Optimal Asset Allocation and Risk Shifting in Money Management - - PowerPoint PPT Presentation

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

Page 2

Fund Flow - Performance Relationship (Chevalier and Ellison (1997))

• 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 relative return year-end flow

Lowest

❨

Highest

✻

fL fH

• 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 = µStdt + σStdwt

• 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 WT

max

θ, WT E[u(WT fT )] = E (WT fT )1−γ

1 − γ

subject to

dWt = [r + θt(µ − r)] Wtdt + θtσWtdwt

• 2. Model

Page 6

Flow-Performance Relationship (Simplest)

relative return

fT η fL fH

How does one measure risk-taking incentives?

• Conventional view:

– sensitivity of the payoff’s value to volatility (vega):

∂V (σW

t

; RW

t

−RY

t )

∂σW

t

• This paper:

– optimal volatility ˆ

σW

t = ˆ

θtσ. That is,

∂V (σW

t

; RW

t

−RY

t )

∂σW

t

= 0 ⇒ ˆ σW

t .

• 2. Model

Page 7

Manager’s Optimal Risk Exposure

• 1.5
• 1
• 0.5

0.5 1 2 3 4

η θY θN ˆ θt RW

t

− RY

t

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

• 0.2

0.2 0.4 0.6 0.8 1

η θY θN ˆ θt RW

t

− RY

t

(a) Economies with θN > θY (b) Economies with θN < θY

θN : risk exposure in Merton’s problem, θY : risk exposure of the index

• 2. Model

Page 8

An Alternative Flow-Performance Relationship (Collar-Type)

relative return

fT ηH ηL fL fH

Can also be reinterpreted as an 80/120 annual bonus plan.

• 2. Model

Page 9

Manager’s Optimal Risk Exposure (Collar-Type)

• 1.5
• 1
• 0.5

0.5 1 2 3 4

ηH θY θN ˆ θt RW

t

− RY

t

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

• 0.2

0.2 0.4 0.6 0.8 1

ηH θY θN ˆ θt RW

t

− RY

t

(a) Economies with θN > θY (b) Economies with θN < θY

θN : risk exposure in Merton’s problem, θY : risk exposure of the index

• 2. Model

Page 10

Further Alternative Flow-Performance Specifications

• Linear-convex (Sirri and Tufano (1998))

relative return year-end flow

fL

• Linear-linear (asymmetric fee structure)

relative return year-end flow

fL

• 2. Model

Page 11

Manager’s Optimal Risk Exposure: Dynamics

• 1.5
• 1
• 0.5

0.5 1 3 5 7 9

η θY θN ˆ θt RW

t

− RY

t

T -tlow T -tmed T -thigh

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

• 0.2

0.2 0.4 0.6 0.8 1

η θY θN ˆ θt RW

t

− RY

t

T -tlow

✸

T -tmed T -thigh

(a) Economies with θN > θY (b) Economies with θN < θY

• Manager engages in risk shifting well before the year-end
• Risk shifting more pronounced as the year-end approaches

• 2. Model

Page 12

Multiple Stocks

• 1

. 5 . 5 1 2 3

Stock 1

η θY

1

θN

1

ˆ θ1t RW

t

− RY

t

• 1

. 5 . 5 1 2 3

Stock 2

η θY

2

θN

2

ˆ θ2t RW

t

− RY

t

(a) Economies with θN

1 > θY 1 and θN 2 > θY 2

• 1

. 5 . 5

.2 . 2 . 4 . 6

Stock 1

η θY

1

θN

1

ˆ θ1t RW

t

− RY

t

• 1

. 5 . 5 1 2 3 4

Stock 2

η θY

2

θN

2

ˆ θ2t RW

t

− RY

t

(b) Economies with θN

1 < θY 1 and θN 2 > θY 2

• 2. Model

Page 13

Idiosyncratic versus Systematic Risk

Economic setup:

dS1t = µ1S1tdt + σ11S1tdw1t + σ12S1tdw2t dS2t = µ2S2tdt + σ21S2tdw1t + σ22S2tdw2t µ = @ µ1 r 1 A , σ = @ σ11 σ22 1 A , Y = S1

• 1

. 5 . 5 1 2 3 4

η θY

1

θN

1

ˆ θ1t RW

t

− RY

t

• 1

. 5 . 5

.2 . 2 . 4 . 6

η θN

2

ˆ θ2t RW

t

− RY

t

• 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(RW − 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. LHS: σm(RW

i, t − RY t )

LHS: σm(RW

i, t)

Point Estimate t-Statistic Point Estimate t-Statistic OVERi,m × 103

• 0.1819
• 4.73

0.1058 2.38 Fund-year FE Yes Yes

R2

0.39 0.36 N 40721 40721

σm(RW

i, t − RY t ) – standard deviation of tracking error for month m; Y is S&P 500

σm(RW

i, 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.

RW

i, t − RF t = a + (bF und−Y ear1F Y + bMonth1M + bU UNDERi,w)(RY t − RF t ) + εi, t

Dependent Variable: RW

i, t − RF t

BetaT below 1 BetaT −1 below 1 (1) (2) (3) (4) UNDERi,w × (RY

t − RF t )

• 0.017

(-6.61)

• 0.020

(-7.72)

• 0.018

(-6.99)

• 0.021

(-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 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, ˆ

λ, in units of investor’s initial wealth: V

I((1 + ˆ

λ)W0) = ˆ V (W0)

– V I(·) is investor’s indirect utility under θI – ˆ

V (·) is investor’s indirect utility under delegation

• Decompose ˆ

λ into two components: 1 + ˆ λ = (1 + λN)(1 + λY )

– λN: gain/loss due to explicit incentives, solves

V I((1 + λN)W0) = ˆ V (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) (θN > θY )

Fixed parameter values: γ = 1, γI = 2, fL = 0.8, fH = 1.5, fL + fH = 2.3, β = 1,

ηL = −0.08, ηH = 0.08, ηL + ηH = 0, µ = 0.06, r = 0.02, σ = 0.29, W0 = 1, T = 1.

Cost-benefit measures Effects of

λY , λN ˆ λ (%)

Managerial risk

γ

0.5 1.0 2.0 3.0 4.0 aversion

• 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

Implicit reward

fH -fL

0.3 0.5 0.7 0.9 1.1 for outperformance

• 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

Risk exposure

θY

0.50 0.75 1.00 1.25 1.50

• f the benchmark
• 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

Flow threshold

ηH-ηL

0.08 0.12 0.16 0.20 0.24 differential

• 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) (θN < θY )

Fixed parameter values: γ = 1, γI = 2, fL = 0.8, fH = 1.5, fL + fH = 2.3, β = 1,

ηL = −0.08, ηH = 0.08, ηL + ηH = 0, µ = 0.06, r = 0.02, σ = 0.29, W0 = 1, T = 1.

Cost-benefit measures Effects of

λY , λN ˆ λ (%)

Managerial risk

γ

0.5 1.0 2.0 3.0 4.0 aversion

• 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

Implicit reward

fH -fL

0.3 0.5 0.7 0.9 1.1 for outperformance

• 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

Risk exposure

θY

0.50 0.75 1.00 1.25 1.50

• f the benchmark
• 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

Flow threshold

ηH-ηL

0.08 0.12 0.16 0.20 0.24 differential

• 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.