On the Size of the Active Management Industry
Ľuboš Pástor
Booth School of Business
University of Chicago
NBER, CEPR
Robert F. Stambaugh
The Wharton School
University of Pennsylvania
NBER
Q-Group Presentation, March 23, 2010
The “Active Management Puzzle”?
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Track record of the active management (AM) industry is poor
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active equity mutual funds in aggregate: α̂ :< 0 with t ≈ −2
Nonetheless, active management remains popular:
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e.g., 87% of mutual funds assets are actively managed
The “Active Management Puzzle”?
I
Track record of the active management (AM) industry is poor
I
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Nonetheless, active management remains popular:
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active equity mutual funds in aggregate: α̂ :< 0 with t ≈ −2
e.g., 87% of mutual funds assets are actively managed
Puzzle?
The “Active Management Puzzle”?
I
Track record of the active management (AM) industry is poor
I
I
Nonetheless, active management remains popular:
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active equity mutual funds in aggregate: α̂ :< 0 with t ≈ −2
e.g., 87% of mutual funds assets are actively managed
Puzzle? not necessarily
The “Active Management Puzzle”?
I
Track record of the active management (AM) industry is poor
I
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Nonetheless, active management remains popular:
I
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active equity mutual funds in aggregate: α̂ :< 0 with t ≈ −2
e.g., 87% of mutual funds assets are actively managed
Puzzle? not necessarily
Decreasing returns to scale in the active management industry
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industry’s α depends on industry’s size:
α becomes more elusive as more money chases it
past underperformance ⇒ industry should shrink, but
uncertainty about decreasing returns ⇒ large confidence
interval for the size we should expect
The “Active Management Puzzle”?
I
Track record of the active management (AM) industry is poor
I
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Nonetheless, active management remains popular:
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e.g., 87% of mutual funds assets are actively managed
Puzzle? not necessarily
Decreasing returns to scale in the active management industry
I
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active equity mutual funds in aggregate: α̂ :< 0 with t ≈ −2
industry’s α depends on industry’s size:
α becomes more elusive as more money chases it
past underperformance ⇒ industry should shrink, but
uncertainty about decreasing returns ⇒ large confidence
interval for the size we should expect
In contrast, industry’s size would seem puzzling if it were
known that returns to scale are constant
What We Do
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Develop an equilibrium model of active management (AM)
with
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competing utility-maximizing investors
competing fee-maximizing fund managers
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Solve for equilibrium size, alpha, and fees in AM industry
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Relate size of AM industry to past performance
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Analyze learning about returns to scale
What We Find
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AM industry can be large even if the track record is poor
Investors’ learning about returns to scale is endogenous
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what they invest depends on what they’ve learned
what they learn depends on what they invest
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Investors never learn the degree of returns to scale exactly
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Industry size can be suboptimal for a long time
Other features of the model
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industry size crucially depends on the degree of competition
α>0
“investment externality”
Model
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Two types of agents:
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M managers of active funds (have skill but no capital)
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N investors in active funds (have capital but no skill)
Benchmark-adjusted return to investors from fund
i = 1, . . . , M:
ri
= αi + ui
ui
= x + i
σx > 0 ⇒ cannot fully diversify risk by buying many funds
Model: Expected Profits
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Benchmark-adjusted expected total profit to fund i’s manager
and investors:
S
πi = s i a − b
W
si . . . size of manager i’s fund
PM
S = i=1 si . . . aggregate size of the AM industry
W . . . total investable wealth of the N investors
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Manager i charges a proportional fee fi ⇒ investors expect
benchmark-adjusted rate of return
αi = a − b
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S
− fi
W
Decreasing returns to aggregate scale: b > 0
Expected
Return
a
⎛S ⎞
a−b ⎜ ⎟
⎝W ⎠
α+ f
α
0
S
W
S
W
Active Allocation
Figure 1. Decreasing returns to scale in the active management industry.
