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Evolution of returns to scale and investor flows during the life cycle of active asset management

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Evolution of returns to scale and investor flows during the life cycle of active asset management

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Content Evolution of Returns to Scale and Investor Flows During the
Life Cycle of Active Asset Management
by
Geor gios Magkot sios
A Disserta tion Presented to the
F ACUL TY OF THE GRADUA TE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCT OR OF PHILOSOPHY
(Business Administration)
August 2018
Dedication
“…It’ s when you know you’re licked before you begin but you begin anyway and you
see it through no matter what. Y ou rarely win, but sometime s you do…”
Harper Lee, T o Kill a Mockingbir d
Dedicated to family and friends for their encourageme nt
and constant support during my years in graduate school.
i
Acknowledgements
I am grateful to my supervisors W ayne Ferson, Kevin Murphy , Kenneth Ahern, Odilon Câmara,
Arthur Korteweg, and many other faculty at USC. I have received valua ble feedback from Thierry
Foucault, Raman Uppal, and t he participants of the 2016 T rans-Atlantic Doctoral Conference in
London, the 2017 FMA European meeti ng in Portugal, the 2017 EFMA meeting in Greece, and the
2017 FMA meet ing in Boston. I a m also thankful to John Doukas for the best PhD paper a ward at
the 2017 EFMA meeting.
ii
Abstract
I show evidence of returns to scale at the industry leve l and investor fund flow asymmetries
across the best and worst performing ma nagers in asset management. I introduce a new model of
competition among fund managers that features the collecti ve ef fect of the competitive environment
on the performance of the average fund. This theoretical fra mework can explain the endogenous
emer gence and dynamic evolution of returns to scale at the industry level and t he flow-performance
relation. The model shows that competition i n the asset management industry ha s positive and neg-
ative ef fects on fund performance. When funds have increasing (decreasing) returns to scale at the
industry le vel, the flow-performance relation is concave (convex). Active funds outperform their
benchmark initially , but the average returns from ac tive investing are not persistent. The com-
petition among funds ra ises the cost of active management and gradually deplete the profita ble
opportunities in the aggregate. Eventua lly , the total surplus declines to zero and the average a c-
tive manager falls behind the benc hmark. Aggregate risk i s reduced over time through “closet
indexing”, until all active funds form a scal able pool of passively invested capi tal. The equi lib-
rium results of the model are veri fied in the data across multiple fund classes that invest actively ,
including equity , bond, money market, and hedge funds.
iii
T able of Contents
Dedication i
Acknowledgements ii
Abstract iii
List of Figur es vi
List of T ables vii
Chapter 1: Intr oduction 1
Part I Theory 6
Chapter 2: The basic model 7
2.1 Main setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
Chapter 3: Learning with a truncated distribution 14
Chapter 4: The life c ycle of investment management 20
4.1 Investor flows and competition . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The early stage of the life cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 The late stage of the life cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Empirical implications of the life cycle . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 5: T oday’ s alpha is tomorr ow’ s beta 32
5.1 Closet indexing and endogenous benchmark . . . . . . . . . . . . . . . . . . . 32
5.2 Closed-end funds and improving skill . . . . . . . . . . . . . . . . . . . . . . . 35
Part II Empirical Analysis 37
Chapter 6: The data 38
6.1 CRSP data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Hedge fund data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iv
6.3 Factor -mimicking portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 V ariable definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 7: Returns to scale at the industry level 51
7.1 Diminishing returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Increasing returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 The life cycle for returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.4 The ef fect of competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 8: Flow-perform ance r elation 60
8.1 Addressing potential endogeneity problems . . . . . . . . . . . . . . . . . . . . 63
8.2 Dif ferentiation by fund activeness . . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 9: Robustness tests 68
9.1 Investment style ef fects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2 Other robustness tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Part III Epilogue 73
Chapter 10: Conclusion 74
Bibliography 76
Appendix A: Pr oofs 84
Appendix B: Differ entiated Be rtrand competition 95
v
List of Figur es
Figure 1: Comparative statics on performance. . . . . . . . . . . . . . . . . . . . 23
Figure 2: Contour plot for network ef fect. . . . . . . . . . . . . . . . . . . . . . 26
Figure 3: T ime evolution of key variables. . . . . . . . . . . . . . . . . . . . . . 40
Figure 4: Hedge fund databases. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 5: T ests for main comparative statics on performanc e. . . . . . . . . . . . . 56
Figure 6: Flow-performance main results. . . . . . . . . . . . . . . . . . . . . . 62
vi
List of T ables
T able 1: Summary statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
T able 2: Investment style frequency . . . . . . . . . . . . . . . . . . . . . . . . . 100
T able 3: Evolution of returns to scale. . . . . . . . . . . . . . . . . . . . . . . . 101
T able 4: Determinants of returns to scale. . . . . . . . . . . . . . . . . . . . . . 103
T able 5: Evolution of the relation between fund fl ows and pa st performance. . . . . 106
T able 6: Determinants of the flow-performance re lation ( Fund Fees ). . . . . . . . 108
T able 7: Determinants of the flow-performance re lation ( Industry Size ). . . . . . . 109
T able 8: Determinants of the flow-performance re lation ( T otal Funds ). . . . . . . . 1 10
T able 9: Determinants of money market fund flows. . . . . . . . . . . . . . . . . 1 1 1
T able 10: Dynamics among investor flows, fund fees, and returns to scale. . . . . . . 1 12
T able 1 1: Instrumental variables regression for Industry Size . . . . . . . . . . . . . 1 13
T able 12: The ef fect of fund activeness on fees and ret urns t o scale. . . . . . . . . . 1 15
T able 13: The ef fect of fund activeness on the flow-perform ance relation. . . . . . . 1 17
T able 14: Comparison between industry and style-wide size rati os. . . . . . . . . . 1 19
vii
Chapter 1
Intr oduction
A questi on of pa ramount importance to the investor is whether active fund management is more
profitable than passive investing. The bulk of the empirical research has focused on equity mut ual
funds to address this que stion, because of their predominant size in assets under management. The
average ac tively mana ged equity fund underperforms relative to passive indexes, but a minority
of ta lented managers outperform their benchmark and receive more capital from investors. A par -
tial list of papers showing poor track records for the average equity fund in US markets i ncludes
Jensen (1968), Malkiel (1995), Gruber (1996), Carhart (1997), W ermers (2000), Kosowski et al.
(2006), French ( 2008), Fama a nd French (2010), and Guercio and Reuter (2014). Dyck et al.
(2013) find that a ctive managers outperform passive indexes in non-US markets. Grinblatt and T it-
man (1989, 1993), Hendricks et al. (1993), W ermers (2000), and Kosowski et al. (2006) document
stock-picking ability among equity fund managers, but with short-lived persistence in performance.
Fama and French (2010) also find a sma ll fraction of manage rs with negative and positive tr ue alpha
before fees and costs. Chen and Ferson (2016) find no equity funds with positive alpha net of costs.
Ippolito (1992), Chevalier and Ellison (1997), Sirri and T ufano (1998), L ynch and Musto (2003),
Huang et al. (2007), and Brown and W u (2016) among others find a convex flow-performance
relation f or equity funds, indicating that investor flows of capital are more sensitive to good per -
formers.
1
More recent e vidence reveals a heterogeneity in the re lations between scale, performance, and
investor flows of capital that is puzzling for existing theories. For instance, Pástor e t al. (2015)
show decreasing returns to scale at the industry le vel for equity mutual funds, while Magkotsios
(2017a) finds increasing returns to scal e for hedge funds and fixed inc ome funds. Harvey and
Liu (2017) propose instead that the per formance of the average equity fund declines mostly with
the fund’ s own assets. Kaplan and Schoar (2005), Getmansky (2012), a nd Goldstein et al. (2017)
find concave flow-performance relations for private equity , hedge funds, and corporate bond funds
respectively . Magkotsios (2017a) also shows that the returns of index funds and E TFs are inert to
variations in fund or industry scale.
Returns to scale have been studied across multiple literatures. A partia l list includes Krugman
(1980, 1991) in inter national trade and Hall (1988) in macroeconomics. The main empirical chal-
lenges in measuring the ef fect of returns to scale are similar a cross all industries, namely to separate
the ef fect from shocks in the total factor productivity . However , asset management is uni que rel-
ative to other industries, because the input capital to funds is chosen by investors rather than the
funds themselves. This feature implie s a connection between capital and labor as factors of pro-
duction. The investors react to fund performance with capital flows across funds. The connection
between the fa ctors of production suggests that returns to scale at the industry level are r elated to
the sensitivity of investor flows to fund performa nce. Therefor e, I also test for time va riation in the
flow-performance relation along with returns to scale.
I provide a theoretical model of the competition among active managers that can explain these
empirical results. The main predictions inc lude the conditions for succ essful active asset manage-
ment, the endogenous emer gence of increasing and decreasing returns to scale at the industry level,
and an empirically testable connection bet ween returns to scale and investor flows across funds.
Active funds can outperform passive funds and ope rate with incr easing returns to scale at the in-
dustry level, or be less profitable than passive funds and have diminishing returns to scale. The
ef fect of scale on fund performance and the subsequent flows by the investor have been studied
separately in the empirical literature. My mode l combines these two literatures under a common
2
framework. I also provide a potentia l e xplanation for t he popula rity of equit y mutua l funds that
involves a mix of diversified and scalable investment with portfolio services that are unavailable
by passive funds.
The model introduces a life cycle for asset management with time-varying R&D and trading
costs for every manager . The key feature is that competition has positive and negative ef fects on
fund performance. The ir balance in equilibrium is af fected by the cross-sectional distribution of
talent in active investing am ong the fund mana gers. The distribution of tale nt reflects the impact of
competition on individual fund performance. The cost of active management is low when competi-
tion is moderate, and the average active fund outperforms passive benchmarks. On the other hand,
strong competition increases costs and erodes performance until the net returns of the avera ge active
fund fall behind the returns of passive indexes. Despite the poor performance in risk-adjusted re-
turns (alpha), the inve stor doesn’ t withdraw all his ca pital from active funds. Instead, he dive rsifies
optimally across managers on the ef ficient frontier .
T alent is unobservable to the investor and managers. The market learns a bout every mana ger ’ s
talent and gradually resolves the uncertainty by observing fund track records. Le arning allows the
investor to filter out the worst performing managers from his portfolio and augment with capital
inflows the top pe rformers. This process rai ses the average level of managerial talent in the cross-
section, and it creates a barrier to fund entry for future cohorts. The rising level of average talent in
the cross-section expresses t he positive ef fect of competition. The positi ve ef fect dominates during
the early stage of the life c ycle. The supply of managers and aggregate investor demand e xpand
simultaneously in equilibrium. Active management is more profitable to the investor than passive
investing during this stage, and the funds have increasing returns to scale at the industry le vel.
Investor flows are more sensitive to bad-performing funds and less sensitive to top-performing
ones. This implies a concave flow-performanc e relation.
During the late stage of the l ife cycle, the empirica l predictions of the model are the opposite.
Although the incumbent managers are on average more talented over ti me, they also become more
homogeneous in their strategies and crowd into a diminishing opportunity set. Investing in the
3
same direction incre ases trading costs and reduces the value of investment in the a ggregate. The
impact on prices from crowded trading is the negative ef fect of competition. The average active
fund underperforms re lative to passive indexes. Funds have diminishing returns to scale at the
industry level, beca use the increased cost of active management limit s the opportunities to obtain
alpha. Investor flows are more sensitive to the mar ginally good-performing funds, because most
managers are equally talented on average and only a few of them can outperform their rivals. The
flows are less sensitive to bad-performing funds, because expected losses are small. This implies
a convex flow-performance relation.
Fund managers may choose to index a fraction of thei r assets to mitigate thei r costs. Over
the life cycle of asset management, I show that ac tive funds transform from a risky investment
vehicle that is rich in opportunities for alpha to a set of funds whose pe rformance and risk are
similar to those of passive funds. This is the concept of “today’ s alpha is tomorrow’ s be ta”. This
expression was coined by Andrew Lo. See also Cho (2017) for a costly arbitrage-based ar gument
where alphas turn to betas. It provides a potential explana tion for the high investor demand for
mature asset classes such as equity mutual funds. Assets in passive f unds are subject to market risk
only , while act ive management late in its life cycle provides diversification ac ross multiple funds
on the ef ficient frontier . In addition, the portfolios of active funds are customizable and allow the
investor to impose constraints to the manager ’ s choices. This servic e is not of fered by passive funds
that simply track an index.
Previous the oretical work has assumed various relations between scale and performanc e. Berk
and Green (2004) assume diminishing returns with fund size for a monopolist manager . The man-
ager can extract all surplus from investment by optimal ly increasing his fe e in equilibrium. T heir
model implies positive returns be fore fees and zero net-of-fe e returns in excess of passive bench-
marks. In reality , managers compete for investor capital and superior perform ance. W ith competi-
tion, it is not obvious that managers can extract a ll the surplus. Pástor and Stambaugh (2012) and
Feldman e t al. (2016) assume that fund returns decline with industry sca le to explain the persistence
of poor track rec ords for active ma nagers. Theoretical models in the international trade literature
4
also make similar assumptions. For instance, Krugman (1980, 1991) assumes a fixed cost of labor
to generate increasing returns to scale at the firm level, while Grossman and Rossi-Hansber g (2010)
assume external economies of scale. My model derives endogenously increasing and decreasing
returns to scale at the industry level, depending on t he level of competition among managers with
symmetric information.
The compe tition among managers with asymmetric information in García and V anden (2009)
and Gârleanu and Pedersen (2016) gives rise to diseconomies of scale in equilibrium. Diseconomies
of scale at the industry level arise in their models as the market becomes more ef ficient and atten-
uates the compara tive a dvantage of informed managers. My model is dif ferent, because it also
derives increa sing returns to scale at the industry level for moderate competition. In addition, the
unique link that I show between returns to scale and the sensitivity of investor flows to performance
has not been discussed before in the theoretical and empirica l literature.
5
Part I
Theory
6
Chapter 2
The basic model
The model describes t he com petition among active managers for superior pe rformance and
investor flows of capital. The marke ts have rational expectations, and there is no moral hazard or
adverse selection. Funds have a dual cost structure from active investing. The first cost involves
R&D to identify profitable investm ent opportunities within capital markets. The second cost is
the fund’ s trading cost to impleme nt the strategie s that are the outcome of the R&D process. For
instance, funds can recruit quantitative financial analysts to search for mispriced assets in the ma rket
and recommend ne w strategies to the portfolio m anager . The compensation of the analysts is part of
the R&D costs. The portfolio manager will then exe cute trades that incorporate the new strategies
and are subject to trading c osts. Klepper (1996) a lso introduces a dual cost structure during the
life cycle of i ndustries in the real economy . This includes product R&D costs for innovation and
process R&D costs for more ef ficient production.
I show that competition has positive and negati ve ef fects on the performance of the a verage
fund. The positive ef fect arises from economies of agglomeration. The discovery of a ne w prof-
itable opportunity increa ses the average performance in the cross section, because it attracts more
talented managers that attempt to exploit it. The investor augments the posit ive ef fect through an
assortative matching between talent and fund assets in equilibrium. The negative ef fect is related
to crowded trading, when a lar ge number of managers cluster around a small amount of profitable
7
opportunities. This ef fect increases the cost of trading for eve ry manager and reduces the net va lue
of investment in the aggregate. The balance between R&D and t rading costs determines in equi-
librium whether the positive or negative ef fect of competi tion dominates.
2.1. Main setup
A ma nager ’ s talent reflects his ability in exploit ing mispriced investment opportunities and
achieving risk-adjusted returns in excess of a benchmark. Letτ
i
be an exogenous l evel of talent for
manageri ,f
it
andq
it
the fund’ s fee a nd size of assets under management at timet respectively , and
Q
t
the aggregate size for a total numberN
t
of competing funds. The fund fee represents the price
that the investor pays for asset management. I do not make a distinc tion between management and
performance fees. The gross risk-adjusted return for fundi that is realized at timet+1 is
R
it+1
= 1+τ
i
−R
bt+1
+ε
it+1
, (2.1)
whereR
bt+1
is the benchmark return during the same period. The benchmark is exogenous in the
basic setup, but will be extended in later sections. The noise terms ε
it+1
are jointly distributed
over time and across m anagers with z ero mean. These shocks reflect the component of luck in
the realized return. Passive funds (labeled with i = 0 hereafter) have z ero alpha by definition,
implying E
t
[R
0t+1
] ≡ 1 . Depending on the distribution of talent among active mana gers, the
average fund may have either positive or negative alpha .
The expected total surplus from fundi ’ s investment ne t of costs at timet is
TS
it
=q
it


E
t
[R
it+1
]−C
tot

q
it
HH
t
!


, (2.2)
8
where C
tot
(q
it
/HH
t
) is the c ost of active management and HH
t
∈ [0,1] is the Her findahl –
Hirschman index. It is defined as
HH
t
≡
N
t
X
i=1

q
it
Q
t
!
2
, (2.3)
and it is a measure of concentration across the industry . The ratio q
it
/HH
t
in equation (2.2) im-
plies that a fund’ s cost increases eithe r when its own size increases or when the Herfindahl index
decreases. The numerator captures the price impact of a single fund’ s tradi ng, while the denomina-
tor embeds the price impact from correlated trades across the industry . Crowded trading emer ges
when a la r ge numbe r of similarl y ta lented managers trade in the same direction to compete for a
specific investment opportunity . This increases prices and reduces the return from profitable trades.
Therefore, crowded trading destroys net inve stment value and make s alphas more elusive in the ag-
gregate. The intuition is similar to Foster and V iswanathan (1996), who show that trading is less
profitable when many agents chase the same information signal. Intuitivel y , the price impa ct from
a single fund following a specific strategy that manages all a ssetsQ
t
is identical to the price impact
ofN
t
funds each with sizeQ
t
/N
t
following the same strategy . The Herfindahl index in the cost
function embeds the impact of crowded trading.
The fund ma nagers pa rticipate in monopolistic competition. Managers are vertically dif ferenti-
ated based on ta lent in obt aining alpha from arbitrage opportunities. The y ar e a lso horizont ally dif-
ferentiated based on their risk-return tradeof f on the e f ficient fronti er . For instance, some ma nagers
provide high-risk and high-return strategies, while others provide low-risk and low-re turn strate-
gies. Monopolistic competition implie s that perturba tions to the expecte d ret urn and fees of a single
fund cannot af fect the aggregate indices of returnsSR≡
P
j
E
t
[R
jt+1
] and feesSf ≡
P
j
f
jt
respectively . The manager of fundi extracts a fraction of the total surplus through fees
MS
it
=f
it
TS
it
=f
it
q
it


E
t
[R
it+1
]−C
tot

q
it
HH
t
!


, (2.4)
9
whereMS
it
is the manager surplus. Define the following function i n reduced form
C(q
it
,HH
t
)≡f
it
C
tot

q
it
HH
t
!
=
c
2

q
it
HH
t
!
2
(1−HH
t
)+h

q
it
HH
t
!
, (2.5)
where c and h are constants. T he quadratic term describes the trading cost, and it implies di-
minishing returns to scale at the fund level. The linear te rm is the fund’ s R&D cost, reflec ting the
manager ’ s ef fort in discovering mispriced opportuniti es for alpha in the market. The R&D cost also
increases with the ratioq
it
/HH
t
. This implies that lar ger funds ne ed to expe nd more resource s to
discover new opportunities, so that they ca n distribute their portfolio over a broader range of strate-
gies to mitigate trading costs. In addition, less concentrated industries overburden the discovery of
new opportunities in the aggrega te due to competition. The term (1−HH
t
)/2 in the trading cost
is only included to simplify the algebraic calcula tions for the equilibrium solut ion and comparative
statics. It enforces zero trading costs in the asymptotic limitHH
t
→ 1 where c orrelated trades are
absent.
Therefore, the mana ger sets his feef
it
at ti met to maxi mize hi s expecte d profits from active
investing
max
f
it
E
t
[Π
it+1
] =f
it
q
it
E
t
[R
it+1
]−
c
2

q
it
HH
t
!
2
(1−HH
t
)−h

q
it
HH
t
!
, (2.6)
whereq
it
is the investor demand that clears the mar ket. The investor has finite wea lth and mean-
variance preferences. His portfolio of funds at timet involvesN
t
active managers and one passive
fund. The investor ’ s problem is
max
q
t
U =E
t
[q
0
t
(r
t+1
−f
t
)]−
γ
2
q
0
t
V
t
q
t
s.t. q
0
t
1 =W
t
, (2.7)
whereγ is a parameter related to the investor ’ s risk aversion, andW
t
is the exogenous total wealth
invested among the active funds and passive index at time t . All vectors haveN
t
+ 1 elements,
whereq
t
,f
t
, andr
t+1
are the vectors for fund size s, fees, and nominal returns realized att+1 .
10
In addi tion, 1 is a vect or of ones, and V
t
is the covari ance matrix for the ret urns of the passive
index and incumbent active managers.
2.2. Equilibrium
T alent is unobservable a nd imperfectly known to the market, including the managers them-
selves. All market pa rticipants observe the realized returns over time and update their estim ates
about every manager ’ s ta lent. The learning process i nvolves Bayesian updating under symmetric
information. Potential entrants draw their exogenous talent from a normal distributionH(τ
i
) ∼
N(μ,σ
2
) that is comm on to all entry cohorts over time. Let μ
τ,t
andσ
τ,t
be the cross-sectiona l
weighted average and dispe rsion of talent respectivel y . The conditional estimates for the cross-
sectional meanμ
τ,t
and di spersionσ
τ,t
among incumbent managers are ˆ μ
τ,t
and ˆ σ
τ,t
respectively .
The Appendix discusses the details of the stochastic learning process and derives expressions for
the conditional estimates on the mean and dispersion of talent. As the uncertainty about each man-
ager ’ s talent is resolved over time, the estimates conver ge to the popul ation mean and dispersion.
Define the vector of expected risk-adjusted returns that are ne t of fees and costs
E
t
[R
t+1
−f
t
]≡V
−1
t
E
t
[r
t
−f
t
] . (2.8)
Each element of t his vector corresponds to a fund’ s net alpha, andR
it+1
is the gross return give n
by equation (2.1). The solution to the investor ’ s problem in equa tion (2.7) is the following:
LEMMA 1. The equilibrium size for fundi at timet is given by
q
∗
it
=
γW
t
b
it
N
t
+1
+b
1
E
t
[R
it+1
−f
it
]−
b
it
N
t
+1
N
t
X
j=0
E
t
[R
jt+1
−f
jt
] , (2.9)
wher eE
t
[R
jt+1
−f
jt
] is the expected net alpha for fundj , whileb
1
andb
it
ar e
b
1
≡
1
γ
,
b
it
N
t
+1
≡
1
γ
V
−1
1
1
0
V
−1
1
. (2.10)
1 1
The investor values returns net of fees and costs. The demand for a fundi depends on the fund’ s
own net-of-fee alpha , but also includes the average of net-of-fee alphas from every c ompeting fund.
The termb
it
embeds the correlations in performance bet ween fundi and its ri val funds (inc luding
the passive fund). The relationb
1
> b
it
/(N
t
+ 1) in equation (2.10) implies that investor flows
are al ways more sensitive to the fund’ s own performance than the average performance among the
competing funds. Lemma 1 impli es that any arbitrary attempt by a m anager to raise his own fees
may trigger outflows that are redistributed to rival funds in equilibri um.
The focus of this model is on the competition a mong managers and f und fees are the prices. The
investor takes fees a t timet as given, and he clears the market simultaneously by allocat ing capital
according to equation (2.9). The managers on the other hand take capital as give n and choose their
fee in equilibrium at the same time. The Nash equilibrium for equations (2.6) and (2.9) gives the
fund’ s response function
f
∗
it
=

b
1
(N
t
+1)(hHH
t
+ ˆ σ
2
τ,t
/2)HH
t
−cb
1
b
it
(1−HH
t
)

