0 if at ≤ 0(12)

where/0 denotes the empty set.

Proof. See the discussion preceding the proposition.

Several features of the equilibrium in the open-end fund market are worth mentioning.

First, the equilibrium fund sizeAt is a convex function of the conditional mean of man-

agerial abilityat . It is independent of a fund’s dividend policyδt , its systematic riskσm, the

market price of riskλ, and the risk free rater. Since the management fee is proportional to

fund size, this implies that these factors have no influence on the fee income of the manage-

ment company.

Second, the expected abnormal return to the open-end fund investors is zero (see Equa-

tion (10)). This does not imply that managerial ability does not exist, it just means that the

investors cannot benefit from it because the provision of capital is competitive. All the eco-

nomic rents generated by managerial ability are captured by the manager and the management

company. Therefore manager ability is better measured by the fund size and management fee

than by the return to fund investors.

Third, the level of idiosyncratic riskσit is negatively related to the conditional mean

of managerial abilityat . This is counter-intuitive because one might expect managers with

higher estimated ability will take higher idiosyncratic risk. However it is a natural result of the

convex relation between fund size and managerial ability and the negative relation between

fund size and idiosyncratic risk. From Equation (11) and (12) we can easily get

σit =at

2γAt, (13)

which shows that controlling for fund size, idiosyncratic risk is positively correlated with

managerial ability while controlling for managerial ability, idiosyncratic risk is negatively

14

correlated with fund size.17

2.4 The dynamics of fund size and fund flows

From Proposition 1 we can derive the dynamics of the fund size in a straightforward way. By

Ito’s Lemma, we know from Equation (11) that the percentage change of fund size is given

by

dAt

At=

14bγAt

(dat)2 +at

2bγAtdat (14)

=v2

0

4bγAtdt+

v0√bγAt

dZt (15)

=v2

0

4bγAtdt+

v0

bσit dZt , (16)

where the second equality is derived using Equation (9) and Equation (11), and the third

equality is derived using Equation (13). Equation (15) shows that the fund has a positive

expected growth rate, which is positively related to uncertainty about the managerial ability,

and negatively related to the fee ratio and fund size. The positive expected growth rate is

due to the convex relation between the equilibrium fund size and the estimated managerial

ability. The convexity in this relation implies that the fund size is more responsive to the

upward movement of the conditional mean of the managerial ability than to the downward

movement. Since the conditional mean follows a driftless Wiener process, this asymmetric

response leads to a positive drift of the fund size.

SincedZt follows a standard Wiener process, the exposure of the fund growth rate todZt

in Equation (15) , i.e, v0√bγAt

, is the volatility of fund growth rate. It increases withv0 and

decreases withb andAt .

Equation (16) tells us the sensitivity of fund growth to fund performance. To see this,

note thatσit dZt can be interpreted as a performance measure. Substituting Equation (10) into

(8), we get

dNAVt

NAVt= (r +λσm−δt)dt+σmdWmt +σit dZt . (17)

17A negative relation between idiosyncratic risk and fund size has been documented by Golec (1996), Cheva-lier and Ellison (1999b) and Boyson (2003)

15

This shows thatσit dZt is the market adjusted abnormalNAV return. It equals the unexpected

NAV return, adjusted by the unexpected market return:

σit dZt = (dNAVt

NAVt+δtdt)− (r +λσm)dt−σmdWmt. (18)

Therefore, the coefficient ofσit dZt , i.e., v0b , is the sensitivity of fund growth to fund perfor-

mance. It increases withv0 and decreases withb.

Another implication of Equation (16) is that a fund’s size is determined by its history of

performance. Today’s largest funds are usually yesterday’s best performing funds. However,

as Equation (17) shows, since the current fund size has fully reflected the current assessment

of managerial ability, no expected abnormal return exists at any fund. Therefore, fund size is

uncorrelated with future performance. This is exactly what Ciccotello and Grant (1996) find

in their investigation of the size-performance relation.

We summarize our results in the following corollary:

Corollary 1. The dynamics of the fund size is characterized by the following features:

(i). Both the expected rate and the volatility of fund growth are negatively correlated with

the management fee ratio and the fund size, and positively correlated with the uncer-

tainty about managerial ability.

(ii). The fund growth rate is positively related to fund performance; the sensitivity of the

fund growth rate to fund performance decreases with the management fee and increases

with the uncertainty about managerial ability.

(iii). Fund size is positively correlated with past performance but uncorrelated with future

performance.

In the empirical literature, fund flows are normally defined as the fund’s asset growth rate

minus its NAV return. Since the NAV return equals the sum of capital gain and dividend

yield, this suggests the following definition for fund flows:

FLOW :=dAt

At− (

dNAVt

NAVt+δtdt) (19)

16

This definition represents the percentage asset growth rate in excess of the growth that would

have occurred if no new funds had flowed in and if all dividend had been reinvested in the

fund.