Model: Optimization
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Each manager i chooses fi to maximize fee revenue:
max fi si
fi
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Investors know fi ’s before making investment decisions
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Each investor j chooses weights δj on the M funds to
maximize
o
n
γ
max δj0 E(r |D) − δj0 Var(r |D)δj
δj
2
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All investors have same risk aversion γ > 0, same wealth,
and same information set D
Unrestricted allocations to benchmarks and T-bill
No short sales of funds (δj ≥ 0)
Equilibrium with Known a and b
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We solve for a symmetric Nash equilibrium
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First for investors’ allocations, as a function of fees, then for
managers’ fees
In equilibrium, fees and α’s are equal across funds (fi = f ,
αi = α):
aγσ2
2γσ2 + (M − 1)p
Mb
γσ2
1−
α = a 1−
2γσ2 + (M − 1)p
γσ2 + Mp
γσ2
S
Ma
=
1−
,
W
γσ2 + Mp
2γσ2 + (M − 1)p
f
=
where
p=
N +1
b + γσx2
N
Equilibrium Fee
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M ↑ ⇒ f ↓, due to competition among managers
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Note: f is the discretionary component of the total fee
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For M = 1, we have f = a/2
For M → ∞, we have f → 0
f = fee the manager sets while considering its effect on fund
size
(any competitive proportional fee is part of a)
Also, f ↑ when a ↑, b ↓, σ ↑, σx ↓, and N ↑
Equilibrium Alpha
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In general, the equilibrium alpha is positive,
α>0
because investors
1. demand compensation for risk (σx and possibly also σ )
2. internalize some of the “investment externality”
Equilibrium Alpha & Risk
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Investors demand compensation for two kinds of risk:
σ : diversifiable if M → ∞
σx : non-diversifiable even if M → ∞
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When M → ∞, diversifiable risk σ drops out:
(1/N)b + γσx2
α=a
[(N + 1)/N]b + γσx2
When N → ∞ as well,
α=a
γσx2
b + γσx2
Equilibrium Alpha & Number of Investors (N)
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“Investment externality”:
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new investors impose a negative externality on existing
investors by diluting their returns
When N is finite, investors internalize some of the reduction
in profits resulting from their own investment
⇒ α > 0 even if there is no risk
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When M → ∞ and σx → 0,
α=
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Note: α ↓ when N ↑
a
N +1
Equilibrium Alpha & Number of Managers (M)
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The effect of M on α is ambiguous; two opposing effects:
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M ↓ ⇒ α ↓ because fees ↑
M ↓ ⇒ α ↑ because investors demand compensation for σ
Equilibrium Size of the AM Industry
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When N → ∞, we obtain a familiar mean-variance result:
S
E (rA |D)
=
W
γVar(rA |D)
where rA is the aggregate benchmark-adjusted return
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When N → ∞ and M → ∞,
S
α
a
=
=
W
b + γσx2
γσx2
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When N → ∞ and M = 1,
S
α
=
2
W
γ(σx + σ2 )
Equilibrium Size of the AM Industry (cont’d)
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S
Let S ∗ = size maximizing expected total profit, S a − b W
a
S∗
=
W
2b
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The equilibrium size S ≤ S̄ = 2S ∗
Expected
Return
a
⎛S ⎞
a−b ⎜ ⎟
⎝W ⎠
α+ f
α
0
S*
W
S
W
S
W
Active Allocation
Equilibrium Size of the AM Industry (cont’d)
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S
Let S ∗ = size maximizing expected total profit, S a − b W
S∗
a
=
W
2b
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The equilibrium size S ≤ S̄ = 2S ∗
When M = 1, there is underinvestment, S ≤ S ∗
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Manager-monopolist charges high fee
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S = S ∗ only if N → ∞ and σx2 + σ2 = 0
Equilibrium Size of the AM Industry (cont’d)
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When M → ∞,
S
=
S∗
2b
+ γσx2
N+1
N b
⇒ underinvestment (S < S ∗ ) or overinvestment (S > S ∗ )
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When M → ∞ and σx2 → 0, there is overinvestment:
S
2N
=
∗
S
N +1
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N → ∞ ⇒ S → S̄ = 2S ∗
Unknown a and b
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Now suppose a and b are unknown
ã
a
|D
=
E
b
b̃
2
σa σab
a
|D
=
Var
σab σb2
b
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For simplicity, let M → ∞ and N → ∞
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Then f → 0 and α = a − b(S/W )
Solve for a symmetric Nash equilibrium among investors
Unknown a and b (cont’d)
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In equilibrium, S/W is the (unique) real positive solution to
3
i S 2
S
S h
2
2
2γσab −
γσb2
0 = ã −
b̃ + γ(σa + σx ) +
W
W
W
as long as ã > 0. If ã ≤ 0, then S/W = 0.
