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What determines mutual fund size?
August 17, 2009
Abstract
The mutual fund industry manages about a quarter of the assetsin the U.S. stock market and thus plays an important role in the U.S.economy. The question of how much control is concentrated in thehands of the largest players can be quantitatively discussed in termsof the tail behavior of the mutual fund size distribution. We study thedistribution empirically and show that the tail is better described bya log-normal than a power law, indicating less concentration than, forexample, personal wealth. We postulate that the reasons for this stemfrom market efficiency and the stochastic nature of fund returns; theyotherwise have very little to do with investor choice. To demonstratethis we study mutual fund entry, exit and growth empirically and de-velop a stochastic model. Under simplifying assumptions we obtaina time-dependent analytic solution. The distribution evolves from alog-normal into a power law only over long time scales, suggesting thatlog-normality comes about because the industry is still young. Numer-ical solutions under more realistic conditions support this conclusionand give good agreement with the data. Our study shows that, whilemutual funds behave in many respects like other firms, there some re-spects in which they are quite unusual. Surprisingly, it appears thattransaction costs and investor choice play only a minor role in deter-mining fund size.
1
Introduction
In the past decade the mutual fund industry has grown rapidly, moving from3% of taxable household financial assets in 1980, to 8% in 1990, to 23% in20071. In absolute terms, in 2007 this corresponded to 4.4 trillion USD and24% of U.S. corporate equity holdings. Mutual funds account for a signifi-cant fraction of trading volume in financial markets and have a substantialinfluence on prices. Moreover, large players such as institutional investorsare known to play an important role in the market [Corsetti et al., 2001]and it was recently hypothesized that the fund size distribution is of fun-damental importance in explaining both the heavy tailed trading volumedistribution and the heavy tailed price return distribution observed acrossdifferent assets2. This raises the question of who has this influence: Are mu-tual fund investments concentrated in a few dominant large funds, or spreadacross many funds of similar size? What are the economic mechanisms thatdetermine this?
This question can be addressed quantitatively in terms of the upper tailsof the mutual fund size distribution. The two competing hypotheses usuallymade in studies of firms in general are a Zipf (power law) distribution,which is so heavy tailed that none of its moments exist, vs. a log-normaldistribution, for which all the moments exist. For mutual funds we resolvethis empirically using the CRSP data set and find that the equity fund sizedistribution is much better described by a log-normal distribution. Thisimplies less dominance by the largest funds.
This naturally leads to the question of why this is true. One wouldnaively think (as we did) that investor choice would be a key determinantof fund size. At the level of individual funds this is almost certainly true.Investors make their choice based on factors such as past performance, ad-vertising, fees, and investment fads, that are potentially fertile ground forbehavioral economics. This is surely important in determining the size ofindividual funds. Our results, in contrast, suggest that for the distributionof fund sizes the details of individual choice are not important.
A hypothesis put forth by Berk and Green [2004] is that the sizes ofmutual funds are determined by a combination of the distribution of skill
1Data is taken from the Investment Company Institute’s 2007 fact book available atwww.ici.org.
2The equity fund size distribution was argued to be responsible for the observed distri-bution of trading volume [Levy et al., 1996; Solomon and Richmond, 2001], and Gabaixet al. have argued that it is important for explaining the distribution of price returns[Gabaix et al., 2003; Gabaix et al., 2006].
2
of mutual fund managers and transaction costs. All else being equal, largerfunds must make bigger transactions. Due to market impact, bigger trans-actions incur larger costs per share, and at sufficiently large size this shoulddiminish a fund’s performance and make it less able to attract funds. Un-der the assumption of market efficiency, a fund with superior performancebefore transaction costs will grow until its after transaction cost returns areequivalent to those of other funds. This hypothesis is compelling and plau-sible, but as we discuss in more detail later, the empirical evidence suggeststhat it is not consistent with the data.
We offer the competing hypothesis that fund size is a diffusive process,and can be well-understood by treating changes in fund size as random. Weare working in the tradition of stochastic models for firm size initiated byGibrat, Simon and Mandelbrot, and elaborating on earlier work by Gabaixet al.3. Our treatment is new in that we find a time-dependent solution. Weargue that this is essential for understanding the empirical data. In a sensethat we make precise, the diffusion of fund size is slow, and gets even sloweras a fund gets bigger. As we show, since funds are entering and exiting,under stationary conditions the distribution will eventually become a powerlaw, but this takes a very long time. This is exacerbated by the fact thatthe diffusion slows down as a fund gets larger. As a result the typical timeto approach the asymptotic heavy tailed power law distribution is the orderof a century. In the meantime the transient distribution has an essentiallylog-normal upper tail.
Our theory suggests that fund size is a robust property for which investorchoice plays at most a minor role. A critic might say that this implies thatthe whole phenomenon of fund size has “no economic content”. We viewthis as a positive attribute of our theory: It implies that its conclusions arevery robust, depending neither on rationality, nor on the idiosyncrasies ofhuman behavior.
Our results are interesting in the broader context of the literature on firmsize. Mutual funds provide a particularly good type of firm to study becausethere are a large number of funds and their size is accurately recorded. Itis generally believed that the resulting size distribution from aggregatingacross industries has a power law tail that roughly follows Zipf’s law, but
3A similar model yielding a steady state power law distribution, was originally proposedby Gabaix et al. [Gabaix et al., 2003]. Their steady state treatment lacks both an empiricalverification of the power law hypothesis and a justification for the t→∞ limit taken whilesolving the model. A similar argument for the power law steady state in a simple entryand exit process is described in [Mitzenmacher, 2004]. Neither of these solved the moregeneral time dependent case, as we have here.
3
for individual industries the tail behavior is debated4. A large number ofstochastic process models have been proposed to explain this5. Our em-pirical results show that in many respects mutual funds behave like typicalfirms. In particular, they show the same power law decay in the diffusionrate vs. size that has been observed for other types of firms6. However,mutual funds are distinctive in that, unlike other types of firms that havebeen studied, in the large size limit the mean and standard deviation of thegrowth rate do not decay to zero, but rather approach a positive limit. Thishas a simple economic interpretation: In the large size limit, a fund contin-ues to grow in size simply due to its own performance, even without furtherinflows of capital7. As our analytic treatment of this problem shows, thismakes a crucial difference in the asymptotic behavior. It implies that afterthe passage of a sufficient length of time, the size distribution for mutualfunds should approach a power law, whereas other firms should approach astretched exponential, which has relatively thin tails (thinner than those ofa log-normal). Thus for most firms the tails should get thinner with time,but for mutual funds they should become fatter. This does not have mucheffect on the current empirical distribution, however, since because of theslow diffusion mentioned earlier, over timescales less than a century bothmutual funds and other firms should remain roughly log-normal.
The paper is organized as follows. Section 1 describes the data usedfor the empirical study described in Section 2. The underlying dynamicalprocesses responsible for the size distribution are discussed in Section 3 andare used to develop the model discussed in Section 4. Section 4.1 presentsthe solution for the number of funds and Section 4.2 presents the solutionfor the size distribution. In Section 4.3 we present simulation results ofthe proposed model and compare them to the empirical data. In section5 we discuss the incompatibility of the Berk and Green [2004] hypothesis
4Some studies have found that the upper tail is a log-normal [Simon, Herbert A. andBonini, Charles P., 1958; Stanley et al., 1995; Y. and A., 1977; Stanley et al., 1996; Amaralet al., 1997a; Bottazzi and Secchi, 2003a; Dosi, 2005] while others have found a power law[Axtell, 2001; Bottazzi and Secchi, 2003a; Dosi, 2005]
5For past stochastic models see [Gibrat, 1931; Simon, 1955; Simon, Herbert A. andBonini, Charles P., 1958; Mandelbrot, 1963; Y. and A., 1977; Sutton, 1997; Gabaix et al.,2003; Gabaix et al., 2003]
6For the main body of work on the size dependence of firm growth rate fluctuationssee[Stanley et al., 1995, 1996; Amaral et al., 1997a; Bottazzi, 2001a; Bottazzi and Secchi,2003a, 2005a; Dosi, 2005; De Fabritiis et al., 2003a]
7Under the Berk and Green theory [2004], this should be irrational – investors shouldremove their funds and reallocate them to smaller funds. Thus in this sense there is somebehavioral dependence on the solution, though since the convergence is so slow, it makesno difference in the current mutual fund distribution. See the discussion in Section 5.