Substituting (16) and (17) into (19), we see that the fund flow is given by:

FLOW = (v2

0

4bγAt− r−λσm)dt−σmdWmt +(

v0

b−1)σit dZt . (20)

Equation (20) shows that the net fund flows can be decomposed into three parts: the

expected inflows, the response to unexpected market return, and the response to the market

adjusted abnormalNAV return. As we will discuss later, under reasonable parameterization,

both the expected inflows,( v20

4bγAt− r−λσm), and the sensitivity of fund flows to the market

adjusted abnormalNAV return,(v0b −1), are positive (see footnote 23). However, the relation

between fund inflows and market return is always negative.

Sinceσit = b√bγAt

, we also have

FLOW = (v2

0

4bγAt− r−λσm)dt−σmdWmt +

v0−b√bγAt

dZt , (21)

This equation shows that the volatility of fund flows is positively related to the fund’s expo-

sure to systematic risk and uncertainty about the managerial ability, and negatively correlated

with fund size and management fee.

We summarize our results in Corollary 2.

Corollary 2. The net fund flows into a specific fund are characterized by the following fea-

tures:

(i). The expected fund inflows decrease with the management fee ratio and fund size, but

increase with the uncertainty about managerial ability;

(ii). The net fund inflows are positively correlated with the fund’s idiosyncratic return but

negatively correlated with the market return;

(iii). The sensitivity of net fund inflows to market adjusted abnormalNAV return decreases

with the management fee ratio and increases with the uncertainty about managerial

17

ability.

(iv). The volatility of net fund inflows decreases with management fee and fund size, and

increases with the exposure to systematic risk and the uncertainty about managerial

ability.

The predictions of Corollary 2 are generally consistent with the empirical literature on

mutual fund flows, as discussed in the introduction.18 In particular, our model rationalize

a puzzling phenomenon documented in Warther (1995): The negative short-run relation be-

tween aggregate fund inflows to the mutual fund industry and past market return. Within

our framework this is quite intuitive. A fund will attract net inflows if and only if its inter-

nal growth rate, namely itsNAV return, is less than its equilibrium growth rate described by

Equation (16). While the positive unexpected market returndWmt has a positive impact on a

fund’s internal growth rate, it has no influence on its equilibrium growth rate, because it does

not affect the beliefs about managerial ability. Therefore, its influence on fund size must be

offset by corresponding fund outflows. Things are different when the market adjusted abnor-

mal returnσit dZt is high. Higher market adjusted return not only increases the asset value

but also results in a higher evaluation of managerial ability, and the latter effect generally

dominates the former. Therefore fund flows will respond positively to it. A direct implication

of our analysis is that funds delivering abnormal returns in a downward market will attract

money inflows most dramatically. Testing this prediction is an interesting topic for further

empirical examinations of the performance-flow relation.

Although we have not yet introduced manager change into our model formally, the im-

pact of a given manager change on fund flows can be easily derived from the Proposition

1. By assumption, all new managers have the same unconditional mean abilitya0. There-

fore, when a manager with abilityat is replaced by a new manager with abilitya0, the fund

size will adjust from a2t

4bγ to a20

4bγ .19 Whether this change is positive or negative depends on

whether the conditional mean of the departing manager’s ability,at , is higher or lower than

the unconditional meana0. We summarize this result in Corollary 3:

18Note that the convexity in the flow-performance relation is reflected in the positive drift of the flow dynamicsin our continuous-time model.

19Note that investors have no incentive to respond earlier in our simplified world without transaction costs,even if the manager change is fully anticipated.

18

Corollary 3. Manager changes in the open-end fund will be accompanied by a discrete

change in the fund size. The magnitude of this change is given bya2

0−a2t

4bγ , whereat denotes the

outgoing manager’s expected ability.

The prediction that fund size will adjust immediately to reflect the new manager’s ability

is admittedly quite strong, because in reality, managerial ability is not the only determinant

of fund performance. However some anecdotic evidence do show that investors are quite

sensitive to fund manager change. For example, it was reported that when William von

Mueffling, a star manager running the hedge-fund business of Lazard Asset Management

company, resigned in January 2003, Lazards4 billion hedge-fund business dwindled to less

than1 billion in just a few weeks (The Economist (2003)).

Corollary3 also has important implications on the flow-performance relation. Since man-

agers with extremely bad performance will be replaced, thus inducing money inflows, bad

performance is not necessarily associated with money outflows. This introduces additional

convexity in the flow-performance relation. It is in sharp contrast with the prediction of the

Berk and Green (2004) model, which assumes that funds will shut down when its perfor-

mance toughes a lower boundary.

3 The optimal manager replacement rule

3.1 The optimal replacement threshold

We now examine the optimal manager replacement rule of the management company, taking

the relation between the fund size and the beliefs about managerial ability as given. To elim-

inate the horizon effect and simplify our analysis, we assume for the moment that manager

tenure can in principle be infinitely long. However, a manager can be fired by the manage-

ment company or quit at his own will. Whenever the incumbent manager departs, either

because he is fired or he leaves voluntarily, the management company has to incur a cost

when hiring a new manager. This cost can either be interpreted as a search cost or a cost for

training the new manager. The ability of the new manager is again believed to be normally

distributed with meana0 and variancev0. The cost is assumed to be a fixed proportionk of

19

the initial value of the management company.