SR−Sf−γW
t

+
E
t
[R
it+1
]

b
2
1
c(N
t
+1)(1−HH
t
)+b
1
(N
t
+1)HH
t
−b
it
HH
t
·

SR−Sf−γW
t

·

b
1
(N
t
+1)

b
1
c(HH
t
−1)+2E
t
[R
it+1
]HH
t

−1
, (2.1 1)
where SR and Sf are the aggregate indices of returns and fees re spectively . Equation (2.1 1)
involves the approximationR
2
it+1
≈ R
it+1
+ ˆ σ
2
τ,t
/2 . The fund’ s response function depends on
the sum of equil ibrium fees for al l incumbent managers. Adding up equations (2.1 1) for all funds
gives the solution for the sum of fees. Then the fee for each fund can be retrieved by substitution
to equation (2.1 1). The equilibrium fund size is given by equation (2.9). The equilibrium industry
sizeQ
∗
t
is the sum of the fund sizes for theN
t
active managers.
Every endogenous variable in the model is linked to expectations about manage rial talent and
its distribution in the cross section. Therefore, the distribution of talent is the fundamental quantity
of this model. Fund entrie s and exits over time change the distribution of talent, and also impact
12
the equilibrium in prices and qua ntity . Potential entra nt managers enter if their expe cted profits
are positive, while incumbents with negative expected profit s sustain capital outflows until they
liquidate and exi t. In equilibrium, the mar ginal incumbent at time t is the manager who has zero
expected profits. As a result, the numbe r of inc umbent managersN
t
at timet is determined by the
following condition
E
t
[Π
mt+1
] =f
∗
mt
q
∗
mt
E
t
[R
mt+1
]−
c
2

q
∗
mt
HH
∗
t
!
2
(1−HH
∗
t
)−h

q
∗
mt
HH
∗
t
!
= 0 , (2.12)
wherem is the mar ginal incumbent fund, and the asterisks denote equilibrium values tha t are spec-
ified by the solution to equations (2.6) a nd (2.7). The solution to equation (2.12) gives N
∗
t
, the
equilibrium number of funds at timet .
13
Chapter 3
Learning with a truncated distribution
The learning process is identical to all funds, and it is independent from the time of entry .
The prior probability density for the talent of potential entrants is a normal distributionH(τ
i
)∼
N(μ,σ
2
) . The investor update s his estimates for the talent of every fund by observing their returns.
During any period, the mar ginal incumbent whose profits satisfy the zero-profit conditi on (equation
(2.12)) is the least talented a mong the competing managers. Therefore, the conditional distri bution
for the cross secti on of talent a mong incumbentsH
t
1
(τ
i
) at timet
1
is a normal distribution that is
truncated at its left tail. T he truncation point ˆ τ
min,t
1
is defined as the talent of the mar ginal fund at
timet
1
. Any potent ial entrant who draws at timet
1
from the unrestricted prior distributionH(τ
i
)
a value of talentτ
i
< ˆ τ
min,t
1
would not ente r successfully . The investor considers this ef fect in
equilibrium, and the prior distribution for a c ohort of managers who enter succe ssfully at t
1
will
also be truncated at ˆ τ
min,t
1
.
In order to ge t a closed-form solution for learning, the noise termsε
it+1
only need to be jointly
distributed over time and across managers with zero mean. I simplify the notation by assuming
independence among those shocks, i.e. ε
it+1
∼ N

0,σ
2
ε
i

in equation (2.1). This is a strong
assumption, but the formulas are scalar rathe r than in matri x format. The results from learning are
unaf fected by correlated returns. I focus on a single cohort and reset the time of entry at t
1
= 1 .
As a result, all quantities that are indexed by timet refer tot periods after the fund’ s entry .
14
The normality and independence assumptions for the entrant tale nt and noise terms allow the
use of a sim ple Kalman filter for learning. What is unusual about the learning process in this model,
is that the prior distribution of talent a mong incumbents is truncated. The standard Kalman filter
solution gives estimates for every period that are unbiased but suboptimal in terms of variance.
On the othe r ha nd, the constrained Kalman filte r for trunca ted normal priors by Simon and Simon
(2010) gives estimates that are optimal in terms of variance, but potentially biased for every period.
This extension to the standard filter optimally includes the hard inequality c onstraintτ
i
> ˆ τ
min,t
1
.
Alternative versions of constrained Kalman filters use projection methods and gi ve estimators that
are both optimal and unbiased. Their disadva ntage is the lack of a closed-form solution.
Once the manager e nters, his talent is fixed over time. This simplification al lows to write the
learning problem in the following state space notation
τ
it+1
=τ
it
(3.1)
y
it
=τ
it
+ε
it
, (3.2)
where (3.1) is the state transition equation, and (3.2) is the observation equation. The investor
observes over time the processy
it
, which is defined as
y
it
≡R
it
−1+R
bt
. (3.3)
The recursive equations from the unconstrained Kalman filter a re
K
t
=
σ
2
σ
2
ε
i
+tσ
2
(3.4)
e
τ
it+1
= (1−K
t
)
e
τ
it
+K
t
y
it
(3.5)
e
P
2
it+1
=σ
2
ε
i
K
t
, (3.6)
15
whereK
t
is the Kalman gain, and
e
τ
it
is the conditional expectation for tal ent during the periodt
tot+1 , using the full history of observations until timet . Also,
e
P
2
it
is the conditional variance of
the talent estimator
e
τ
it
.
The solutions for the posterior mean and variance in talent a re
e
τ
it
=
σ
2
ε
i
μ+tσ
2
¯ y
i
σ
2
ε
i
+tσ
2
(3.7)
e
P
2
it
=
(σσ
ε
i
)
2
σ
2
ε
i
+tσ
2
, (3.8)
where
¯
R
i
is the sample average of all innovations onR
it
for fundi until t imet . The estimator
e
τ
it
is optimal when the mea n and variance of an unconstrained normal distribution a re used. However ,
it ca n also be used for the constrained problem. The estimator
e
τ
it
is unbiased for the posterior , but
as I show be low it is not ef ficient in terms of variance.
The constrained Kalman filter of Simon and Simon (2010) uses the posterior mean and vari ance
for truncated normal priors. Their algorithm involves transforming the unconstrained state variable
into a standard normal variable, the n enforce the constraint with a renormalization of the density
function, and apply the reverse transformation to retrieve the constrained posterior for talent. The
resulting estimate is the mean of the truncated posterior with a variance tha t is smaller than the one
from the unconstrained Kalman filter . However , the estimator can be biased, and the dire ction of
the bias is hard to assess.
The prior for the talent of potential entrants at timet
1
is distributed asH
t
1
(τ
i
)∼ N

μ,σ
2

but forτ
i
> ˆ τ
min,t
1
. Ignoring the constraint for now , define z
i
≡ (τ
i
−μ)/σ as the auxiliary
transformation of talent to a standard normal variable. Applying the constraint implies that the
minimumz
i
is
z
min
≡
ˆ τ
min,t
1
−μ
σ
. (3.9)
16
The normalization constant of the truncated standard norm al is
1
M
Z
∞
z
min
e
−
ζ
2
2
dζ = 1↔M =
s
π
2
"
1−erf

z
min
√
2
!#
, (3.10)
whereerf(·) is the error function. The mean and vari ance of the truncated distribution forz
i
are
E[z
i
] =
1
M
Z
∞
z
min
ζe
−
ζ
2
2
dζ =
1
M
e
−
z
2
min
2
> 0 (3.1 1)
Var(z
i
) =
1
M
Z
∞
z
min
(ζ−E[z
i
])
2
e
−
ζ
2
2
dζ = 1−E[z
i
]
2
−z
min
E[z
i
]< 1 . (3.12)
The reverse transformation on the constrainedz
i
gives the constrained Kalman filter estimator
for the manager ’ s talent with a truncated prior . The reverse transformation is simply
τ
i
=σz
i
+μ , (3.13)
whereμ andσ are the mean and standard deviation respectively of the unconstrained density func -
tion, implying that their posterior values could be retrieved from the standard Kalman filter solution
(equations (3.7) and (3.8) respectively). As a result, the posterior estimates for the constrained sta te
variable at timet are
ˆ τ
it
=
q
e
P
2
it
E[z
i
]+
e
τ
it
(3.14)
ˆ
P
2
it
=Var(z
i
)
e
P
2
it
, (3.15)
where the conditional estimates use all history of observations until timet . Interestingly ,
ˆ
P
2
it
<
e
P
2
it
which makes the constrained estimator more ef ficient in terms of variance . In the asymptotic limit
t→∞ the variances tend to zero and the two estimators are identical. As a result, the uncertainty
about the manager ’ s true talent is resolved over time through the l earning process.
On the other hand, the posterior estimate for talent can be biased for eve ry period, sinc e the
estimator ’ s mean va lue is not necessarily equal to the true mean of the population for potential
17
entrants. The direction of thi s bias is hard to assess. The potential bia s in the posterior estimates
does not stem from the existence of the lower bound for talent, but rather is an artifact of the specific
algorithm for the learning process. Alternative algorithms for constrained Kalman filters provide
both ef ficient and unbiased estimators, but lack a closed-form solution (Simon and Simon, 2010).
However , the artificial bias does not af fect any features of the model, because it is related to the
true mean of the distribution for the potential entrantsH
t
1
(τ
i
) , not the distribution of talent among
the incumbents who have successfully entered the industry .
For the purpose of clarity , the recursive equations for the constrained Kal man filter are
K
t−1
=
σ
2
σ
2
ε
i
+(t−1)σ
2
(3.16)
ˆ τ
it
=σ
ε
i
E[z
i
]K
3/2
t−1
+(1−K
t−1
)ˆ τ
it−1
+K
t−1
y
it−1
(3.17)
ˆ
P
2
it
=Var(z
i
)σ
2
ε
i
K
t−1
. (3.18)
The true value of talentτ
i
for fundi is subject to estimation errorsu
it
. This implies that the true
value of talent is equal to
τ
i
= ˆ τ
it
+u
it
, (3.19)
with the conditional varianceVar
t
(u
it
) =
ˆ
P
2
it
from equation (3.18).
The estimate s for the cross-sectional wei ghted mean and dispersion of talent among the incum-
bent managers are
ˆ μ
τ,t
=
1
Q
t
N
t
X
i=1
q
it
ˆ τ
it
(3.20)
ˆ σ
τ,t
=


1
Q
t
N
t
X
i=1
q
it
(ˆ τ
it
− ˆ μ
τ,t
)
2


1/2
. (3.21)
18
From equation (3.19), the cross-sectional dispersion of talentσ
τ,t
is equal to
σ
τ,t
=


ˆ σ
2
τ,t
+
1
Q
t
N
t
X
i=1
q
it
ˆ
P
2
it


1/2
. (3.22)
As the uncertainty about each value of talent is resolved over time and
ˆ
P
2
it
tends asympt otically
to zero for all fundsi , the cross-sectional estimates ˆ μ
τ,t
and ˆ σ
τ,t
conver ge to the population mean
μ
τ,t
and dispersionσ
τ,t
respectively .
19
Chapter 4
The life cycle of investment management
A compara tive statics analysis on the endogenous va riables f
∗
it
, q
∗
it
, and Q
∗
t
reveals the life
cycle for act ive management. I show below that the two dri ving forces for the evolution of this
life cycle are the performance-based flows of capital across funds by the investor , and the balance
between R&D and trading costs (see equation (2.5)).
4.1. Investor flows and competit ion
The equil ibrium alloc ations of capital to funds are assortative, with more talented managers
administering lar ger portions of the assets. The foll owing lemma shows a monotonically increasing
relation between investor flows and expected fund performance.
LEMMA 2. Investor flows to each fund ar e monotonically inc r easing in pe rformance, but decline
to zer o asHH
t
diminishes. Specifically ,
∂q
∗
it
∂E
t
[R
it+1
]
> 0 ∀i,t , (4.1)
with the equality satisfied whenHH
t
→ 0 .
The investor forms e stimates about every manager ’ s talent and expected returns by observing
their track record. The fund flows are his response to i nnovations about performance, where an
20
innovation is the dif ference between expected and real ized returns. He suppli es inflows to funds
with positive realizations in returns, in anticipation of lar ger returns in the future. On the contrary ,
the investor removes c apital from funds when realized returns are lower than expected. Therefore,
Lemma 2 i mplies that l earning and inve stor flows filter out over time the bad managers and favor
the most talented ones in equilibrium.
Competition and the investor ’ s response to perform ance influence the cross-sectional distribu-
tion of talent . The less talented managers experienc e outfl ows from investors until they liquidate
the fund and exit. Their capita l is reallocated to other incumbents or new cohorts of more tale nted
managers. The transfer of capital increases the estimate for the weighted a verage talent in the cross
section ˆ μ
τ,t
(equation (3.20)), a nd also the population average talentμ
τ,t
as the uncertainty about
talent is gradually resolved. The rising average tale nt suggests that many incumbents who attain
positive abnormal returns at a c ertain time will eventually be surpassed by more talented rival s in
the future. For instance, Pástor et al. (2015) docume nt empirically a rising average talent over ti me
among equity mutua l funds, and explain it in terms of changes to the population of managers. This
feature of competition is similar to “cr eative destruction” (Schumpeter , 1942).
Furthermore, the increasing μ
τ,t
and the exogenous prior distributi onH(τ
i
) for the talent of
potential ent rants imply that the cross-sectional dispersionσ
τ,t
for incumbent managers must de-
crease over time. The ef fects of manage rial turnover during the life cycle of a ctive managem ent
are summarized below .
PROPOSITION 1 ( Creative destruction ) . The competition among managers and investor flows
concentrate the cr oss-sectional distribution of talent at lar ger values, i.e.
dμ
τ,t
dt
> 0 and
dσ
τ,t
dt
< 0 . (4.2)
The concentration of talent at lar ger values over time can improve the average performance in
the cross section, but it also make s active management more competiti ve over time. Proposition 1
implies that the barrier to fund entry is ta lent, and this barrier is raised over t ime. This equili brium
21
mechanism induces fund exits. Incumbent managers whose exogenous talent is a bove avera ge and
harvest positive alphas may be forc ed to exit lat er in the life cycle if their tal ent be comes much
lower tha n the rising average talent. Therefore, the creative destruction mechanism implies that
returns from active management a re not persistent. The rising level of com petition is also reflected
by the evolution of the cross-sectional dispersion of fund fees as the following Corollary shows.
COROLLAR Y 1. The dispersion of fund fees declines over time as the managers become mor e
alike in talent.
The dynamic evolution of talent in the cross section af fects the investor ’ s flows in equilibrium.
Intuitively , the unce rtainty about t alent in the cross section is lar ger early in the life cycle. For two
funds of the same talentτ
i
, the one that enters e arlier will experience lar ger and more volatile flows.
Asμ
τ,t
andσ
τ,t
evolve, the talent range for successful entrants narrows. The prior for subsequent
entrant cohorts has smaller dispersion, and the investor resolves the uncertainty about talent faster
and with smaller flows in absolute magnitude.
COROLLAR Y 2. Funds in earlier entry cohorts r eceive comparativel y lar ger flows than funds
in subsequent entry cohorts. Spe cifically , bet ween two managers of identical talent and r ealized
r eturns who enter ed at differ ent cohorts, the manager who ente r ed first will r eceive lar ger flows
over time.
The following le mma and Figure 1 show some comparative statics. The independent variable
in Figure 1 is the Herfindahl inde x. The green curve is the correlation between the industry size
Q
∗
t
and expected fund performance. When this curve is positive, the funds operate under inc reas-
ing returns to scale at the industry level and vice versa. The blue curve describes the sensiti vity
of i nvestor flows to perform ance. When this curve is positive, the flow-performance relation is
convex. This means that inve stor flows are more sensitive to good performers and less sensitive to
bad performers. The opposite is true when the blue c urve is negative.
22
Figur e 1
Comparative statics of fees on performance (red solid line), returns to scale at the industry le vel
(green dashed line), and curvature of the flow- performance relation (blue dot-dashed line) as func-
tions of the Herfindahl index. The vertical dashed line marks the threshold valueHH
thr
where the
returns to scale change from inc reasing to decreasing, and the curvature of the flow-performance
relation switches from concave to convex. The graph also shows limiting values for all curves
whenHH
t
→ 0 andHH
t
→ 1 . These are:
(Point A) lim
HH
t
→0
∂f
∗
it
∂E
t
[R
it+1
]
= +∞
(Point B) lim
HH
t
→1
∂f
∗
it
∂E
t
[R
it+1
]
=−

(N
t
+1)b
1
+b
it

h+ ˆ σ
2
τ,t
/2

2(N
t
+1)b
1
E
t
[R
it+1
]
2
(Point C) lim
HH
t
→1
∂Q
∗
t
∂E
t
[R
it+1
]
=
b
0t

h+ ˆ σ
2
τ,t
/2

(N
t
+1)E
t
[R
it+1
]
2
(Point D) lim
HH
t
→0
∂Q
∗
t
∂E
t
[R
it+1
]
=−∞
(Point E) lim
HH
t
→1
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
=−

h+ ˆ σ
2
τ,t
/2

(N
t
+1)b
1
−b
it

(N
t
+1)E
t
[R
it+1
]
3
23
LEMMA 3. The following equations
∂f
∗
it
∂E
t
[R
it+1
]
= 0 ,
∂Q
∗
t
∂E
t
[R
it+1
]
= 0 , and
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
= 0 (4.3)
have a common and unique positive r ootHH
thr
in terms of the Herfindahl index.
Lemma 3 and Figure 1 suggest that the life cycle of active ma nagement may be separated into
two stages, depending on the value of the Herfindahl index relati ve to its threshold value HH
thr
.
Each stage has dif ferent impl ications for re turns to scale, the sensitivity of flows to performance,
and the performance of the ave rage active fund relati ve to passive benchmarks. I discuss these
implications below for each stage of the life cycle.
4.2. The early stage of the life cycle
Let’ s assume a small number of funds early in the life cycle. This implies a highly concentrated
asset class. The Herfindahl index is near 1, and it satisfies this relation: HH
t
> HH
thr
. The com-
bination of smallN
t
and relatively lar ge ˆ σ
τ,t
(Proposition 1) implies a less competitive landscape
for potential entrants, and the cost of trading is low .
The fi rst cohort of managers discover unexploited investment opportunities and obtain alphas.
Their performance attracts more managers subsequently , and the supply of funds is expanding.
The i nvestor bene fits from the agglomerat ion of labor , because he can diversify his wealth across
funds to mitigate risk. He is learning about every manager ’ s talent over time, and distributes his
capital among the best performing managers (Lemma 2). The concurrent incre ase in the supply and
demand for active management improves the performance of the average fund (Proposition 1). The
improved performance at the industry level further amplifies the expansion of supply and demand
through a positive feedback loop. As a result, the early stage of the life cycle features a posit ive
network external ity similar to Katz and Shapiro (1985). T he agglomerat ion of funds and investor
flows creat e suitabl e conditi ons for the growth of active management and its success re lative to
24
passive investing. The following lemma shows t he conditions for the outset of the life cycle and
the dominance of active asset management agai nst passive investing.
LEMMA 4. The number of fundsN
∗
t
incr eases over time, the Herfindahl inde xHH
∗
t
decr eases,
and the average active fund outperforms passive benchmarks when its r eturn satisfies the following
conditions
1+
1
2


h+
ˆ σ
2
τ,t
2


+
γW
t
2(N
t
+1)
< ˆ μ
τ,t
<
1
2


h+
ˆ σ
2
τ,t
2


+
γW
t
N
t
+1
(4.4)
Figure 2 illustrates the conditions for the initiation of the network ef fect. The contour plot
between the average performance in the cross-sectionμ
τ,t
and investor total wealthW
t
is divided
into three areas. Area ( A) corresponds to the case where a small num ber of managers have positive
alphas, but inve stor demand is small. T he investment opportunities a re relatively undiscovered by
the m arket. The network ef fect does not start, because the aggregate capital invested is insuf ficient.
Area (C) corresponds to the case where the available capital for investment is lar ge, but the average
performance of active funds is below that of passive indexes. The poor performance among a
small number of funds does not attra ct any ne w managers. Area (B) corresponds to the case where
the conditions in Lemma 4 hold true. The number of fundsN
t
increases when there is suf ficient
investor capita l availabl e. The network ef fect initiates the life cycle , a nd the rising number of funds
coincides with an expansion in the aggregate investor demand.
The network externality , also known as demand-side economies of scale, is the positive ef fect
from the competition among ma nagers. The following proposition summarizes its implications
about the life cycle of active management .
PROPOSITION 2 ( Network ef fect ) . During the early stage of the l ife cycle, (a) the net-of-f ee
r eturns of the average acti ve fund outperform passive be nchmarks, (b) the funds have incr easing
r eturns to scale at the industry level, (c) the flow-performance r elation is concave, (d) fund fees ar e
negatively corr elated with expected r eturns, and (e) the investor ’ s surplus fr om act ive management
incr eases.
25
Figur e 2
A verage performance and inve stor demand conditions f or network ef fect. The coef ficients for the
curves marking dif ferent areas in the contour plot area
1
= 1/2