4
with the empirical data and our hypothesis. Finally Section 6 presents ourconclusions.
1 Data Set
We analyze the CRSP Survivor-Bias-Free US Mutual Fund Database. Be-cause we have daily data for each mutual fund, this database enables usnot only to study the distribution of mutual fund sizes in each year butalso to investigate the mechanism of growth. We study the data from 1991to 20058. We define an equity fund as one whose portfolio consists of atleast 80% stocks. The results are not qualitatively sensitive to this, e.g. weget essentially the same results even if we use all funds. The data set hasmonthly values for the Total Assets Managed (TASM) by the fund and theNet Asset Value (NAV). We define the size s of a fund to be the value of theTASM, measured in millions of US dollars and corrected for inflation rela-tive to July 2007. Inflation adjustments are based on the Consumer PriceIndex, published by the BLS.
2 The observed distribution of mutual fund sizes
Despite the fact that the mutual fund industry offers a large quantity ofwell-recorded data, the size distribution of mutual funds has not been rig-orously studied. This is in contrast with other types of firms where the sizedistribution has long been an active research subject. The fact that the dis-tribution is highly skewed and heavy tailed can be seen in Figure 1, wherewe plot the cumulative distribution of sizes P (s > X) of mutual fund sizesin three different years.
The exact functional form for the distribution of fund size is still debated.The two principal competing hypotheses are that the tail of the size distri-bution p(s) is either log-normal or a power law. Log-normality means thatlog s has a normal distribution, while power law means that the cumulativedistribution P for large s is of the form9
P (s > X) ∼ X−ζs ,8There is data on mutual funds starting in 1961, but prior to 1991 there are very few
entries. There is a sharp increase in 1991, suggesting incomplete data collection prior to1991.
9We denote scaling by f(x) ∼ g(x) which means that the two terms are proportionalfor large x, i.e. f(x) ∝ g(x) for large x.
5
10−2
100
102
104
100
10−2
10−4
X
P(s
>X
)
103
104
105
10−3
10−2
10−1
Figure 1: The CDF for the mutual fund size s (in millions of 2007 dollars)is plotted with a double logarithmic scale. The cumulative distribution forfunds existing at the end of the years 1993, 1998 and 2005 are given by thefull, dashed and dotted lines respectively.Inset: The upper tail of the CDF for the mutual funds existing at the endof 1998 (dotted line) is compared to an algebraic relation with exponent −1(solid line).
6
where ζs > 0. In the special case ζs ≈ 1 a distribution P is said to obeyZipf’s law. Power laws have the property that moments of higher orderthan α do not exist, so if a distribution follows Zipf’s law its right skewnessis so extreme that the mean is on the boundary where it becomes infiniteand there is no such thing as an “average fund size”. In contrast, for alog-normal all the moments exist. From the point of view of extreme valuetheory this distinction is critical, since it implies a completely different classof tail behavior10.
A visual inspection of the mutual fund size distribution suggests thatit does not follow Zipf’s law11. To see this, in the inset of Figure 1 wecompare the tail for funds with sizes s > 102 million to a power law s−ζs ,with ζs = −1. Whereas a power law corresponds to a straight line whenplotted on double logarithmic scale, the data show substantial and consistentdownward curvature. In the remainder of this section we make more rigoroustests that back up the intuitive impression given by this plot, indicating thatthe data are not well described by a power law.
2.1 Is the tail a power law?
To test the validity of the power law hypothesis we use the method developedby Clauset et al. [2007]. They use the somewhat strict definition12 that theprobability density function p(s) is a power law if there exists an smin suchthat for sizes larger than smin, the functional form of the density p(s) canbe written
p(s) =ζssmin
(s
smin
)−(ζs+1)
, (1)
where the distribution is normalized in the interval [smin,∞). There are twofree parameters smin and ζs. This crossover size smin is chosen such that it
10According to extreme value theory a probability distribution can have only four pos-sible types of tail behavior. The first three correspond to distributions with finite support,thin tails, and tails that are sufficiently heavy that some of the moments do not exist, i.e.power laws. The fourth category corresponds to distributions that in a certain sense donot converge; it is remarkable that most known distributions fall into one of the first threecategories [Embrechts et al., 1997].
11Previous work on the size distribution of mutual funds by Gabaix et al. [Gabaix et al.,2003; Gabaix et al., 2003; Gabaix et al., 2006] argued for a power law while we argue herefor a log-normal.
12In extreme value theory a power law is defined as any function that in the limit s→∞can be written p(s) = g(s)s−(ζs+1) where g(s) is a slowly varying function. This means itsatisfies lims→∞ g(ts)/g(s) = C for any t > 0, where C is a positive constant. The testfor power laws in reference [Clauset et al., 2007] is too strong in the sense that it assumesthat there exists an s0 such that for s > s0, g(s) is constant.
7
minimizes the Kolmogorov-Smirnov (KS) statistic D, which is the distancebetween the CDF of the empirical data Pe(s) and that of the fitted modelPf (s), i.e.
D = maxs≥smin
|Pe(s)− Pf (s)| .
Using this procedure we estimate ζs and smin for the years 1991- 2005 asshown in Table I. The values of ζs computed in each year range from 0.78to 1.36 and average ζs = 1.09 ± 0.04. If indeed these are power laws thisis consistent with Zipf’s law. But of course, merely computing an exponentand getting a low value does not mean that the distribution is actually apower law.
To test the power law hypothesis more rigorously we follow the MonteCarlo method utilized by Clauset et al. Assuming independence, for eachyear we generate 10, 000 synthetic data sets, each drawn from a power lawwith the empirically measured values of smin and ζs. For each data-set wecalculate the KS statistic to its best fit. The p-value is the fraction of thedata sets for which the KS statistic to its own best fit is larger than the KSstatistic for the empirical data and its best fit.
The results are summarized in Table I. The power law hypothesis isrejected with two standard deviations or more in six of the years and rejectedat one standard deviation or more in twelve of the years (there are fifteenin total). Furthermore there is a general pattern that as time progressesthe rejection of the hypothesis becomes stronger. We suspect that this isbecause of the increase in the number of equity funds. As can be seen inTable I, the total number of equity funds increases roughly linearly in time,and the number in the upper tail Ntail also increases.
We conclude that the power law tail hypothesis is questionable but can-not be unequivocally rejected in every year. Stronger evidence against itcomes from comparison to a log-normal, as done in the next section.
2.2 Is the tail log-normal?
The log normal distribution is defined such that the density function pLN (s)obeys
p(s) =1
sσ√
2πexp
(−(log(s)− µs)2
2σ2s
)and the CDF is given by
P (s′ > s) =12− 1
2erf(
log(s)− µs√2σs
).
