Optimal Asset Allocation and Risk Shifting
in Money Management
Suleyman Basak (LBS), Anna Pavlova (LBS), and Alex Shapiro (Azimuth Trust)
(1) Motivation and Objective
(2) Model
(3) Empirical Analysis
(4) Costs of Active Management to Investors
1. Motivation and Objective
Page 1
1. Motivation and Objective
• Mutual fund managers’ compensation is linked to the value of assets under
management
• Implicit incentives due to fund flows to performance relationship
• The flow-performance relationship is
– positive
– exhibits convexities
• Question: How does a fund manager respond to these incentives?
1. Motivation and Objective
Fund Flow - Performance Relationship (Chevalier and Ellison (1997))
Page 2
1. Motivation and Objective
Page 3
Summary of Main Testable Implications
• Taking risk = increasing volatility of portfolio
• Gambling entails either an increase or a decrease in portfolio volatility
– a sufficiently risk averse manager decreases volatility
– manager manipulates systematic risk rather than idiosyncratic
– gambling intensifies towards year-end
• Incentives to gamble are state-dependent. For example, the manager’s risk-taking
incentives are
year-end flow
fH
fL
0
6
Highest
Y
relative return
Lowest
1. Motivation and Objective
Page 4
Related Literature
• Risk-taking of fund managers in response to fund flows:
Chevalier and Ellison (1997)
• Managerial incentives and portfolio choice:
Brennan (1993), Carpenter (2000), Cuoco and Kaniel (2000), Berk and Green (2005),
Hugonnier and Kaniel (2002), Gomez and Zapatero (2003), Ross (2003)
• Empirical literature on risk-taking by mutual fund managers:
Brown, Harlow, and Starks (1996), Busse (2001), Reed and Wu (2005)
2. Model
Page 5
2. Model
• Finite horizon, [0, T ], Black-Scholes economy
• Assets:
– Money market account with rate r
– Stock follows
dSt = µSt dt + σSt dwt
• Fund manager:
– evaluated relative to the index Yt (fraction β in stock)
– receives flows at T at rate fT
– chooses a trading strategy θ and terminal portfolio value W T
(WT fT )1−γ
max E[u(WT fT )] = E
θ, WT
1−γ
subject to
dWt = [r + θt (µ − r)] Wt dt + θt σWt dwt
2. Model
Page 6
Flow-Performance Relationship (Simplest)
fT
fH
fL
0
η
relative return
How does one measure risk-taking incentives?
• Conventional view:
– sensitivity of the payoff’s value to volatility (vega):
Y
∂V (σtW ; RW
t −Rt )
∂σtW
• This paper:
W
– optimal volatility σ̂t = θ̂t σ . That is,
Y
∂V (σtW ; RW
t −Rt )
∂σtW
=0
⇒
σ̂tW .
2. Model
Page 7
Manager’s Optimal Risk Exposure
θ̂t
θ̂t
1
4
θY
0.8
3
θN
0.6
2
1
0.4
0.2
θY
-0.3
-1.5
-1
-0.5
η
(a) Economies with θ
θN
0.5
N
Rt − Rt
W
> θY
θN : risk exposure in Merton’s problem,
Y
-0.2
-0.1
0.1
-0.2
η
0.2
0.3
N
< θY
(b) Economies with θ
θY : risk exposure of the index
RtW − RtY
2. Model
Page 8
An Alternative Flow-Performance Relationship (Collar-Type)
fT
fH
fL
0
ηL
ηH
relative return
Can also be reinterpreted as an 80/120 annual bonus plan.
2. Model
Page 9
Manager’s Optimal Risk Exposure (Collar-Type)
θ̂t
θ̂t
1
4
θY
0.8
3
θN
0.6
2
1
0.4
0.2
θY
-0.3
-1.5
-1
-0.5
ηH
0.5
N
> θY
(a) Economies with θ
θN
Rt − Rt
W
θN : risk exposure in Merton’s problem,
Y
-0.2
-0.1
0.1
-0.2
ηH
0.2
0.3
N
< θY
(b) Economies with θ
θY : risk exposure of the index
RtW − RtY
2. Model
Page 10
Further Alternative Flow-Performance Specifications
• Linear-convex (Sirri and Tufano (1998))
year-end flow
fL
0
relative return
• Linear-linear (asymmetric fee structure)
year-end flow
fL
0
relative return
2. Model
Page 11
Manager’s Optimal Risk Exposure: Dynamics
9
T -tmed
T -thigh
-0.5
θY
7
0.8
5
0.6 θ N
3θ
1
-1
θ̂t
1
T -tlow
-1.5
θ̂t
0.4
N
T -tmed
θY
η
(a) Economies with θ
3
0.2
-0.4 -0.3 -0.2 -0.1
0.5
N
T -thigh
Rt − Rt
> θY
W
Y
-0.2
T -tlow
0.1
η
(b) Economies with θ
•
Manager engages in risk shifting well before the year-end
•
Risk shifting more pronounced as the year-end approaches
0.2
N
0.3
RtW − RtY
< θY
2. Model
Page 12
Multiple Stocks
Stock 1
Stock 2
θ̂1t
θ̂2t
3
2
3
θ1N
2
1
Y
Y
θ1
-1
1
θ2
η
-0.5
θ2N
0.5
RtW − RtY
N
(a) Economies with θ1
-1
η
-0.5
0.5
RtW − RtY
> θ1Y and θ2N > θ2Y
Stock 1
Stock 2
θ̂1t
θ̂2t
θ1Y 0.6
4
0.4
3
N
θ1
θ2N2
0.2
-1
η
-0.5
-0. 2
0.5
1
Y
Rt − Rt
W
Y
θ2
-1
N
(b) Economies with θ1
-0.5
η
0.5
< θ1Y and θ2N > θ2Y
RtW − RtY
2. Model
Page 13
Idiosyncratic versus Systematic Risk
Economic setup:
dS1t
=
µ1 S1t dt + σ11 S1t dw1t + σ12 S1t dw2t
dS2t = µ2 S2t dt + σ21 S2t dw1t + σ22 S2t dw2t
1
0
1
0
0
µ1
σ
A , Y = S1
A , σ = @ 11
µ=@
0
σ22
r
θ̂1t
θ̂2t
4
0.6
3
0.4
θ1N 2
0.2
θY 1
θ2N
1
-1
-0.5
-1
η
0.5
RtW − RtY
-0.5
η
-0. 2
0.5
RtW − RtY
3. Empirical Analysis
Page 14
3. Empirical Analysis
Existing Work
• Brown, Harlow, and Starks (1996)
– find that underperforming managers increase volatility towards the year-end
• Busse (2001)
– shows that the above test fails on daily data
• Chevalier and Ellison (1997)
– look at σ(R
W
− RY ) towards the year-end; find an increase
– use monthly data
• Reed and Wu (2005)
– test the results of this paper on daily data
– distinguish between tournaments- vs. benchmarking-induced incentives
3. Empirical Analysis
Page 15
Data
• Daily mutual fund returns from Will Goetzmann and Geert Rouwenhorst (International
Center for Finance at Yale SOM).
• Data range: 1970 through 1998 (comparable with Chevalier-Ellison, Sirri-Tufano).
• Merged with CRSP to find out mutual funds objective codes
– left only actively managed US equity mutual funds in the aggressive growth, growth
and income, and long-term growth categories.
• Used the S&P 500 index as the benchmark.
3. Empirical Analysis
Page 16
Tracking error and standard deviation tests
Hypothesis 1: Tracking error variance is higher for underperforming managers.
W
LHS: σm (Ri, t
OVERi,m
× 103
− RtY )
W
LHS: σm (Ri, t )
Point Estimate
t-Statistic
Point Estimate
t-Statistic
-0.1819
-4.73
0.1058
2.38
Fund-year FE
Yes
Yes
R2
0.39
0.36
N
40721
40721
W
Y
σm (Ri,
t − Rt ) – standard deviation of tracking error for month m; Y is S&P 500
W
σm (Ri,
t ) – standard deviation of fund returns for month m
OVERi,m – relative performance indicator prior to month m
3. Empirical Analysis
Page 17
Beta tests
Hypothesis 2: Sufficiently risk-averse managers decrease their portfolio betas when
underperforming the market.
