Localized competition in the French mutual fund market:
Participation costs and performance
Linh TRAN DIEU1
(May 2009)
Abstract:
The localized competition observed in French mutual fund market leads to
important participation costs for investors. This reduces the intensity of
performance competition, then gives funds some “market power” and thus
increases the conflict of interest between investors and funds. In this article, we
present a model that takes into account these specificities of the French market.
Using an address model approach, we show that the performance proposed by
funds is lower than their marginal performance corresponding to the management
fees paid by investors and the marginal costs of funds’ performance. The model
yields some testable implications.
Key words: Localized competition, address model, French mutual fund market,
participation costs, performance.
Classification JEL: G10, G20, L20, L10
1
I thank Bellando Raphaëlle, Fontaine Patrick, Gallais-Hamonno Georges, Pollin Jean-Paul, Yusupov
Nurmukhammad for helpful comments.
Laboratoire d’Economie d’Orléans. Faculté de Droit, d’Economie et de Gestion. Rue de Blois, BP 6739, 45067
Orléans Cedex 2. Mail : linh.tran-dieu@univ-orleans.fr
The convex relationship between mutual fund flows and performance found in
literature (Ippolito (1992), Chevalier and Elison (1997), Sirri and Tufano (1998),
Kempf and Ruenzi (2004)), where the best performing funds obtain great inflows
whereas the worst performing funds do not experience outflows, have been
explained in different ways. It is intuitive to understand why high performance
funds have important inflows. However, it is not easy to understand why low
performance ones are not penalized by significant outflows. Lynch and Musto
(2003) explain this by the fact that funds replace “lost” strategies by other ones or
merely change managers. Accordingly, past performance cannot predict the
perspective of funds any more and investors will thus not reward their money
from these funds. Huang, Wei and Yan (2007) explain the fact that low
performance funds do not have significant outflows by the presence of
participation costs. In fact, the investment decision of investors depends on the
difference between expected performance and total costs they have to pay.
Consequently, even if they are attracted by high performance funds, they could
not freely move money from low performance to higher performance funds.
Participation costs defined in their study contain two components: transaction
costs and search costs. Before investing in a fund, investors should seek for
information about the fund, search costs then represent the costs of collecting and
treating information2. Huang, Wei and Yan (2007) argue that greater inflows in
high performance funds are explained not only by the record performance
obtained by these funds, but also by the fact that at this performance level, more
investors could exceed participation cost barrier. If performance is always
considered as a crucial criterion for investors to evaluate funds, the presence of
participation costs can reduce the competition in terms of performance among
funds.
In the case of the French market, participation costs are even more important due
to the specificity of this market. In contrast to the American market, the majority
2
For an investor, search costs are often different across funds, for example, a fund that has important marketing
activities could relatively reduce search costs for investors. Analogously, for a fund, search costs can be different
across investors, it depends on investors’ financial sophisticated degree. In general, search costs are relatively
less important for institutional investors than for individual ones.
2
of French funds are created, managed and distributed by banks3. In addition, in
France, distribution channels are almost integrated in fund families4. On one hand,
this mode of distribution represents some advantages for investors. In fact, many
investors already have a relationship with a bank and thus investing in funds
proposed by the bank appears to be “simpler” than opening a new account in
competing families’ funds. On the other hand, this mode of distribution might
sometimes lead to more important search and transaction costs. It could be more
costly for investors, being clients of banks, to invest in the competing families’
funds. Moreover, banks frequently offer some privileged services to investors as
long as they invest in their funds. Otten and Schweitzer (2002) mention that in
Europe investors seem to value service and convenience at least as much as
performance. Korpela and Puttonen (2005), examining the effect of distribution
channels on fund flows in the Finnish market which has some similar
characteristics in comparison with the French market, show that the existing
customer relationship and convenience contribute to high inflows of funds
managed by banks. Frey (2001), in her study about American bank funds,
demonstrates that banks may attract investors different from those attracted to
non-banks. Specifically, she finds evidence that banks may market more to
individual investors than to institutional ones. In a larger sense, participation costs
borne by investors could be considered as total costs they have to pay once they
decide to invest in a fund. Thus, its can contain not only search and transaction
costs, but also the disutility of non-usage of all privileged services offered by
financial institutions in the case of investing in concurrent families’ funds. Indeed,
clients of a bank, in particular individual investors, tend to invest in funds offered
by their company even if these funds generate a modest performance because
participation costs of the rival families’ funds are too important. As a result, in the
French market, funds seem to be less competitive in terms of performance.
3
82% flows of French mutual funds are realized via distribution channels integrated in banks (BCG-rapport
(2003) and FFSA (2004)). In addition, Otten and Schweitzer (2002) show that European mutual funds
predominantly use banks as the principal distribution channel with a market share of 53%, whereas in the
American market, only 8% of funds are commercialized through banks.
4
A fund family is a group of funds managed by the same management company (e.g. in the case of the French
market, a group of funds created and managed by the same bank is often considered as a family).
3
The specificity of the French market could enhance the conflict of interest
between investors and funds. In fact, if investors would like funds to maximize
risk-adjusted return, the funds’ objective is to maximize total inflows. Although it
is difficult to observe manager’s actions, investors could control funds in the
sense that they could reward their money if funds obtain a low result. However,
participation costs do not always allow them to penalize low performance funds
by moving their money to higher performance ones. Jondeau and Rockinger
(2004) empirically analyze the flow-performance relationship in the case of
French mutual funds market. Their results show that investors are not sensitive to
the past performance of funds. Thus, they conclude that, in the French market,
investors seem to react like “passive” clients. They explain this by the fact that
there exists a “bank bias” effect, issued to the specificity of the French market, in
which investors do not sufficiently diversify across banks’ funds.
Additionally, a part of management fees might not serve to get information in
order to maximize risk-adjusted return of funds, but is used to attract new
investors, such as marketing activities or expenses for brokers. This also can lead
to a conflict of interest between investors and managers. Sirri and Tufano (1998),
Jain and Wu (2000), Khorana and Servaes (2004) and Pagani (2006) argue that an
increase in marketing expenses could reduce search costs for potential investors
and, thus, attract more clients for funds. Of course, these expenses are not usually