Since the focus of this paper is the optimal firing decision of the management company,

we do not endogenize the job quitting decision of the manager. Instead we assume that the

manager’s job quitting can be described by a Poisson processqt with a constant mean arrival

rateµ:

dqt =

0 with probability1−µdt

1 with probabilityµdt(22)

Theqt process is assumed to be uncorrelated with either theWmt or theZt process.

The management company’s problem is to determine an optimal threshold of manager

replacement. This is similar to optimizing the exercise decision for the owner of an American

option and can be analyzed using dynamic programming techniques. In the setup we consider

so far, the Bellman equation can be written down as follows:

F(at) = max{(1−k)F(a0), bt(1−s)Atdt+e−rdtE[F(at)+dF(at)]}. (23)

where the value functionF(at) is the market value of the management company, which is

essentially the present value of all future net profits under the optimal replacement rule. On

the right-hand side, the first term is the value of the management company when the manager

is replaced. We call it the replacement value. The second term is the continuation value,

which consists of the immediate net profit in the period fromt to t + dt and the expected

company value att + dt, discounted back at the risk free rate. The risk free rate is used

because the change in the market value of management company is driven bydZ anddq,

both of which are idiosyncratic. Since the owners of the management company bear no

systematic risk, the fair expected return is the risk free rate.

Note that there is only one state variable, namelyat , in the Bellman equation (23). This

is due to two special features of the model we consider so far: (1) The conditional variance

of the managerial ability is constant so it can be omitted; (2) There is no upper limit on

manager tenure, consequently there is no horizon effect. We will consider the case of a

tenure-dependent value function in Section 4.

20

Since forat ∈ (0,+∞), which is the relevant range for our model,bt(1− s)At − r[(1−k)F(a0)] is monotonically increasing withat (note thatAt is a quadratic function ofat and

F(a0)−K is a constant), there will be a single cutoffa, with continuation optimal ifat

is abovea and replacement optimal otherwise.20 This cutoff value is exactly the optimal

threshold that we wish to identify.

In the continuation region, the Bellman equation is

rF (at)dt = bt(1−s)Atdt+E[dF(at)], (24)

which is essentially a no-arbitrage condition.

Suppose that the functionF(a) is continuous and twice-differentiable. Using Ito’s Lemma

and taking into account the possibility of job quitting, we have

dF(at) =

12Faa(at)v2

0dt+Fa(at)v0dZ if dq= 0

(1−k)F(a0)−F(at) if dq= 1(25)

whereFa(at) andFaa(at) denote the first and second order derivative ofF(at) with respect to

at .

From Equation (25) and (22), we can see that the expected change ofF(at) is

E[dF(at)] =12

Faa(at)v20dt+µ[(1−k)F(a0)−F(at)]dt. (26)

Substituting Equation (11) and (26) into (24), we get the following ordinary differential

equation forF(at):

(r +µ)F(at) =12

Faa(at)v20 +µ(1−k)F(a0)+

(1−s)a2

4γ. (27)

The general solution of this ordinary differential equation is

F(at) = F0 +C(1)e−√

2(r+µ)at/v0 +C(2)e√

2(r+µ)at/v0, (28)

20See Dixit and Pindyck (1994), pp. 128-130.

21

where

F0 =µ(1−k)F(a0)

r +µ+

(1−s)[v20 +a2

t (r +µ)]4γ(r +µ)2 .

andC(1) andC(2) are two constants that need to be determined by boundary conditions.

Note thatF0 is the value of the management company with managerial abilitya if it had

no option to fire the manager. The rest of the company value, i.e.,C(1)e−√

2(r+µ)at/v0 +

C(2)e√

2(r+µ)at/v0, arises from the option to fire managers with low ability. Whena goes

to plus infinity, the manager will never be replaced so the option becomes worthless. This

asymptotic consideration implies that the undetermined constant associated with the positive

square root, namely,C(2), must be zero.

Furthermore, the functionF(at) has to satisfy the following boundary conditions:

(i). Value matching condition. At the replacement thresholda, we must have

F(a) = (1−k)F(a0) (29)

(ii). Optimality condition. As a first order condition of optimality, we require21

Fa(a)≥ 0, Fa(a)a = 0. (30)

(iii). Starting valueF(a0). The starting valueF(a0) should satisfy the general solution (28).

21This is a generalized version of the smooth-pasting condition frequently used in option pricing literature,i.e.,

Fa(a) = 0.

We use the Kuhn-Tucker optimality condition instead of the standard smooth-pasting condition becausea can-not be negative. When the replacement cost is very high and uncertainty about the managerial ability is alsovery high, it might be optimal for the management company not to replace the manager even if the conditionalmeanat is zero, because there is still some chance that he will turn out to be or become a good manager later.However, since the fund size would be zero whenat = 0, the manager has to be replaced as long asat = 0,otherwise the fund will cease to exist.In reality, funds may face some additional constraints, for example, the minimum size or maximum risk con-straint, which may potentially result in non-optimal manager replacement. However, these types of constraintswill never be binding in our model, since funds can always satisfy such constraints by lowering their fee ratio.Note, however, that although fee adjustment can help to get round those constraints, it cannot be a substitute forthe manager replacement in our model, since it has no influence on the net profit of the management company.