8
vari
able
91
92
93
94
95
96
97
98
99
00
01
02
03
04
05
mea
nst
d
R-0
.50
-1.3
5-1
.49
-1.7
1-3
.29
-18.4
2-2
.25
-1.2
9-6
.57
-4.9
6-2
.63
-2.9
5-2
.00
-1.0
5-0
.99
-3.4
34.4
5
N372
1069
1509
2194
2699
3300
4253
4885
5363
5914
6607
7102
7794
8457
8845
--
E[s
](m
n)
810
385
480
398
448
527
559
619
748
635
481
335
425
458
474
519
134
Std
[s]
(bn)
1.9
80.9
91.7
1.6
61.6
82.4
12.8
23.3
84.0
53.3
72.6
91.8
72.4
52.6
42.6
52.4
20.8
E[ω
]5.5
84.4
04.4
03.8
63.8
63.9
13.8
43.8
54.0
63.9
73.6
03.3
73.5
53.5
13.5
93.9
60.5
4
Std
[ω]
1.5
11.9
82.0
92.4
32.5
02.4
62.5
02.5
12.4
62.4
52.6
32.4
22.4
92.5
92.5
02.3
40.2
9
ζ s1.3
31.3
61.1
91.1
51.1
10.7
81.0
81.1
00.9
50.9
71.0
11.0
71.0
71.1
01.1
41.0
90.1
4
s min
955
800
695
708
877
182
1494
1945
1147
903
728
836
868
1085
1383
974
408
Ntail
81
129
232
256
280
1067
290
283
557
662
717
494
652
630
550
--
p-v
alu
e0.5
80.4
80.0
20.4
50.0
70
0.0
10.1
15
10−
40.0
40.0
30.0
70.0
80.0
80.1
50.1
50.1
9
µ(1
0−
3)
26
42
81
46
72
67
58
39
39
20
-3-1
050
18
30.5
38.5
26
σ(1
0−
1)
0.7
82.0
2.6
3.0
2.3
2.8
2.9
3.0
2.8
2.8
2.5
2.4
2.4
2.4
2.5
2.5
0.5
5
Nexit
041
45
61
139
115
169
269
308
482
427
660
703
675
626
--
Nenter
185
338
581
783
759
885
1216
1342
1182
1363
1088
1063
1056
796
732
891
346
Tab
leI:
Tab
leof
mon
thly
para
met
erva
lues
for
equi
tyfu
nds
defin
edsu
chth
atth
epo
rtfo
lioco
ntai
nsa
frac
tion
ofat
leas
t80
%st
ocks
.T
heva
lues
for
each
ofth
em
onth
lypa
ram
eter
s(r
ows)
wer
eca
lcul
ated
for
each
year
(col
umns
).T
hem
ean
and
stan
dard
devi
atio
nar
eev
alua
ted
for
the
mon
thly
valu
esin
each
year
.R
-th
eba
se10
log
likel
ihoo
dra
tio
ofa
pow
erla
wfit
rela
tive
toa
log-
norm
alfit
asgi
ven
byeq
uati
on(3
).A
nega
tive
valu
eofR
indi
cate
sth
atth
elo
g-no
rmal
hypo
thes
isis
alik
elie
rde
scri
ptio
nth
ana
pow
erla
w.
For
all
year
sth
eva
lue
isne
gati
vem
eani
ngth
atth
elo
g-no
rmal
dist
ribu
tion
ism
ore
likel
y.N
-th
enu
mbe
rof
equi
tyfu
nds
exis
ting
atth
een
dof
each
year
.E
[ω]
-th
em
ean
log
size
offu
nds
exis
ting
atth
een
dof
each
year
.Std
[ω]
-th
est
anda
rdde
viat
ion
oflo
gsi
zes
for
fund
sex
isti
ngat
the
end
ofea
chye
ar.
E[s
]-
the
mea
nsi
ze(i
nm
illio
ns)
offu
nds
exis
ting
atth
een
dof
each
year
.Std
[s]
-th
est
anda
rdde
viat
ion
ofsi
zes
(in
billi
ons)
for
fund
sex
isti
ngat
the
end
ofea
chye
ar.
ζ s-
the
pow
erla
wta
ilex
pone
nt(1
).s m
in-
the
low
erta
ilcu
toff
(in
mill
ions
ofdo
llars
)ab
ove
whi
chw
efit
apo
wer
law
(1).
Ntail
-th
enu
mbe
rof
equi
tyfu
nds
belo
ngin
gto
the
uppe
rta
ils.
t.s≥s m
in.
p-v
alue
-th
epr
obab
ility
ofob
tain
ing
ago
odne
ssof
fitat
leas
tas
bad
asth
eon
eca
lcul
ated
for
the
empi
rica
ldat
a,un
der
the
null
hypo
thes
isof
apo
wer
law
uppe
rta
il.µ
-th
edr
iftte
rmfo
rth
ege
omet
ric
rand
omw
alk
(10)
,co
mpu
ted
for
mon
thly
chan
ges.
σ-
the
stan
dard
devi
atio
nof
the
mea
nze
roW
iene
rpr
oces
s(1
0),
com
pute
dfo
rm
onth
lych
ange
s.N
exit
-th
enu
mbe
rof
equi
tyfu
nds
exit
ing
the
indu
stry
each
year
.N
enter
-th
enu
mbe
rof
new
equi
tyfu
nds
ente
ring
the
indu
stry
inea
chye
ar.
9
3 3.5 4 4.5 5 5.53
3.5
4
4.5
5
5.5
Empirical Quantiles
Log−
norm
al Q
uant
iles
3 3.5 4 4.5 5 5.53
3.5
4
4.5
5
5.5
6
Empirical Quantiles
Pow
er L
aw Q
uant
iles
(a) (b)
Figure 2: A Quantile-Quantile (QQ) plot for the upper tail of the size distri-bution of equity funds. The quantiles are the base ten logarithm of the fundsize, in millions of dollars. The empirical quantiles are calculated from thesize distribution of funds existing at the end of the year 1998. The empiricaldata were truncated from below such that only funds with size s ≥ sminwere included in the calculation of the quantiles. (a) A QQ-plot with theempirical quantiles as the x-axis and the quantiles for the best fit power lawas the y-axis. The power law fit for the data was done using the maximumlikelihood described in Section 2.1, yielding smin = 1945 and α = 1.107. (b)A QQ-plot with the empirical quantiles as the x-axis and the quantiles forthe best fit log-normal as the y-axis, with the same smin as in (a). The log-normal fit for the data was done used the maximum likelihood estimationgiven smin (2) yielding µ = 2.34 and σ = 2.5.
A visual comparison between the two hypotheses can be made by looking atthe Quantile Quantile (QQ) plots for the empirical data compared to each ofthe two hypotheses. In a QQ-plot we plot the quantiles of one distributionas the x-axis and the other’s as the y-axis. If the two distributions are thesame then we expect the points to fall on a straight line. Figure 2 comparesthe two hypotheses, making it clear that the log-normal is a much better fitthan the power law. For the log-normal QQ plot most of the large valuesin the distribution fall on the dashed line corresponding to a log-normaldistribution, though the very largest values are somewhat above the dashedline. This says that the empirical distribution decays slightly faster than alog-normal. There are two possible interpretations of this result: Either thisis a statistical fluctuation or the true distribution really has slightly thinnertails than a log-normal. In any case, since a log-normal decays faster than
10
a power law, it strongly suggests that the power law hypothesis is incorrectand the log-normal distribution is a better approximation.
A more quantitative method to address the question of which hypothesisbetter describes the data is to compare the likelihood of the observation inboth hypotheses [Clauset et al., 2007]. We define the likelihood for the tailof the distribution to be
L =∏
sj≥sminp(sj).
We define the power law likelihood as LPL =∏sj≥smin pPL(sj) with the
probability density of the power law tail given by (1). The lognormal likeli-hood is defined as LLN =
∏sj≥smin pLN (sj) with the probability density of
the lognormal tail given by
pLN (s) =p(s)
1− P (smin)
=√
2s√πσ
[erfc
(ln smin − µ√
2σ
)]−1
exp
[−(ln s− µ)2
2σ2
]. (2)
The more probable that the empirical sample is drawn from a givendistribution, the larger the likelihood for that set of observations. The ratioindicates which distribution the data are more likely drawn from. We definethe log likelihood ratio as
R = ln(LPLLLN
). (3)
For each of the years 1991 to 2005 we computed the maximum likelihoodestimators for both the power law fit and the log-normal fit to the tail,as explained above and in Section 2.1. Using the fit parameters, the loglikelihood ratio was computed and the results are summarized graphicallyin Figure 3 and in Table I. The ratio is always negative, indicating that thelikelihood for the log-normal hypothesis is greater than that of the powerlaw hypothesis in every year. It seems clear that tails of the mutual funddata are much better described by a log-normal than by a power law.