h+ˆ σ
2
τ,t
/2

+1 ,a
2
= 1/2

h+
ˆ σ
2
τ,t
/2

,d
1
=γW
t
/(2(N
t
+1)) , andd
2
=γW
t
/(N
t
+1) .
The green curve in Figure 1 shows that forHH
t
> HH
thr
the correlation between aggregate
sizeQ
∗
t
and the expected return of a fund is positive. Although each manager has diminishing re-
turns to scale from R&D and trading costs, the industry as a whole opera tes under increa sing returns
to scale because of the network ef fect. This result highlights the dif ference between fund-leve l and
industry-level returns to scale, and demonstrate s the benefit to the investor from competition. The
performance of a monopolist mana ger deteriorates from diseconomies of scale. However , compe-
tition and the investor ’ s response to performance cre ate conditions where the average performanc e
and aggregate demand increase simultaneously in a positive feedback loop.
The fl ow-performance relation during the ea rly stage of the life cycle is concave. Figure 1
shows that the second derivat ive of fund size with expected returns (blue curve) is negative when
HH
t
> HH
thr
. Lar ge value s of ˆ σ
τ,t
during this stage of the life cycle (Proposition 1) imply the
possibility of lar ge realized losses from managers on the left t ail of the distribution. A concave
flow-performance rela tion means that investor flows are m ore sensitive to ba d performance and
26
less sensitive to good performance. In equil ibrium, the investor anticipates more talented managers
to enter the industry in the future, because of the dynamic network ef fect a nd increasing returns
to scale. Therefore, his optimal response is to remove the worst-performing managers from his
portfolio, rather than invest more capital into the curre nt top performers.
The red curve in Figure 1 suggests that fund fees are negativel y correlated with expected returns
during the early stage of the life cycle. The negative corre lation stems from the increase in returns
because of the ne twork ef fect, and the downward pressure on fees from competition. Specifically ,
the competition a mong managers provides a way for the investor to counteract rising fee s in equi-
librium. The investor diversifies his wealth across multiple funds with allocations proportional to
net-of-fee returns. If a manager attempts to increase his fees, the investor r eacts by redirec ting
flows to rival ma nagers of simil ar talent. A manager cannot af fect the aggregate indices of re-
turns and fees under monopolistic com petition. If he decreases his fee he can attract lar ger capit al
inflows by the investor . The discount of fees from incumbent manager s is optimal, because the
market anticipates the entry of more talented managers in the future (Proposition 1). For instance,
Christof fersen (2001) shows that money market funds relinquish fees to increase future investor
flows.
The results of Proposition 2 a re based on monopolistic c ompetition. The Appendix shows
an extension of t he model with di f ferentiated Bertrand competition that has the same empirical
predictions for the early stage of the life cycle.
4.3. The late stage of the life cycle
The growth of acti ve m anagement during the early stage of its l ife cycle is unsustai nable. As the
competition intensifi es and funds harvest alphas, the managers graduall y crowd into a diminishing
set of investment opportunities. Crowded trading emer ges from the combination of rising N
t
,
rising ˆ μ
τ,t
, and decreasing ˆ σ
τ,t
. T he declining Herfindahl indexHH
∗
t
reduces the R&D costs ove r
time, and facilitat es the discovery of profitable investment opportunities by managers. However ,
27
it increases trading costs increase and af fects prices in the aggregate. The rising trading costs
attenuate the network ef fect, diminish the net va lue of the available investment opportuniti es, and
make alphas more elusive at the industry level. When the Herfindahl index falls below the critical
thresholdHH
thr
, the life cycle of active management tra nsitions to its late stage . Managers who
outperform their peers and benchmark net of costs by a wide mar gin are rare during this stage,
because the incumbents are alike in talent.
The rising trading costs and crowded investing reflect the negative aspe ct of competition. In-
vestment opport unities deple te and value is destroyed as i t becomes more strenuous to outperform
rivals over time. The following proposition summarizes the empirical im plications from act ive
management for this stage of the life cycle.
PROPOSITION 3 ( Depletion of alpha ) . During the late stage of the life cycle, (a) the net-of-fee
r eturns of the average active fund fall behind passive benchmarks, (b) the funds have diminishing
r eturns to scale at the industry level, (c) the flow-performance r elation is convex, (d) fund fe es ar e
positively corr elated with expected gr oss r eturns, and (e) the total surplus fr om active investing
decr eases asymptotically to zer o.
Proposition 3 suggests that lar ge and very c ompetitive groups of funds will have only few man-
agers who achieve positive net alpha. The lack of profitable opportuniti es and managerial compe-
tition impose negative net alphas for most incumbents. This result is consistent with evide nce for
equity mutual funds in Kosowski et al. (2006) and Fama and French (2010), who find t hat only a
small fraction of managers at the right t ail of the distribution attain risk-adjusted returns i n excess
of their benchmark. It is also consistent with a voluminous empiric al literature that documents the
poor average performance of equity funds relative to passive funds.
The gree n curve in Figure 1 shows that for HH
t
< HH
thr
the correl ation bet ween the to-
tal activel y managed assets Q
∗
t
and the expected return of a fund is negative. T his implies that
managers have diminishing returns t o scale at the industry l evel, relate d to the lack of profita ble
investment opportunities. An increase in a ggregate investor demand destroys investment value,
because more managers chase the remaining opportunities for alpha at higher trading costs. The
28
correlation between fund fee s and performance becomes positive. Although managers are very
talented on average, those who can obtain positive alphas after costs are rare and increase their fees
in equilibrium.
The flow-performance relation during the late stage of the life cycle is convex. The blue curve
in Figure 1 shows that the second derivati ve of fund size with expected returns is positive for 0<
HH
t
< HH
thr
. Convexity implies that investor flows are more sensitive to good performance, and
less sensitive to bad performance. The investor rewards with inflows the few managers who earn
positive alphas when the opportunities in the aggregate are limite d. Berk and Green (2004) also
derive a convex flow-performance relation when managerial talent is scarce. In addition, investor
outflows from poor -performing managers are moderate, because all incumbe nts are alike in talent
and realized losses are smaller compared to the earl y stage of the life cycle.
4.4. Empirical implications of the life cycle
Recent evide nce by Pástor et al. (2015) show that e quity mutual funds have diminishing returns
to sca le at the industry le vel. My model provides a pot ential theoretical explanation for the origin
of this result, which is the depletion of the opportuni ties for alpha (Propositions 1 and 3). Gârle anu
and Pedersen (2016) provide an alterna tive explanation f or diseconomies of scale, based on the
competition among manage rs and informational inef ficiency in investment management. However ,
their model does not predict increasing returns to scale at the industry level. Proposition 2 of this
model suggests t hat the equity mutual fund industry must have operated under increasing returns
to scale during the early growth phase of its life cycle. The same prediction applies for other fund
classes too, such as bond mutual funds, hedge funds, and other alternative investment funds.
W ithin mature fund classes like equity mutual funds, the empiric al literature shows insignificant
net abnormal returns and low gross abnormal returns on average. These results have been inter -
preted through rent-seeking managers that successfully absorb all surplus from investment (e.g.
Berk and Green (2004)). I ar gue that mature fund classes ha ve little alpha to of fer . This should
29
make it hard to measure alpha with statistical precision. This is consistent with the empiri cal results
of Kosowski et al. (2006) and Fama and French (2010), who show that most equity mutual funds
have zero alpha and managers at the tails have small positive and negative alphas.
My model shows that fund pe rformance is not pe rsistent at the industry level, and predicts a
non-monotonic evolution over time for the return of the average fund. It increases initiall y , but
eventually declines. Depending on the choice of sample period, ave rage performance may seem
persistent due to the ef fect of the network externality . However , the mode l predicts that performance
must eventually erode, along with the profitable investment oppor tunities at the industry level.
The model predicts that the aggregat e size of equi ty mutual funds will continue to grow in the
long run even with poor abnormal returns, as they have evolved into a relatively safe investment
vehicle that can manage lar ge pools of capital at low cost. This result is similar to Glode (201 1),
who justifies the nega tive expected performance as an insurance premium tha t investors pay to
protect themselves against bad states of the economy . The reduced aggrega te risk in m y m odel
during strong competition stems from a lar ge fracti on of indexed assets. This is consistent with
Cremers and Petajisto (2009), who show an increase by 30% for the fraction of closet indexers
among equity mutual fund managers since 1980.
The competition among managers and the availability of investment opportunities link the re-
turns to scale at the industry level with the curvature of the flow-performa nce rel ation. Funds with
limited opportunities have diminishing returns to scale and conve x flows. Equity mutual funds
are an asset class that fits this description. The literature on the convexity of flows within equity
mutual funds is ext ensive. For more details, see also a survey of this literature by Christof fersen
et a l. (2014). The model predicts that this fund class has diminishing returns to scale at the industry
level, a nd Pástor et al. (2015) verify this prediction. However , when managers have suf ficient in-
vestment opportunities they should have increasing returns to scale and concave flow-performance
relations. Goldstein et al. (2017), Kaplan and Schoar (2005), and Getma nsky (2012) find con-
cave flows among corporate bond funds, priva te equity , and hedge funds respectively . These fund
classes are good candidates to test for increasing returns t o scale at the industry level. Sensoy et al.
30
(2014) show a maturing of the private equity industry that resembles the life cyc le evolution of my
model.
The mechanism for the li fe cycle of active ma nagement allows the creation of “mega funds”,
namely funds wit h significantly lar ger assets under management than rivals. The most successful
managers over the life cyc le will be those who entered early with a talent level that is deep in the
right tai l of the prior distributionH(τ
i
) . These managers receive the lar gest inflows over time
(Corollary 2), and their long tenure allows them to ama ss lar ge amounts of capital. The existence
of mega funds does not af fect the final number of funds.
31
Chapter 5
T oday’ s alpha i s tom orr ow’ s beta
5.1. Closet indexing and endogenous benchmark
The main setup shows that active manageme nt is more profitable than passive investing during
the early stage, but less profitable during the late stage of the life cycle. Area (C) in the contour
plot of Figure 2 shows that poor avera ge performance whenN
t
is small would prevent the network
ef fect from starting. However , the number of funds N
t
is lar ge during the late stage of the life
cycle. Why is ac tive management popular during the l ate stage? For instanc e, why are equity
mutual funds popular , even though they underperform passive benchmarks on average? T o answer
these que stions, I extend the main setup by introducing an endogenous benchmark and allowing
active managers to invest a fraction of their assets into passive stra tegies.
The gross risk-adjusted return for fundi is
R
it+1
= 1+τ
i
−

(1−HH
t
)μ
τ,t
+HH
t
·R
Mt+1

+ε
it+1
. (5.1)
The term in bra ckets is the endogenous benchmark, which is a weighted average between the ma rket
returnR
Mt+1
and the ave rage talentμ
τ,t
in the cross-section of active managers. The weight is the
Herfindahl indexHH
t
, and it de creases over the life cycle. I ntuitively , the benchmark during the
early stage is an exogenous factor , such as the market return. The asset class coul d in principle grow
32
until the managers hold the full unde rlying market of securities. In this case, the active managers
would be the benchmark themselves and the average gross alpha would be zero. The endogenous
benchmark at timet is a weighted average between these two extremes, withHH
t
as the weight.
Equation (5.1) implies that compe tition and investor learning gradual ly assimilate alpha-earning
strategies into the benchmark. For instance, equity fund managers could obtain alphas when their
benchmark is R
Mt+1
by trading on small stocks. Thi s strategy would qualify as alpha, because
only a few managers use the strategy a t low trading cost. As the Herfindahl index decre ases over
time and more managers adopt the strategy , net alpha is compe ted away . The ave rage talentμ
τ,t
in
equation (5.1) captures this ef fect, and size-related strategies become a r isk factor in the benc hmark.
The cross-sectional distribution for managerial talent af fects investment value. A very talented
manager can attain lar ge alphas when he competes aga inst mediocre rivals with lowμ
τ,t
. However ,
when the same manager competes against similarly talented rivals andμ
τ,t
is lar ge, then everyone
has the potential to exploit the same opportunities for alpha, and this af fects adversely the perfor -
mance of all incumbents. Therefore, a ma nager ’ s performance before fees and costs is af fected by
his talent relative to the environment where he competes, rather than his individual level of tal ent
only .
The asymptotic limit where funds manage the full market portfolio is consistent with Sharpe’ s
arithmetic of active manageme nt (Sharpe, 1991). The ac tive manage rs compete in a zero-sum
game, and for every winner there i s an of fsetting loser . Therefore , the average gross alpha is zero
by definition. However , this ident ity is valid if the market portfolio does not change a nd no new se-
curities are issued (Pedersen, 2017), or active m anagers maintain full control of the ma rket portfolio
for every period. For any other case, the avera ge gross a lpha is not zero.
Each manage r may reduce his trading and R&D costs if he invests a fraction of his assets into
passive or previously assimilated strategies. This is termed “c loset indexing”, because the investor
cannot monitor whether all capital is actively traded or not. The drawback of closet indexing for
the manager is that the investor observe s a smalle r realized alpha and reduces the fund’ s assets in
equilibrium. The manager trades actively x
it
q
it
of his assets under management, and invests the
33
rest of his capital (1−x
it
)q
it
in passive strategies that have zero alpha and E
t
[R
0t+1
] = 1 by
definition. The optimal fracti onx
∗
it
< 1 is found by maximizing the following objective:
max
x
it
(
f
∗
it
q
∗
it

x
it
E
t
[R
it+1
]+(1−x
it
)E
t
[R
0t+1
]

−
c
2

x
it
q
∗
it
HH
∗
t
!
2
(1−HH
∗
t
)−h

x
it
q
∗
it
HH
∗
t
!)
. (5.2)
The asterisks in equa tion (5.2) denote equil ibrium values for ca pital and fees t hat are determined
by equations (2.6) and (2.9). The investor allocate s c apital to the fund based on its expected return
x
∗
it
E
t
[R
it+1
]+(1−x
∗
it
)E
t
[R
0t+1
] in equilibrium.
LEMMA 5. In equil ibrium, each manageri trades acti vely at timet only a fraction of his assets
under management given by
x
∗
it
=
h
f
∗
it