22

SinceC(2) = 0, we have

F(a0) =µ(1−k)F(a0)

r +µ+

(1−s)[v20 +a2

0(r +µ)]4γ(r +µ)2 +C(1)e−

√2(r+µ)a0/v0 (31)

These conditions allow us to prove the following proposition:

Proposition 2. The optimal manager replacement thresholda and the initial value of the

management companyF(a0), are the solution to the following system of equations22:

a = Max[0, − v0√2(r +µ)

+

√4rγ(1−k)F(a0)

1−s− v2

0

2(r +µ)], (32)

F(a0) =

(1−s)[v20 +a2

0(r +µ)+√

2(r +µ)v0ae−√

2(r+µ)(a0−a)/v0]4γ(r +µ)(r +µk)

if a > 0,

(1−s)[v20 +a2

0(r +µ)−v20e−

√2(r+µ)a0/v0]

4γ(r +µ)[(r +µk)− r(1−k)e−√

2(r+µ)a0/v0]if a = 0.

(33)

Proof. See Appendix A.

3.2 The distribution of manager tenure

Given the optimal manager replacement threshold and the job quitting process, we can derive

the following proposition about the manager tenure.

Proposition 3. Under the optimal replacement thresholda and the constant job quitting

densityµ, the distribution of the manager’s tenure has the following properties:

(i). The expected tenure of the manager is given by

T =1µ[1−e

−√

2µv0

(a0−a)]; (34)

22Generally, there are two roots for this equation system, but only one of them makes economic sense, becausethe larger root ofa is bigger thana0. This root can be excluded because it implies that no manager will beemployed at all.

23

(ii). The cumulative distribution function of the manager’s tenure is given by

P(t) = 1−e−µt[2Φ(a0−a

v0√

t)−1]; (35)

Proof. See Appendix B.

We can also derive the conditional density of manager departure and the expected man-

agerial ability given the manager’s tenure. These results are summarized in Proposition 4:

Proposition 4. Under the optimal replacement thresholda and the constant job quitting

densityµ, we have:

(i). For a manager who has survived untilt, the probability density of departure is given

by

f (t) = µ+(a0−a)e

− (a0−a)2

2v20t

√2πv0t

32 [2Φ(a0−a

v0√

t)−1]

. (36)

(ii). For a manager who has survived untilt, the expected ability is given by

E(at |t) =a0−2a[1−Φ(a0−a

v0√

t)]

2Φ(a0−av0√

t)−1

. (37)

Proof. See Appendix C

3.3 Results and Comparative statics

Table 1 summarizes the base case parameter values. Suppose that the idiosyncratic riskσit

is fixed at0.10, then under this set of parameter values, a newly hired manager can generate

on average2 percent abnormal rate of return for the first dollar he manages. However, the

expected rate of abnormal return goes to zero if the fund size increases to200million dollars.

A manager whose ability is one standard deviation above the mean can generate on average

24

5.5 percent abnormal rate of return for the first dollar he manages. He can be expected to

outperform the market as long as his fund size is less than550million dollars.23

Panel A of Table 2 summarizes the initial value of the management company, the replace-

ment threshold, and the expected tenure of the base case under the current model specifica-

tion, i.e., the true abilityθt follows a driftless Wiener process and manager tenure can be

infinitely long. Note that the replacement threshold,a = 0.0732, is significantly below the

expected ability of a new manager,a0 = 0.20. Since the downward movement ofat before

it hits the replacement threshold is associated with bad performance and decrease in fund in-

flows, our model is consistent with the empirical findings of Khorana (1996), and (Chevalier

and Ellison 1999a), who document that managerial turnover is usually preceded by bad per-

formance and a lower asset growth rate.

Figure 1 plots the cumulative distribution function of manager tenure,P(t); the condi-

tional density of manager departure as a function of tenuref (t); and the expected ability

of a manager with tenuret, E(at |t). TheP(t) curve shows that about70%of the managers

will have a tenure less than 5 years. The conditional density of manager departure,f (t),

increases rapidly when the manager starts his tenure, and then keeps going down. The non-

monotonicity of the conditional departure density reflects the effect of learning and firing.

Without learning and firing, the conditional departure density will always be constant due

to the assumption of constant quitting density. Since it takes some time for the manage-

ment company to learn about the manager’s ability, the initial density of firing is low. But

it goes up very quickly because many new managers may be not good enough to survive.

As more and more incompetent managers are fired over time, managers who survive longer

will generally have a higher ability, as shown in theE(at |t) curve. Therefore, consistent with

the empirical evidence discussed in the introduction, they tend to manage larger funds, have

lower probability of being fired, and their departures are more likely due to non-performance

reasons.