desired by existing investors, who would like funds to focus on performance
activities. Consequently this might create another sort of interest conflict between
investors and funds. The French mutual fund market strengthens this type of
conflict because funds can easily pursue this practice since distribution channels
are often integrated in the family.
Using an original approach, particularly “address” model, in which funds differ
not only in terms of performance but also in participation costs, this article
presents a theoretical model where funds determine their performance level
according to the management fees paid by investors and costs borne by funds for
performance services. We show that performance produced by funds could be
inferior to the marginal performance required because participation costs give
funds some “market power”. This model describes French mutual fund market, in
4
which participation costs are higher, the competition in performance would then
be less intensive, consequently funds could exercise their “market power” by
lowering performance without losing their market share. Participation costs
considered in this model are slightly different from those in Huang, Wei and
Yan’s (2006) model. They contain not only search and transaction costs but also
the disutility of non-usage of all privileged services offered by banks to their own
clients. This could also be characterized as the total costs borne by investors to
transfer cash from one financial institution to another one.
Massa (2003a, 2003b) argue that fund families, in the American market, try to
reduce competition intensity by changing their structure. Precisely, fund families
choose either to increase product differentiation in order to reduce performance
competition intensity or to focus on performance services if their performance is
not too low in comparison with their rivals. In other words, in the American case,
funds would intervene in the performance competition intensity. In the French
situation, the market specificities make performance competition naturally less
intensive, funds would thus take this advantage and propose a relatively lower
performance level.
The rest of the paper is organized as follows. In section I, we present a general
address approach. This section reports some characteristics in which market
competition might be considered as a localized competition. Also we bring up the
link between localized competition and the French mutual fund market. Section II
explains the construction of the model, where a symmetric equilibrium is
presented. The address approach is used to construct mutual fund demand side. A
symmetric equilibrium performance will be determined as a solution of fund
profits’ maximization. The conditions of the existence of such a symmetric
equilibrium are examined in section III. Section IV analyzes proprieties of the
long-run equilibrium in comparison with the welfare optimum. Some direct
empirical issues are presented in section V. Finally, we conclude with some
principal results of the model.
5
I. Address model and localized competition approach.
This section presents the link between French mutual fund market and localized
competition. A brief introduction of localized competition and address model is
also brought up.
Competition in mutual fund industry is a particular case of product differentiation
competition studied in industrial economies, where there exist two principal
approaches: non-localized competition (Chamberlin (1933)) and localized
competition (Kaldo (1935)). Chamberlinian approach suggests that a product is in
competition with all other products in market. If products are homogeneous, the
difference between two products is their price. Competition determines then the
equilibrium price (Bertrand model). Kaldor (1935) argues that a product is in
more intensive competition with its “closer" products, rather than with all other
products in the market. Indeed, competition is, in this case, considered as
“localized”. A classical example of localized competition is the competition
among supermarkets. A consumer would choose a supermarket which not only
offers various types of products with reasonable prices, but also because it is close
to his home, given the transportation costs. Thus, a consumer might prefer a
proximity supermarket which proposes products even with a relatively superior
price to a further supermarket because of transportation costs. This proximity
supermarket is then in direct competition with other supermarkets localized in the
same area. The above example highlights the concept of localized competition.
The specificities of French mutual fund market explained in the introduction refer
to some characteristics of localized competition. Participation costs mentioned in
mutual fund competition could be considered as transportation costs in the case of
supermarket competition. Yet, in the case of mutual fund market, the term
“localized” should not be related to the geographic area. Instead, it could be
simply understood that investors seem to be attracted by funds proposed by their
banks. Localized competition of the French mutual fund market presents
especially in the case of individual investors, who have, in general, more
important search and transaction costs relatively to institutional ones. Thus, being
clients of banks, they tend to invest in funds offered by their banks, unless funds
6
proposed by competing families generate a record performance. This rejoins the
argument proposed by Huang, Wei and Yan (2007).
An “address” model is frequently used to explain demand side in localized
competition. It is defined as one in which both commodities and consumers can be
described by a particular point, or address, in a characteristics space. In the case of
mutual fund market, funds can be considered like substitute products that are
different in many characteristics such as investment categories, risk level, fund
family. Then, each fund has its own address determined by these characteristics.
The address of an investor is an optimal point determined by funds’
characteristics. In this article, we are only interested in one dimension of funds’
characteristics, participation costs. Thus, funds are assumed to be homogenous in
all characteristics, but participation costs.
Let us take a simple example, in which two funds are assumed homogenous in all
characteristics but are offered by two different banks (A and B). An investor, who
is a client of bank A, is a priori “closer” to the fund offered by bank A than the
one proposed by bank B. The “address” of the investor is then closer to the
“address” of the fund of bank A than the one of the fund offered by bank B.
Participation costs are considered as “transportation costs” borne by investors to
go to funds’ address. Shorter is the distance between investor’s address and fund’s
address, lower are the participation costs. If two funds are homogenous, then
investors will prefer the “closest” fund, provided that the difference between two
performance levels does not matter.
Hotelling’s model is a type of address model that is often used to explain demand
side in localized competition. In this model, commodities and consumers are
supposed to be distributed in a circle. For commodity, the distance between two
variants is assumed to be equal and consumers are supposed to be uniformly
distributed in the same circle. Consumers choose a variant according to the
difference between price and “transportation costs”. It is demonstrated that a
variant is in competition with at most two other variants. Transportation costs are
often in quadratic form of the distance between the consumer’s address and the
variant’s one. Mathematically, this is the square of the Euclidean distance between
the consumer’s ideal point and the location of the variant. In this paper, we adopt
7
a simple version of circle model and a more general form of transportation costs
to model the demand side of mutual fund market.
II. The Model
Participation costs would reduce the intensity of competition in terms of
