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Activism and Indexing in Equilibrium∗
Steven D. Baker† David A. Chapman‡
Michael F. Gallmeyer§
March 15, 2020
Abstract
We study a general equilibrium production economy with a financial sector that con-tains both actively managed and passively managed funds. Activists in the activelymanaged sector are socially valuable because they improve productive efficiency bymonitoring firm managers; i.e., through the use of “voice.” Quants in the activelymanaged sector displace activists by buying previously identified efficient firms. A rep-resentative household allocates its wealth to activist, quant, and index funds. Giventhe model structure, we conduct a series of experiments to characterize the impacton the equilibrium of changes in the matching function between active managers andfirms and changes in the costs of intermediation. For a wide range of parameters, anon-degenerate stationary distribution for each manager type exists. While a decreasein the passive index fee is beneficial for cheaper diversification, it can negatively impactthe overall efficiency of the economy due to the existence of fewer activists.
Keywords: Activism, delegated fund management, index funds
∗We would like to thank Wei Li (discussant), conference participants at the Johns Hopkins Carey SchoolFinance conference, and seminar participants at the University of Melbourne for comments.†The McIntire School of Commerce, University of Virginia, [email protected].‡The McIntire School of Commerce, University of Virginia, [email protected].§The McIntire School of Commerce, University of Virginia and the Faculty of Business and Economics,
University of Melbourne, [email protected].
1 Introduction
If a firm’s managers are destroying shareholder value, an asset manager faces two primary
choices: voice or exit. Voice, or activism, is generally understood to describe a broad range of
shareholder actions designed to influence a firm’s real or financial policy decisions. As such,
it is recognized as one important component of corporate governance. We build a model
of the asset management industry, embedded in general equilibrium, with “activism” at its
heart. Our goal is to understand how activism coexists with alternative asset management
channels, with a particular emphasis on the implications of fee compression for household
welfare and for the relative sizes of the different fund channels.
The foundation of the model economy is a set of productive technologies or “firms.”
The firms are owned by a household sector that is characterized by a representative agent
who can invest in firms through three different types of intermediaries: a passive index,
“activists,” and “quants.” The passive fund allows the representative household access to an
equal-weighted portfolio of all firms at low or zero fees. Its existence captures the intuition
that the household does not require a skilled manager to spread capital evenly across firms,
but the household does required a skilled manager to identify the most efficient firms.
An activist creates a “fund” by incurring a search cost to match with a firm, exercising
“voice,” and forcing the firm to use its capital efficiently. The firm remains efficient as
long as it is matched with an activist. This is the direct benefit of voice. However, the
activist manager also generates a positive externality because the increase in firm efficiency
contributes to increased performance for the passive index. A quant manager can identify
and invest in an efficient firm, but the quant does not engage in using voice. The arrival of
the quant destroys the match between an activist and a firm, and the resulting fund may
operate less efficiently than an activist fund and with different fees. In the absence of a
match with an activist, a firm held by the quant will revert over time, at some rate, to an
1
inefficient use of its capital.1 Both activists and quants choose fees optimally in a symmetric
Nash equilibrium, given the state of competition in the managed fund market.
The household determines the demand for each of the three types of funds by solving an
optimal consumption and portfolio choice problem. In equilibrium, the markets for goods
and fund shares all clear, determining the aggregate size of the sector of each fund type in
the economy. For a wide range of parameter values, there is a non-degenerate stationary
distribution for each manager type in the steady state of the model. Although the equilibrium
characteristics of activists and quants are symmetric in many regards, their responses to some
changes to equilibrium parameters are not necessarily identical. There is a “predator/prey”
component to the dynamics of the relative sizes of the activist and quant sectors that can
lead to some interesting interaction effects.
Given the solution to the model, we are able to perform a number of experiments that
examine the key factors determining the interactions of agents in the model. In particular,
how do search costs and fee structures affect the relative sizes of the active management
sectors and market efficiency? How sensitive are the sizes of active and passive sectors to
changes in household demand for diversification benefits offered by active managers? Finally,
what can we learn about the relationship between active management and household welfare?
In addition to demonstrating that a non-degenerate distribution of activists and quants
co-exists in equilibrium, our baseline parameters demonstrate how fund market competition
impacts assets under management for the household through diversification and spillover
effects. As the number of activist funds grow, the household invests more in the sector overall
as it becomes better diversified. However, more managed funds also make the passive index
more efficient leading to a spillover effect that eventually makes assets under management
in the active sector fall. Quants also generate a spillover effect. If more of them are in
the market, activist assets under management fall even sooner. This dynamic between
1The quant fund is effectively serving the role of the free rider in Shleifer and Vishny [1986]
2
activists and quants also impacts differentially their present value of fees collected. When
quant-owned firms are less efficient than activist-owned firms, quants charge lower fees to
households. However, these lower fees are not necessarily beneficial to households due to a
reduction in the quality of the household investment opportunity set.
The dynamics of the model are particularly sensitive to variations in the cost of interme-
diation. If quant search costs fall too much, say through better technology to detect efficient
firms, the size of the activist sector can crash ultimately leading to a crash in the quant
sector too. This renders the market less efficient leading to lower household welfare. Across
the board increases in the cost of intermediation through a fee on the passive index lead
to an increase in fees charged by activists and quants. This has a direct negative impact
on household welfare, but this increase in fees can also adversely impact the composition of
activists relative to quants again leading to lower household welfare through a reduction in
the quality of the investment opportunity set.
Interestingly, while declining passive index fees and fee compression have become com-
monplace in the asset management industry,2 understanding their role on investor welfare
requires a dynamic assessment of their impact on the incentives for activists to participate
in the asset management industry. While a decrease in the passive index fee is beneficial
to provide cheaper diversification options, it can be harmful to overall efficiency if the re-
duction in fees seriously curtails the number of activists willing to participate in the fund
management industry as we document in a simple example.
In focusing on the role of voice in equilibrium, we have largely abstracted from the
conventional focus of mutual fund and hedge fund research: does active management deliver
alpha through the selection of mispriced securities? We say “largely” because quant managers
do engage in a form of identification of mispriced assets. Our purpose is not to argue that
2See for example the 2019 Morningstar Fund Fee Study available at https://www.morningstar.com/
lp/annual-us-fund-fee-study.
3
the conventional focus is unimportant but rather to argue that activism may play a more
important role in understanding the role of asset managers in general equilibrium than has
generally been acknowledged. This view is consistent with the recent survey evidence in
McCahery et al. [2016] and the empirical literature on activism summarized in Brav et al.
[2015b].
The rest of the paper is organized as follows: After placing our work in the context
of the relevant (large) literature on both shareholder monitoring and active management,
we present the model structure and then describe the equilibrium. We then examine how
the model equilibrium responds to practically relevant shocks, and we conclude in the final
section of the paper.
2 Our Contribution Relative to the Existing Literature
In order for any shareholder (or group of shareholders) to effectively monitor firm managers,
they must solve a free-rider problem; i.e., the costly efforts of shareholders who monitor the
firm also benefit other shareholders who exert no effort. This problem might be overcome
by large shareholders via takeovers, as in Shleifer and Vishny [1986]. Yet ownership that is
too highly concentrated could lead to sub-optimally tight control by shareholders [Burkart
et al., 1997], or to weaker monitoring incentives stemming from reduced liquidity and price
informativeness [Holmstrom and Tirole, 1993]. Assuming that the right number of share-
holders are paying the right amount of attention and providing improved incentives to the
firm’s managers remains a nontrivial problem [Core et al., 2003].
The results in Admati, Pfleiderer, and Zechner [1994] provide theoretical arguments
against the use of voice by demonstrating that the equilibrium level of monitoring is well
below the socially optimal level of monitoring, and DeMarzo and Urosevic [2006] extend this
result to show that, over time, an activist will actually hold a perfectly diversified portfolio.