3 Empirical investigation of size dynamics
Our central thesis in this paper is that the mutual fund size distributioncan be explained by a stochastic process governed by three key underlyingprocesses: the size change of existing mutual funds, the entry of new funds
11
−8 −6 −4 −2 0 2 4 6 80
1
2
3
4
5
log−normal tail power law tail
Figure 3: A histogram of the base 10 log likelihood ratios R computedusing (3) for each of the years 1991 to 2005. A negative log likelihood ratioimplies that it is more likely that the empirical distribution is log-normalthen a power law. The log likelihood ratio is negative in every year, inseveral cases strongly so.
12
variable 1991- 1998 1991- 2005
ω0 0.14 −0.37σω 3.02 3.16σ0 0.35± 0.02 0.30± 0.02β 0.31± 0.03 0.27± 0.02σ∞ 0.05± 0.01 0.05± 0.01µ0 0.15± 0.01 0.08± 0.05α 0.48± 0.03 0.52± 0.04µ∞ 0.002± 0.008 0.004± 0.001
Table II: Model parameters as measured from the data in different timeperiods. ω0 and σ2
ω are the mean and variance of the average (log) size ofnew funds described in (4). σ0, β and σ∞ are the parameters for the sizedependent diffusion and µ0, α and µ∞ are the parameters of the averagegrowth rate (9). The confidence intervals are 95% under the assumption ofstandard errors. The time intervals were chosen to match the results shownin Fig. 9.
and the exit of existing funds. In this section we empirically investigate eachof these three processes, providing motivation for the model developed inthe next section.
3.1 Fund entry process
We begin by examining the entry of new funds. We investigate both thenumber of funds entering each year Nenter(t) and the sizes with which theyenter. We perform a linear regression of Nenter(t) against the number ofexisting funds N(t − 1), yielding slope α = 0.04 ± 0.05 and intercept β =750± 300. The slope is not statistically significant, and so we approximatefund entry as a Poisson process with a constant rate ν.
The size of entering funds is more complicated. In Figure 4 we comparethe distribution of the size of entering funds f(s) to that of all existingfunds. The distribution is somewhat irregular, with peaks at round figuressuch as ten thousand, a hundred thousand, and a million dollars. Theaverage size13 of entering funds is almost three orders of magnitude smaller
13When discussing the average size one must make sure to account for the differencebetween the average log size and the average size, since due to the heavy tails the differenceis striking. The average entry log size E[ωc] ≈ 0, corresponding to a fund of size one million,
13
−4 −2 0 2 4 60
0.1
0.2
0.3
0.4
log10
(s)
p(s)
Existing funds
Entering funds
Figure 4: The probability density for the size s of entering funds in millionsof dollars (solid line) compared to that of all funds (dashed line) includingall data for the years 1991 to 2005. The densities were estimated using agaussian kernel smoothing technique.
than that of existing funds, making it clear that the typical surviving fundgrows significantly after it enters. We compared the distribution of enteringfunds to a log-normal and found that the tails are substantially thinner thanthose of the log-normal. When we consider these facts (small size and thintails) it is clear that the distribution of entering funds cannot be importantin determining the upper tail of the fund size distribution.
To conclude, the fund entry process is reasonably well approximated asa Poisson process in which an average of ν funds enter per month, with thesize of each fund drawn from a distribution f(ω, t). The distribution of newfund sizes is approximated as a log-normal distribution in the fund size s,that is a normal distribution in the log size ω
f(ω, t) =1√πσ2
ω
exp
(−(ω − ω0)2
σ2ω
)θ(t− t0), (4)
where ω0 is the mean log size of new funds and σ2ω is its variance. θ(t− t0)
is a unit step function ensuring no funds funds enter the industry before the
while if we average over the entry sizes E[sc] = E[eωc ], we get an average entry size ofapproximately 30 million. For comparison, both the average size and the average log sizeof existing funds are quoted in Table I.
14
1000 2000 3000 4000 5000 6000 7000 8000
0
100
200
300
400
500
600
700
800
900
N(t−1)
Nex
it (t
)
Figure 5: The number of equity funds exiting the industry Nexit(t) in theyear t as a function of the total number of funds existing in the previousyear, N(t− 1). The plot is compared to a linear regression (full line). Theerror bars are calculated for each bin under a Poisson process assumption,and correspond to the square root of the average number of funds exitingthe industry in that year.
initial time t0. The value of the mean log size and its variance are calculatedfrom the data for different periods, as summarized in Table II.
3.2 Fund exit process
As we will show later, fund exit is of critical importance in determining thelong-run properties of the fund size distribution. In Figure 5 we plot thenumber of exiting funds Nexit(t) as a function of the total number of fundsexisting in the previous year, N(t−1). The linearly increasing trend is clear,and in most cases the number of exiting funds is within a standard deviationof the linear trend line. This suggests that it is a reasonable approximationto assume that the probability for a given fund to exit is independent oftime, so that the number of exiting funds is proportional to the number ofexisting funds, with proportionality constant λ(ω), where ω is the logarithmof the fund size. Furthermore, it seems to be reasonable to assume thatλ(ω) = λ is independent of size; where λ is the slope of the linear regressionin Figure 5. On an annual time scale this gives λ = 0.092 ± 0.030. Underthe assumption that fund death is a Poisson process the monthly rate isjust the yearly rate divided by the number of months per year. Thus, we
15
approximate the fund exit process as a Poisson process where funds of sizeω exit the industry with a constant rate λ.
3.3 Fund growth
Fund growth is a complicated process as it involves not only size changesdue to fund performance but also changes due to investors depositing andwithdrawing money. The investor behavior induces correlations and sizedependence in the random process. It is convenient to define the relativechange in a fund’s size ∆s(t) as
∆s(t) =s(t+ 1)− s(t)
s(t). (5)
The relative change can be decomposed into two parts: the return ∆r andthe fractional investor money flux ∆f (t), which are simply related as
∆s(t) = ∆f (t) + ∆r(t). (6)
The return ∆r represents the return of the fund to its investors, defined as
∆r(t) =NAV (t+ 1)−NAV (t)
NAV (t), (7)
where NAV (t) is the Net Asset Value at time t. The fractional money flux∆f (t) is the change in the fund size by investor deposits or withdrawals,defined as
∆f (t) =s(t+ 1)− [1 + ∆r(t)]s(t)
s(t). (8)
Since the data set only contains information about the size of funds andtheir returns we are not able to separate deposits from withdrawals, butrather can only observe their net.
Figure 6 gives an overview of the size dependence for both the returns ∆r
and the money flux ∆f . The two behave very differently. The returns ∆r areessentially independent of size14. This is expected based on market efficiency,
14The independence of the return ∆r on size is verified by performing a linear regressionof µr vs. s for the year 2005, which results in an intercept β = 6.7 ± 0.2 × 10−3 and aslope α = 0.5± 8.5× 10−8. This result implies a size independent average monthly returnof 0.67%.
16
100
101
102
103
104
105
10−4
10−3
10−2
10−1
100
101
s
σr
µr
σf
µf
Figure 6: The average monthly return µr and its volatility σr, and themoney flux µf and its volatility σf , are plotted as a function of the fundsize (in millions) for the year 2005 (see Eqs. (5 - 8)). The data are binnedbased on size, using bins with exponentially increasing size. The averagemonthly return µr is compared to a constant return of 0.008 and the monthlyvolatility σr is compared to 0.03. The average monthly flux µf is comparedto a slope of -0.5 and the money flux volatility σf is compared to a linearslope of -0.35. On a double logarithmic scale a line of slope α correspondsto a power law relation f(s) = sα.
17
as otherwise one could obtain superior performance simply by investing inlarger or smaller funds [Malkiel, 1995]. This implies that equity mutual fundscan be viewed as a constant return to scale industry [Gabaix et al., 2006].Both the mean µr = E[∆r] and the standard deviation σr = Var[∆r]1/2
are constant; the latter is also expected from market efficiency, as otherwiseit would be possible to lower one’s risk by simply investing in funds of adifferent size.
In contrast, the money flow ∆f decreases with size. Both the meanmoney flux µf = E[∆f ] and its standard deviation σf = Var[∆f ]1/2 roughlyfollow a power law over five orders of magnitude in the size s. This is similarto the behavior that has been observed for the growth rates of other types offirms [Stanley et al., 1995, 1996; Amaral et al., 1997a; Bottazzi and Secchi,2003a]. Several theories have been put forth to explain this for firms ingeneral, but at this point there is no consensus view as to its cause15. Formutual funds one would expect that in absolute terms, all else being equal,it is harder to raise large amounts of money than small amounts of money,and thus the relative growth rate should decrease with size. This does not,however, explain why the functional form should be a power law. The factthat similar behavior is observed for other types of firms suggests that thereshould be a common explanation that holds for firms in general, and is notspecific to mutual funds.
These two effects can be combined to model the mean total size change µsand the standard deviation σs. The standard deviations are related throughthe approximate relation
σ2s ≈ σ2
f + σ2r .
This is only an approximation since we neglected the correlation term be-tween ∆r and ∆f . Based on our empirical studies the correlation is close tozero and this is a good approximation. In Figure 7 we plot both µs and σsas a function of size.
The decomposition illustrated in Figure 6 suggests that µs and σs canbe reasonably well approximated as
σs(s) = σ0s−β + σ∞ (9)
µs(s) = µ0s−α + µ∞.
15There has been a significant body of work attempting to explain the observed growthrate of firms. For the main body of work see [Amaral et al., 1997b; Buldyrev et al., 1997;Amaral et al., 1998; De Fabritiis et al., 2003b; Matia et al., 2004; Bottazzi, 2001b; Sutton,2001; Wyart and Bouchaud, 2002; Bottazzi and Secchi, 2003b, 2005b; Fu et al., 2005;Riccaboni et al., 2008; Podobnik et al., 2008].
18
10−3
10−1
101
103
105
10−1
100
s
σ
10−3
10−1
101
103
105
10−2
10−1
100
s
µ
(a) (b)
Figure 7: (a) The standard deviation σ in the logarithmic size change ∆s =∆(log s) of an equity fund as a function of the fund size s (in millions ofdollars). (b) The mean µ logarithmic size change ∆s = ∆(log s) of an equityfund as a function of the fund size s (in millions of dollars). The data forall the funds were divided into 100 equally occupied bins. µ is the mean ineach bin and σ is the square root of the variance in each bin for the years1991 to 2005. The data are compared to a fit according to (9) in Figures(a) and (b) respectively.
As illustrated in Figure 7, these functional forms fit the data reasonablywell, with only slight variations of parameters in different periods, as shownin Table II.
To summarize: In one respect, mutual funds are like other firms, in thattheir money flux shows the same power law decay with size that has beenobserved for the growth rates of other firms. At the same time, mutual fundsare unusual in that a component of their growth comes from the returns ofthe fund itself; this component is independent of size, as expected frommarket efficiency. Thus mutual funds are unusual in that, due to marketefficiency, their relative growth rate does not go to zero in the limit s →∞. The upshot is that for small funds money flux is the dominant growthprocess and for large funds the return is the dominant growth process. Thismeans that for the largest funds we can approximate the mean and standarddeviation of the growth rate as being size independent.
19
−6 −4 −2 0 2 4 6 10
−6
10−5
10−4
10−3
10−2
10−1
100
∆ω
p(∆ ω
)
Figure 8: The PDF of aggregated monthly log size changes∆w = log10(1 + ∆s) for equity funds in the years 1991 to 2005. The logsize changes were binned into 20 bins for positive changes and 20 bins fornegative changes. Monthly size changes were normalized such that the av-erage log size change in each month was zero.
3.4 Diffusion model
We approximate the growth of existing funds as a multiplicative Gibrat-likeprocess16 satisfying a stochastic evolution equation of the form
ds = s [µs(s)dt+ σs(s)dWt] , (10)
where Wt is a mean zero and unit variance normal random variable, µs isthe drift and σs is the volatility. The use of an underlying Wiener process inthe stochastic evolution equation (10) assumes implicitly that the relativesize change distributions of funds of a given size is an i.i.d normal randomvariable.
We first discuss the normality assumption. As can be seen in Figure 8,the relative size change distribution has a tent shape when plotted in semi-logarithmic scale17. It can be approximated as a double exponential, alsocalled a Laplace distribution, which has heavier tails than a normal distri-bution. Nonetheless, due to the fact that the logarithmic increments are
16A Gibrat-like process is a multiplicative process in which the size of the fund at anygiven time is given as a multiplicative factor times the size of the fund at a previous time.In Gibrat’s law of proportionate effect [Gibrat, 1931] the multiplicative term dependslinearly on size while here we allow it to have any size dependence.
17A similar phenomenon was originally observed by Stanley et al for the distribution offirm growth [Stanley et al., 1996; Axtell, 2001; Dosi, 2005].
20
date 12/2005 9/2005 6/2005 3/2005 12/2004β1 0.10± 0.16 0.40± 0.98 0.27± 0.68 1.17± 4.68 −0.23± 1.24β2 0.14± 0.27 0.36± 1.20 0.48± 0.83 −0.79± 3.13 −0.65± 2.31β3 0.28± 0.45 0.01± 1.07 0.33± 0.83 1.79± 3.24 0.60± 2.57β4 0.56± 0.40 −0.28± 0.85 0.24± 1.27 −0.28± 1.65 0.44± 2.32β5 0.24± 0.43 −0.25± 1.13 0.21± 0.90 −0.24± 2.95 0.43± 2.49β6 0.48± 0.38 −0.02± 1.03 0.30± 0.92 1.27± 3.50 0.31± 2.09β −0.02± 0.02 0.03± 0.05 0.01± 0.05 0.14± 0.21 0.06± 0.15
Table III: Cross-sectional regression coefficients of the monthly fund flow,computed for several months, against the performance in past months, asindicated in Eq. 11. The regression was computed cross-sectionally usingdata for 6189 equity funds. For example the entry for β1 in the first (fromthe left) column represents the linear regression coefficient of the money fluxat the end of 2005 on the previous month’s return. The errors correspondto the 95% confidence intervals.
additive, after several steps the central limit theorem causes this to quicklyconverge to a normal distribution. We have explicitly verified this by track-ing a group of funds in a given size range over time. Thus even thoughthe normality assumption is not true on short timescales it rapidly becomesvalid on longer timescales.
The second approximation is that of an independent process. Strictlyspeaking this approximation is also violated, due to the fact that investorsreact to past performance18. This causes correlations in the money flux ∆f ,which in turns induces correlations in the total size change ∆s. We find,however, that these correlations are small. This can be seen in Table III.We perform cross sectional regressions for several different times t of theform
∆f (t) = β + β1∆r(t− 1) + β2∆r(t− 2) + . . .+ β6∆r(t− 6). (11)
The results are extremely noisy; for example, eight of the thirty possiblecoefficients βi are negative and only two of them are significant at the twostandard deviation level. The independent process approximation is partic-ularly true for large funds where the size growth is dominated by the return
18For empirical evidence that investors react to past performance see [Remolona et al.,1997; Busse, 2001; Chevalier, Judith and Ellison, Glenn, 1997; Sirri, Erik R. and Tufano,Peter, 1998; Guercio, Diane Del and Tkac, Paula A., 2002; Bollen, 2007].
21
∆r, which is essentially uncorrelated due to market efficiency [Bollen andBusse, 2005; Carhart, 1997].
4 An analytical model for the mutual fund sizedistribution
The empirical evidence described in the previous section motivates a simplestochastic growth model for the equity mutual fund industry. The aim ofthe model is to describe the time evolution of the size distribution, that is,to solve for the probability density function p(ω, t) of funds with (log) sizeω existing at time t. The size distribution can be written as
p(ω, t) =n(ω, t)N(t)
, (12)
where n(ω, t) is the number of funds at time with logarithmic size ω andN(t) =
∫n(ω, t)dω is the total number of funds at time t. To simplify the
analysis we solve separately for the total number of funds N(t) and for thenumber density n(ω, t).
4.1 Dynamics of the total number of funds
Based on the empirical observations of Section 3, we have argued that it isnatural to model the total number of funds as a function of time as
dN
dt= ν − λN (13)
where ν is the rate of creating new funds and λ is the exit rate of exist-ing funds. Under the assumption that ν and λ are constant in time andindependent of N this has the solution
N(t) =ν
λ
(1− e−λt
)θ(t), (14)
where θ(t) is a unit step function at t = 0, the year in which the first fundsenter. This solution has the surprising property that the dynamics onlydepend on the fund exit rate λ, with a characteristic timescale 1/λ. Forexample, for λ ≈ 0.09, as estimated in Section 3, the timescale for N(t) toreach its steady state is only roughly a decade. An examination of Table Imakes it clear, however, that ν = constant is not a very good approximation.Nonetheless, if we crudely use the mean creation rate ν ≈ 900 from Table I
22
and the fund exit rate λ ≈ 0.09 estimated in Section 3, the steady statenumber of funds should be about N ≈ 10, 000, compared to the 8, 845 fundsthat actually existed in 2005. Thus this gives an estimate with the rightorder of magnitude.
The important point to stress is that the dynamics for N(t) operate on adifferent timescale than that of n(ω, t). As we will show in the next sectionthe characteristic timescale for n(ω, t) is much longer than that for N(t).
4.2 Analytical solution for the number density n(ω, t)
We define and solve the time evolution equation for the number densityn(ω, t) using the empirically motivated approximations described earlier.These approximations are: The entry process is modeled as a Poisson processwith rate ν, such that at time t a new fund enters the industry with aprobability νdt and (log) size ω drawn from a distribution f(ω, t). Theentry size distribution is approximated as normal distribution in log sizegiven by equation (4). The exit process is modeled as an individual Poissonprocess such that at any time time t a fund exits the industry with a sizeindependent probability λdt. The size change is approximated as a (log)Brownian motion with a size dependent drift and diffusion term given byequation (9). Under these assumptions the forward Kolmogorov equation(also known as the Fokker-Plank equation) defining the time evolution ofthe number density [Gardiner, 2004] is given by
∂
∂tn(ω, t) = νf(ω, t)−λn(ω, t)− ∂
∂ω[µ(ω)n(ω, t)] +
∂2
∂ω2[D(ω)n(ω, t)], (15)
where D(ω) = σ(ω)2/2 is the size diffusion coefficient. The first term on theright describes the entry process, the second describes the fund exit processand the third and fourth terms describe the change in size of an existingfund. Under the change of variables s = exp(ω) in (9) the size dependentdrift and diffusion terms are given by
µ(ω) = µ0e−αω + µ∞ (16)σ(ω) = σ0e−βω + σ∞.
4.2.1 Approximate solution for large funds
Eqs. (15) and (16) cannot be solved analytically, but a great deal of insightcan be gained by going to the large size limit. As previously described, due
23
to the dominance of the returns, in this limit ∆s ≈ ∆r, and µ(ω) and σ(ω)are constant, i.e. µ = µ∞ and σ = σ∞. The evolution equation becomes
∂
∂tn(ω, t) = νf(ω, t)− λn(ω, t)− µ ∂
∂ωn(ω, t) +D
∂2
∂ω2n(ω, t), (17)
where we used the notation D = σ2∞/2. The upper tail behavior of the dis-
tribution described by (17) strongly depends on the exit and entry process.If for example, funds only change in size and no funds enter or exit, thenthe resulting distribution is normal
n(w, t) =1√
4πDtexp
[−(ω − µt)2
4Dt
], (18)
which corresponds to a size distribution p(s) with a lognormal upper tail.The exit process plays a major role and is responsible for the thickening
with time of the upper tail of the distribution. The intuition is as follows:Since each fund exits the industry with the same probability and since thereare more smaller sized funds than larger ones the number of small fundsexiting the industry is larger. This results in an increasing fraction of largerfunds. As we will now show this results in the distribution evolving froma log-normal upper tail to a power law upper tail. In contrast, the entryprocess is not as important and determines mainly the number of funds inthe industry. This is true as long as the entry size distribution f(ω, t) is notheavier-tailed than a lognormal, which is supported by the empirical data.
In the large size limit the solution for an arbitrary entry size distributionf is given by
n(ω, t) = ν
∫ ∞−∞
∫ t
0exp−λt
′ 1√4πDt′
exp
[−(ω − ω′ − µt)2
4Dt′
]f(ω′, t−t′) dt′dω′.
(19)Stated in words, a fund of size ω′ enters at time t − τ with probabilityf(ω′, t− τ). The fund will survive to time t with a probability exp(λτ) andwill have a size ω at time t with a probability according to (18).
If funds enter the industry with a constant rate ν beginning at t = 0,with a log-normal entry size distribution f(ω, t) centered around ωs withwidth σs as given by (4), the size density can be shown to be
n(ω, t) =νµ
4√γD
exp
[(γ +
14
)σ2s
2−√γ
∣∣∣∣∣σ2s
2+µ
D(ω − ωs)
∣∣∣∣∣+ µ
2D(ω − ωs)
]
×(A+ exp
[√γ|σ2
s + 2µ
D(ω − ωs) |
]B
), (20)
24
where the drift and diffusion terms are approximated as µ = µ∞ and D =σ2∞/2. The parameters A, B and γ are defined as
γ =
√14
+λD
µ2, (21)
A = Erf
∣∣∣σ2s2 + µ
D (ω − ωs)∣∣∣−√γσ2
s√2σs
− Erf
∣∣∣σ2s2 + µ
D (ω − ωs)∣∣∣−√γ (σ2
s + 2µ2
D t)
√2√σ2s + 2µ
2
D t
(22)
and
B = Erf
√γ(σ2s2 + µ2
D t)
+ |σ2s2 + µ
D (ω − ωs) |√
2√σ2s + 2µ
2
D t
−Erf
√γσ2s +
∣∣∣σ2s2 + µ
D (ω − ωs)∣∣∣
√2σs
,(23)
where Erf is the error function, i.e. the integral of the normal distribution.
4.2.2 Steady state solution for large funds
Making a further approximation simplifies the solution considerably. Let usdefine a large fund as one with ω � ωs, where ωs is the logarithm of thetypical entry size of one million USD. For large funds we can approximatethe lognormal distribution as having zero width, in other words we assumethat all new funds have the same size ωs. The number density is then givenby
n(ω, t) =νD
4√γµ2
e12µD
(ω−ωs)[e−√γ µD|ω−ωs|
1 + erf
√γµ2t
D− |ω − ωs|
2√Dt
− e
√γ µD|ω−ωs|
(1− erf
[µ
√t
D(12
+√γ)
])],
where we have approximated the drift and diffusion as µ = µ∞ and D =σ2∞/2. Since γ > 1/4 (21), the density vanishes for both ω → ∞ andω → −∞.
The steady state solution for large times is achieved by taking the t→∞limit of (24), which gives
n(ω) =ν
2µ√γ
expµ
D
(ω − ωs
2−√γ|ω − ωs|
). (24)
25
Since the log size density (24) has an exponential upper tail p(ω) ∼ exp(−ζsω)and s = exp(ω) the CDF for s has a power law tail with an exponent19 ζs,i.e.
P (s > X) ∼ X−ζs . (25)
Substituting for the parameter γ using Eq. (21) for the upper tail exponentyields
ζs =−µ+
√µ2 + 4Dλ
2D. (26)
Note that this does not depend on the creation rate ν. Using the averageparameter values in Table II the asymptotic exponent has the value
ζs = 1.2± 0.6. (27)
This suggests that when the distribution will reach its steady state the tailswill be in agreement with those expected under Zipf’s law, ζs ≈ 1.
4.2.3 The importance of µ∞ and σ∞
The non vanishing drift µ∞ > 0 and diffusion terms σ∞ > 0 are essentialfor the distribution to evolve towards a power law. As can be seen from thefit parameters, based on data for ∆s alone, we cannot strongly reject thehypothesis that the drift and diffusion rates vanish for large sizes, i.e. µ∞ →0 and σ∞ → 0. However, because the size change ∆s can be decomposedinto ∆r and ∆f , and because the mean of ∆r > 0 due to the growth of themarket, we are confident that neither µ∞ nor σ∞ are zero.
This distinguishes mutual funds from other types of firms, which aretypically observed empirically to have σ∞ = 0 [Stanley et al., 1996; Matiaet al., 2004]. It suggests that in a sufficiently stationary situation and afterthe passage of sufficient time their distributions should be different. Previousempirical studies for other firms found µ∞ = 0 and σ∞ = 0. Assuming thatother types of firms obey similar diffusion equations to those used here, itcan be shown that the resulting distribution has a stretched exponentialupper tail, which is much thinner than a power law20.
19To calculate the tail exponent of the density correctly one must change variablesthrough p(s) = p(ω)dω
ds∼ s−ζs−1. This results in a CDF with a tail exponent of ζs.
20A stretched exponential is of the form p(x) ∼ exp(ax−b), where a and b are positiveconstants. There is some evidence in the empirical data that the death rate λ also decayswith size. However, in our simulations we found that this makes very little difference forthe size distribution as long as it decays slower than the distribution of entering funds,and so in the interest of keeping the model parsimonious we have not included this in ourmodel.
26
4.2.4 Timescale to reach steady state
The most useful aspect of solving for the time dependent solution is thatit provides the ability to estimate the timescale to reach the steady statesolution. The time dependence in Eq. 24 is contained in the arguments ofthe error function terms on the right. When these arguments become large,say larger than 3, the solution is roughly time independent, and can bewritten as
t >9D
4γµ2
1 +
√1 +
29
√γµ2
D|ω − ωs|
2
. (28)
Using the values obtained by fitting the whole data set, in units of monthsµ = µ∞ ≈ 0.005, D = σ2
∞/2 and σ∞ ≈ 0.05. This gives
t > 180(
1 +√
1 + 0.7 |ω − ωs|)2
,
where the time is in months. Plugging in some numbers makes it clear thatthe time scale to reach steady state is very long. For instance, for funds ofa billion dollars it will take about 170 years for their distribution to comewithin 1 percent of its steady state. This agrees with the observation thatthere seems to be no significant fattening of the tail over nearly two decadessince 1991. Note that the time required for the distribution n(ω, t) to reachsteady state for large values of ω is much greater than that for the totalnumber of funds N(t) to become constant.
4.3 Comparisons to empirical data
We are unable to find an analytic solution for the general case of Eq. (15)including the size dependence of the diffusion and drift terms, and so weemploy a simulation. This was done on a monthly time scale, averaging over1000 different runs to estimate the final distribution. As we have emphasizedin the previous discussion the time scales for relaxation to the steady statedistribution are long. We thus take the evidence for the huge increase inthe number of new funds seriously. We begin the simulation in 1991 andsimulate the process for varying periods of time, making our target theempirical data at the end of the period. In each case we assume the sizedistribution for injecting funds is log-normal, as discussed in Section 3.1.
To compare our predictions to the empirical data we measure the pa-rameters for each of the processes of fund creation, entry and exit usingdata from the same period as the simulation, summarized in Table II. A
27
10−2
100
102
104
100
10−2
10−4
−2 −1 0 1 2 3 4 5−4
−2
0
2
4
6
10−2
100
102
104
100
10−2
10−4
X
P(s
>X
)
−2 −1 0 1 2 3 4 5−4
−2
0
2
4
6
Simulation Quantiles
Em
piric
al Q
uant
iles
Figure 9: The model is compared to the empirical distribution at differenttime horizons. The left column compares CDFs from the simulation (fullline) to the empirical data (dashed line). The right column is a QQ-plotcomparing the two distributions. In each case the simulation begins in 1991and is based on the parameters in Table II. The first row corresponds to theyears 1991-1998 and the second row to the years 1991-2005 (in each case weuse the data at the end of the quoted year).
key point is that we are not fitting these parameters on the target data forfund size21, but rather are fitting them on the individual creation, exit anddiffusion processes and then simulating the corresponding model to predictfund size. One of our main predictions is that the time dependence of thesolution is important. In Figure 9 we compare the predictions of the simula-tion to the empirical data at two different ending times. The model fits quitewell at all time horizons, though the fit in the tail is somewhat less goodat the longest time horizon. This indicates that even for this model thereis a slight tendency to estimate a tail that is heavier than that of the data.Note, however, that the simulations also make it clear that the fluctuationsin the tail are substantial, and this deviation is very likely due to chance –many of the individual runs of the simulation deviate from the mean of the
21It is not our intention to claim that the processes describing fund size are constantor even stationary. Thus, we would not necessarily expect that parameters measured onperiods outside of the sample period will be a good approximation for those in the sampleperiod. Rather, our purpose is to show that these processes can explain the distributionof fund sizes.
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1000 simulations much more than the empirical data does.
5 Comparison with the hypothesis of Berk andGreen
As mentioned in the introduction, Berk and Green [2004] have offered analternative hypothesis to ours, based on the distribution of fund managerskill and the size dependence of transaction cost. Under market efficiencythe after-transaction returns of funds should be independent of size. Trans-action costs are known to increase with trading size. Thus, they argue, thedifference between a large fund and small fund is that larger funds havehigher before-transaction cost returns, corresponding to superior skill. Thisskill allows the funds to grow larger and make better profits by collectingmore fees.
Given a distribution P (α) of investor skill α and a functional form g(s)for transaction costs, the Berk and Green model makes a prediction aboutthe distribution of fund sizes. Berk and Green somewhat arbitrarily assumedthat P (α) is normal and that g(s) is linear. This implies a normal distribu-tion of fund size, which is clearly very different than what is observed. Amore realistic assumption for transaction costs would be22 g(s) ∼ sb, withb ≈ 0.5. Under this assumption it is possible to show that P (α) must have alog-normal upper tail in order to be consistent with the data. (This is truefor any b > 0 including the Berk and Green choice of b = 1). Whether ornot this is plausible is not clear; certainly it means that the distribution ofskill must be very heavy tailed. What is clear is that their theory has littlepredictive value for fund size as it stands, since the prediction of fund sizeis itself based on an unknown distribution for skill, which requires as muchinformation to specify as the prediction itself.
In contrast, the prediction made by our hypothesis is very robust. Whilegetting a good fit to the true fund size distribution requires several assump-tions, as we have spelled out here, under our model the log-normal uppertail depends only on the fact that returns are essentially random, and thatthe timescales to achieve the asymptotic distribution are extremely long.The essence of the answer emerges from the simple fact that returns aremultiplicative.
22For a select body of work on the functional form of transaction cost see [Torre, 1997;Kempf and Korn, 1999; Farmer and Lillo, 2004; Gabaix et al., 2003; Gabaix et al., 2006;Hopman, 2007; Bouchaud et al., 2008] and references therein.
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This brings up the question of whether or not our theory can be com-patible with the Berk and Green hypothesis. In our theory the aspect thattransaction costs might affect is the mean inflow of money to funds as afunction of size, µf (s). This is where investor choice might enter our model.For example, one can hypothesize that the net rate of deposits to largerfunds decreases with size due to the fact that investors correctly understandthat there are increased transaction costs. Alternatively, fund managersmay understand that they have higher costs and close their funds or takeless relative effort in marketing them as they get bigger23.
While both of these effects are plausible, they do not qualitatively changethe results of our model. The lognormal upper tail is very robust undervariations of the investor money flux – the upper tail of the distributionis essentially independent of µf (s) and σf (s), as long as they stay withinreasonable bounds24. Thus, it seems that the diffusive nature of fund sizedominates investor choice in determining the fund size distribution.
We are in the process of making a more detailed study quantitativelytesting the Berk and Green hypothesis, under several additional criteriathat we do not have the space to explain here. We believe, though, that thedata presented here already strongly suggest that their hypothesis cannotprovide the correct explanation for fund size, while ours does.
6 Conclusions
In this paper we have shown that the empirical evidence strongly favorsthe hypothesis that the upper tail of the distribution of fund size is not apower law and is better approximated by a log-normal. Even though thelog-normal distribution is strongly right skewed, and thus in some respectsis a heavy-tailed distribution, it is still much thinner-tailed than a powerlaw, as evidenced by the fact that all its moments exist. As a consequencethere are fewer extremely large funds than one would expect if it were apower law.
We have argued that the mutual fund size distribution is naturally ex-plained in terms of the stochastic nature of the growth process. The essentialelements of the growth process are multiplicative random changes in the size
23One must bear in mind that the rate only drops relative to size; in absolute terms therate is increasing, so perhaps what one needs to explain is why there is an increase withfund size.
24The requirement is that µs(s) remain greater than zero. It fails if either µf (s) or σ(s)increases with size, which seems implausible, of if µf (s) becomes sufficiently negative thatµs(s) < 0. Neither of these are consistent with the data
30
of existing funds, entry of new funds, and exit of existing funds as they goout of business. We find, however, that entry plays no role at all other thansetting the scale; exit plays a role in thickening the tails of the distribution,but this acts only on a very slow timescale. The log-normality comes aboutbecause the industry is young and still in a transient state. In the future, ifthe conditions for fund growth and removal remain stationary for more thana century, the distribution should become a power law. The thickening ofthe tails happens from the body of the distribution outward, as the powerlaw tail extends to successively larger funds.
There is also an interesting size dependence in the growth rate of mutualfund size, which is both like and unlike that of other types of firms. Mutualfunds are distinctive in that their overall growth rates can be decomposed asa sum of two terms, ∆s = ∆f + ∆r, where ∆f represents the flow of moneyin and out of funds, and ∆r the returns on money that is already in the fund.The money flow ∆f decreases as a power law as a function of size, similarto what is widely observed in the overall growth rates for other types offirms. Furthermore the exponents are similar to those observed elsewhere.The returns ∆r, in contrast, are essentially independent of fund size, asthey must be under market efficiency. As a result, for large sizes the meanand variance of the overall growth are constant – this is unlike other firms,for which the mean and variance appear to go to zero in the limit. As wediscuss here, this makes an important difference in the long-term evolution:While mutual funds should eventually evolve toward a power law, other firmsshould evolve toward a thinner-tailed, stretched exponential distribution.
Our analysis here suggests that investor preference has a negligible in-fluence on the upper tail of the mutual fund size distribution. Investorpreference enters our analysis only through ∆f , the flow of money in andout of the fund. Since ∆f becomes relatively small in the large size limit, inthis limit the growth of funds is dominated by the returns ∆r, whose meanand variance are constant. Thus the upper tail of the size distribution isdetermined by market efficiency, which dictates both that returns are essen-tially random, and thus diffusive, and that there is no dependence on size.Thus for large fund size investor preference doesn’t have much influence onthe growth process; this is reinforced by the fact that the statistical prop-erties of the money flux ∆f are essentially like those of the growth of otherfirms. This, as well as other arguments presented in the previous section,indicates that transaction costs do not play a major role in determining fundsize, as suggested by Berk and Green (2004).
Our results present a puzzle as to what determines the distribution oftrading volume. The existence of a power law for large trading volumes
31
is now fairly well documented25. A possible explanation for the volumedistribution assumes that fund trading is proportional to its size and relieson the assumption that funds size is described by a power law obeying Zipf’slaw. Our analysis here shows that this argument fails for two reasons: (1)Mutual funds sizes do not follow Zipf’s law, and are not even power lawdistributed and (2) transaction costs are not important in determining fundsize. Thus the puzzle of what determines the distribution of trading volumeremains an open question.
We have argued here that the cause of the log-normal upper tail of themutual fund size distribution is essentially market efficiency, combined withthe fact that fund growth is a slow-moving, diffusive process that takes placeon time scales measured in decades. Investors are not making rational allo-cations as posited by Berk and Green; rather they move their money in andout of funds slowly, and with a great deal of inertia. This explanation mightbe disappointing to some as having“little economic content”; its strengthis that it explains an important economic phenomenon in a manner that isvery robust, and largely independent of the details of human choice.
25For work arguing for a power law volume distribution see [Gopikrishnan et al., 2000;Gabaix et al., 2003; Lillo et al., 2005; Gabaix et al., 2006], while Eisler and Kertesz [2006]argue against one.
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A Simulation model
We simulate a model with three independent stochastic processes. Theseprocesses are modeled as Poisson process and as such are modeled as havingat each time step a probability for an event to occur. The simulation usesasynchronous updating to mimic continuous time. At each simulation timestep we perform one of three events with an appropriate probability. Theseprobabilities will determine the rates in which that process occurs. Theprobability ratio between any pair of events should be equal to the ratio ofthe rates of the corresponding processes. Thus, if we want to simulate thismodel for given rates our probabilities are determined.
These processes we simulate are:
1. The rate of size change taken to be 1 for each fund and N for theentire population.Thus, each fund changes size with a rate taken to be unity.
2. The fund exit rate λ which can depend on the fund size.
3. The rate of creation of new funds ν.Each new fund enters with a size ω with a probability density f(ω).
Since some of these processes are defined per firm as opposed to thecreation process, the simulation is not straightforward. We offer a briefdescription of our simulation procedure.
1. At every simulation time step, with a probability ν1+λ+ν a new fund
enters and we proceed to the next simulation time step.
2. If a fund did not enter then the following is repeated (1 + λ)N times.
a. We pick a fund at random.
b . With a probability of λ1+λ the fund enters.
c. If it is not annihilated, which happens with a probability of 11+λ ,
we change the fund size.
We are interested in comparing the simulations to both numerical andempirical results. The comparisons with analytical results are done for spe-cific times and for specific years when comparing to empirical data. In orderto do so, we need to convert simulation time to ”real” time. The simulationtime can be compared to ’real’ time if every time a fund does not enterwe add a time step. Because of the way we defined the probabilities each
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simulation time step is comparable to 1/(1 + λ) in ”real” time units. Theresulting ”real” time is then measured in what ever units our rates weremeasured in. In our simulation we use monthly rates and as such a unittime step corresponds to one month.
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