performance among funds. The flow-performance relationship mentioned in
literature suggests that even modest performance funds could maintain their
market share. This produces some “market power” for funds and increases the
conflict of interest between investors and managers. Theoretically, funds have to
generate a “marginal” performance corresponding to management fees, paid by
investors and marginal costs supported by funds. However, in the presence of
participation costs, funds might use a part of management fees to cover
commercialisation activities in order to attract new investors and thus produce an
inferior performance without having market loss. In this model, we try to
determine a symmetric equilibrium performance chosen by funds in the situation
where participation costs are not negligible. For this, we first describe demand and
offer sides, symmetric equilibrium performance will then be determined by the
maximization of funds’ profit.
A. Demand side
The “address” model approach is utilized to model investors’ demand for mutual
funds. Some assumptions are necessary for the construction of mutual fund
demand side in an address model.
Assumption 1: We assume that funds are homogenous in all characteristics but
participation costs. Thus, the investment decision of investors depends on the
difference between expected performance and participation costs.
Assumption 2: For commodity, it is assumed that funds are equally localized on a
straight line, the distance between two funds is equal to L. The distance among
funds is not a geographical distance, it can rather be interpreted as a difference in
participation costs of funds. Recall that, for a given fund, participation costs vary
across investors (i.e. individual investors bear more participation costs than
institutional ones). For a given performance level, investors would invest in the
8
“closest” funds. The distance between two funds is assumed to be exogenous.
This hypothesis will be relaxed later when socially optimal equilibrium is
compared with the free-entry optimum.
i 1
i 1
i
L
L
Assumption 3: Investors are assumed to be uniformly distributed on the same line
with a density, noted by a. This assumption does not mean that all clients of a
financial institution have the same localization. In fact, different clients of a given
bank, for example, could have a different “distance” relative to the fund proposed
by the bank. The localization of investors depends on many characteristics of
investors such as: the degree of financial sophistication of investors. Thus, ceteris
paribus, the “address” of a sophisticated investor is different from that of a
“naïve” investor even though both of them are clients of the same financial
institution.
Assumption 4: It is also assumed that each investor invests only in a fund with an
identical amount normalized to 1.
Since the decision of investors depends on the difference between anticipated
performance and participation costs, a marginal investor is indifferent between
fund i and fund (i+1) if the following equation is satisfied:
ri   ( x  zi )   ri 1   ( zi 1  x ) 
(1)
Where ri is the expected performance of fund i. It is assumed that investors use
past performance to anticipate the performance of funds. Participation costs of this
marginal investor when she invests in fund i is  ( x  z i )  . We utilized a general
form of “transportation costs” proposed by Anderson, De Palma and Thisse
(1994), in which x is the location of the investor, z i is the location of fund i.
Hence, x  z i is the distance between the investor and fund i. We assume that
  1 so that participation costs are convex. Other things equal, an investor, who
is a client of a bank, is a priori “closer” to funds proposed by his bank than
9
competing families’ funds. Hence, the convexity of participation cost function
would imply that it is very costly for investors to move from a financial institution
to other ones. The parameter of participation cost intensity,  , is supposed to be
positive. The greater is  , the higher are participation costs. It is obvious that all
investors located between x and z i will invest in fund i.
Analogously, a marginal investor, who is indifferent between fund i and fund (i1), has a location x satisfying:
ri   ( z i  x)   ri 1   ( x  z i 1 ) 
(2)
All investors located between x and z i will invest in fund i. We will demonstrate
later that a fund is in competition with at most two other funds. Thus, the demand
of fund i contains all investors located between x and x .
Finally, demand of fund i is determined as follow:
Di  a( x  x)
(3)
Where x and x are respectively determined by equation (1) and (2).
B. Supply side
Fund i’s profits are determined as:
 i  ( pi  li  g i  ci ri ) Di  K
(4)
Where p i are management fees, l i are load fees, g i is the marginal cost of nonperformance services, such as expenses for the functioning of funds or
commercial activities, ci is the marginal cost of fund i to generate one level of
performance. Finally, K is fixed costs.
In general, management fees are in percentage of asset under management.
Theoretically, management fees are used not only to cover the costs of
performance services, but also to guarantee the functioning of funds. Accordingly,
10
a part of management fees could serve to pay for non-performance services5 such
as conservation of actions, publication of asset value, accountancy, audit. The
functioning expenses can represent more than 50% of management fees declared
by funds6. Other expenses that are not related to fund’s operation like the
remuneration of distribution channels or expenses for marketing activities are also
partially financed by management fees. Thus this could violate the principle of
fees equality across shareholders and increase the conflict of interest between
investors and funds. It means that money of existing investors must not serve to
pay for “new” investors (e.g. expenses for marketing activities may reduce search
costs for “new” investors). Knuutila, Puttonen and Smythe (2006) find an
important effect of distribution channels on the inflows of the Finnish mutual fund
market. They show that for funds distributed by bank channels, investors seem not
to react to past performance. Thus, instead of focusing on performance services,
funds could have incentives to spend for commercial activities. Such a situation
especially reflects the specificities of the French market, in which many fund
families are subsidiaries of banks, which in turn charge to distribute funds7.
Theoretically, load fees must cover costs incurred in the reconstruction of the fund
portfolio, issued to redeeming or purchasing fund shares, and partially finance
distribution channels.
C. Symmetric equilibrium performance
Equilibrium performance is a resolution of a non-cooperative game. A pure
strategy of each fund is its performance level, funds’ payoff is their profits
determined by equation (4). In equilibrium, each fund chooses its performance
level in order to maximize its profits given the performance levels chosen by other
funds.
However, we aim to find only a symmetric Nash equilibrium where each fund
chooses the same performance level and the performance chosen by a fund is the
5
This ambiguity might lead to flawed comparison of management fees across funds.
6
Source: COB rapport (2002).
7
In reality, about 2/3 market assets are distributed by banks and 10 greatest bank groups represent 60% market
(Source: Europerformance (2003)).
11
best reply to the performance proposed by other funds. The conditions of the
existence of such an equilibrium will be presented in the next section. In this
section, we suppose the existence of such a symmetric equilibrium and then solve
for the equilibrium performance.
Each fund solves the following program :
Max  i   pi  li  g i  ci ri Di  K
ri
(5)
With Di  a( x  x)
Proposition 1: A symmetric equilibrium performance of funds is determined as:
r 
pi  l i  g i
  21  L
ci
(6)
Proof: see Appendix A.
The proposition 1 leads directly to the proposition 2.
Proposition 2: Since  21  L  0 , the equilibrium performance, r  , is always