4
Marinovic and Varas [2019] extend these findings by showing that whether or not the activist
monitors and whether or not that monitoring increases firm value depends critically on the
presence of information asymmetry about the activist’s ability.3
Despite these substantial incentive and information obstacles, McCahery et al. [2016]
present survey evidence that over half of institutional investors engage directly with the
management and boards of the firms in which they hold stock, presumably to the benefit of
shareholders at large. The survey article of Brav et al. [2015b] provides additional empirical
evidence of the impact of activism on firms. In particular, Brav et al. [2015a] show that
activism empirically has a positive impact on plant-level productivity. We reconcile the
theoretical results with the empirical findings by abstracting from the specific details of how
activists change firm policies – and by extension firm value. Instead, we assume a matching
technology that pairs fund managers (both activists and quants) to units of productive
capital. Once the match is formed the monitoring requires no costly effort by managers,
and the representative household has perfect information about which units of capital are
matched to each type of manager. We use voice and monitoring as a metaphor for how
active managers can deliver value to households in equilibrium in the absence of asymmetric
information. We are interested in understanding the factors that affect the relative sizes of
the active and passive sectors of the asset management industry and how the structure of the
asset management industry affects household welfare in general equilibrium under perfect
information.
Pastor and Stambaugh [2012] provides an alternative explanation for the relative sizes
of the active and passive sectors of the fund industry. Their focus is on explaining the
poor performance of active funds relative to a passive benchmark (the “active management
3This additional layer of delegation also introduces new agency problems that have been the focus of alarge literature on optimal managerial contracting; see, for example, the theoretical work of Bhattacharyaand Pfleiderer [1985], Stoughton [1993], Heinkel and Stoughton [1994], Starks [1987], and the empirical workin Almazan et al. [2004].
5
puzzle”) in an equilibrium model where funds face decreasing returns to scale; i.e., when
fund performance erodes because too many managers trade in related strategies. Their
explanation of active vs. passive sector sizes is complementary to our model. Their model
is more general than ours in the sense that “alpha” in their analysis might come from a
variety of manager actions whereas we focus exclusively on voice. Our model is more general
than theirs in that we focus on a production economy and our household explicitly solves
a multiperiod (infinite-horizon) optimization problem. Finally, in Pastor and Stambaugh
[2012], investors are learning about managerial skill while our framework assumes complete
information.
We assume that passive managers provide diversification services at low cost, and they
do not engage in any use of voice or (by definition) other active strategies. Azar et al. [2018]
purport to provide evidence that even passive managers can facilitate collusion in concen-
trated industries, to the detriment of households. However, there is a growing literature that
contradicts the findings in Azar et al. [2018]; see, for example Dennis et al. [2018].
3 Model
We study a continuous-time, infinite-horizon economy in which the fund management indus-
try intermediates between households and productive investment opportunities. We build
the model from the bottom up, beginning with the productive technologies, or firms. In-
vestment managers who are either activists or quants search for opportunities to match with
firms, thereby forming funds. A lower cost passive index fund also exists which invests
equally in all firms. Finally, a representative household optimally allocates capital across
funds. Having closed the model, we solve and analyze fund and capital market dynamics in
general equilibrium.
6
3.1 Firms
Capital may be productively invested in N firms. All firms have identical productivity of
capital µ, such that a firm j with capital Kj,t produces output at gross rate µKj,tdt. However
capital in firm j also depreciates at a rate
Kj,t
[−δj,tdt+ σdW t + σdWj,t
]. (1)
The Brownian motion W t captures a capital depreciation shock common to all firms, whereas
the Brownian motion Wj,t captures firm-specific depreciation, which is independent of W t
or Wi,t, i 6= j. At any given time, the deterministic depreciation rate δj,t takes one of three
values. When a firm is in an activist fund, depreciation takes its lowest value δj,t = δA,
reflecting efficiency gains from monitoring. When a firm previously monitored by an activist
becomes part of a quant fund, depreciation becomes δj,t = δQ ≥ δA, allowing for an immediate
drop in efficiency following cessation of activist scrutiny. A firm that is not part of any
actively managed fund is assumed to be inefficiently run, with δj,t = δU > δQ. This simple
reduced-form specification captures the idea that active management produces economic
benefits through monitoring, particularly in the case of activists.
3.2 Funds
Activists seek out inefficient firms to monitor, thereby rendering them efficient. Firms are
efficient as long as they are combined with an activist. How these managers effect change
or “voice” is modeled in reduced form. However, we have in mind mechanisms for voice as
discussed in the survey evidence in McCahery et al. [2016]. We also assume that activism
is always successful. If an activist chooses to invest in a firm, that firm is always efficiently
7
run as modeled by the lower depreciation rate δj,t = δA.4
By contrast quantitative fund managers, or quants, are able to invest in efficient firms
that are currently monitored by activists. Quant managers are the manifestation, in the
model, of the free-rider problem of Shleifer and Vishny [1986]. Quants perform little or no
monitoring themselves. Instead, they displace incumbent activists, which immediately leads
to a partial loss of efficiency: δj,t = δQ ≥ δA. In addition, once the activist stops monitoring
the firm, there is a chance that it reverts to full inefficiency δj,t = δU > δQ. If and when this
occurs, the quant fund dissolves.
A large but finite number of ex-ante identically skilled managers search for opportunities
in the labor market for asset managers. Managers match with firms to form investment
funds, which may be either quant funds or activist funds. A fund consists of one firm
and one manager. An unemployed (potential) manager may search for opportunities as
either an activist or a quant, or may choose not to search at all. The choice is sensitive to
the current competitiveness of the fund market. At any moment, there are nt incumbent
activist funds and mt incumbent quant funds. Potential managers who choose not to search
can be thought of as remaining in the general household pool, earning reservation utility
with certainty equivalent value cKt, where Kt is the aggregate capital stock, or equivalently
aggregate wealth.5
If a potential manager chooses to search, he pays flow cost ζAKtdt while searching for
activist opportunities or ζQKtdt for quant opportunities. New matches are formed at rates
n1−νt (N − nt −mt)
ν , (2)
m1−νt nνt , (3)
4As noted earlier, reality dictates that activism will not always be successful and can be curtailed by freerider problems [Grossman and Hart, 1980, Shleifer and Vishny, 1986] or liquidity issues [Coffee, 1991].
5The number of potential managers is not generally important to the setup, but we assume that thereare a large number of potential managers relative to firms.
8
for activists and quants, respectively, where nt is the number of potential managers search-
ing for activist opportunities and mt the number of potential quants searching. Although
empirical work on fund manager labor market dynamics is comparatively scant, the con-
stant returns to scale matching technology is standard in the broader labor literature; see
for example Petrongolo and Pissarides [2001] and Rogerson et al. [2005]. In our experiments,
we assume that elasticity parameter ν = 1/2 and that there is no difference in ν between
activists and quants. This corresponds to a prior belief that matching is equally difficult for
both manager types.
Constant returns to scale imply that individual potential managers find matches at rates
(N − nt −mt
nt
)ν, (4)(
ntmt
)ν. (5)
That is, the chance a searching manager succeeds depends on the tightness of his labor
market, a ratio of available target firms to searching managers. For most of our analysis, the
rate at which new quant funds are formed is also the rate at which existing activist funds
dissolve.6 Finally, quant funds dissolve at an exogenous rate θQ individually, or θQmt in the
aggregate. When a quant fund dissolves, an efficient firm reverts to being inefficient.
Potential managers take their earnings present value as given when making search deci-
sions. An incumbent activist manager earns fee income with time t present value Φ(nt,mt, Kt).
An incumbent quant manager has present value of expected earnings Ψ(nt,mt, Kt). Fee in-
come depends on funds under management, which reflects household asset allocation, for
which we later solve in equilibrium.
Potential managers will enter the labor market (search) until the net present value of a
6We allow for the possibility that some activists also exit for exogenous reasons, at rate θA individuallyor θAnt in the aggregate. We set θA = 0 in our baseline parameters.