E
t
[R
it+1
]−1

HH
∗
t
−h
i
HH
∗
t
cq
∗
it
(1−HH
∗
t
)
. (5.3)
Lemma 5 shows that the managers with the lar gest alpha s are more active, and tend t o increase
the fraction of actively managed assets when fees and the Herfindahl index are lar ge. However ,
managers tend to become more passive for increasing fund size or average R& D costh . In equi-
librium, a declining HH
t
implies a rising number of funds, increasing talent among competitors
on average, and decreasing dispersion of talent. As the competition inte nsifies and opportunities
to obtain alpha deplete, active managers become more passive. Indexing is le ss risky than active
management, suggesting that aggregate risk is reduced asx
∗
it
decreases for every fund. The funds
transform over the life cycle from investment vehicles that of fer positive surplus from active trad-
ing to a lar ge pool of capital where risk and performa nce are similar to indexing at the mar gin. This
34
is the intuition of “today’ s alpha” becoming inevitably “tom orrow’ s beta”, where the opportunities
for alpha are gradually competed away and assimilate d into the benchmark.
1
PROPOSITION 4 ( T oday’ s alpha is tomorrow’ s beta ) . Manager gr oss alphas decline in absolute
magnitude, and the average acti ve fund becomes the benchmark as the funds hold an incr easing
fraction of the market portfolio. The aggr egate risk is r educed over time. At the end of the li fe
cycle, the asset class becomes a perfectly competitive market and net entry is zer o asHH
∗
t
→ 0 .
The investor ’ s aggrega te de mand for ac tive management remains lar ge during the late stage
of the life cycle despite the poor track records on average. Although a very competitive group of
funds is deficient in alpha, it provides a pla tform to investors for lar ge managed pools of capital
that are scalable a nd well diversified. The scalabi lity stems from indexing the bulk of managed as-
sets, because indexed capital is not subject to diminishing returns to scale. The investor diversifies
his wealth optimally across managers on the ef ficient frontier , instead of a llocating all capital to
the pa ssive fund and be subject to a single source of ma rket risk. In principle, the life cycle will
evolve as long as there are investors who chase alpha, until alpha is fully depleted in the aggregate.
However , Proposition 4 implies that investors also value t he benefits of diversification and cus-
tomized portfoli o services that are ava ilable only by active funds. This is a potential explanation
for the popularity of equity mutual funds, despite the lack of superior returns on average relative to
passive funds.
5.2. Closed-end funds and impr oving skill
The quadra tic tr ading cost in the main setup implies diminishing returns t o scale at the fund
level. This implies that capital inflows following lar ge realize d returns may erode the fund’ s future
performance. Some managers may choose to close their fund to new investme nt and cap their
assets, so that they can maintain lar ge returns. This practice does not cha nge the dynamics of the
1
For instance, Stulz (2007) predicts that the performa nce gap between hedge funds and mutual funds will narrow ,
and hedge funds will become more regulated and less risky in the future.
35
model. The closed-e nded funds are still subject to diminishing returns to scale at the industry level,
because the tra ding cost includes the Herfindahl index. The performance for every fund depends
on the environment where managers compete, in addition to the fund’ s own size. A closed-end
fund that invests its assets actively may still be forced to exit as the average talent increases over
time andHH
∗
t
declines. As a result, e nforcing a fund size cap is not a dominant strategy in the long
term, because it is optimal to be open to new investm ent and char ge fees while indexing a fraction
of the assets under management.
Another possible extension is to allow improvements in managerial skill with experience. It
is likely that ma nagers of long-lived funds have improved their ability in identifying profitable
opportunities or reducing c osts. Thi s can be modeled by extending the exogenous talent level τ
i
with the sum of a fixe d component and a linear trend. This would extend the manager ’ s lifeti me,
but i t doe sn’ t change the dynamic s of the life cycle at the industry level. The results in Propositions
1 to 4 involve aggregated variables, and are invariant to the identity of incumbent managers. Other
extensions to the functional form of talent, such as models with manager ef fort and career concerns
(Holmström, 1999) cannot change the evolution of the life cycle either for the same reason.
36
Part II
Empirical Analysis
37
Chapter 6
The data
The data involve an ensemble of multiple databases that cover six distinct fund classes. Each
fund class is distinguished by its unique set of ri sk factors, reflec ting the unde rlying assets that the
managers invest in. The classes are equity funds, hedge funds, bond funds, money market funds,
index funds, a nd exchange-traded funds (ETFs). Summary statistics are shown in T ables 1 and 2.
Figure 3 also shows the time series of variables related to competition.
The Survivor -Bias-Free Mutual Fund Databa se from the Center for Research in Security Prices
(CRSP) is the data source for all fund classes except hedge funds. The starting year for equit y funds
is 1961, for bond funds is 1976, for money market funds is 1972, for inde x funds is 1979, and for
ETFs is 1998. The last month in sample is December 2015. The CRSP summary file contains
supplementary fund information such as fund na me, investment style, and fees. I consider only the
expense ratio for the fees of m utual funds. CRSP also contains the time seri es for the US total stock
market c apitalization. I collect the time series for the US total issued debt (governme nt, munici pal,
and corporate) from the Bank for International Settlement s (BIS) stat istical file.
The features of hedge fund data are significantly dif ferent than those of mutual funds. Hedge
funds are subject to fewer regulatory restrictions, and reporting of information to commercial
databases is voluntary . Contrary t o mutual funds, there is no unique ve ndor that covers the ma-
jority of the hedge fund universe. For instance, Lipper T ASS c overs approximately one third of
38
this universe (Agarwal et al., 2009). I mer ge the hedge fund data from HFR, Lipper T ASS, and
Morningstar CISDM tha t are available through the W harton Research Data System (WRDS). The
sample period ranges from January 1994 to December 2015.
The empirical literature has documented mul tiple biases in hedge fund data (Fung and Hsieh,
2000). Biases arise due to sampling from an unobserva ble population of funds that refl ects the
underlying structure of the industry and the voluntary nature of disclosure. Biases can also be
injected from the process of information collec tion by the data vendors. The mer ging of multiple
databases mitigates some of these biases (see bel ow for more details). The hedge fund data vendors
also list funds of funds (FOF) in the databa ses. The assets of FOFs reflect capital that investors
allocate indirectly to the hedge fund industry . I exclude t hese funds from the m ain data set, because
their business model and fee structure are dif ferent than that of regular hedge funds. In addition,
their assets are not included in the estimation of aggregate variables to avoid double-counting.
I assume a unique set of factor -mimicking portfolios for each fund class. I use theq -factor model
of Hou e t al. (2015) for equity funds. Theq -factor data extend from 1967. For the period 1962 to
1966 I substitute the fourq factors with the four factors from Carhart (1997). The market return
and size factors are common between the two da tabases. However , the c orresponding market return
factors dif fer by 10% on average. I use an aggregate bond index for bond funds, and the market
return wit h the FED funds rate for money market funds. Simil arly to Fung and Hsieh (2000), I
construct an equally-we ighted index of FOF returns that serves as a ri sk factor to estimate hedge
fund a lphas. I descri be be low in more detail the main risk factors for each fund class, as well as
alternative benchmarks that are used for robustness tests.
6.1. CRSP data
The time series for each fund includes its mont hly total returns per sharemret , net asset value
per share mnav , and total net assets mtna as of month end. The first date is December 1961.
However , many funds before March 1993 have only quarterly values for their total net assets and
39
Figur e 3
T ime e volution of Industry Si ze (top l eft), the number of f unds T otal Funds in 1000s (top right), the
Herfindahl Index (middle left), the dispersion of fees in the cross-secti on Std(Fund Fee s) (middle
right), the A verage T alent (bottom left), and Std(T alent) (bottom right). Industry Size is the aggre-
gate size within the fund class di vided by the total m arket capitalization (US stock market for equity
funds; US debt market for bond and money m arket funds; weighted mean of US stock and debt
markets for ETFs, inde x, and hedge funds based on fund investment styles in T able 2). A verage
T alent and Std(T alent) are the cross-sectional mean and dispersion of tale nt in act ive i nvesting. A
manager ’ s talent is measured by the estimated fund dummy from OLS fixed-ef fect similar to those
in T able 3. The shaded are as indicate the NBER recession periods in US.
40
monthly for their net asset value. I use linear i nterpolation with mnav to fill in the regularly-
gridded missing values for mtna during periods with quarterly reporting. I ac cept interpolated
mtna values when the result lies between the two actual data values at the start and end of the
quarter , otherwise I assume constant assets. In addition to the r egularly-gridded missing values
from quarterly reporting for early funds in the data, CRSP has multiple scattered missing values
of the total assets. If the time series includes values formtna in the previous month and a return
mret for the current month, then I fill in the current assets by assuming no monthly flows from
investors. If the return is also missing, then I assume constant assets. Finally , funds with missing
or duplicate dates in their time series are exclude d.
CRSP ut ilizes the 4-letter objective c odecrsp _obj _cd to unify the W iesenber ger , Stra tegic In-
sight, Lipper , and Thomson Reuters objective and class codes that characterize the fund styles. The
updates on the supplemental information have varying frequency among funds. I mer ge the time
series a nd summar y file s by assuming constant values over time for the supplementary informa-
tion in between updates. In addition, CRSP assigns a dif ferent fund identifier for each of the share
classes of a single fund (opera ted by the same fund ma nager). I use the fund name to consolidate
all the share classes under a single fund, and reassign to the fund a new and unique identifier . The
separators for share classes include the rightmost semicolon (;) or slash (/) c haracters in the fund
name. Previous studies have used the colon (:) as a share class separator too. However , the colon
character in CRSP is also used to designate fund family names, and its use may result in incorre ct
consolidations of dif ferent fund managers within the same family of f unds. The fund le vel returns
are calculated as weighted aver ages of the returns from each share cl ass, where the weights are the
total net assets of each class. The total assets at the fund l evel is simply the aggregate of all assets
from the share classes.
The 4-letter CRSP objective code crsp _obj _cd by itself is not always ade quate to classify a
fund within a fund class, because of data errors and sporadic changes in investment style over time.
Certain keywords in the fund nam e are used to properly cl assify funds. I inc lude all CRSP codes
withED as the first two letters for US equity mutual funds, including cap-based, style-oriented,
41
and sector funds. The only exception are rea l estate funds (EDSR ), because a subset of them
trade REIT s and they tend to be riskier on average than the stock market. I remove funds with the
following keywords in thei r nam e: index, ETF , real estate, mortgage bond, fixed, yield, “ m uni” (the
leading blank allows equity funds that focus on the communications sector , but excludes municipal
bond funds that have erroneously an equity objective code), money , m/m, cash, liq, and f oreign
place keywords such as euro, asia, Japan, “Canad”, pacific, world, foreign, int’l, and i nternational.
In addition, I require that the majority of a fund’ s assets be invested in equity , to exclude alternative
mutual funds from the equity class.
The fixed income-rela ted keywords above are used t o classify funds in the domesti c bond fund
class, except those that imply short-term debt. These are: money , m/m, cash, liq, and “short term”.
The objective codes for the bond class are: I , IU , IUS , IUI , IUH , IC , ICQH , ICQM ,
ICQY , ICDS , ICDI , IG , IGT , IGD , IGDS , andIGDI . The keywords that are related
to short te rm debt a nd the obj ective codes IM and IMM are used to identify money market
funds. Index funds are identified by the keywords “index” , while ETFs are found with the use of
the dummy variable et
f
lag and the keywords “etf”, “exchange traded”, and “exchange-traded”
respectively . There are no restrictions on the objective code for index funds and ETFs. I require a
minimum expense ratio of 1 basis point for all funds from CRSP .
6.2. Hedge fund data
The voluntary disclosure and the dif ferent criteria for inclusion among hedge fund data ven-
dors results in selection biases. A selection bias implies that the sample in a database does not
fully reflect the features of the true population of hedge funds. It is li kely that the best pe rformers
voluntarily report their da ta to signal their performance to investors and attract new capital, while
the worst performers would choose to conceal their returns. On the other hand, a fraction of the
top performers may choose to stop re porting once the fund’ s goals have been reache d a nd no new
capital is needed. Selection biases may also arise from the vendors’ inclusion criteria. For instance,
42
not all databases accept mana ged futures funds. In addition, there exists geographical clustering for
regions ot her tha n US and Europe. Most of the funds in Brazil are listed in T ASS, while HFR and
Morningstar CISDM cover the ma jority of Chinese funds. HFR also covers the majority of funds
in Japan and South Africa . The mer ging of these databases mitigates the selection bias problem,
but it does not eliminate it.
The data licensing in WRDS is dif ferent than the direct comm ercial licenses from the se vendors.
Fund name and other supplementary information is either limited or unavailable. T o a void double-
counting of funds that are listed in more than one da tabase, I compare the corresponding time
series of tot al net assets and retur ns across all funds from dif ferent databases. I acce pt a ma tch
across databases when the overlapping series for both assets and returns dif fer by le ss than 10%.
Figure 4 shows a V enn dia gram that illustrates the outcome of the mer ging process among these
three databases. Once a match is found, I extend the tim e series to the longest available for the new
fund, including data available for non-overlapping months too.
T ASS
33.47%
HFR
39.30%
CISDM
26.02%
0.21%
0.09%
0.27% 0.64%
Figur e 4
The total number of unique funds at the 10% level of accuracy among CISDM, HFR, and T ASS
is 57659. The perc entages in the V enn diagram illustrate the fra ctions of single-, dual-, and tri ple-
listed funds across these databases.
The hedge fund data from every vendor before 1994 included only active funds at the end of
the sample period, a c ollection m ethod that resulted in data with survivorship biases. The direction
43
of the bias is hard to assess, because some funds become de funct after poor performance while
others may cease to report once they don’ t need investor capital. The survivorship bias cannot be
fully mitigated, because it is a linked wit h t he underlying structure and evolution of the hedge fund
industry and voluntary reporting (Fung and Hsieh, 2002; Aiken et al., 2013). I accept fund data
after January , 1994, when the vendors began including defunct funds in their data .
Another bias in hedge fund data i s the backfill or “instant history” bias. It involve s the back-
dating of the fund’ s performance before it started reporting to a vendor . The backfill bias most
often stems from an incubation period for the fund, where seed capital is usually provided by the
manager ’ s close circ le of contacts. Evans (2010) shows that incubation in mutual funds may also
result in backfill bias. Fung a nd Hsieh (2002) find that the ave rage incubation period in T ASS is
12 months. However , there exist cases where the gap between t he fund’ s inception and the date
that it was first added to the database exceeds 10 years. Previous studies have considered fund data
only after the date that it started to report, to eliminate the ba ckfill bias. However , this practice may
result in a substantial loss of informat ion Fung and Hsieh (2009). In addition, t he WRDS version
of CISDM does not list the date that the funds were added to the sample. I apply the following rule
to balance the tradeof f between the backfill bias and the informational l oss from removing obser -
vations. I begin with a test date that is 12 mont hs past the first date of the fund’ s time series. If the
test date is subsequent to the date that the fund was added to the database, then the latter becomes
the of ficial starting date for my sample. Otherwise, the test date becomes the starting date. This
ensures tha t the discarded observations either comprise the full time period before the fund was
added to the database, or are a maximum of one year aft er the fund’ s inc eption.
Hedge funds char ge a manage ment fee as a perc entage of assets under manage ment, a nd may
also inc lude inc entive fees tha t are continge nt on performance . T he ince ntive f ees may also include
a high-water mark and a hurdle rate. Incentive fees in my sample are colle cted after the fund has
completed a full calendar year , with the proceeds spread throughout the next year from January to
December . Missing values for the management fee, incentive fee, and the hurdle rate are replac ed
by 2%, 20%, and 0.5% respectively . I also use the currency exchange rates from the H.10 Relea se by
44
the FED to convert the values of the fund assets from foreign denomina tions to US dollar amounts.
I consolidate the fund styles in seven broad categories, each involving the following sub-styles
from the various databases:
1. Arbitrage : event driven, rela tive value, convertible arbitrage, fixed income arbitrage, debt
arbitrage, mer ger arbitrage, diversified arbitrage, c urrency , and distressed securities
2. Hedging : equity hedge, dedicated short bias, long/short e quity hedge, bear market equity ,
biased, and any instance of “hedged” or “long/short” in the style fie ld
3. Multi-strategy : multistrategy , CT A, managed futures, systematic futures, options, volatility
4. Emer ging Marke ts
5. Fund of Funds
6. Macr o
6.3. Factor -mimicking portfolios
The empirical factors for asset cl asses other than equity mutual funds are not well established
in the lit erature. Fama and French (1993) showed that a credit risk and a term structure factor
explain bond returns better than the thre e equity factors. The term structure factor is equivalent
to the level factor in traditional term structure models. However , the se two factors cannot explain
well the returns of junk bonds. Elton et al. (1995) showed t hat the ef fect of the term structure factor
is captured by an aggregate bond inde x and the stock market excess return. I use the market return
and relative dif ferences of the Barclays US Aggregate Bond Index for investment-grade bonds,
which is an update to the Lehman index that is used in Elton et al. (1995). The inception date for
the Barclays index i s January , 1976. I substitute the index with a term risk factor for dates before
that. The te rm risk factor is similar to that in Chen et al. (2010), namely the dif ference bet ween
the T reasury Constant Maturity 10-year and 1-year rates from the H.15 Relea se by the FED. The
45
default risk fact or is the credit spread between the Baa and Aaa bonds, also available by the H.15
Release.
I also use relative dif ferences of the Barcl ays US Corporate High Y ield Bond Inde x for junk
bonds. Its inception date is July , 1983. I exte nd the junk bond factor back to January , 1977 with
the High Y ield Bond Index return of Blume et al. (1991). They created the ir high-yield factor
with data from the Drexel Burnham Lambert investment bank. The DBL bank pioneered the junk
bond market starting in 1977. Since junk bonds did not exist before that, I use two sepa rate factor
models for bond funds. The first spa ns the period 1962-1976 and does not use a high-yield factor .
The second ranges from 1977 to the end of the sample, and uses the c onsolidated hi gh-yield fac tor .
Similarly to Elton et al. (1995), I use a macro factor to capture the une xpected changes in
expectations about the US economy that could af fect the interest rates. The factor com bines the
monthly relative changes in the Conference Board Composite of 4 Coincident Indicators Index
with the lagged relative cha nges in the survey of Expe cted Business Conditions for Next Y ear by
the University of Michigan. The Conference Board (CB) composite index comprises the ef fects of
coincident indicators on non-agricultural labor , personal income less transfer payments, industri al
production, and manufacturing and trade sal es. The factor capture s the surprise ef fect t o investors
and consumers, defined as the percent changes in the realized econom y m inus t he lagged percent
changes in t he expectations about the economy for the following ye ar . The bene fit of using CB
indices is the monthly frequency of data, contrary to re al GNP , inflation, or other macro factors that
are available only at a quarterly frequency .
The factor model for the money market funds uses t he Federal Funds rate as a proxy to money
market indices, the stock mar ket exce ss return, and the macro factor defined above. The index fund
and ETF cla sses are the de facto representatives of passive investment strategies. A regression of
their return on the return of the index that they track should have a zero i ntercept. For simplici ty , I
use the stock market excess return for all index and ETF funds.
I consider all hedge funds as a single class, and use a single set of fa ctors to eval uate their perfor -
mance. I use two benchmarks for robustness t ests. One benchma rk is the equally-weighted avera ge
46
index of returns from FOFs in my sample. The second benchmark involves a set of exogenous risk
factors. This set is a variation of the seven-factor model of Fung and Hsieh (2004). I use the stock
market excess return and size factors of Hou e t al. (2015) for equity strategies, the investment-grade
aggregate bond inde x and default risk factors as defined previously for fixed income strategies, the
commodity and currency trend-following factors from Fung and Hsieh (2001) for the corresponding
strategies, and the macro factor for the unexpected cha nges in the US economy .
The choice of a single set of factors for hedge funds is not unique in the empirical lite rature.
Patton and Ramadorai (2013) use the Bayesian Information Criterion to fit the best two out of
four Fung-Hsieh factors for each style. However , the fund style is also voluntaril y reported by the
manager to the data vendors, and is not updated over time along with the return and asset time series
for the fund. The absence of updates may re sult in stale information and induce a “sta le information
bias” to the fund’ s alpha if a style-specific set of factors is used. In addition, it is possible that the
reported style does not encompa ss the full set of strategies that the fund is using. For these cases, the
self-reported style may induce the cherry-picking bias that Sensoy (2009) document s for mutual
funds, where the manager attempts to choose a niche benchmark that he can easily outperform.
The che rry-picking bia s may be coupled with selection biases for hedge fund alphas. For instance,
a manager may start reporting to HFR while he is already outperforming the HFRI index. The
addition of his fund to the database would signal to inve stors his superior performance in the HFR
universe of funds.
6.4. V ariable definitions
The data for all fund classes have monthly frequency . The variables at the fund leve l include
Fund Age (in years), the fund’ s assets under management Fund Size (in billion dollars), and the
net return plus fees Gr oss Return . Fund Fee is the expense ratio for all fund classes exc ept hedge
funds. The fee structure of hedge funds includes mana gement, incenti ve, high-water mark, and
hurdle fees. Funds in every class are included in the sample only after they exceed for the first ti me
47
a threshold of $10 million in December 2015 dollars. The fund’ s size is allowed to fall be low this
threshold subsequently , but all observations before the first time that the fund exc eeds the threshold
are ignored. Removing funds from the data that neve r surpassed a certain threshold is recommended
to eliminate many smal l hedge funds are not really open to investors, but rather manage the assets
of a small group related to the manager . Dropping such funds from mutual fund data also mitigates
survivorship bia ses from reporting conventi ons Elton et al. (1996) or from funds that never were
open to investors because of incubation Evans (2010).
The net return plus fees minus the benchmark return is the fund’ s Gr oss Alpha . The fund alpha
reflects a fund manager ’ s potential in outperforming the benchmark return. Gr oss Alpha is the
intercept from regre ssions of the fund’ s Gr oss Return on the class-specific benchmark portfolios.
The regressions have a rolling window of five years to account for potentially time-varying fund
betas on benchmark portfolios.
T able 1 shows that most active fund classes have a small positive a verage for fund Gr oss Alpha
in sample. Jensen (1968) suggested a potential upward bias in the estimation of al pha when fund
betas on benchmark portfolios are tim e-varying (equati on (12) of his paper). The upward bias in
alpha follows from a downward bias on estimated fund betas. According to Jensen, the bias in beta
estimates arises when managers with true skill capitalize on price movements in the aggregate that
are related to the portfolios used for benchmarking. This bias is likely to appear in empirical studies
that use the same factors throughout the full sample. Magkotsios (2017b) shows that the benc hmark
itself can evolve over time along with the fund class, as i t grows from a small set of managers to
a lar ge number of funds that hold a signi ficant fraction of a vailable assets in the market. Positive
true alphas could exist early in the life cycle when only a few managers capitalize on strat egies
that become part of the benchma rk subsequently . The separa tion between true skill relative to the
benchmark and measurement error is quite challenging when the sample a verage gross alpha is
positive. Such a task for every fund class is beyond the scope of my anal ysis, although I perform
robustness tests to mitigate the ef fect of outliers.
48
The flow of a fund is defined as the percentage gr owth in the assets under management that
stems purely from new investment. This definition isola tes the capital transfers by i nvestors to and
from the fund, while it excludes investor capital gains and the growth of the fund’ s assets from the
returns earned by the fund. The flow of a specific f undi at timet is defined as
Fund Flow
it
≡
q
it
−(1+r
it
)q
it−1
q
it−1
, (6.1)
whereq
it
is Fund Size andr
it
is the observed fund return net-of-fees. I winsoriz e Fund Flow at
the 1% a nd 99% to mitigat e t he e f fect of outl iers from dat a e rrors on fund size and return. Fund
managers with superior performance can dif ferentiate themselves from rivals and attract lar ger
flows of capital. However , funds can dif ferentiate themselves in other ways too. For instanc e,
Nanda et al. (2009) show that funds with addit ional share s classes (each shares class of fering a
dif ferent fee structure) attract significantly lar ger inflows than funds wit h a single shares class,
after controll ing for performance and other fund attributes. These types of dif ferentiation can imply
market power and barriers t o entry . The latter can stem from rising levels of ave rage talent in the
cross-section (Magkotsios, 2017b), investor participation costs (Sirri and T ufano, 1998; Huang
et al., 2007), or sticky investor flows (Sialm et al., 2015).
Some aggregate variables for each f und class include the t otal number of funds T otal F unds ,
and the percentage growth in the number of funds Fund Gr owth . The Herfindahl Index is defined
as the sum of squared ratios of Fund Size over the aggregate size across all incumbent funds. It is
the weighted average of the market share for each fund relative to the total assets managed across
all funds, where the weight is the market share itself. Therefore, the Herfindahl Index measures
concentration within the fund class. Further aggregate va riables include the cross-sectional disper -
sion, skewness, and kurtosis for Fund Size . A verage T alent and Std(T alent) are the cross-sectional
mean and dispersion of talent in active inve sting. A manager ’ s talent is m easured by the fixed ef fect
from OLS regressions similar to the specifica tions in T able 3, which is the ma in re gression ta ble
for returns to scale.
49
The Industry Size is defined by the ratio of total assets managed within the fund class relat ive to
the size of the market where the funds operate. The market for equity funds is the US stock ma rket,
while for bond funds and money market funds it is the US total debt market. The ma rket for index
funds, ETFs, and hedge funds is a weighted mean of US stock and debt markets based on fund
investment styles. T able 2 shows that 75% of index funds and ETFs focus on equi ty indices. The
weights for hedge fund strategies are approximately 40% debt-rela ted (arbitrage, m ulti-strategy ,
and macro) and 60% equity-related (emer ging markets, hedging, and m arket-neutral).
The panel regressions use recursively demeaned (RD) variables to control for lagge d stoc hastic
regressor bias (Hjalmarsson, 2010; Pástor e t al., 2015). Similar to OLS with fixed ef fects, the RD
estimations control for the unobserved component of each fund, including managerial talent. OLS
with fixed ef fects is equivalent to demeaning every variable by its c orresponding full-sample mean.
Each va riable in the RD estimations is de meaned onl y by the average of future or past observations
for forward- or backward-demeaned time-series respectively . I cluster the standard errors in two
dimensions by fund and style-month.
50
Chapter 7
Returns to sc ale at the industry level
7.1. Diminishing r eturns to scale
T able 3 shows the relation between Gr oss Alpha with Industry Size and Fund Size . The corre-
lation between the average fund’ s return and Industry Size describes returns to scale at the industry
level. The first regression in thi s t able corresponds to the full sample with 2015 as the last ye ar of
observations. The estimat e for Industry Size in equity funds is negative, implying that the se funds
have dimini shing returns to scale. T his result is also consistent with Pástor et al. (2015). The co-
ef ficient of Fund Size corresponds to fund-level returns to scal e. The estimate for equity funds is
positive, implying increasing returns with the fund’ s size. This result seems inconsistent with the
assumption of di minishing returns to sca le in most the oretical models. Previous empiri cal evidence
on the existence of diseconomies of scale among equity mutual funds is mi xed.
The impact of Industry Size on the ret urns of the average equity fund is lar ger than that of Fund
Size . T he average total assets managed by equity funds and total US stock market capitalization
during the sample period are $2.32 and $13.36 trillion respect ively . The average number of funds
every month is 1 195. T he first regression im plies that an increase in every fund’ s assets by $0.1
billion would increase t he monthly return of each fund by 3.9· 10
−5
∗ 0.1 = 0.039 bp (0.47 bp
annually) from fund-leve l returns to scale. On the contrary , the Industry Size would increase by
51
0.00733 and the monthly return of each fund would decrease by 0.018∗ 0.00733 = 1.3 bp (16
bp annually) from i ndustry-level returns to sca le. T he same ar gument a pplies for hedge, bond, and
money market funds. Therefor e, economies of scale in active asset management operate mostly at
the industry level, rather than at the fund level.
The model of Magkotsios (2017b) and the m ain hypothesis in Pástor et al. (2015) suggest that
the profitable investment opportunities within a fund class are in finite supply . The diminishing
returns at the industry le vel emer ge from the depletion of these opportunities. As the fund class
grows and the competition among managers intensifies, more funds will chase the same investment
opportunities. Although each manager is atomistic and cannot i mpact prices on his own, the prof-
itable investments become costlier and more elusive as an increasing number of managers push
prices in the same direction.
The opportunities to outperform in the information hypothesis of Gârleanu and Pedersen (2016)
are deple ted as a lar ger amount of capital is allocated to informed managers and markets become
more ef ficient. The aggregate size wit hin the fund class grows when the cost of active investing
decreases. The cost of active investing includes both the searc h cost for the investors and the
managerial cost of obtaining information about profitable opportunities. The investors gradually
make market s more ef ficient and at tenuate the comparative advantage of informed mana gers by
allocating to them more capital. This mechanism also implie s diminishing returns to scale for all
incumbent managers.
The second and thi rd pane ls in T able 3 show the ef fect of size on returns for index mutual funds
and exchange-traded funds. These funds represent passive strategies and are not expected to seek
abnormal returns. The dependent variable is Gr oss Return , defined as the fund’ s net return with the
expense ratio added back. The coef ficients on Fund Size and Industry Size are statistically insigni f-
icant, suggesting that the returns of passive funds are unaf fected by sca le. Passive strategies nei ther
require managerial talent nor do they depl ete any opportunit ies for alpha. Therefore, any increases
in size should have no ef fect on fund returns. This result is also compatible with the informa tion
52
hypothesis. The passive managers have no ef fect on the market’ s inf ormational ef ficiency , and any
increase in their capital should not af fect their returns.
7.2. Incr easing r eturns to scale
The first column of T able 3 shows positive coef ficients for Industry Size among hedge funds and
fixed income funds. This striking result indicates that t hese funds e xperience increasing returns to
scale at the industry level. In particular , the aggregate growth of these industri es implies an increase
in the returns of the average fund. Increasing re turns to scale in Magkotsios (2017b) arise within
less competitive fund classes with plenty of opportunities for abnormal returns. The growth of
aggregate investor demand in such fund classes induces the entry of more talented managers over
time, resulting in increased returns for the average fund.
On the other hand, incre asing returns to scale a t the industry level a re puzzling for the informa-
tion hypothesis of Gârleanu and Pedersen (2016). Increasing ret urns in their setup would suggest
that either the market becomes less ef ficient as the fund class grows and informed managers oversee
a lar ger fraction of the total size, or that more ef ficient markets imply l ar ger returns for t he average
fund. However , an increasing fraction of capital among informed managers cannot plausibl y make
the marke ts less ef ficient. The acquisition of information is the comparative advantage of active
managers against passive indexing. More e f ficient markets weaken this advantage, and should thus
decrease the excess returns from active investing. If increased returns were t o coexist with more
ef ficient markets, then the re must be an additional underlying mechanism that compensate s for the
loss of informat ional value. As a result, the informationa l ef ficiency hypothesis by itself cannot
explain the increasing returns to scale at the industry level. For instance, Cao et al . (2016) show that
the holdings of levered hedge funds that were broke red by Lehman Brothers e xperienced dec lines
in price ef ficiency during the financial crisis. However , they also show that hedge fund ownership
of stocks results in improved ef ficiency on average.
53
T alent in active investing and alpha for bond funds and money market funds is defined just as
in any other fund class. Interest rate risk exists even for short-term debt, and credit risk becomes
important for long-term debt. In addit ion, bond funds and money market funds tend to hold deriva-
tives in their portfolios (Schulte et al., 2016). As a result, some funds may be able to provide lar ger
yields to investors than others. Increasing r eturns to scale within the bond and money market fund
industries indicate the existence of profitable opportunities that can be exploited through active
investing (Magkotsios, 2017b).
7.3. The life cycle for r eturns to scale
The discussion of the first column in T able 3 has highlighted a heterogeneity of industry-level
returns to scale across fund classes. The rest of the c olumns of this table show that returns to sca le
are time-varying. Magkotsios (2017b) predic ts that every fund c lass starts with increasing returns
to scale that gradually decrease over time, until the fund c lass operates with dimini shing returns.
Each column i n T able 3 involve s a dif ferent end year in the sample, corresponding to a dif ferent
stage of the life cycle. The coef ficients for Industry Size outline t he evoluti on for the returns to scale
at the industry level. For every fund class the c oef ficient increa ses when the sample is restricted to
earlier stages of its life cycle.
Figure 5 extends the recursive regressions in T able 3 t o annual frequency for all types of funds.
The top graph depicts the evolution of returns to scale at the industry level for ea ch fund class.
The blue curve shows that the returns of equity funds were initially increasing with Industry Size ,
but that sensitivity was declining as the fund class was growing during the stock market rally in
the 1990s. The coef ficient for Industry Size becomes negative after the dot-com crisis, implying
that the returns of equity funds started to decline. This curve illustrates clea rly a full life cycle for
equity funds that is consistent with the model of Magkotsios (2017b). The evolution for bond and
money market funds is sim ilar . Bond funds still have increasing returns to scale. This suggests
that profitable opportunities for abnormal returns within these fund classes have not been depleted
54
yet. Money market funds had increasing returns to scale early in their life cycle, but the correlation
between returns and Industry Size has been near zero since 2000.
Hedge funds have had a more complex evolution. Returns to scale at the industry level are ini-
tially increasing until mid-2000s, and become decreasing afterward. This would imply that their life
cycle has already matured. However , the number of funds during the period 2012-2015 decreases
by 3000 and Industry Si ze decreases from 0.12 to ne arly 0.06. The positive correlation between
fund returns and Industry Size in Figure 5 implies that fund ret urns also decrease on average. The
competition hypothesis of Magkotsios (2017b) im plies that investors gradually filte r out the worst
performers and augment the re-emer gence of increasing returns to scale . It i s plausible that the
transition from decreasing to increasing returns to scale within a two-year peri od i s c aused by a
shock to the long-term lif e cycle of hedge funds. Many hedge funds experienced lar ge outflows of
capital in the aftermat h of the financial crisis of 2007. This i s also consistent with the hypothesis
that investors filter out the worst performers from their portfolio.
Hedge funds have existed for ma ny years before the dot-com crisis, but they started to grow
substantially following the crash. This growth was fue led by the transition of the inve stor base from
high-net-worth individuals and family funds to lar ge institutiona l investors that were transferring
capital fr om equity funds Fung and Hsieh (2012). A plausibl e explanati on for the initially increas-
ing returns to scale is that hedge funds provided fresh liquidity on investm ent strategies tha t could
help institutional investors hedge against market crashes. For i nstance, Kruttli et al. (2015) mea-
sure the aggregate liquidity of the hedge fund industry and show that these funds supply liquidity
to asset markets. The liquidity that was originated by i nstitutional capital inc reased the returns for
all hedge funds by decreasing their cost of active trading. Jame (2016) shows supporting evidence
of lar ger returns for liquidity-supplying hedge funds.
The hedge fund industry in my sample peaks in size and number of funds by 2008, and has been
shrinking following the financial crisis. The maximum Industry Size that the fund class achieved
with respect to the US stock ma rket is 0.1 16, which is lower than that of domestic equity funds.
This number could be smaller if the sample is re stricted to hedge funds with a domicile within
55
Figur e 5
T ime e volution of industry- level returns to scale (top) and the correlation between Fund F ees and
fund performance Gr oss Alpha (bottom). The graphs depict t he estimated coef ficients for Industry
Size and Fund Fees from regressions similar to the specifications in T able 3 for equity funds (blue),
bond funds (green), money market funds (yellow), a nd hedge funds (red). Ea ch estimate stems
from recursive regressions with varying cutof f ye ar . The x-axis shows the cut of f year , i.e. the
last year in sample. The plots show only t he statistically significant estimates. The shaded areas
indicate the NBER recession periods in US.
56
the US. The re-emer gence of increasing ret urns to scale may imply that the new opportunities for
investing in this industry appear as the previously bad-performing funds gradually exit ove r time.
Figure 5 also shows the ef fect of re cession periods other than the cri sis of 2007. Market crashes
may involve investor demand shocks that can impact Industry Size through fund flows. In addition,
a market crash may a lso trigger many fund exits, which would af fect the distribution of talent
among the remaining incumbents. Figure 5 shows only tra nsitory variations in the coef ficient of
Industry Size during re cession periods. This implies that returns to scale at the industry are relativel y
unaf fected by business cycle shocks in the long term.
The bottom graph in Figure 5 shows the correlation between fund fees a nd expected perfor -
mance. Berk and Green (2004) predict that this correl ation is always positive, because managers
can increase their fees in expectation of good performance. Gârleanu and Pedersen (2016) also
derive a positive correlation, since informed managers can char ge higher fees because of their
comparative adva ntage against uninformed rivals. Magkotsios (2017b) predicts that this correla-
tion is ne gative when funds have increasing returns to scale at the industry level, and vice versa.
The graph shows a negative correlation for equi ty funds be fore the dot-com crisis and a positive one
after 2001. Fixed income funds have ne gative correlations between fees and performance. These
results are consistent with Magkotsios (2017b). Hedge funds fees a re positively correlated with
performance until the mid-2000s, but they are negatively correlated afterward. T he negative cor -
relation after the financial crisis of 2007 is consistent with the reappearance of increasing returns
to scale at the industry level.
7.4. The effect of competition
In the monopolist manager setup of Berk and Green (2004) there are no ef fects from the compe-
tition among ma nagers. Their model implies that industry-level returns to sc ale should be explained
by fund-level diseconomies of scale. However , T able 3 shows that the coef ficients for Fund Size
57
and Industry Size have quite dif ferent values for each fund class, making it unlikely that the super -
position of fund-level scale ef fects can explain the economie s of scale in the aggregate.
Returns to scale at the industry level emer ge from the competition among managers for a lim-
ited set of profitable investment opportunities in Magkotsios (2017b). The empirical proxies for
competition and the availability of investment opportunities include the Herfindahl Index , the num -
ber of incumbent managers T otal Funds , and the cross-sectional distribution of managerial talent.
The evolution of the se variables gives rise to economies and diseconomies of scale at the industry
level. The empirical prediction is that returns to sca le increase with the Herfindahl Index and the
dispersion of t alent in the cross-section Std(T alent) , while t hey decrease with A verage T alent and
T otal Funds . T able 4 tests this hypothesis for equity , hedge, money market, and bond funds. The
dependent variable in every re gression is Gr oss Alpha . The inter action terms with Industry Size
show the ef fect of competition on returns to scale. The table includes estimates for the goodness-
of-fit in models with and without interacti on terms. The adjusted-R
2
for every asset class increases
when an interaction term is included.
Equity funds satisfy all the pre dictions for the Herfindahl Index , A verage T alent , and T otal
Funds . The coe f ficients that include Std(T alent) are statistical ly insignificant. A potential expla-
nation that is consistent with the hypothesis above is that competition has already deple ted the
majority of investment opportunities for equity funds. For instance , T able 1 shows that Std(T al-
ent) has been persistently low throughout the sample pe riod for equity funds, and their number has
reached nearly 2800. Such a lar ge number of similarly talented managers imply that they crowd
into the same investment opportunities and average performance declines. The lack of opportu-
nities for managers is also consistent with diminishing returns to scale for this fund class, and an
overall poor performance of the average equity fund rela tive to passive funds.
The distribution of talent within hedge funds af fects adversel y their performance, because re-
turns decline as the A verage T alent increases and its dispe rsion Std(T alent) declines. However , the
returns to scale for this fund class are mostly af fected by Std(T alent) and T otal Funds , with oppo-
site signs than those pre dicted by the competition hypothesis. This result is still consistent with the
58
hypothesis of a long-term life cycle, as it may stem from a shock that has temporarily disrupted its
evolution.
Money market and bond funds operate under increasing returns to scale at the industry level.
The coef ficient signs of the interaction terms with Herfindahl Index and A verage T alent for bond
funds are t he sam e a s those for equity funds and consistent with the competition hypothesis. This
is an indication that the determinants of returns to scale at the industry level are common across
dif ferent fund classes. The coef ficient for Std(T alent) is negat ive, i.e. opposite than predi cted.
The returns to scale in money mar ket funds are near zero by 2015 and relatively insensitive to the
proxies for competition. The int eraction coef ficient with T otal Funds is negative as predicted, and
the interaction with Std(T alent) is nega tive and signi ficant during earlier stages of the life cycle (not
shown i n T able 4).
59
Chapter 8
Flow-performance r elation
Berk and Green (2004) derive a convex flow-performa nce relation from the competition among
investors for a manager with diminishing returns to scale a t the fund level. On the other ha nd, the
competition among managers in Magkotsios (2017b) results in a time-varying flow-performance
relation that is concave for increasing returns to scale and convex for diminishing returns to scale
at the industry level. The following tests for investor flows introduce a reduced form for the flow-
performance relation that can be either concave or convex locally . Its functional form for fundi at
timet is the following:
Fund Flow
it
=b
L
ln