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In this setting,
E(rA |D) = ã − b̃
Var(rA |D) =
⇒
S
W
=
σa2
+
S
W
σx2
−2
E(rA |D)
γVar(rA |D)
S
W
σab +
S
W
2
σb2
Prior Beliefs for a and b
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One prior for a, two priors for b
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Prior 1: b = 0, known (constant returns to scale)
Prior 2: b ≥ 0, unknown (decreasing returns to scale)
Assume
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S/W = 0.9 is optimal under Prior 2
Prior mean of α = 10% per year at S/W = 0.9
Risk aversion of γ = 2
Volatility of aggregate active return σx = 2% per year
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Close to empirical estimates for active equity mutual funds
Prior for a
Priors for b
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
−0.5
0
0.5
b≥0
b=0
0
1
0
0.2
0.6
0.4
0.4
0.2
0
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
−0.2
−0.4
0
0.2
0.4
0.6
0.8
1
0.8
1
Priors for α when b ≥ 0
0.6
Percentiles of α
Percentiles of α
Priors for α when b=0
0.4
0.2
0
−0.2
−0.4
0.6
0.8
1
0
0.2
0.4
S/W
Figure 2. Prior distributions.
0.6
S/W
Implied Prior Beliefs for α
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Implied prior for α depends on S/W when b ≥ 0
α = a − b(S/W )
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Prior 2 is more pessimistic about α than Prior 1
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α is smaller under Prior 2 for any S/W > 0
Nonetheless, we’ll see that Prior 2 investors invest more
in AM than Prior 1 investors after a negative track record
Updated Beliefs
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300,000 samples of simulated AM returns and allocations
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For each sample, randomly draw a and b from their priors
In each year t, beginning with t = 1, we perform three steps:
1. Solve for equilibrium allocation to AM
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Restrict (S/W )t between 0 and 1
2. Construct AM return
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rA,t = a − b(S/W )t + xt , where xt ∼ N(0, σx2 )
3. Update beliefs about a and b
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Regress rA on S/W and constant ⇒ intercept a, slope −b
. . . then back to step 1
b ≥ 0; Posterior std dev of a
b = 0; Posterior std dev of a
Std(a)
0.05
0
Std(b)
0.1
0
10
20
30
40
0
50
20
30
40
Year
0.15
0.1
0.1
50
0.05
0
10
20
30
40
0
50
0
10
20
30
Year
b = 0; Posterior std dev of α
b ≥ 0; Posterior std dev of α
Std(α)
0.05
10
20
30
Year
40
50
0.05
0
50
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
0.1
0
40
Year
0.1
0
10
b ≥ 0; Posterior std dev of b
0.15
0
0
b = 0; Posterior std dev of b
0.05
Std(α)
0.05
Year
Std(b)
Std(a)
0.1
0
10
20
30
Year
Figure 3. Posterior standard deviations.
40
50
Perceived a minus true a
Perceived b minus true b
0.2
0.2
0.15
0.1
0.1
0.05
0
0
−0.1
−0.05
−0.1
−0.2
−0.15
−0.2
0
10
20
30
40
50
−0.3
0
10
20
Year
30
40
50
Year
Perceived α minus true α
Equilibrium S/W minus true S/W
0.08
0.3
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
0.06
0.04
0.02
0.2
0.1
0
0
−0.02
−0.04
−0.1
−0.06
−0.08
0
10
20
30
Year
40
50
−0.2
0
10
20
30
Year
Figure 4. Deviations from true values.
40
50
Learning About Returns to Scale
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“Endogeneity”: learn ⇒ invest ⇒ learn ⇒ invest ⇒ . . .