9
new fund is zero. For activists, nt is the number of searchers7 such that
(Φ(nt + 1,mt, Kt)− cKt)
(N − nt −mt
nt
)ν= ζAKt. (6)
When a potential activist matches with a target firm, he exchanges fee revenues for his
household consumption stream. The exchange must be favorable enough, and the intensity
of matching high enough, to justify the search costs. The equivalent condition for quants is
(Ψ(nt + 1,mt, Kt)− cKt)
(ntmt
)ν= ζQKt. (7)
We later show that the present value of fees is homogeneous of degree one in capital Kt,
such that they can be written Φ(nt,mt, Kt) = φ(nt,mt)Kt for activists, and Ψ(nt,mt, Kt) =
ψ(nt,mt)Kt for quants. This allows us to solve for nt, the number of potential managers
seeking to start activist funds, such that
nt =
(φ(nt + 1,mt)− c
ζA
) 1ν
(N − nt −mt) , (8)
whereas mt, the number of potential managers searching to be quants, satisfies
mt =
(ψ(nt,mt + 1)− c
ζQ
) 1ν
nt. (9)
Finally, a third type of investment fund exists alongside quants and activists: a passive
index fund, which invests equally in each of the N firms. This fund is unique, requires no
manager, and charges a small, exogenous, and constant fee rate π proportional to capital
7We allow nt to take nonnegative real values, rather than restricting it to nonnegative integer values,hence Equation (6) holds with equality. nt, and mt in Equation (7), can be interpreted more generally assearch intensities. As a practical matter, continuously valued search intensities facilitate convergence of thenumerical solution routine.
10
under management.8 In addition to capturing competition from low-cost index funds, the
passive fund reflects the idea that households do not require skilled managers in order to
spread capital evenly across all firms, but they do require skilled managers to identify the
most efficient firms. In a simple way, this captures the idea that fund managers are better
informed than households.
3.3 Households
A representative household allocates capital to investment funds in order to maximize ex-
pected lifetime utility. While the state of competition among funds matters to households,
there is no reason for households to differentiate between funds of a given type. For example,
each activist charges an identical fee proportional to capital, and each invests in a unique —
but equivalent — efficient firm. Hence it is optimal, for purposes of diversification, for the
household to invest the same amount in each activist fund. The relevant question is how
much the household should invest in activist funds collectively.
Without loss of generality, let the first nt firms be activist funds, the next mt firms be
quant funds, and the remaining N − nt − mt firms belong exclusively to the the passive
index.9 We define composite Brownian motions
WA,t =nt∑j=1
Wj,t, (10)
WQ,t =nt+mt∑j=nt+1
Wj,t, (11)
WI,t =N∑
j=nt+mt+1
Wj,t, (12)
representing risk specific to activists, quants, and the passive fund, respectively.
8In our baseline parameters, it is zero.9Recall that funds consist of non-overlapping firm-manager pairs but the passive index invests equally in
every firm.
11
While we have in mind that the passive fund fee π is small, a fraction N−nt−mtN
of its
holdings are inefficient, with higher depreciation rate δU . Both activists and quants invest
exclusively in relatively efficient firms, with deterministic capital depreciation δA for activists
and δQ for quants. However these funds also charge commensurately higher fees proportional
to capital under management. Fee rates may vary depending on the amount of competition
in the fund market. The quant’s fee rate is ψt > 0, and the activist’s fee rate is φt > 0.
Households and potential managers take these fees as given. Later we explain how incumbent
managers strategically set fees to maximize revenues.
Households allocate aggregate capital, Kt, across the three types of funds, and also choose
the consumption-capital ratio ct. Let at be the activists’ fraction of capital, qt the quants’
fraction of capital, and the residual 1−at−qt is invested in the passive index. The aggregate
capital accumulation process is
dKt
Kt
=(µ− ct)dt+ σdW t
+
at
qt
(1− at − qt)
·
−(φt + δA)dt+ σ√ntdWA,t
−(ψt + δQ)dt+ σ√mtdWQ,t
−(π + δI,t)dt+ σN
(√N − nt −mtdWI,t +
√ntdWA,t +
√mtdWQ,t
) ,
(13)
where
δI,t =1
N((N − nt −mt)δU + ntδA +mtδQ) (14)
is the mean depreciation rate of firms in the index.
Households have CRRA utility, and choose consumption and capital allocation to maxi-
mize their lifetime expected utility,
maxct,at,qt∞t=0
E0
[∫ ∞0
e−ρtu(ctKt)dt
], (15)
12
subject to Equation (13).
4 Equilibrium
The economic state is summarized by the tuple ωt = (mt, nt), a Markov process capturing
current competition in the fund markets, and by the aggregate capital stock Kt. Since the
economy is time-homogeneous, we omit time subscripts in the solution.
4.1 The Fund Managers’ Problems
We begin by solving for a manager’s entry decision taking fees as given. Expectation under
the risk-neutral measure is denoted EQ, and the instantaneous risk free discount rate is
r(w), i.e., a function of the Markov state only. Define the space of Markov states Ω, and
for current state ω ∈ Ω, the instantaneous rate of transition to ω′ ∈ Ω is λω,ω′ , and Λ the
transition matrix, with λω,ω = −∑
ω′∈Ω,ω′ 6=ω λω,ω′ . The equilibrium fee and the manager’s
entry decision are determined simultaneously, but we discuss them sequentially.
The instantaneous hazard rate β(ω), serves two distinct purposes in the model: (i) It
captures the possibility that a particular activist is driven out of the market. This occurs
when the activist’s firm becomes part of a quant fund in our baseline example, or because
an activist exits for exogenous reasons in one of our extensions. (ii) The hazard rate also
limits the conditions under which potential activists will search to form new funds. It is
possible for the present value of activist fees to be positive even if the current flow of fees
is non-positive, i.e., if the household would choose a zero or short position in activist funds.
To prevent this, we impose that activists immediately exit if their current fee flow turns
13
negative. It follows that
β(ω) =
λ(m,n),(m+1,n−1)
nif a(ω) > 0,
∞ otherwise .
(16)
The present value of an incumbent activist’s fees is
Φ(ωt, Kt) = EQt[∫ ∞
t
e∫ st β(ωv)+r(ωv)dv φ(ωs)a(ωs)
nsKsds
]. (17)
We use a generalized Feynman-Kac theorem to express the value of the stochastic integral
above as the solution to a PDE, which reduces to a simple system of equations in the
Markov state variable due to homogeneity of the valuation in K.10 Let A be the infinitesimal
generator for (ωt, Kt). Then the function Φ satisfies
AΦ(ω,K)− (β(ω) + r(ω))Φ(ω,K) +φ(ω)a(ω)
nK = 0. (18)
Recall our supposition Φ(ω,K) = φ(ω)K. Note that ΦKK = 0 and the drift of dKt is
(r(ωt)− c(ωt)Kt under the risk-neutral measure, where c(ωt) is the aggregate consumption-
capital ratio, interpretable as a dividend yield. Following equation (2.9) in Zhu et al. [2015],
we have
AΦ(ω,K) = φ(ω)(r(ω)− c(ω))K +∑ω′∈Ω
λω,ω′ [φ(ω′)− φ(ω)]K. (19)
10See Zhu et al. [2015], Theorem 3.2.
14
The optimal φ(ω) solves
φ(ω)(r(ω)− c(ω))K +∑ω′∈Ω
λω,ω′ [φ(ω′)− φ(ω)]K (20)
− (β(ω) + r(ω))φ(ω)K +φ(ω)a(ω)
nK = 0,
⇒∑ω′∈Ω
λω,ω′ [φ(ω′)− φ(ω)]− (β(ω) + c(ω))φ(ω) +φ(ω)a(ω)
n= 0. (21)
The solution for Ψ(ω,K) = ψ(ω)K, the present value of quant fees, is similar. Recall
that a quant exits the market if his firm reverts to inefficiency, hence his hazard rate is θQ.