α
it−1
+1

+b
H
α
2
it−1
+γControls
it−1
+ε
it
, (8.1)
whereb
L
,b
H
, andγ are constants andα
it−1
is the fund’ s Gr oss Al pha during the previous period.
This functional form embe ds the limited liabi lity of fund managers by restric ting the maximum loss
for a fund atα
it−1
=−1 , where the Fund Flow diver ges to−∞ asymptotically . The top-le ft graph
in Figure 6 shows a schematic of the flow-performa nce rel ation for various combinations of the
coef ficientsb
L
andb
H
. When the logarithmic term dominates the quadratic (b
L
b
H
) the flow-
performance relation is concave near zero Gr oss Alpha , and vice versa. When the two coe f ficients
are positive and comparable in magnitude, the flow-performance relation has an inverted S-shape
where it is concave for negative alpha and convex for positive a lpha.
60
T able 5 shows fits of equa tion (8.1) for active equity , hedge, bond, and money market funds.
The table demonstrates the evolution of the flow-performance relat ion with recursive regressions.
Each column corresponds to a dif ferent end year for the sample, re flecting dif ferent stages of the
life cycle for each fund class. The quadratic term is stronger than the log term for equity funds after
the year 2000, implying a convex flow-performance relation. On the c ontrary , this term is statisti-
cally insignificant until 1990 and the relation i s concave. This result is consiste nt with Magkotsios
(2017b). The relation switches curvature from concave to convex in 1993. Ippolito (1992) was
the first to suggest that the flow-performance relation for equity funds is non-linear , but the earli-
est studies that showed convexity are those of Sirri and T ufano (1998) and Chevalier and Ellison
(1997). Both studies used sam ples that ended after 1995. Further studies that verify the convexity
for this fund class are L ynch and Musto (2003), Huang et al . (2007), and Brown and W u (2016)
among others. See also the literature survey by Christof fersen et al. (2014). I show that t he relation
was concave until the early 1990s.
Getmansky (2012) includes a sample of hedge funds until 2003 and shows a concave flow-
performance relation for this fund class. T able 5 verifies this result for 2015, which is also consistent
with increasing returns to scale as Figure 5 suggests. The re gression for year 2000 is the first that
yields estimates, because the sample starts sinc e 1994. The flow-performance relat ion appears
strongly convex onl y for this year , and becomes concave from 2001. This result could either be a
statistical artifac t from the lack of data points, or simply c oincide with the transition of the investor
base to lar ger pension funds that started in 2001. The rela tion remains concave after that year until
2015, implying an inconsistency with the short-term appearance of diminishing returns to scale
in the mid-2000s (see figure 5). A potential explanation for the relati vely inert monthly investor
flows to economies of scale at the industry level is that hedge fund strategies tend to have a longer
horizon compared to other types of funds.
The flow-performance relation for bond funds is concave. This is consistent with the compe-
tition hypothesis, because this fund class has increasing re turns to scale at the industry level. The
concavity is also consistent with the main result in Goldstein et al. (2017) for corporate bond funds.
61
-1 -0.5 0 0.5 1
Lagged Gross Alpha
-1
-0.5
0
0.5
1
Fund Flow
-1 -0.5 0 0.5 1
Lagged Gross Alpha
-1
-0.5
0
0.5
1
Equity Fund Flow
2015
2000
1990
-1 -0.5 0 0.5 1
Lagged Gross Alpha
-1
-0.5
0
0.5
1
Hedge Fund Flow
2015
2008
2000
-1 -0.5 0 0.5 1
Lagged Gross Alpha
-1
-0.5
0
0.5
1
Bond Fund Flow
2015
2005
1995
Figur e 6
The top-left graph shows a schematic of the flow-performance relation for various parameter val-
ues. The dashed gree n line is the tangent at zero alpha for t he solid green curve, showing that the
inverted S-shaped flow-perform ance curve is less (more) sensitive to past performance for nega-
tive (positive) alpha compared to a linear flow-performanc e relation. The rest of the graphs show
the time evolut ion of the flow-performance relation in sample for equity funds (t op right), hedge
funds (bottom left), and bond funds (bottom right). The graphs show ext rapolations of the fitted
curve from regressions similar to the specifications in equation (8.1) and T able 5. The curves in
each graph stem from recursive regressions with varying cutof f year . Dif ferent colors correspond
to a dif ferent last year in sample. The dashed red line is the tangent at zero Gr oss Alpha of the
flow-performance relation for the full sample (cutof f at 2015).
62
However , this relation was convex until 2008, which is a puzzling result for all current theories.
The transition to a conc ave relation after 2008 may be related to the FED’ s quantitative easing
programs that were initiated that year , or a short-term flight to liquidity by investors during the
financial crisis. T able 5 implies that the investor flows to money market funds are unrelated to
performance.
The following tests add detail about the ef fect of the competition among managers on the sen-
sitivity of investor flows to fund per formance. T ables 6, 7, and 8 show the ef fect of Fund Fees ,
Industry Size , and T otal Funds respectively for equit y , hedge, and bond funds. I separat e manage rs
with positive a nd negative lagged Gr oss Alpha in the sample for eac h fund class, a nd perform two
separate regressions. The top panel i n every table reflects the impac t of competition fundamentals
on inflows to good performers and the bottom one on outflows from bad perform ers. An increase
in fees among funds with positive alpha increases the flows into hedge and bond funds, but it does
not change the outfl ows from funds with negative ret urns. Thi s implies a sharper concentration
of capita l to the most talented fund managers and a more convex flow-performance relation. An
increase of Industry Size reduces inflows to winning equity and hedge funds and increases outflows
from losing e quity funds. Finally , a rising numbe r of funds intensifies the competition among man-
agers. This results in reduced flows into winning equity and hedge funds, but increased flows into
bond funds.
8.1. Addr essing potential endogeneity pr oblems
The regressions for returns t o scale have performanc e as a function of assets. The relation
between flows and performance assumes that the growth in fund assets is a function of past per -
formance. Previous studies on the evaluation of fund performance and the sensitivity of investor
flows were developed in separate literatures. I relate returns to scale with fund flows in my analysis.
Interchanging the roles of the dependent and independent variables may result in endogenei ty prob-
lems and misspecified m odels. I introduce a time lag when testing for returns to scale relative to
63
investor flows, to mitigate the concerns about endogeneity between the two relati ons. The expected
return a t tim et in my setup is a function of fund-level and industry-level size at timet−1 for the
estimation of returns to scale, while the investor flow at timet+1 is a func tion of pa st per formance
at time t . The intuition for this time lag is that the current size of assets af fects the expe ctations
about future returns, but once those returns are realized the investor will react subsequently and
reallocate his capital across funds.
The regressions in T able 10 show tests for the dynamic connection across the three relations
that I measure: the flow-performance relation, the correlation between fees and performance, and
returns to scale at the industry level. The evolution of each relation is based on time series of
regression estimates for the main regressions in T ables 3 and 5. Each column in the T able 10
corresponds to an acti ve fund class, such as equity , hedge, bond, and money market funds. The
time-series of estimate s have monthly fre quency for all types of mutual funds. Hedge funds have
annual frequency , because they tend to be more illiqui d than mutual funds.
The first panel shows the dynami c connection between t he flow-performance relation and re-
turns to scale at the industry level. The dependent variable i s the dif ference β
Flow
≡ b
H
−b
L
,
whereb
H
andb
L
are the time series of re cursive estimates for the quadratic and logarithmic terms
in Gr oss A lpha respectively for the flow-performance relation in equation (8.1) and T able 5. Re-
turns t o Scale are the time series of recursive estimates in returns to scale at the i ndustry level, i.e.
the estimates for the correlation between Gr oss Alpha and Industry Size from the regressions in
T able 3 and the top graph in Figure 5. The second panel tests the connection between the flow-
performance and fee-performance relations. The dependent vari able is t he same as in the first
panel. Fee-Performance r elation are the time series of recursive estimates for t he correlation be-
tween Gr oss Alpha and Fund Fees from the regressions in T able 3 and the bottom graph in Figure
5. The third panel tests the connection between the Fee-Performance r elation (dependent variable )
and returns to scale at the industry level.
The estimates for Returns to Scale are lagged by one period compared to those for the fee
and performance relation and by two periods compared to β
Flow
. The estimates for the Fee-
64
Performance r elation are lagged by one period compare d toβ
Flow
. These lags imply a sequence
among the ef fects of size in expe cted returns, fund fees, and investor flows, and mitigate the con-
cerns about endogeneity bet ween the size of assets and performance. Magkotsios (2017b) makes
the following predictions:
1. Returns to Scale should be correlated negatively withβ
Flow
.
2. The Fee-Performance r elation should be correlated positively withβ
Flow
.
3. Returns to Scale should be correlated negatively with the Fee-Performance r elation .
T able 10 shows that ea ch of these predictions are satisfied for every asset class. The only exception
is the correlation between fees and returns to sca le for bond funds in Panel C. These results show
that fund performance, fees, aggregate assets, and investor flows are rela ted and evolve dynamica lly
under a specific pattern that is common across all types of funds.
Statistical models can suf fer from omitted-variable bias when the residual of the regression
is correlated to some regre ssors. Models similar to those in T able 3 measure returns to scale and
correspond to production functions. In these models, the dependent variable is the firm’ s output, the
independent va riables repre sent the input factors of production, and the residual can be associa ted
with shocks in the total fa ctor productivity (T FP). Input factors of production can include firm
capital, labor , aggregat e capital, etc. A dif ference between asset management and other industries
in the real economy is that the i nput capital is chosen by the investor instea d of t he firm. The
investor flows respond to performance, which is related to labor input by funds in terms of ef fort to
outperform. Therefore, changes in c apital also reflect changes in labor too. When the TFP shocks
in the model (i.e . the residual) is uncorrelated to the unobserved components of l abor input, the
correlation between fund returns and lagged Industry Size measures industry-level returns t o scale.
When TFP shocks are correlated wit h Industry Size the model can be misspecified. One remedy
is to find an instrumental variable that is correlated with Industry Si ze but not with the residual.
T able 1 1 shows such a test that accounts for potential biases in the main regression for returns to
scale in T able 3. Similarly to Pástor et a l. (2015), I choose the backward-demea ned Industry Size
65
as an instrument for the forward-demeaned Industry Size in the main regression. This instrument
satisfies the relevance condition because both stem from the nominal size of total assets within a
fund class. It plausibly satisfies the exclusion condition too, because backward-looking informa tion
is unlikely to predict the forward-looking return information from TFP in the residual term. The
results on returns to scale are qua litatively the same as those in the main regression without an IV
for Industry Size .
8.2. Differ entiation by fund act iveness
T able 3 suggests that index funds and ETFs are not subject to va riations in performance from
the sca le of passively managed assets. An active manager could also invest a fraction of his assets
into passive strategies, so that he can alleviate the cost of active trading. This process is called
“closet indexing”, because the investors cannot observe what frac tion of assets is invested passively .
Amihud and Goyenko (2013) provide a measure for fund activeness. This measure is e qual to1−
R
2
i
, whereR
2
i
the goodness-of-fit for fundi from the rolling-window re gressions on the benchmark
factors that are used to estimate the monthly time seri es for Gr oss Alpha .
T ables 12 and 13 show the ef fect of fund activeness on ret urns to scale and the flow-performance
relation respectively . The funds in each asset class are divided into quintiles based on their active-
ness. The most active funds belong to the top two quintiles, while the le ast active funds belong to
the bottom two quintiles. The coef ficient on Industry Size measures returns to scale. The returns to
scale at the industry level for e quity and bond funds are drive n by the most active funds. This result
is consistent with the earlier placebo test in T able 3 for returns to scale a mong passive funds, where
the null hypothesis of fund re turn scalability cannot be rejected. There are minimal dif ferences
among the most and least active hedge funds. T he increasing returns to scale for money market
funds are driven by the most active funds until the mi d-2000s, but the least active funds dominate
afterward.
66
The correlation between fund fees and performance among equit y funds i s positi ve for the most
active and negative for the least act ive funds. There are minimal dif ferences among the most and
least a ctive hedge funds for this correlation. T he fee-pe rformance relation is more sensitive for the
most active bond funds and the least active mone y market funds. The flow-performance relation
is driven by the m ost active managers for equity , hedge, and bond funds. Similarly to T able 13,
money marke t flows are not driven by performanc e. This implies that the least a ctive funds for
every asset class have a flatter flow- performance re lation, which is consiste nt the hypothesis that
investors chase superior performance relative to a benchmark.
67
Chapter 9
Robustness tests
9.1. Investment style effects
The main results suggest that the excess returns of the average f und are more sensitive to in-
dustry size than the fund’ s own size. T o i dentify the origin of the economies of scale , it is useful to
examine whether the ef fects of competition within a particular investment style are e conomically
more important tha n the industry ef fects across the full fund class. My classification of funds into
classes stems from the risk of the traded securities, similarities in the fee structure, and the reg-
ulatory regime that applies to the funds. Based on these features, I use a common set of factors
for all funds within a class to adjust their returns for risk. This classification implie s that the dis-
tinction among inve stment styles within a fund class is irrelevant, and all m anagers share the same
opportunities in generating abnormal returns.
I test below cases where some investment styles involve niche strategies that restrict the invest-
ment opportunitie s to a subset of managers. For instance, the main results consider hedge funds
as a single fund class. It is plausible though that ma cro funds are inhere ntly dif ferent than long-
short equity hedge funds, and the opportunities for alpha between these styles do not overlap. The
investment styles for each fund class are summarized in T able 2.
68
T able 14 shows the sensitivity of Gr oss Alpha on Industry Size as before, and also inc ludes the
sensitivity on Style Size . The latter is defined similarly to Industry Size , but the aggregate size spans
the funds of a specific inve stment style instead of all the funds within t he class. For instance, all
equity funds shar e a c ommon Industry Size , but funds in each of the cap-based, growth, growth and
income, or sector investment styles have their own Style Size . The ef fect of Style Size for equity
funds is statistically insignificant i n regressions (1) and (3). This implies that diseconom ies of
scale operate at the industry l evel for this class of funds, and the dif ferent styles do not af fect the
distribution of the opportunities for alpha across managers.
The results for inde x and exchange traded funds are a lso statistically insignificant across styles.
These fund classes represent passive investing. The style separates t he funds that track equity
indices from those that track fixe d income indices. As a result, the gross returns of the average
passive fund are not sensitive to economies of scale.
The fourth panel of T able 14 demonstrates the dominance of industry scale over style ef fects
for hedge funds. The positive and significant coef ficients for Style Size in re gression (1) become
insignificant in regressions (3), while the ef fect of Industry Size remains relatively same (compare
regressions (2) and (3)). This is an indica tion that the economies of scale operate at the industry
level for this class, and all funds share the same opportunities for alpha regardless of investment
style. This is a qui te surprising result, because the inve stment style of hedge funds cover a wide
range of strategies and underlying assets of varying risk. It also suggests that a common factor
model to benchmark hedge fund returns is more appropriate than style-spec ialized factor models.
The Industry Size dominates Style Size for bond funds too. The ef fect of the former is similar
across regre ssions (2) and (3), while the coef ficient for Style Size is reduced by half when compared
to the Industry Size . The interpretati on of this result is simil ar to the intuiti on about hedge funds.
On the other hand, the comparison between Style Size and Industry Size for money market funds
gives mixed results. Each of the estimate s are statistically significant in regressions (1) and (2).
When both Style Size and Industry Size are used in the regression, their estimates are insignifi cant.
69
I group the government, municipal, and prime money market funds i n a single class f or my main
analysis. It is plausible though that prime funds are inherently dif ferent than funds that r estrict their
assets to government securitie s only . For instance, Kacperczyk and Schnabl (2013) assume t hat
government money market funds are riskless and unaf fected by business c ycle ef fects. They restri ct
their analysis to prime money market funds, and show that these funds experienced a huge growth
in their investment opportunities during the period 2007-2010. However , the implications of the
competition hypothesis for money market funds are valid, irrespective of the le vel where economies
of scale operate (industry or investment style). The excess returns of the average money market
fund i ncrease with aggregate size, reflecting the availability of profitable investment opportunities.
9.2. Other r ob ustness tests
Let’ s assume for simplicity that the data are generated by the main process for true talent i n ac-
tive i nvesting and a secondary noisy process that contaminates the data. OLS estimates are based
on conditional weighted averages of these two processes. A ranking of Gr oss Alpha would popu-
late t he top and bottom deciles not with the best and worst managers respectively , but with those
having the greatest estimation error . Therefore , OLS estimates for Gr oss Alpha are misspeci fied
(Mamaysky et al., 2007, 2008). I estimate alphas using the MM-estimator of Y ohai (1987) to
mitigate measurement e rrors in fund alphas. The MM-estimat or belongs to the c lass of robust re-
gressions that are based on medians instead of conditional means, and it is less sensitive to out liers
than the OLS esti mator . Compared to other robust regression methods, the MM-estima tor aims
for an optimal mix between robustness and ef ficiency . The results are similar to those in the m ain
analysis.
I exa mine below the robustness of the mai n result s on returns to scale and fund flows (tables
not shown). First, I subtra ct trading costs from net returns to define Gr oss Alpha , in additi on to
the fund’ s monthly expense ratio. The trading cost is given by the round-trip cost per transaction
multiplied with the fund’ s turnover ratio. I assume a round-trip cost of 1%. This is t he same as in
70
Pástor and Stambaugh (2002), and it is very close to the estimate of 0.95% in Car hart (1997). The
results are similar to those in the main analysis, where the trading c osts a re ignored.
The choice of Industry Size is a proxy for liquidity at the aggregate level, because it compares the
aggregate assets under management with the relevant total market tha t is accessible to the managers
and investors. T o test the sensitivity of the re sults on historical market conditions, I perform two
robustness tests. First, I repl ace Industry Size with Industry Size , the aggregate size within the fund
class in $trillions. This removes the ef fects of the market capitalization from sample. In the second
test, I restore Industry Si ze and set a comm on start ing yea r for all fund classes. Thi s guarant ees
that all fund types experience the same market conditions ove r time. The results on economies of
scale remain the same. Equity mut ual funds ha ve diminishing returns, while hedge funds and fixed
income funds have increasing returns at the industry level.
I esti mate Gr oss Alpha as t he i ntercept from expanding window regressions of the fund’ s Gr oss
Return on the benchmark for the fund class. The expanding window method utilizes t he full in-
formation set t hat is reveal ed over time for each fund, which is similar in principle t o Bayesian
updating in learning models. An additional estimation of alphas utilizes rolling regressions with a
24-month window . The disadva ntage of rolling window regressions is the loss of data compared to
the expanding window method, espe cially for funds with short track records. On the ot her hand,
the expanding window regressions for young funds may result in lar ge measurement errors for the
alphas. The main results are robust to the estim ation method for fund alphas.
The choice of benchmark portfolios for eac h class in the estimation of Gr oss Alpha could also
af fect the results. I replac e the q -factor mode l of Hou et al . (2015) for equity mutual funds with
the four -factor model of Carhart (1997), and the results remain robust. However , Berk and van
Binsber gen (2015) ar gue that factor portfolios are not an appropriate benchmark, and use i nstead
a combination of index funds that are managed by the V anguard Group. Similarly to Berk and
van Binsber gen, I benchmar k the classes of equit y and hedge funds using the Capital Asset Pricing
Model (Sharpe, 1964; Lintner , 1965). The benefit of adjusting for risk with the CAPM is that it
captures the profitable opportunities t hat investors would receive at low cost, had they not invested
71
in active funds. The disadvantage is that, unlike the fa ctor models, the CAPM fails i n explaining
adequately the cross section of stock returns (Fama and French, 1993; Hou et al., 2015). As a result,
passive benchma rks could induce spuriously lar ge alphas for the active managers. The economies
of scale for the CAPM-adjusted equity mutual fund and hedge fund excess returns remain the same
as in the main results.
72
Part III
Epilogue
73
Chapter 10
Conclusion
This dissertation provides a mode l and empirical analysis for the life cycle of investment man-
agement. Fund managers compete for investor flows and profitable opportunities. The markets
learn about ma nagerial talent from the fund’ s track record. The driving f orces for the life cycle are
the competition among funds and the investor ’ s response to performance.
During the ea rly stage of the life cycle, the competition among managers triggers a network
externality for the investor and funds operate under increasing returns to industry scale. The flow-
performance relat ion is concave. The investor surplus from alpha increases during this stage. As
the competiti on among managers intensifies, the opportunities for abnormal returns are curtailed
and the funds operat e under diminishing re turns to industry scale. The flow-performance rela tion
becomes convex, and the total surplus from active inve sting is depleted by the end of the life cycle.
Active investing outperforms passive strategies during the early stage, although managers become
more passive over time to mitigate trading costs. By the end of the life cycle, the managers index
all their assets.
The returns of t he average fund are mostly af fected by the aggregat e demand for a ctive man-
agement, rather than the fund’ s own size. Act ive returns are not persistent, because the competi tion
among managers for superior performance erodes the investment opportunities within a fund class.
The returns to scale at the industry level are increasing for less competitive fund classes and decreas-
74
ing for very competit ive ones. The sensitivity of investor flows to fund performance is inherently
related to economies of scale. Equity mutual funds and hedge funds have diminishing returns to
scale at the industry level and a convex flow-performance relation. Fixed i ncome mutual funds
have increasing returns to scale and concave flow-performance relation.
75
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83
Appendix A
Pr oofs
Pr oof of LEMMA 1. T o simplify the notation for this proof, I omit the time subscriptt from vectors.
The Lagrangian for the investor ’ s problem (see equation (2.7)) is
L =E
t
[q
0
(r−f)]−
γ
2
q
0
Vq−λ(q
0
1−W
t
) , (A.1)
whereq = (q
0t
,q
1t
,...,q
Nt
)
0
is the fund siz e vector , r = (r
0t+1
,r
1t+1
,...,r
Nt+1
)
0
is the
fund nominal re turn vector ,f = (f
0t
,f
1t
,...,f
Nt
)
0
is the fund fee vector , 1 is a (N + 1)× 1
vector of ones,V is the(N +1)×(N +1) covariance matr ix for the retur ns of the passive index
and the incumbent active managers, whileγ ,W
t
, andλ are constants.
The first-order condition forq
0
is
E
t
[r−f]−γVq =λ1 (A.2)
and the optimal fund sizes are given by
q
∗
=
1
γ
V
−1
h
E
t
[r−f]−λ1
i
≡
1
γ
h
E
t
[R−f]−λV
−1
1
i
, (A.3)
84
where the net-of-fee alpha is defined as the net-of-fee Sharpe ratio
E
t
[R−f]≡V
−1
E
t
[r−f] (A.4)
withR = (1,R
1t+1
,...,R
Nt+1
) the vector of ri sk-adjusted fund ret urns in excess of the bench-
mark, i.e. the fund gross alphas. The first element ofR corresponds to the passive index, and it
has zero alpha by definition. Multiplying equation (A.3) by 1
0
allows to reconstruct the budget
constraint and solve forλ
1
0
q
∗
=
1
γ
h
1
0
E
t
[R−f]−λ