inferior to the marginal performance of funds,
pi  l i  g i
.
ci
The symmetric equilibrium performance determined by equation (6) shows that
the performance level chosen by funds depends on two principal sources:
marginal performance (which is equal to
pi  l i  g i
) and participation cost term
ci
(reflected in  21  L ).
Intuitively, in the presence of participation costs, funds could propose an
equilibrium performance inferior to their marginal performance without losing
their market share. This suggests that participation costs strengthen the “market
power” of funds and weaken the competition among funds.
12
The first term of the right-hand side of equation (6) suggests that performance
increases with net expenses for performance service, ( pi  li  g i ) . Accordingly, a
fund that charges significant management fees, does not necessarily generate high
performance since a part of management fees is used to finance non- performance
services. The higher are the expenses for non-performance services, the lower is
the equilibrium performance. If some expenses for non- performance services are
essential for the functioning of funds, commercial activity expenses, on the other
hand, are not always desired by “existing” investors. However, it is not easy for
investors to verify whether funds use correctly management fees. If in the case of
the American market, funds have to declare their percentage of expenses
transferred to distribution channels, in France, this information is not always
accessible for investors.
In contrast, the equilibrium performance of funds decreases with the marginal
costs to produce one level of performance. High marginal costs mean that
performance services become more expensive, thus funds have to either reduce
their performance level, or increase management fees. Both ways are not desired
by investors and could lead to a market loss for funds. Nevertheless, it is not easy
to measure marginal costs of funds. In reality, the marginal costs of funds’
performance might depend on many characteristics of funds. For instance,
performance costs can be different according to investment categories of funds,
the ability of fund managers or the size of fund families. In general, funds
belonging to big families, might take advantages of economies of scale or
“learning by doing” to reduce the costs of their performance. As a result, these
funds could offer higher performance to investors without reducing expenses for
non-performance services.
The second term of the right-hand side of equation (6) implies that funds can
reduce their performance inferior to their marginal performance thanks to
participation costs. The greater is the distance among funds (L), the lower is the
equilibrium performance. Intuitively, if switching costs from a financial
institution to another one are high, investors will not want to move their money.
Therefore, funds would take advantage of the situation to lower their performance.
On the contrary, when L tends to 0, it means that funds are so “similar” in terms
13
of participation costs, the competition in performance thus becomes more
intensive. Consequently, funds must produce a performance level close to the
marginal one if they do not want to lose their market share. Performance
decreases also with the intensity of participation costs (  ) and the degree of
convexity of participation cost function (  ). The effect of these parameters is
similar to the one generated by the distance. To sum up, when participation costs
are not negligible, funds can under-perform without losing their market share.
III. Existence of symmetric equilibrium
In this section, we examine conditions for the existence of such a symmetric
equilibrium. The method used for analyzing the existence of equilibrium
performance is similar to the one used in a classical address model (See Anderson,
De Palma and Thisse (1994)). First, we demonstrate that a fund is in direct
competition with at most two funds. Second, we determine the interval of fund i’s
equilibrium performance. Finally, we determine the equilibrium strategy chosen
by fund i. If r  is the unique best response of fund i for r  chosen by other funds,
we can conclude that the symmetric Nash equilibrium exits.
It is obvious that a fund does not want to produce a performance level superior to
its marginal one,
pi  l i  g i
, because in this case it will generate negative
ci
profits. Indeed, we can demonstrate a classical result of an address model,
translated in our model as below:
Proposition 3: A fund does not have incentives to take all market share of its two
closest neighbors. In other words, a fund is in direct competition with at most two
funds.
Proof: See Appendix B
Moreover, we can determine the interval of the equilibrium performance proposed
by fund i. Our intuition is that a fund could not freely reduce their performance
inferior to a certain level without losing their market share even though
participation costs are important. Similarly, a fund would not like to raise
14
significantly their performance to attract clients of other funds because profits
made by an increased market share would be inferior to expenses for a
supplementary performance level.
Proposition 4: The best reply performance chosen by fund i given that all other
funds choose r  must belong to [r   L , r   L ] .
Proof: See Appendix C
For each level of performance proposed by fund i, there exists only one position
of a marginal investor, who is indifferent between fund i and (i-1). Therefore,
determining ri is equivalent to determining the position of this marginal investor.
Consider the distance between the marginal investor and fund i, l  z i  x  hL .
From the proposition (4), r   [r *  L , r *  L ] , the marginal investor must
locate between fund i and fund (i-1), thus 0  h  1 and x  z i 1  1  h L .
Proposition 5: The position of the marginal investor determines all the roots of
the H (h)  0 in [0,1]:
H (h)  (   1)h   h(1  h)  1  (1  h)    21  0
(7)
Proof: See Appendix D.
It is obvious that the position of the marginal investor only depends on the degree
of the convexity of participation cost function,  . Moreover, it follows that h =
0.5 is a root of H (h)  0 , which means that there exists a marginal investor
located exactly at the middle between fund i and fund (i-1). This implies directly
that r  is one of the best responses of fund i to r  chosen by other funds.
However, it might not be the unique best reply of fund i since H (h)  0 can have
more than one solution. Thus, the existence of the symmetric Nash equilibrium
depends on the condition at which H (h)  0 has a unique root in [0,1]. The detail
of this analyze is presented in the appendix E, we summarize here principal
results.
15
Proposition 6: For 1    5,9688 H (h)  0 has an unique root, h=0.5, this means
that the symmetric Nash equilibrium exists. For   5,9688 , H (h)  0 has at
least two solutions, consequently r  is not the only best reply of fund i to r 
chosen by other funds. In other words, such a symmetric equilibrium does not
exist.
Intuitively, for sufficiently high value of  , participation costs become too
expensive for investors. Consequently, a fund could lower their performance
under the “equilibrium” performance without reducing much its demand.
IV. Long-run vs. welfare optimum equilibrium
In this section, we characterize the long-run equilibrium in comparison with the
welfare optimum. So far the distance between two funds has been exogenous. In
this section, L is endogenously determined in the long-run equilibrium and in the
welfare optimum. Precisely, we compare the long-run “distance” between two
funds with the socially optimum. This allows us to better understand the behavior
of funds in long term. Indeed, in the long run equilibrium, funds would entry in
the market until their profits equal to zero.
The socially optimal distance between two funds is obtained by minimizing the
total resource cost borne by investors and funds. The total costs are determined as:
L/2
TC  K  ( g i  ci r )aL  2a  x  dx