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Learning about the intercept and slope from
rt = a − b(S/W )t + t
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(S/W )t+1 depends on beliefs about a and b at time t
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If (S/W )t stops changing, learning about a and b stops
⇒ Never learn a and b
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(S/W )t converges to optimal level quickly when b is high,
but it can stay suboptimal for a long time when b is low
Learning When b ≥ 0
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Investors learn differently under Priors 1 and 2 because
(S/W )t affects learning when b ≥ 0 but not when b = 0
Representative examples of learning paths: Figure 5
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Three values of b: “low”, “median”, “high”
(5th, 50th, 95th percentiles of the prior distribution)
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Given b, pick a such that the “true” S/W = 0.5
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Use (a, b) to generate random samples of returns
Results:
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When b is high, investors find the optimal S/W quickly
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When b is low, investors can get stuck at “wrong” S/W for a
long time
Return
Low b, example 1
Median b, example 1
0.1
0.1
0
0
0
−0.1
−0.1
0
0.5
1
−0.1
0
Return
Low b, example 2
0.1
1
−0.2
0
0
0
0.5
1
High b, example 2
0.1
0
−0.1
0
0.5
1
−0.1
0
Low b, example 3
Return
0.5
Median b, example 2
0.1
−0.1
0.5
1
−0.2
0
Median b, example 3
0.1
0.1
0
0
0.5
1
High b, example 3
0.1
0
−0.1
−0.1
0
0.5
1
−0.1
0
Low b, example 4
Return
High b, example 1
0.1
0.5
1
−0.2
0
Median b, example 4
0.1
0.1
0
0
0.5
1
High b, example 4
0.1
0
−0.1
−0.1
0
0.5
S/W
1
−0.1
0
0.5
S/W
1
−0.2
0
Figure 5. Examples of learning paths.
0.5
S/W
1
Role of Historical Performance
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Plot the posterior of the equilibrium S/W conditional on t(α̂)
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How much is invested in AM when α̂ is significantly negative?
Prior 1 (b = 0) investors invest nothing
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√
Note: t(α̂) = α̂ T /σx
t(α̂) is a sufficient statistic for S/W
Prior 2 (b ≥ 0) investors can invest a lot
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despite Prior 2 being more pessimistic about α than Prior 1
t(α̂) is NOT a sufficient statistic for S/W
S/W after 50 years when b=0
1
0.8
S/W
0.6
0.4
0.2
0
−4
−3
−2
−1
0
Sample t−statistic
1
2
3
4
Distribution of equilibrium S/W after 50 years when b ≥ 0
1
Percentiles of S/W
0.8
0.6
0.4
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
0.2
0
−4
−3
−2
−1
0
Sample t−statistic
1
2
3
4
Figure 6. Posterior distribution of the equilibrium allocation to active
management conditional on the sample t-statistic.
Prior for a
Priors for b
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
−1
−0.5
0
0.5
1
b≥0
b=0
0
1.5
0
0.2
0.6
0.4
0.4
0.2
0.2
0
−0.2
−0.4
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
−0.6
−0.8
−1
0
0.2
0.4
0.8
1
0.8
1
0
−0.2
−0.4
−0.6
−0.8
−1
0.6
S/W
0.6
Priors for α when b ≥ 0
0.6
Percentiles of α
Percentiles of α
Priors for α when b=0
0.4
0.8
1
0
0.2
0.4
0.6
S/W
Figure 7. Alternative prior distribution.
S/W after 50 years when b=0
1
0.8
S/W
0.6
0.4
0.2
0
−4
−3
−2
−1
0
Sample t−statistic
1
2
3
4
Distribution of equilibrium S/W after 50 years when b ≥ 0
1
Percentiles of S/W
0.8
0.6
0.4
95th pctile
75th pctile
50th pctile
25th pctile
5th pctile
0.2
0
−4
−3
−2
−1
0
Sample t−statistic
1
2
3
4
Figure 8. Equilibrium allocation to active management under the alternative prior.
Differences from Berk and Green (2004)
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Focus
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Decreasing returns to scale
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BG: At individual fund level
We: At aggregate industry level
Parameter uncertainty
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BG: Capital flows across funds
We: Size of the AM industry
BG: a unknown, b known
We: a and b both unknown
Fund managers setting fees
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BG: Monopoly
We: Competition
Differences from Berk and Green (2004) (cont’d)
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Equilibrium alpha
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BG: α = 0; no explicit investor optimization
We: α > 0; equilibrium outcome for optimizing investors
Our model comes closest to BG when . . .
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M = 1 (a single fund manager) ⇒ same fee setting
N → ∞ (many investors) ⇒ no investment externality
σ = σx = 0 (no risk) ⇒ no compensation for risk
Equilibrium then features α = 0, S = S ∗ , and f = a/2, as in
BG
Conclusions
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Size of AM industry can be large even if the track record is
poor
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Due to decreasing returns to scale
Learning about returns to scale is “endogenous” and slow
Never learn the degree of returns to scale exactly
Industry size can be suboptimal for a long time
Interesting features of the model
1. Industry size crucially depends on the degree of competition
2. α > 0
3. “Investment externality”