1
0
V
−1
1
i
=W
t
⇒ (A.5)
λ =
1
0
E
t
[R−f]
(1
0
V
−1
1)
−
W
t
γ
(1
0
V
−1
1)
. (A.6)
Substituting forλ in equation (A.3) gives the equilibrium demand function
q
∗
=
W
t
(1
0
V
−1
1)
V
−1
1+
1
γ
E
t
[R−f]−
1
0
E
t
[R−f]
γ(1
0
V
−1
1)
V
−1
1 . (A.7)
As a result, the demand function for fundi at timet is given by
q
∗
it
=
W
t
N
t
X
j=0
ω
ij
N
t
X
k=0
N
t
X
j>k
ω
kj
+
1
γ
E
t
[R
it+1
−f
it
]−
N
t
X
j=0
ω
ij
γ
N
t
X
k=0
N
t
X
j>k
ω
kj
N
t
X
j=0
E
t
[R
jt+1
−f
jt
] , (A.8)
where ω
ij
the matrix element ofV
−1
at row i and column j . The following ratio has order of
magnitude
N
t
X
j=0
ω
ij
N
t
X
k=0
N
t
X
j>k
ω
kj
=O

1
N
t
+1
!
. (A.9)
85
As a result, the optimal fund size may be written as
q
∗
it
=
γW
t
b
it
N
t
+1
+b
1
E
t
[R
it+1
−f
it
]−
b
it
N
t
+1
N
t
X
j=0
E
t
[R
jt+1
−f
jt
] (A.10)
withb
1
>b
it
/(N
t
+1) for allN
t
> 1 . The covariance matrixV is positive defi nite, implying that
its inverse is positi ve definite too. Therefore, the coef ficientb
it
is positive for al l fundsi , because
it involves summations of matrix elements along a dimension ofV
−1
.
Pr oof of LEMMA 2. The correlation between size and expected perfor mance describes the flow-
performance relation. The deri vative
∂q
∗
it
∂E
t
[R
it+1
]
(A.1 1)
has no root for positive values ofHH
t
, and its limiting values are
lim
HH
t
→0
"
∂q
∗
it
∂E
t
[R
it+1
]
#
= 0 (A.12)
lim
HH
t
→1
"
∂q
∗
it
∂E
t
[R
it+1
]
#
=
b
2
1
(N
t
+1)

2E
t
[R
it+1
]
2
+ ˆ σ
2
τ,t
/2+h

2b
1
(N
t
+1)E
t
[R
it+1
]
2
−
b
1
b
it

ˆ σ
2
τ,t
/2+h

2b
1
(N
t
+1)E
t
[R
it+1
]
2
> 0 , (A.13)
becauseb
1
(N
t
+1)>b
it
. As a result,
∂q
∗
it
∂E
t
[R
it+1
]
> 0 for HH
t
> 0 , (A.14)
which implies a monotonically increasing flow-performa nce. The equality is sati sfied when HH
t
declines to zero.
86
Pr oof of PROPOSITION 1. The weighted average talent among the inc umbent managers μ
τ,t
at
timet is
μ
τ,t
≡
1
Q
t
N
t
X
i=1
q
it
τ
i
, (A.15)
using fund assets under management as weights. Lemma 2 shows that managers with lar ger ex-
pected returns are assigned more capital. The worst-performing managers lose capital, while the
best ones rec eive more capital. This implies that the investor flows gradually decrease the weights
in equation (A.15) from the le ft tail of the distribution for talent. On the other ha nd, the flows
increase the weights on the right tail of the distribution over time. As the uncertainty about tal ent
decreases over time, the estimate ˆ μ
τ,t
conver ges to μ
τ,t
. Therefore, the weighted average talent
shifts upward
dμ
τ,t
dt
> 0. (A.16)
Chapter 3 shows that the a verage talent creates a barrier to entry , and the cross-sectional distri-
bution of talent among incumbents is truncated at the left tail. The barri er to entry during a period
is the talent value for the mar ginal inc umbent with zero profits. The rising μ
τ,t
implies that the
mar ginal inc umbent with talent ˆ τ
min,t
at time t is also more talented over time. All incumbents
before timet with talentτ
i
< ˆ τ
min,t
will start losing capital from timet and beyond. In the long
run, these managers are forced to exit by liquidation, and t he distributi on of talent among the in-
cumbents is truncat ed within the range τ
i
∈ [ˆ τ
min,t
,∞) . Since the lower bound increases over
time, the cross-sectional dispersionσ
τ,t
among the incumbent managers must decline, i.e.
dσ
τ,t
dt
< 0. (A.17)
87
Pr oof of COROLLAR Y 1. Equation (2.1 1) shows that a fund’ s fee depends on the risk-adjusted
return of every incumbent fund, t he sum of correlations for the fund’ s nominal return with that of
all rival funds through the coef ficient b
it
, and the Herfindahl index. The decli ning dispersion of
talent implies that managers become similarly talented over time. As a result, their returns will
have simi lar value s and be more correlated. This implies that the fund fee s will follow the same
pattern, and their dispersion will decrease over time.
Pr oof of COROLLAR Y 2. Let t
1
< t
2
. Proposition 1 shows that σ
τ,t
1
> σ
τ,t
2
. Therefore, the
dispersion of the prior distri bution for successful entrants at t
1
is lar ger than the corresponding
prior for the cohort att
2
. Equations (3.16) to (3.18) in t he Appendix show that a smaller variance
in the prior (σ
2
in that notation) reduces the Kalma n gain, and the marke t puts more wei ght on the
previous estimate rather than the observations from realized returns. This implies that the smaller
uncertainty for every fund in the cohort att
2
gets re solved faste r . The fund flows by the investor
are his response to the innovations about pe rformance. Since the weight on observations is smaller
for the cohort at timet
2
, the flows to every manager will be smaller too. The smaller Kalman gain
implies that two manage rs with the same talent a nd identical track re cords of realized returns will
receive flows of dif ferent magnitude if they entered at dif ferent time periods.
Pr oof of LEMMA 3. The equation
∂f
∗
it
∂E
t
[R
it+1
]
= 0 (A.18)
is a polynomial of degree six. Ac cording to the Abel–Ruf fini theorem, polynomials with abstract
coef ficients of degree five or higher may lac k a closed-form solution. Depending on parameter
values, equation (A.18) has either of the following forms
−a
0
−a
1
HH
t
−a
2
HH
t
2
+a
3
HH
t
3
+a
4
HH
t
4
+a
5
HH
t
5
+a
6
HH
t
6
= 0 (A.19)
88
−a
0
−a
1
HH
t
−a
2
HH
t
2
−a
3
HH
t
3
−a
4
HH
t
4
+a
5
HH
t
5
+a
6
HH
t
6
= 0 , (A.20)
where the coef ficientsa
0
toa
6
are positive. Descartes’ rule of signs implies that this polynomi al
has a unique positive rootHH
thr
, and five negative or complex roots.
Define the following auxiliary variables
I
t
≡
N
t
X
j=0
b
jt
E
t
[R
jt+1
]
b
1
c(1−HH
t
)+2E
t
[R
jt+1
]HH
t
(A.21)
J
t
≡
N
t
X
j=0
b
jt
b
1
c(1−HH
t
)+2E
t
[R
jt+1
]HH
t
. (A.22)
The c orrelations f or industry-level returns to scale and the curvature of the flow performa nce re la-
tion are
∂Q
∗
t
∂E
t
[R
it+1
]
=−[b
1
b
0t
(b
1
c+2E
t
[R
it+1
]y
t
)]·

b
2
1
c((N
t
+1)−cJ
t
)−y
t
(2I
t
y
t
−b
it
)E
t
[R
it+1
]+
b
1

2(N
t
+1)E
t
[R
it+1
]y
t
−c

I
t
y
t
−b
it
+
2J
t
E
t
[R
it+1
]y
t

−1
·
∂f
∗
it
∂E
t
[R
it+1
]
(A.23)
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
= 2b
1
y
t

2b
4
1
c
2
(N
t
+1−cJ
t
)
2
+
4I
t
y
3
t
(2I
t
y
t
+b
it
)E
t
[R
it+1
]
2
+b
3
1
c(N
t
+1−cJ
t
)·
(8(N
t
+1)E
t
[R
it+1
]y
t
−c(4I
t
y
t
−b
it
+8J
t
E
t
[R
it+1
]y
t
))+
b
2
1
y
t
(8(N
t
+1)
2
E
t
[R
it+1
]y
2
t
−16c(N
t
+1)E
t
[R
it+1
]y
t
(I
t
+
J
t
E
t
[R
it+1
])+c
2
(2I
2
t
y
t
−I
t
b
it
+
16J
t
I
t
E
t
[R
it+1
]y
t
+8J
2
t
E
t
[R
it+1
]y
2
t
))−b
1
E
t
[R
it+1
]y
t
·

4(N
t
+1)y
t
(4I
t
y
t
+b
it
)E
t
[R
it+1
]−c

8I
2
t
y
2
t
+16J
t
I
t
y
2
t
E
t
[R
it+1
]
89
−b
it
(b
it
−4J
t
E
t
[R
it+1
]y
t
)

·



b
1
(N
t
+1−J
t
)+I
t
y
t

b
1
c+
2E
t
[R
it+1
]y
t

3
b
0t


−1
·
∂f
∗
it
∂E
t
[R
it+1
]
, (A.24)
whereI
t
andJ
t
are defined in equations (A.21) and (A.22) respectively , andb
0t
is the coef ficientb
it
for the passive fund (i = 0 ). The equations above show that these two c orrelations are proportional
to the sensitivity of fund fe es to performance. As a result, they have the sam e positive rootHH
thr
.
Pr oof of LEMMA 4. The number of funds in equi librium is given by the zero-profit c ondition in
equation (2.12). The solution of this equation for lar ge ˆ σ
τ,t
is
N
∗
t
=
"
E
t
[R
mt+1
]

1+γb
mt
(−1+2SR−(h+ ˆ σ
2
τ,t
/2)SRinv−2γW
t
)

+h+ ˆ σ
2
τ,t
/2
−2E
t
[R
mt+1
]
2
#
·
"
E
t
[R
mt+1
](2E
t
[R
mt+1
]+γb
mt
−1)−h− ˆ σ
2
τ,t
/2
#
−1
, (A.25)
wherem is the mar ginal incumbent fund,SR≡
P
N
t
j=0
E
t
[R
jt+1
] is the aggregate index of fund
returns, andSRinv≡
P
N
t
j=0
1/E
t
[R
jt+1
] . T he first condition for the ne twork ef fect to initiate is
that the number of funds increases with performance in the aggregate, including the performanc e
of the mar ginal incumbent
∂N
∗
t
∂E
t
[R
jt+1
]
=γb
mt

−2SR+2γW
t
+

h+ ˆ σ
2
τ,t
/2

SRinv

h+ ˆ σ
2
τ,t
/2
+E
t
[R
mt+1
]
2

·
"

E
t
[R
mt+1
](2E
t
[R
mt+1
]+γb
mt
−1)−h− ˆ σ
2
τ,t
/2

2
#
−1
> 0 .
(A.26)
Using the approximationSRinv≈ (N
t
+1) , the condition becomes
SR
N
t
+1
≈ ˆ μ
τ,t
<
1
2


h+
ˆ σ
2
τ,t
2


+
γW
t
N
t
+1
. (A.27)
90
The return of the benchmark has zero alpha by definition, implyingE
t
[R
0t+1
] = 1 for the passive
fundi = 0 (see equation (2.1)). T he average active fund will outperform net-of-fees the passive
benchmark if
P
N
t
j=0
(E
t
[R
jt+1
]−f
∗
jt
)
N
t
+1
> 1 , (A.28)
which gives
SR
N
t
+1
≈ ˆ μ
τ,t
> 1+
1
2


h+
ˆ σ
2
τ,t
2


+
γW
t
2(N
t
+1)
. (A.29)
The Herfindahl index is negatively correlated with the num ber of funds
∂HH
∗
t
∂N
t
=γW
t
b
1
b
0t


2

SR
N
t
+1
−
γW
t
N
t
+1
!
−


1+h+
ˆ σ
2
τ,t
2




< 0 , (A.30)
because of condition (A.27). As a result , the number of funds increases and the Herfindahl index
decreases during the early stage of the life cycle.
Pr oof of PROPOSITION 2. Lemma 4 shows that the average active fund has positive alpha after
the network ef fect begins. The returns to scale at the industry level, the c urvature of the flow-
performance relation, and the sensitivity of a fund’ s fee relative to its pe rformance at the asymptotic
limit forHH
t
→ 1 are given by
lim
HH
t
→1
∂f
∗
it
∂E
t
[R
it+1
]
=−

(N
t
+1)b
1
+b
it

h+ ˆ σ
2
τ,t
/2

2(N
t
+1)b
1
E
t
[R
it+1
]
2
< 0 (A.31)
lim
HH
t
→1
∂Q
∗
t
∂E
t
[R
it+1
]
=
b
0t

h+ ˆ σ
2
τ,t
/2

(N
t
+1)E
t
[R
it+1
]
2
> 0 (A.32)
lim
HH
t
→1
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
=−

h+ ˆ σ
2
τ,t
/2

(N
t
+1)b
1
−b
it

(N
t
+1)E
t
[R
it+1
]
3
> 0 , (A.33)
91
for all funds i . As a result, the returns to scale are increasing at the industry level, the fl ow-
performance relation is concave, and the fees are negatively correlated with expected re turns within
the rangeHH
t
∈ (HH
thr
,1] . The investor ’ s surplus increases over time, because the network ef-
fect increases gross returns and competition decreases fund fees. As a result, the net-of-fee returns
increase in the aggregate.
Pr oof of PROPOSITION 3. Let’ s assume that the average fund has negative net a lpha. This implies
the following inequality
P
N
t
j=0
(E
t
[R
jt+1
]−f
∗
jt
)
N
t
+1
< 1 . (A.34)
The net-of-fees return for the average fund at smallHH
t
is
P
N
t
j=0
(E
t
[R
jt+1
]−f
∗
jt
)
N
t
+1
=
P
N
t
j=0
(E
t
[R
jt+1
]x−cW
t
)
(N
t
+1)x
, (A.35)
where x ≡ 2HH
t
E
t
[R
it+1
]/(b
1
c(1− HH
t
)) a small quantity . Solving f or the average gross
return gives
P
N
t
j=0
E
t
[R
jt+1
]
N
t
+1
< 1+
cW
t
x
, (A.36)
which is always true becausex is small. As a re sult, the assumpti on that the ave rage active fund
underperforms relative to the benchmark holds true.
Lemma 3 shows that HH
thr
is the unique root for the industry-level returns to scale, the cur -
vature of the flow-performance relation, and the correla tion between fees and gross returns. The
Herfindahl index isHH
t
< HH
thr
during the late stage of the life cycle. Proposition 2 shows that
∂Q
∗
t
∂E
t
[R
it+1
]
> 0 ,
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
< 0 and
∂f
∗
it
∂E
t
[R
it+1
]
< 0 (A.37)
92
for all fundsi whenHH
t
> HH
thr
. As a result, each of these c orrelations has the opposite sign
within the range HH
t
∈ (0,HH
thr
) . Therefore, the funds have diminishing returns to scale at
the industry level, the flow-performance is conve x, and fees are positively correlated with gross
returns during the late stage of the life cycle.
The equilibrium demand for an active manager is zero when the profitable opportunities are
depleted nearHH
t
→ 0 . As a result, the total surplus from active investing
TS
t
≡
N
t
X
j=0

q
∗
jt
E
t
[R
jt+1
]−
cq
2
it
(1−HH
t
)
2HH
t
−hq
it
!
(A.38)
also declines asymptotically to zero.
Pr oof of LEMMA 5. The first-order condition from equation (5.2) for fundi at timet is
f
∗
it
q
∗
it
(E
t
[R
it+1
]−1)−
cx
it
(q
∗
it
)
2
(1−HH
∗
t
)
HH
∗
t
−hq
∗
it
= 0 , (A.39)
which implies a fraction of assets under management
x
∗
it
=
h
f
∗
it

E
t
[R
it+1
]−1

−h
i
HH
∗
t
cq
∗
it
(1−HH
∗
t
)
. (A.40)
that is actively invested in equilibrium.
Pr oof of PROPOSITION 4. Equation (5.1) shows that eve ry ma nager ’ s a lpha declines asHH
∗
t
de-
creases and μ
τ,t
increases. The weighted a verage return for the endoge nous benchmark shifts
progressively toward the average talent in the cross-section. The a ggregate risk is reduced over the
life cycle, because managers index progressively l ar ger portions of their assets (Lemma 5). The
profits from active investing decline asymptotically to zero as HH
∗
t
→ 0 . At this limit, the net
entry is zero because managers are indif ferent between entering or exiting. Since all managers have
93
the same ta lent asσ
τ,t
→ 0 and e xpected profits a re ze ro, they have a common feef
∗
t
. Therefore,
the funds participate in a perfect competition.
94
Appendix B
Differ entiated Bertrand competition
The sensitivity of fund fees t o performance for a dif ferentiated Bertrand competition and lar ge
values of ˆ σ
τ,t
is
lim
ˆ σ
τ,t
→∞
∂f
∗
it
∂E
t
[R
it+1
]
=
((N
t
+1)b
1
−b
it
)

E
t
[R
it+1
]
2
−h

(2(N
t
+1)b
1
−b
it
)E
t
[R
it+1
]
2
> 0. (B.1)
The first and second derivatives of fund size on performance are
lim
ˆ σ
τ,t
→∞
∂q
∗
it
∂E
t
[R
it+1
]
=
"
((N
t
+1)b
1
−b
it
)