As to the comparative static analysis, we first state some analytical results, which are

23Under this base case parameter set, consider a fund with a median size of100million dollars, and a typ-ical fee ratio of1.5% (taking into account the front- and back-load fee), as well as a typical expected an-nual NAV return of 14%. According to Equation (20), the expected net fund inflows of this fund will be10% (= 0.122

4∗0.015∗10−8∗108 −0.14), and the sensitivity of fund flows to market adjusted abnormal return will be

7(= 0.120.015−1). Both terms are strongly positive.

25

derived from Proposition 2 and summarized in Proposition 5:

Proposition 5. Whena > 0,24 the optimal replacement thresholda, the initial value of the

management companyF(a0), and the expected manager tenureT, have the following prop-

erties:

∂a∂a0

>0,∂a∂k

<0,∂a∂γ

=∂a∂s

=0; (38)

∂F(a0)∂a0

>0,∂F(a0)

∂k<0,

∂F(a0)∂γ

<0,∂F(a0)

∂s< 0; (39)

∂T∂k

>0,∂T∂γ

=∂T∂s

=0. (40)

Proof. See Appendix D.

All the results stated in Proposition 5 are very intuitive and self-explanatory. In particular,

the optimal replacement thresholda is independent of the diseconomy of scale parameterγ

and the variable costs. This is because both the continuation value and the replacement value

are proportional to these two parameters, so the replacement decision is not affected by them.

The influence ofv0 andµ are more difficult to sign analytically. Therefore, we resort to

numerical methods. We examine the influence of one parameter at a time by fixing the value

of all other parameters at the base case level. We then plotF(a0), a, T as a function of the

parameter under consideration. The parameters under consideration includev0, µ as well as

a0 andk, although analytical results are available for the last two parameters.

Figure 2 plots the initial value of the fund management company,F(a0), as a function of

a0, k, v0, andµ respectively. Consistent with the analytical results,F(a0) increases witha0

and decreases withk. The figure also shows that the relation betweenF(a0) andµ is negative,

this is so because high frequency of manager quitting not only results in high replacement

cost, but also means the loss of good managers. The relation betweenF(a0) andv0 is more

subtle. On the one hand, higher uncertainty about managerial ability makes the replacement

decision more difficult to make; on the other hand,the management company essentially holds

24We skip the less-realistic case wherea= 0 for brevity, although many similar results can be derived for thatcase as well.

26

an option to fire the manager, and the value of this option increases with the heterogeneity

and volatility of the managerial ability. The positive relation plotted in Figure 2 shows that

the latter effect is dominating.

Figure 3 plots the optimal replacement threshold,a, as a function ofa0, k, v0, andµ. As

expected,a increases with thea0 and decreases withk. As a0 increases,a increases almost

linearly, but the slope is slightly less than one. This is because highera0 is associated with

higher value of the management company, which in turn implies higher replacement cost

since the replacement cost is proportional to the company value. The figure also shows thata

is negatively related tov0. There are several reasons why this is the case: Whenv0 is high, (1)

the company is less sure whether a manager is good or not; (2) there is a high probability that

a bad manager can become a good manager later; (3) the replacement cost is higher because

the company value increases withv0. Another message conveyed by the figure is thata

is relatively insensitive to the change of the manager quitting densityµ. The reason is as

follows. Higher frequency of manager quitting reduces the value of a management company,

thus lowering the replacement cost; however, it also lowers the incentive to fire the incumbent

manager, because even if the new manager is very good, his high quitting probability makes

him less valuable to the company. These two effects tend to offset each other, leaving the

replacement threshold largely unchanged.

Figure 4 plots the expected tenureT as a functiona0, v0, k, andµ respectively. It shows

that T increases significantly withk and decreases withµ. It also shows thatT increases

slightly with a0. This should be expected because as we have seen, the increase ofa0 does not

lead to a one-to-one increase ina. The slightly negative relation betweenv0 andT suggests

that although the management company tends to adopt a less stringent threshold when the

uncertainty is higher, the higher volatility of the mean ability process still leads to higher

probability of manager firing.

27

4 Constant managerial ability

In this section we consider the case where the true managerial ability remains constant. In

other words, we replace Equation (4) with:

dθt = 0. (41)

An immediate consequence of this change in assumption is that both the mean and vari-

ance of the conditional beliefs about the managerial ability will change over time. More

specifically, according to Theorem12.1 in Liptser and Shiryayev (1978), the conditional dis-

tribution of θ is normal at every point of time, and the conditional meanat and variancevt

follow the differential equations:

dat =vt

σi[dπt −atdt] = vtdZt , (42)

dvt =−v2t dt. (43)

The conditional variance is a deterministic function of the manager tenure, and can be

solved explicitly as:

vt =v0

v0t +1. (44)

It can be easily seen thatvt shrinks over time. It goes to zero ast goes to infinity. This implies

that all agents in this model will eventually learn the true value ofθ if the manager’s tenure

is infinitely long.25

In a world of no transaction cost, the manager’s portfolio choice and the equilibrium

fund size are independent of the conditional variancevt and the manager tenure. As a re-

sult, proposition 1 continues to hold under our new specification. However, the dynamics of

the equilibrium fund size and fund flows do depend on the learning process. More specif-

ically, under the new assumption, we have to replace the constant conditional variancev0

in Equations (15),(16), (20), and (21) by the time-varying conditional variancevt . Sincevt

25However, the decrease ofvt is slow, if v0 = 0.12 as in our base case parameterization, then after10 years,vt will be 0.0545, which still implies a strongly positive flow-performance relation.

28

decreases over the tenure, our model predicts that the fund growth and fund inflows will tend

to slow down, and become less volatile, less responsive to past performance as the tenure of

the incumbent manager increases over time.

This version of the model does capture some important empirical features that are not

captured by the model with constant conditional variance. For example, both Chevalier and

Ellison (1997) and Boudoukh, Richardson, Stanton, and Whitelaw (2003) find that fund flows

are more sensitive to the past performance for young funds than for old funds. Boudoukh,

Richardson, Stanton, and Whitelaw (2003) also find that young funds have higher expected

percentage inflows than the old funds. If we interpret young funds as funds managed by man-

agers with shorter tenure, then these findings are consistent with our model with decreasing

conditional variance.

The determination of the optimal manager replacement rule is now more complicated,

since both conditional mean and variance are varying over time, i.e., we have two state vari-

ables. Since the conditional variance is a deterministic function of tenure, the Bellman equa-

tion for the value of the management company can be written as

F(at , t) = max{(1−k)F(a0,0), bt(1−s)Atdt+e−rdtE[F(at , t)+dF(at , t)]}. (45)

Since forat ∈ (0,+∞), b(1− s)At − r(1− k)F(a0,0) is monotonically increasing withat at

any t, there exists a tenure-dependent thresholda(t), with continuation optimal whenat >

a(t) and replacement optimal whenat < a(t). In the continuation region, we must have

rF (at , t)dt = bt(1−s)Atdt+E[dF(at , t)]. (46)

This leads to the following partial differential equation:

(r +µ)F(at , t) = bt(1−s)At +Ft(at , t)+12

Faa(at , t)v2t +µ(1−k)F(a0,0), (47)

whereFt denotes the partial derivative ofF with respect tot, Faa denotes the second order

partial derivative ofF with respective toa.

29

Substituting Equation (11) and (44) in (47), we get

(r +µ)F(at , t) =(1−s)a2

t

4γ+Ft(at , t)+Faa(at , t)

v20

2(v0t +1)2 +µ(1−k)F(a0,0), (48)

The partial differential equation (47) has to satisfy the following boundary conditions:

(i). Value matching condition. At the replacement boundary, we must have

F(a(t), t) = (1−k)F(a0,0) (49)

(ii). Optimality condition. At the replacement boundary, we should also have

Fa(a(t), t)≥ 0, Fa(a(t), t)a(t) = 0 (50)

Similarly, we can get the partial differential equation for the expected tenureT(at , t):

µT(at , t) = 1+Tt(at , t)+Taa(at , t)v2

0

2(v0t +1)2 , (51)

with the boundary conditionsT(a(Tmax),Tmax) = 0 andT(a(t), t) = 0.

Unfortunately, the partial differential equations (47) and (51) have no closed-form solu-

tion. We need to resort to numerical procedures. We perform the valuation on a binomial

tree. Since the state variableat has a nonconstant volatility, we use the approach developed

by Nelson and Ramaswamy (1990) to construct a recombining tree. We set the maximum

tenure that a single manager can have to be30 years, after which he has to be replaced by a

new manager. We find the value of the management company and the replacement threshold

recursively.

Using the same parameter values as in Table 1, we get the initial value of the management

company and the expected tenure as reported in panel B of Table 2. We see that the value

of the management company is substantially lower than in the case of randomly evolving

managerial ability. This is because the value of the option to replace the manager is lower

due to decreasing volatility of the conditional mean. The expected manager tenure is also

shorter.

30

Figure 5 plots the tenure-dependent manager replacement threshold. The threshold starts

at approximately0.08 and keeps going up with manager tenure. It exceedsa0, the uncondi-

tional mean of the managerial ability, after about6 years. This could never be the case in the

specification with constantly changing ability, because in that case, the incumbent manager

is always better than the new manager as long asat > a0. However, in the current version

of the model, the management company is not satisfied with a manager with average ability.

It will fire a manager when it becomes increasing likely that he is just an average manager.

Since the replacement threshold is increasing with tenure, this version of the model predicts

a stronger size-tenure relation than our previous version.

Figure 6 plots the cumulative distribution of manager tenure. Due to the higher replace-

ment threshold, the fraction of managers whose tenure is shorter than5 years is higher than

in the case of continuously changing ability.

5 Conclusion

This paper has developed a continuous-time model in which the portfolio manager’s ability is

not known precisely. All parties involved, including the manager, learn about the manager’s

ability from past portfolio returns. In response, investors can move capital into or out of a mu-

tual fund, portfolio managers can alter the risk of the portfolio and the management company

can replace incumbent managers with new managers. Thus, the model formalizes simultane-

ously the external governance of the product market and the internal governance mechanism

in a fully-dynamic framework, thereby generating a rich set of empirical predictions on both

mutual fund flows and manager turnover.