(8)
0
The first two terms of the right-hand side of equation (8) are the costs borne by
fund i. The last term is the participation costs of investors. Since in the symmetric
equilibrium, the demand of fund i is aL, ( g i  ci r  )aL is the variable costs. As
investors will invest in their closest fund and there are aL/2 investors, located
between fund i and fund (i-1) will invest in fund i, the participation costs born by
L/2
these investors are: a  x  dx , where x is the distance between an investor and
0
fund i. By symmetry, the participation costs of aL/2 investors located between
16
L/2
fund i and fund (i+1) are a  x  dx . Accordingly, the total participation costs of
0
L/2
investors are 2a  x  dx . Per unit of market area, the total resource cost, which
0
we denote by TC (L) , is:
TC ( L) 
K
a
 ag i  aci r   
L
L
2 (   1)
(9)
Using r  from equation (6) in (9), we get:
TC ( L) 
K
a
 ( pi  li )a  
L  aci  21  L
L
2 (   1)
(10)
The first order condition:
TC ( L)
K
a
0 2  
L 1  aci  2 21  L 1  0
L
L
2 (   1)
The socially optimal distance between two funds is thus:

K

LO  
ac  21 

 i
1 /(  1)
1

 
1

  

 2(   1)ci
 

(11)
In the long-run equilibrium with free-entry, funds generate a zero profit:
  0  ( pi  li  g i  ci r  ) La  K  0
The distance between two funds in the long-run equilibrium, LF , is:

K
LF  
1 
 aci  2
1 /(  1)