An incumbent quant has fee present value
Ψ(ωt, Kt) = EQt[∫ ∞
t
e∫ st θQ+r(ωv)dv ψ(ω)q(ωs)
ms
Ksds.
](22)
Therefore the function Ψ satisfies
AΨ(ω,K)− (θQ + r(ω))Ψ(ω,K) +ψ(ω)q(ω)
mK = 0, (23)
where
AΨ(ω,K) = ψ(ω)(r(ω)− c(ω))K +∑ω′∈Ω
λω,ω′ [ψ(ω′)− ψ(ω)]K. (24)
After simplification this leaves
∑ω′∈Ω
λω,ω′ [ψ(ω′)− ψ(ω)]− (θQ + c(ω))ψ(ω) +ψ(ω)q(ω)
m= 0. (25)
Subject to φ(ω) and ψ(ω) satisfying Equation (21) and Equation (25), respectively, manage-
rial search intensities n and m satisfy Equation (8) and Equation (9), respectively.
15
Therefore instantaneous transition rates are
λ(m,n),(m,n+1) = n1−ν(N − n−m)ν , (26)
λ(m,n),(m+1,n−1) = m1−νnν + θAn, (27)
λ(m,n),(m−1,n) = θQm, (28)
and zero to all other states, subject to the additional restriction that transition intensity to
states outside of 0 . . . N is always zero. θA is an exogenous activist exit rate, which we set
to zero in our baseline examples.
Given the entry decision, we now endogenize how fee rates are set by incumbent managers.
We assume managers behave strategically, albeit in a limited sense. For each state ω, we solve
for a symmetric Nash equilibrium in which each manager chooses his fee rate to maximize his
flow of revenues, i.e., the product of his fee rate and the household’s allocation to his fund. In
doing so the manager takes into account the response of the household to a potential change in
fee rate, given the fees charged by the manager’s competitors. However incumbent managers
do not consider how their fee strategy alters fund market dynamics through its effects on
the labor market search behavior of potential managers. This assumption is primarily for
tractability, since it decouples the determination of fee rates from equilibrium fund market
dynamics. However it seems reasonable that while fund managers might respond to current
competitive pressures, they may not anticipate how their behavior will alter the evolution
of the fund market in the future.
For some state ω = (m,n) with n > 0, consider an individual activist who chooses his
fee rate φ′(ω), while the remaining n− 1 activists charge fee φ(ω) and m quants charge fee
16
ψ(ω). Given his choice of fee, the individual activist will attract capital
a′(ω) =N(δI(ω) + π)− (N −m− n+ 1)(φ′(ω) + δA)−m(ψ(ω) + δQ)− (n− 1)(φ(ω) + δA)
γσ2(N −m− n).
(29)
The individual activist solves revenue maximization problem
maxφ′(ω)
φ′(ω)a′(ω), (30)
which has solution
φ′(ω) =N(δI(ω) + π − δA)−m(ψ(ω) + δQ − δA)− (n− 1)φ(ω)
2(N −m− n+ 1). (31)
Similarly, for some state ω = (m,n) with m > 0, an individual quant choosing fee rate
ψ′(ω), while n activists charge fee φ(ω) and the remaining m − 1 quants charge fee ψ(ω),
attracts capital
q′(ω) =N(δI(ω) + π)− (N − n−m+ 1)(ψ′(ω) + δQ)− n(φ(ω) + δA)− (m− 1)(ψ(ω) + δQ)
γσ2(N −m− n).
(32)
The individual quant solves revenue maximization problem
maxψ′(ω)
ψ′(ω)q′(ω), (33)
which has solution
ψ′(ω) =N(δI(ω) + π − δQ)− n(φ(ω) + δA − δQ)− (m− 1)ψ(ω)
2(N − n−m+ 1). (34)
The symmetric Nash equilibrium is one in which all activists choose identical state-
contingent fee φ(ω), all quants choose identical state-contingent fee ψ(ω), and no individual
17
fund manager has an incentive to deviate from the rate schedule. The general solution is
φ(ω) =(2(N − n)−m+ 1)(N(δI(ω) + π − δA)−m(δQ − δA))
(2N −m− n+ 1)(2(N −m− n) + 1)
− m(N(δI(ω) + π − δQ)− n(δA − δQ))
(2N −m− n+ 1)(2(N −m− n) + 1), (35)
ψ(ω) =(2(N −m)− n+ 1)(N(δI(ω) + π − δQ)− n(δA − δQ))
(2N −m− n+ 1)(2(N −m− n) + 1)
− n(N(δI(ω) + π − δA)−m(δQ − δA))
(2N −m− n+ 1)(2(N −m− n) + 1). (36)
In the special case where activist and quant firms are equally efficient (δA = δQ), activist
and quant funds will optimally charge identical rates. In this case, equilibrium fee rates
reduce to
φ(ω) = ψ(ω) =N(δI(ω) + π − δA)
2N −m− n+ 1. (37)
We focus on this case in our numerical examples.
4.2 The Household’s Problem
The household’s problem, in Equation (15), is essentially a conventional planner’s problem
with affine production technologies and no capital adjustment costs. The novelty is that the
risk and productivity characteristics of the production technologies arise from equilibrium
in the fund manager labor market.11 We make the usual conjecture that the value function
can be written
V (ω,K) =1
1− γH(ω)K1−γ, (38)
for a function H(ω) to be determined.
11Our solution of the household’s problem follows Sotomayor and Cadenillas [2009], who establish opti-mality conditions in a setting similar to Merton [1969], but with market opportunities and utility contingentupon a continuous-time Markov process. See in particular Theorem 3.2, which covers the case where utility isnot directly contingent on the Markov state. Although we generalize to a production setting and endogenousinvestment opportunities, the household’s problem remains very similar to Sotomayor and Cadenillas [2009].
18
The value function satisfies Hamilton-Jacobi-Bellman (HJB) equation
1
1− γ[ρH(ω)−
∑ω′∈Ω
λω,ω′H(ω′)]
= maxc,a,q
u(c) + (µ− c− a(φ(ω) + δA)− q(ψ(ω) + δQ)− (1− a− q)(π + δI(ω))H(ω)
−γH(ω)
2
[σ2 +
σ2
N
(1− 2aq +
(N
n− 1
)a2 +
(N
m− 1
)q2
)].
(39)
The solution to the asset allocation problem is similar to Merton [1969]: it reflects the
mean returns and covariance matrix for the funds. Because transitions in the number of
fund managers are unpredictable, they do not introduce a hedging component to the asset
allocation problem. Optimal c, a, and q are
c(ω) = H(ω)−1/γ, (40)
a(ω) =n((N −m)(δI(ω) + π − φ(ω)− δA) +m(δI(ω) + π − ψ(ω)− δQ)
)γσ2(N −m− n)
, (41)
q(ω) =m((N − n)(δI(ω) + π − ψ(ω)− δQ) + n(δI(ω) + π − φ(ω)− δA)
)γσ2(N −m− n)
. (42)
To condense notation, write the drift and variance of the household’s optimal portfolio
returns, respectively, as
µ(ω) = µ− a(ω)(φ(ω) + δA)− q(ω)(ψ(ω) + δQ)− (1− a(ω)− q(ω))(δI(ω) + π), (43)
σ(ω) = σ +σ2
N
(1− 2a(ω)q(ω) +
(N
n− 1
)a(ω)2 +
(N
m− 1
)q(ω)2
). (44)
The drift function for the household’s optimal portfolio has a straightforward interpre-
tation: In the absence of activists and quants, the aggregate capital stock grows at µ less
the high depreciation rate of inefficient firms, since all firms are inefficient in the absence of
monitoring. The drift term is then adjusted by the allocation to each manager type mul-
19
tiplied by the net return effect of each manager type. The manager’s effect on expected
returns consists of the fee paid to the manager in order to achieve the lower depreciation
rate. The diffusion similarly shows the effect of allocating household wealth outside of the
passive fund. If there were no activists or quants, the diffusion would consist of the diffusion
of the aggregate shock, σ, plus the (common) individual diffusion divided by N .
Substituting the optimal portfolio choice and consumption and making use of the con-
densed notation, the HJB equation is
γH(ω)(γ−1)/γ +
((1− γ)(µ(ω)− γσ(ω)
2)− ρ
)H(ω) +
∑ω′∈Ω
λω,ω′H(ω′) = 0. (45)
The risk-free rate under the household’s pricing kernel is
rf (ω) = ρ+ γ(µ(ω)− c(ω))− 1
2γ(1 + γ)σ2(ω). (46)
5 Results
We characterize model behavior via a series of propositions and numerical examples. We
begin with some analytical comparative static results conditional on the state of the managed
fund market ω. These provide intuition for some mechanisms at work in the model, and
inform our numerical experiments. We then characterize the dynamic model for a set of
baseline parameters. Finally we study comparative static examples unconditional of the
state of the managed fund market, including a case in which the intuition from one of our
conditional propositions is overturned by dynamic effects.
20
5.1 Conditional Comparative Statics
We begin by establishing some properties of fees when quants run their firms less efficiently
than activists, i.e., δQ > δA. We show that this is a necessary condition for quants to
undercut activist fees in our model. We further show that transitions to states in which
quants replace activists are not desirable to the household when quants charge lower fees,
because the portfolio opportunity set worsens when activists are replaced by comparatively
inefficient quants. It is for this reason, in addition to parsimony, that we later focus on
numerical examples in which δQ = δA: this allows the greatest scope for quants to have a
positive implact upon household welfare in the dynamic equilibrium, not by undercutting
activists on fees, but by potentially increasing the overall state of competition in the managed
fund market.
The results in this section condition on the state of fund market competition, ω =
(m,n). They reflect what would occur in a static version of our model, or one in which state
transitions were exogenous and unanticipated.
Proposition 1. For any state ω such that n + m < N , quants charge lower fees than
activists, ψ(ω) < φ(ω), if and only if quant firms are run less efficiently than activist firms,
δQ > δA.
Proof. From Equations (35) and (36),
φ(ω)− ψ(ω) =(N − n−m)(δQ − δA)
2(N −m− n) + 1.
Since 2(N − m − n) + 1 > 0 always and N − n − m > 0 by assumption, the sign of the
difference in fees is given by δQ − δA.
Although the details are specific to our setting, Proposition 1 exemplifies a general point,
and may be useful to corral potential explanations for fee heterogeneity. Absent frictions or
21
behavioral biases, quants offering a substitute product for “traditional” actively managed
funds should not find it optimal to charge relatively lower fees unless their product is inferior
in some way. Hence if we observe quant funds offering their strategies at a discount, it is
unlikely that the quant fund possesses an advantage in efficiency — more likely the opposite.12
There are, of course, other potential explanations not present in our model: perhaps the
household requires time to learn that the quantitative strategy offers equivalent returns, for
example, a form of information friction from which we abstract.
We can also relate efficiency, fees, and expected returns conditional on state ω.
Proposition 2. Suppose quant firms are less efficient than activist firms, δQ > δA. Then for
any state ω such that n+m < N , quant funds have lower net expected returns than activist
funds: µ− δQ − ψ(ω) < µ− δA − φ(ω).
Proof. Since δQ > δA and n+m < N , the result follows immediately because
φ(ω)− ψ(ω) =N − n−m
2(N −m− n) + 1(δQ − δA) < δQ − δA.
Although Proposition 2 conditions on our choice of parameters δA and δQ, it would be
equivalent to condition on relative fees, per Proposition 1. In combination, these propositions
imply that any discount in fees charged by quants relative to activists does not fully offset
the relative inefficiency of the firms quants manage. It could be detrimental to households
if “low cost” quants replaced activists, even if they charged lower fees, because expected
returns for quants net of fees are still lower than for activists.
The following proposition shows that this is indeed the case.
Proposition 3. Assume δQ > δA, and consider a transition from state ω = (m,n) to state
12Our assumption that the difference in efficiency arises at the firm (due to inferior monitoring) rather thanat the fund (due to, e.g., higher overhead) is important for the aggregate productivity of capital and expectedreturns to the passive index, but not for the results on relative fees. However we do rule out sophisticatedintertemporal strategic behavior by incumbent funds, oriented to affect entry and exit dynamics.
22
ω′ = (m+∆m,n−∆m), for some integer ∆m > 0, and m+n < N . This transition replaces
∆m activists with quants. The transition increases activist fees, increases quant fees, and
decreases average firm efficiency: φ(ω′) > φ(ω), ψ(ω′) > ψ(ω), and δI(ω′) > δI(ω). The
household’s portfolio opportunity set worsens when quants replace activists if quants are less
efficient than activists.
Proof. After simplification,
φ(ω′)− φ(ω) = ψ(ω′)− ψ(ω) =(N −m− n)∆m(δQ − δA)
(2N −m− n+ 1)(2(N −m− n) + 1)≥ 0,
with strict inequality when m+ n < N , and
δI(ω′)− δI(ω) =
∆m(δQ − δA)
N> 0.
Although the above result may seem obvious in light of the quants’ relative inefficiency,
when considered in terms of fees it is less intuitive: quants charge lower fees than activists,
yet transitioning to a state in which quants replace activists will increase fees, and will
immediately make the household worse off.
Because of the results above, we focus on numerical examples in which there is no imme-
diate loss in efficiency when a quant displaces an activist, because δA = δQ. Although it is
not optimal for quants to undercut activist fees in such examples, we believe these param-
eters offer quants the best chance to provide social value, perhaps by increasing the overall
level of competition (number of funds) in the managed fund market.
In addition to competing with each other, activists and quants also compete with the
passive index fund. In contrast to activist and quant fees, the passive index fee π is exogenous
and constant. Although modeled in reduced form, we have in mind a fully competitive
market for identical index funds, such that π represents labor and capital costs entailed to
23
run an index fund at zero profit. In keeping with this interpretation, a reduction in π can
be interpreted as a form of increased productivity. The following proposition summarizes its
effects.
Proposition 4. For any state ω, decreasing the passive index fee π decreases both activist
fee φ(ω) and quant fee ψ(ω). Therefore a reduction in π improves the household’s portfolio
opportunity set conditional on state ω.
Proof. This simply requires examination of the partial derivatives of optimal fees with respect
to π:
∂φ
∂π=∂ψ
∂π=
(2(N −m− n) + 1)N
(2N −m− n+ 1)(2(N −m− n) + 1)> 0.
Because we assume time-additively-separable household utility, Proposition 4 implies
that a reduction in index fees π is conditionally welfare improving for households, in the
sense that expected utility within a given state ω increases due to the improvement in the
household’s portfolio opportunity set. Although it is still possible that a reduction in π could
unconditionally reduce household welfare, such a result must hinge on the dynamic affects
of the fee reduction on the incentives of activists and quants to search in their respective
labor markets. We present relevant examples in Section 5.3.
5.2 Dynamic Characterization for Baseline Parameters
We characterize the dynamic model via numerical examples, with baseline model parame-
ters in Table 1. Our baseline parameters are chosen to be reasonable relative to observed
economic quantities. It is worth noting a few of the parameters. First, we assume that
quant and activist firms are equally efficient, with δQ = δA = 0. Therefore they offer equiva-
lent products, and charge identical fees. By contrast inefficient firms have a depreciation of
δU = 4%. Given the mean productivity of capital µ is 11%, the efficiency gain from moni-
toring is economically meaningful. Second, our number of firms, N = 25, is chosen for fast
24
equilibrium computational time. Experiments with larger numbers of firms lead to similar
results. Finally, the search parameters chosen, while consistent with the labor literature, are
largely ad hoc in a managed money setting. Our baseline is to start with identical search
parameters for both activists and quants.
Figure 1 illustrates the stationary distribution of activists and quants under the Table 1
parameters. The top panels summarize the joint density, while the bottom panels summarize
the marginal densities. In equilibrium, a non-degenerate distribution exists where the modal
percentage of firms managed by activists is 36% and the modal percentage of firms managed
by quants is 28%. In separate experiments, the stationary distribution for activists and
quants is non-degenerate across a wide range of parameters. Given fees are endogenous, it is
optimal for activists and quants to set fees such that they remain in the fund management
sector.
While Figure 1 conveys information about the number of activists and quants in equilib-
rium, Figures 2, 3, and 4 demonstrate how fund market competition impacts assets under
management (AUM) through the household’s portfolio problem. Figure 2 plots AUMs as
fraction of wealth across the three fund types as a function of fund market competition as
summarized by the number of activists (n) and quants (m). Figures 3 and 4 provide different
slices of Figure 2 conditional on 1, 5, or 10 quants or activists respectively. These two slices
along the quant and activist dimensions are symmetric as the households are indifferent
between holding similarly sized activist and quant sectors as both sectors provide the same
expected returns at the same fee.
Figure 3 is a good place to understand how competition among funds has diversification
and spillover effects. As the number of activist funds in the market increases from, for
example, a single fund, the household will initially invest more in the activist sector overall,
because it becomes better diversified. But more managed funds also implies more efficient
firms in the passive index, a spillover effect that eventually causes AUM among activists to
25
decline with the entry of additional activists. Since quant funds also produce the spillover
effect, the point at which AUM begins to decline with additional activists comes sooner
when there are many quants in the market. Passive investment is also high when both the
number of activists and quants are low. Given activist and quant funds match on single
non-overlapping firms, these funds are not well-diversified when small, so the household has
less incentive to invest in them relative to the passive fund even though it largely delivers
inefficient returns.
Equilibrium fees are summarized in Figures 5 and 6. The top panel of Figure 6 plots the
fee rate, common for both activists and quants, for different amounts of fund competition.
Equilibrium fees range between slightly above 0 basis points and slightly below 200 basis
points. Increased competition, either through more activists or quants, lowers the fee rate.
But, the more pertinent measure of fees is not the fee rate. Instead, the present value of fees
collected over a fund’s lifetime is presented as a density for activists and quants in Figure 5
and conditional on the current amount of fund competition in the lower two panels of Figure
6.
The present value of fees in Figure 5 highlights a big difference between activist and
quant dynamics. While the present value of fees per activist fund are decreasing with the
number of activists (left panel), the same is not true for quants (right panel). On a per-fund
basis, the value of quant fees is relatively flat in the number of quants, rising slightly as the
number of quants increases. The slight increase reflects a deterrent effect from a market
dominated by quants: new funds would attract low AUM, hence fewer managers search for
new opportunities. A high number of quants also implies strong spillovers in efficiency to
the index, which leads to relatively flat fee present value overall. What explains the activist
fee present value generally declining in the number of activists? While a large number of
incumbent activists may deter new activists from entering, it does not deter new quants,
who have a higher chance of matching with an efficient firm with many activists in the
26
market. Since both this effect and the spillover effect operate in the same direction, activist
fee present values are declining in n.
Figure 7 analyzes how household welfare is driven by competition in the fund management
sector. Analogous to the early plots, household welfare is represented for differing levels of
activists and quants. Even after fees, all things equal, more activists and quants are preferred
to less from a welfare perspective as the market for firms is more productive after fees. The
plots for some activist, quant combinations are truncated due to that combination having
zero probability in the steady state distribution. For example, for m = 10 quants, the
maximal number of activists that still has a positive probability is n = 13.
However, from the left panel, the households would prefer a high number of activists and
a low number of quants in equilibrium from a welfare perspective. Given activists actively
find the inefficient firms and quants free ride, a high number of activists is desirable as they
make the market for firms more efficient, including the passive fund. Given free entry by
quants though, the second best is to have a setting where activists and quants jointly make
the market more efficient.
5.3 Unconditional Comparative Statics
To better understand the key factors driving the dynamics of the fund management sector
and household choice in equilibrium, we consider three different comparative statics: (1)
lower quant search costs relative to activists, (2) a higher cost to invest in the passive fund,
and (3) a lower household risk aversion. The first comparative static is meant to capture
a scenario where entry by quants is easier relative to activists. The second comparative
static is meant to demonstrate the trade-off between investing in the passive sector versus
the managed money sector. Finally, the last comparative static is meant to capture the role
of lower diversification costs on equilibrium quantities.
In contrast to Section 5.1, here our results average over possible states ω according to the
27
stationary distribution implied by each set of model parameters, and reflect the incentives
for fund managers to search for opportunities in equilibrium.
5.3.1 Quant Search Costs
Figure 8 summarizes a comparative static exercise where the quant’s search cost ζQ is varied
around the baseline parameter choice of ζQ = 0.001. Instead of presenting the stationary
distributions, we plot mean equilibrium quantities to more conveniently summarize the im-
pact of a change in the quant’s search cost. From the two top left plots, the mean number
of activists and mean activist AUM are increasing in the quant’s search cost. As a quant’s
search cost increases, the probability drops that an activist will be picked off implying more
activists at higher quant search costs.
The mean number of quants and mean quant AUM, in the top right panels, are nonmono-
tonic. Starting from a high enough quant search cost, the mean number of quants increases
as search costs fall. Given more quants, activists are picked off at a higher rate. Eventually,
this is detrimental to the number of quants. As search costs continue to fall, this has such a
detrimental impact on the supply of activists that the mean number of quants also fall along
with the mean quant AUM.
Interestingly, the mean fee rate for activists and quants is decreasing in the quant’s
search cost as seen in the bottom left panel. As a quant’s search cost falls, more activists
are driven from the market implying less competition. The remaining activists are then able
to charge higher fees. However, this does not translate into a higher present value of fees
to the surviving activists as seen in the panel above. This perverse impact on the activist
sector as quant search costs fall seriously impacts household welfare due to the bulk of the
firms now being inefficient as can be seen in the bottom right panel.
28
5.3.2 Higher Passive Investing Costs
In our baseline parameters, the cost of investing in the passive index π is zero. One way to
impact the overall costs of intermediation in the fund management sector is to raise the cost
of investing in the index which is then internalized in the optimal fee set by the activists and
quants as can be seen in equation (37). Figure 9 summarizes comparative statics for mean
equilibrium outcomes as the index fee rises from 0 to 150 basis points.
As expected, the mean fee rate and the present value of fees for activists and quants
increase as the substitute asset, the passive index, becomes more expensive as can be seen
in the bottom panels. Surprisingly, increasing index fees has an opposite effect on the mean
number of activists and quants. As the index fee increases, the mean number of activists
falls while the mean number of quants increases. This is driven by a reconfiguration of the
steady state between activists and quants. From a competition stand point, higher index fees
mean that quants can also charge higher fees, incentivising more potential quant managers
to search, and leading to more quants in equilibrium. This in turn then lessens the desire for
more activists to enter the fund management space, because they are displaced by quants at
a higher rate, leading to fewer activists and a non-monotonic activist AUM as a function of
index fees.
From a welfare perspective, households are worse off when index fees are higher, if we
leave our baseline parameters otherwise unchanged. First, higher index fees simply make
investing in the index more expensive. Second, in line with Proposition 4, higher index
fees lead to higher managed fund fees. Third, higher index fees also reduce the number of
activists in the market, because competition from quants intensifies as higher fees increase
the incentive for quants to enter.
Although higher index fees are typically bad for households, there are counterexamples.
In Figure 10, we repeat the experiment after modifying the baseline parameters by increasing
the quant exit rate to θQ = 1, increasing fund manager reservation utility to c = 0.08%, and
29
introducing exogenous activist exit, at a rate of θA = 0.2. The first two parameter changes
essentially shut down labor search by quants, by reducing the present value of fees below
the level of reservation utility. Introducing an exogenous activist exit rate ensures that any
activist fund will eventually exit the market even though no quants enter.
In combination with these changes, higher index fees have a nonmonotonic effect on
household welfare. Very low index fees are bad for households in Figure 10, because they
deter activists from entering the market, and the reduction in average firm efficiency is
greater than the value of the fee reductions. This emphasizes the importance of examining
the effect of reduced passive fees on market dynamics, which may overturn the static effects
summarized in Proposition 4.
5.3.3 Lower Household Risk Aversion
Figure 11 summarizes a comparative static exercise where the household’s risk aversion γ is
varied from 2.5 to 4.0. Given γ = 4.0 was our baseline risk aversion, this comparative static
is meant to demonstrate how the desire for less diversification and more expected returns
through lower risk aversion impacts the fund management sector. When risk aversion is
lower, the mean number of activists falls while the mean number of quants rises. AUMs for
both sectors increase though risk aversion falls. With the drop in risk aversion, households
are willing to hold less diversified activist positions.
6 Conclusion
Traditional active fund management has been transformed over the past two decades with
both significant changes in the costs of running a fund, with the rise in competition from
index products, delivered through both open-end funds and ETFs, and competition from
managers building new forms of quantitative strategies. At the same time, following the
30
financial crisis, there has also been an argument in the political arena about the size of the
financial service sector as a whole.
In order to better understand the competitive pressures between these different types of
funds and their impact on household welfare, we construct a stylized general equilibrium
model of the fund industry. In our model, active managers have social value by exercising
“voice” to improve the operating efficiency of the assets that they own. This does not
preclude the possibility that managers deliver value through the identification of mispriced
securities, but we do not need to assume mispricing skill for managers’ services to improve
household welfare.
Although the model is simple enough to clearly delineate the important margins affecting
the decisions of managers and households, it is rich enough to deliver nontrivial implications
for the impacts of fees and costs on the composition of the fund industry and household wel-
fare. The model can deliver both endogenous fee structures and nondegenerate steady state
distributions of the sizes of the different sectors. Through simple quantitative experiments,
we are able to use the model to understand the impact on the fund industry and household
welfare of changes in search costs, managerial efficiency, and fee compression. We highlight
interesting endogenous nonlinearities in the model equilibrium and provide some insight into
the spillover effects created by active management.
We do not view our results as the final word on fund industry dynamics and household
welfare in general equilibrium. However, by providing a fully articulated example of the
importance of general equilibrium considerations, we hope that it will stimulate further
research in this area.
31
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Parameter Value
Mean productivity of capital µ 0.11Inefficient deterministic cap. depreciation δU 0.04Quant deterministic cap. depreciation δQ 0.00Activist deterministic cap. depreciation δA 0.00Firm-specific std. dev. of cap. depreciation σ 0.3Systematic std. dev. of cap. depreciation σ 0.13Number of firms N 25Household relative risk aversion γ 4Household subjective discount rate ρ 0.02Fund labor market share coefficient ν 0.5Activist exogenous exit rate θA 0Quant exogenous exit rate θQ 0.35Activist search cost ζA 0.001Quant search cost ζQ 0.001Passive index fee π 0Manager reservation utility coeff. c 0
Table 1: Parameter values. The table reports the baseline parameter values used in ournumerical examples.
34
Figure 1: Stationary Joint Distribution of Activists and Quants: Base Param-eters. The top figure shows the joint stationary density of the number of quant (m) andactivist (n) funds in the market, under example parameter values in Table 1. The bottomfigure represents the same information as a contour plot.
24
0 5 10 15 20 25
Quants (m)
0
0.05
0.1
0.15
0.2
Sta
tionary
Densi
ty
0 5 10 15 20 25
Activists (n)
0
0.05
0.1
0.15
Sta
tionary
Densi
ty
0 5 10 15 20 25
Quants (m)
0
1
2
3
4
5
6
PV
Fees
per
Fund
10 -4
0 5 10 15 20 25
Activists (n)
0
1
2
3
4
5
6
PV
Fees
per
Fund
10 -4
Figure 5: Stationary Density and Fees: Base Parameters. In the top row, the figureshows the marginal stationary density of the number of quant (m) and activist (n) fundsin the market, under example parameter values in Table 1. The bottom row illustrates thepresent value of fees, per fund, conditional on the number of funds of that type in the market.Multiple values along the y-axis reflect potential levels of competition from the other fundtype, and the size of each circle indicates the likelihood (conditional density) of being in eachstate of competition.
28
Figure 1: Stationary Distribution of Activists and Quants: Base Parameters. Thetop left figure shows the joint stationary density of the number of quant (m) and activist(n) funds in the market, under example parameter values in Table 1. The top right figurerepresents the same information as a contour plot. The bottom figures show the marginalstationary densities of the number of activist (n) and quant (m) funds in the market.
35
Figure 2: AUM: Base Parameters. The figure shows assets under management (AUM)in each of the three fund sectors, as a fraction of aggregate wealth, conditional on the stateof competition in the fund manager market. Fund market competition is summarized by thetuple (m,n), where m is the number of quant funds, and n the number of activist funds. Thetop panel summarizes total activist AUM, the middle panel summarizes total quant AUM,and the bottom panel summarizes passive AUM. Parameter values are per Table 1.
36
0 5 10 15 20
Activists (n)
0
0.1
0.2
0.3
0.4
Activis
t A
UM
m=1
m=5
m=10
0 5 10 15 20
Activists (n)
0
0.1
0.2
0.3
0.4
Quant A
UM
m=1
m=5
m=10
0 5 10 15 20
Activists (n)
0.5
0.6
0.7
0.8
0.9
Passiv
e A
UM
m=1
m=5
m=10
Figure 3: AUM Conditional on Quants: Base Parameters. The figure shows assetsunder management (AUM) as a fraction of aggregate wealth in each of the three fund sectorsas a function of the number of activists (n). The information here summarizes three differentquant levels from Figure 2. The top panel summarizes total activist AUM, the middle panelsummarizes total quant AUM, and the bottom panel summarizes passive AUM. Parametervalues are per Table 1.
37
0 5 10 15 20
Quants (m)
0
0.1
0.2
0.3
0.4
Activis
t A
UM
n=1
n=5
n=10
0 5 10 15 20
Quants (m)
0
0.1
0.2
0.3
0.4
Quant A
UM
n=1
n=5
n=10
0 5 10 15 20
Quants (m)
0.5
0.6
0.7
0.8
0.9
Passiv
e A
UM
n=1
n=5
n=10
Figure 4: AUM Conditional on Activists: Base Parameters. The figure shows assetsunder management (AUM) as a fraction of aggregate wealth in each of the three fund sectorsas a function of the number of quants (m). The information here summarizes three differentactivist levels from Figure 2. The top panel summarizes total activist AUM, the middle panelsummarizes total quant AUM, and the bottom panel summarizes passive AUM. Parametervalues are per Table 1.
38
0 5 10 15 20 25
Quants (m)
0
0.05
0.1
0.15
0.2
Sta
tion
ary
De
nsi
ty
0 5 10 15 20 25
Activists (n)
0
0.05
0.1
0.15
Sta
tion
ary
De
nsi
ty
0 5 10 15 20 25
Quants (m)
0
1
2
3
4
5
6
PV
Fe
es
pe
r F
un
d
10 -4
0 5 10 15 20 25
Activists (n)
0
1
2
3
4
5
6
PV
Fe
es
pe
r F
un
d
10 -4
Figure 5: Present Value of Fees Densities: Base Parameters. The figure illustratesthe present value of fees, per fund, conditional on the number of funds of that type in themarket. Multiple values along the y-axis reflect potential levels of competition from theother fund type, and the size of each circle indicates the likelihood (conditional density) ofbeing in each state of competition. Parameter values are per Table 1.
39
5 10 15 20
Activists (n)
0.005
0.01
0.015
0.02
Fee
Ra
te (
%/A
UM
)
m=1
m=5
m=10
5 10 15 20
Quants (m)
0.005
0.01
0.015
0.02
Fee
Ra
te (
%/A
UM
)
n=1
n=5
n=10
5 10 15 20
Activists (n)
0
1
2
3
4
5
PV
Activis
t F
ee
s p
er
Fu
nd
10 -4
m=1
m=5
m=10
5 10 15 20
Quants (m)
1
2
3
4
5
PV
Activis
t F
ee
s p
er
Fu
nd
10 -4
n=1
n=5
n=10
5 10 15 20
Activists (n)
0
1
2
3
4
5
PV
Qu
ant
Fe
es p
er
Fu
nd
10 -4
m=1
m=5
m=10
5 10 15 20
Quants (m)
0
1
2
3
4
5
PV
Qu
ant
Fe
es p
er
Fu
nd
10 -4
n=1
n=5
n=10
Figure 6: Fees: Base Parameters. The figure shows properties of fees conditional on thestate of competition in the fund manager market. Fund market competition is summarizedby the tuple (m,n), where n is the number of activist funds, and m the number of quantfunds. The top panels summarize the fee rate for both activists and quants, the middle panelssummarize the present value of activist fees per fund, and the bottom panels summarize thepresent value of quant fees per fund. The left panels plot relative to the number of activistsn, while the right panels plot relative to the number of quants m. Parameter values are perTable 1.
40
0 5 10 15 20
Activists (n)
-7.5
-7
-6.5
-6
-5.5
Ho
use
ho
ld W
elfa
re
104
m=1
m=5
m=10
0 5 10 15 20
Quants (m)
-7.5
-7.4
-7.3
-7.2
-7.1
-7
-6.9
-6.8
-6.7
Ho
use
ho
ld W
elfa
re
104
n=1
n=5
n=10
Figure 7: Household Welfare: Base Parameters. The figure shows properties ofhousehold welfare conditional on the state of competition in the fund manager market. Fundmarket competition is summarized by the tuple (m,n), where n is the number of activistfunds, and m the number of quant funds. The left panel plots relative to the number ofactivists n, while the right panel plots relative to the number of quants m. Parameter valuesare per Table 1.
41
0 0.5 1 1.5 2
Quant Search Cost 10-3
0
5
10
15
Mean A
ctivis
ts (
n)
0 0.5 1 1.5 2
Quant Search Cost 10-3
4
6
8
10
Mean Q
uants
(m
)
0 0.5 1 1.5 2
Quant Search Cost 10-3
0
0.1
0.2
0.3
0.4
Mean A
UM
, A
ctivis
ts (
a)
0 0.5 1 1.5 2
Quant Search Cost 10-3
0.1
0.2
0.3
0.4
Mean A
UM
, Q
uants
(q)
0 0.5 1 1.5 2
Quant Search Cost 10-3
0
1
2
3
Mean P
V o
f F
ees (
), A
ctivis
ts (
a)
10-4
0 0.5 1 1.5 2
Quant Search Cost 10-3
2
4
6
8
Mean P
V o
f F
ees (
), Q
uants
(q)
10-4
0.5 1 1.5 2
Quant Search Cost 10-3
0.008
0.01
0.012
0.014
0.016
0.018
Mean F
ee R
ate
, A
ctivis
ts a
nd Q
uants
0.5 1 1.5 2
Quant Search Cost 10-3
-3
-2.5
-2
-1.5
-1
-0.5
Household
Welfare
105
Figure 8: Vary Quant Search Costs. The effect of changing the quant’s search cost isillustrated in the above figure. With the exception of quant search costs, given on the x-axis,all parameters are per Table 1. The quant search cost parameter ζQ is varied from 0.0001 to0.002.
42
0 0.005 0.01 0.015
Index Fee Rate
5
6
7
8
9M
ean A
ctivis
ts (
n)
0 0.005 0.01 0.015
Index Fee Rate
7
8
9
10
11
12
Mean Q
uants
(m
)0 0.005 0.01 0.015
Index Fee Rate
0.29
0.3
0.31
Mean A
UM
, A
ctivis
ts (
a)
0 0.005 0.01 0.015
Index Fee Rate
0.2
0.3
0.4
0.5
0.6
0.7
Mean A
UM
, Q
uants
(q)
0 0.005 0.01 0.015
Index Fee Rate
3
4
5
6
Mean P
V o
f F
ees (
), A
ctivis
ts (
a)
10-4
0 0.005 0.01 0.015
Index Fee Rate
4
6
8
Mean P
V o
f F
ees (
), Q
uants
(q)
10-4
0 0.005 0.01 0.015
Index Fee Rate
0.01
0.012
0.014
0.016
0.018
0.02
Mean F
ee R
ate
, A
ctivis
ts a
nd Q
uants
0 0.005 0.01 0.015
Index Fee Rate
-1.4
-1.2
-1
-0.8
Household
Welfare
105
Figure 9: Vary Index Fee. The effect of increasing the passive index fee is illustratedin the above figure. With the exception of passive index fee π, given on the x-axis, allparameters are per Table 1. The passive index fee π is varied from 0.00 to 0.015.
43
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
5
10
15M
ean A
ctivis
ts (
n)
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
5
10
15
Mean Q
uants
(m
)
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
0.2
0.4
0.6
0.8
1
Mean A
UM
, A
ctivis
ts (
a)
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
0.2
0.4
0.6
0.8
1
Mean A
UM
, Q
uants
(q)
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
0.5
1
Mean P
V o
f F
ees (
), A
ctivis
ts (
a)
10-3
0 0.005 0.01 0.015 0.02
Index Fee Rate
0
0.5
1
Mean P
V o
f F
ees (
), Q
uants
(q)
10-3
0 0.005 0.01 0.015 0.02
Index Fee Rate
0.018
0.02
0.022
0.024
Mean F
ee R
ate
, A
ctivis
ts a
nd Q
uants
0 0.005 0.01 0.015 0.02
Index Fee Rate
-3
-2.5
-2
Household
Welfare
105
Figure 10: Vary Index Fee, Modified Parameters. The effect of increasing the passiveindex fee is illustrated in the above figure. To illustrate nonmonotonic welfare effects dueto changes in index fees, this example uses fund manager reservation utility c = 0.08%,exogenous quant exit rate θQ = 1, and exogenous activist exit rate θA = 0.2. The passiveindex fee π is varied from 0.00 to 0.015. All other parameters are per Table 1.
44
2.5 3 3.5 4
Household Risk Aversion
6.5
7
7.5
8
8.5
Me
an
Qu
an
ts (
m)
2.5 3 3.5 4
Household Risk Aversion
8
8.5
9
9.5M
ea
n A
ctivis
ts (
n)
2.5 3 3.5 4
Household Risk Aversion
0.2
0.25
0.3
0.35
0.4
0.45
Me
an
AU
M,
Qu
an
ts (
q)
2.5 3 3.5 4
Household Risk Aversion
0.28
0.3
0.32
0.34
0.36
0.38
Me
an
AU
M,
Activis
ts (
a)
2.5 3 3.5 4
Household Risk Aversion
2.5
3
3.5
4
Me
an
PV
of
Fe
es (
), Q
ua
nts
(q
)
10-4
2.5 3 3.5 4
Household Risk Aversion
2.6
2.8
3
3.2
3.4
3.6
Me
an
PV
of
Fe
es (
), A
ctivis
ts (
a)
10-4
2.5 3 3.5 4
Household Risk Aversion
9.8
9.9
10
10.1
Me
an
Fe
e R
ate
, A
ctivis
ts a
nd
Qu
an
ts
10-3
2.5 3 3.5 4
Household Risk Aversion
-6
-4
-2
0
Ho
use
ho
ld W
elfa
re
104
Figure 11: Vary Household Risk Aversion. The effect of changing household riskaversion γ is illustrated in the above figure. With the exception of household risk aversion,given on the x-axis, all parameters are per Table 1. The household risk aversion γ is variedfrom 2.5 to 4.0.
45
