Role of Reputation For Mutual Fund Flows

Apoorva Javadekar1

September 2, 2015

1Boston University, Department of Economics

Broad Question

1. Question:What causes investors to invest or withdraw money frommutual funds?

I In particular: what is the link between fund performance andfund flows?

2. Litarature:Narrow focus on ”Winner Chasing” phenomenon

I link between recent-most performance and fund flows ignoringrole for reputation of fund

3. This paper: Role of Fund ReputationI Investor’s choicesI Risk Choices by fund managers

Why Study Fund Flows?

1. Important Vehicle of Investment

I Large: Manage 15 Tr $ (ICI, 2014)I Dominant way to equities: (ICI -2014, French (2008))

I HH through MF: owns 30% US equitiesI Direct holdings of HH: 20% of US equities

I Participation: 46% of US HH invest

2. Understand Behavioral Patterns:I Investors learn about managerial ability through returnsI =⇒ fund flows shed light on learning, information processing

capacities etc.

3. Fund Flows Affect Managerial Risk Taking

I Compensation ≈ flows: 90% MF managers paid as a % ofAUM

I =⇒ flow patterns can affect risk takingI =⇒ impacts on asset prices

Literature Snapshot

1. Seminal Paper: Chevallier & Ellison (JPE, 1997)

Returns(t)

Flows(t+1)

=⇒ Convex Fund Flows in Recent Performance!

2. Why Interesting? Non-Linear Flows (could) mean

I Bad and extremely bad returns carry same information !I Non-Bayesian LearningI Behavioral BiasesI Excess risk taking by managers given limited downside

Motivating Role of Reputation

1. No Role For Reputation: Literature links time t returns (rit)to time t + 1 fund flows (FFi ,t+1)

2. Why a Problem? The way investor perceives currentperformance depends upon historic performance

Why? History of Returns ≈ reputation

Manager 1: {rt−3, rt−2, rt−1, rt} = {G ,G ,G ,B}Manager 2: {rt−3, rt−2, rt−1, rt} = {B,B,B,B}

3. What it means for estimation?

FFi ,t+1 = g(rit , ri ,t−1, ...) + errori ,t+1

where g(.) is non-separable in returns

4. Useful For Studying Investors Learning

FFi ,t+1︸ ︷︷ ︸=decision

= g( rit︸︷︷︸=signal

, ri ,t−1, ri ,t−2, ...︸ ︷︷ ︸=priors

)

Data

1. Source: CRSP Survivor-Bias free mutual fund dataset

2. Time Period: 1980-2012.

3. Include:I Domestic, Open ended, equity fundsI Growth, Income, Growth&Income, Small and Mid-Cap, Capital

Appreciation funds (Pastor, Stambaugh (2002))

4. ExcludeI Sectoral, global and index or annuity fundsI Funds with sales restrictionsI young funds with less than 5 yearsI small funds (Assets < 10 Mn $)

5. Annual Frequency: Disclosures of yearly returns, ratings arebased on annual performance

Performance Measures

1. Reputation: Aggregate performance of 3 or 5 years prior tocurrent period

2. How to Measure Performance?I Factor Adjusted: CAPM α or 3-factor α (Fama,French

(2010), Kosowski (2006))I Peer Ranking (Within each investment style):

(Chevallier,Ellison (1997), Spiegel (2012))

3. Which Measure?I Not easy for naive investor to exploit factors like value,

premium or momentum =⇒ factor-mimicking is valued(Berk, Binsbergen (2013))

I Flows more sensitive to raw returns (Clifford (2011))I Peer ranking within each style control for bulk of risk

differentials across fundsI CAPM α wins the horse race amongst factor models (Barber

et.al 2014)

4. I use both the measures: CAPM α and Peer Ranking but not3-factor model.

Main Variables

1. Fund Flows: Main dependent variable is % growth in Assetsdue to fund flows

FFi ,t+1 =Ai ,t+1 − (Ait × (1 + ri ,t+1))

Ait

Ait : Assets with fund i at time trit : Fund returns for period ended t

Empirical Methodology

1. Interact Reputation With Recent Performance: Tounderstand how investors mix signals with priors

FFi ,t+1 = β0 +K∑

k=1

βk

[Z ki ,t−1 × (rankit)

]+

K∑k=1

ψk

[Z ki ,t−1 × (rankit)

2]

+ controls + εi ,t+1

2. Variables:I Z k

i,t−1: Dummy for reputation category (k) at t − 1I rankit ∈ [0, 1]

3. Structure:I Capture learning technologyI No independent effects of reputation(t-1) on flows(t+1):

I Reputation affect flows only through posteriors

Results 1: OLS Estimation

Table: Reputation And Fund Flows

Only Short Term Reputation

Dep Var:FFit+1 Peer CAPM Peer CAPM

Time Effects Yes Yes Yes YesStandard Errors Fund Clustered Fund Clustered Fund Clustered Fund Clustered

N 13512 13512 11468 11468Adj R-sq 0.137 0.135 0.158 0.148

Constant -0.088*** -0.109*** -0.098*** -0.126***(0.021) (0.021) (0.022) (0.022)

Rank(t+1) 0.216*** 0.202*** 0.207*** 0.193***(0.010) (0.010) (0.011) (0.011)

Risk(t) -0.894*** -0.808*** -0.830*** -0.761***(0.183) (0.178) (0.193) (0.188)

Log Age (t) -0.031*** -0.027*** -0.010 -0.006(0.005) (0.005) (0.005) (0.005)

Log Size(t) -0.002 -0.002 -0.011*** -0.008***(0.001) (0.001) (0.001) (0.001)

∆ Style(t+1) 0.045 0.039 0.039 0.035(0.049) (0.038) (0.038) (0.033)

FFit+1 Peer CAPM Peer CAPM

Unconditional Estimates

Rank(t) 0.043 0.117**(0.041) (0.041)

Rank-Sq(t) 0.296*** 0.223***(0.043) (0.043)

Low Reputation (Bottom 20%)

Rank(t) -0.031 -0.031(0.054) (0.060)

Rank-Sq(t) 0.210*** 0.250***(0.062) (0.074)

Medium Reputation (Middle 60 %)

Rank(t) -0.023 0.077(0.046) (0.044)

Rank-Sq(t) 0.374*** 0.260***(0.052) (0.049)

Top Reputation (Top 20%)

Rank(t) 0.308*** 0.285***(0.059) (0.061)

Rank-Sq(t) 0.116 0.124-0.0693 -0.0741

Mean Estimates Graph

-.2

0.2

.4

0 .5 1 0 .5 1 0 .5 1

Low reputation (t-1) Med reputation (t-1) Top Reputation(t-1)

95% Confidence Interval Mean Flow Growth%(t+1)

Flo

w G

row

th(%

)

Rank (t)

Flow Sensitivities In Response to Reputation

Unconditional Estimates

-.1

0.1

.2.3

0 .2 .4 .6 .8 1Rank(t)

95 % Confidence Interval Flow Growth % (t+1)

Flo

w g

row

th (

t+1)

%Short Term Performance And Flow Growth

Mean Estimates Graph

-.2

0.2

.4

0 .5 1 0 .5 1 0 .5 1

Low reputation (t-1) Med reputation (t-1) Top Reputation(t-1)

95% Confidence Interval Mean Flow Growth%(t+1)

Flo

w G

row

th(%

)

Rank (t)

Flow Sensitivities In Response to Reputation

Piecewise Linear Specification

-.2

0.2

.4

0 .5 1 0 .5 1 0 .5 1

Low Reputation Medium Reputation Top Reputation

95 % CI Flow Growth %

Rank ( t)

Reputation And Fund Flows (Piecewise Linear)

Implications

1. Shape:

I Convex Fund Flows For Low ReputationI Linear Flows for Top Reputation

2. Level:

I Flows% increasing in reputation for a given short-term rank

I Break Even Rank: 0.90 for Low reputation funds Vs 0.40 forTop repute funds

3. Slope:I Flow sensitivity is lower for low reputation, even at the extreme

high end of current performance.

Robustness Checks

1. Reputation: 3 or 5 or 7 years of history

2. Performance Measure: CAPM or Peer Ranks

3. Standard Errors:I Clustered SE (cluster by fund) with time effects controlled

using time dummiesI Cluster by fund-year (Veldkamp et.al (2014))

4. Institutional Vs Individual Investors

5. Fixed Effects Model: To control for fund family effects

Robustness With Fixed Effects

Only Short Term ReputationDep Var:FFit+1 Peer CAPM Peer CAPM

Unconditional EstimatesRank(t) 0.0345 0.0871*

(0.0435) (0.0430)

Rank-Sq(t) 0.276*** 0.232***(0.0453) (0.0448)

Low ReputationRank(t) -0.0978 -0.140*

(0.0592) (0.0630)

Rank-Sq(t) 0.244*** 0.339***(0.0682) (0.0776)

Medium ReputationRank(t) -0.0566 0.0270

(0.0496) (0.0491)

Rank-Sq(t) 0.389*** 0.308***(0.0553) (0.0542)

Top ReputationRank(t) 0.323*** 0.359***

(0.0585) (0.0585)

Rank-Sq(t) 0.100 0.0528(0.0671) (0.0691)

Section II:

Risk Shifting

Evidence on Risk Shifting: Background

1. Do mid-year losing funds change portfolio risk?I Convex flows =⇒ limited downside in payoff

2. Previous Papers:

I Brown, Harlow, Starks (1996): Mid-Year losing fundsincrease the portfolio volatility

I Chevallier, Ellison (1997): marginal mid-year winnersbenchmark but marginal losers ↑ σ

I Busse (2001):I Uses daily data =⇒ efficient estimates of σI No support for ∆σ(rit)

I Basak(2007):I What is risk? σ or deviation from benchmark/peers?I Shows that mid-year losers deviate from benchmarkI Portfolio risk can be up or down (σ ↓ or ↑)

3. But Flows Are Not Convex For All Funds !

Measuring Risk Shifting

1. Consider a simplest factor model

Rit = αi + βi︸︷︷︸= loading

× Rmt︸︷︷︸

=price

+ εt

2. Fact: Factors (e.g market) explain substantial σ(rit)

3. σ(rit) Flawed meaure: Lot of exogenous variation formanager

4. Factor Loadings (β): Within manager control =⇒ goodmeasure of risk-shifting

5. Measure of Devitation:

∆Risk = | βi ,2t︸︷︷︸β for 2ndhalf

− β2t︸︷︷︸median β for 2nd half

|

I Median β for funds with same investment style

Some Statistics

Table: Summary Statistics For Risk Change

Reputation Category

Variables Low Med Top

Annual BetaMean 1.04 1.02 1.02

Median 1.03 1.00 1.00Dispersion 0.19 0.15 0.20

∆ RiskMean 0.12 0.09 0.12

Median 0.084 0.066 0.091Dispersion 0.14 0.09 0.11

First Pass: Polynomial Smooth

Regression Results

Table: Risk Shifting

Unconditional Control For Reputation

Dep: Beta Devitation Peer CAPM Peer CAPM

Time Effects Yes Yes Yes YesStyle Effects Yes Yes Yes Yes

Standard Errors Fund Clustered Fund Clustered Fund Clustered Fund Clustered

Constant 0.298*** 0.291*** 0.304*** 0.305***(0.015) (0.015) (0.015) (0.016)

∆Risk Rank (H1) 0.542*** 0.543*** 0.534*** 0.532***(0.008) (0.008) (0.008) (0.008)

Log Age(t) -0.007 -0.007 -0.006 -0.006(0.004) (0.004) (0.004) (0.004)

Log Size(t) -0.000 -0.000 -0.000 -0.002(0.001) (0.001) (0.001) (0.001)

Unconditional Beta Deviation

Perf. Rank(H1) -0.243*** -0.228***(0.029) (0.029)

Perf. Rank(H1)2 0.229*** 0.226***(0.028) (0.028)

Result ContinuedPeer CAPM Peer CAPM

Low Reputation(t)

Perf. Rank(H1) -0.355*** -0.378***(0.042) (0.043)

Perf. Rank(H1)2 0.377*** 0.410***(0.048) (0.049)

Medium Reputation(t)

Perf. Rank(H1) -0.250*** -0.227***(0.032) (0.034)

Perf. Rank(H1)2 0.219*** 0.208***(0.033) (0.034)

Top Reputation(t)

Perf. Rank(H1) -0.0573 -0.0472(0.042) (0.043)

Perf. Rank(H1)2 0.034 0.044(0.046) (0.047)

Observations 15720 15720 14434 13406

Adj. R2 0.308 0.308 0.306 0.304

Mean Estimates For Risk-Shift

Discussion of Results

1. Low Reputation FundsI Severe career concerns

I Low Mid-Year Rank: Gamble for resurrection

I High Mid-Year Rank: Exploit convexity of flows as risk ofjob-loss relatively low

2. Top Reputation Funds:I No immediate career concerns =⇒ Level of deviation slightly

higher

I Flows Linear =⇒ No response to mid-year rank

Section III

Model Of Fund Flows

Model Overview

1. Question: What explains the heterogeniety in observedFund-Flow schedules

2. Possible Answer:I Investor-Base is heterogenous for funds with different

reputation or track record.

3. Basic Intuition:I A model with loss-averse investors + partial visibility

I Rational investors shift out of poor perfoming funds butloss-averse agents stick

I =⇒ Bad fund performs poor again: No outflows

I =⇒ Poor fund perform Good: Some inflows as fundbecomes ’visible’

Model Outline

1. Basic Set-Up:I Finite horizon model with T <∞I Two mutual funds indexed by i = 1, 2

I Two types of investors (N of each type)I Rational Investors (R): 1 unit at t = 0I Loss-Averse Investors (B): has η units at t = 0

2. At t = 0: Each fund has N2 of each type of investors

3. Partial Visibility:I Fund is visible to fund insiders at year end

I Fund visibility at t to outsiders increases with performance attime t

I visible =⇒ entire history is known

Returns and Beliefs

1. Return Dynamics:

ri ,t+1 = αi + εit+1

εit+1 ∼ N(

0, (σε)2)

where αi = unobserved ability of manager i

2. Beliefs:I Iit = Set of investors to whom i is visibleI For every j ∈ Iit , priors at end of t are

αi ∼t N(α̂it , (σt)

2)

I All investors are Bayesian =⇒ Normal Posteriors with

α̂it+1 = α̂it + (ri,t+1 − α̂it)

[(σt)

2

(σt)2 + (σε)2

]

Loss-Averse Investors

1. Assumptions:I Invest in only one of the visible funds at a timeI Solves Two period problem every t as if model ends at t + 1

2. Preferences: Following Barberis, Xiong (2009)I πt = accumulated loss/gain for investor of B type with iI Instantaneous Utility realized only upon liquidation

u (πt) =

{δπt1 (πt < 0) + πt1 (πt ≥ 0) If sell

0 If no sell

I Evolution of πt

πt+1 =

πt+1 + ri,t+1 If no sell

rj,t+1 If shift to fund j ∈ Ii0 If exit from industry

3. Trade-off: =⇒ B can mark-to-market loss today and exitfund i or carry forward losses in hope that rit+1 is large enough

4. Why? Loss hurts more: δ > 1

Motivation For Loss-Averse Investors

1. Strong Empirical Support:I Shefrin, Statman (1985), Odean(1998): Investors hold on

to losses for long but realize gains earlyI Calvet,Cambell, Sodini(2009): Slightly weaker but robust

tendency to hold on losing mutual fundsI Heath (1999): Disposition effect present in ESOP’sI Brown (2006), Frazzini (2006): Institutional traders exhibit

tendency to hold losing investments

2. Why Realized Loss-Aversion?

I Barberis, Xiong (2009): Realization Loss Averse preferencescan generate disposition effect

I Usual Prospect utility preferences over terminal gain/loss neednot generate tendency to hold losses

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}

In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Problem of Loss-Averse InvestorKeep Vs Sell Decision: B type invested in fund i = 1

V(πt , {α̂it}i=1,2

)= max

{V sellt ,V keep

t ,V exitt

}In turn

Vkeep (α̂1t , πt) = Et [u (πt + ri ,t+1) |α̂1t ]

= P (πt + r1t+1 ≥ 0)Et [πt + r1t+1|πt + r1t+1 ≥ 0]

+P (πt + r1t+1 < 0) δEt [πt + r1t+1|πt + r1t+1 < 0]

= Q (πt + α̂1t)

Vsell (πt , α̂2t) = u (πt) + Et [u (r2t+1) |α̂2t ]

= u (πt) + P (r2t+1 ≥ 0)Et [r2t+1|r2t+1 ≥ 0]

+δP (r2t+1 < 0)Et [r2t+1|r2t+1 < 0]

= u(πt) + Q (α̂2t)

Properties of Q(µ)

1. Expression for Q(µ), µ ∈ R

Q (µ) = µ+ (δ − 1)[µΦ(−µσ

)− σφ

(µσ

)]2. Q(µ) is increasing in µ. In particular, one unit rise in µ

changes Q(µ) by more than 1 unit

∂Q (µ)

∂µ= 1 + (δ − 1) Φ

(−µσ

)∈ (1, δ)

3. Q(µ) is concave, with limµ→∞

∂Q(µ)∂µ = 1

∂2Q (µ)

∂µ2= −(δ − 1)

σφ(−µσ

)< 0

Optimal Policy For Loss Averse Investor

1. Result 1: Participation Premium

I For any πt , liquidation of current fund is optimal if α̂1t < 0.

I In fact, break-even skill is positive. That is ifVkeep(α1,min(πt), πt) = Vexit(πt), then α1,min(πt) > 0, for anyπt

I Similarly, break-even level for manager 2 skill α2,min > 0. ElseB will exit but not shift to fund 2

2. How to interpret ”LOW reputation then?

I Relative: Low relative to Top, but still with positive expectedexcess returns.

I Replacement Theory: Bad managers are replaced or badfunds merge with good funds. Hence expectation about ”fundreturns” never go negative (e.g Lynch,Musto 2003)

3. Assumption: α̂2t > α2,min and α̂1t(πt) > α2,min(πt)

Optimal Policy For Loss-Averse Investor1. Result 2: Hold Losses Unless Fund is Extremely Bad

I If Q (α̂2t) < δα̂1t , then B holds if πt < π∗ (α̂1t , α̂2t), for someπ∗ (α̂1t , α̂2t) < 0

2. Understanding Why?

∂Q(µ)

∂µ︸ ︷︷ ︸=Marginal value to skill

< δ = u′(π)︸ ︷︷ ︸=Marginal Loss

I =⇒ realizing loss is costly if ∆µ is small or πt < 0 is large inmagnitude.

I Note If shifted

Gain =∂Q(µ)

∂µ× (α̂2t − α̂1t)

Loss = δπt

Optimal Policy

1. Result 3: Loss-Holding Region Increases in α̂1t

I Why? Relative gain from shifting (α̂2t − α̂1t) decreases as α̂1t

increases

2. Result 4: Policy For GainsI If Q(α̂2t) < α̂1t , hold gains if greater than someπ∗(α̂1t , α̂2t) > 0

I If Q(α̂2t) > α̂1t , liquidate any gain.

I Why? Hold large gains in some cases as current gains reducesprobability that πt+1 = πt + rit+1 < 0

3. Result 5: No Liquidation If Manager Is Better

I No liquidation is optimal if α̂1t > α̂2t for any given πt ∈ RI Why? If α̂1t > α̂2t , then sticking with same manager is the

best chance to recover losses (given participation is satisfied)

Illustration Of Optimal Policy

Figure: Hold Losses Even if α̂1 < α̂2

Illustration Of Optimal Policy

Figure: Loss-Holding Region

Optimal Policy For Rational Investor

1. Objective: Mean-Variance Optimization

V Rt = max

ω∈HRt

[ω′α̂t −

γ

2ω′Σω

]2. Solution:

ωi =α̂it

γσ2it

3. Discussion:I Simplification: General time consistent policy under learning

is complicated

I Lynch&Musto (2003): Similar simplification assumptionwith exponential utility and one-period investors

I Alternative: Assume exponential utility and one-periodagents, so that policy of old and new agent coincide giveninformation

Dynamics Of Investor-Base

Figure: Dynamics Of Investor-Base

I Sequence of poor performce =⇒ Higher fraction ofLoss-Averse Investors in Fund

Equilibrium Fund Flows

Figure: Fund Flow Schedules

Alternative Theories

1. Lynch & Musto (JF,2003):I Optimal replacement of manager by company below a cut-off

performanceI =⇒ Magnitude of shortfall has no information contentI =⇒ asset demand similar below cut-off

2. Berk & Green (JPE,2004)I Decreasing returns to scaleI Quantities (size of fund) adjust so that expected excess returns

on all funds are equalized to zeroI Return chasing, differential abilities and lack of persistence are

all consistent with each other

3. Lynch & Musto For Current Evidence?I P(firing) and hence convexity decreasing in reputationI Consistent with empirics? Some manager firing even for ’Top’

categoryI =⇒ Some insensitivity should have been observed if firing

mechanism was true

Conclusions

1. Lack of Flow Convexity for Reputed Funds (or for 40% ofIndustry money)

2. No Risk Shifting For Top funds in response to Mid-Year rank

3. Some 2nd half risk-sfiting for bad repute funds

4. Fund Flow heterogeniety could be explained through presenceof loss-averse investors

Thank You !