W
F
Y
F
Ri,
t − Rt = a + (bF und−Y ear 1F Y + bM onth 1M + bU UNDERi,w )(Rt − Rt ) + εi, t
Dependent Variable:
BetaT below 1
W
F
Ri,
t − Rt
BetaT −1 below 1
(1)
(2)
(3)
(4)
-0.017
-0.020
-0.018
-0.021
(-6.61)
(-7.72)
(-6.99)
(-8.22)
Month fixed effects
No
Yes
No
Yes
Fund-year fixed effects
Yes
Yes
Yes
Yes
R2
0.37
0.37
0.37
0.37
Number of observations
808642
UNDERi,w
× (RtY − RtF )
816783
3. Empirical Analysis
Page 18
Robustness
• Tried several alternative definitions of the OVER/UNDER indicator
• Included lagged dependent variables to deal with autocorrelation
• Clustered errors by month/day and fund objective code
4. Costs of Active Management to Investors
Page 19
4. Costs of Active Management to Investors
• Define a measure of gain/loss, λ̂, in units of investor’s initial wealth:
V I ((1 + λ̂)W0 ) = V̂ (W0 )
–
V I (·) is investor’s indirect utility under θ I
–
V̂ (·) is investor’s indirect utility under delegation
• Decompose λ̂ into two components: 1 + λ̂ = (1 + λN )(1 + λY )
–
λN : gain/loss due to explicit incentives, solves
V I ((1 + λN )W0 ) = V̂ (W0 ; fT =1)
–
λY : gain/loss due to implicit incentives
4. Costs of Active Management to Investors
Page 20
Costs of Active Management in Economies (a) (θ N
> θY )
γ = 1, γI = 2, fL = 0.8, fH = 1.5, fL + fH = 2.3, β = 1,
= 0.08, ηL + ηH = 0, µ = 0.06, r = 0.02, σ = 0.29, W0 = 1, T = 1.
Fixed parameter values:
ηL = −0.08, ηH
Cost-benefit measures
λY , λN
λ̂ (%)
Effects of
Managerial risk
γ
aversion
Implicit reward
fH -fL
for outperformance
Risk exposure
θY
of the benchmark
Flow threshold
differential
ηH -ηL
0.5
1.0
2.0
3.0
4.0
-8.13, -4.19
-5.12, -0.47
-3.31, 0.00
-2.56, -0.05
-2.15, -0.11
-11.98
-5.61
-3.31
-2.61
-2.27
0.3
0.5
0.7
0.9
1.1
-3.33, -0.47
-4.29, -0.47
-5.12, -0.47
-6.01, -0.47
-6.88, -0.47
-3.79
-4.74
-5.61
-6.46
-7.32
0.50
0.75
1.00
1.25
1.50
-5.43, -0.47
-4.63, -0.47
-5.12, -0.47
-6.69, -0.47
-8.45, -0.47
-5.88
-5.08
-5.61
-7.13
-8.88
0.08
0.12
0.16
0.20
0.24
-4.33, -0.47
-4.70, -0.47
-5.12, -0.47
-5.67, -0.47
-6.21, -0.47
-4.78
-5.15
-5.61
-6.12
-6.65
4. Costs of Active Management to Investors
Page 21
Costs of Active Management in Economies (b) (θ N
< θY )
γ = 1, γI = 2, fL = 0.8, fH = 1.5, fL + fH = 2.3, β = 1,
= 0.08, ηL + ηH = 0, µ = 0.06, r = 0.02, σ = 0.29, W0 = 1, T = 1.
Fixed parameter values:
ηL = −0.08, ηH
Cost-benefit measures
λY , λN
λ̂ (%)
Effects of
Managerial risk
γ
aversion
Implicit reward
fH -fL
for outperformance
Risk exposure
θY
of the benchmark
Flow threshold
differential
ηH -ηL
0.5
1.0
2.0
3.0
4.0
-8.13, -4.19
-5.12, -0.47
-3.31, 0.00
-2.56, -0.05
-2.15, -0.11
-11.98
-5.61
-3.31
-2.61
-2.27
0.3
0.5
0.7
0.9
1.1
-3.33, -0.47
-4.29, -0.47
-5.12, -0.47
-6.01, -0.47
-6.88, -0.47
-3.79
-4.74
-5.61
-6.46
-7.32
0.50
0.75
1.00
1.25
1.50
-5.43, -0.47
-4.63, -0.47
-5.12, -0.47
-6.69, -0.47
-8.45, -0.47
-5.88
-5.08
-5.61
-7.13
-8.88
0.08
0.12
0.16
0.20
0.24
-4.33, -0.47
-4.70, -0.47
-5.12, -0.47
-5.67, -0.47
-6.21, -0.47
-4.78
-5.15
-5.61
-6.12
-6.65
5. Conclusion
Page 22
5. Conclusion
• Characterized the manager’s optimal behavior in response to incentives induced by the
fund flow-performance relationship.
• Identified circumstances in which the manager would like to gamble.
• Gambling may be associated with a decrease in the fund’s volatility.
• Adverse incentives of the manager result in an economically significant cost to the
investor.