The model shows that product market forces introduce strong incentives to monitor and

replace managers of open-end funds. At the same time, this internal governance also influ-

ences how investors respond to past performance. Our analysis rationalizes many empirically

documented findings (e.g., convex flow-performance relation; negative relation between fund

risk and fund size; positive relation between fund risk and management fees; negative relation

between fund inflows and market return; positive relation between manager tenure and fund

size, etc.) At the same time, our analysis also generates new empirical hypotheses. For ex-

31

ample, our model predicts that fund inflows respond most positively to good performance in

downward markets; that the convexity of the flow-performance relation and the sensitivity of

flows to performance decrease with manager tenure; and that the probability of management

replacement first increases and then decreases with manager tenure.

The framework derived in this paper can be extended in a number of ways. In particular,

we have assumed that capital can be freely moved in an out of the fund. In practice this is

associated with costs. In the limit, capital cannot be withdrawn or injected in a fund at all.

This would be true for the case of closed-end funds. Applying this analysis to the case of

closed-end funds and analyzing the resulting discounts or premia is our current subject of

research.

6 Appendix

A. Proof of Proposition 2

Proof. SinceC(2) must be zero by the usual asymptotic argument, from Equation (28) and

the value-matching condition (29) we have

µ(1−k)F(a0)r +µ

+(1−s)[v2

0 +a2(r +µ)]4γ(r +µ)2 +C(1)e−

√2(r+µ)a/v0 = (1−k)F(a0). (52)

If a > 0, then by the optimality condition (30) we will haveFa(a) = 0, which implies

(1−s)a2γ(r +µ)

+C(1)−

√2(r +µ)v0

e−√

2(r+µ)a/v0 = 0. (53)

Therefore we have

C(1) =(1−s)v0a

(2(r +µ))32 γ

e√

2(r+µ)a/v0. (54)

Substituting Equation (54) into (52) and collecting terms, we get

a =− v0√2(r +µ)

+

√4rγ(1−k)F(a0)

1−s− v2

0

2(r +µ).

32

Substituting Equation (54) into (31), we get

F(a0) =(1−s)[v2

0 +a20(r +µ)+

√2(r +µ)v0ae−

√2(r+µ)(a0−a)/v0]

4γ(r +µ)(r +µk).

If a = 0, then by Equation (52) we have

C(1) =r(1−k)F(a0)

r +µ− (1−s)v2

0

4γ(r +µ)2

Substituting this equation into (31) and noting thata = 0, we have

F(a0) =(1−s)[v2

0 +a20(r +µ)−v2

0e−√

2(r+µ)a0/v0]

4γ(r +µ)[(r +µk)− r(1−k)e−√

2(r+µ)a0/v0].

This completes our proof of Proposition 2.

B. Proof of Proposition 3

Proof. We first derive the unconditional mean of the manager’s tenure.

Since the only state variable in our model is the conditional mean of managerial ability

at , we can write the expected tenure T as a function ofat . Starting from anyat greater than

a, the expected remaining tenureT(at) must satisfy the following Bellman Equation:

T(at) = dt+E[T(at)+dT(at)]. (55)

Since with probabilityµdt the manager will quit att +dt, it follows that

T(at) = dt+(µdt∗0)+(1−µdt)∗ [T(at)+E(dT(at)|dqt = 0)]. (56)

Using Ito’s Lemma and Equation (9), we have

µT(at) = 1+1dt

E[dT(at)|dqt = 0] = 1+12

Taa(at)v20. (57)

One can easily see that the differential equation forT(at) is the same as that forF(at)

33

with only two adaptations: the flow term is 1 (because the flow is the time itself) and the

discount rate isµ. Solving this second order ordinary differential equation, using the usual

asymptotic argument and the boundary conditionT(a) = 0, we get

T(at) =1µ[1−e−

√2µ(at−a)/v0] (58)

Settingat to bea0 yields the expected tenure of a new manager, i.e., Equation (34).

To calculate the cumulative distribution function of the manager’s tenure, we first note

that for the manager’s tenure to be longer thant, it must be true that: (1) he has not quit

before timet; (2) he has not been fired, i.e., the conditional mean of his ability has not hit the

replacement thresholda before timet.

Since the job quitting follows a Poisson process, the probability that the manager did not

quit before timet is simplye−µt. And since the conditional meanat follows a Wiener Process

with volatility v0, the probability thatat stays abovea until t is given by

Pa0( in f0<s<t

as≥ a) = 2Φ(a0−a

v0√

t)−1 (59)

whereΦ(·) denotes the cumulative standard normal probability.

Due to the assumption that the quitting processqt is uncorrelated with the conditional

mean processat , the joint probability that neither quitting nor firing happens beforet is

simply the product of two marginal probabilities. Therefore, the probability that the manager

survives untilt is

Psurvive(t) = e−µt[2Φ(a0−a

v0√

t)−1], (60)

and the cumulative distribution function of the manager’s tenure is

P(t) = 1−Psurvive(t) = 1−e−µt[2Φ(a0−a

v0√

t)−1],

whereP(t) denotes the probability that the manager’s tenure is shorter thant.