(12)
The distance among funds in the long-run equilibrium increases with the fixed
costs of funds and decreases with the marginal costs of performance and the
parameters of participation costs. In fact, if fixed costs are significant, the number
17
of funds offered to investors would be smaller and the distance among funds
would be greater. Consequently, the competition in terms of performance is less
intensive. On the contrary, a low marginal cost of performance enhances the
competition in performance among funds. Since no funds would like to compete
in performance, they try to maintain their “market power” by keeping a
sufficiently high distance from each other. Similarly, the density of investors
reduces the long-run distance. In fact, when the market size increases, more funds
would be offered to investors.
Let us now compare the long-run equilibrium distance with the welfare optimal
one. From (11) and (12) we have:

LF 
1
 
  
LO  2(   1)ci

It follows that
1 /(1  )
(13)


LF
1
1
. It leads straight
 1  
    1  ci 
2
LO
21   
 2  1ci

forward to the following proposition.
Proposition 7: The long-run equilibrium distance is greater than the welfare
optimal one if ci  1 / 2(1   ) 2 .
Intuitively, if marginal costs of performance services are sufficiently low given
the “power” of participation costs,  , it is less expensive to produce performance.
Therefore, funds would have more incentive to generate higher performance, as a
result it would strengthen the competition in performance. In order to maintain the
“market power”, funds would keep their distance above the welfare optimum
level.
From (13) we have:
 LF

 LO



 1

1
 .
2(   1)ci
(14)
18
It follows that the right-hand side of equation (14) is a decreasing function of  8.
For  sufficiently great so that LF
LO
is inferior to 1, the long-run equilibrium
distance could decrease to the level below the social optimum. Intuitively, when
participation cost “power”,  , is too high, the participation barriers become very
important so that even with a small distance among funds, they could still keep
their “market power”. Consequently, funds could reduce the distance below the
welfare optimum level without raising the intensity of performance competition.
V. Empirical implications
In this section, we discuss some of the model’s empirical implications:
Implication 1: The role of participation costs in investors’ decision could be
empirically analyzed in the sense that individual investors, who have relatively
more important participation costs, would less actively react to performance.
The model presented in this paper describes especially the situation in which it is
very costly for investors, being clients of banks, to invest in competing families’
funds. Jondeau and Rockinger (2004), cited in the introduction, have empirically
analyzed the response of investors to past performance in the French mutual fund
market. They show that investors seem not to be sensitive to past performance of
funds and conclude that investors react like passive clients. In contrast to
individual investors, institutional investors, who have relatively smaller
participation costs, can react more actively to past performance of funds. Del
Guercio and Tkac (2002) note that in the case of pension fund market, where most
clients are institutional investors, poor performance funds are punished by
significant outflows.
Implication 2: Do funds managed by banks under-perform in comparison with
non-bank funds?
8
The first derivate is 
1
2ci (   1) 2
1  0
19
Theoretically, bank funds have relatively more “market power” than non-bank
funds, which allows them to reduce their performance without losing their market
shares. However, some empirical results from the U.S market as well as the
European market are quite mixed. For instance, Frye (2001) shows that in the
American market, bank managed mutual funds do not under-perform relative to
non-bank funds. Whereas, Korkeamaki and Smythe (2004), in their study on the
Finnish market, find that bank managed funds charge higher expenses but
investors are not compensated with higher risk-adjusted returns.
Implication 3: Investors might tend to switch from funds to others belonging to
the same family rather than moving their money to competing families, because of
high participation barrier.
Yet, this implication is not easy to empirically verify since we do not always have
access to data in which each movement of investors is observed. Instead, previous
studies deal with this question by verifying whether high performance funds of a
family can experience more inflows (Kempf and Ruenzi (2004), Jondeau and
Rockinger (2004)). However, this method sometimes leads to some biased
conclusions since the best performing funds of a family are often “stars” of
market. Consequently, inflows in these funds are usually issued not only from
“existing” investors of the family but also from clients of competing families as
record performance allows these investors pass participation barrier. Additionally,
Massa (2003a) suggests that each family responds to investors’ heterogeneity by
offering various types of funds. It allows the family to avoid moving money into
rival families.
Implication 4: The performance mark-up of funds, defined as the difference
between the performance generated by funds and their marginal performance, can
be different with respect to client types.
We can imagine that funds which serve individual investors could have a larger
performance mark-up than institutional funds, as individual investors bear higher
participation costs than institutional ones. However, performance mark-up is not
easy to calculate, it might be more convenient to simply compare performance
across funds. Yet, the empirical evidence on this issue needs to be interpreted with
20
some care because institutional funds often have lower marginal costs thanks to
economies of scale, so higher absolute performance of institutional funds might be
a result of their lower marginal costs due to economies of scale.
Implication 5: Funds’ market power may lead to, ceteris paribus, relatively
inferior average performance in French in comparison with the American market.
However, it would be complex to verify and interpret this implication because
funds are considered in different context. Instead, previous studies have tried to
verify whether funds could beat the market. In the American market, Grinblatt and
Titman (1992), Ibbotson and Goetzmann (1994) and Brown and Goetzmann
(1995) find an evidence of “winner” repetition. On the contrary, Bergeruc (2001),
Aftalion (2001) and De Marchi (2006) show that French mutual funds do not
generate results superior to market return9.
Implication 6: Ceteris paribus, funds charging high management fees should
generate high performance. This implication follows directly from equation (6).
Implication 7: Given an amount of net expenses on performance activities,
different funds might produce different performance; this depends on the marginal
costs of funds’ performance.
How to understand the marginal costs of funds’ performance? On one hand, the
marginal costs of performance can depend on the ability of funds’ managers.
Funds that are managed by experience and competent managers could invest more
efficiently and thus obtain high performance. On the other hand, the marginal
costs of performance could depend on funds’ characteristics like size of the
family, size of funds, or the investment specialty of family. For instance, funds
that belong to a big family might have relatively lower marginal costs thanks to
9
Others argue that French funds should beat the market as they are managed by banks, which have possibility to
access to non-public information of enterprises and thus use this information to make abnormal return
(McDonald (1973)).
21
economies of scale10. Accordingly, they could offer higher performance without
imposing greater management fees.
Implication 8: The principal-agent problem could be empirically analyzed in the
sense that management fees could not be properly used by funds.
In fact, funds might spend an sufficient amount on commercial activities in order
to experience more inflows even if it leads to modest performance. Gruber (1996)
and Cahart (1997) show that higher expenses are associated with inferior rather
than superior management. As a result, investors should compare net management
expenses for performance activities across funds rather than “gross” management
fees.
Implication 9: Even though investors are interested in “price-quality”,
performance still plays a crucial role in their decision. Fund families could use it
as a commercial argument by reducing management fees of certain funds so that
these funds could display high performance, which is, in reality, calculated after
subtracting management fees from gross result of funds.
Moreover, fund families might sometimes use fees collected by a fund to finance
other funds’ activities. This “cross-fund subsidization” strategy allows the family
to have some “flagship” funds which can attract more attention of investors.
Guedj and Papastaikoudi (2005) and Gaspar, Massa and Matos (2006) argue that
families strategically “transfer” performance across member funds to favour those
that are more likely to increase overall family profits.
VI. Discussion and Conclusion
This model has been developed to describe the specificities of the French mutual
fund market. These specificities are reflected in the fact that French funds have
relatively more “market power”, due to higher participation costs borne by
investors. This allows funds to lower their performance below their marginal level
10
This proxy needs to be used with some care since big families propose in general different types of funds.
Also, there is a trade-off with the advantage of “learning by doing”, due to the investment deepening in one
category and the risk diversify, issued to the different types of funds proposed to investors. Hence, performance
marginal costs of big families might not necessarily be lower than the one of other families’ funds.
22
without being penalized. In this model, we adopt an original approach, the address
model, to explain the demand of funds. This allows us to take into account the
characteristic of localized competition in the French market. Finally, this model
may also be applied in some European markets like Spain or Finland, where
mutual fund distribution channels are dominated by banks.
This model describes a situation, in which, being clients of a financial institution,
it is very costly for investors to invest in competing families’ funds. What happen
with investors who would like to invest in mutual funds but they are not clients of
any banks or they have different accounts in different financial institutions? How
can we place these types of investors in the localized competition? For instance,
Huang, Wei and Yan (2007) distinguish two types of investors: “new” investors,
who never invest in the fund considered and “existing” investors, who already
hold their shares in the fund. The latter type is assumed to incur lower search costs
than new investors do. Nevertheless, this distinction is not necessary in our model
because it is supposed that investors are able to compare their participation costs
once they decide to invest in funds. Therefore, investors could feel “closer” to
certain funds than others. Indeed, the decision of investors always depends on the
difference between anticipated performance and participation costs.
23
Appendix A
We will determine the symmetric equilibrium performance r * .
Let us suppose that all funds but i choose the same performance level r . The
marginal investor located at x is indifferent between fund i and fund (i-1) if x is a
solution of equation (2) rewritten as:
ri   ( z i  x)   r   ( x  z i 1 ) 
Take the total derivative of the above equation ( r fixed):
dri   ( z i  x)  1 d x    ( x  z i 1 )  1 d x
The above equation leads straight forward to:

dx
1

( z i  x)  1  ( x  z i 1 )  1
dri


1
(A.1)
Moreover, by symmetry, we have zi  x  x  zi . The demand of fund i is then
equal to:
Di  a( x  x)  a[( x  zi )  ( zi  x)]  2a( zi  x)
The profit of fund i is as follow:
 i  2a( zi  x)( pi  li  g i  ci ri )  K
Assume that the second order condition is satisfied, the first order condition leads
to:
 i
dx
 0  2a
( pi  li  g i  ci ri )  2a( z i  x)ci  0
ri
dri
(A.2)
A symmetric equilibrium in which fund i generating the same performance level,
r , implies that
24
r   ( z i  x)   r   ( x  z i 1 )  .
It means that the marginal investor is located exactly in the middle between fund i
and fund (i-1). This leads to:
z i  x  x  z i 1 
L
2
(A.3)
By plugging (A.1) and (A.3) into (A.2), we have:
2a
1

[2( L / 2)  1 ]1 ( pi  li  g i  ci r )  2a
L
ci  0
2
(A.4)
Solving directly for r from equation (A.4) leads to:
r 
pi  l i  g i
  21  L
ci
▄
Appendix B
We will show that a fund is in competition with at most two funds.
In fact, to take all market shares of two neighboring funds, fund i must propose a
performance at least equal to:

3 
L
r   L    
2 
2


L / 2 i 1
i2
i
(B.1)
i 1
A
i2
B
3L / 2
3L / 2
At this performance level, the furthest investor (A) of fund (i-1), located in the
middle between fund (i-2) and fund (i-1), is indifferent between fund i and fund (i1).
We can verify that this performance is higher than marginal performance. By
plugging r  from equation (6) into equation (B.1), we will demonstrate that:
25


pi  l i  g i
p  l  gi
3 
 L
  21  L    L       i i
ci
ci
2 
2

.L
2

(3   2  1)  0
The equation (B.2) is always hold for   1
(B.2)
▄
11
Appendix C
We will demonstrate that the equilibrium performance of fund i, ri , should belong
to the following interval [r   L , r   L ] .
If fund i chooses a performance inferior to r   L , it will have a zero demand
since the closest investors of the fund, locating at the same point of the fund,
would like to invest in one of two neighboring funds (fund (i-1) or (i+1)) as
performance proposed by fund i is too low. As a result, fund i should propose a
performance ri  r   L .
Now let us show that fund i does not want to increase its performance superior to
r   L since in this case it will generate a profit lower than the profit obtained if
it chooses r  . Intuitively, r   L is the performance level, at which fund i could
certainly attract all investors located between fund (i-1) and fund (i+1). If fund i
proposes a performance superior to r   L , the maximum market share that
fund i could obtain are 3La, while its performance has to be at least equal
to r   L .
The maximum profit generated in this case is:
 1  3La[ pi  li  g i  ci (r   L )]  K
11
Note that
(C.1)
g ( )  3  2  1 is an increasing function as its first derivate, g   3  ln( 3)  2 , is always
superior to zero with
  1 . Moreover, we have: g (1)  0 . As a result, 3  2  1  0
for
  1.
26
We show that this profit is lower than the profit generated with r  , which is equal
to:
 2  La( pi  li  g i  ci r  )  K
(C.2)
Using (6), (C.1) and (C.2), we can calculate the difference between two profits:
 2   1  La(ci  21  L )  3La(ci  21  L  ci L )
 L 1aci  (3  2 2   )
We have ( 2   1 )  0 since (3  2 2   )  0 for   1
12
.
Finally, the best reply performance of fund i, ri , to r  chosen by other funds should
belong to [r   L , r   L ] .
▄
Appendix D
From (A.1) and (A.2):
( pi  li  g i  ci ri )   [( z i  x)  1  ( x  z i 1 )  1 ]( z i  x)ci
(D.1)
Moreover, a marginal investor is indifferent between fund i and fund (i-1) if the
following equation is hold:
ri   ( z i  x)   r    ( x  z i 1 ) 
 ri  r    ( z i  x)    ( x  z i 1 ) 
 ri 
12
pi  l i  g i
  21  L   ( z i  x)    ( x  z i 1 ) 
ci
Let us analyse the function :
f (  )  2 2  (  ln 2  1) ,
f ( )  3  2 2  . The first derivate of this function is equal to
which
is
f (1 / ln 2)  211 / ln 2 ln 2  0 . Consequently,
Otherwise f min  f (1 / ln 2)  3 
equal
to
f (  ) has
zero
a
at
unique
  1 / ln 2 .
minimum
at
Moreover,
  1 / ln 2 .
1 ( 21 / ln 2)
2
 0 . This leads to (3  2 2   )  0 .
ln 2
27
Replacing ri in equation (D.1) and eliminating  in the two hands of the above
equation we have:
 21  L  ( z i  x)   ( x  z i 1 )    ( z i  x)[( z i  x)  1  ( x  z i 1 )  1 ] (D.2)
Recall that l  z i  x  h.L and x  z i 1  1  h L .
From (D.2), we have:
(   1)h   h(1  h)  1  (1  h)    21   0
(D.3)
Set H (h)  (  1)h   h(1  h)  1  (1  h)    21  . Thus, determining the best
reply performance of fund i is equivalent to determine all the roots of H (h)  0
▄
in [0,1].
Appendix E
Let analyze numerically the below function:
H (h)  (  1)h   h(1  h)  1  (1  h)    21 
with   1 and h  [0,1]
It is easy to find that h=0.5 is a solution of the function H(h)=0. Plot the function
H(h) for different values of beta with 2-11 step (The below figure illustrates the
function H(h) for some values of  ). We can find that for 1    4 , H(h)=0 has
only a solution h=0.5. For   8 , H(h)=0 has more than one solution. We will try
to find  0  [4,8) at which H(h)=0 has more than one turning point. Beside, we
have H (h)  0 for all h  0.5 and   1 . Then, if the function has more than one
solution, there must be a solution that belongs to (0, 1/2). Moreover,
H (0)  (1)    21   0 for all positive values of beta, thus  0 is the value at
which there exist a value
h   (0,1 / 2)
satisfying
H (h  )  0
since if
H (h  ) H (0)  0 there always exists a solution belonging to (0,1/2). Programming
with Matlab, we obtain  0 equal to 5,9688. We can thus conclude that for 
superior or equal to 5,9688 the function H(h)=0 has at least two solution. For 
inferior to 5,9688, H(h)=0 has only one solution h=0.5.
28
beta = 1
beta = 4
0.5
0.5
0
0
-0.5
0.4
0.5
0.6
0.7
-0.5
0
0.2
beta = 8
0.5
0
0
0
0.2
0.4
0.6
0.8
0.6
0.8
beta = 10
0.5
-0.5
0.4
0.6
0.8
-0.5
0
0.2
0.4
Figure: Function H(h) for some values of 
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