(N
t
+1)b
1

E
t
[R
it+1
]
2
+h

−
b
it
E
t
[R
it+1
]
2

#
·
"
(N
t
+1)

2(N
t
+1)b
1
−b
it

E
t
[R
it+1
]
2
#
−1
> 0 (B.2)
lim
ˆ σ
τ,t
→∞
∂
2
q
∗
it
∂E
t
[R
it+1
]
2
=−
2b
1
h((N
t
+1)b
1
−b
it
)
(2(N
t
+1)b
1
−b
it
)E
t
[R
it+1
]
3
< 0 , (B.3)
which imply that the flow-performance relation is monotonically increasing and concave. The
correlation between aggregate demand and fund performa nce is
lim
ˆ σ
τ,t
→∞
∂Q
∗
t
∂E
t
[R
it+1
]
=b
1
−
1
N
t
+1
N
t
X
j=1
b
jt
> 0 , (B.4)
95
and the funds opera te under increasing returns to aggregate scale. A fund’ s fee must be non-
negative, giving the following constraint for the index of returns
SR≡
N
t
X
j=0
E
t
[R
jt+1
]>
γW
t
b
it
E
t
[R
it+1
]+b
1
(N
t
+1)(E
t
[R
it+1
]
2
+h)−b
it
h
b
it
E
t
[R
it+1
]
,>
b
1
(N
t
+1)
b
it
>N
t
+1 (B.5)
proving that the aggregate gross al pha is positive, and the average active fund outperforms passive
funds. As a result, the results from a dif ferentiated Bertrand competition are consistent with those
from monopolistic competition during the early stage of the li fe cycle.
96
T able 1
Summary statistics for all fund classes.
The starting year for equity funds is 1961, for hedge funds is 1994, for bond funds is 1976, for
money market funds is 1972, for index funds is 1979, and for ETFs is 1998. The sample end date is
December , 2015. T he unit of observation is the fund-month for all classes. All returns, the investor
flows, and fund fees are in units of fraction per month. Fund Age is the num ber of yea rs since the
fund’ s star ting date in the database. A fund’ s age may be lar ger than the sample durat ion if the fund
preexisted in the database. Fund Size is the fund’ s assets under management in $bill ions. Gr oss
Return is the fund’ s return before fees. Gr oss Alpha is the fund’ s return before fees in excess of
the class benc hmark. Fund Flow is the investor capital t hat is deposited to or redeemed from a
fund excluding capital gains. T otal Funds is the number of incumbent funds for a given month, and
Fund Gr owth is the monthly percent growth in T otal Funds . A verage Size and A verage Fee are the
cross-sectional mean of the incumbe nts’ Fund Size and fund fees for each month. Industry Size is
the aggregat e size of managed assets withi n a fund class di vided by the total market capitalization,
where the total market depends on the type of funds and it includes the US stock and/or debt m arkets.
The Herfindahl Index is the sum of squared ratios of Fund Size over t he a ggregate size of assets
across the incumbent funds. A verage Gr oss Alpha and Std(Gr oss Alpha) are the cross-secti onal
mean and dispersion respectively for Gr oss Alpha among all incumbents during a month. A verage
T alent and Std(T alent) are the cross-sectional mean and dispersion of estimated managerial talent.
The summary stat istics columns from left to right correspond t o the number of observations N, the
sample mean, standard deviation, minimum, maximum, and the first-order autocorrelation AR1.
N Mean SD Min Max AR1
Equity F unds
Fund Age (years) 775723 10.0462 9.5204 0.0000 54.0000 0.2419
Fund Size ($bn) 769388 0.9613 5.1478 0.0000 295.7608 0.0702
Gr oss Return 760661 0.0057 0.0545 − 0.7530 1.6960 0.7279
Gr oss Alpha 479320 0.0005 0.0041 − 0.0420 0.0494 0.1584
Fund Flow 753367 0.0097 0.0872 − 0.9182 6.1006 0.1 131
Expense Ratio/Fees 768887 0.001 1 0.0008 0.0000 0.1219 0.1389
Std(Gr oss Return) 479320 0.0512 0.0181 0.0001 0.2091 0.3992
T otal Funds (1000s) 649 1.1953 1.0401 0.1250 2.7790 1.0000
Fund Gr owth (%) 648 0.4820 1.9238 − 6.2500 24.0741 − 0.0400
A verage Si ze ($bn) 649 0.5481 0.5357 0.0766 2.1427 0.9980
A verage F ee 649 0.0007 0.0001 0.0005 0.0009 0.9974
Industry Size 649 0.0955 0.0637 0.0254 0.2147 0.9990
Herfindahl Index 649 0.0168 0.0104 0.0061 0.0471 0.9974
A verage Gr oss Alpha 589 0.0003 0.0008 − 0.0024 0.0019 0.9788
Std(Gr oss Alpha) 589 0.0040 0.0009 0.0020 0.0082 0.9729
A verage T alent 649 0.0012 0.0008 − 0.0001 0.0023 0.9981
Std(T alent) 649 0.0016 0.0003 0.0009 0.0022 0.9839
Continued on next page
97
T able 1 – continued from previous page
N Mean SD Min Max AR1
Hedge Funds
Fund Age (years) 1588787 6.3358 4.8390 0.0000 56.9167 0.4440
Fund Size ($bn) 151 1600 0.1969 0.8706 0.0000 72.3500 0.0614
Gr oss Return 1558752 0.0102 0.31 17 − 1.0008 68.3530 0.0071
Gr oss Alpha 548162 0.0003 0.0155 − 0.1790 0.9972 0.2093
Fund Flow 1457019 0.0051 0.1028 − 1.3956 3.5262 0.0801
Expense Ratio/Fees 1588787 0.0055 0.3028 0.0000 68.2512 0.0000
Std(Gr oss Return) 548263 0.0430 0.2228 0.0000 27.4926 0.0142
T otal Funds (1000s) 264 6.0181 2.8696 0.7000 9.4770 0.9998
Fund Gr owth (%) 263 0.8782 2.2870 − 1.4999 26.7442 0.2396
A verage Si ze ($bn) 264 0.1704 0.0699 0.0741 0.3141 0.9980
A verage F ee 264 0.0069 0.0093 0.0026 0.0674 0.8292
Industry Size 264 0.0536 0.0284 0.0107 0.1 165 0.9985
Herfindahl Index 264 0.0033 0.0027 0.0009 0.0165 0.9962
A verage Gr oss Alpha 205 0.0006 0.0022 − 0.0053 0.0041 0.9891
Std(Gr oss Alpha) 205 0.0142 0.0035 0.0092 0.0282 0.9624
A verage T alent 264 0.0163 0.0013 0.0135 0.0181 0.9919
Std(T alent) 264 0.0103 0.0015 0.0080 0.0136 0.9828
Bond Funds
Fund Age (years) 586815 10.4699 8.5654 0.0000 54.0000 0.4163
Fund Size ($bn) 584045 0.7021 3.9060 0.0000 292.8760 0.0652
Gr oss Return 581781 0.0025 0.0166 − 0.8995 0.8617 0.5177
Gr oss Alpha 389427 − 0.0016 0.0033 − 0.0488 0.0330 0.4751
Fund Flow 576961 0.0086 0.0781 − 1.0051 1.0000 0.1 157
Expense Ratio/Fees 583892 0.0007 0.0003 0.0000 0.0071 0.3857
Std(Gr oss Return) 389427 0.0142 0.0087 0.0000 0.1292 0.2954
T otal Funds (1000s) 479 1.2251 0.7199 0.0640 2.0010 0.9999
Fund Gr owth (%) 478 0.7260 3.3287 − 1.3472 38.6047 − 0.0045
A verage Si ze ($bn) 479 0.5622 0.5127 0.0564 1.8735 0.9995
A verage F ee 479 0.0006 0.0001 0.0005 0.0008 0.9949
Industry Size 479 0.0437 0.0236 0.0021 0.0872 0.9982
Herfindahl Index 479 0.0139 0.0086 0.0043 0.0409 0.9941
A verage Gr oss Alpha 420 − 0.0027 0.0026 − 0.0079 0.0039 0.9961
Std(Gr oss Alpha) 420 0.0028 0.0008 0.0018 0.0051 0.9779
A verage T alent 479 0.0076 0.0007 0.0067 0.0090 0.9985
Std(T alent) 479 0.0024 0.0003 0.0018 0.0030 0.9958
Money Market Funds
Fund Age (years) 28901 1 10.6460 7.9786 0.0000 42.0000 0.5774
Fund Size ($bn) 279301 2.0015 6.5071 0.0000 176.0757 0.2002
Continued on next page
98
T able 1 – continued from previous page
N Mean SD Min Max AR1
Gr oss Return 275747 0.0002 0.0048 − 0.8995 0.4275 0.1034
Gr oss Alpha 192594 0.0005 0.0022 − 0.0946 0.1 141 0.0822
Fund Flow 272048 0.0089 0.1 144 − 0.5000 1.0000 0.0754
Expense Ratio/Fees 279239 0.0005 0.0002 0.0000 0.0040 0.4459
Std(Gr oss Return) 192594 0.0009 0.0039 0.0000 0.1 162 0.0301
T otal Funds (1000s) 517 0.5590 0.3628 0.0010 1.0350 0.9997
Fund Gr owth (%) 516 1.5599 1 1.4654 − 14.7124 200.0000 − 0.0149
A verage Si ze ($bn) 517 1.6612 1.6064 0.0004 5.9882 0.9994
A verage F ee 517 0.0004 0.0001 0.0001 0.0008 0.9830
Industry Size 517 0.0590 0.0331 0.0000 0.1216 0.9972
Herfindahl Index 517 0.0689 0.1824 0.0058 1.0000 0.9718
A verage Gr oss Alpha 458 0.0004 0.0006 − 0.0016 0.0018 0.9410
Std(Gr oss Alpha) 446 0.0022 0.0015 0.0004 0.0106 0.9171
A verage T alent 517 − 0.0006 0.0003 − 0.0010 − 0.0002 0.9992
Std(T alent) 517 0.0005 0.0001 0.0000 0.0009 0.9748
Index F unds
Fund Age (years) 61606 7.7608 6.7145 0.0000 54.0000 0.4604
Fund Size ($bn) 61291 2.5048 13.0461 0.0000 353.7367 0.2993
Gr oss Return 60907 0.0065 0.0453 − 0.3771 0.5286 0.7124
Expense Ratio/Fees 61232 0.0005 0.0004 0.0000 0.0022 0.5242
T otal Funds (1000s) 444 0.1388 0.1 146 0.0030 0.3220 0.9998
Fund Gr owth (%) 444 1.0809 4.541 1 − 25.0000 50.0000 − 0.0155
A verage Si ze ($bn) 444 1.4063 1.5366 0.0227 5.9134 0.9990
A verage F ee 444 0.0003 0.0002 0.0001 0.0007 0.9978
Industry Size 444 0.0172 0.0181 0.0001 0.0620 0.9999
Herfindahl Index 444 0.2227 0.1864 0.0651 0.6686 0.9970
ETFs
Fund Age (years) 87610 4.3251 3.4269 0.0000 20.5833 0.6590
Fund Size ($bn) 87528 1.2710 5.7457 0.0000 215.9085 0.1720
Gr oss Return 86705 0.0034 0.0698 − 0.6774 0.9760 0.3727
Expense Ratio/Fees 87455 0.0004 0.0003 0.0000 0.0028 0.7838
T otal Funds (1000s) 216 0.4056 0.3519 0.0030 1.0220 0.9999
Fund Gr owth (%) 216 3.9265 22.1203 − 3.5294 300.0000 0.0334
A verage Si ze ($bn) 216 1.3595 0.5133 0.6430 4.6676 0.8576
A verage F ee 216 0.0002 0.0000 0.0001 0.0002 0.9927
Industry Size 216 0.0227 0.0186 0.0005 0.0598 0.9995
Herfindahl Index 216 0.1547 0.2021 0.0206 0.8093 0.9984
99
T able 2
Investment style frequency .
The table shows the frequency of investment styles in sample, including the time-series average
of the total assets under management in $trillion for a parti cular style.
Frequency Percent A verage Size ($tn)
Equity F unds
Cap-based 177465 22.88 0.477
Growth 334164 43.08 0.896
Growth and Income 188805 24.34 0.756
Sector 75289 9.71 0.160
T otal 775723 100.00 0.695
Hedge Funds
Arbitrage 327547 20.62 0.401
Emer ging Markets 198069 12.47 0.143
Hedging 621222 39.10 0.418
Macro 178109 1 1.21 0.290
Market Neutral 40860 2.57 0.025
Multistrategy 175001 1 1.01 0.194
Unknown 47979 3.02 0.029
T otal 1588787 100.00 0.319
Bond Funds
Corporate 84780 14.45 0.1 17
Generic 1691 14 28.82 0.725
Government 85027 14.49 0.130
Municipal 247894 42.24 0.368
T otal 586815 100.00 0.400
Money Market Funds
Government 90293 31.24 0.522
Municipal 108849 37.66 0.239
Prime 89869 31.10 0.766
T otal 28901 1 100.00 0.492
Index F unds
Equity 47343 76.85 0.530
Fixed Income 14263 23.15 0.127
T otal 61606 100.00 0.436
ETFs
Equity 64002 73.05 0.691
Fixed Income 23608 26.95 0.245
T otal 87610 100.00 0.571
100
T able 3
Evolution of returns to scale.
This table shows the correlation between fund performance with the size of managed assets and
fees over time. Each column cor responds to a recursive regression with dif ferent end year for the
sample. The dependent variable for equity , bond, money market, and hedge funds is Gr oss Alpha ,
the fund’ s net return plus Fund Fees minus the benchma rk return. For ETFs and index funds it
is Gr oss Return , the fund’ s net ret urn plus Fund Fees . Fund Fees are the expense ratios for all
types of mutual funds. For hedge funds the fee structure incl udes management, incentive, high-
water mark, and hurdle fe es. Fund Size is the the fund’ s assets under management in $billions.
Industry Size is the aggregate size within the fund class divide d by the total ma rket capitalization
(US stock market for equity funds; US debt market for bond and money market funds; weighted
mean of US stock and de bt markets for ETFs, index, a nd hedge funds based on fund investment
styles in T able 2). This variable i s simi lar to the industry size in Pástor et al. (2015), but normali zed
over the securities market that is rele vant to the fund class. The control variables include l agged
values for the percentage growth i n the number of funds Fund Gr owth , Fund Age (in years), and
the cross-sectional dispersion, skewness, and kurtosis for Fund Size . Additional control variables
for active fund classes are lagged values of the serial dispersion of a fund’ s Gr oss Return within
a rolling-window of the pre vious 5 years, and lagged investor flows of capital Fund Flow . The
regressions for hedge funds also include Style Size , the aggregate size of funds wit h a common
investment style divide d by the total market c apitalization for hedge funds. The coef ficients stem
from recursive demeaning estimations. All variables are forward-demeaned, except Fund Size that
is backward-de meaned t o mitigat e correl ations with aggregate variables such as its cross-sectional
distribution mom ents and Industry Size . The forward-demeaned Fund Fee is instrumente d by its
backward-demeaned counte rpart. The t -statistics in parentheses are clustered i n two di mensions
by fund and style-month.
2015 2010 2000 1990
Equity F unds
Industry Size − 0.019 − 0.018 − 0.030 0.018
(− 9.897) (− 9.137) (− 0.870) ( 2.470)
Fund Size 3.850e− 05 6.417e− 05 9.373e− 05 1.086e− 03
( 16.285) ( 18.244) ( 5.040) ( 17.936)
Fund Fees 0.660 0.603 13.623 − 3.164
( 1.333) ( 1.296) ( 1.721) (− 12.160)
Observations 469807 354993 148530 58130
Number of Funds 5521 4958 3307 1051
Controls Y es Y es Y es Y es
Index F unds
Industry Size 0.506 0.927 8.085
( 1.013) ( 0.791) ( 2.470)
Fund Size − 1.1 17e− 05 6.881e− 05 − 5.877e− 05
(− 0.217) ( 0.337) (− 0.512)
Fund Fees 28.296 − 288.689 83.027
( 0.166) (− 0.748) ( 1.108)
Continued on next page
101
T able 3 – continued from previous page
2015 2010 2000 1990
Observations 59369 42540 13144
Number of Funds 526 430 260
Controls Y es Y es Y es
ETFs
Industry Size 0.558 2.033 19.491
( 1.226) ( 2.832) ( 1.003)
Fund Size 9.083e− 05 − 6.237e− 04 − 6.771e− 03
( 0.777) (− 2.805) (− 3.760)
Fund Fees − 54.528 − 246.609 − 245.875
(− 0.580) (− 1.368) (− 3.399)
Observations 83791 33192 494
Number of Funds 1212 767 57
Controls Y es Y es Y es
Hedge Funds
Industry Size 0.072 0.037 − 0.214
( 6.383) ( 3.993) (− 0.889)
Fund Size 3.1 10e− 04 5.621e− 04 4.954e− 03
( 13.420) ( 12.547) ( 14.633)
Fund Fees − 0.272 − 0.142 0.100
(− 12.544) (− 5.544) ( 2.085)
Observations 492714 288981 16775
Number of Funds 24879 21097 5450
Controls Y es Y es Y es
Bond Funds
Industry Size 0.035 0.026 0.036 0.085
( 9.230) ( 6.859) ( 16.778) ( 9.134)
Fund Size 1.143e− 05 1.672e− 05 3.954e− 05 − 5.335e− 07
( 7.084) ( 5.412) ( 4.593) (− 0.031)
Fund Fees − 2.616 − 2.245 − 0.565 − 2.275
(− 7.332) (− 6.973) (− 1.946) (− 2.881)
Observations 383071 307170 134491 18231
Number of Funds 3676 3333 2731 1052
Controls Y es Y es Y es Y es
Money Market Funds
Industry Size 0.013 0.005 0.015 0.050
( 5.266) ( 4.208) ( 3.978) ( 1 1.139)
Fund Size − 2.888e− 06 9.609e− 07 5.143e− 06 − 8.781e− 05
(− 2.491) ( 0.983) ( 1.514) (− 6.930)
Fund Fees − 5.641 − 1.408 0.623 0.447
(− 4.644) (− 2.800) ( 0.988) ( 0.441)
Observations 189047 159680 75199 1 171 1
Number of Funds 1532 151 1 1316 697
Controls Y es Y es Y es Y es
102
T able 4
Determinants of returns to scale.
This table focuses on the connection between returns to scale and empirical proxies for t he com-
petition among m anagers. The dependent variable is Gr oss Alpha . Industry Size is the normalize d
aggregate size within t he fund class relative to its underlying securitie s market. The Herfindahl In-
dex is the sum of squared ratios of Fund Size over the aggregat e size of assets across all incumbent
funds. A verage T alent and Std(T alent) are t he cross-secti onal me an and dispersion of talent in active
investing. A manager ’ s talent is measured by the estimated fund dummy from the specifica tion in
Column 1 of T able 3, but with OLS fixed-ef fect estimation instea d of recursive de meaning. T otal
Funds is the total number of incumbent m anagers (in thousands). The control variables include all
controls from the regressions in T able 3, and also Fund Size and Fund Fees . The coef ficients stem
from recursive demeaning estimations. All variables are forward-demea ned, except Fund Size and
Fund Fees that are backward-demeaned to mitigate correlations with aggregate va riables such as its
cross-sectional distribution moments and Industry Size . Thet -statistics in parentheses are clustered
in two dimensions by fund and style-month.
(1) (2) (3) (4) (5)
Equity F unds
Industry Size − 0.019 − 0.022 − 0.038 − 0.015 0.004
(− 9.806) (− 1 1.273) (− 12.306) (− 6.991) ( 0.859)
Industry Size× Herfindahl Index 2.017
( 5.130)
Industry Size× A verage T alent − 47.762
(− 7.829)
Industry Size× Std(T alent) 13.612
( 2.509)
Industry Size× T otal Funds − 0.009
(− 3.283)
Herfindahl Index 0.1 18
( 5.578)
A verage T alent − 2.1 13
(− 9.948)
Std(T alent) 0.313
( 1.079)
T otal Funds − 0.002
(− 16.341)
Observations 469807 469807 469807 469807 469807
Adj. R
2
0.033 0.034 0.036 0.033 0.039
Controls Y es Y es Y es Y es Y es
Hedge Funds
Industry Size 0.038 0.090 0.038 0.033 − 0.046
( 8.818) ( 5.243) ( 7.251) ( 6.478) (− 2.673)
Industry Size× Herfindahl Index − 21.160
(− 3.033)
Industry Size× A verage T alent − 0.433
(− 0.077)
Industry Size× Std(T alent) − 9.970
Continued on next page
103
T able 4 – continued from previous page
(1) (2) (3) (4) (5)
(− 2.607)
Industry Size× T otal Funds 0.033
( 8.541)
Herfindahl Index 2.493
( 5.409)
A verage T alent 0.060
( 0.395)
Std(T alent) 0.205
( 1.750)
T otal Funds 0.002
( 6.089)
Observations 492714 492714 492714 492714 492714
Adj. R
2
0.180 0.182 0.180 0.180 0.183
Controls Y es Y es Y es Y es Y es
Bond Funds
Industry Size 0.032 0.042 − 0.012 0.01 1 0.045
( 8.156) ( 7.149) (− 1.394) ( 2.749) ( 4.864)
Industry Size× Herfindahl Index 3.836
( 7.155)
Industry Size× A verage T alent − 68.527
(− 5.844)
Industry Size× Std(T alent) − 165.744
(− 1 1.165)
Industry Size× T otal Funds 0.004
( 0.455)
Herfindahl Index 0.1 15
( 5.328)
A verage T alent − 0.207
(− 1.546)
Std(T alent) − 0.660
(− 3.287)
T otal Funds − 0.000
(− 3.212)
Observations 383071 383071 383071 383071 383071
Adj. R
2
0.41 1 0.415 0.414 0.427 0.412
Controls Y es Y es Y es Y es Y es
Money Market Funds
Industry Size 0.002 − 0.000 − 0.001 0.000 0.013
( 1.931) (− 0.190) (− 0.471) ( 0.193) ( 5.854)
Industry Size× Herfindahl Index 0.020
( 0.156)
Industry Size× A verage T alent 29.710
( 1.705)
Industry Size× Std(T alent) − 3.487
(− 0.206)
Industry Size× T otal Funds − 0.030
(− 5.573)
Continued on next page
104
T able 4 – continued from previous page
(1) (2) (3) (4) (5)
Herfindahl Index − 0.019
(− 2.715)
A verage T alent 2.443
( 5.169)
Std(T alent) − 0.640
(− 1.076)
T otal Funds − 0.001
(− 6.044)
Observations 189047 189047 189047 189047 189047
Adj. R
2
0.006 0.007 0.008 0.006 0.008
Controls Y es Y es Y es Y es Y es
105
T able 5
Evolution of the relation between fund flows and past performance.
This table shows the response of i nvestors to fund perf ormance over time. Each column c orre-
sponds to a recursive regression with dif ferent end year for t he sample . The dependent variabl e is
Fund Flow , the investor capital that is deposited or redeemed from a fund excluding capital gains.
Gr oss Alpha is the fund’ s net return plus Fund Fees minus the benchmark return. Fund Fees are
the expense ratios for all types of mutual funds. For hedge funds the fee structure includes man-
agement, incentive, high-water mark, and hurdle fees. The control variables include all controls
from t he regre ssions in T able 3, and also Fund Size and Industry Size . The coef ficients stem from
recursive demeaning estimations. All variables are forward-demeaned, e xcept Fund Size that is
backward-demeaned to mitigate correlations with aggregate variables such as its cross-sectional
distribution mom ents and Industry Size . The forward-demeaned Fund Fee is instrumente d by its
backward-demeaned counte rpart. The t -statistics in parentheses are clustered i n two di mensions
by fund and style-month.
2015 2010 2000 1990
Equity F unds
Log(Gr oss Alpha + 1) 1.240 1.277 1.502 1.71 1
(20.686) (20.065) (12.430) (9.846)
(Gr oss Alpha)
2
15.734 17.235 38.960 13.331
(3.568) (3.782) (3.591) (1.127)
Fund Fees 0.293 -0.063 -1 1.873 0.322
(0.271) (-0.056) (-1.141) (0.088)
Observations 468553 354154 148267 58055
Number of Funds 5521 4958 3307 1051
Controls Y es Y es Y es Y es
Hedge Funds
Log(Gr oss Alpha + 1) 0.316 0.291 0.851
(13.448) (10.121) (7.819)
(Gr oss Alpha)
2
-0.522 -0.662 13.525
(-1.789) (-4.360) (5.872)
Fund Fees 0.148 0.763 0.610
(1.571) (7.181) (1.710)
Observations 487451 286189 16628
Number of Funds 24879 21097 5450
Controls Y es Y es Y es
Bond Funds
Log(Gr oss Alpha + 1) 0.389 0.481 0.610 0.673
(3.651) (4.139) (4.389) (1.451)
(Gr oss Alpha)
2
-0.780 18.808 98.050 103.381
(-0.098) (2.180) (6.856) (2.553)
Fund Fees -18.226 -14.545 -24.284 34.765
(-4.135) (-3.701) (-6.216) (1.906)
Observations 382367 306661 134279 1821 1
Number of Funds 3676 3333 2731 1052
Continued on next page
106
T able 5 – continued from previous page
2015 2010 2000 1990
Controls Y es Y es Y es Y es
Money Market Funds
Log(Gr oss Alpha + 1) 0.075 0.054 0.121 0.063
(0.680) (0.532) (0.969) (0.319)
(Gr oss Alpha)
2
0.206 -0.273 -1.169 2.212
(0.136) (-0.264) (-0.997) (1.072)
Fund Fees 17.418 -6.126 -14.366 -84.636
(0.31 1) (-0.314) (-0.718) (-2.169)
Observations 1881 14 158870 74879 1 1663
Number of Funds 1532 151 1 1316 697
Controls Y es Y es Y es Y es
107
T able 6
Determinants of the relation between fund flows and past performance.
This table examines the ef fect of competit ion among managers to the sensitivity between investor
flows and fund performance. Each column corresponds t o a dif ferent fund class. The depe ndent
variable is Fund Flow , the growth of fund assets from capital that is deposited or redeemed by the
investor exc luding capi tal gai ns. Gr oss Alpha is the fund’ s net return plus Fund Fee s minus the
benchmark ret urn. Fund Fees are the expense ratios for all types of mutual funds. For hedge funds
the fee structure includes management, incentive, high-water mark, and hurdle fees. The control
variables are i dentical to those in T able 3. The sample for the regressions in the top (bottom) panel is
restricted for funds with positive (negative) realized returns in the previous period. The coef ficients
stem from recursive demeaning estimations. The t -statistics in parentheses are clustered in two
dimensions by fund and style-month.
Equity Funds Hedge Funds Bond Funds
Positive alpha
Gr oss Alpha 1.234 0.192 0.766
( 10.000) ( 4.398) ( 2.849)
Gr oss Alpha× Fund Fees 16.503 25.899 2548.866
( 0.373) ( 3.330) ( 2.312)
Fund Fees 1.599 − 0.389 − 3.564
( 1.913) (− 2.528) (− 1.024)
Observations 238046 337687 89321
Controls Y es Y es Y es
Negative alpha
Gr oss Alpha 0.901 − 0.074 − 0.066
( 6.276) (− 0.734) (− 0.355)
Gr oss Alpha× Fund Fees − 21.355 0.273 − 633.571
(− 0.551) ( 1.240) (− 0.723)
Fund Fees 0.754 0.027 − 5.627
( 1.314) ( 0.395) (− 2.594)
Observations 230507 149764 293046
Controls Y es Y es Y es
108
T able 7
Determinants of the relation between fund flows and past performance.
Same a s in T able 6, but with Industry Size replacing the Fund Fe es . Industry Size is the normal-
ized aggregate size within the fund class relative to i ts unde rlying securities market.
Equity F unds Hedge Funds Bond Funds
Positive alpha
Gr oss Alpha 0.878 0.224 1.063
( 5.156) ( 5.447) ( 2.770)
Gr oss Alpha× Industry Size − 9.259 − 4.634 30.695
(− 2.280) (− 2.361) ( 1.048)
Industry Size 0.041 0.022 0.196
( 0.917) ( 0.491) ( 1.903)
Observations 238046 337687 89321
Controls Y es Y es Y es
Negative alpha
Gr oss Alpha 1.384 − 0.122 − 0.083
( 6.552) (− 0.782) (− 0.350)
Gr oss Alpha× Industry Size 13.243 − 4.546 − 12.878
( 3.123) (− 1.416) (− 0.788)
Industry Size 0.073 − 0.102 − 0.242
( 1.604) (− 2.019) (− 3.171)
Observations 230507 149764 293046
Controls Y es Y es Y es
109
T able 8
Determinants of the relation between fund flows and past performance.
Same as i n T able 6, but with the total number of incumbent managers T otal Funds (in thousands)
replacing the Fund Fees .
Equity F unds Hedge Funds Bond Funds
Positive alpha
Gr oss Alpha 0.985 0.215 0.573
( 7.549) ( 5.322) ( 2.100)
Gr oss Alpha× T otal Funds − 0.570 − 0.059 2.246
(− 2.658) (− 2.445) ( 2.314)
T otal Funds 0.003 0.000 − 0.005
( 1.739) ( 0.818) (− 1.932)
Observations 238046 337687 89321
Controls Y es Y es Y es
Negative alpha
Gr oss Alpha 1.070 − 0.149 0.098
( 6.583) (− 1.013) ( 0.541)
Gr oss Alpha× T otal Funds 0.521 − 0.141 0.203
( 2.444) (− 1.782) ( 0.435)
T otal Funds 0.001 − 0.002 − 0.010
( 0.535) (− 2.272) (− 4.165)
Observations 230507 149764 293046
Controls Y es Y es Y es
1 10
T able 9
Determinants of money market fund flows.
This table examines the ef fect of the aggregat e investor demand and competition among man-
agers on money market fund flows. The de pendent variable is Fund Flow , the growth of fund assets
from capital that is deposited or re deemed by the investor excluding capital gains. Lag Fund Flow
is the fund’ s flow during the pre vious period. Gr oss Re turn is the fund’ s nominal net ret urn with
the expense ratio Fund Fees added back. Gr oss Alpha is Gr oss Re turn minus the benchmark return.
Fund Si ze is the fund’ s total assets in $billions. Industry Si ze is the aggre gate size of mone y ma rket
funds divided by the total debt market capitalization in US. Fund Age is the fund’ s age in years.
Std(Fund Size) , Skew(Fund Size) , and Kurt(Fund Size) are the cross-sectional dispersion, skew-
ness, and kurtosis respectively of Fund Si ze . The Herfindahl Index is the sum of squared ratios
of Fund Size over the aggregate size of assets across all incumbent funds. T otal Funds is the total
number of incumbent managers (in thousands). Fund Gr owth is the percentage growth of T otal
Funds . Std(Gr oss Return) is the serial dispersion of a fund’ s Gr oss Return within a rolling-window
of the previous 5 years. Thet -statistics in pare ntheses are cluste red in two dimensions by fund and
style-month.
(1) (2)
Gr oss Alpha 0.108
( 0.768)
Gr oss Return 0.095
( 1.281)
Lag Fund Flow − 0.1 12 − 0.1 12
(− 1 1.682) (− 1 1.680)
Industry Size 0.278 0.278
( 3.258) ( 3.255)
Fund Age 0.001 0.001
( 5.570) ( 5.538)
Std(Fund Size) − 0.003 − 0.003
(− 5.464) (− 5.465)
Skew(Fund Size) − 0.032 − 0.032
(− 2.892) (− 2.883)
Kurt(Fund Size) 0.002 0.002
( 3.1 10) ( 3.103)
Herfindahl Index 0.508 0.503
( 1.765) ( 1.749)
Fund Size 0.000 0.000
( 0.057) ( 0.052)
Fund Fees − 3.786 − 3.799
(− 0.693) (− 0.696)
T otal Funds − 0.006 − 0.006
(− 0.430) (− 0.440)
Fund Gr owth − 0.096 − 0.096
(− 1.321) (− 1.321)
Std(Gr oss Return) − 0.034 − 0.041
(− 0.181) (− 0.227)
Observations 1881 14 1881 14
1 1 1
T able 10
Dynamics among investor flows, fund fees, and returns to scale.
This table shows the dynamic connection across the three re lations that I measure: the flow-
performance rel ation, the correl ation between fees and perform ance, and returns to scale. Panel A
tests the connec tion between the flow-performance a nd returns to scale. The dependent variable is
β
Flow
≡ b
H
−b
L
, whereb
H
andb
L
are the time series of re cursive estimates for the quadratic
and logarithmic te rms in Gr oss Alpha respectively for the flow-performance relation in equation
(8.1) and T able 5. Returns to Scale are the time series of recursive estimates in returns to scale
at the industry level, i.e. the estimates for the correlat ion betwee n Gr oss Alpha and Industry Size
from the regressions in T able 3 and the top graph in Figure 5. Panel B te sts the connection between
the flow-performance and fee-performance rel ations. The dependent vari able isβ
Flow
as in panel
A. Fee-Performance r elation are the time series of recursive estim ates for the correla tion between
Gr oss Alpha and Fund Fee s from the regressions in T able 3 a nd the bottom graph in Figure 5. Panel
C tests the connection between the Fee-Performance r elation (dependent variable) and returns to
scale. The time series of estima tes for all mutual funds have monthly frequency , while for hedge
funds they have annual f requency . The estimates for Returns to Scale are lagged by one period
compared to those for the Fee-Performance r elation and by two periods compared toβ
Flow
. The
estimates for the Fee-Performance r elation are lagged by one period compared to β
Flow
. These
lags imply a sequence among the ef fects of size in expected returns, fund fees, and investor flows,
and m itigate the concerns about endogeneity between the size of a ssets and performance . The
t -statistics in parentheses are clustered within a calendar year for al l mutual funds, a nd within a
decade for hedge funds.
Equity F unds Hedge Funds Bond Funds Money Funds
A. Flow-performance on r eturns to scale
Returns to Scale − 410.867 − 33.178 − 76.301 − 1226.277
(− 8.634) (− 3.477) (− 2.610) (− 1.363)
Inter cept 10.765 1.672 21.1 1 1 − 38.310
( 1.619) ( 45.070) ( 0.655) (− 0.506)
Observations 538 14 394 418
R
2
0.512 0.1 16 0.01 1 0.094
B. Flow-performance on fee-performance
Fee-Performance r elation 1.840 22.214 22.180 14.681
( 8.907) ( 2.421) ( 1.394) ( 2.457)
Inter cept − 0.448 3.032 74.687 15.218
(− 0.048) ( 26.133) ( 2.514) ( 0.261)
Observations 539 15 395 419
R
2
0.290 0.61 1 0.282 0.535
C. Fee-performance on r eturns to scale
Returns to Scale − 1 14.380 − 1.948 3.723 40.259
(− 1.803) (− 4.050) ( 3.320) ( 0.582)
Inter cept 0.702 − 0.044 − 2.163 − 2.641
( 0.582) (− 3.037) (− 2.855) (− 0.649)
Observations 539 15 395 419
R
2
0.475 0.558 0.046 0.044
1 12
T able 1 1
Instrumental variables regression for Industry Size .
This table is the same as T able 3, but with the backward-demeaned Industry Size used as an
instrument to the forward-demeaned Industry Size . The control variables are identical to those in
T able 3. Thet -statistics in parentheses are clustered in two dimensions by fund and style-month.
2015 2010 2000 1990
Equity F unds
Industry Size − 0.034 − 0.026 − 0.060 0.054
(− 8.429) (− 6.637) (− 1.308) ( 4.259)
Fund Size 4.028e− 05 6.540e− 05 9.977e− 05 1.101e− 03
( 17.305) ( 19.065) ( 6.015) ( 17.942)
Fund Fees 0.688 0.616 13.761 − 3.177
( 1.343) ( 1.304) ( 1.710) (− 12.128)
Observations 469807 354993 148530 58130
Number of Funds 5521 4958 3307 1051
Controls Y es Y es Y es Y es
Hedge Funds
Industry Size 0.086 0.039 − 0.233
( 6.287) ( 3.701) (− 0.670)
Fund Size 3.019e− 04 5.583e− 04 4.955e− 03
( 13.624) ( 12.607) ( 14.670)
Fund Fees − 0.271 − 0.141 0.100
(− 12.541) (− 5.534) ( 2.100)
Style Size 0.032 0.002 − 0.206
( 0.800) ( 0.049) (− 3.064)
Observations 492714 288981 16775
Number of Funds 24879 21097 5450
Controls Y es Y es Y es
Bond Funds
Industry Size 0.123 − 0.026 0.024 0.071
( 8.470) (− 1.733) ( 5.597) ( 6.601)
Fund Size 7.856e− 06 1.649e− 05 4.338e− 05 5.555e− 06
( 6.014) ( 5.155) ( 5.024) ( 0.328)
Fund Fees − 3.864 − 1.448 − 0.564 − 2.376
(− 8.050) (− 3.351) (− 2.002) (− 2.992)
Observations 383071 307170 134491 18231
Number of Funds 3676 3333 2731 1052
Controls Y es Y es Y es Y es
Money Market Funds
Industry Size 0.016 0.007 0.012 − 0.032
( 5.835) ( 5.091) ( 2.935) (− 1.518)
Fund Size − 3.688e− 06 3.451e− 07 6.471e− 06 − 8.943e− 05
(− 3.025) ( 0.347) ( 1.872) (− 6.880)
Fund Fees − 5.870 − 1.417 0.646 0.495
(− 4.708) (− 2.819) ( 1.032) ( 0.486)
Continued on next page
1 13
T able 1 1 – continued from previous page
2015 2010 2000 1990
Observations 189047 159680 75199 1 171 1
Number of Funds 1532 151 1 1316 697
Controls Y es Y es Y es Y es
1 14
T able 12
The ef fect of fund activeness on fees and returns to scale.
This tabl e shows the main regressions for returns to scale in T able 3, but for sub-samples within
each fund class in terms of fund activeness. The measure of fund activene ss is the one in Amihud
and Goyenko (2013). Funds in each asset class are divided in quintiles. The most (least) active
funds belong to the top (bottom) two qui ntiles. Similarly to T able 3, the estimate s on Industry Size
and Fund Fees measure returns to scale at the industry level and the e f fect of fees on performance
respectively . The t -statistics in parentheses are clustered in two dimensions by fund and style-
month.
2015 2010 2000 1990
Equity F unds (most active)
Industry Size − 0.030 − 0.032 − 0.004 0.030
(− 8.278) (− 8.879) (− 0.223) ( 3.478)
Fund Fees 0.468 0.409 7.790 − 2.919
( 1.198) ( 1.142) ( 2.221) (− 1 1.397)
Observations 187532 141640 59123 23080
Controls Y es Y es Y es Y es
Equity F unds (least active )
Industry Size − 0.000 0.003 0.027 0.001
(− 0.173) ( 1.306) ( 5.697) ( 0.179)
Fund Fees − 1 1.637 − 12.065 − 8.164 − 7.225
(− 8.579) (− 9.081) (− 8.419) (− 5.541)
Observations 188149 142223 59605 23342
Controls Y es Y es Y es Y es
Hedge Funds (m ost ac tive)
Industry Size 0.017 − 0.004 − 0.181
( 2.683) (− 0.758) (− 1.560)
Fund Fees − 0.270 − 0.148 − 0.148
(− 12.603) (− 4.547) (− 2.308)
Observations 195991 1 15553 6629
Controls Y es Y es Y es
Hedge Funds (l east active)
Industry Size 0.068 0.075 − 0.045
( 4.860) ( 4.664) (− 0.161)
Fund Fees − 0.21 1 − 0.157 0.155
(− 4.604) (− 3.260) ( 2.700)
Observations 19851 1 1 15951 6789
Controls Y es Y es Y es
Bond Funds (most active)
Industry Size 0.044 0.033 0.055 0.090
( 6.677) ( 4.915) ( 10.228) ( 5.842)
Fund Fees − 4.354 − 3.905 − 1.273 − 5.321
(− 8.824) (− 8.854) (− 4.960) (− 5.031)
Continued on next page
1 15
T able 12 – continued from previous page
2015 2010 2000 1990
Observations 153094 122798 53743 7282
Controls Y es Y es Y es Y es
Bond Funds (least active)
Industry Size 0.015 0.004 − 0.016 0.044
( 5.162) ( 1.354) (− 4.609) ( 7.069)
Fund Fees − 1.193 − 0.493 − 0.422 1.769
(− 5.798) (− 2.867) (− 1.81 1) ( 2.246)
Observations 153033 122689 53750 731 1
Controls Y es Y es Y es Y es
Money Market Funds (most active)
Industry Size 0.005 − 0.002 0.013 0.024
( 1.449) (− 1.029) ( 2.792) ( 5.51 1)
Fund Fees − 3.951 0.230 4.470 − 3.997
(− 2.512) ( 0.266) ( 2.893) (− 4.913)
Observations 75579 63829 30096 4713
Controls Y es Y es Y es Y es
Money Market Funds (least active)
Industry Size 0.022 0.01 1 0.008 0.081
( 6.51 1) ( 7.152) ( 1.801) ( 10.628)
Fund Fees − 8.058 − 2.812 − 1.733 3.804
(− 5.287) (− 5.167) (− 2.514) ( 2.486)
Observations 75580 63845 29971 4606
Controls Y es Y es Y es Y es
1 16
T able 13
The ef fect of fund activeness on the flow-performance relation.
This ta ble shows the main regressions for the flow-performance relation in T able 5, but for sub-
samples within each fund class in terms of fund activeness. The measure of fund ac tiveness is the
one in Amihud and Goyenko (2013). Funds in each asset class are divided in quinti les. The most
(least) active funds belong to the top (bottom) two quintiles. Similarly to T able 5, the estimates on
the logarithmic and quadra tic terms for Gr oss Al pha measure the curvature of the flow-performance
relation. Thet -statistics in parentheses are clustered in two dimensions by fund and style-month.
2015 2010 2000 1990
Equity F unds (most active)
Log(Gr oss Alpha + 1) 0.946 1.01 1 1.123 1.459
( 14.472) ( 14.51 1) ( 9.635) ( 8.961)
(Gr oss Alpha)
2
21.924 23.058 30.070 4.185
( 4.377) ( 4.387) ( 3.470) ( 0.371)
Fund Fees − 0.152 − 0.228 − 6.707 2.790
(− 0.136) (− 0.201) (− 1.149) ( 0.829)
Observations 186883 141 191 58946 23030
Controls Y es Y es Y es Y es
Equity F unds (least active )
Log(Gr oss Alpha + 1) 1.960 1.997 2.266 2.373
( 21.278) ( 20.804) ( 14.571) ( 7.215)
(Gr oss Alpha)
2
− 1.022 − 3.765 89.374 164.484
(− 0.105) (− 0.393) ( 4.31 1) ( 3.632)
Fund Fees − 33.893 − 14.977 − 1.173 − 38.329
(− 2.925) (− 1.527) (− 0.1 1 1) (− 1.402)
Observations 187743 141969 59550 23331
Controls Y es Y es Y es Y es
Hedge Funds (m ost ac tive)
Log(Gr oss Alpha + 1) 0.391 0.396 0.606
( 7.523) ( 7.852) ( 3.465)
(Gr oss Alpha)
2
− 0.724 − 0.742 22.106
(− 3.108) (− 3.015) ( 2.968)
Fund Fees 0.069 0.714 0.830
( 1.073) ( 5.480) ( 1.306)
Observations 193363 1 14201 6541
Controls Y es Y es Y es
Hedge Funds (l east active)
Log(Gr oss Alpha + 1) 0.190 0.224 1.886
( 6.348) ( 5.257) ( 7.031)
(Gr oss Alpha)
2
− 0.608 − 0.797 32.675
(− 4.185) (− 3.819) ( 5.694)
Fund Fees 0.798 0.440 − 0.241
( 4.522) ( 2.387) (− 0.431)
Observations 196847 1 15026 6744
Continued on next page
1 17
T able 13 – continued from previous page
2015 2010 2000 1990
Controls Y es Y es Y es
Bond Funds (most active)
Log(Gr oss Alpha + 1) 0.291 0.312 0.713 1.391
( 2.384) ( 2.332) ( 4.892) ( 3.1 18)
(Gr oss Alpha)
2
− 4.747 13.864 109.682 137.144
(− 0.539) ( 1.452) ( 5.719) ( 3.520)
Fund Fees − 25.804 − 21.478 − 21.459 0.398
(− 4.829) (− 4.654) (− 4.520) ( 0.018)
Observations 152797 122595 53659 7272
Controls Y es Y es Y es Y es
Bond Funds (least active)
Log(Gr oss Alpha + 1) 1.089 1.353 0.471 − 0.645
( 5.869) ( 6.577) ( 1.570) (− 0.825)
(Gr oss Alpha)
2
49.137 62.683 45.656 − 13.375
( 4.048) ( 4.982) ( 2.640) (− 0.152)
Fund Fees − 7.805 − 4.490 − 26.941 63.193
(− 1.132) (− 0.721) (− 3.964) ( 2.090)
Observations 152708 122447 53649 7309
Controls Y es Y es Y es Y es
Money Market Funds (most active)
Log(Gr oss Alpha + 1) 0.249 0.133 0.094 0.185
( 1.814) ( 1.152) ( 0.481) ( 0.253)
(Gr oss Alpha)
2
− 2.032 0.270 − 0.071 1.550
(− 0.725) ( 0.171) (− 0.031) ( 0.140)
Fund Fees − 31.783 27.140 20.529 − 185.346
(− 0.334) ( 0.771) ( 0.516) (− 2.068)
Observations 75182 63482 29963 4694
Controls Y es Y es Y es Y es
Money Market Funds (least active)
Log(Gr oss Alpha + 1) − 0.572 − 0.306 0.075 − 0.133
(− 1.835) (− 1.319) ( 0.369) (− 0.291)
(Gr oss Alpha)
2
− 2.599 − 1.643 − 1.836 1.089
(− 0.668) (− 0.61 1) (− 0.873) ( 0.252)
Fund Fees 64.620 − 8.941 − 34.313 − 25.51 1
( 1.323) (− 0.422) (− 1.339) (− 0.492)
Observations 75249 63565 29846 4585
Controls Y es Y es Y es Y es
1 18
T able 14
Comparison between industry and style-wide size ratios.
This table shows regressions that ar e similar to those in t he first column of T able 3, but with Style
Size replacing Industry Size . Style Size is the aggregate size within an investme nt style divided by
the relevant total market capitalization (US stock market for equi ty funds; US debt market for bond
and money market funds; weighted mean of US stock and debt markets for ETFs, index, and hedge
funds based on fund investment styles in T able 2). Columns (1) and (2) use Style Size and Industry
Size respectively as the normalized aggregate a ssets. Column (3) compares the ef fect of both Style
Size and Industry Size . The control variables are identical to those in T able 3. The t -statistics in
parentheses are clustered in two dimensions by fund and style-month.
(1) (2) (3)
Equity F unds
Style Size − 0.015 0.001
(− 4.728) ( 0.284)
Industry Size − 0.019 − 0.019
(− 9.897) (− 7.637)
Observations 469807 469807 469807
Controls Y es Y es Y es
Index F unds
Style Size 0.156 − 0.073
( 0.558) (− 0.186)
Industry Size 0.506 0.540
( 1.013) ( 0.886)
Observations 59369 59369 59369
Controls Y es Y es Y es
ETFs
Style Size 0.090 − 0.248
( 0.209) (− 0.514)
Industry Size 0.558 0.713
( 1.226) ( 1.406)
Observations 83791 83791 83791
Controls Y es Y es Y es
Hedge Funds
Style Size 0.273 0.072
( 13.710) ( 2.139)
Industry Size 0.090 0.072
( 13.145) ( 6.383)
Observations 492714 492714 492714
Controls Y es Y es Y es
Bond Funds
Style Size 0.029 0.015
( 4.872) ( 2.752)
Industry Size 0.035 0.030
( 9.230) ( 9.442)
Observations 383071 383071 383071
1 19
T able 14 – continued from previous page
(1) (2) (3)
Controls Y es Y es Y es
Money Market Funds
Style Size 0.019 0.010
( 7.843) ( 5.008)
Industry Size 0.013 0.010
( 5.266) ( 3.704)
Observations 189047 189047 189047
Controls Y es Y es Y es
120