This completes our proof of Proposition 3.

34

C. Proof of proposition 4

Proof. From the cumulative distribution function (35) we can derive the conditional proba-

bility that the manager departs in the time interval[t, t +dt], given that he has survived until

t. This conditional probability is given by[P(t +dt)−P(t)]/[1−P(t)]. Dividing the condi-

tional probability bydt, we get the conditional density of manager departuref (t), which is

given by Equation (36).

For the expected ability of a manager with tenuret, first note that based on the reflec-

tion principle of the Wiener process, the unconditional density of mean managerial ability

at tenuret, given the optimal replacement thresholda and the constant quitting densityµ, is

given by

g(at) =

e−µt

v0√

2πt[e

−(at−a0)2

2v20t −e

− (at+a0−2a)2

2v20t ] if at > a,

0 if at ≤ a.

(61)

From the unconditional density, we can calculate the unconditional mean ofat , which equals

e−µt[a0−2a(1−Φa0−a

v0√

t)],

Dividing the unconditional mean obtained above by the survival probability given by Equa-

tion (60), we get the conditional expectation ofat for a manager with tenuret, which is given

by Equation (37).

This completes our proof of proposition 4.

D. Proof of Proposition 5

Proof. Substituting the first equality of Equation (33) into (32), we see immediately that the

optimal replacement thresholda is independent of the diseconomy of scale parameterγ and

the variable costs. Using the implicit function theorem, it is not difficult to prove that

∂a∂a0

> 0,∂a∂k

< 0.

35

Since by Equation (34), the expected tenureT is negatively related toa, it follows from

the above results that

∂T∂k

> 0,∂T∂γ

=∂T∂s

= 0.

Noting thata is independent ofγ ands, we can easily see from Equation (33) that

∂F(a0)∂γ

< 0,∂F(a0)

∂s< 0.

Substitutinga = − v0√2(r+µ)

+

√4rγ(1−k)F(a0)

1−s − v20

2(r+µ) into the first equality of Equation

(33), and applying the implicit function theorem, it is not difficult to show that

∂F(a0)∂a0

> 0,∂F(a0)

∂k< 0.

This completes our proof of Proposition 5.

36

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Table 1: base case parameters value

Notation Definition Base Case Valuer risk free rate 0.05s variable cost as a percentage of fee income 0.5k percentage replacement cost 0.05γ measure of diseconomy of scale 10−8

a0 unconditional mean of managerial ability 0.2v0 unconditional variance of managerial ability 0.12µ mean arrival rate of job quitting 0.05

Table 2: Base case results

Panel A: Results whenθt evolves randomlyF(a0) 4.96∗107

a 0.0732T 5.68 (years)

Panel B: Results whenθt is constantF(a0) 3.20∗107

a tenure-dependentT 4.16 (years)

41

5 10 15 20

0.2

0.4

0.6

0.8

t

P (t)

E(at|t)

f(t)

Figure 1. The base case results with constant conditional variance.

The figure plots the cumulative distribution of manager tenureP(t), the conditional density ofmanager departuref (t), and the expected ability of the manager for a given tenure,E(at |t).The parameter values are given in Table 1.

42

0.18 0.21 0.24 0.27 0.3

2·1073·1074·1075·1076·1077·1078·107

a0

F (a0)

0.1 0.12 0.14 0.16

2·1073·1074·1075·1076·1077·1078·107

v0

F (a0)

0.02 0.04 0.06 0.08 0.1

2·1073·1074·1075·1076·1077·1078·107

k

F (a0)

0.02 0.04 0.06 0.08 0.1

2·1073·1074·1075·1076·1077·1078·107

µ

F (a0)

Figure 2. Comparative statics: The value of the management company

43

0.18 0.21 0.24 0.27 0.3

0.02

0.04

0.06

0.08

0.1

0.12

0.14

a0

a

0.1 0.12 0.14 0.16

0.02

0.04

0.06

0.08

0.1

0.12

0.14

v0

a

0.02 0.04 0.06 0.08 0.1

0.02

0.04

0.06

0.08

0.1

0.12

0.14

k

a

0.02 0.04 0.06 0.08 0.1

0.02

0.04

0.06

0.08

0.1

0.12

0.14

µ

a

Figure 3. Comparative statics: The optimal manager replacement threshold

44

0.18 0.21 0.24 0.27 0.3

4

5

6

7

8

a0

T

0.1 0.12 0.14 0.16

4

5

6

7

8

v0

T

0.02 0.04 0.06 0.08 0.1

4

5

6

7

8

k

T

0.02 0.04 0.06 0.08 0.1

4

5

6

7

8

µ

T

Figure 4. Comparative statics: The expected manager tenure

45

5 10 15 20

0.1

0.15

0.2

0.25

t

a

Figure 5. The optimal replacement threshold: The case of constant ability

5 10 15 20

0.2

0.4

0.6

0.8

1

t

P (t)

Figure 6. The cumulative distribution of manager tenure: The case of constant ability

46