Abstract (if available)

Abstract I show evidence of returns to scale at the industry level and investor fund flow asymmetries across the best and worst performing managers in asset management. I introduce a new model of competition among fund managers that features the collective effect of the competitive environment on the performance of the average fund. This theoretical framework can explain the endogenous emergence and dynamic evolution of returns to scale at the industry level and the flow-performance relation. The model shows that competition in the asset management industry has positive and negative effects on fund performance. When funds have increasing (decreasing) returns to scale at the industry level, the flow-performance relation is concave (convex). Active funds outperform their benchmark initially, but the average returns from active investing are not persistent. The competition among funds raises the cost of active management and gradually deplete the profitable opportunities in the aggregate. Eventually, the total surplus declines to zero and the average active manager falls behind the benchmark. Aggregate risk is reduced over time through ``closet indexing'', until all active funds form a scalable pool of passively invested capital. The equilibrium results of the model are verified in the data across multiple fund classes that invest actively, including equity, bond, money market, and hedge funds.

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University of Southern California Dissertations and Theses

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Creator Magkotsios, Georgios (author)

Core Title Evolution of returns to scale and investor flows during the life cycle of active asset management

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 12/28/2018

Defense Date 04/24/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag alpha,bond funds,ETF,flow-performance relation,hedge funds,index funds,investment management,money market funds,mutual funds,network externality,OAI-PMH Harvest,returns to scale

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ferson, Wayne (committee chair), Ahern, Kenneth R. (committee member), Camara, Odilon (committee member), Korteweg, Arthur (committee member), Murphy, Kevin (committee member)

Creator Email Georgios.Magkotsios.2017@marshall.usc.edu,gmagkots@alumni.nd.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-14363

Unique identifier UC11671828

Identifier etd-Magkotsios-6357.pdf (filename),usctheses-c89-14363 (legacy record id)

Legacy Identifier etd-Magkotsios-6357.pdf

Dmrecord 14363

Document Type Dissertation

Format application/pdf (imt)

Rights Magkotsios, Georgios

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA