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TESIS de DOCTORADO

IInstituto N S T I de T Economía U T O D E

DOCUMENTO DE TRABAJO

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The Delegated Portfolio Management Problem: Reputation and Herding por Félix Fernando Villatoro Godoy

Licenciado en Economía y Negocios, Escuela Superior de Economía y Negocios, 2001 Magíster en Economía, Ponti…cia Universidad Católica de Chile, 2003

Esta tesis se presenta como requerimiento parcial para optar al grado de Doctor en Economía

Instituto de Economía PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE

Comité: Felipe Zurita (profesor guía) Jaime Casassus Rodirgo Harrison Bernardita Vial 2008

La tesis de Félix Fernando Villatoro Godoy fue aprobada:

Felipe Zurita (profesor guía)

Jaime Casassus

Rodirgo Harrison

Bernardita Vial

Ponti…cia Universidad Católica de Chile

30 de diciembre de 2008

Abstract

The Delegated Portfolio Management Problem: Reputation and Herding by Félix Fernando Villatoro Godoy Doctor en Economía, Ponti…cia Universidad Católica de Chile

Felipe Zurita (Chair)

This work studies the e¤ects of the possibility of building a reputation in a delegated portfolio management context where …nancial intermediaries may herd. Reputation is modelled as investors’ Bayesian beliefs regarding the ability of intermediaries given their past performance. Unlike previous works, we characterize reputational equilibria in which intermediaries’ decision to invest in reputation depends on their current reputation. We …nd that intermediaries with good reputation are prone to invest in information, whereas those with poor reputation herd. Also, the presence of implicit incentives provided by reputation allows this market to operate using simple remuneration schemes (i.e. a percentage of assets under management). The empirical predictions of the model are discussed and are found to be broadly consistent with previous evidence.

Felipe Zurita Chair

Resumen

The Delegated Portfolio Management Problem: Reputation and Herding por Félix Fernando Villatoro Godoy Doctor en Economía, Ponti…cia Universidad Católica de Chile

Felipe Zurita (profesor guía)

Este trabajo estudia los efectos de la posibilidad de construir una reputación en un contexto de administración delegada de portafolio, en el cual los intermediarios …nancieros pueden imitar a otros. La reputación se modela como las creencias Bayesianas de los inversionistas respecto a la habilidad de los intermediarios, dado su desempeño pasado. A diferencia de trabajos previos, caracterizamos equilibrios en los que la decisión del intermediario de invertir en reputación depende de su nivel actual de reputación. Encontramos que intermediarios con buena (mala) reputación tienden a invertir en reputación (imitar). La presencia de incentivos implícitos provistos por la reputación hace que este mercado pueda funcionar usando remuneraciones simples (i.e. un porcentaje del valor de la cartera administrada). Se discuten las implicancias empíricas del modelo y se encuentra que estas son en su mayoría consistentes con evidencia previa.

Felipe Zurita Profesor Guía

I would like to dedicate this thesis to Claudia, Sebastián and Juan Pablo. Without your love, support and encouragement I couldn’t have gotten this far. I love you and I’m thankful for having you with me. This work is for you, with all my love.

Contents List of Figures

List of Tables

1 Introduction

2 Stylized Facts and Literature Review 2.1 Delegated Portfolio Management Stylized Facts . . . . . . 2.1.1 Fees and Expenses . . . . . . . . . . . . . . . . . . 2.1.2 Herding and Impact on Prices . . . . . . . . . . . . 2.1.3 Flows and Performance . . . . . . . . . . . . . . . 2.1.4 Trading Volume . . . . . . . . . . . . . . . . . . . 2.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Delegated Portfolio Management Problem Literature 2.2.1 Reputational Herding Literature . . . . . . . . . . 2.2.2 Non-Finance Reputation Literature . . . . . . . . . 2.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

3 The Relation Between Reputation and Herding in a Delegated Management Problem Context 3.1 A Static Model in a Risk Neutral Economy . . . . . . . . . . . . . 3.1.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Pooling Equilibria . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Separating Equilibria . . . . . . . . . . . . . . . . . . . . . . 3.2 A Dynamic Model in a Risk-Neutral Economy . . . . . . . . . . . . 3.2.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Steady-State Equilibria . . . . . . . . . . . . . . . . . . . . 3.2.3 Importance of the Investment Cost . . . . . . . . . . . . . . 3.2.4 Absorbing States . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Reputation E¤ects under other Remuneration Schemes . . . 3.2.6 On the Existence of Sticky Fees . . . . . . . . . . . . . . . . 4 Numerical Examples 4.1 The Case with Flexible Fees . . . . . . . . 4.1.1 Determination of . . . . . . . . 4.1.2 Comparative Statics . . . . . . . . 4.2 The Case with Sticky Fees . . . . . . . . . 4.2.1 Determination of . . . . . . . . 4.2.2 Comparative Statics . . . . . . . . 4.3 The Dynamic versus the Static Equilibria

. . . . . . .

. . . . . . . . . .

9 10 10 13 16 18 19 20 21 31 42

. . . . . . . . . .

Portfolio . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

44 47 47 55 57 62 63 65 100 100 102 104

. . . . . . .

110 110 111 116 126 128 129 130

iii 5 Empirical Discussion 5.1 Empirical Predictions 5.2 Empirical Strategy . . 5.2.1 The Variables . 5.2.2 The Estimation

. . . .

137 137 141 142 150

6 Conclusions

152

Bibliography

159

A Delegated Portfolio Management Literature Review A.1 Optimal Contracts . . . . . . . . . . . . . . . . . . . . A.2 Asymmetric Contracts . . . . . . . . . . . . . . . . . . A.3 Churning . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Use of Benchmarks . . . . . . . . . . . . . . . . . A.5 E¤ects on Securities’Prices . . . . . . . . . . . . . . . A.6 Reputation . . . . . . . . . . . . . . . . . . . . . . . . A.7 Herding . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Other Topics . . . . . . . . . . . . . . . . . . . . . . . A.8.1 Mutual Fund Performance and Persistence . . A.8.2 Multiple Agency Layers . . . . . . . . . . . . . A.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 179 183 186 189 194 207 222 222 224 225

B MATLAB Code

. . . . . . . . . . .

228

C Additional MATLAB Functions 236 C.1 Mug Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 C.2 Mub Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 C.3 Fee Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

List of Figures 2.1

NYSE Turnover Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19

Pooling Equilibrium . . . . . . . . . . . . . . . . . Separating Equilibria . . . . . . . . . . . . . . . . . Investment Decisions with Investment . . . . . . . Investment Decisions with Herding . . . . . . . . . Evolution of Reputation . . . . . . . . . . . . . . . Evolution of Reputation with Policy Function . . . Evolution of Reputation: Two Decision Sequences Function w(mu) . . . . . . . . . . . . . . . . . . . . Function w(mu) for Policy Function . . . . . . . . Function v(mu) . . . . . . . . . . . . . . . . . . . . Function v(mu) for Policy Function . . . . . . . . . Equilibrium with (almost) no Investment . . . . . Equilibrium with Investment 1 . . . . . . . . . . . Equilibrium with Investment 2 . . . . . . . . . . . Equilibrium with nonmonotonic Policy Function . Absorbing States . . . . . . . . . . . . . . . . . . . Reputation E¤ects with Alternative Contract . . . Function w(mu) with Sticky Fee . . . . . . . . . . Equilibrium with Investment and Sticky Fees . . .

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18

Price Function: Baseline Case . . . . . . . . . . . . . . . . Reputation Evolution: Baseline Case . . . . . . . . . . . . Functions w(mu) and v(mu): Baseline Case . . . . . . . . Value Function: Baseline Case . . . . . . . . . . . . . . . Price Function: Baseline Case with Updated Mu* . . . . . Reputation Evolution: Baseline Case with Updated Mu* . Functions w(mu) and v(mu): Baseline Case with Updated Value Function: Baseline Case with Updated Mu* . . . . Mu* Comparative Statics: Eta . . . . . . . . . . . . . . . Mu* Comparative Statics: PH . . . . . . . . . . . . . . . Mu* Comparative Statics: c . . . . . . . . . . . . . . . . . Mu* Comparative Statics: W . . . . . . . . . . . . . . . . Mu* Comparative Statics: Lambda . . . . . . . . . . . . . Mu* Comparative Statics: Delta . . . . . . . . . . . . . . Mu* Comparative Statics: rH . . . . . . . . . . . . . . . . Mu* Comparative Statics: rL . . . . . . . . . . . . . . . . Mu* Comparative Statics: q . . . . . . . . . . . . . . . . . Mu* Comparative Statics: pi . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

57 59 70 72 74 75 76 89 90 91 92 93 94 95 99 101 103 108 109

. . . . . . . . . . . . . . . . . . Mu* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

112 112 113 113 114 114 115 115 117 119 120 120 121 122 123 123 124 125

v 4.19 4.20 4.21 4.22 4.23 4.24 4.25

Mu* Comparative Statics: Theta . . . . . . . . . . . . . . . . Mu*max and Mu*min Comparative Statics: Theta . . . . . . Functions w(mu) and v(mu): Baseline Case with Sticky Fees Functions w(mu) and v(mu): Updated mu* with Sticky Fees Mu* Comparative Statics with Sticky Fees: Lambda . . . . . Mu* Comparative Statics with Sticky Fees: pi . . . . . . . . . Mu* Comparative Statics with Sticky Fees: Theta . . . . . .

. . . . . . .

126 127 129 130 131 131 132

List of Tables 1.1 1.2 1.3

Financial Intermediaries’Assets Under Management. . . . . . . . . . . . . . Mutual Funds’Assets Under Management . . . . . . . . . . . . . . . . . . . Chilean Mutual and Pension Funds Assets Under Management . . . . . . .

4.1 4.2 4.3 4.4

Flexible Fees Model’s Parameters: Baseline Case . . . Comparative Statics Summary: Flexible Fees . . . . . Sticky Fees Model’s Parameters: Baseline Case . . . . Summary: The Static Equilibrium versus Reputational

. . . .

111 127 128 136

5.1

Mutual Fund Managers Ranking by Market Share. . . . . . . . . . . . . . .

151

. . . . . . . . . . . . . . . . . . Equilibria

. . . .

2 2 3

vii

Acknowledgments I would like to thank my committee for all their help throughout this project, specially my advisor Felipe Zurita, and Bernardita Vial. I am also grateful for helpful discussions and comments received by Reinaldo Arellano, Ricardo Caballero, Rodrigo Cerda, Borja Larraín, Andrés Lehuede, Claudia Martínez and Salvador Valdés. Finally, I would like to thank comments received by seminar participants at the PUC economics seminars, SECHI 2008 and LAMES 2008 meetings.

Chapter 1

Introduction

Financial intermediaries (FI) play a very important role in the economy as they channel resources from agents with liquidity surpluses (both individuals and …rms) towards those with liquidity needs. Banks are one of the more traditional types but there are other intermediaries whose importance has increased over time, namely insurance companies, pension funds and investment companies (such as open and closed end mutual funds and hedge funds). Table 1.1 shows how the stock of delegated assets under management for the second group of intermediaries in OECD countries has averaged an annual growth rate of 9.2% during the last …ve years. Some analysts (see BIS, 2003, IMF, 2004) have suggested that one of the main factors behind this considerable growth would be the social security reforms undertaken during the past few years in Latin American and Central Europe countries. Other important factors that explain this industry’s growth would be …nancial liberalization, technological advances and an economic environment characterized by low in‡ation, which increases the attractiveness of …nancial asset holdings.

As a result of the previously mentioned reforms there has been a rise in demand for portfolio management services to invest the sizable funds that have been accumulated.

2 (USD Billions) Investment Funds Insurance Firms Pension Funds Others Total

2002 11,546 10,100 9,696 868 32,186

2003 13,910 12,034 11,876 986 38,771

2004 15,922 13,877 13,387 1,257 44,400

2005 18,239 15,141 14,782 1,480 49,586

2006 19,712 15,781 13,837 1,636 50,966

% 10.7 10.6 5.8 14.2 9.3

S o u rc e : A u th o r’s c a lc u la tio n s b a se d o n O E C D d a ta . O E C D c o u ntrie s a re : A u stra lia , A u stria , B e lg iu m , C a n a d a , C z e ch R e p u b lic , D e n m a rk , F in la n d , Fra n c e , G e rm a ny, G re e c e , H u n g a ry, Ic e la n d , Ita ly, J a p a n , K o re a , L u x e m b o u rg , M e x ic o , N e th e rla n d s, N o rw ay, P o la n d , P o rtu g a l, S lova k R e p u b lic , S p a in , S w e d e n , S w itz e rla n d , Tu rke y, U n ite d K in g d o m a n d U n ite d S ta te s.

Table 1.1: Financial Intermediaries’Assets Under Management. (USD Billions) World Americas Europe Asia and Paci…c Africa

2002 11,324 6,776 3,463 1,064 20.9

2003 14,048 7,970 4,683 1,361 34.5

2004 16,165 8,792 5,640 1,678 54.0

2005 17,771 9,764 6,002 1,939 65.6

2006 21,823 11,485 7,804 2,457 78.0

% 14.0 11.1 17.6 18.2 30.2

S o u rc e : Inve stm e nt C o m p a ny In stitu te 2 0 0 8 Fa c tb o o k .

Table 1.2: Mutual Funds’Assets Under Management

Additionally, in countries with de…ned-bene…ts pension schemes there has been a tendency by sponsors (e.g. the State or the a¢ liates’ companies) to professionalize the portfolio management of accumulated funds as means to meet in a better way future obligations with a¢ liates. Table 1.2 shows the world-wide evolution of assets under management for the mutual fund industry.

As Table 1.3 shows, the Chilean mutual fund market has also experienced a remarkable growth in the last years, managing assets in the order of USD 18 billions by December 2006. This …gure is considerably smaller than the amount of assets managed by

3 (USD Billions) Mutual Funds Pension Funds

2002 6.349 36.358

2003 8.295 48.940

2004 11.496 58.818

2005 13.564 74.491

2006 17.964 89.440

% 23.1 19.7

S o u rc e : A u th o r’s c a lc u la tio n s u sin g d a ta fro m th e C h ile a n M u tu a l Fu n d s A sso c ia tio n , th e S u p e rinte n d e n c y o f P e n sio n Fu n d s a n d th e C e ntra l B a n k o f C h ile .

Table 1.3: Chilean Mutual and Pension Funds Assets Under Management

pension funds, which were introduced in the early 80’s. Both types of intermediary have experienced sustained growth in the 2002 - 2006 period, with growth rates surpassing those of European and American countries.

The growth of these intermediaries has been perceived as a positive development for several reasons: they have greater diversi…cation capacity than individual investors, plus they reduce transaction costs; if insurance companies and pension funds have long term liabilities, they can help to develop and give stability to long-term …nancial assets markets; the presence of intermediaries can help to improve transparency and the corporate government of …nancial markets (IMF, 2004); and, due to their information processing capacity, FI can improve the e¢ ciency of …nancial markets, exploiting arbitrage opportunities and preventing …nancial securities’prices to deviate from their fundamentals. However, there are also reasons to monitor and study carefully the development and behavior of FI given that the size of the portfolios managed by these institutions imply that their trading decisions may have signi…cant impact on …nancial stability and resource allocation. Some …nancial intermediaries, such as pension funds, tend to show similar investment strategies and portfolios, a phenomenon that is called herd behavior and which

4 could increase excessively volatileness in …nancial markets1 ; legal restrictions to investment in foreign assets for some FI, such as pension funds, could create price distortions in some domestic securities in countries where …nancial markets are small in relation to the size of these funds, besides limiting the investors’diversi…cation opportunities; in a …nancial market where managers worry about their perceived ability by the market; this is, their labor market reputation, it’s possible that they ignore useful information to make investment decisions and decide to imitate the decisions taken by other managers (Dornbusch et al, 2000). This possibility, which has theoretical grounds in the works of Scharfstein and Stein (1990), Avery and Chevalier (1999) and Graham (1999), sharply contrasts with the view that reputation can act as a market mechanism to align incentives between investors and intermediaries, therefore allowing the delegated portfolio management market to operate even in the presence of informational asymmetries. This latter argument is made by Heinkel and Stoughton (1994), Arora and Ou-Yang (2001) and Farnsworth (2003). We make a contribution by studying the relationship between reputation and herding in a delegated portfolio management market applying a new methodology from the reputation literature which considers aspects left aside by previous works. In particular, we study how the incentives to herd change as a …nancial intermediary’s reputation evolves over time, explicitly accounting for the fact that there must be permanent uncertainty about the intermediary’s characteristics for a steady state reputational equilibrium to be feasible. This point is made by Hölmstrom (1999), Mailath and Samuelson (1998), (2001), Cripps et al (2004) and Vial (2008), although in a di¤erent context. 1

See the following Chapter for a survey on this and other delegated portfolio management market stylized facts.

5 Our main results are that as an intermediary’s reputation improves his incentives to herd decrease. As we show in Chapter 3 there are two situations in which an intermediary may disregard the e¤ects of his actions on his reputation; when his reputation is very bad or when it’s very good. In these cases it is possible that the intermediary may try to cheat investors by shirking. Moreover, if the remuneration scheme is given by a percentage fee of the …nal value of assets under management (which as we will show in the stylized facts Section, is the case for most mutual funds); and if this fee is increasing in the intermediaries’ reputation, then an intermediary with bad reputation that chooses to herd instead of incurring the costs of gathering private information will experience an expected loss in …nal value of assets under management but since his pro…ts are given by a small percentage of this value, he will herd. On the other hand, an intermediary with good reputation that decides to herd will experience an important loss in expected pro…ts, since these are given by a larger fee of the …nal value of assets under management. In order to avoid this loss the intermediary will acquire private information. This prediction is also made by Avery and Chevalier (1999). However, in their case an agent with good reputation in fact chooses a contrarian strategy, disregarding his private information and making the opposite decision from other agents in order to signal to principals that he is skilled. Of course, this behavior is ine¢ cient from the investors’point of view. The work by Graham (1999) makes the opposite prediction: as the initial reputation of agents improves they will herd more because they want to avoid the large drop in pro…ts associated with a fall in reputation, which in this model occurs if an agent’s decision is di¤erent from that of other agents. However, in a long run equilibrium the initial reputation of an intermediary may be of limited importance

6 in terms of determining his reputation several periods ahead. Therefore, this work is unsuitable to study how an agent’s incentives to herd change as his reputation endogenously evolves over time. We view our work as an alternative rationalization for the empirical evidence found in Chevalier and Ellison (1999) and Hong et al (2000) regarding the existence of a negative relationship between reputation and herding. However in our setup the mechanisms operating in equilibrium are di¤erent. In particular due to our modelling decision of using a continuum of intermediaries, the portfolio choice of a particular intermediary contains no information regarding the possible type of another intermediary. This is the basic mechanism a¤ecting the behavior of managers in Scharfstein and Stein (1990), Avery and Chevalier (1999) and Graham (1999). Additionally, while our model makes predictions regarding reputation and herding like those of Avery and Chevalier (1999), our …ndings are much more optimistic in the sense that lack of herding by intermediaries with high reputation is associated with e¢ cient investment and use of private information. This is important because the works by Avery and Chevalier and Scharfstein and Stein assume a positive relationship between reputation and pro…ts for intermediaries in a two period setup. However, in the presence of the pathological behavior implied by these models, endogenously deriving a long-term, positive relation between reputation and willingness to pay seems like a harder task (Ottaviani and Sørensen (2006) also make a similar observation). Moreover, rationalizing the increasing importance of institutional investors in …nancial markets is di¢ cult if all intermediaries, regardless of their reputation, make little or no use of private information. While we believe that the cases described by these authors may be of great relevance

7 in determined time periods or situations, we argue that it is di¢ cult to imagine that the delegated portfolio market could have experienced such successful growth if pathological behavior was always present, since intermediaries would have a hard time competing with investors who trade on their own behalf and presumably always make good use of their private information. We also show how the size of the percentage fee that must be paid to intermediaries in order to align incentives can be considerably smaller if investing in reputation is possible relative to a situation in which this isn’t feasible. Moreover, we illustrate how the possibility of investing in reputation may allow the delegated portfolio management market to operate when more sophisticated remuneration schemes cannot be used. However, there is a cost involved since in equilibrium the intermediaries’types are never revealed to investors. Therefore, it is possible that some skilled unlucky intermediaries are punished by investors through low fees while some lucky unskilled intermediaries may be paid high fees. Nevertheless, since skilled intermediaries that acquire information have a greater probability of making good investment decisions, which results in having a better reputation, this type of undesirable situations are rather unlikely to occur. Additionally, we show that for a reputational equilibrium to be feasible, the gains from investing in reputation can either be obtained through higher fees or through larger assets under management. In both cases the intermediaries’expected pro…ts are increasing in their reputation and the main features of the equilibria remain unchanged. The rest of this work is organized as follows. Chapter 2 presents a series of stylized facts regarding the delegated portfolio management market and presents a selective review

8 of the literature studying this problem. Chapter 3 presents the theoretical model, …rst in a static setup and then in a dynamic one. Su¢ cient conditions are given for reputational equilibria (i.e. equilibria in which at least some intermediaries acquire information) to be feasible and we partially characterize such equilibria. Chapter 4 presents numerical exercises used to study the comparative statics properties of the model and to compare some of the properties of the static and dynamic economies. Chapter 5 discusses the models’empirical predictions and a comparison is made with previous evidence from other authors. We also discuss an estimation strategy in order to validate our predictions. Finally, Chapter 6 concludes and discusses some possible research areas.

Chapter 2

Stylized Facts and Literature Review

In this Chapter we introduce from a theoretical perspective the features of the delegated portfolio management market. We then proceed to survey the existing literature, both theoretical and empirical, summarizing the main …ndings and highlighting some interesting research areas. The delegated portfolio management problem is an agency situation in which an investor (the principal), who has some initial wealth level W , chooses to delegate the task of managing and investing his resources to a …nancial intermediary (the agent). The investor would like the portfolio to be managed so as to maximize his expected utility for some future period. However, the intermediary will take his investment decisions with the objective of maximizing his own expected utility. This con‡ict of interests is worsened because investors don’t know with certainty whether an intermediary has the skills needed to make investment decisions and he can’t verify that the intermediary has made e¤ort to take sound investment decisions. Additionally, in most cases intermediaries have superior knowledge about …nancial markets compared to that of the investors whose portfolio they manage. Moreover, investors usually don’t have a reliable mechanism to evaluate the manager’s performance. Thus, an investor who observes a poor result (e.g. a low return of the portfolio) won’t be able to know with certainty if this was due to incompetence, negligence or bad luck.

2.1

Delegated Portfolio Management Stylized Facts The delegated portfolio management market has been the subject of research by

several studies. These works have found some stylized facts in the data, which we brie‡y summarize.

2.1.1

Fees and Expenses In the USA the types of remuneration schemes that can be used by investment

companies is regulated by the Investment Advisers Act of 1940. Under this Act the use of performance based compensation is allowed as long as this is a "fulcrum" type compensation. This means that the compensation must include a bonus if the investment company’s performance exceeds that of a benchmark and also a penalty if the performance is below the benchmark. This type of compensation is also referred to as symmetric. Even though the USA law allows for the use of compensation-based fees, in practice such fees are not always used. Golec (1992) presents data on 476 open end mutual funds for 1985. Out of this funds only 29 used incentive or performance based fees, which represents a 6.9% of the sample. Also, Blake et al (2003) report that in 1999 only 108 out of 6,716 bond and stock mutual funds, that is only 1.6%, used this type of fee. Also, in the 108 funds using compensation based fees, 44% used the S&P 500 index as their relevant benchmark. Finally, Cuoco and Kaniel (2007) report that as 2004 50% of USA corporate pension funds with assets of more than US$5 billion; 35% of all USA pension funds and 9% of all USA mutual funds employed performance based fees. Overall the data suggests that the use of performance-based fees is not uniform across institutional investors. Also, larger funds seem

11 to be more likely to use such fees. However, for a large share of pension and mutual funds performance-based fees are not widely used. Fung and Hsieh (1999) describe how hedge funds are allowed by the SEC to use asymmetric performance fees. The authors show that at the end of 1997 83% of the hedge funds in their sample used an incentive-based fee. For 51% of these hedge funds this fee ranged between 1500 to 2000 basis points, while average management fees range between 100 and 200 basis points. Surveying data from 1994 to 2006, Ang et al (2008) …nd hedge funds charge an average management fee of 150 basis points and a mean performance fee of 2000 basis points. All these authors document that one type of incentive based fee that is widely used is the “high-water mark”. Under this arrangement the hedge fund manager receives a percentage of the increase in assets under management in excess of the last registered maximum. If the fund value doesn’t exceed this high-water mark then the manager only gets the management fee. In the introduction we documented how the number of mutual funds has increased over recent years. In many cases there is a large number of funds competing in the same category. Given this, it could be possible to expect fees in this narrowly de…ned markets to show little to no dispersion if the o¤ered service is fairly hom*ogenous. However, the work by Khorana et al (2008) challenges this view. The authors gather data on mutual fund fees for 46,580 mutual fund classes o¤ered for sale in 18 countries. These funds account for 86% of the worldwide fund industry in 2002. The authors’ measure of fund fees includes two components. The …rst one is the expense ratio, which encompasses management fees and charges made to cover investment management, administration, servicing, transfer agency,

12 etc. The second element are distribution fees, such as front-end or back-end loads. Using this variable the authors …nd substantial dispersion in funds fees from country to country. For example mean fees vary from 63 basis points in Sweden to 189 basis points in Dublin for bonds funds. In the United States this fee averages 105 basis points, while the full sample mean is 121 basis points. For equity funds, fees range from 82 basis points in the Netherlands to 300 basis points in Canada. In the United States the average fee is 153 basis points, while the full sample mean is 180. To determine the causes of this dispersion the authors explore fund, sponsor an national characteristics. While the investment objective of funds1 seems to be important for the size of fees charged, it seems that fees tend to be lower in countries with stronger investor protection. For the Chilean mutual funds market, Maturana and Walker (1999) report that the average fee charged by equity mutual funds from 1990 to 1997 was 600 basis points, while long term and short term bonds mutual funds charged average fees of 310 and 240 basis points, respectively, for the same time period. The authors don’t provide evidence on fees’dispersion. While the degree of fee dispersion across countries may not seem surprising as the mutual funds market conditions vary from nation to nation, Hortaçsu and Syverson (2003) …nd evidence of dispersion of fees even in narrowly de…ned categories such as index funds replicating the S&P 500 index. For instance, the authors study fees charged by 1,267 mutual funds operating in the International Equities sector. Funds in this sector charged an average annual fee of 225.5 basis points. The 90th to 10th percentile ratio is 3.2 while the 75th to 25th is 1.9. Even in a narrowly de…ned sector, such as S&P 500 index funds 1

The authors de…ne 122 investment objectives based on fund category (e.g. bonds, bonds and cash, equities, money market, real state, etc.) as well as the region where the funds invest (e.g. Danish equities or Eurozone bonds) and the type of securities held (e.g. small cap stocks).

13 the authors …nd that in 2000 there were 85 funds in this category, charging an average fee of 97.1 basis points. For this sector the 90th to 10th percentile ratio is an outstanding 8.2 while the 75th to 25th is 3.1. These numbers certainly seem to be large, specially given all the information available for investors in the mutual fund market. The authors …nd evidence suggesting that this dispersion could be related to the existence of search costs, which would prevent investors from investing in the cheapest funds. Overall, the evidence the evidence suggests that intermediaries such as mutual funds make little use of performance-based fees, although this type of fees is widely used in the hedge fund industry. Moreover, there is a considerable degree of fee dispersion in the mutual fund industry, even for narrowly de…ned categories.

2.1.2

Herding and Impact on Prices There is substantial evidence regarding the existence of a certain degree of correla-

tion in portfolio decisions amongst institutional investors who seem to make similar buying and selling decisions. This phenomenon has been called "herding". For USA pension funds Lakonishok et al (1992b) …nd weak evidence of funds buying or selling in herds. Grinblatt et al (1995) study the behavior of 155 mutual funds over the 1975 to 1984 period, …nding evidence of momentum investment strategies. This is, funds tend to buy stocks with good recent performance while also selling stocks with poor recent performance (although in less degree). The authors also …nd evidence of herding examining stock prices. This …nding could be explained by the use of momentum strategies. Nevertheless, the authors …nd that the level of herding, this is funds buying and selling the same stocks at the same time, is rather small. Wermers (1999) performs a comprehensive study of mutual fund behavior for

14 virtually all mutual funds based on the USA which held equities from 1974 through 1994. The author …nds evidence of low levels of herding consistent with those documented by Grinblatt et al (1995) and by Lakonishok et al (1992). Also, the evidence suggests that mutual funds are equally likely to herd as buyers or sellers of stocks. Additionally, the stocks that funds buy in herds seem to have higher abnormal returns that stocks that funds sell in herds. Finally, Wermers …nds that most observed stock price adjustments following trading by herds appear to be permanent which favors the hypothesis that mutual funds herds speed the price adjustment process and are not destabilizing. This is interpreted as evidence supporting theories of herding based on private information on fundamentals (see Bikhchandani et al, 1992) as opposed to herding based on reputational concerns (see Scharfstein and Stein, 1990) and the works on reputational herding surveyed in the following Section). Regarding evidence for other countries, Maturana and Walker (2002) …nd evidence of herd behavior in Chilean mutual funds for the 1990-1998 period. Walter and Moritz Weber (2006) …nd evidence suggesting the existence of moderate herding by German mutual funds in the 1998-2002 period. Wylie (2005) documents the existence of moderate herding for United Kingdom mutual funds covering from 1986 through 1993. Following the methodology developed by Lakonishok et al (1992b), Lobao and Serra (2002) study the Portuguese mutual funds market over 1998 through 2000 period. The authors …nd evidence of strong herding behavior in order of magnitude of 4 to 5 times stronger than herding documented in the USA and United Kingdom markets. Finally, Voronkova and Bohl (2005) study a sample of Polish pension funds from 1999 to 2001. They …nd stronger evidence of herding

15 than for mature markets. The authors attribute this …nding to more stringent investment regulation and high market concentration. The works surveyed above focus mainly on …nding evidence of herding by studying the behavior of stocks’prices rather than the decisions of individual funds. The works by Chevalier and Ellison (1999), Hong el al (2000) and Graham (1999) document how herding changes over the agents’ careers2 . Chevalier and Ellison study the behavior of a mutual fund managers sample from 1992 to 1994. The authors …nd that the probability of being terminated due to bad performance is higher for younger managers. Also, they …nd that these managers tend to take on less unsystematic risk than older mangers and they also hold more conventional portfolios. Hong et al (2000) report similar …ndings using a sample of 8,421 security analysts producing earnings forecasts from 1983 to 1996. The authors …nd that more inexperienced analysts are likelier to be terminated if their forecasts di¤er too much from the consensus and that these analysts tend to make predictions closer to the consensus. These two studies proxy reputation by agents’age. Finally, Graham (1999) studies the relation between reputation and herding for investment newsletters. Contrary to Chevalier and Ellison and Hong et al, this author …nds that investment newsletters with higher reputation tend to herd more. Overall, there is evidence of herding from institutional investors, as well as analysts, although it doesn’t seem to be too pervasive. Also, incentives to herd seem to change over time for individuals but there isn’t a consensus on whether herding increases or not with agents’age or experience. 2 This three studies cite the work by Scharfstein and Stein (1990) which explains how an agent’s incentives to copy others’decisions instead of using his own information may be related to reputational concerns. This is, the degree of ability the agent is perceived to posses by the rest of the market. We discuss this and other related literature in the following Section.

2.1.3

Flows and Performance Starting with Ippolito (1992), several works such as Gruber (1996), Chevalier and

Ellison (1997), Sirri and Tufano (1998), Lynch and Musto (2003), and Olivier and Tay (2008) …nd evidence of an asymmetric relationship between past performance and net in‡ows in the mutual fund industry. For instance, Ippolito studies 143 mutual fund performance starting in 1965 and up until 1984. He …nds that funds which perform better than the market by 100 basis points experience a growth in assets under management equal to 0.90% in the following year, while funds which under perform the market by 100 basis points su¤er a decline of 0.35% in assets under management. Chevalier and Ellison (1997) use a mutual fund sample spanning from 1989 to 1994. The authors are able to derive the performance‡ow relationships for young funds (funds that have operated for less than 2 years) and old funds (which have existed for more than 10 years). In the case of young funds, having a return 10 basis points above the market return means an average growth of 55% in assets under management, while having a return 10 basis points below the market means an expected reduction of less than 30% in fund size. For older funds, beating the market by 10 basis points leads to an expected increase of almost 15% in assets under management while trailing the market by 10 basis points means an expected decrease of little more than 10% in assets under management. Therefore, the performance-‡ow relationship appears to be convex for all funds and gets ‡atter for older funds. Sirri and Tufano (1998) use mutual fund data from 1971 through 1990. They …nd that while performance is positively associated with ‡ows, this relationship is signi…cant only for good performers, while it is statistically weak for the lower quintiles. Finally, the evidence provided by Gruber (1996), Lynch and

17 Musto (2003) and Olivier and Tay (2008) is also consistent with the existence of a convex relation between funds performance and ‡ows. This …ndings imply that fund managers face asymmetric incentives since even if their pro…ts are a fee of assets under management that doesn’t change over time, they can raise their earnings if the size of the portfolio they manage grows. Given the response of investors to mutual fund performance it is possible that managers take on excessive risk in an attempt to outperform the market since succeeding would greatly increase their pro…ts, while failing would reduce their gains by a smaller amount. Regarding the value of active investment by mutual funds, Gruber (1996) …nds evidence suggesting that average mutual funds performance between 1985 and 1994 is 65 basis points below that of market indexes. However, Wermers (2000) …nds evidence supporting the theory that active investment by mutual funds is valuable. The author decomposes the returns and costs of mutual funds into: stock-picking skills; stocks holdings; trade related costs of stock-picking; fund expenses and management fees; and di¤erences between gross stock portfolio returns and net fund returns that are due to holdings of cash and bonds. Wermers …nds that mutual funds hold stocks that outperform the market index by 130 basis points, which amounts to the sum of their expenses rate and transaction costs. Of this 130 basis points 71 are due to stock-picking skills, while 55 to 60 are explained by stock holdings. For the Chilean mutual funds market Maturana and Walker (1999) report that equity mutual funds underperformed the authors’ benchmark by 80 basis points between 1990 and 1997. These works are meant to be indicative of this topic, but do not constitute and exhaustive list. For further papers on this area see the references on Gruber (1996)

18 and Wermers (2000). For more recent papers see Lo (2007), who suggests the existence of value in active investment. Also, see French (2008) who presents evidence supporting the opposite view. While the existing evidence suggests that there is some value in active management, some authors have found evidence of little persistence in mutual fund performance. Cuthbertson et al (2006) survey the extensive literature that studies mutual fund performance and persistence. The existing evidence suggests that there is some persistence amongst the top decile of USA funds. Using a risk-adjusted gross returns measure persistence may last up to four years for a small number of growth funds and for up to one year when the top decile is formed using all funds categories. Also, there is strong evidence that poor performance persists across deciles.

2.1.4

Trading Volume Dow and Gorton (1997) document that there appears to be a consensus in that

the trading volume by institutional investors in inexplicably high, although it’s di¢ cult to rigorously prove this assertion as there is a lack of models that predict just how big trading volume should be. The authors gather data on this topic for foreign exchange market and for the New York Stock Exchange. For the …rst market, daily trading volume of foreign exchange transactions in all currencies in 1992 was US$880 billion. Meanwhile, the total value of annual world trade in 1992 was $3,646 billion which means that 24% of the annual trade was traded each day in this market. On the other hand, turnover for the NYSE was 49% in 1992. Moreover, Glaser and Weber (2007) report that annualized monthly turnover on the NYSE was roughly 100% in 2004. During this year 367,098,489,000 shares were

19 NYSE Annual Turnover Rate

Turnover

0.8

0.6

0.4

0.2

1950

1960

1970

1980

1990

2000

Figure 2.1: NYSE Turnover Rate S o u rc e : N Y S E w e b site

traded in this market. The …gure below shows the evolution of the NYSE turnover rate since 1944 until 2003. The existing turnover rate in this market seems to be large given the existing consensus that a 49% turnover rate was high back in 1992.

2.1.5

Summary We have presented studies documenting several stylized facts in the delegated

portfolio management market. We summarize these facts in the following list.

Fact 1 Performance based fees and bonuses are not widely used (at least in the mutual fund industry). Fact 2 There is considerable dispersion in mutual fund fees, even for narrowly de…ned categories.

20 Fact 3 There is a certain degree of correlation between institutional investors’ portfolio decisions. Fact 4 The degree of correlation changes with the fund managers’age. Fact 5 There seems to be some value -although with low persistence- in active investment management. Fact 6 The use of asymmetric contracts has been restricted by regulators in the USA. Fact 7 There seems to be an excessive volume of …nancial transactions. Fact 8 There is a convex relation between institutional investors’performance and in‡ows. This relationship turns ‡atter for older funds.

2.2

The Delegated Portfolio Management Problem Literature The literature studying the delegated portfolio management problem is vast and

deals with many issues present in this agency problem. Some of these works attempt to rationalize one or several of the stylized facts discussed in the previous section, while others attempt to derive closed form solutions for optimal contracts between investors and intermediaries. A selective review of these and other works is made in the Appendix where we also select and discuss works that belong to the reputation and the herding literature. Some authors such as Heinkel and Stoughton (1994), Arora and Ou-Yang (2001) and Farnsworth (2003) argue that if managers are able to build a reputation this may help to lessen the problems caused by moral hazard. In particular, by being diligent in making investment decisions managers may favorable in‡uence investors’opinions about their ability which may

21 allow them to charge higher fees or manage larger portfolios in the future. On the other hand, some authors attempt to rationalize the …nding that sometimes agents’decisions seem to be correlated; this is, that agents exhibit herd behavior. In particular, Scharfstein and Stein (1990) suggest that one possible reason for managers to herd is that they could opt to ignore private information and follow others if investors doubt more about their skill when they make a bad decision that is di¤erent from others’as opposed to a bad decision that is equal to others’. This phenomenon has been called reputational herding and it has been explored by Avery and Chevalier (1999), Graham (1999) and Ottaviani and Sørensen (2006). Given the goals of our work, we discuss the reputation and herding literature in the Appendix and in this section we focus on the reputational herding literature. We also review a strand of the reputational literature that focuses on long-run equilibria in which agents have incentives to invest in their reputation. For this, we discuss the works by Hölmstrom (1999), Mailath and Samuelson (1998), (2001), Cripps, Mailath and Samuelson (2004) and Vial (2008) as they will be methodologically important for our work. Throughout the literature review we will use the terms principals and investors interchangeably and the same applies to the terms agents, …nancial intermediaries and managers.

2.2.1

Reputational Herding Literature Scharfstein and Stein (1990) Scharfstein and Stein suggest that the presence of reputation in the delegated

portfolio management market could cause phenomena like herd behavior, distorting the investment decisions of agents. The basic argument is that when there is uncertainty regarding a FI’s skill he may choose to ignore his private information when taking investment

22 decisions and instead, he would imitate the decisions of other managers if investors doubt more the skill of a manager when he makes a bad choice, that is di¤erent from the one made by the rest of agents, as opposed to a bad choice that is similar to the one made by other agents. In fact, this is one of the explanations suggested by Dornbusch et al (2000) for the contagion observed during the Asian crisis: during this period there was and exit of mutual funds from the majority of emerging markets, even though these markets’fundamentals did not justify such a decision. In this model there is uncertainty regarding the managers’ true type: neither investors nor managers know if they are skilled or unskilled. Managers must decide between investing or not in a new technology for a …rm. It’s possible that the investment turns out to be bene…cial for the …rm, although it may result useless too. Once the investment decisions are made and their results are known, agents update their beliefs about the managers’type given the quality of the investment decision they made and also based on how di¤erent the decision is relative to the decision made by other managers. The managers’remuneration is assumed to be a positive function of his reputation. The authors make a critical assumption about the information received by managers. An unskilled manager receives no useful information about the convenience of adopting the new technology. On the other hand, skilled managers receive an informative signal useful to predict the investment’s payo¤. This information is correlated (e.g. skilled managers receive a noisy observation about the new technology’s real value). Therefore, if a manager makes a wrong investment decision and his decision is di¤erent from other managers, the principal will be inclined to think that he is unskilled (i.e. he received a signal

23 uncorrelated with the rest of managers which made him take a wrong decision that is di¤erent to the one taken by the rest of managers). On the other hand, if the agent takes a bad investment decision but he acts like the rest of managers, the principal will think that he is skilled (i.e. he received a signal correlated with that of other agents and therefore took a wrong investment decision) but was unlucky. This beliefs updating process will cause managers to ignore private information (which is available to them free of charge) useful to predict the investment’s returns and rather choose to copy the investment decisions of other agents since doing this enhances their reputation. The authors point out that herd behavior could be partially avoided if: managers’utility functions include the pro…ts of the …rms they manage; they have limited liability; their remuneration depends on their relative rather than absolute perceived ability; or alternative de…nitions of ability are used by investors. It is important to emphasize that the work by Scharfstein and Stein studies the behavior of …rm’s managers who must decide sequentially between undertaking an investment project or not3 , so it is possible that their conclusions are not entirely valid in a DPMP context. In particular there are two important considerations in this context. Namely, the investment opportunities prices are assumed to be exogenous by the authors, while in a portfolio management setup this isn’t necessarily the case (particularly for large investors). Also, portfolio managers may not be able to perfectly observe and copy other managers’ decisions as this implies knowing the share of the portfolio invested in hundreds of assets which is a di¢ cult task to achieve in real time. Regulation in the USA and Chile requires 3

Zwiebel (1995) develops a similar model, which studies the relationship between reputation and herd behavior among …rm’s managers who must decide whether to adopt an innovation with uncertain results.

24 some intermediaries such as mutual fund managers to report their portfolios’composition, but this is done with a lag of three months in the USA and several weeks in Chile. Avery and Chevalier (1999) This work builds on Scharfstein and Stein’s model in an attempt to rationalize the empirical …nding that herding and reputation are negatively related (see Chevalier and Ellison, 1999 and Hong et al, 2000). Unlike Scharfstein and Stein, the authors assume that managers learn about their abilities as they make investment decisions. In this case, once a manager learns enough about his type the herding equilibrium will give way to a signaling equilibrium in which managers who have learned enough about their type will take a di¤erent decision from the rest of managers in order to show their self-con…dence to the rest of agents. The reason for this is that if a manager receives a private signal suggesting to deviate, i.e. making a di¤erent decision from other managers, this will be taken by the rest of agents as a signal that he is a promising manager (the authors argue that this interpretation is the most intuitive one for out-of-herding-equilibrium beliefs). On the other hand, by herding the manager can hide the fact that he received a di¤erent signal and therefore either he or one of the other managers is unskilled. At the beginning of the manager’s career he knows little about his own type, so the incentives to herd will be stronger. However, as time goes by he will come to be more con…dent about his abilities (given that he is indeed skilled) and he will decide to take a contrarian behavior. It should be emphasized that both the herding and signaling equilibria are ine¢ cient in this model, since in both cases managers disregard private information and follow

25 others decisions in the former equilibrium or take the opposite decision in the latter. Graham (1999) Graham further builds on the Scharfstein and Stein’model, studying it’s comparative statics properties and empirical predictions. The author studies pure strategy Bayesian Nash equilibria in a sequential decision setup. Emphasis is placed on equilibria in which the …rst manager to make his investment decision chooses according to the private signal received, investing if he receives a bad signal and doing nothing if he receives a bad signal. The second manager, on the other hand, dismisses his private information when this suggests to make a di¤erent decision from the …rst manager. In other words, the second manager herds due to reputational concerns. The author demonstrates that there are parameter values for which these type of equilibria exist. The model has four key parameters. First, the intermediaries’ ability, which is measured as the probability of receiving a good (bad) signal given that the investment opportunity is good (bad) and given that the manager is skilled. The second parameter is the informative signal correlation, which is assumed to be strictly positive.4 Third is the managers’ initial reputation; and last is the strength or prior information or the ex-ante (unconditional) probability that the investment opportunity is attractive. The author studies how both managers’incentives are a¤ected by changes in the model’s parameters. Remarkably the e¤ects are di¤erent for both managers, even though they are both ex-ante alike and di¤er only in the order in which they decide. For the …rst manager ,incentives to make e¢ cient use of his private information are increasing in 4 As Scharfstein and Stein (1990) show, for reputational herding to arise it is crucial that skilled agents’ information is at least partially correlated. This is the reason that explains a smaller fall in reputation for managers or intermediaries that make bad investment decisions that are identical to that of other managers. Scharfstein and Stein assume that the skilled managers’information is perfectly correlated.

26 his ability since as his information is more accurate it is more likely that the investment outcome will be consistent with his private information. Also, if the correlation degree is higher this manager’s incentive to use his private information increase, since it’s likelier that the second manager will not make a di¤erent choice thus a¤ecting his reputation. If the manager’s initial reputation is higher his incentives to use his private information increase. This is so because the manager doesn’t know his true type but if initial reputation is higher he will be more con…dent that the signal received is indeed informative about the investment opportunity’s true value. Finally the incentives to use private information increase with the strength of the unconditional probability that the investment is attractive if this is consistent with the managers information and are lower otherwise. For the second manager, who observes the …rst manager’s decision before making his own, the incentives to use his private information also increase with his ability; with the informative signal’s correlation; and with the strength of the unconditional probability that the investment is attractive given that this is consistent with the managers information. However, the incentives to herd increase with the manager’s initial reputation. The reason for this is that this parameter determines the manager’s pro…ts if he herds, since in equilibrium the rest of agents will be aware of his behavior and therefore his investment decision won’t a¤ect his initial reputation. If initial reputation is low the manager’s pro…ts will be low in case he doesn’t make use of his private information. However, when this parameter is high the manager will prefer to abstain form making di¤erent investment decisions, as this may cause him to have a lower reputation therefore losing high pro…ts. Using data for investment news letters Graham …nds evidence consistent with the model’s predictions.

27 This work makes an important contribution as it makes a prediction regarding the relationship between a managers’reputation and his incentives to herd. We emphasize how this prediction is actually the opposite of that provided by Avery and Chevalier (1999). Dasgupta and Prat (2005) This work’s objective is to study the dynamics of assets’ prices in a delegated portfolio management context in a setup in which managers care about their reputation.5 . Speci…cally, the authors study an economy with multiple discrete time periods. There is an asset whose …nal liquidation value is unknown. There is also a large number of fund managers and noise traders or agents that take random decisions due to exogenous liquidity shocks. Every period a manager or noise trader is randomly selected to submit orders to a market maker to either buy or sell one unit of the asset. This market maker adjusts the price using all available public information. A manager may be one of two types. A skilled manager receives private information about the asset’s …nal value, while an unskilled manager receives less accurate information. Neither the manager nor investors are aware of the former’s type. After several periods have passed the asset’s true value is known. Investors update their beliefs about managers’abilities and each manager is paid. The remuneration scheme is assumed to be exogenous and is given by a weighted average between the trading pro…ts made by the manager and a reputational payo¤. Under this setup Dasgupta and Prat show that prices never converge to the asset’s true liquidation value. The reason for this is that as more information about the true value is gathered, incentives appear that make managers stop using their private information. 5

Other works that study asset pricing in a delegated portofolio managment context are Cuoco and Kaniel (2007) and Goldman and Slezak (2003). However, this authors do not explore the subject of reputational concerns.

28 First, as prices become more precise, there are less opportunities for managers to pro…t from trading. Also, when prices are close to their true value, managers who receive a signal that contradicts current beliefs (e.g. getting a good signal when the asset’s price is very low) may choose to ignore their private information because it they use it and the investment decision turns out to be bad the reputational cost will be high. This leads to a situation in which private information stops ‡owing in to the market and the asset’s true value in never fully known. However, as long as the asset’s price is not too low or too high, it is possible that managers trade on their private information, since there is no reputational cost associated to trading for intermediate asset prices. This …ndings are used to make predictions about the long term return for assets: if an asset has been persistently bought (sold) by managers then it is likely to experience negative (positive) corrections when uncertainty is resolved, leading to low (high) long-term returns. The authors report that these predictions are consistent with existing empirical evidence. Finally, it is shown that the model’s results are robust when signals are continuous rather than binary and if managers care about their relative reputation. However, the mispricing doesn’t survive if managers are aware of their type. Ottaviani and Sørensen (2006) Ottaviani and Sørensen study the incentives faced by an expert who cares about his reputation when he is evaluated on the basis of the advice given and the realized state of the world. The authors show that in equilibrium no more than two messages are reported by the expert, even though he has a continuum of possible messages to report. This means that the expert will only be able to truthfully report the direction of his private information, e.g.

29 the state will be "good" or it will be "bad", but he is unable to give information regarding how good or bad the state is. Also, in the long run there will be incomplete learning and herding. In this model there is an expert and an evaluator. The expert’s ability is unknown even by himself and he receives a private signal that is informative regarding the true state of the world. The authors generalize the commonly used binary signal (see for example Scharfstein and Stein, 1990 and Avery and Chevalier, 1999). The expert’s signal is assume to be a multiplicative linear experiment, this is, a mixture between an informative and uninformative experiment. The expert’s ability determines the probability that his signal is from the informative experiment. This allows to consider a continuum of states, signals and ability types in an analytically tractable way. Once the signal is observed the expert sends a message, chosen as to maximize his posterior reputation since it is assumed, as in Scharfstein and Stein (1990), that pro…ts are an increasing function of reputation. Once the expert sends the message and the true state of the world is revealed, the evaluator updates his prior beliefs about the expert’s ability and the expert is rewarded. Given the model’s structure the authors show that if the evaluator believes that the expert is truthfully reporting his signal the latter will have incentives to lie. In particular, if prior beliefs about the true state are biased towards a bad realization, then experts will want to report messages that are ex-ante likely to be similar to the prior. This is due to the fact that the evaluator will lower his assessment of the expert’s reputation if his reported message is di¤erent from the realized state. Therefore, regardless of the signal received, the expert’s reported message will be biased downwards. For similar reasons, when prior beliefs

30 about the true state are biased upwards, the expert will be over optimistic with his report. On the other hand, if the prior beliefs are in a middle range any message reported is likely to result in a positive or negative impact on the expert’s ex-post evaluation. Therefore, the expert will bias his report upwards or downwards depending on the message observed. Using the previous results the authors show that it is possible that no informative equilibria exist if priors about the state are too biased upwards or downwards. Moreover, when priors are not too biased informative equilibria exist but this will be binary. Moreover, the lower message sent by experts will be negative and the higher message will be positive. This means that the expert will be able to report the direction of his information, but not the intensity. The authors also …nd that there are cases in which experts with better initial reputation report messages that are less informative than experts with lower initial reputation. In this case it would be di¢ cult for monotonic reputational pro…ts to exist in a dynamic version of the model with more than two periods. Regarding the robustness of their results, the authors argue that all informative equilibria continue to be binary even if experts pro…ts are not only a¤ected by their reputation but also through their decisions, provided that reputational payo¤s are su¢ ciently important to experts. However, if experts can learn about their own ability there will always be informational equilibria. Finally, the model is extended to a multiple period setup in which di¤erent experts sequentially give their report about the same state of the world. The ordering is exogenous. After all experts give their messages the evaluator observes the true state and updates reputations. Unlike Scharfstein and Stein (1990), Ottaviani and Sørensen assume that experts’ private signals are independent, which means that each expert message carries

31 information about his ability but not about the ability of others. However, the authors show that once the beliefs about the true state become su¢ ciently concentrated experts are no longer informative and learning stops before the true state is revealed.

2.2.2

Non-Finance Reputation Literature There is a group of works that study the e¤ects of reputation, although not in a

delegated portfolio management context. The works by Hölmstrom (1999), Mailath and Samuelson (1998, 2001), Cripps, Mailath and Samuelson (2004) and Vial (2008) di¤er from the previously discussed papers because they recognize that investing and building a reputation is a slow process, and if there isn’t some source of permanent uncertainty about the FI’s characteristics the existence of a long-run equilibrium with investment in reputation wouldn’t be feasible. The reason for this is that investors would be eventually convinced that the FI is competent, so he would lose interest in making e¤ort to maintain his reputation, because investors will attribute a bad outcome to bad luck, rather than to the intermediary being negligent. This characteristic is not present in the existing models of reputation in a DPMP context reviewed in the Appendix, such as Heinkel and Stoughton (1994) and Farnsworth (2003). Hölmstrom (1999) Hölmstrom builds upon the argument of Fama (1980), who claims that in a relationship in which there is moral hazard, time should have a bene…cial impact since it allows to have more information about agents’behavior , thus allowing the principal to make more accurate inferences about agents’actions. This work’s contribution lays in that it studies under what circ*mstances the possibility of investing in reputation will have a long term

32 impact on agents’behavior. The author models an economy in which a manager sells labor services to a principal. The manager has the possibility of making costly e¤ort to help obtain higher output. However, it is not possible to write contracts contingent on results. Therefore, in a one period context it won’t be possible to induce the manager to make e¤ort. If there are multiple time periods the manager could be willing to make e¤ort if his present performance transmits information about future performance thus leading to higher wages being paid by the principal. Since the agent’s true ability is unknown, the principal will infer this variable through his observation of each period’s output. Hölmstrom shows that as time goes by, in the limit, the agent’s type will be fully known. In the meantime, the agent will make a lot of e¤ort (more than he would make in a …rst base case without moral hazard) as he will try to make a good impression in order to be paid higher wages. However, as his ability is revealed his actions will have little impact on the principal’s beliefs, so he will make no e¤ort and his labor supply will fall below that of a …rst base case. Therefore, in the long run there will be no role for the possibility of investing in reputation. However, if there is permanent uncertainty about the agent’s true ability, then his actions will always have an e¤ect on his reputation, thus a long run equilibrium with investment in reputation will be feasible. Hölmstrom assumes that the agent’s ability is not …xed, but rather changes over time. This modi…cation allows him to show that there will be a higher level of e¤ort in the long run compared to a case where ability is eventually fully known. Moreover, if the agent doesn’t discount utility from future periods, this e¤ort

33 level is e¢ cient, but if the agent has a discount factor of less than one, the long-run e¤ort level will be lower than the …rst-best level. Hölmstrom’s comparative statics results show that reputation will work more effectively if there is greater uncertainty about the agent’s ability and if the observations on results are less noisy. In these two cases learning will be faster and investing in reputation will be more pro…table. Also, the steady-state equilibrium will be stable, and will feature an overinvestment in labor supply for earlier periods since investing in reputation is more attractive when the principal’s information about ability is still obscure. Since early labor supply levels are higher than the …rst-base case this transition to the steady state will be ine¢ cient. Finally, even though the possibility of investing in reputation is bene…cial in terms of aligning incentives, the author points out that if there is little alignment between higher reputation and higher outputs, the fact that agents’care about their reputation may introduce ine¢ ciencies, since they could take actions to improve the principal’s opinion about their ability but this could be detrimental for output. One way in which this problem could be solved is by giving the agent some participation on output. Mailath and Samuelson (1998) Mailath and Samuelson study a model in which a …rm’s reputation is gradually built, can be managed, and slowly disappears if it isn’t maintained. There is a continuum of consumers buying an experience good from the …rm, whose type is unknown. There is a moral hazard problem since a skilled or competent …rm can incur in costly e¤ort in order to raise the probability that the consumers’receive high utility from buying the good.

34 Therefore, outcomes act as signals that help consumers infer whether the …rm is making e¤ort or not; however, they are imperfect signals since it’s possible that a …rm makes high e¤ort but bad luck causes the consumer to receive low utility. Also, each consumer’s experience is unique and unobservable to other consumers. This means that this is a model of imperfect private monitoring. The authors focus on this kind of models instead on those of perfect public monitoring (such as Fudenberg et al, 1990) and imperfect public monitoring (such as Abreu et al, 1990 and Fudenberg and Levine, 1992) because in this games the means by which incentives are aligned are trigger strategies. This is, since there is public monitoring, coordination between consumers is feasible. As a result, equilibria exist in which consumers initially believe that the …rm is competent and makes e¤ort and the will …rm behave accordingly if making e¤ort is not too costly and the …rm is patient enough. Any deviation on part of the …rm (intentional or not) triggers a punishment from all consumers, thus aligning incentives. This means that reputations spring to life and end suddenly, which prevents the study of a situation in which …rms gradually invest in reputation and this asset’s value slowly changes over time. Also, in previous reputation literature like Kreps et al (1982) and Kreps and Wilson (1982) there is a good or Stackelberg type of …rm who always chooses high e¤ort and ordinary …rms, who may choose to make e¤ort in order to make consumers believe that the …rm is good. Again, the existence of equilibria with e¤ort by ordinary …rms relays on the use of trigger strategies which means that reputations spring to life and may steady decline later on. Mailath and Samuelson make the assumption that there is an additional type of …rm: unskilled or inept ones, who never make e¤ort. Now an ordinary …rm makes e¤ort in order to make consumers believe that they are not bad …rms,

35 thus they must gradually build and then manage their reputation. An additional key ingredient in this model is the introduction of a permanent source of uncertainty about the …rm’s characteristics. In this sense, this work is related to Hölmstrom (1999). However, Mailath and Samuelson assume the existence of a continual possibility that an existing …rm might be replaced by a new …rm (consumers cannot observe when this replacement takes place). Also, Hölmstrom’s model is one of imperfect public monitoring where neither the market nor the agent himself knows his true ability. This means that the agent’s e¤ort cannot depend on his talent, so the agent’s evaluation about the pro…tability of e¤ort re‡ects only market beliefs and he doesn’t think that his e¤ort will a¤ect the average market beliefs about his ability. As stated before, the authors study an economy with a continuum of in…nitely lived consumers and a single …rm. The consumers buy an experience good from the …rm and may receive a high or low utility from this purchase. Each consumer’s experience is private information. The …rm may be competent or inept. A competent …rm can choose to make costly e¤ort in order to raise the probability of consumers obtaining a good outcome. The …rm knows its true type, but consumers can only infer it from the history of private results obtained when buying the good. Each period there is an exogenous probability that the current owner of the …rm is replaced and the new owner’s type will be competent with some positive exogenous probability. Consumers do not observe if a replacement occurred. This means that they will never be completely sure that a …rm is competent or incompetent, so there will always be incentives for the …rm to make e¤ort in order to invest in reputation. Like Hölmstrom (1999), the authors show that if the replacement probability is

36 zero, then the only possible long-run equilibrium features the …rm always making low e¤ort. The reason for this is that once consumers are convinced that the …rm is competent, the realization of bad outcomes will be attributed to bad luck rather than to negligence. This induces the …rm to stop making e¤ort since it’s reputation won’t be substantially a¤ected, thus destroying the reputational equilibrium. With a positive replacement probability bad (good) outcomes will always have a negative (positive) impact on …rm’s reputation and the authors show that if the cost of making e¤ort is not too high there will be an equilibrium in which the competent …rms always choose to make e¤ort. The authors then extend their analysis to consider the existence of multiple e¤ort levels. In particular, they study how the equilibrium properties change when …rms can make intermediate e¤ort levels, which improve their reputation but are ine¢ ciently low, and excessive e¤ort levels, which have high impact on their reputation but are ine¢ ciently high. The possibility of the …rm making high e¤ort is negatively a¤ected by the fact that this choice implies a resource expenditure now but only future rewards, since in this models the …rm’s actions a¤ect its pro…ts only by changing consumers’perception about its type. Therefore, if a …rm is to choose an ine¢ cient e¤ort level it’s likelier that it will involve too little e¤ort. Cripps, Mailath and Samuelson (2004) This work provides further results showing that in long-run equilibria with imperfect public monitoring it’s impossible for players to maintain a permanent reputation unless there is some mechanism by which the uncertainty about types is continually replenished, as in Hölmstrom (1999) and Mailath and Samuelson (1998). The authors argue that the

37 assumption of imperfect public monitoring is crucial for their results. The reason for this is that if monitoring was perfect then it is not di¢ cult to construct equilibria that exhibit permanent reputations. In this case, any deviation from the commitment strategy reveals the type of the de‡ector and triggers a punishment, which prevents the deviation from occurring. However, under imperfect monitoring, any deviation by the long run player doesn’t reveal his type nor triggers a punishment. Rather, as beliefs about the long-run player type converge over time, this guarantees that any deviation will have only small e¤ects on the short-lived players’beliefs. Therefore, there won’t be a cost from deviating for the long-run player and the …nal e¤ect of this situation will be to eliminate uncertainty from the equilibrium, thus revealing the long-run player’s true type. The authors prove this result under the use of simple Markov strategies and under more complicated commitment types. Mailath and Samuelson (2001) This work is similar to Mailath and Samuelson (1998). There is, however, a key di¤erence since the authors now turn to study the properties of a market for reputations. Each period there is an exogenous probability that the …rm exits the market. While the probability that the new …rm is competent is exogenous as in Mailath and Samuelson (1998) in this model potential entrants include both competent and incompetent …rms who compete to buy the right to use the existing …rm’s name and reputation. Therefore the authors are able to study what kind of …rms will buy which kind of reputations. In order to have a tractable model the authors assume that consumers’experiences after buying the good from the …rm are observed by all participants. This means that the model is one of public imperfect monitoring. Since this introduces the existence of multiple

38 equilibria, such as ones featuring trigger strategies which may depend on the history of consumers’results and where reputations are not gradually built, the authors require that all behavior is Markov, guaranteeing that all strategies are based only on history’s length and the current value of state variables. The rest of the model maintains the assumptions and structure of Mailath and Samuelson (1998). The authors show that the result of no reputational equilibrium in the long run in the absence of positive replacement probability continues to hold with public monitoring. When there is a positive replacement probability reputational equilibria are feasible. The authors assume that, when a …rm exogenously abandons the market, the owner sells the name to a new …rm. The potential entrants include both competent and incompetent …rms, where the former type is relatively scarce and has higher opportunity costs than the latter type. The exiting …rm sells the right to use its name using a second price auction, which guarantees that the …rm’s name is sold to the entrant with the highest valuation. If the cost of making e¤ort and the probability that there is a competent …rm with no opportunity costs amongst the potential entrants are small enough then the existence of a reputational equilibrium is feasible. That the cost of making e¤ort shouldn’t too high has a clear intuition. On the other hand, the requisite that there is a small probability that of one of the potential entrants is competent and has no opportunity cost is necessary to avoid situations in which, for some values of the current …rm’s reputation, consumers and potential entrants coordinate on an equilibrium in which entrants are likely (unlikely) to be competent because the value of a competent …rm is high (low), only because consumers expect entrants to be competent (incompetent). This would render meaningless the notion that higher

39 reputations are good and not just a product of the coordination between consumers’ and …rm’s beliefs. With regards to what kind of …rms buy which kind of names in a reputational equilibrium, the authors …nd that average reputations are likelier to be bought by competent …rms, while incompetent …rms are likelier to buy very good or very bad reputations. The intuition behind this result is that competent …rms …nd it too expensive to build a good name "from scratch" and, while getting a good name is attractive, they …nd it more convenient to buy a cheaper, more average reputation, and then make e¤ort to improve it. On the other hand, inept …rms won’t value average reputations too much since they don’t have the means to improve them. Very good names, however, are much more attractive since they can guarantee high pro…ts while slowly depleting the value of reputation. Finally, the authors discuss the implications of allowing …rms to announce consumers that a replacement has occurred. If this announcement is costless, it will be ignored by consumers, since they know that both competent and incompetent …rms with low reputations will be interested in announcing a change. On the other hand, if the announcement is costly (e.g. the …rm remodels or introduces a limited-time o¤er for costumers) then it’s possible that it modi…es consumers’beliefs. Speci…cally, if the …rm can choose how much to spend on sending a costly signal after the consumers’utility is received, but before the replacement is realized, then an equilibrium exists in which costly signals will be sent only by competent …rms. In this equilibrium competent …rms always make e¤ort. However, if bad luck causes the competent …rm’s reputation to fall below some critical value, then the signal is sent. This signal convinces consumers that the …rm has been replaced by a com-

40 petent …rm and thus they adjust their beliefs accordingly. Incompetent …rms don’t make e¤ort and eventually end up having the lowest reputation possible, but do not send a signal unless they are eventually replaced by a competent …rm. Vial (2008) This work studies the properties of reputational equilibria in an imperfect public monitoring context using a similar setup as Mailath and Samuelson (1998), (2001). A key di¤erence between this and previous works is that instead of the …rm being a monopoly, there is a continuum of …rms. This raises the issue of whether the existence of a competitive equilibrium can be reconciled with the fact that …rms investing in reputation should be able to charge higher prices for their goods or services. The author addresses this question and also studies the properties of the distribution of …rms reputations …nding that in the long run the aggregate distribution for reputations is constant even though the reputation of each particular …rm changes each period. This makes it possible to study the steady-state equilibrium of the model, where it’s feasible to analyze which consumers will buy from which …rms and how large is the improvement in pro…ts associates with having a higher reputation.6 In this model there is a continuum of short-lived consumers and a continuum of long-lived …rms, each capable of serving at most one consumer. Firms can be either competent or inept, with the former type being able to make e¤ort in order to improve the odds of its consumer having a good experience, while the latter doesn’t have this option. If a …rm makes e¤ort it ensures that the quality of the good provided is high. This is observable 6 Hörner (2002) also studies reputational equilibria with many …rms and consumers. However, he uses a di¤erent framework in which all …rms charge identical prices and consumers stop buying from a …rm after obtaining a poor result. In order to avoid losing consumers, competent …rms make e¤ort. Under this conditions all …rms share the same reputation.

41 only to the consumer. However, at the end of the period a public signal is observed by all agents. The chances of a …rm getting a high signal are increased if e¤ort was made. As in Mailath and Samuelson (1998) there is an exogenous replacement probability for …rms, which guarantees the existence of permanent uncertainty about the …rms’characteristics so long run reputational equilibria are feasible. A key element of the model is that consumers’willingness to pay for a …rm’s good is increasing in the …rm’s reputation, if they conjecture that such …rms make e¤ort. The author points out the existence of a low quality equilibrium in which no …rm makes e¤ort and consumers adjust their believes accordingly. Therefore, reputations are irrelevant since competent …rms o¤er the same quality as incompetent ones. This means that all …rms charge the same price and thus there are no incentives to invest in reputation. However, if the cost of making e¤ort is bounded from above, there is also an equilibrium in which all competent …rms make e¤ort. In this case Vial proves the existence of a steady state distribution of reputations for …rms. This is, even though each …rm’s reputations changes over time, improving after delivering a high-quality good and declining after bad results, the distribution of aggregate reputation for …rms, which evolves deterministically due to the continuum assumption, is invariant. This makes it possible to study a steady-state equilibrium, where prices and assignments are independent of time. In particular, since the quality of the good is appreciated by consumers, an interior solution to their decision problems requires prices to be increasing in reputation. This provides …rms with the incentives to invest in reputation. Moreover, if consumers are heterogenous in wealth and a personal attribute which negatively a¤ects the costs of providing the good, then there will be strat-

42 i…cation by wealth and personal attributes. More precisely, holding the personal attribute constant, richer consumers will be served by …rms with higher reputations. Also, holding wealth constant, consumers with higher endowment of the personal attribute will be served by …rms with higher reputations. Vial applies this …ndings to the schooling markets, where there is evidence of the existence of strati…cation by wealth and ability (if a student is more able then it’s cheaper for the school to educate him).

2.2.3

Summary As we have discussed, one of the reasons for intermediaries to herd that has received

attention in the literature is that of reputational concerns. The works by Scharfstein and Stein (1990), Avery and Chevalier (1999), Graham (1999) and Ottaviani and Sørensen (2006) show how intermediaries worried about their reputation may herd instead of using their private information. In fact, Dornbusch et al (2000) suggest that this could be one of the contagion mechanisms that operated during the Asian crisis. This view contrasts with that of Heinkel and Stoughton (1994), Chemmanur and Fulghieri (1994) and Farnsworth (2003), who argue that the presence of implicit incentives provided by reputation may alleviate the ine¢ ciencies caused by informational asymmetries even without the use of bonus of performance-based fees (we discuss these three papers in the Appendix). Also, the predictions about the relationship between reputation and incentives to herd are mixed; Avery and Chevalier (1999) predict a negative relationship while Graham (1999) makes the opposite prediction. There is also mixed evidence with Chevalier and Ellison (1999) and Hong et al (2000) validating the prediction by Avery and Chevalier (1999) and Graham (1999) presenting evidence supporting his own predictions.

43 Given the existent lack of consensus regarding the e¤ects of the possibility of investing in reputation in a delegated portfolio management context, we make a contribution by studying the relationship between reputation and herding in such a context, recognizing that investing in reputation is a slow process that takes place over several periods and that, absent some source of permanent uncertainty about the intermediaries’characteristics, steady-state reputational equilibria cannot exist. We thus follow the methodology developed by Mailath and Samuelson (1998), (2001) and Vial (2008), which hasn’t been applied before in a delegated portfolio management context with herding.

Chapter 3

The Relation Between Reputation and Herding in a Delegated Portfolio Management Problem Context

The fact that the delegated portfolio management remuneration schemes -at least in the mutual funds case- tend to be rather simple and do not exhibit sophisticated properties suggested by the literature leads us to explore the possibility that reputation building by …nancial intermediaries plays a key role as an incentive-aligning device that substitutes for more sophisticated remuneration schemes. Also, we seek to shed light on the e¤ects of reputation on agency problems in the existing delegated portfolio management problem literature. In this sense this work makes a contribution by studying the relationship between reputation and herd behavior, applying the methodology of reputation models such as Mailath and Samuelson (1998) and to Vial (2008). Our work is also related with those of Heinkel and Stoughton (1994), Arora and Ou-Yang (2001) and Farnsworth (2003) with regards to their basic argument that views reputation as an implicit incentive that can help to align incentives. In addition, like Scharfstein and Stein (1990), Avery and Chevalier (1999) and Graham (1999) we seek to unite the reputation and herd behavior literature, but in a delegated portfolio management context. In building our model we improve over some of the limitations of the existent

45 literature, which we discuss in the Appendix. For example, Arora and Ou-Yang (2001) assume the existence of a linear relationship between the agent’s future revenue and his present performance. While this may be consistent with the existence of reputation as a market mechanism to align incentives, it may also be product of pre-speci…ed contracts between investors and intermediaries. Moreover, investors beliefs are not explicitly modeled. On the other hand, the work by Heinkel and Stoughton (1994) does model investors’beliefs but makes extreme assumptions about how the negotiation power shifts form investors to intermediaries. Also, by modeling a two-period economy they overlook the steady state issues studied by Hölmstrom (1999) and Mailath and Samuelson (1998) and (2001). The work by Farnsworth (2003) makes exogenous assumptions about investors’pre-commitment to delegate more wealth to intermediaries’if they make good investment decisions and, like Heinkel and Stoughton, doesn’t study how the presence of permanent uncertainty about intermediaries’types a¤ects the strategies and nature of the long-run reputational equilibria. Like most of the reputational literature, we will refer to a …nancial intermediary’s reputation to the probability assigned by agents (investors and possibly other intermediaries) to the possibility that this intermediary is skilled1 . Furthermore, we will state that an intermediary presents herd behavior if he prefers to imitate the decisions of others instead of obtaining private information to make his investment decisions. We emphasize that in this sense our de…nition of herding is di¤erent from the traditional reputational herding literature such as Scharfstein and Stein (1990), Avery and Chevalier (1999) and Graham (1999) and is closer to the de…nition used by Calvo and Mendoza (2000), because in the former agents imitate the decisions of others even though they have free access to private 1

Later we will specify the characteristics of a skilled intermediary.

46 information. We predict the existence of a negative relationship between reputation and herding. This prediction is also made by Avery and Chevalier (1999). However, in their case an agent with good reputation not only doesn’t herd, but chooses a contrarian strategy, ignoring his private information and making the opposite decision from other agents in order to signal to principals that he is skilled. Of course, this behavior is ine¢ cient from the investors’ point of view. The work by Graham (1999) makes the opposite prediction: as the initial reputation of agents improves they will herd more because they want to avoid a large drop in pro…ts associated with a fall in reputation, which in this model occurs if an agent’s decision is di¤erent from those of other agents. However, in a long-run equilibrium the initial reputation of an intermediary may be of limited importance in terms of determining his reputation several periods ahead. Therefore, this work doesn’t focus on studying how an agent’s incentives to herd change as his reputation endogenously changes over time. We also show how the size of the percentage fee that must be paid to intermediaries in order to align incentives can be considerably smaller if investing in reputation is possible as opposed to a situation in which this isn’t feasible. Moreover, we illustrate how the possibility of investing in reputation can allow the delegated portfolio management market to operate when the use of more sophisticated remuneration schemes is not possible. Of course, there is a cost involved since in the reputational equilibrium the intermediaries’types are never revealed to investors. Therefore, it is possible that some skilled unlucky intermediaries are punished by investors through low fees while some lucky unskilled intermediaries may be paid high fees. Nevertheless, since skilled intermediaries who acquire information have

47 greater probability of making good investment decisions, this type of scenario is unlikely to occur. Additionally, we show that for a reputational equilibrium to be feasible, the gains from investing in reputation can either be obtained through higher fees or larger assets under management. In both these two cases intermediaries’expected pro…ts are increasing in their reputation.

3.1

A Static Model in a Risk Neutral Economy We proceed to model a static delegated portfolio management problem where

…nancial intermediaries cannot build a reputation. We explore what type of remuneration schemes would be needed in order for skilled intermediaries to separate out form unskilled ones. As we will show, a remuneration scheme with both a …xed monetary payment and a percentage of the …nal value of assets under management is needed for a separating equilibrium to be feasible. However, this type or scheme calls for a negative …xed pay (i.e. …nancial intermediaries pay investors in order to have the right to invest their wealth). This is not con…rmed by the stylized facts. We now proceed to describe the model’s setup.

3.1.1

The Economy

Financial Securities In the economy there is a risk-free asset, that pays rf for each unit of wealth invested and has a price qf . There is also a risky asset, which pays r for each unit of wealth invested. This return can take on two values: rG with probability

, or rB with

48 probability (1

), where rG > rB . The risky asset has a price of q. The life of both assets

lasts one period; their short sale is not allowed for investors nor for intermediaries; and their prices are exogenous, in the sense that all investors and intermediaries are price-takers and prices don’t reveal the intermediaries’ private information. If this assumption is not ful…lled it is possible that, in equilibrium, intermediaries would not be willing to obtain information, because they wouldn’t be able to bene…t from this action if their investment decisions reveal their information to the rest of intermediaries and investors through price changes in assets. This assumption could be justi…ed by the existence of unmodeled noise traders in the economy (on this see Grossman and Stiglitz, 1980). We make the following assumption: R

rG q

+ (1

rB q

)

rf qf

(3.1)

We note that if (3:1) holds, a risk-neutral investor will be indi¤erent (ex ante) between buying the risky or the risk-free asset. Note that rather than an assumption, (3:1) could be seen as an equilibrium condition for the prices of both assets in an economy in which investors are risk-neutral. Also, Equation (3:1) implies: rG q

rf qf

rB q

(3.2)

Therefore, in the good (bad) state the risky asset’s gross return is higher (lower) than that of the risk-free asset. This means that the agent’s optimal portfolio composition would be di¤erent if he knew which state was to materialize. We have assumed a Bernoulli distribution for the risky assets return. This is nonstandard in the DPMP literature, which tends to assume a normal distribution. However,

49 our choice allows us to have an analytically tractable multi-period model.2

Investors There is a continuum of risk neutral investors with measure 1, indexed by i. These investors live for one period. At the beginning of their lives, in period t, they are endowed with initial wealth Wt , identical for all i. Investors may choose to manage their own investment portfolio or they may choose to delegate the portfolio’s management to a FI. Also, let Rj represent the gross expected return for the portfolio managed by intermediary j. Therefore, if D represents the delegated wealth to a FI we have that the decision problem faced by investor i is: max E (Wt+1 ) D

Since investors are risk-neutral we have: 8 > > < Wt if Rj R D= > > : 0 if Rj < R

(3.3)

(3.4)

An investors delegates the management of all his wealth only if Rj is greater than the gross expected return that he would have if he managed the portfolio by himself, denoted by R. If this condition is not met, the amount of delegated wealth is 0. The expression for Rj is given by (3:50), whereas R is given by (3:1).

Financial Intermediaries There is a continuum of risk-neutral …nancial intermediaries, with measure 1, indexed by j. These …nancial intermediaries may be hired by investors to delegate their 2

We will return to this point in the dynamic version of the model.

50 wealth management. There are two types of intermediary. Skilled FI ( = s) may pay for information that is useful to predict the risky asset’s return. An unskilled FI ( = u) doesn’t have the ability to acquire information. Although the FI’s type is not known to investors, they know there is a mass

of skilled FI in the population. Additionally, the e¤ort made to obtain

information can only be observed by the FI that makes it. This investment has a cost c and allows the intermediary to receive a signal , which can take two values: good ( = bad ( =

B ).

That is to say,

B g,

where:3

Pr ( =

G jr

= r G ) = PG

Pr ( =

G jr

= r B ) = PB

(3.5)

Furthermore, PG > PB

(3.6)

Therefore, if intermediary j receives a good signal, he will revise upwards his previous estimate of the risky asset’s return, because the Bayesian update of r given

Pr (r = rG j =

PG + PB (1

is:

(3.7)

)

On the other hand, if he receives a bad signal, he will revise downwards his previous estimate, since: Pr (r = rG j = 3

(1 PG ) PG ) + (1 PB ) (1

)

(3.8)

The information structure se use di¤ers from standard DPMP models, such as Bhattacharya and P‡eiderer (1985) and is similar to the one used by Chemmanur and Fulghieri (1994). Also, note that we do not index the FI’s signal, because all skilled FI that gather information receive the same signal. In other words, skilled FI’s information is perfectlly correlated.

51 Based on the previous equations, we can see that the intermediary decides to invest all the portfolio in the risky asset if he receives a good signal, whereas if he receives a bad signal he decides to invest all in the risk-free asset. We can see that acquiring information is useful, because it is more likely to receive a good signal when the return of the asset is higher. Therefore, letting

represent the portfolio share that the skilled intermediary

invests in the risky asset when he acquires private information, the optimal investment strategy is given by: I

8 > > < 1 if

> > : 0 if

= =

(3.9)

Equation (3:9) incorporates the fact that short selling is not allowed. From the discussion above we can conclude that the gross expected return of a portfolio managed by a skilled FI that acquires information is given by: RI = PG

rG q

+ (1

rB q

) PB

This is so because with probability

+ ( (1

PG ) + (1

) (1

PB ))

rf qf

(3.10)

the risky asset has a high return. Nevertheless, this

asset will be bought only if the intermediary receives a good signal, which will happen with probability PG . In that case, the gross rate of return received by the investor will be the other hand, with probability (1

rG q .

) the risky asset’s return is low. Given this, there is

a probability PB that the intermediary receives a good signal and buys the risky assets so the return to the investor will be

rB q .

The third term corresponds to the probability that

the intermediary receives a bad signal and buys the risk-free asset, so gross return is

rf qf .

Using (3:1) it’s possible to rewrite the previous equation: Ri = R + (1

) (PH

PL )

rH rL q

(3.11)

52 In (3:11) we show how the expected return of the portfolio managed by a FI that acquires information is greater than the one of a portfolio managed without information (e.g. managed by the investor himself). Also, each …nancial intermediary has a …xed capacity to serve one investor. This assumption is consistent with the decentralization observed in the market of …nancial intermediation4 . Berk and Green (2004) develop a model in which the …nancial intermediaries (mutual funds) choose to limit the size of the portfolios they manage, because larger funds limit the intermediary’s capacity to generate above normal returns. In this sense, our assumption of …xed capacity is consistent with the work of these authors, but it is not totally equivalent, because Berk and Green’s limited capacity refers to the amount of assets under management, whereas in our model it is related to the mass of investors served.

The Herding Possibilities There is an exogenous probability

2 (0; 1) that any given intermediary succeeds

in copying the portfolio decision of an intermediary j 0 . That is to say, if j 0 buys the risky (risk-free) asset, with probability

the intermediary j will observe that j 0 bought the

risky (risk-free) asset, and with probability (1

) j will not be able to observe the other

manager’s portfolio. This assumption is introduced to consider, in a simple manner, the fact that in a DPMP context it is not possible to observe with certainty the decisions of other intermediaries.5 4

According to our own calculations, the Her…ndahl-Hirschman Index for the stocks category mutual fund market in Chile is 1,268, whereas for USA emerging markets category mutual fund it’s 1,262. According to the Department of Justice of the United States, markets with indices between 1,000 and 1,800 present a "moderate" degree of concentration. 5 An alternative assumption would be that an intermediary makes a noisy observation of the portfolio composition of another intermediary. However, this does not seem very suitable in our context of binary investment decisions.

53 Additionally, we will limit the number of intermediaries at which j can try to imitate in the following way: every period j can try to observe the portfolio of only one intermediary, and if j is competent, once he decides to imitate he cannot obtain private information during that time period. The fact that j can only try to observe the portfolio of one intermediary per period could be justi…ed by means of the existence of a cost of observing the portfolio of rivals. On the other hand, the assumption of the irreversibility in the decision to imitate for a competent intermediary could be due to the fact that once j decides to imitate, he doesn’t have enough time left to collect information on the risky asset. These two assumptions considerably simplify the model. Given that there is a continuum of intermediaries a FI that decides to imitate must also decide from whom to do so. Since we limit to one the number of intermediaries to imitate and since the type of each FI is known to the rest of intermediaries, the imitator will try to observe the portfolio of any given skilled FI. Finally, given (3:1), we will assume that, in the event that an intermediary doesn’t acquire information and isn’t able to observe the portfolio of a rival, he will randomize between investing his portfolio in the risky or risk-free asset with 50% probability.6 Therefore, following the same steps used to derive (3:11), the expected gross return of the portfolio managed by an intermediary who herds is: RH = R + In view of the fact that

) (PG

PB )

rG rB q

(3.12)

< 1 the expected return for the portfolio managed by a

skilled intermediary who acquires information will be greater than the expected return for 6 This assumption is important since the investment strategy of an uninformed intermediary will a¤ect the way in which investors update their beliefs regarding intermediaries’characteristics in the dynamic version of the model. We will return to this point in Section 3:2.

54 the portfolio managed by an imitator. Also, RH will be greater than the expected return for the portfolio managed by an uninformed FI that doesn’t imitate:

RH = RH

) (1

) (PG

PB ) (rG q rB ) > 0

(3.13)

(3.14)

This leads us to the following proposition.

Proposition 3.1 Whenever

2 (0; 1), for any linear remuneration scheme consisting of a

…xed monetary payment plus a percentage of the …nal value of the assets under management, a …nancial intermediary who does not acquire information will prefer to imitate the portfolio of another intermediary over randomly making his investment decisions.

Proof. For any given remuneration scheme investors’ willingness to pay will be increasing in his portfolio’s expected return. Therefore, a FI prefers to imitate rather than to invest without information if RH > R, which holds by Equation (3:14). It’s important to emphasize that, although the assumptions made about the portfolio imitation are ad hoc, they allow to have an analytically tractable model. In addition, the results obtained later are robust, in the sense that the only necessary condition for some FI to be willing to acquire information is that his choice can’t be perfectly imitated by other intermediaries. If this were true, the expected return of the portfolio that he manages will be greater than those of a FI that imitates. Otherwise investors would not be willing to pay more by their services and they would not be able to compete with imitators, who don’t incur in information acquisition costs.

55 The Remuneration Scheme We will assume that the FI’s revenue consists of a …xed monetary payment p plus a percentage p of the …nal value of assets under management, W . Therefore, the expected utility for an intermediary who acquires information will be given by the expected di¤erence between the revenue from fees and the investment costs.

3.1.2

Pooling Equilibria We begin by characterizing an equilibrium in which there is a single …xed payment

and percentage fee for both types of intermediaries. We study a situation in which investors o¤er the same contract to all intermediaries. If the intermediaries accept the o¤er they are hired and those who are skilled acquire private information to make portfolio decisions, while those that are unskilled copy the portfolio of some skilled intermediary. After that, assets’ return are realized and payo¤s are made to all participants. We emphasize that in order for an equilibrium to be feasible investors should not be tempted to hire an intermediary di¤erent from the one that is serving them. The feasible values for p and p must satisfy all agent’s participation constraints. Additionally, skilled FI should have incentives to acquire information. These constraints are given by: Wt+1 = (1 I

p) R W

= pRI W + p H

= pRH W + p

pRI W + p

0 0

pRH W + p

(3.15) (3.16) (3.17) (3.18)

56 where R = (RI

RH ) + RH

(3.19)

Equations (3:15) through (3:17) state that the contract o¤ered must be such that investors and FI (both skilled and unskilled) receive at least their reservation utility. Equation (3:18) states that skilled FI must be better of investing than herding, if this Equation does not hold, there won’t be spending in private information and therefore there will be no imitation either, and investors would be indi¤erent between delegating their wealth management or investing themselves. Figure (3:1) depicts the participation and incentive constraints and the combination of fees p and p that satisfy investors, skilled and unskilled FI’s participation constraints; and the incentive compatibility constraint. Since investors’ expected utility is higher for schemes closer to the origin, if there is competition among FI the equilibrium contract should be given by point A, where intermediaries’expected pro…ts are at their lowest. With this contract investors would have an expected utility of R W

c, while

both skilled and unskilled FI would have zero expected utility. We note that the o¤ered contract features a …xed payment p =

RH RI RH

c and a percentage fee

c W (RI RH ) .

focus on a case in which c is not too high relative to W so that this fee is strictly less than 1. The commission is an increasing function of the investment cost and of the probability of making a successful imitation, since if these parameters are higher, skilled FI would …nd it more attractive to herd. Also, the commission is a decreasing function of investors wealth, the quality of the information received by skilled FI (measured by (PG

PB )) and

of the di¤erence in gross expected return between the good and the bad state (measured

Investors participation

(Rθ − R ) Rθ

c W (RI − RH

)

Incentive compatibility

c WR I

Unskilled FI participation

p Skilled FI participation

Figure 3.1: Pooling Equilibrium

rG rB q

). An increase in any of these variables makes acquiring information more

attractive for skilled FI. However, a pooling equilibrium will not be feasible since skilled FI will have incentives to signal their type to investors it they can o¤er a contract lying on their participation constraint and that is to the left of point A in …gure (3:1). Note that such a contract, which features a higher percentage fee and a lower and negative …xed payment would not be attractive for unskilled FI since they would have negative expected utility.

3.1.3

Separating Equilibria Having established that a pooling equilibrium is not feasible we proceed to char-

acterize equilibria in which there is a menu of remuneration schemes for intermediaries to choose from. Again, contracts o¤ered must satisfy investors’ and FI’s participation con-

58 straints. Additionally, skilled FI must be given incentives to obtain information, and no type of FI should envy contracts designed for other types of FI. The constraints are given by: I Wt+1 = (1 H Wt+1 = (1

pI ) R I W pH ) R H W

= pI R I W + pI

pI pH c

= pH R H W + pH

pI R I W + pI

pH R H W + pH p I R I W + pI

(3.20)

(3.21)

(3.22)

(3.23)

pH R H W + pH

(3.24)

p I R H W + pI

(3.25)

pH R I W + pH

(3.26)

Equations (3:20) through (3:23) are participation constraints for investors and FI. Equation (3:24) states that skilled FI must be better o¤ investing than herding. If Equation (3:24) does not hold, there will be no spending in private information and therefore there will be no imitation either and investors would be indi¤erent between delegating their wealth management or investing themselves. Finally, equations (3:25) and (3:26) are self-selection or no-envy constraints which state that unskilled and skilled FI should prefer the contracts designed for they own types Figure (3:2) shows the participation and incentives constraints. Feasible contracts should o¤er skilled FI a fee pI of no less than

c W (RI RH )

so that they are willing to acquire

information. Also, the self-selection constraints imply that unskilled FI should be better o¤ by selecting the pair (pH ,pH ) than selecting the contract showed in point A. This is

p A

1 Investors’ utility (skilled)

Investors’utitlity (unskilled)

c W (RI − RH

)

B − RH W

Incentive compatibility Skilled FI participation

Unskilled FI participation

Figure 3.2: Separating Equilibria

accomplished by o¤ering the unskilled FI a contract that lies on their isopro…t curve passing trough point A. Therefore a contract o¤ered to skilled FI given by A and a contract o¤ered to unskilled FI given by zero …xed payment and zero fee (shown in point B in …gure (3:2)) satisfy the intermediaries’participation and self-selection constraints, while also giving incentives to skilled intermediaries to acquire information and make investors indifferent regarding which intermediary they hire. Thus, these contracts form part of a feasible separating equilibrium. Although there are in…nite pairs of contracts (pI ,pI ) (pH ,pH ) that could achieve a separating equilibrium, the pair proposed has the property of giving all FI their reservation utility, while also giving incentives to skilled FI to acquire information and satisfy the self-selection constraints. Moreover, the proposed contracts leave all investors with the same expected utility, whether they hire a skilled or unskilled FI. The contract o¤ered to unskilled FI features no …xed payment (pH = 0) and zero percentage commission (pH = 0). On the other hand, the contract o¤ered to skilled FI has

60 a …xed payment pI =

RH W and a percentage fee pI = 1. Given this, it is easy to see

that, in equilibrium, all investors will have expected utility of RH W . Unskilled FI’s will have zero utility, and skilled FI will have positive expected utility as long as the investment cost isn’t too large. Namely: Wt+1 = RH W H

= (RI

(3.27)

=0 RH ) W

(3.28) c

(3.29)

We note that, in order for the described separating equilibrium to exist, it must be true that pI < 0. If for some reason the FI are not allowed to make payments to investors in exchange for managing their portfolios, then the existence of a separating equilibria will not be possible. The reason for this is that for investors to be indi¤erent between both types of intermediaries and for the contracts to satisfy the self-selection constraints it must be true that pI = 1. But this implies that pI must be negative, since otherwise the investors’ participation constraints will not be satis…ed. We formalize this argument below. Proposition 3.2 For a separating equilibria to be feasible, skilled intermediaries must charge a negative …xed fee to investors. This is, pi must be negative. Proof. In a separating equilibrium it must be true that the self-selection constraints are satis…ed for both types of intermediary. Also, investors must be indi¤erent between both types of intermediary, since otherwise they would have incentives to outbid for the intermediaries they …nd more attractive. Additionally, the participation constraints of all market participants must be met, as well as the incentive compatibility constraint for skilled intermediaries.

61 We note that for an unskilled intermediary to (weakly) prefer the contract (pH ,pH ) rather than the contract (pI ,pI ) it must be true that their expected utility under the former contract is at least as high as he would get under the latter. This is:

pH R H W + pH pH

= pI R H W + p I = pI R H W

pH R H W + pI

(3.30)

On the other hand, for investors to be indi¤erent between both types of intermediary, it must be true that:

pH ) R H W

= (1

pI ) R I W

= (1

pH ) R H W

pI (1

pI ) R I W + pI

(3.31)

pI ) R I W + pI

making the replacement probability

[

;

+ (1

)]

[0; 1]. However, note that by

arbitrarily small all these intervals can be arbitrarily

similar.

It will be useful to consider that for policy functions in which the FI invests if his reputation is equal or greater than a critical level will look like the one in the Figure (3:9).

, and herds otherwise, the Function w

w(β = 0 )

µˆ

µ*

w(β = 1)

Figure 3.9: Function w(mu) for Policy Function

Change in Expected Future Utility This term is equal to: h

v ( ) = (PI

PH ) V

(d=g)

(d=b)

(3.83)

The …rst two terms are positive and their analytical expressions are simple.14 However, the last term depends on the value function V ( ), whose analytical form is not known. We will partially characterize this term. Assuming term

(d=g)

= 1 and since V is an increasing function of V

, we know that the

will be positive or equal to zero. The reason is that if

(d=b)

2 (0; 1), and if investors assign probability 1 to the skilled FI making e¤ort, reputation for the following period will always be greater if a good investment decision is taken (d = g) as opposed to a bad decision (d = b). Therefore, given the dynamic system that describes 14

These terms are given by: 1 2

) (1

) (2PG

1) > 0

v(β = 1) 0

1 µ

v(β = 0 )

Figure 3.10: Function v(mu)

the evolution of beliefs, V

(d=g)

(d=b)

> V , so

(d=b)

(d=g)

. If

(d=b) .

= 0 or

In addition, since V is increasing, we will have = 1, then

(d=g)

(d=b)

and V

(d=g)

( ) will be equal to 0 in these cases.

Below we graph v ( ) for the case in which investors assign probability 1 to the skilled FI acquiring information (

= 1) and for the case in which this probability is 0

( = 0). In the latter function v is equal to 0 for every .

As with function w, it will be useful to consider that for policy functions in which the FI invests if his reputation is equal or greater than a critical level

and imitates

otherwise, function v will look like Figure (3:11). It is important to note that v ( ) is a discontinuous function of

, due to the fact that the dynamic system describing the

evolution of beliefs is discontinuous for the previously described policy functions, which will

v(β = 1)

0 v(β = 0 )

1 µ

µ*

Figure 3.11: Function v(mu) for Policy Function

cause function p ( ) to be discontinuous. Although it is not trivial to predict the way in which v ( ) will be a¤ected by changes in

, the fact that it presents discontinuities will

not a¤ect the feasibility of the equilibria we study below.15

Equilibria with (almost) no Investment in Reputation Let us suppose that investors assign probability 0 to the skilled FI acquiring information. That is to say,

= 0 for

2 [0;

max ).

For this equilibria to be feasible it must

be true that no skilled FI prefers to acquire information, given the investors’beliefs. From …gure (3:12) we see that, if there is a positive cost of acquiring information, w will be negative and v will be equal to 0, so in equilibrium no skilled FI will obtain information, validating the investors’beliefs. 15

In the numerical examples section we will graph Function v ( ) for various parameters values.

v 0

Figure 3.12: Equilibrium with (almost) no Investment

The following lemma formalizes this …nding.

Lemma 3.4 Equilibria with almost no investment in reputation are feasible in the economy F; GS ; GU . Proof. Suppose investors assign ~ = 0 for all

2 [0;

max ).

From equations

(3:80) and (3:83) we know that w ( ) > v ( ), whenever ~ = 0 for . This means that the incentive compatibility constraint fails to hold and skilled intermediaries with reputation will choose to herd, validating investors beliefs.

Equilibria with Investment in Reputation We now explore a situation in which investors believe that all skilled FI acquire information and proceed to verify if these beliefs are consistent.

µ*

Figure 3.13: Equilibrium with Investment 1

Starting from a situation in which

= 1 for all values of

, we consider the

following policy function: a = I if

2 [0; 1]

(3.84)

According to (3:84) the FI would acquire information independent of its reputation level. However, observing …gure (3:13) we see that this policy function doesn’t constitute an equilibrium, because for reputation values

, FI would prefer not to acquire infor-

mation.

Based on these results we conjecture the existence of equilibria in which FI with high reputation will obtain information, whereas those of low reputation will herd. We have established that the existence of equilibria in which all FI invest in reputa-

µ’

Figure 3.14: Equilibrium with Investment 2

tion is not feasible even for investment cost values such that ^ 2 (0; 1). Now we demonstrate the feasibility of equilibria in which FI whose reputation is higher than a certain level acquire information. We assume that investors beliefs are such that

= 0 for every

and

otherwise. Furthermore, we propose the following policy function for skilled FI:

8 > > < I

> > : H if

(3.85)

From …gure (3:14) we can see that a FI with reputation greater than deed obtain information, whereas those whose reputations are smaller than Therefore, the proposed policy function is optimal, given the investors beliefs.

will in-

will herd.

96 There is a numerous set of this type of equilibria. Indeed, if we lower the value from which investors think that FI will invest in reputation we will …nd that FI with reputation greater than that level will indeed acquire information. In fact, the smaller level of reputation for which we can be sure that the previous conclusions holds is ^ , since we know that from this value onwards w will always be positive, whereas v is always negative or equal to 0. Therefore it should be possible to apply some re…nement criterion, such as evolutionary stable equilibria or the intuitive criterion to reduce the size of the equilibria set. It’s possible that intermediaries whose reputation are smaller than ^ choose to obtain information. This is clear from …gure (3:13). In this …gure we see that for values of between

and ^ function w is negative, but v is yet more negative, meaning that FI

with reputation located in this interval would prefer to acquire information rather than to imitate. Thus, if investors assign probability 1 to a FI with reputation between

and ^

acquiring information, the optimal policy function would be given by:

8 > > < I

> > : H if

(3.86) <

This is to say, for reputation levels greater than a critical level acquire information, whereas if reputation is below

they will imitate. The characteristics

of this type of equilibria depend on the form in which v changes if moves toward the horizontal axis when true that

skilled FI will

is modi…ed. Thus, if v

> 0, the critical level will increase but it must be

< ^ . On the other hand, if v moves towards the horizontal axis when

the critical level will fall. However we know that

> 0,

will never be greater than ^ , because

97 w is zero for this value, whereas v is always positive for

2 (0; 1). Also, we know that

will never be equal to 0 either because even if investors assign

= 1 for

= 0, function w

will be positive for such reputation, whereas function v will be equal to zero. Therefore it must be true that

2 (0; ^ ).

The following proposition proves the existence of equilibria in which skilled intermediaries with reputation

< ^ acquire information.

Proposition 3.4 In the economy F; GS ; GU

for suitable c values such that ^ 2 (0; 1),

there are equilibria in which skilled …nancial intermediaries with reputation

2 [0; ^ ) acquire

information.

Proof. If c is such that ^ 2 (0; 1) then we know that the intercept of function w ( ) will be positive. Moreover, for also know that, given ~ = 1, for all v ( ) = 0 when

< ^ , w ( ) > 0, while for

2 (0; 1) function v ( ) will be strictly positive, while

2 f0; 1g. This means that, given ~ = 1 for all , functions w ( ) and v ( )

will cross at least once in the (0; ^ ), interval. Let left of

< ^ , w ( ) < 0. We

, w ( ) < v ( ) and to the right of

denote the value of

such that to the

, w ( ) > v ( ). Additionally, note that by

construction w (^ ) < v (^ ). This means that for all reputation values above or equal to ^ the incentive compatibility constraint holds. Also, w (0) > v (0) so the incentive compatibility constraint doesn’t hold for this reputation value. This means that the reputation value for which functions w ( ) and v ( ) cross will be in the interval (0; ^ ). That is

2 (0; ^ ).

The existence and characterization of these steady state equilibria doesn’t depend on the existence of a stationary cdf for reputations since the price function p ( ) is such that investors are indi¤erent between intermediaries. Moreover, since investors are hom*ogeneous

98 in their initial wealth level intermediaries are indi¤erent between investors. This implies that the assignment rule is not determined in the model. Note that the equilibrium we study has the implication of a negative relationship between herding and reputation. This is, intermediaries with higher reputation are less prone to herd. This is in contrast with Graham (1999) who makes the opposite prediction. Graham argues that the reason for this is that if an intermediary’s initial reputation is high and he herds, the rest of agents will not revise their opinion regarding the intermediary’s type since they know that he herds and he will enjoy high pro…ts, which are a linear function of his reputation. In our model, on the other hand, even if an intermediary herds investors will revise upward the intermediary’s reputation due to the existence of a replacement probability. On the other hand, the dynamic system that describes the evolution or reputation shown in Figure (3:5) shows that the intermediary’s decision has low impact on his reputation when his initial reputation is low and when it’s high. Also, his remuneration will be low when

t 1

is low, and therefore he will herd. However, if

t 1

is high and the

intermediary herds he will forfeit expected return of a larger percentage of assets under management. In order to avoid this loss the intermediary with high reputation will choose to acquire private information instead of herding.

Non Monotonic Policy Functions Under certain conditions, the policy function given by (3:86) may be non-monotonic in the interval

2 [0; ^ ). This is shown in …gure (3:15).

v µ

’µ’ ’ ’ µ’µ’

Figure 3.15: Equilibrium with nonmonotonic Policy Function

In this case for very low values of reputation ( 2 [ 0;

00 ;

00 ],

000 )),

the FI imitates, whereas

he prefers to acquire information. Moreover, if his reputation increases ( he will imitate again.

In order to avoid this situation it’s necessary that the value function V doesn’t display slope changes that are too steep for values of

smaller than ^ . In terms of the price

function p ( ), this shouldn’t have slope changes that are too steep for reputation values smaller than ^ . This would be ful…lled if the value of the parameters PG and q ( ) are not too great (small).16 16

The second derivative of p ( ) is given by: @2p = @ 2

2 (RI RH )2 K ( (RI RH ) + RH )3

and it is a negative (positive) function of parameters PG and q ( ). Thus, for example, if we increase the value of PG the second derivative’s value falls, that is to say, it becomes more negative, and so function p ( ) becomes more concave. The concavity of p ( ) also depends negatively on the value of the integration constant K, which doesn’t have a readily interpretation as the parameters previously mentioned, but whose value is limited by RH since otherwise the fee charged by a FI with reputation 0, or that does not invest in

100 While we cannot rule out this type of situations from a theoretical perspective, the numerical exercises showed in the following Chapter suggest that this wouldn’t be a relevant case, at least for the set of parameters studied.

3.2.3

Importance of the Investment Cost As we discussed before, the cost of acquiring information plays an important part

in the determination of ^ . In addition, this cost will also a¤ect the value of

and the

feasibility of the policy function (3:86). If c increases, function w’s intercept will be lower, whereas v will change slightly. Therefore, to

K RH

[RI

will be greater. If c is 0, w will be equal

RH ] W . If this term is positive,

= 0. That is to say, all skilled

FI will obtain information since they obtain greater expected utility for period t and a greater probability of having a better reputation (thus charging higher fees) in the future. Therefore, it’s always possible to guarantee the existence of equilibria with spending in information if c is not too high.

3.2.4

Absorbing States The equilibria in which FI with reputation greater than a critical level obtain

information whereas those with lower reputation herd may not be sustainable if the dynamic system that describes the evolution of reputations presents absorbent states. Speci…cally, if the critical reputation These levels are called while

max

is too low or too high, the proposed equilibria will not be feasible. min

= .

information, would be negative.

and

max ,

and are shown in Figure (3:16). Note that

min

101

µt+1 1

45º

λθ + (1 − λ )

d=g

θ β=0

d=b

λθ ∗ µ max

∗ µ min

1 µt

Figure 3.16: Absorbing States

In the …rst case, if

min

once a FI leaves the "punishment zone" (i.e. the

zone where he earns a constant fee given by Equation (3:58)), he will never return to it, because even if he always makes bad investment decisions, his reputation will never be lower than

min

and therefore it will not fall below the critical level. In this case the equilibria

with policy function given by (3:86) will be irrelevant because, even though there is a punishment zone in which the fee falls, the FI will always acquire information regardless of his reputation. Therefore herd behavior will not exist among skilled FI. On the other hand, if

max ,

once a FI enters the punishment zone he never

leaves it because his reputation will evolve deterministically until it is equal to

max ,

but it

will never increase above this level, independent of the FI’s investment decisions. Therefore

102 the FI will never reach the zone where he prefers to invest in reputation instead of herding.

3.2.5

Reputation E¤ects under other Remuneration Schemes In this model, unlike Mailath and Samuelson (1998), (2001), (2004) and Vial

(2008), changes in the strategy of the FI will a¤ect not only their utility in future periods through changes in reputation (impact captured by v), but also their payments for period t, which is re‡ected in the term w. Indeed, if the intermediary decides to herd, it’s more probable that his reputation falls lowering his expected utility for future periods, and also his expected revenue for the present period is lower, since this is a percentage p ( ) of the …nal value of assets under management. If the FI’s revenue consisted of a …xed payment a ( ), the set of feasible equilibria would be di¤erent. In this case, it’s possible to demonstrate that FI’s revenue will be given by: a ( ) = (RI where

RH ) W +

(3.87)

is an integration constant. Under this scheme, if a FI herds this will a¤ect his

expected utility for future periods (the term v corresponding to this situation would still be given by (3:83)) but their revenue in period t would not be a¤ected; in fact, his expenses fall in c, so his utility for this period would increase. Now w would be equal to

c, and we

can no longer assure that a policy such as (3:86) is optimal. In the best case, as seen in …gure (3:17), we would have a situation in which the e¤ects of reputation on future utility are su¢ ciently important to guarantee the existence of a reputation interval for which the FI invest in reputation. Nevertheless, for values of

that are too low ( 2 [0;

0 ])

or too

103

c µ’

µ’ ’µ

Figure 3.17: Reputation E¤ects with Alternative Contract

high ( 2 [

00 ; 1])

the FI will herd.

On the other hand, if the remuneration scheme contemplates the payment of a percentage p ( ) of the initial value of assets under management, W , the results in terms of optimal strategies would also be di¤erent, because once again function ! would be equal to

c, so equilibria in which all FI of high reputation obtain information wouldn’t be feasi-

ble.17 This possibility is present in Farnsworth (2003) who argues that in this case explicit incentives would be needed to solve the moral hazard problem between intermediaries and investors. Nevertheless, it’s remarkable that the possibility of constructing a reputation substantially modi…es the FIs’ strategies and therefore, the type of feasible equilibria in this economy. Indeed, if it weren’t possible to construct reputation and investors assigned prob17 In the case of the Chilean mutual funds, the law establishes that the remuneration will be given by a percentage of the value of assets under management. This percentage is accrued daily.

104 ability

to any FI being skilled, function w would be equal to (RI

RH ) pW

c, whereas

v would be equal to 0 since the investment decisions would not modify investors’beliefs. In this case, depending on how expensive it is to acquire information, either all or none of the skilled FI would make this investment. Thus, for this market to exist contracts would have to include the use of explicit performance bonds or a more complex remuneration scheme would have to be used in order for skilled FI to obtain information.

3.2.6

On the Existence of Sticky Fees So far we have focused on a situation in which changes in reputation a¤ect …nancial

intermediaries’pro…ts through changes in the fees they charge to investors. However, even if fees do not change across intermediaries and/or over time, for instance due to the existence of price controls or menu costs, a reputational equilibrium may still be feasible if intermediaries with good reputation manage larger portfolios than intermediaries with poor reputation. In this case, investing in reputation would be worthwhile because it increases the expected future utility through increases in assets under management. Sirri and Tufano (2008) study how investors react to changes in mutual fund fees but do not provide evidence of whether there is substantial variation in fees charged by individual funds in their sample. ICI (2008) presents evidence of falling mutual fund fees over time but again, at an aggregate level. Even though we are not aware of any works studying the evolution of fees over time for mutual funds or other institutional investors, the available data suggests that fees do not present substantial variation. For example, Chilean mutual funds are forced by regulation to establish fees in their prospectuses which usually are changed every four to …ve years. This would suggest that fees remain …xed for substantial time periods. However, the fees

105 reported in the prospectuses are maximum fees to be charged by funds and not necessarily e¤ective ones. Therefore, this is a topic that requires further research. In order to asses the feasibility of reputational equilibria under sticky fees we modify our model by assuming that all intermediaries charge the same percentage of assets under management p. Now this will be a parameter rather than a function of intermediaries’ reputation. Also, we assume that investors are heterogeneous in their initial wealth level. Thus there is a distribution of wealth across investors, with cdf given by F (W ) which has a support W; W . We maintain the rest of our previous assumptions. ¯ We make the following proposition regarding the assignment rule in the economy with sticky fees and heterogeneous investors.

Proposition 3.5 In an economy with …xed fee p

R RH

and heterogenous initial wealth

across investors, if a stationary distribution G for reputations exist, then the assignment rule will be given by: F (W ( )) = G ( )

(3.88)

where F (W ( )) is the cdf of initial wealth across investors, whereas G ( ) is de cdf of intermediaries’reputation; each intermediary will serve one investor and the intermediaries as a whole will serve all investors. Furthermore, there will be positive matching between investors and intermediaries. This is, investors with high levels of wealth will be served by intermediaries with good reputation.

Proof. If fees are such that p

R RH

then investors will be willing to hire any

…nancial intermediary, even those who are known to herd in equilibrium. This can be seen

106 from the investors’participation constraint: (1

p) RH W p

RW 1

R RH

(3.89)

Therefore, in equilibrium intermediaries will all investors. Given the existence of capacity constraints, each intermediary will serve one investor. Also, investors won’t be indi¤erent between all intermediaries due to the fact that their expected utility is increasing in the intermediaries’ reputation and p is …xed. This means that there will be an excess demand for the services of the intermediary with the highest reputation, . However, this intermediary can only attend one investor which means that he will choose to manage the portfolio of the investor with the highest wealth endowment, W . Note that intermediaries will not be indi¤erent between investors. However, the intermediary with the second highest reputation will have to conform with attending the investor with the second highest wealth level. There is no way to persuade the investor with the highest wealth level to hire him since discounts can’t be o¤ered (recall that p is …xed). This is also true for the investor with the second highest level of wealth: he cannot persuade the intermediary with the highest reputation to work for him o¤ering him a higher fee since p is …xed. Therefore, the investor with the second highest level of wealth will hire the intermediary with the second highest reputation and so no. This means that the assignment rule will be given by equation (3:88). Finally, note that: @W f (W ) = >0 @ g( )

(3.90)

where f and g denote the pdf for investors’initial wealth level and intermediaries’ reputation, respectively. This is, there will be positive matching between investors and

107 intermediaries. In this case, the skilled intermediaries’Bellman equation will be given by:

V ( t ) = max

a2fI;Hg

pRI W ( )

c+ E V

ja = I ; pRH W ( ) + E V

t+1

ja = H (3.91)

This equation is very similar to equation (3:60), indeed, the set of feasible equilibria will resemble that of the case with hom*ogeneous investors and variable fees. In particular, functions w ( ) and v ( ) will now be given by:

w( ) = c

pW ( ) (RI

RH )

(3.92)

Without loss of generality we assume that the lower bound of the wealth distribution is equal to zero. Then we have that:

w (0) = c

w (1) = c ^=

(3.93)

pW (RI

RH )

c pW (RI

RH )

(3.94) (3.95)

For suitable investment cost c, it will be true that ^ 2 (0; 1). We plot function w ( ) in Figure (3:18). Note that unlike the case with ‡exible fees, function w doesn’t depend on the probability assigned by investors to the intermediary acquiring information. Therefore, w will be continuous regardless of the intermediary’s strategy. On the other hand, function v ( ) is still given by (3:83) and looks like the one in …gure (3:10). This is due to the fact that the dynamic system that describes the evolution

108

µˆ

Figure 3.18: Function w(mu) with Sticky Fee

of beliefs will present a discontinuity if the skilled intermediaries’ policy function is given by (3:86). Finally, note that all the arguments given for the case with ‡exible fees still apply when we have sticky fees, in the sense that for all reputation values for which investors assign = 0, intermediaries will …nd it optimal to herd, validating investors’beliefs. Also, there will be some value

2 (0; ^ ) such that, for reasonable investors’beliefs, intermediaries will

herd whenever their reputation is less than

and will acquire in information otherwise.

Figure (3:19) below shows this equilibrium. We have shown how a reputational equilibria may arise even if fees are …xed and equal for all intermediaries provided that investors are heterogeneous in their initial wealth level so that the reward for investing in reputation is managing larger funds thus obtaining higher expected utilities.

109

v µ*

Figure 3.19: Equilibrium with Investment and Sticky Fees

110

Chapter 4

Numerical Examples

We use a MATLAB routine, which is showed in the Appendix, to study the model’s comparative statics properties. Speci…cally, we are interested in establishing how herd behavior changes when the model’s parameters are modi…ed. We …rst focus on the case with ‡exible fees and later on the case with sticky fees. Later on we compare some of the equilibrium properties for the static version of the model with those of the ‡exible and sticky fee versions.

4.1

The Case with Flexible Fees We will explain the procedure to determine the value of

, the critical reputation

level for which skilled intermediaries are just indi¤erent between acquiring information and herding. The parameters for our baseline case are speci…ed below. The values chosen for the model’s parameters imply an ex-ante net expected return of 60% for both the risk free and risky asset. Also, the critical reputation values for which the model presents absorbing states are

min

= 0:1251 and

in equilibrium no skilled FI will herd, whereas if

max

= 0:5. This is, if

< 0:1251

> 0:5, once a skilled FI lands in the

"punishment zone" he will no longer acquire information. The implied investment costs for intermediaries are given by the ratio 1

c W

and is equal to 50 basis points1 . For the parameters

Wermers (2000) reports that mutual funds have mean investment costs of 100 basis points but we choose

111 Parameter PG c W

rG rB q

Description Successful imitation probability Information quality Information cost Assets under management Replacement probability Discount factor Risky asset’s return in good state Risky asset’s return in bad state Risky asset’s price Good state (unconditional) probability Mass of skilled FI

Value 0.45 0.85 3 600 0.15 0.75 2.2 1 1 0.5 0.5

Table 4.1: Flexible Fees Model’s Parameters: Baseline Case

chosen

= 0:1792. Also, we set K = RH , which implies that a FI that herds will charge

a fee of zero. Also, our parameters imply that ^ = 0:6638.

4.1.1

Determination of First, we plot the price function in Figure (4:1) and the beliefs updating rule in

Figure (4:2). In order to determine

we initially assume it’s value to be equal to zero. We

then proceed to obtain the optimal policy function given our guess. The MATLAB routine plots the functions w and v in Figure (4:3) and the value function V in Figure (4:4). We see that

= 0:1792 which is di¤erent from our initial guess. Therefore we

update our guess and determine that

is indeed equal to 0:1792. Also, we update our

plots to re‡ect this. The updated plots are shown in Figures (4:5) through (4:8). our values so as to maintain reasonable fees range and interesting

values.

112

Fee 600

500

Basis Points

400

300

200

100

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.1: Price Function: Baseline Case

Bayes Rule 1 0.9 0.8

Reputation in t+1

0.7 0.6 0.5 0.4 0.3 0.2 Good decision Bad decision 45º

0.1 0

0.1

0.2

0.3

0.4 0.5 0.6 Reputation in t

0.7

0.8

0.9

Figure 4.2: Reputation Evolution: Baseline Case

113

Functions w(mu) and v(mu) 4 w(mu) v(mu)

3.5 3 2.5 2

1.5 1 0.5 0 -0.5 -1 0

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.3: Functions w(mu) and v(mu): Baseline Case

Value Function 160

140

120

100

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.4: Value Function: Baseline Case

114

Fee 600

500

Basis Points

400

300

200

100

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.5: Price Function: Baseline Case with Updated Mu*

Bayes Rule 1 0.9 0.8

Reputation in t+1

0.7 0.6 0.5 0.4 0.3 0.2 Good decision Bad decision 45º

0.1 0

0.1

0.2

0.3

0.4 0.5 0.6 Reputation in t

0.7

0.8

0.9

Figure 4.6: Reputation Evolution: Baseline Case with Updated Mu*

115

Functions w(mu) and v(mu) w(mu) v(mu)

4 3.5 3 2.5

2 1.5 1 0.5 0 -0.5 -1 0

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.7: Functions w(mu) and v(mu): Baseline Case with Updated Mu*

Value Function 160

140

120

100

60 40

20 0

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.8: Value Function: Baseline Case with Updated Mu*

116

4.1.2

Comparative Statics The following expressions will be useful in explaining our results: R

RH = P

PI b

) (1

PH = =

1 2

1) (rG q rB ) > 0

) (2PG (1

) (2PG

(1 ) P (1 ( P + Ph ) (1 P

(4.1) (4.2)

) Ph )

(4.3)

Equation (4:1) shows the di¤erence in expected return between a portfolio managed by a skilled intermediary who acquires information and one managed by an intermediary who herds. Equation (4:2) shows the di¤erence in the probability of making a good investment decision between an intermediary who invests and one who herds. Finally, Equation (4:3) shows the di¤erence in next period’s reputation if the intermediary makes a good investment decision as opposed to a bad one. We now proceed to report the results of our comparative statics analysis, detailing how changes in the model’s parameters a¤ect the incentives faced by FI and how this, in turn, a¤ects the degree of herd behavior in the delegated portfolio market. Throughout our exercises we change the parameter values, thus a¤ecting

, however, we focus on equilibria

in which ^ 2 (0; 1), so functions w ( ) and v ( ) cross only once in the [0; 1] interval.2 Changes in the Probability of Successful Imitation An increase in

will have a negative e¤ect on

R. This, in turn, will lower

investors’ willingness to pay, causing a decrease in the price function p ( ). This means that skilled FI will …nd acquiring information less attractive since they would get a smaller 2 For all the following excercises was found using the iteration procedure described above. Although the numer of iterations needed to …nd the critical reputation value varied from case to case we always obtained convergence.

117

Mu* Comparative Statics: Eta 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 Eta

0.6

0.7

0.8

0.9

Figure 4.9: Mu* Comparative Statics: Eta

percentage of end-of-period assets under management. In terms of w ( ), this function will move towards the horizontal axis. On the other hand, the increase in

will also a¤ect

P , since now the probability

of making a good investment decision if a FI herds will be higher. Finally, be reduced because given a lower

will also

P , the result of the investment decision will be less

informative to investors in terms of guessing if a given type of decision was made by a skilled intermediary with information or by an intermediary who herd. These changes will a¤ect function v ( ), reducing its value, which will make herding more attractive than before in terms of the change in expected utility for future periods. Therefore, both in terms of this period and future periods’expected utility, skilled investors will …nd herding more attractive than before, resulting in an increase in shows how changes in

a¤ect

. Figure (4:9)

118 Changes in Information Quality An increase in PG will have a positive e¤ect on

R. This, in turn, will raise

investors’willingness to pay, causing an increase in p ( ). This means that skilled FI will …nd acquiring information more attractive since they will get a higher percentage of endof-period assets under management. In terms of w ( ), this function will move away from the horizontal axis. Also,

P will change since now the probability of making a good investment

decision if a FI invests will be higher. Finally, higher

will also be increased because given a

P , the result of the investment decision will be more informative to investors These changes will a¤ect function v ( ), rasing its value, which will make herding

less attractive than before in terms of the change in expected utility for future periods. Therefore, both in terms of this period and future periods’expected utility, skilled investors will …nd herding less attractive than before, resulting in a reduction in

. This is shown

in Figure (4:10). Changes in Information Cost An increase in c won’t a¤ect neither

R or p ( ). However, function w ( ) will

decrease, meaning that skilled FI will …nd acquiring information less attractive since it’s costlier to do so. Also, the increase in c won’t change

P or

. Nevertheless, function v ( ) will

move slightly towards the horizontal axis, re‡ecting the impact of higher investment costs on pro…ts. Therefore, both in terms of this period and future periods’expected utility, skilled

119

Mu* Comparative Statics: PH 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 PH

0.6

0.7

0.8

0.9

Figure 4.10: Mu* Comparative Statics: PH

investors will …nd herding more attractive than before, resulting in an increase in

. This

is shown in Figure (4:11) below. Changes in Assets Under Management An increase in W doesn’t a¤ect

R or p ( ). Function w ( ) will move towards

the horizontal axis since now if the skilled FI herds he will obtain some percentage p ( ) of a larger portfolio. For the same reason, function v ( ) will move away from the horizontal axis, even though

P and

haven’t changed. Therefore, both in terms of this period

and future periods’expected utility, skilled investors will …nd herding less attractive than before, resulting in a reduction in

, which is shown in Figure (4:12).

Changes in Replacement Probability If

increases, it is clear from equations (4:1) - (4:3) that the only impact on inter-

mediaries’behavior will come through a reduction in

. Indeed, with a higher replacement

120

Mu* Comparative Statics: c 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0 0.5

1.5

2.5

3.5

Figure 4.11: Mu* Comparative Statics: c

Mu* Comparative Statics: W 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

450

500

550

600 W

650

700

750

Figure 4.12: Mu* Comparative Statics: W

800

121

Mu* Comparative Statics: Lambda 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 0.6 Lambda

0.7

0.8

0.9

Figure 4.13: Mu* Comparative Statics: Lambda

probability, building a good reputation becomes less attractive, since it’s more likely that the intermediary won’t be in the market in future periods to bene…t from his investment. This e¤ect will cause function v ( ) to move towards the horizontal axis, increasing

, as

Figure (4:13) illustrates. Changes in Discount Factor If

increases then function v ( ) will be increased, since pro…ts in future periods

will have a higher present value. Therefore, in equilibrium there will be less herding, as Figure (4:14) shows. Changes in Good State Return An increase in rG will have a positive e¤ect on

R. This, in turn, will raise in-

vestors’willingness to pay, causing an increase in p ( ) and w ( ), which will make acquiring information more attractive.

122

Mu* Comparative Statics: Delta 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 Delta

0.6

0.7

0.8

0.9

Figure 4.14: Mu* Comparative Statics: Delta

Even though

P will not change, function v ( ) will increase, since pro…ts in future

periods will be higher due to the increase of the risky asset’s pay in the good states, which will also make herding less attractive. Figure (4:15) illustrates the results. Changes in Bad State Return Even though an increase in rB raises the risky asset’s ex-ante expected return, it decreases the attractiveness of acquiring information since the di¤erence in payment in the good and bad states is reduced. Therefore investors’willingness to pay, re‡ected in p ( ), is reduced, which has a positive e¤ect on w ( ), making herding more attractive. For the same reason function v ( ) will move towards the horizontal axis, even though haven’t changed. The result will be an increase in

P and

, which is showed in Figure (4:16).

Changes in Asset’s Price Our numerical simulations show that an increase in q has no signi…cant e¤ect

123

Mu* Comparative Statics: rH 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

2.2

2.4

2.6

2.8

3 rH

3.2

3.4

3.6

3.8

Figure 4.15: Mu* Comparative Statics: rH

Mu* Comparative Statics: rL 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.1

Figure 4.16: Mu* Comparative Statics: rL

124

Mu* Comparative Statics: q 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.2

0.4

0.6

0.8 q

1.2

1.4

Figure 4.17: Mu* Comparative Statics: q

on p ( ). However,

R will be reduced, which will increase w ( ), making herding more

attractive relative to acquiring information. The reduction in

R will also bring about a

reduction in v ( ), further increasing the incentives to herd. As a result herd behavior will be more frequent, as Figure (4:17) shows. Changes in States Probabilities In this case, the e¤ects on

are not straightforward to analyze. This can be seen

from (4:1) which shows that an increase in

will raise

R only if

e¤ect will be negative. The numerical simulations show that for low

1 2,

values, an increase in

this parameter will a¤ect negatively p ( ) (recall that this is a function of both However,

otherwise the

R and Rh ).

R will increase, o¤setting the …rst e¤ect and causing function w ( ) to move

towards the horizontal axis. For similar reasons, function v ( ) will be slightly increased. The result in this case will be a reduction in

. However, as

increases, both p ( ) and

125

Mu* Comparative Statics: pi 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 pi

0.6

0.7

0.8

0.9

Figure 4.18: Mu* Comparative Statics: pi

R will fall, therefore increasing function w ( ). Accordingly v ( ) will be reduced and the result will be an increase in

. Figure (4:18) summarizes these results.

Changes in Mass of Skilled Intermediaries A change in

will have no e¤ect on p ( ) or w ( ). Also, it won’t a¤ect the proba-

bility of making good investment decisions or the di¤erence in reputation between making good or bad investment decisions. However,

will a¤ect the intermediaries reputation’s ab-

solute value, thus a¤ecting his incentives through changes in v ( ). Our results suggest that the …nal e¤ect on

will be small and non-monotonous, as Figure (4:19) shows. However,

even if this parameter doesn’t have substantial direct e¤ects on the amount of herd behavior, its role is important since it determines the reputation values for which the dynamic system describing the evolution of beliefs presents absorbing states. Figure (4:20) shows how increases in

widen the range of reputation values for which our proposed equilibrium

126

Mu* Comparative Statics: Theta 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 Theta

0.6

0.7

0.8

0.9

Figure 4.19: Mu* Comparative Statics: Theta

is feasible. We summarize our …ndings regarding the e¤ects of changes in the model’s parameters in Table (4:2) below.

4.2

The Case with Sticky Fees Now we proceed to solve for

in the case where intermediaries charge a …xed fee

and investors di¤er in their initial wealth level. For the purposes of our example we assume that intermediaries’reputation follow an uniform distribution with support [

min ;

max ]

be determined below. Also we assume that investors’initial wealth level also has uniform distribution with supports [Wmin ; Wmax ] speci…ed below. The parameters for our baseline case are given in Table (4:3). We maintain or previous parameter values from the ‡exible fee case in order to

127

Mumin and Mumax Comparative Statics 1 Mumin Mumax

0.9 0.8 0.7

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 Theta

0.6

0.7

0.8

0.9

Figure 4.20: Mu*max and Mu*min Comparative Statics: Theta

Parameter PG c W

rG rB q

E¤ect on + + + + + Non-monotonous Non-monotonous

Table 4.2: Comparative Statics Summary: Flexible Fees

128 Parameter p PG c Wmin Wmax

rG rB q

Description Fee (basis points) Successful imitation probability Information quality Information cost Lower support W Upper support W Replacement probability Discount factor Risky asset’s return in good state Risky asset’s return in bad state Risky asset’s price Good state (unconditional) probability Mass of skilled FI

Value 330 0.45 0.85 3 100 1100 0.15 0.75 2.2 1 1 0.5 0.5

Table 4.3: Sticky Fees Model’s Parameters: Baseline Case

make both cases as comparable as possible. As before, the model’s parameters imply an ex-ante net expected return of 60% for both the risk free and risky asset. Also, the critical reputation values for which the model presents absorbing states are max

min

= 0:1251 and

= 0:5. The sticky fee p is set equal to 330 basis points. This means that for an

investor with initial wealth level of 600 that is served by an intermediary with average reputation ( = ) expected utility will be the same as for investors in the ‡exible fee case. For the parameters chosen the critical information is

4.2.1

value that determines whether FI herd or acquire

= 0:1862.

Determination of The procedure to determine

is the same as in the sticky fee case. The plots

also turn out to be very similar to the previous case; therefore we only show the graphs of functions w and v for the initial guess

= 0 and for the correct guess

= 0:1862. The

129

Functions w(mu) and v(mu) 4

w(mu) v(mu)

-1 0

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.21: Functions w(mu) and v(mu): Baseline Case with Sticky Fees

results are shown in Figures (4:21) and (4:22).

4.2.2

Comparative Statics The comparative statics exercises for the sticky fee case shows that

behaves

in a similar way to the ‡exible fees case. The results suggests that herding increases with investment costs; the probability of successful imitation; the risky asset’s payment on the bad state; and the risky asset’s price. On the other hand, herding decreases with information accuracy; the discount factor and the risky asset’s return in the good state. However, we obtain somewhat di¤erent results for the rest of parameters. Namely, the replacement probability, the good state unconditional probability and the mass of skilled intermediaries. For the replacement probability we …nd a non-monotonous e¤ect: increases in this parameter lead to an increase in further increases reduce

. However, for high replacement probabilities (more than 80%) . This is shown in Figure (4:23). In the case of the good state

130

Functions w(mu) and v(mu) 4

w(mu) v(mu)

-1 0

0.1

0.2

0.3

0.4

0.5 0.6 Reputation

0.7

0.8

0.9

Figure 4.22: Functions w(mu) and v(mu): Updated mu* with Sticky Fees

unconditional probability, for the range of parameters used we …nd a negative e¤ect of increases in

. But as Figure (4:24) shows, the e¤ect becomes very small as

increases. Finally, for the mass of skilled intermediaries the results showed in Figure (4:25) suggest a positive, although weak relationship between

4.3

and

The Dynamic versus the Static Equilibria In our static version of this economy we demonstrated that, if a separating equi-

libria exists, then investors’expected utility will be given by:

Wt+1 = RH W

(4.4)

We also noted that, for a separating equilibria to be feasible, it was necessary that skilled intermediaries charged a negative …xed fee to investors in order for them to credibly "reveal" their type. In this way, skilled intermediaries o¤ered a …xed monetary payment to investors

131

Mu* Comparative Statics: Lambda 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 0.6 Lambda

0.7

0.8

0.9

Figure 4.23: Mu* Comparative Statics with Sticky Fees: Lambda

Mu* Comparative Statics: pi 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 pi

0.6

0.7

0.8

0.9

Figure 4.24: Mu* Comparative Statics with Sticky Fees: pi

132

Mu* Comparative Statics: Theta 1 0.9 0.8 0.7

Mu*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.1

0.2

0.3

0.4

0.5 Theta

0.6

0.7

0.8

0.9

Figure 4.25: Mu* Comparative Statics with Sticky Fees: Theta

and in return kept all the portfolios returns, i.e. the charged percentage fee was 100%. An unskilled intermediary would not bene…t form o¤ering such a contract, since he has lower probability of making good investment decisions and therefore he would have negative expected utility under such an arrangement. However, if this type of contracts, which contemplate the use of negative monetary payments is not available, for example due to liquidity constraints for intermediaries or due to legal restrictions, the separating equilibrium would unravel. An additional reason for remuneration schemes that present a negative …xed payment to be prohibited could be the existence of limited liability as in Dow and Gorton (1997) and Bhattacharya (1999). These authors emphasize how limited liability prevents investors from screening skilled and unskilled intermediaries. In this sense reputation could be seen as a mechanism to allow portfolio delegation to take place even without perfect screening.

133 The stylized facts suggest that the type of remuneration schemes just described isn’t used by …nancial intermediaries (at least by mutual funds). Therefore, an additional mechanism may be in place so as to make some type of equilibria feasible in the delegated portfolio management market. It turns out that in a dynamic equilibrium of the model where it is possible for …nancial intermediaries to build a reputation, a partially separating equilibria is feasible. Indeed, investors are never completely sure about the type of intermediary they hire. However, skilled intermediaries with reputation higher than a critical value will obtain information. All that is needed for this is the use of a contract featuring a percentage fee of the …nal value of assets under management. This fee is given by:

p( ) = 1

(RI

K RH ) + RH

(4.5)

Moreover, if K = RH , which implies that an intermediary who doesn’t acquire information (or one with reputation equal to zero) will charge a fee p = 0; then investors’ expected utility will be equal to RH W , just as in the separating static equilibrium. In this sense the possibility of investing in reputation acts as a substitute for the use of more sophisticated remuneration schemes. Comparing expected utility for intermediaries between the static and dynamic equilibria is not straightforward since we have to account for the replacement probability , which is not present in the former. Moreover, we would also have to compute the numerical value of the Bellman equation for intermediaries in the dynamic model which was made using numerical examples. In our baseline case RH = 1:6945 and W = 600, implying that investors’expected utility will be equal to 1016:7 both in the static and dynamic equilibria. Also, for a skilled

134 intermediary the present value expected utility in the static case equals 116:6, while for a unskilled intermediary it is 0.3 In the dynamic case, a skilled intermediary with average reputation (

= 0:5) has expected utility of 95:4, while an unskilled intermediary’s

present value of expected utility is strictly positive since p (0) = 0. We highlight the fact that, given the parameters’ values, in the static case, for skilled investors to be willing to obtain information, the percentage fee of assets under management paid must be no less than

c W (RI RH ) .

This turns out to be 433 basis points.

On the other hand, in the dynamic case with investment in reputation and ‡exible fees, paid fees will vary between 121:3 basis points for an intermediary in the brink of falling in the "punishment zone" (i.e. an intermediary whose reputation is

= 0:1792) and 486:8

basis points for an intermediary with the highest (theoretically) possible reputation (which is equal to 0:75 for the baseline case). For an intermediary with average reputation (equal to

= 0:5) the fee paid is 329:9 basis points. Therefore we see that a skilled intermediary will acquire information if he is paid

121:3 basis points of assets under management, which is considerably less than the fee that he would have to receive in the static case, i.e. a situation in which investing in reputation is not possible. In fact, the minimum required fee in the static case is 3.5 times the minimum required fee in the reputational equilibrium. Of course, investors are no better in the dynamic scenario, since they don’t receive a …xed payment from intermediaries in return from managing their portfolios. However, it’s remarkable that the possibility of building a reputation makes aligning incentives "cheaper" in the sense of requiring remuneration schemes 3

In order to calculate the expected present value for the skilled intermediaries we use the adjusted discount factor , which takes into account both the intermediaries’discount factor and the replacement probability .

135 that require less power (i.e. a lower share of assets under management is paid to investors) and are far simpler (i.e. they don’t require negative …xed monetary payments to intermediaries). Nevertheless, there is a cost involved, since in the reputational equilibria a complete separation between types isn’t achieved, hence some skilled (unskilled) intermediaries are paid less (more) than they should; and also some skilled intermediaries herd whereas in the static equilibrium they always acquire information. Mailath and Samuelson (1998) also point out that the use of implicit incentives such as reputation may be a cheaper way to align incentives instead of …rms o¤ering guarantees to compensate consumers receiving bad outcomes or poor service. They argue that this could be prohibitively expensive if exerting high e¤ort didn’t always guarantee a good result. This argument is particularly valid in a delegated portfolio management context in which the …nal value of an investment portfolio is beyond the …rm’s control. In the case of the sticky fee equilibrium, fees are …xed at 330 basis points. This implies that an investor with initial wealth equal to 600 and that is served by an intermediary with average reputation (equal to 0:5) will have an expected utility of 1016:7. Also, a skilled intermediary with average reputation has an expected utility equal to 90:8, while an unskilled intermediary’s present value of expected utility will be positive. We note that in this case intermediaries with reputation as low as 0:1862 acquire information. Such intermediaries manage portfolios with an initial value of 176:3. In the static equilibrium, if fees were …xed at 330 basis points, the minimum amount of assets under management that a skilled intermediary must receive in order to obtain information is equal to 788:1. Therefore, skilled intermediaries must have 4:5 times more assets under management in the

136 Type of Equilibrium Investors’Expected Utility Skilled FI’s Expected Utility Unskilled FI’s Expected Utility Minimum required fee Ratio of minimum fees Minimum required AUM Ratio of minimum AUM

Static 1016:7 116:6 0 433 1 788:1 1

Rep. w/‡exible fees 1016:7 95:4 >0 121:3 3:6 NA NA

Rep. w/sticky fees 1016:7 90:8 >0 NA NA 176:3 4:5

Table 4.4: Summary: The Static Equilibrium versus Reputational Equilibria

static case compared to the reputational equilibrium with sticky fees in order to be willing to acquire information. Table 4.4 summarizes the comparison between the static and the reputational equilibria with ‡exible and sticky fees.

137

Chapter 5

Empirical Discussion

In this Chapter we summarize the models’ empirical predictions. We discuss to what extent these predictions are supported by previous evidence and point out areas in which further empirical research is needed. Then we propose an estimation strategy in order to obtain further evidence to evaluate the validity of our predictions.

5.1

Empirical Predictions Based on the theoretical results and numerical exercises obtained in the previous

Sections we obtain the following empirical predictions:

Prediction 1. There will be a negative relationship between imitation or herd behavior and the FIs’reputation. Prediction 2. There will be cross-sectional dispersion in fees charged by the FI and/or the amount of their AUM. Prediction 3. There will be a positive relationship between the FIs’ fees/AUM and his reputation. Prediction 4. There will be variation in the time series of fees charged by any given FI and/or the amount of their AUM.

138 Prediction 5. Herding increases with higher investment costs. Prediction 6. Herding increases when imitation is easier. Prediction 7. Herding increases with higher replacement or exit probabilities. Prediction 8. Herding decreases with more accurate private information. Prediction 9. Herding decreases with more uncertainty about the value of assets. Prediction 10. Herding decreases when intermediaries are more patient. Prediction 11. An increase in the mass of skilled intermediaries in the market will have a nonnegative e¤ect on herding.

Regarding the …rst prediction, the work by Chevallier and Ellison (1999) studies the USA mutual fund managers market. Using age of manager as a proxy for reputation, the authors …nd that older managers tend to imitate less the decisions of the rest. As theoretical support for this …nding the authors cite the work by Scharfstein and Stein (1990). However, this work simply postulates the existence of a relationship between herd behavior and reputation, whereas our work predicts that this relation will be negative (i.e. a greater the reputation, lowers the probability of incurring herd behavior). In a similar line, Hong et al (2000) …nd a negative relationship between herd behavior amongst …nancial analysts and their age (proxy for reputation). We emphasize how Graham (1999) presents evidence contradicting this prediction for a sample of investment newsletters’recommendations. Hortaçsu and Syverson (2004) study the dispersion in fees charged by mutual funds in the USA. The authors …nd evidence of substantial dispersion in fees even in closely de…ned

139 categories (e.g. mutual funds that replicate the S&P 500 index). The authors …nd that these di¤erences can be explained partially by the existence of search costs for investors. Although this study’s objective is not to determine if reputation plays a part in explaining the existing dispersion, it’s suggestive that one of the authors’…ndings is that investors are willing to pay larger fees to older mutual funds (age is a commonly used proxy for reputation). It’s worth emphasizing that our prediction of dispersion in fees charged is not shared by the model of Heinkel and Stoughton (1994), who, in fact, make the opposite prediction, sustaining it with evidence provided by Lakonishok et al (1992). However, see Khorana, Servaes and Tufano (2008), who …nd evidence that suggests the existence of considerable dispersion in fees charged by mutual funds world-wide. Regarding the existence of cross-sectional dispersion in AUM, Sirri and Tufano (1998) provide some supportive evidence for this prediction; the mutual funds in their sample have a mean value of AUM of $588.2 millions with a rather large standard deviation of $2267.7 millions. However, Sirri and Tufano’s focus isn’t to relate the size of AUM with mutual funds’reputation. On the other hand, the work by Carter and Manaster (1990) presents preliminary evidence suggesting the existence of a positive relationship between the fees charged by investment banks when making IPOs and their reputation. The measure of reputation used in this case is built on the basis of the investment bank’s relative position within publicity made to advertise the IPO, since generally there are several banks participating simultaneously and it’s presumed that the most prestigious one occupies the …rst place in the list.

140 Concerning the prediction of AUM dispersion over time, in the stylized facts Section of Chapter 2 we already discussed some works such as Chevalier and Ellison (1997), and Sirri and Tufano (1998), who document how the size of mutual funds varies over time. While these authors show that the relationship between ‡ows and performance is asymmetric and less steep for more experienced funds, our prediction simply states that as intermediaries’ reputation changes over time the size of AUM should also change. In our model, whether these changes are of bigger magnitude depends on the change in reputation resulting from making good or bad investment decisions. In particular, if the improvement in reputation resulting from good decisions is higher than the fall resulting from bad decisions, then there will be an asymmetric impact in the change of AUM. However, we cannot guarantee that this occurs for all parameter con…gurations. The work by Graham (1999) …nds supportive evidence for our 8th and 9th predictions. Namely, as the quality of intermediaries’ information improves, herding should decrease as investing in reputation becomes more attractive. Also, when there is increased uncertainty about assets’value having private information is more useful, therefore reducing herding.1 Regarding the relationship between reputation and herding, we …nd theoretical support for Graham’s evidence and are able to reconcile his …ndings with our …rst empirical prediction. Indeed, Graham’s evidence suggests that as a manager’s initial reputation increases he will tend to herd more. However, as time goes by, the agents’reputation will endogenously evolve and change, and as it gets better, his incentives to herd will weaken. Therefore, prediction number 1 should be interpreted as relating herding with the agent’s 1

In our setup it’s di¢ cult to disentangle the e¤ects of changes in assets’ unconditional expected return and variance, since for a Bernoulli distribution this two variables are intrinsecally correlated. Therefore we state our prediction in terms of assets’volatility instead of expected return, as done by Graham (1999).

141 current reputation, whilst prediction 11 relates herding with the agents’initial reputation. Lastly, according to our knowledge no systematic empirical evidence exists regarding the existence of a positive relationship between reputation and fees/AUM for …nancial intermediaries who manage portfolios such as mutual funds. Also, there is lack of evidence regarding the existence of time-series dispersion for …nancial intermediaries’ charged fees; ICI (2008) presents evidence of falling mutual fund fees over time but at an aggregate level. Also, even though we are not aware of any works studying the evolution of fees over time for mutual funds or other institutional investors, the existing data seems to suggest that fees do not present substantial variation. For example, Chilean mutual funds are forced by regulation to establish fees in their prospectuses which usually are changed every four to …ve years. This would suggest that fees remain …xed for substantial time periods. However, the fees reported in the prospectuses are maximum fees to be charged by funds and not e¤ective fees2 . Also, there is further empirical work needed to asses how herding changes when acquiring private information is costlier; when imitation is easier; and when intermediaries are more patient due to higher discount factors or lower exit probabilities.

5.2

Empirical Strategy This Section discusses an empirical methodology to obtain new evidence regard-

ing the relationship between reputation and herding in delegated portfolio management markets. We will propose variables to proxy for …nancial intermediaries’ reputation; the extent to which they herd; as well as information regarding investment costs, assets’return uncertainty, successful imitation probability, replacement or exit probability and discount 2

For evidence on fee-waiving by money managers see Chirsto¤ersen (2001).

142 factor. We will …rst discuss how these variables could be proxied and then we brie‡y sketch an empirical strategy to …nd evidence which allows us to validate or reject our models’ predictions.

5.2.1

The Variables

Reputation Usually, reputation is proxied by a manager’s age. This variable is used for example by Chevalier and Ellison (1999) and Arora and Ou-Yang (2001). The use of this variable is justi…ed by arguing that it closely proxies a manager’s stage in his career and the amount of information the market has gathered about his skills. In this same spirit, Hong et al (2000) proxy analyst’s reputation by their experience. One disadvantage of these proxies is that it’s possible for low skilled managers to keep their jobs for long time periods, for instance alternating between good and bad investment decisions. According to our model this kind of intermediary should have lower reputation than, say one that keeps making good investment decisions all periods. However, if tenure or age is used as a proxy, then both managers will seem to have the same reputation. One way to avoid this could be the use of cumulative returns to proxy for reputation. This measure is used by Ippolito (1992) as a measure of quality for mutual funds. The model predicts that, on average, managers investing in private information should make better investment decisions than those who herd, thus obtaining better reputation. Of course, it’s possible that in any given time period a lucky manager who herds makes a larger return than one who acquires information. However, over larger time intervals this should not be

143 the case. The use of cumulative returns, for example for the last 3 or 5 years, could then be a reasonable proxy for reputation. In the spirit of Graham (1999) it could even be possible to construct a synthetic variable by de…ning that a manager has made a good investment decision if his portfolio return on this period exceeded some threshold, such as a stock index or the sample’s mean return. This kind of proxies have the advantage of improving only if the manager shows a good performance.

Herding One of the statistical measures of herding used by numerous authors is the Lakonishok, Shleifer and Vishny (LSV) measure, presented in Lakonishok, Shleifer and Vishny (1992b). This measure tries to identify the degree of correlation in trading patterns for a group of traders, such as pension or mutual funds, assessing if they tend to either buy or sell some particular stock in groups. If there is no herding between investors one could expect that trading is independent between them so, as the number of investor increases, the number of sellers of the stock should be roughly equal to the number of buyers. In this case the value of the LSV measure will be close to zero; otherwise the results are interpreted as suggesting the existence of herding. Bikhchandani and Sharma (2001) further elaborate on this measure and possible improvements. In our case the LSV measure won’t be of much use since we are interested in assessing the degree of herding at the intermediary level and not only …nding evidence of the existence of herding among a group of intermediaries. Graham (1999) studies the recommendations of Investment Newsletter which consist in advising to increase or decrease the portfolio weighting of a certain stock. The author argues that the Value Line Investment Survey is the best known investment newsletter,

144 whose advice is freely observable to investors. Moreover, this Survey has been well studied by the literature and would have a high stature. Therefore, it’s assumed that this Survey has no reputational concerns and acts as the “market leader”, which means that other newsletters may want to imitate its advice. Thus herding is modeled as a dummy variable taking the value of one if a newsletter makes the same recommendation as the leader and zero otherwise. Chevalier and Ellison (1999) use three distinct measures to evaluate the degree in which a mutual fund manager’s actions di¤er from the average. The …rst variable measures boldness in the sense of a manager having concentrated his portfolio in sectors that di¤er from those that are more popular at the time. This variable is the square root of the sum of squared di¤erences between the share of fund i’s assets in each of the industry sectors de…ned in this study and the mean share in each sector in year t among all funds in fund i’s objective class (the authors study the growth and growth and income classes). The second variable measures boldness in terms of the unsystematic risk level in fund i’s portfolio versus that of a typical fund. The variable is equal to the absolute value of the di¤erence between unsystematic portfolio risk for fund i in period t and the mean of this variable for the rest of funds in i’s category in period t. Additionally, the third variable measures whether fund i takes a large bet on the direction of the market and is de…ned as the absolute value of the di¤erence between fund i’s beta in year t and the average beta in that year for the rest of funds in i’s category. Hong et al (2000) measure the degree in which security analysts herd when making earnings forecasts by estimating the absolute value of the di¤erence between analyst’s i

145 earnings estimate for stock j in year t and the average value of the forecast for the rest of analysts. Since analysts cover several stocks at the same time, the authors use the previously describe measure of deviation for all stocks and then estimate a score to summarize the information. The …rst step to build the score is to rank all analysts covering stock j in year t according to the degree of deviation. The boldest analyst is assigned the highest rank and so on. After that, the authors scale each analyst’s rank for a stock by the number of analysts who cover that stock since analysts who cover stocks that are thinly followed are more likely to have lower average ranks than those who follow stocks with high coverage. Finally the deviation measure is the average of deviation scores for analyst’s boldness scores in year t and the previous two years. The authors also explore the use of two additional measures of herding. The …rst variable measures whether an analyst is the …rst person to issue an earnings estimate for stock j in year t. This analyst is assume not to herd, since there wouldn’t be anyone else to imitate. By doing this the authors then estimate the probability of an analyst making the …rst forecast as a function of their independent variables. The second proxy for herding is the frequency of revisions of earnings forecasts. A higher frequency of revisions may be interpreted as evidence of the analyst changing his mind many times to accommodate the opinions of others (although as the authors recognize this could also be due to new private information arriving to the analyst). Speci…cally, the revision variable is estimated counting the number of times in a year an analyst revises his earnings forecast of a stock and comparing that to how often other analysts covering the same stock in the year change

146 their estimates. Arora and Ou-Yang (2001) proxy the degree of herding by a mutual fund manager with two variables. The authors make the assumption that a manager’s performance is measured against the average performance of funds in his objective group. Given this, the …rst variable measures the degree of correlation of fund i’s monthly returns in year t with the monthly mean returns in the objective group. The proxy is then estimated as the absolute value of the di¤erence between fund i’s correlation and the average correlation for all funds in the objective group. The more a fund herds, the closer this variable should be to one. The second proxy for herding is the correlation coe¢ cient between the monthly returns of fund i in year t and the monthly mean returns of the objective group. Higher values of correlation are considered as evidence of a higher degree of herding. Maturana and Walker (2002) take a di¤erent approach to measure herding. They assume that there will be mutual funds that act as leaders in the market and conjecture that the rest of mutual funds will try to imitate the leaders’portfolio decisions. Therefore, the authors make pair-wise estimations of Granger causality between funds buying/selling decisions. For example, the authors determine whether fund i’s portfolio decisions led the portfolio decisions of fund j or viceversa. As the previous summary shows, there are several ways to proxy for herding. In our current context, perhaps the most suitable variables would be those that study the degree of correlation of portfolio weights across intermediaries’portfolios. Since portfolios are composed of hundreds of assets it’s very unlikely to …nd correlation at the individual asset’s level. Rather than that, it could be possible to assess the degree of correlation at

147 broader levels such as 3 or 4 digits NAICS classi…cations for industries or at country levels for international portfolios. There is also an issue of how herding should be interpreted and de…ned. In our theoretical model we assumed that …nancial intermediaries copied the portfolio decisions of another intermediary. In empirical work however, herding is usually de…ned as the degree of correlation between an intermediaries’ portfolio or decisions and those of the rest of intermediaries. We note that since we study a situation in which all skilled intermediaries receive exactly the same information, the intermediary that herds will be e¤ectively copying the behavior of a group of peers rather than just one rival. It’s also important to stress the fact that in a situation in which …nancial intermediaries are aware of their own type but not of their rivals’type there is a non trivial decision to be made regarding which intermediary or which group of intermediaries should they attempt to imitate if it isn’t feasible to try to observe all intermediaries’actions. In this case it can be shown that the probability of making good investment decisions and therefore having larger expected returns and better reputation in future periods is increasing in the reputation of the intermediary being copied. In this case one could study whether one intermediary or group of intermediaries in the sample seems to act as a leader in the market. An alternative to examine this possibility is testing for Granger causality as Maturana and Walker (2002).

Other Variables While the reputation and herding variables are the main focus of our model, in order to fully evaluate if the existing data validates our predictions we also need to come

148 up with proxies for the rest of variables. Below we brie‡y discuss how to construct these proxies. Successful imitation probability A candidate for this variable would be legal requirements on portfolio disclosure. As argued by Wermers (2001), an increase in the frequency of the portfolio disclosure could have the negative e¤ect of making imitation easier to free-riders. One disadvantage is that it seems unlikely that these requirements are changed frequently. Therefore we would have to …nd the latest regulatory change on this subject and proxy the imitation probability as a dummy variable taking the value of one for periods in which regulation is tighter. Private information quality Chan et al (2005) study the relationship between herding and private information quality for analysts’earning forecasts. The authors proxy the quality of private information by the dispersion in analysts’forecasts. A stock with high dispersion would be one in for which there is little reliable information that can help analysts forecast the future, therefore suggesting poor quality of private information. Since we are interested in studying …nancial intermediaries this proxy cannot be directly applied. Nevertheless it could be possible to identify intermediaries that belong to a larger …nancial group which includes analysts forecasts. If the branches of this group support each other and share information, then the dispersion in forecasts made by the corresponding unit could be used to proxy for quality of private information. Given that analysts’forecasts follow multiple stocks, an option for summarizing these information into a single variable could be used to build a quality score using the methodology employed by Hong et al (2000) for forecasts’boldness.

149 Private information cost This variable could be proxied with management costs reported by …nancial intermediaries in their balance sheets and income statements. Before using this proxy, the data should be expressed as a percentage of assets under management in order to be comparable with variables such as fees, which are also usually reported in this fashion. Discount factor While this variable is widely used in calibration exercises and various estimates are available, these estimates are built at a more aggregate level. Given the di¢ culty in …nding a reasonable proxy to be used in econometric applications this variable could be omitted as it plays a similar role to the replacement probability, whose estimation we discuss below. Replacement probability In our model we have assumed the existence of an exogenous replacement probability which takes the same value for all managers. However, Chevalier and Ellison (1999) and Hong et al (2000) estimate actual survival probabilities for samples of mutual fund managers and security analysts. If we de…ne the unit of our analysis as managers then a similar methodology could be used, employing probit models to estimate this variable as a function of, say, manager’s performance measured by generated excess return. Uncertainty about the value of assets The uncertainty about the value of assets has a rather natural empirical counterpart. Namely, the historic volatility of assets. However, since portfolios are composed of several …nancial assets, an aggregate measure of volatility should be used, such as last period’s portfolio’s standard deviation. Alternatively, a more forward-looking variable could

150 be used, predicting the portfolio’s future risk given its current composition using a GARCH methodology. Mass of skilled intermediaries Our model assumes a constant mass of skilled …nancial intermediaries. The empirical counterpart of this variable could be the proportion of intermediaries with returns in excess of the fees they charge. This is the de…nition of a skilled manager used by Berk and Green (2004). According to their calibration about 80% of managers in their sample satisfy this criterion.

5.2.2

The Estimation Once all the previous variables are constructed the empirical estimation strategy

would consist in running regressions relating the herding variable to the reputation proxies and the rest of variables. Since …nancial intermediaries such as mutual funds usually di¤er in their investment objectives and styles it’s important to include controls for this in the regressions. Additionally, the possibility of an intermediary belonging to a larger …nancial group must also be taken into account. For example, many Chilean mutual funds are associated with commercial banks. In this case there are tied sales considerations that are absent for stand-alone mutual funds. As an example, of the relevance of this possibility, Table (5:1) below shows the top nine mutual fund companies by market share and for private commercial banks (when applicable). These rankings turn out to be fairly similar.

151

Firm B. de Chile B. Santander BCI BICE B. Security L. Vial BBVA Scotiabank Penta

Ranking in Market MF Banking 1 2 2 1 3 3 4 9 5 8 6 NA 7 4 8 5 9 NA

S o u rc e : A u th o r’s c a lc u la tio n s b a se d o n d a ta o f th e C h ile a n S u p e rinte n d e n c y o f S e c u ritie s a n d In su ra n c e a n d th e C h ile a n S u p e rinte n d e n c y o f B a n k s a n d F in a n c ia l In stitu tio n s.

Table 5.1: Mutual Fund Managers Ranking by Market Share.

152

Chapter 6

Conclusions

We have studied the delegated portfolio management market using a simple model in which …nancial intermediaries have the possibility of investing in reputation. The existence of equilibria with investment in reputation would allow this market to subsist even with the use of simple remuneration schemes. It was demonstrated that in our model di¤erent types of equilibria may exist. In particular, there are equilibria in which no FI invest in information/reputation. Nevertheless, we also demonstrated that there are equilibria in which all FI with reputation higher than certain level

will acquire information, whereas those with smaller reputation may

imitate. Although from a theoretical point of view it’s not possible to rule out that the FIs’policy function is non monotonic in this interval, our numerical exercises suggest that this isn’t a common case. Therefore, one of our main results is that as an intermediary’s reputation improves his incentives to herd decrease. As we showed in Chapter 3, there are two situations in which an intermediary may disregard the e¤ects of his actions on his reputation. Namely, when his reputation is really bad or when it’s really good. In these cases it is possible that the intermediary may try to cheat investors and shirk. Additionally, if the remuneration scheme is given by a percentage fee of the …nal value of assets under management (which is the case for most mutual funds, as we argued in the stylized facts Section); and if this fee is increasing in the intermediaries’reputation, then an intermedi-

153 ary with bad reputation that chooses to herd instead of acquiring private information will experience an expected loss in …nal value of assets under management, but since his pro…ts are given by a small percentage of this value, he will choose to herd. On the other hand, an intermediary with good reputation that decides to herd will experience an important loss in expected pro…ts, since these are given by a larger fee of the …nal value of assets under management. In order to avoid this loss the intermediary will acquire private information. This prediction is also made by Avery and Chevalier (1999). However, in their case an agent with good reputation actually chooses a contrarian strategy, disregarding his private information and making the opposite decision from other agents in order to signal to principals that he is skilled. Of course, this behavior is ine¢ cient from investors’point of view since it implies a misuse of private information. The work by Graham (1999) makes the opposite prediction: as the initial reputation of agents improves they will herd more because they want to avoid a large drop in pro…ts associated with a fall in reputation, which in this model occurs if an agent’s decision is di¤erent from that of other agents. Moreover, we reconcile Graham’s …ndings with our own by noting that in our model there is a clear di¤erence between an intermediary’s initial reputation and his incentives to herd in the current period as opposed to his reputation in this period and his incentives to herd today. We …nd that it’s possible that as intermediary’s initial reputation increases herd behavior also increases. However, as his current period reputation gets better, the incentives to herd always decrease. The in…nite time horizon of our model allows us to neatly point out the distinction between these two cases. In this sense we view our work as an alternative rationalization for the evidence found in Chevalier and Ellison (1999) and

154 Hong et al (2000) regarding the relationship between reputation and herding. However, in our setup the mechanisms operating in the reputational equilibria are di¤erent. In particular due to our modelling decision of using a continuum of intermediaries, the portfolio choice of a particular intermediary contains no information regarding the possible type of another intermediary. This is the basic mechanism a¤ecting the behavior of managers in Scharfstein and Stein (1990), Avery and Chevalier (1999) and Graham (1999). Additionally, while our model makes similar predictions regarding reputation and herding as that of Avery and Chevalier (1999), our …ndings are much more optimistic in the sense that lack of herding by intermediaries with high reputation is associated with e¢ cient investment and use of private information. This is important because the works by Avery and Chevalier and Scharfstein and Stein assume a positive relationship between reputation and pro…ts for intermediaries in a two-period setup. However, in the presence of the pathological behavior implied by these models, endogenously deriving a long-term positive relationship between reputation and willingness to pay seems a harder task (Ottaviani and Sørensen, 2006 make a similar observation). Moreover, rationalizing the increasing importance and presence of institutional investors in …nancial markets documented in Chapter 1 is di¢ cult if all types of intermediaries, regardless of their reputation, make little or no use of private information. While we believe that the cases described by these authors may be of great relevance in determined time periods or situations, we argue that it’s di¢ cult to imagine that the delegated portfolio markets could have experienced such strong growth if pathological behavior was always present, since intermediaries would have a hard time competing with investors who trade on their own behalf and presumably always make good use of their

155 private information. We also show how the size of the percentage fee that must be paid to intermediaries in order to align incentives can be considerably smaller if investing in reputation is possible as opposed to a static model. Our numerical exercises suggest that the minimum percentage fee necessary to align incentives in a static model is 3:6 times the minimum fee required in a reputational model. Moreover, we illustrate how the possibility of investing in reputation can allow the delegated portfolio management market to operate when the use of more sophisticated remuneration schemes is not possible. However, there is a cost involved since in the reputational equilibrium the intermediaries’types are never revealed to investors. Therefore, it’s possible that some skilled unlucky intermediaries are punished by investors through low fees while some lucky unskilled intermediaries may be paid high fees. Nevertheless, since skilled intermediaries who acquire information have a greater probability of making good investment decisions, the risk of this type of scenario is bounded. Additionally, we show that for a reputational equilibrium to be feasible, the gains from investing in reputation can either be obtained through higher fees or through larger assets under management. In both cases the intermediaries’expected pro…ts are increasing in their reputation. If higher reputation is rewarded with more assets under management most of the qualitative characteristics of the reputational equilibrium are maintained. In this case, for a …xed fee, the minimum amount of assets under management that skilled intermediaries must receive in order to obtain information in a static equilibrium turns out to be 4:5 times the minimum amount of assets under management necessary to align incentives in the reputational equilibrium.

156 On the other hand, since there is no complete separation between intermediaries in the reputational equilibria there is some loss in e¢ ciency since unskilled intermediaries will have strictly positive expected utility, while skilled intermediaries will tend to have lower expected utility. However, the probability of these types of situation occurring are small since intermediaries who acquire private information are more likely to make good investment decisions in our model. From a policy standpoint, our model points out the existence of a policy trade-o¤ between demanding more transparency from institutional investors, such as requiring more frequent portfolio disclosure by mutual funds and making portfolio imitation easier, which encourages herd behavior. This point has been made before by Wermers (2001) and our model provides theoretical support for this argument. We have also identi…ed some areas for future empirical research such as the degree of time-series variation in mutual fund fees and we brie‡y outlined an empirical estimation strategy that would allow us to obtain further evidence to validate our models’predictions. Regarding areas for future research, the most straightforward extension for our model would be to consider the case where asset’s prices are endogenously determined. The methodology typically used is the one developed by Kyle (1985) and Glosten and Milgrom (1985) which features a risk-neutral market maker who receives trading orders from investors and sets bid and ask prices using all available information. In order to maintain incentives to acquire private information, it would be necessary to introduce noise traders in the economy in order to avoid prices re‡ecting all private information. This extension would allow us to study equilibrium price dynamics and properties for assets in the long-run reputational

157 equilibrium. An interesting extension would be to include the possibility of an unexpected negative shock in our model and study how …nancial intermediaries are a¤ected. For example, consider the occurrence of a shock that a¤ects the economy before the risky asset’s return is due. Further, suppose that this is an event whose materialization was assigned probability zero by all agents. If this shock takes the form of an alternative investment opportunity for investors, then those …nancial intermediaries with lowest reputation would be most a¤ected, since the portfolios they manage are a less attractive investment opportunity compared to those managed by intermediaries with high reputation. This would have direct empirical predictions regarding, for instance, the relationship between the amount of out‡ows from an intermediary during a …nancial crisis or panic and its reputation. In this sense, having a good reputation would be valuable not only because it means charging higher fees and managing larger portfolios, but also because it acts as a shock absorber for intermediaries. In our model we have assumed that entry and exit probabilities for …nancial intermediaries are exogenous. Relaxing this assumption would allow us to gain insights upon the behavior of survival rates for intermediaries in a reputational equilibrium. Mailath and Samuelson (2001) study a market for reputations for a monopoly and provide a characterization of what types of …rms buy good, average and bad reputations. However, an analysis for the case of perfect competition among many …rms remains to be done. On the other hand, considering a model with risk-averse investors and intermediaries would make our model more comparable to previous delegated portfolio management literature. In particular, it would be possible to study whether the irrelevance result pro-

158 posed by Stoughton (1993) holds if investors have information about an intermediary’s behavior in the past. Moreover, it would also be possible to evaluate whether implicit reputational incentives are a cheaper and more e¢ cient way of aligning incentives as opposed to more sophisticated (e.g. quadratic) remuneration schemes. The di¢ culty of working with risk-averse investors and assets whose returns follow a normal distribution is that the Bayes’ rule that describes the evolution of beliefs may no longer have a tractable form. Consider for example an intermediary who chooses the portfolio weights of two risky assets conditional on receiving stochastic private information. If this information follows a normal distribution and the intermediary has a lineal investment strategy, then the portfolio weights will follow a normal distribution. However, the return of the portfolio is the result of the sum of products of portfolio weights and assets returns. Such product need not have a normal distribution. Moreover, to our knowledge the sum of these products has no closed-form solution. Therefore, if investors update beliefs based on the portfolio return of the intermediaries, as in our current model, numerical methods would then have to be used in order to solve the model. Finally, an overlooked topic is the existence of related markets for …nancial intermediaries. As Table (5:1) showed, often …nancial intermediaries are part of a larger group o¤ering many services to investors. It would be interesting to characterize the types of incentives that arise in these situations. It’s possible that multiple agency layers arise, a topic studied by Gervais et al (2005), or that the decisions made in one market such as commercial banking have an impact on other markets such as mutual fund management market.

159

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171

Appendix A

Delegated Portfolio Management Literature Review

We know proceed to discuss some of the delegated portfolio management literature. Some of these works attempt to rationalize one or several of the stylized facts discussed in Section 2, while others attempt to derive closed-form solutions for optimal contracts between investors and intermediaries. We also make a selective survey of the herding literature and the reputation literature, which are phenomena we are interested in exploring. For an alternative survey of the theoretical delegated portfolio management problem literature see Stracca (2006); for works that survey the theoretical and empirical research on herding see Bikhchandani and Sharma (2001) and Hirshleifer and Hong Teoh (2003). Throughout the survey we will use the terms principals and investors interchangeably and the same applies to the terms agents, …nancial intermediaries and managers.

A.1

Optimal Contracts Finding closed-form solutions for optimal contracts between investors and inter-

mediaries is attractive since it facilitates gaining insights on whether the use commonly observed remuneration schemes can be rationalized, while possibly providing theoretically based advice on how this schemes could be improved. The following works explore this topic. Bhattacharya and P‡eiderer (1985)

172 The seminal work in the DPMP literature is the study by Bhattacharya and P‡eiderer (1985). The authors’purpose is to determine which characteristics should an optimal contract have in order to align the investors’and …nancial intermediaries’incentives when the latter’s skill is not know by the former. This contract should only attract intermediaries with certain degree of skill, giving them at least their reservation utility and should be incentive compatible (i.e. FI should be better o¤ by reporting accurate information to the investor rather than lying). Moreover, the setup studied is static. The FI has the option of investing in a risk-free asset or a risky asset whose gross rate of return follows a normal distribution. The FI receives a private signal that is informative to predict the risky asset’s return. This signal is received at no cost. The intermediaries di¤er in the quality of the signal they receive, which is measured by their precision (i.e. the inverse of the conditional variance of the risky asset’s gross return). The more accurate the information, the closer the posterior on the risky asset’s return will be to the signal received. On the other hand, if the information is rather noisy, the posterior will be close to the unconditional expected gross return. The investor himself has the ability to receive signals and has some precision level. This means that if the investor randomly hires some intermediary, he may well be worse o¤ than if he managed the portfolio by himself. Therefore, the optimal contract must attract FI whose precision is no lower than that of the investor. Additionally, the authors assume that FI have reservation wages that are increasing in their precision. Finally, both investors and agents have negative exponential Bernoulli utility functions.1 1

This type of preferences is also referred to as Constant Absolute Risk Aversion preferences or CARA,

173 Under this setup the authors examine the incentive alignment properties of a linear contracts. However, this linear remuneration scheme won’t be successful in aligning incentives since given that the intermediary has CARA utility function, he may lie about his precision and cover this lie by reporting a false information. Moreover, it’s possible that under the linear remuneration scheme the investor fails to attract FI with a high precision. The reason is that if an FI’s precision is higher, this has two opposing e¤ects on the minimum ex-ante payment he must receive in order to work for the investor. The …rst e¤ect is an increase in his reservation wage which raises the minimum ex-ante payment he must receive from the investor. The second e¤ect is an increase in expected payment if he works for the investor since if the FI realizes he has high precision, the expected utility from receiving a percentage of assets under management increases, which lowers the minimum ex-ante payment he must receive form the investor. If the former e¤ect is larger than the latter, then any contract o¤ered to a high- precision FI will also be attractive to less skilled intermediaries. This would be the case if the reservation wage increases rapidly with the intermediary’s precision. In this case, the authors show that a non-linear remuneration scheme is necessary to screen out unskilled intermediaries and align incentives. Nonlinearity assures that the agent can no longer cover lies about his private information. The authors show that under the proposed remuneration scheme the payo¤ distribution that the FI obtains if he tells the truth stochastically dominates (in second order) the payo¤ distribution he would get if he lied, meaning that incentives would be aligned for all slightly risk-averse intermediaries. since it has the property that the amount of wealth that the individual wishes to invest in a risky asset is independent of his total wealth level.

174 One drawback of this scheme is that it doesn’t achieve optimal risk sharing between the investor and the FI since given CARA utility functions, in order to achieve optimal risk sharing the remuneration scheme should be linear, as demonstrated by Wilson (1968). There is one important element overlooked by this work: the possibility that it may be necessary for an intermediary to make (costly) e¤ort in order to receive private information. If e¤ort is non observable this would aggravate the informational asymmetries between investors and intermediaries. The following work studies this possibility. Stoughton (1993) The work by Stoughton (1993) addresses the question of whether non-linear remuneration schemes continue to be optimal under a moral hazard setup. The author uses a framework similar to that of Bhattacharya and P‡eiderer (1985). There is one investor and one intermediary. Additionally, e¤ort is modeled as a linear increase in the FI’s posterior precision. Therefore, if the intermediary makes greater e¤ort, his signal’s quality increases, which in turn makes him pay more attention to the signal received than to the risky asset’s ex-ante expected return2 . However, exerting e¤ort is costly for the FI. This is modeled as a monetary cost. Stoughton considers initially the …rst-best problem where e¤ort is observable for the investor. In this case, the optimal remuneration scheme is linear (recall that this is due to the fact that both the investor and the FI have negative exponential utility functions). Also, the FI makes e¤ort up to a point where the marginal disutility is equal to the marginal gain. Moreover, the author shows that as the investor’s risk aversion decreases, the optimal 2

Although the author maintains Bhattacharya and P‡eiderer’s notation, he uses net assets returns instead of gross returns

175 e¤ort level increases, since in this case the value of having more accurate information is higher given that a larger proportion of wealth is invested in the risky asset. If e¤ort is non observable, the linear remuneration scheme is no longer optimal, as there will be underinvestment in e¤ort relative to the …rst-best case. Moreover, as long as the FI is risk averse, the share of the portfolio’s …nal value promised in the remuneration scheme is irrelevant. The reason for this result (sometimes referred to as the irrelevance result) is that in a DPMP context, the FI decides both the e¤ort level, and the portfolio composition. Therefore, a risk averse agent can shirk and then invest all the portfolio in the risk free asset, claiming he received a bad signal and obtaining a risk-free payment.3 The author then proceeds to examine the properties of a nonlinear remuneration scheme. Such a contract manages to align incentives (i.e. the agent doesn’t lie about the signal received). Also, as the investor’s risk aversion approaches zero, the outcome using the nonlinear remuneration scheme resembles the …st-best case.4 Carpenter et al (2001) Carpenter et al (2001) make use of the mechanism design theory and, appealing to the direct revelation principle, …nd mechanisms in which the intermediaries reveal their private information (a private signal useful to predict the risky assets’return); the portfolio choices are then made by a third party using preestablished rules. In order to have an analytically tractable model the authors assume logarithmic utility functions for the investor and the …nancial intermediary. Also, there is a single period, 3

Since the irrelevance result is derived under the assumption of negative exponential utility functions for the investor and the FI it is not clear whether it holds under more general risk averse preferences. See also the discussion on Gómez and Sharma (2005) ahead where under a similar setup the irrelevance result doesn’t hold. 4 However, see the work by Carpenter et al (2001) for a challenge on this claim.

176 and unlike the works by Bhattacharya and P‡eiderer (1985) or Stoughton (1993), the e¤ort made by intermediaries doesn’t a¤ect their precision but rather modi…es the probability of receiving a signal from an "informed" density function as opposed to an "uninformed" density function. This work then proceeds to explore the characteristics of an optimal remuneration scheme that gives incentives to intermediaries to make e¤ort and make good use of the signal received, while sharing risk e¢ ciently. Such contracts are characterized under two di¤erent scenarios. In the …rst one the investor observes both the signal and e¤ort level made by the intermediary. In this case the optimal contract turns out to be a proportional sharing rule (i.e. a …xed fee of assets under management). In the second scenario the investor observes the signal received by the agent, but not his e¤ort level. In this setup the optimal remuneration scheme for the manager consists in a proportion of assets under management plus a fraction of the excess return of the portfolio over a benchmark (in this model the benchmark is given by the portfolio that the investor would choose if he made his own investment decisions). Finally, if investors cannot observe the signal either, an analytical contract cannot be derived. However, numerical examples show that the optimal contract rewards intermediaries for taking extreme or risky portfolio choices. The reason for this is to prevent intermediaries from slacking o¤, failing to receive an informed signal and then making conservative portfolio choices (a practice known as "closet indexing", i.e. intermediaries that pretend to follow active investment strategies charging high fees, when they are actually following passive strategies). The authors point out that even though contracts such as the ones described in

177 cases one and two are used in practice this is not the case for contracts described as optimal for the third case, namely non-observable e¤ort and signals. If this is so, it may be that there is another mechanism in play which guarantees that reasonable good investment decisions are made by intermediaries even if no explicit incentives are given to avoid closet indexing. One mechanism suggested by the authors is that the …nancial intermediaries’ care about their reputation. Exploring this possibility is beyond this paper’s scope as it uses a static framework. Ou-Yang (2003) In this work the author seeks to obtain optimal contracts between investors and intermediaries in a continuous-time economy with multiple periods. Ou-Yang claims that some of the previous principal-agent models in a continuous time setup are not suited for a delegated portfolio management context, since they don’t take into account the fact that intermediaries control both the drift and di¤usion terms of the portfolio process at the same time. This is explicitly modeled in the intermediary’s decision problem. In order to have closed-form solutions the author assumes negative exponential utility functions for investors and intermediaries. It should be stressed that Ou-Yang’s work doesn’t feature asymmetric information regarding stock returns. This is, unlike most of the delegated portfolio management literature surveyed so far, in this model intermediaries don’t have superior information to make investment decisions. Therefore, investors hire them either because they can reduce transactions costs; because they have greater diversi…cation possibilities; or because investor themselves don’t have time to make their own portfolio decisions. In this model investors observe the evolution of the risky asset’s prices, but they

178 cannot observe the evolution of their portfolio’s value. This is the source of asymmetric information between them and intermediaries. Also, intermediaries must incur in costs in order to manage portfolios. The author assumes that this cost is increasing in the size of the portfolio. Under the described setup the author derives the closed form of optimal contracts; i.e. contracts such that intermediaries make portfolio decisions according to investors’preferences, subject to intermediaries’ participation constraints, which are binding in equilibrium. The optimal contracts feature a …xed payment, plus a percentage of assets under management and a bonus or penalty depending on the portfolio’s performance relative to a benchmark. The optimal contract turns out to be symmetric since it rewards (penalizes) the intermediary if his performance is above (below) some benchmark. Also, the appropriate benchmark is constituted by an active rather than a passive index. If the intermediaries’ costs are constant, the optimal contract turns out to be a simple linear sharing rule, i.e. a percentage of assets under management. Finally, the author explores the case of more general preferences for investors. In order to obtain closed form solutions for contracts a simple cost function, consisting of a …xed cost only, is assumed and the optimal contract continues to be symmetric. This work shows how obtaining closed form solutions for optimal contracts is no easy task and requires special assumptions about agents’ preferences, cost functions, etc. In particular, the author assumes that there are no net in‡ows to the portfolio before intermediaries are paid. One important drawback is that no adverse selection or moral hazard problems are present and therefore it is not clear that the remuneration schemes derived continue to be optimal in these cases.

179

A.2

Asymmetric Contracts One justi…cation for the USA regulators to limit the use of asymmetric compen-

sation contracts is the desire to avoid excessive risk taking by fund managers. Indeed, if managers receive a performance-fee in case of high returns, but no penalties in case of low returns they may desire to increase the riskiness of their portfolio, since their potential loses would be bounded. However, the following works challenge this intuition and make a case for the use of asymmetric contracts. Carpenter (2000) In a continuous-time economy the author derives the optimal dynamic investment strategy for a risk averse-manager. Investors pay this manager a …xed fee plus a call option on the assets he controls. Thus, it’s possible that the manager may invest in overly risky positions as his loses are limited due to the characteristics of his remuneration scheme. The author shows that following his optimal investment strategy, the manager will either outperform his benchmark or severely underperform it. However, his convex remuneration scheme wouldn’t necessarily cause him to increase the portfolio’s risk too much since as the value of assets under management grow or if the evaluation date is far, the manager will moderate the portfolio’s risk. This result is robust to di¤erent speci…cations of the manager’s utility function: constant relative risk aversion and hyperbolic absolute risk aversion. In the …rst case the manager determines the level of volatility for his personal portfolio. Investors then can increase the number of options thus increasing the volatility of the manager’s portfolio. This would lead him to reduce the volatility of assets under management to o¤set the increase in borne risk.

180 Ross (2004) Ross also challenges the traditional view that convex or option like contracts increases risk taking by otherwise risk-averse intermediaries or managers beyond e¢ cient levels. The author argues that this common belief is due to option pricing theory, which demonstrates that the value of an option is increasing in its volatility. However, this doesn’t imply that the option is more desirable to a risk averse manager. In a static setup Ross shows that the introduction of an option-like remuneration scheme doesn’t necessarily reduce the managers risk aversion. For example, if the fee schedule is convex then for bets near the strike price the induced utility function for the manager may be less risk averse, but for bets to the right of the strike price the relevant domain of the agent’s utility function changes and therefore his risk aversion can either increase or decrease. It is shown that the total e¤ect on the manager’s risk aversion equals the sum of three e¤ects. Namely: the translation, magni…cation and convexity e¤ect. If the manager is o¤ered a convex remuneration scheme, then the convexity e¤ect is negative, since it makes the manager more risk loving (this is consistent with the option pricing intuition). However, if the managers’utility function exhibits increasing risk aversion, the relevant domain after the remuneration scheme is implemented will feature a higher degree of risk aversion. This means that in this case the translation e¤ect would be positive. Finally, if the increase in remuneration associated to an increase in the value of the managers’output (for example the stock price of the company he runs) is large enough, then even a small gamble at the stock price with certain standard deviation will be magni…ed, thus exposing the risk averse manager to more risk and making him less willing to undertake it. This would also increase

181 the manager’s risk aversion. Therefore, if the sum of the translation and magni…cation e¤ect o¤sets the convexity e¤ect, the introduction of the option like remuneration scheme will make the manager more risk averse. One of the utility functions commonly used in the delegated portfolio management literature is the constant absolute risk aversion function. For this class of utility functions there is no translation e¤ect, since manager’s risk aversion doesn’t change with his wealth level. However, the magni…cation e¤ect is still present and, if the fee schedule increases faster than output value, this e¤ect will be positive and the manager may become more risk averse. On the other hand, if the fee increases slower than output value the magni…cation e¤ect will reinforce the convexity e¤ect and the popular belief of less risk averse managers will hold. To conclude, it’s not obvious that introducing a convex remuneration scheme increases risk taking by managers. This will depend on the characteristics of the managers’ utility function and also on the characteristics of the remuneration scheme itself. Panageas and Wester…eld (2007) Like Carpenter (2000) and Ross (2004), the work by Panageas and Wester…eld (2007) also shows that a convex remuneration scheme doesn’t always imply more risk taking by intermediaries. The authors study a dynamic setup in which a risk neutral intermediary who seeks to maximize the present value of future fees is o¤ered an option-like remuneration scheme called a high-water mark contract. Such contracts are widely used in the hedge funds industry (see Fung and Hsieh ,1999, and Ang et al, 2008). Under this remuneration scheme the intermediary receives a …xed percentage of assets under management plus a fraction

182 of the increase in fund value in excess of the last recorded maximum, which is called the high-water mark. If there is no increase the intermediary only gets the …xed percentage. It would seem that such contracts may induce excessive risk taking by intermediaries, specially if they are risk neutral. The authors show that this intuition is correct in a …nite-time horizon setup. In this case intermediaries will increase the volatileness of the portfolio they manage without bound as the …nal date approaches. However, in an in…nite horizon context even though a bolder portfolio in the current date increases the probability of crossing the last recorded high-water mark, it also increases the probability that the value of assets under management will be substantially lower in the next period, which given the unchanged last maximum, will lower the value of future options. This creates a trade-o¤ for the intermediary making him behave like a risk-averse agent. The authors also study some of the model’s comparative statics properties and …nd that as the manager discounts more the future, he will tend to make bolder portfolio decisions. While this seems intuitive, Panageas and Wester…eld also …nd that an increase in the high-water mark increases the present value of expected fees. The reason for this is that while an increased high-water mark means that it will take more time to receive the performance fee, the expected gain to an intermediary that has just reached the highwater mark is proportional to the value of assets under management, which means that once the manager reaches the high-water mark assets under management will be larger thus increasing the intermediary’s pro…ts. It turns out that for a risk-neutral intermediary the latter e¤ect dominates the former explaining the counter intuitive result. However, the authors show that this result doesn’t always hold for risk-averse intermediaries.

183 It’s important to emphasize that this model doesn’t study the e¤ects of informational asymmetries such as unknown ability or unobservable e¤ort between investors and intermediaries.

A.3

Churning Amidst the works concerned with the delegated portfolio management problem

there is a strand of the literature that studies to what degree the existence of the principalagent relationship in this market is able to explain the relatively high volumes of securities transactions observed in the mutual funds market. Dow and Gorton (1997) Dow and Gorton (1997), argue that the existing volume of transactions seems to surpass the transactions needed to rebalance portfolio or those motivated by hedging needs. The authors suggest that this stylized fact could be due to investors failing to distinguish if a …nancial intermediary that didn’t trade chose to do so out of negligence or because the information he collected suggested that this was the best course of action. If investors believe that the …rst possibility is valid, then intermediaries could trade even if they don’t have real reasons to do so in order to avoid being punished by investors. In Dow and Gorton’s model there are two types of intermediaries, skilled and unskilled. An intermediary’s type is private information. A skilled intermediary has the ability of making e¤ort in order to receive private information about the economy’s risky asset. This asset can give a high return or a low return with equal probability. Although making e¤ort is costless per se, managers do have the option of shirking in their work,

184 which gives them a positive payo¤. This payo¤ is also equal to both types of managers’ reservation wage. Investors will be interested in o¤ering a contract that attracts only skilled intermediaries since unskilled ones would only shirk on their job and make random investment decisions. Also, the contract must give skilled intermediaries incentives to get information. Unlike the works surveyed so far, in Dow and Gorton’s paper it’s not certain that a skilled intermediary that makes e¤ort receives useful information. There is a probability that he will receive information useful to predict the risky asset’s return but it’s also possible that he will …nd himself with no useful information. Since the contract stipulates no payment in case the intermediary doesn’t make any transaction (otherwise unskilled intermediaries would …nd it attractive) the skilled intermediary will make a random investment decision, i.e. he will churn, in case he doesn’t have useful information to make his investment decisions. Nevertheless, in this model the existence of churning may be bene…cial to all agents in the economy. The reason for this is that, in order for skilled intermediaries to make above-normal expected pro…ts, there should be someone making below-normal expected pro…ts; namely hedgers, or uninformed investors who receive random income shocks which they want to hedge against, even if it means buying (selling) overvalued (undervalued) …nancial securities. The existence of churning makes hedging cheaper, since it implies that the transaction’s counterpart will not always be informed about the security’s true value. This means that hedgers’ transaction volume will increase, which will increase the amount of transactions intermediaries can make. This is due to the fact that, in equilibrium

185 skilled intermediaries’volume of transactions will be equal to that of hedgers since otherwise they would reveal their type to the price-setting market maker, thus revealing the asset’s true value and diluting their expected return. Therefore, …nancing cost for hedgers would be lower, expected return for investors would be higher, and intermediaries would be no worse (since the contract gives them their reservation utility) thus improving the economy’s welfare. An important element in this work is the existence of limited liability for …nancial intermediaries. This implies that, if the intermediary churns and makes a good investment decision out of luck he will receive a positive payment, whereas if he churns and makes a bad investment decision, he will receive zero payment, even though it will be clear to investors that he churned. If the limited liability assumption was removed it may be possible that the intermediary was charged with a …ne if he churns, which my prevent this behavior. The following work elaborates on this point. Bhattacharya (1999) This paper is closely related to Dow and Gorton’s (1997) model. The author demonstrates that it’s possible to design contracts in such a way as to screen out unskilled …nancial intermediaries, while giving skilled ones incentives to acquire information and avoiding churning, even with the existence of limited liability for intermediaries. In a setup almost identical to that in Dow and Gorton (1997), Bhattacharya introduces changes regarding the set of alternative activities at the intermediaries’disposal and also about the existence of trading costs. The existence of trading costs implies that unskilled intermediaries won’t necessary …nd it attractive to work for an investor and churn if

186 this means giving up their outside option. This is the assumption made by the author, who goes on to show that screening of skilled intermediaries can be carried out without inducing churning. However, for this to be feasible, it’s necessary that the value of the outside options for skilled and unskilled intermediaries doesn’t di¤er too much, since otherwise it would be too expensive to give skilled intermediaries their reservation utility (which is strictly higher than the value of reservation utility of unskilled intermediaries) while preventing churning. The author acknowledges that the limited liability constraint emphasized by Dow and Gorton does limit investors’ ability to screen agents. The reason is that it prevents investors from punishing bad investment decisions (which are evidence of churning in this model) by, say, charging …nes. Bhattacharya then shows that such a restriction could be relaxed if the investor hires multiple intermediaries. If the skilled intermediaries information is correlated, then they should make very similar investment decisions. Therefore, if one of the intermediaries’ ex-post return is very di¤erent (i.e. higher than the rest of agents ex-post return) then investors would punish such intermediary because the higher-thanaverage return would be evidence of churning. This unorthodox use of a benchmark based compensation would enhance investors’ screening possibilities since it allows investors to e¤ectively raised the expected cost of churning for intermediaries.

A.4

The Use of Benchmarks Up until now we have seen that benchmarks may be useful as part of the remu-

neration schemes for …nancial intermediaries (e.g. Bhattacharya,1999, and Carpenter et al, 2001). However, other authors are critical about rewarding intermediaries’performance in

187 relation to some benchmark, at least the ones used in practice. Admati and P‡eiderer (1997) This work probes the use of benchmarks on the remuneration schemes used to compensate …nancial intermediaries who have superior information that allows them to make (potentially) better investment decisions than investors. Using a model with riskaverse investors and intermediaries the authors …nd that commonly used benchmarks are generally inconsistent with optimal risk sharing between agents and generally fail to align incentives leading to suboptimal portfolio choices. Moreover, these benchmarks tend to lessen or have no e¤ect on the e¤ort level made by the intermediary; they aren’t useful to screen unskilled intermediaries from skilled ones; and they fail to align incentives when the investors don’t know the intermediaries’risk aversion degree. One of the causes behind these …ndings is the irrelevance result present also in Stoughton (1993): if the intermediary has the ability of controlling the scale of his response to the incentives provided by the remuneration scheme, then he will always choose his most preferred portfolio, regardless of the incentives provided by investors. Admati and P‡eiderer conclude that a series of benchmarks can be useful in mitigating the e¤ects of the agency problems present in a DPMP context. However, the form of this benchmarks di¤er from those observed in practice. For example they will depend on the intermediaries’risk aversion degree. Gómez and Sharma (2005) Gómez and Sharma show how the irrelevance result obtained by Stoughton (1993) and Admati and P‡eiderer (1997) fails to hold when intermediaries face short-selling con-

188 straints. The authors study an economy in which investors are assumed to hire an intermediary to make investment decisions. This is, investors’ decision regarding the amount of wealth to delegate is not modelled. In line with previous literature, the intermediary has the ability to make costly e¤ort in order to obtain information useful to make better investment decisions. Also, the intermediary is assumed to have negative exponential utility and is paid a …xed monetary amount plus a percentage of assets under management. If no constraints are placed on the intermediary’s investment opportunities, then his e¤ort level will be independent from the percentage of assets under management that he is paid by the investor and from the …xed amount he receives. Therefore, the irrelevance result holds and the contract will fail to align incentives between investors and intermediaries. Next, a case of bounded short-selling is considered. This means that the intermediary is able to invest an arbitrarily large fraction of the portfolio on either a risky or risk-free asset; however, this fraction must be …nite. Under this conditions, when an intermediary makes e¤ort the quality of his information increases, but the existence of bounded short-selling means that for some signals he will no longer be able to form his preferred portfolio. This would lessen the intermediary’s incentives to make e¤ort since it is possible that in some cases the information received won’t be put to use. Therefore, by increasing the percentage fee paid to the intermediary, the investor will be able to marginally relax the portfolio restrictions faced by the intermediary thus increasing the attractiveness of making e¤ort. Under bounded short-selling and observable e¤ort Gómez and Sharma show that the intermediary’s optimal e¤ort will be smaller than in a …rst base case with no agency concerns. Also, optimal e¤ort will be increasing in the percentage fee paid to the intermediary.

189 Finally, the di¤erence between e¤ort level in the bounded short-selling scenario versus the …rst base cased will be increasing in the bound placed on portfolio decisions and decreasing on the managers’ risk aversion. The …rst result has a straightforward interpretation. As for the second one, if the intermediary is more risk averse then he will make less extreme portfolio decisions, which means that it is less likely that the short-selling bounds will a¤ect him, thus validating the irrelevance result. If there is bounded short-selling and e¤ort is unobservable no analytic solution to the problem is derived, but using numerical methods the authors show that the percentage fee paid to the intermediary will be higher than in a …rst base case, causing an ine¢ cient risk sharing between agents. Nevertheless this is necessary in order to align incentives. The analysis made by the authors show that the level of this distortion will be increasing in the intermediary’s risk aversion and for tighter short-selling bounds.

A.5

E¤ects on Securities’Prices A relevant issue is to what extent the existence of agency problems between in-

termediaries and investors a¤ects assets’ prices. This is of great practical relevance as institutional investors account for a large share of market participation. For instance, ICI (2008) reports that USA mutual funds were the owners of 24% of outstanding equity in 2007. Thus, the portfolio decisions made by these investors are likely to impact the prices of …nancial assets. Understanding this topic is the main objective of the following works (Dow and Gorton, 1997 also provides insights on this subject). Cuoco and Kaniel (2007)

190 This paper studies the asset pricing implications of typical delegated portfolio management contracts which use relative compensation. The authors model a dynamic continuous-time economy which features a risk-free asset and two risky assets. There are three types of agents: active investors, fund investors and fund managers. Active investors make their own portfolio decisions, choosing the investment strategy that maximizes their expected utility of the …nal value of their portfolio. Fund investors delegate the choice of an investment strategy to fund managers. This could be due, for example to the fact that this investors face higher transaction or information costs. In this sense this work di¤ers from the more traditional delegated portfolio management literature in which investors hire a manager because of his superior stock picking abilities. Finally, fund managers are hired by investors to make portfolio decisions. Managers make this decisions so as to maximize the expected utility of the value of their fees. Cuoco and Kaniel assume an exogenous remuneration scheme given by a …xed payment; a percentage of the …nal value of assets under management; and a performance based fee which is relevant whenever the manager’s performance is di¤erent from that of the benchmark. Initially the benchmark is exogenously chosen to be composed of a percentage of each of the two risky assets. By solving the investors’maximization problems the authors are able to show that when the managers’remuneration scheme includes symmetric performance fees and no …xed payment, in equilibrium managers will hold more (less) units of the …rst risky asset than of the second at a given period if and only if they are benchmarked against a portfolio holding more (less) unites of the …rst risky asset than of the second asset. The result of managers investing more heavily in the assets more weighted in the benchmark is to rise

191 the equilibrium price of such assets. This is reported to be consistent with empirical evidence regarding prices of stocks that are included or dropped from the S&P 500 index. Also, if the benchmark portfolio is buy-and-hold, then the equilibrium strategies are also buy-and-holdso in this case the portfolio’s turnover is not increased by the use of performance based fees. Using numerical examples Cuoco and Kaniel also show that under asymmetric performancebased fees risk-averse managers can either over weight their portfolio with assets that have high correlation with the benchmark, in order to make their remuneration less risky, or select assets with low correlation with the benchmark, attempting to maximize the variance of the excess return of the managed portfolio over the benchmark, thus increasing the expected value of their remuneration, which is a convex function of the excess returns. The authors also show that in their model, even though investors and managers share the same preferences and risk aversion, a linear remuneration scheme fails to achieve …rst best decisions, contradicting the results obtained by Ross (1973). The reason for this is that in this model fund investors have direct access to the risk-free investment opportunity and they take the remuneration scheme structure as given when making their investment decisions. For this reason individual fund investors do not choose the level of delegation that ensures that a linear remuneration scheme achieves …rst base. This stems from the fact that individual investors fail to internalize that fees will have to increase if they underinvest in mutual funds in order to guarantee a certain reservation utility level for managers. Moreover, the authors show that for low managers’reservation utilities a contract that uses asymmetric performance fees or …xed payments doesn’t generate welfare improvements compared to a contract that features only a …xed percentage fee. This is due to the fact that for

192 this reservation utility levels, investors delegate enough wealth to managers so that the linear remuneration contract almost matches the …rst-best results. Nevertheless, if managers’ reservation utilities are higher, performance based fees are preferred to …xed payments; and both this two contracts are preferred to contracts that only use a …xed percentage fee. Goldman and Slezak (2003) Goldman and Slezak show how managers’private information mail fail to be re‡ected in assets’prices when there is a mismatch between managers’tenure and the time needed for their private information to become public. This may lead to prolonged mispricing of assets, replicating a bubble. In this model there are three types of risk neutral agents: Fund managers, noise traders and a market maker. There is also a risk-free asset and a risky asset. In the initial period a manager is entrusted to make investment decisions. The manager collects private information about the risky asset’s future value and makes his investment decision. At the same time, noise traders make random investment decisions. The market maker observes only the net order ‡ow and adjusts the asset’s price to its expected value conditional on public information. After this there is a part of the …rst manager’s private information that becomes public. The asset’s price is adjusted and the manager is rewarded a percentage of the di¤erence between the …nal and initial value of the portfolio. Then a second manager inherits the portfolio, receives private information and makes investment decisions. With an exogenous probability, called the revelation probability, the second manager’s private information becomes public before his tenure ends. This means that it’s possible that this doesn’t occur. After this, the manager is paid a percentage of the di¤erence between the

193 portfolio’s value when he leaves and its value before his …rst trade. Under this setup, the change in the portfolio value for the second manager will be given by two parts. The …rst one is the change in the value of the inherited portfolio and the second one is the value of the second manager’s trade pro…t. This means that if the current manager has negative information and inherited a long position on the risky asset, the trade that maximizes his trade pro…t will lower the value of his inherited position. Moreover, if his private information doesn’t become public, the second e¤ect may dominate and thus unless he receives su¢ ciently negative information he will not sell, causing the asset to be mispriced. The authors show that there are four situations in which there will be no prolonged mispricing: if there is no noise in the …rst manager’s signal; if there is no noise trading in the …rst period; if the revelation probability is one; and if, on average, it turns out that securities are priced without bias. When there is mispricing, its size will increase with the inaccuracy of the part of the …rst manager’s private information that never becomes public since in this case this manager will trade more aggressively, thus inheriting a more extreme portfolio to his successor. Mispricing decreases with higher revelation probability, since even though the second manager will trade more aggressively, which increases the probability of mispricing, his current orders will be less dependent on the inherited portfolio so the …rst manager’s signal error will have less persistent e¤ects on asset’s prices. When managers’ decision to acquire costly information is endogenous, Goldman and Slezak use numerical examples to show that for a …xed quality of private information, as the revelation probability falls, the informational e¢ ciency of prices fall, since managers

194 are less likely to bene…t from their private information and therefore have less incentives to acquire it. Also, for a …xed revelation probability, price e¢ ciency improves as the quality of private information decreases. This is so because prices will be less responsive to investment decisions so managers may invest more aggressively and make higher pro…ts, which encourages them to acquire private information.

A.6

Reputation This literature is sometimes referred to as career concerns. Reputation is typi-

cally modeled as the probability assigned by investors and intermediaries to a particular intermediary being skilled, conditional on some information set, which usually includes the intermediary’s previous decisions record. While some of these works emphasize the positive role of reputation as an incentive aligning mechanism (e.g. Heinkel and Stoughton, 1994, Chemmanur and Fulghieri, 1994, and Farnsworth, 2003) there are also studies suggesting that reputational or career concerns may have undesirable side e¤ects (e.g. Scharfstein and Stein, 1990, Huddart, 1994, Prendergast and Stole, 1996, Avery and Chevallier, 1999, Dasgupta and Prat, 2006). Heinkel and Stoughton (1994) So far we have seen that optimal remuneration schemes in a DPMP context are symmetric (e.g. Ou-Yang, 2003), may be non-linear (e.g. Bhattacharya and P‡eiderer, 1985) and contemplate the use of non-standard benchmarks (e.g. Admati and P‡eiderer, 1997, Bhattacharya, 1999). However, Heinkel and Stoughton (1994) argue that, even without the use of remuneration schemes previously described, the delegated portfolio manage-

195 ment market could exist. The reason this is possible would be that the presence of implicit incentives, such as the FI’s reputation, is su¢ cient to align incentives, even if the contracts used are simple (e.g. a …xed percentage of the value of assets under management). Indeed, if the agent’s future remuneration depends on his reputation (e.g. greater fees, more clients), he could be willing to make a greater level of e¤ort to manage the portfolio, thus avoiding possible future loses in revenues. Heinkel and Stoughton’s work is amongst the …rst ones to extend the DPMP literature to a dynamic context. This work considers the existence of adverse selection (i.e. investors are not aware if the intermediary they hire is skilled or unskilled) as well as moral hazard (i.e. it’s not possible for investors to observe whether the intermediary makes e¤ort to manage his portfolio). In this model both investors and intermediaries are risk-neutral. The economy has two periods. In the …rst period the principal o¤ers the agent a menu of contracts for him to choose from. Even though the principal may design this menu as to screen skilled from unskilled agents he will …nd it optimal not to do so, maintaining some degree of doubt about the intermediary’s type. The reason for doing this is that in an intermediate date the investor may hire a third party to evaluate the intermediary’s performance. Moreover, the investor can observe the result of the investment decisions made by the intermediary. With this information the principal decides whether to …re or retain the intermediary. If the intermediary is not …red he o¤ers the investor a contract which gives all the portfolio earnings to the intermediary, while giving the investor a …xed pay, equal to his reservation utility. The intuition behind this is that once the intermediary’s type is known,

196 several investors will compete for their services. However, given the characteristics of the contract o¤ered in the …rst period (i.e. low degree of response of the intermediary’s pay to his performance during the period), the investor cannot be certain about the intermediary’s type. Instead, he uses the end of period performance evaluation to decide if he …res or retains the intermediary. Therefore, a skilled agent will make a greater level of e¤ort in the …rst period, since he knows he will be …red if his performance is poor and he will miss the opportunity of getting attractive contracts in the second period. The authors argue that this model is capable of explaining some stylized facts: long term contracts are not frequently used (Heinkel and Stoughton assume this type of contracts don’t exist since the parts cannot credibly commit to avoid renegotiation of the contract in future periods); there is a weak relation between fees paid to intermediaries and their past performance; most contracts do not feature performance-based fees5 ; and most institutional investors pay professional evaluation services in order to have performance reports about their portfolio managers. This work presents some shortcomings. In particular, it contains some arbitrary assumptions about the principal-agent relation and how negotiation power shifts completely from the former to the latter if he isn’t …red. Although this assumption allows to obtain a tractable model it casts doubt on the general validity of their conclusions. Chemmanur and Fulghieri (1994) This work studies how reputational concerns modify agents’behavior in non DPMP context. In this model there is an entrepreneur who whishes to make his …rm public, i.e. 5

The authors cite Golec (1992) to assert that most mutual funds contracts don’t contemplate the payment of performance-based commissions. However, this type of arrangement is used in hedge funds. On this see Panageas and Wester…eld (2007).

197 he wants to make an IPO. For this he has the option of hiring an investment bank. The investment bank realizes an evaluation about the project’s pro…tability and this information is used by investors to value the project. Moreover, the quality of the investment bank’s evaluation depends on the degree of e¤ort he makes. This e¤ort is non observable to investors. Therefore, if the investment banks concludes that the …rm is pro…table, but the …rm turns out to be a bad investment, investors cannot verify if the investment bank shirked or if he was unlucky. In a two period setup the authors demonstrate that investment banks will make more e¤ort than in a situation in which reputation was absent. This is, the presence of reputation helps to mitigate the ine¢ ciencies caused by moral hazard in this economy. The reason is that banks face a dynamic trade-o¤: making more e¤ort in their evaluation is costly in the short run but it may be bene…cial in the long run, since investors will assign a lower probability to banks recommending bad investment projects. Therefore, in this model reputation is equal to the probability assigned by investors to the event that the …rm recommended by the bank is pro…table, conditional on the result of the bank’s recommendation in the previous period. This will raise investors’ willingness to pay for …rms recommended by investment banks with good reputation, which in turn will increase these banks’pro…ts, as they are given by an exogenous percentage of the IPO’s raised funds. The authors claim that the models predictions are supported by existing evidence. Namely: investment banks with better reputations charge higher fees; IPO’s underperformance due to informational asymmetries will be lower if the investment banks’reputation is higher; and more prestigious investment banks will work with less risky …rms.

198 Finally, Chemmanur and Fulghieri don’t assume a speci…c relation between revenue and reputation; instead, they model reputation as a Bayesian updating process of beliefs regarding the agent’s skill. However, their model studies the behavior of the investment banks, and not a DPMP situation. Arora and Ou-Yang (2001) This work builds on Ou-Yang (2003) and studies the incentives that arise when the …nancial intermediaries’ performance in the present period a¤ects their reserve wage and portfolio in‡ows in following periods. In order to simplify their analysis the authors assume that there is a linear monotonic relation between future portfolio in‡ows and this period performance. They also assume there is a linear monotonic relationship between future reserve wages and present performance. In both cases performance is measured both in absolute terms and in relation with a benchmark. In a multiple period economy the authors characterize the form of the optimal contracts between investors and intermediaries as well as the intermediaries’optimal portfolio policy. Optimal contracts are found to be symmetric; i.e. the agent is rewarded if he has a good result but is punished if the performance is poor. Also, the authors …nd that the agent will invest part of the portfolio actively while the remaining portion is used to mimic the benchmark; i.e. the portfolio will incorporate a herding component, whose existence is due to the fact that the agent is risk averse and in part desires to avoid having a performance worse than that of the benchmark, since his results will be partially measured against the benchmark. An interesting …nding is that the herding component of the portfolio is larger

199 during the …rst periods of the intermediary’s life and its importance declines over time during the last stages of his career. The authors present empirical evidence that supports this prediction using data for USA mutual fund managers. Even though the assumptions about monotonic relationships between reserve wage and fund in‡ows allow to have closed form solutions, they are certainly ad hoc and could be consistent with the existence of mechanisms like reputation (although investors’beliefs are not modeled) but also with the existence of long term binding contracts between intermediaries and investors. Farnsworth (2003) Farnsworth analyses a DPMP problem in a multiple period model similar to that of Carpenter et al (2001) but he allows for …nancial intermediaries to build a reputation. In this model there is a moral hazard problem between investors and intermediaries since the former can’t observe the degree of e¤ort made by the latter when managing their portfolio. The author argues that, even though this problem could be solved using performance bonus, these don’t seem to be frequently used in practice. Instead of this type of explicit incentives, the author suggests that reputation could be a mechanism that manages to align incentives. The model assumes the existence of two types of intermediaries: skilled and unskilled. The …rst type can make e¤ort to receive an informative signal about the pays of the assets in the market; the second type has no such possibility. In this model a signaling equilibria is not feasible since unskilled intermediaries are convinced that they are skilled. This allows reputation to play a part as a mechanism to align incentives. Speci…cally, reputation succeeds in avoiding the intermediaries’incentive compat-

200 ibility constraints to be binding. If this is the case explicit incentives would have to be provided in order to align incentives. Farnsworth assumes that investors’have committed ex ante with intermediaries to increase the amount of assets under management if the intermediaries’reputation improves, therefore increasing intermediaries’pro…ts which are given by a constant percentage of assets under management. However, as the intermediaries make e¤ort and their reputation grows, the subsequent increases in reputation are smaller since there is a limit as how good reputation can be. When this happens the incentive compatibility constraint will bind and additional incentives would have to be given to overcome the moral hazard problem. The author argues that this would be the case of large investment funds, which already have high reputation and must resolve to include explicit incentives in their contracts. One shortcoming of this work is the assumption that the principal has committed ex-ante to increase the ‡ow of delegated funds if the agent’s performance has been good. This suggests that binding contracts should be used between investors and intermediaries when in fact many mutual fund contracts can be ceased on very short notice even if the relationship between the parts has only lasted a few months. Huddart (1994) This work studies the e¤ects that reputation can have on the behavior of skilled …nancial intermediaries. In this model there are risk averse investors who hire …nancial intermediaries to make investment decisions. There are two intermediaries in the economy. One is skilled and the other unskilled and this is private information. The di¤erence between these intermediaries is that the skilled one has private information useful to make better

201 portfolio decisions while the unskilled one has no such information. The economy has two time periods. At the beginning of the …rst period half of the investors invest with one intermediary and the remaining half with the other intermediary. Both intermediaries make their investment decisions in a risky and risk free asset. At the end of this period all agents know whether the decisions made by the intermediaries were good or bad. Also, since it is more likely that a skilled intermediary made a good investment decision, investors will delegate their portfolio management to such an intermediary. Since the intermediaries’ pro…ts are given by a percentage of assets under management changes in their reputation will a¤ect their expected utility. Huddart assumes that investment decisions are taken simultaneously. This means that the unskilled intermediary won’t be able to copy the investment decision of his rival. Since he has no information he should make his investment decision based on the unconditional expected return and variance of the risky asset, which given the author’s assumptions means that he should invest all the portfolio in the risk free asset. However, since the structure of the information received by the skilled intermediary is known to all agents (although the speci…c realization of the signal received is only known to the skilled intermediary) the unskilled intermediary knows that the portfolio choice of his rival can take two possible values. Therefore, by choosing one of this values at random he has a chance of making a good investment decision. It is possible that the unskilled intermediary tries to make this guess rather than choosing to invest all the portfolio in the risk free asset, since under the …rst choice if he gets lucky he won’t be …red by investors while under the second choice he knows for sure his type will be revealed and he will be …red.

202 The author explores the feasibility of two types of equilibria. In a pooling equilibrium the skilled intermediary makes his investment decisions according to his private information, while the unskilled intermediary attempts to copy this behavior. Sometimes the unskilled intermediary will succeed and investors initial beliefs won’t change, while sometimes the unskilled intermediary will fail and his type will be revealed. It is even possible that the skilled intermediary invests according to his information, which turns out to be misleading, while the unskilled intermediary makes a lucky guess and investors mistakenly conclude that he is the skilled intermediary. For this equilibrium to be feasible it is necessary that the unskilled intermediary does indeed prefer to make a random investment decision rather than reveal his type (this decision is not trivial since he is risk averse); also, the skilled intermediary must prefer this type of equilibrium instead of a separating one in which he attempts to signal his type to investors through his investment decisions. In a separating equilibrium the skilled intermediary will choose more extreme portfolio positions (e.g. if his information suggests to invest 40% or 60% in the risky asset he will invest 30% or 70%). By doing this, he makes copying less attractive to his rival, thus revealing his type to investors and obtaining greater pro…ts in the second period. To do this, however, he makes suboptimal use of his private information. The separating equilibrium will prevail if the unskilled intermediary is relatively risk averse (in this case making random investment decisions is less attractive because of the risk involved) and also if the fee received by intermediaries is larger (in this case even if the unskilled intermediary invests all the portfolio in the risk free asset he will receive high pro…ts at the end of this period, while receiving none in the second period). Huddart shows

203 that the ine¢ ciencies caused by reputational concerns can be mitigated using performance fees rather than …xed fees, since the former do a better job at aligning incentives. Also, a commitment by investors not to reallocate their investment among intermediaries would enhance investors’ex ante expected utility. Prendergast and Stole (1996) The work by Prendergast and Stole examines how a manager’s concern for his reputation can have an impact on his investment decisions. In this model each period managers must decide the amount of resources to devote to an investment project. The productivity of the project is unknown, but managers receive an imperfect private information of this parameter. Each manager is di¤erent in their talent, with more (less) skilled managers receiving more precise (imprecise) private information. The managers’type is unknown to the rest of agents in the economy. The authors assume that managers’objective function includes only current pro…ts (which are function of the investment project’s pro…tability) and his immediate end of period reputation. It is also assumed that the market updates its beliefs about a manager’s ability using only information on his investment decisions and not on the pro…ts earned. This allows greater tractability of the model. In this setup the authors show that during the …rst period that the managers are employed, the more talented managers will exhibit a higher variation of the posterior about the market’s prior regarding the productivity of the investment. This happens because managers who are very talented (and have very precise information) will give less attention to the common prior and place more weight on their private information. On the other hand,

204 for later periods the variance of the posterior has two components acting in opposite directions. There is an element that measures the weight placed on the most recent observation. This element is increasing in the manager ability because for skilled managers the current observation is trustworthy and therefore it should considerably change the posterior. This could lead managers to induce excess variability in their investment decisions in an attempt to appear well informed. However, there is a second element related with the variability of the current observation. This is decreasing in ability because as ability decreases this means that the previous observation did not yield a precise measure of true pro…tability and also the noise associated with the current observation is large. This second e¤ect could lead managers to be too conservative since they would want to give the impression of already having made correct decisions. Whether a manager is conservative or overreacts to his private information for later periods depends on which of these two e¤ects dominate. Focusing on separating equilibria, Prendergast and Stole show that managers will exaggerate their decisions with respect to a case with no asymmetric information regarding ability, when they are working for the …rst time. Also, once enough time has passed since they were hired, they will be too conservative, making little use of their private information. This is not due to concavity of the rewards function, which is assumed to be lineal, but rather because the manager wishes to act like he already has made good investment decisions because changing his opinion too much will result in a decrease in reputation. In between the …rst time period and the date when the manager becomes conservative the authors do not fully characterize the manager’s behavior although they …nd conditions under which separation arises.

205 Finally, the authors show that when the parameter re‡ecting the productivity of investments changes over time, managers will not be conservative even in the long run. The reason for this is that if the environment evolves enough managers will be recriminated if they don’t change their investment decisions, no matter how good their previous decisions were. On the other hand, if beliefs about ability are updated using public observation, such as past periods’pro…ts it is possible that managers will not be conservative even in the long run. Dasgupta and Prat (2006) Dasgupta and Prat study the equilibrium features of an economy in which …nancial intermediaries are concerned about their reputation. The authors are able to study the properties of prices and transactions volume, showing how reputational concerns lead to excessive trading -i.e. churning- and a skewed relationship between fund returns and net fund in‡ows. In the economy there are four types of agents. Investors are risk neutral and since they cannot trade directly they have to delegate this task to fund managers. These managers are also risk neutral and they can be skilled or unskilled. The managers’ type is not known by investors nor managers. Managers make portfolio decisions on behalf of investors. At the end of the …rst period all parties observe the result of the investment decisions. Based on this information investors then decide whether to retain or …re the manager. There is also a number of noise or liquidity traders, who make random investment decisions. The last type of agents are risk neutral uninformed rational traders. These act as

206 market makers who face Bertrand competition and therefore set prices equal to the expected value of the asset being bought or sold, conditional on the orders received. The authors assume that trade is anonymous and therefore market makers observe only transaction orders but they are not aware of the issuer’s identity. Skilled managers receive an informative signal, useful to predict the asset’s true liquidation value. An unskilled manager receives no such information. We stress the fact that managers have to make no e¤ort in order to receive their information. This is, the authors do not model a moral hazard situation. The remuneration scheme is assumed to be some percentage fee p of assets under value plus a …xed positive payment p. The values of p and p are exogenously given and therefore, the authors assume that they satisfy investors’ participation constraint. If there are no career concerns, i.e. the probability that the manager is …red at the end of …rst period is exogenous, managers will only trade if they have information. Therefore, only skilled managers will trade and there will be no churning. The reason for this is that skilled managers face positive expected returns if they invest due to the presence of noise traders. As market makers are aware of this they will charge a positive bid-ask price, which means that unskilled managers won’t trade because they face negative expected returns. On the other hand, the authors demonstrate that if career concerns are present, in equilibrium investors will retain their manager if he makes a good investment decision in the …rst period and they …re managers if they make a bad investment decision or do not trade, since either one of this events reveals the manager is unskilled. Faced with this, an

207 unskilled manager will have to choose between no trade, which results in getting …red, or churning (i.e. making a random investment decision), which has an uncertain outcome: if he is lucky and makes a good investment decision he is retained but if the investment decision is a bad one, he will be …red. If the manager’s contract features a low p the unskilled manager will prefer to churn. As a result trading volume will be higher in the presence of career concerns. Moreover, in this setting the reputational reward for good investment decisions will be higher (in absolute terms) than the reputational cost of bad investment decisions, which is consistent with the ‡ow performance relationship documented in the stylized facts Section. Additionally, Dasgupta and Prat show how their results are robust to somewhat more general assumptions about the information received by managers and endogenous contracts. One subject overlooked by this work is how the relationship between reputation and churning evolves over time. That is, as time goes by and managers’reputation improves or get worse, how do the incentives to churn change? Also, the authors do not deal with the possibility that obtaining information is costly to managers.

A.7

Herding The correlation observed in portfolio holdings of …nancial intermediaries such as

mutual funds and pension funds was documented in the stylized facts Section. This phenomenon is called herding. We brie‡y discuss some of the reasons for intermediaries to herd, beginning with earlier papers that explain herding as a result of informational cascades (see Banerjee, 1992, and Bikhchandani et al, 1992); this phenomenon is also called

208 statistical herding. Then we proceed to discuss the works by authors who relax some of the assumptions made by the earlier literature such as exogenous prices and exogenous decision sequences and study whether herding can still occur in equilibrium. Other authors (see Arora and Ou-Yang, 2001, whose work was surveyed above, and Maug and Naik, 1995) discuss how herding can result from compensation schemes that rely on relative performance. Emphasis will be given to works discussing the possibility that intermediaries herd due to reputational concerns (see Scharfstein and Stein, 1990, Avery and Chevalier, 1999, Graham, 1999, and Ottaviani and Sørensen, 2006) . For exhaustive surveys of theoretical and empirical works studying the herding phenomenon in …nancial markets see Bikhchandani and Sharma (2001) and Hirshleifer and Hong Teoh (2003). Banerjee (1992) and Bikhchandani et al (1992) The works by Banerjee (1992) and Bikhchandani et al (1992) rationalize the existence of herding as a result of informational cascades. In a situation in which individuals must sequentially decide whether to take a decision or not, the decisions of early players will have a determinant impact on the behavior of the rest. Indeed, if each individual receives a signal that can take a good or a bad value, he will decide whether to buy a good or abstain from buying it basing his decision on his prior beliefs about the attractiveness of owning the good; his received private information; and also on the decisions made by those individuals who decided earlier. For example, suppose that the individuals share a common prior belief suggesting that they should buy the good when in fact the truth is that they would be better o¤ not buying it. Further, suppose that all the individuals’signals are of the same quality and that the …rst individual receives a good signal. Given his prior beliefs and the signal

209 received he updates his beliefs via Bayes’rule and buys the good. Now suppose the second individual receives a bad signal (which is the correct signal in this example). He knows that the …rst person received a good signal since otherwise he wouldn’t have bought it. Since all signals have equal quality the second individuals’beliefs equal his prior and thus he buys the good. From then on, even if all following individuals receive bad signals, they will ignore their private information and follow the herd. Thus, the informational cascade that forms will prevent private information from reaching the market and the equilibrium outcome will be ine¢ cient. Bikhchandani et al (1992) further discusses how robust herds are and they show that the release of accurate private information can stop individuals from herding. Avery and Zemsky (1998) This work studies the conditions under which herding would arise in …nancial markets focusing on the role market prices have preventing the occurrence of herds. The authors study a model in which a security’s true value is unknown to agents. In the economy there is a continuum of risk-neutral traders who can either be informed or uninformed about the security’s value. An informed trader receives private information and submits orders to the market maker seeking to gain positive expected pro…ts. An uninformed trader is actually a noise trader who randomly submits orders to a market maker. In this model an informed trader herds if he ignores his private information to place orders. The authors show that if the only source of uncertainty in the economy is about the asset’s true value then herding won’t be an equilibrium outcome since adjustments in the security’s price prevents this and guarantees that all private investors will be indi¤erent between buying or selling the asset and once he receives his private information this will be decisive for taking his …nal

210 investment decision. In this fashion private information will keep ‡owing in to the economy. If there is an additional source of uncertainty; namely the quality of the agents’information (or their type) is unknown then it is possible that prices are no longer e¢ cient and herd behavior may surge. The reason for this is that the security’s price has only one dimension and is therefore unsuited to handle uncertainty in more than one dimension (assets’ true value and investors’type). Based on this the authors hypothesize that multiple dimension prices, such as options prices, may be better suited to prevent herding. Beaudry and González (2003) The work by Beaudry and González studies the plausibility of herding in discrete investment decisions in an economy where information is costly to acquire and prices are endogenous in the sense that they react to agents’investment decisions. In this model there are intermediaries who make investment decisions consisting in acquiring some good and selling it at a later date. In deciding the amount of their investment intermediaries form expectations about future demand for the good they sell. They can incur in some cost in order to receive private information useful to determine the state of demand for future periods. This state can be either good (i.e. high demand) or bad (i.e. low demand). The cost of the investment is endogenously determined by the intermediaries’decisions and the same is true for the price at which intermediaries sell in future periods. In the economy there is also a small mass of noise traders. The authors show that in an economy with no private information prices do not convey any information about the state of the economy. This is due to the fact that no agents have information about states. Also, there will be no randomness in aggregate

211 investment decisions even though noise traders are present in the economy. This is due to intermediaries accommodating movements in noise trading since this is the only way in which they will be indi¤erent between investing or staying out of the market. Of course, since noise traders decisions are random, this means that the decisions of the intermediaries will also be random, although the aggregate investment decisions will be deterministic. This result depends on the assumption about noise traders being small, since otherwise intermediaries wouldn’t be able to compensate their decisions. When there is private information the equilibrium price distribution will have two points and the aggregate level of investment will also be given by a two point distribution: one for the good state and one for the bad state. Also, even though prices are informative in this case they will be noisy regardless of the amount of noise trading. The existence of this noisy price function is what gives intermediaries incentives to invest in private information. Those intermediaries who are uninformed will try to extract information about the demand state from prices. However, in addition to the existence of noise traders they must also take into account the presence of informed intermediaries. Therefore, in equilibrium, if the price is high an uninformed intermediary will not be sure if this is due to a high demand by uninformed intermediaries or a large demand by informed ones. Underlying this results is the assumption that investment decisions are discrete and bounded which results in prices never being fully informative. This means that it is possible that sometimes prices are high as a result of the decisions of uninformed intermediaries because many of them will be investing, and they are investing because they believe that the high price is a signal of a high future demand. This phenomenon is interpreted by the authors as herding.

212 Finally, the authors show that as the cost of acquiring information gets smaller, is less likely that intermediaries make wrong investment decisions at the aggregate level. However, in turn, the size of this errors will be larger since they are what give intermediaries incentives to acquire information. Cipriani and Guarino (2003) This work also examines the herding phenomenon in an economy with endogenous prices studying whether herding can be a source of …nancial contagion. Cipriani and Guarino argue that a reason for assets’prices not to re‡ect fundamentals is that information about the fundamentals is spread across investors and if the herd, that is ignore their private information, it won’t be possible for prices to aggregate this information. The authors study an economy in which there are two assets traded in two markets, each one with a risk neutral market maker. Each period an investor is randomly selected from a continuum to trade in one of the two markets. Investors can be either informed or uninformed and trade on their own behalf. Therefore, in this model there are no agency problems. An informed investor receives information about the asset’s value and since the two assets’fundamental values are correlated he also learns something about the asset in the market in which he is not currently trading. A crucial assumption is that informed investors are heterogeneous in the sense that part of them enjoy an extra utility while some su¤er a disutility from holding the asset. The investors’type is private information. Under this setup there will be gains form trade. This means that it is possible that investors who enjoy the asset buy it even when the market makers ask price is higher than the asset’s expected value while investors who experience disutility sell the asset even when it’s expected value

213 is higher than the market makers bid price. Regarding market makers, they set bid and ask prices for the assets according to the information available, which consists on current assets’ prices; the history of prices and transactions for both assets; and the current transaction. Cipriani and Guarino show that informational cascades can occur and in this case the investors’actions will be independent of their private information. Moreover, the authors de…ne herding as a situation in which all investors of the same type choose the same action (i.e. all those who enjoy holding the asset buy and those who dislike holding it sell), ignoring their private information. Herding will occur in this economy whenever private information is not too informative about the assets’true value. This is contrast with Avery and Zemsky (1998) result of no herding with endogenous prices and is due to the existence of gains from trade. Importantly, the authors show that having two markets instead of one is a doubleedged sword. On one hand, if there was only one asset in the economy, once a cascade starts it would never end. However, if information continues to gather in the other market and since assets’value are correlated, information will also arrive to the cascading market and eventually the cascade will break. On the other hand, with two markets the possibility of contagion arises. That is, it is possible that an informational cascade occurs in a market only because agents observe the history of trades in other market and this makes the price of the …rst asset to deviate from fundamentals, thus starting a cascade. Unfortunately, the authors are not able to conclude on which of these two e¤ects is likelier to predominate. Finally, using simulations methods the authors show that in their model the unconditional correlation between prices is always greater than the correlation between funda-

214 mentals, which is consistent with de…ning contagion as an excess correlation between assets’ prices relative to fundamentals. This occurs even in the long run because with gains from trade informational cascades will arise and won’t always break which implies that assets’ true values are never discovered. Chari and Kehoe (2000) One of the criticisms made to the herding literature is that agents make their decisions sequentially and, as shown by Bikhchandani et al (1992) the …nal equilibrium usually depends on the order in which the agents decide, which is treated as exogenous. Chari and Kehoe (2000) relax this assumptions endogenizing the order in which agents act. The authors model an economy in which a group of risk neutral agents must decide between investing in a risk free asset or a risky asset. The risky asset’s pay will be high in the good state and low in the bad state. Each period an informative signal is randomly received by one of the agents, who must make a discrete choice of whether to invest a …xed amount in the risky asset, which is an irreversible decision, or to wait. All agents observe the number of investments made in each period. In this model waiting to invest is bene…cial to agents because they can gather information about the risky asset’s true value from other agents’ decisions. However, there is also a cost of waiting since agents forgo the ‡ow return from investing. The authors show that in this model with endogenous moving order there are equilibria which exhibit informational cascades and herds which are from the same kind of those present in exogenous timing models such as Banerjee (1992) and Bikhchandani et al (1992). In particular, both herds of investment and herds of no investment will be present. Next, Chari and Kehoe relax the discrete investment assumption, allowing agents

215 to make a once and for all investment of any nonnegative amount in the risky asset, which is assumed to have decreasing returns. Not only do the authors prove that herds are possible, but this are likelier to occur compared with a case with exogenous timing. The reason for this is that with continuous investment decisions agents will tend to forego the option of waiting and gathering information before investing because now they can optimally adjust the size of their investment according to what little information they may posses. On the other hand, if timing was exogenous herds of investment would not be present since each agent would have a take it or leave it option to invest and therefore as long as the prior of the risky asset’s attractiveness is above some cuto¤ level the agent will invest some positive amount. However, this investment decision will be used to deduce the agents’ private information. Therefore, with exogenous ordering the only form of herding that could exist are no investment herds. Finally, the authors show that even if agents are given the option of sharing information thew will not have incentives to do so, because truth-telling equilibria will not be feasible. This is the result of assuming that there is an advantage of being early movers for informed investors which gives incentives to mislead other investors. Zhang (2006) Zhang (2006) also relaxes the exogenous ordering assumption of the earlier herding literature and studies whether herding persists and also if it is more or less error prone. In this model individuals must decide between adopting an irreversible investment decision or waiting. If they opt for the latter they can still adopt the decision at any future time period. Thus, the author names this a one-sided commitment decision problem. If both decisions

216 faced by the individual were irreversible then this would be a two-sided commitment decision problem. In this model there is a bene…t from waiting because new information may be released about the attractiveness of the irreversible decision. However, there is also a cost since agents who wait gain only their reservation utility. Zhang shows that there is an equilibrium with the property that each period there will be a critical type of agents who make the irreversible investment decision with probability less than one; also all agents who receive good private signals investment and all others wait. In this equilibrium there is a strategic phase where agents wait or invest, depending on their information, followed by a herding phase in which all remaining agents invest immediately or keep waiting forever regardless of their private information. Interestingly, the author shows that in this case the disclosure of public information is of less use than in a model with exogenous ordering. The reason for this is that with exogenous ordering cascades break down once public information is released. This was established by Banerjee (1992) and Bikhchandani et al (1992) and is due to the fact that public information need only to o¤set the information form the last agent’s action before the cascade began. However, with endogenous ordering, once the herding phase is reached all agents act simultaneously and there is no time for public information to break the cascade. Additionally, Zhang shows that if agents are patient enough, there are equilibria in which almost no one makes decisions, since all prefer to wait for others to reveal their private information. This in turn means that almost no information will be made public. Therefore the expected number of correct decisions will be strictly lower than in a case

217 with exogenous ordering, in which individuals are force to decide once and for all whether to invest or not using their private information and public information available at that moment. This ensures that more private information is revealed to the rest of agents. Maug and Naik (1995) This work seeks to rationalize herding by …nancial intermediaries in a delegated portfolio management setup as a result of performance evaluation contracts. Under this relative performance contract the authors show that it may be optimal for a skilled manager to ignore his private information and herd with other manager even if he knows that this manager is unskilled or less informed. Maug and Naik study a single period economy in which an investor hires an intermediary of manager to make investment decisions. Both agents have CARA utility functions. The manager has the ability of obtaining superior information informative of the risk asset’s return. After receiving his information the manager places an order with a market maker, who sets prices equal to the expected value of the asset, conditional on orders received. In the economy there is also another trader, who manages his own portfolio. The authors assume that the investor o¤ers the manager a linear relative performance contract which stipulates a …xed payment plus a percentage of the portfolio’s …nal value minus a percentage of the other manager’s …nal portfolio value. Optimal parameters for this remuneration scheme are derived maximizing the investor’s expected utility for the case in which there is a moral hazard problem, i.e. the investor cannot verify that the manager makes e¤ort to receive private information; and for cases in which the investor whishes to screen out relatively unskilled managers. In both cases it is shown that the

218 relative performance parameter will be di¤erent form zero. The reason for this is that the manager is risk averse and thus giving him a payment that depends on the uncertain …nal value of assets under management exposes him to risk. For the manager to be willing to work for the investor, the latter must o¤er the former either a signi…cant risk premium or, he can provide partial insurance by making …nal payment a function of the other manager’s pro…ts. Since both managers’information is partially correlated this rules out the possibility of punishing the manager in circ*mstances in which all managers perform poorly due to for example misleading private information. However, it is shown that giving the manager a relative performance contract will lead him to change his investment strategy. Indeed, now when making his portfolio choices he will give a disproportionately small weight to his private information and a disproportionately bigger weight to the information he shares in common with the other manager. The authors show that even if the manager who trades on his own behalf has less information, the manager that works for the investor may herd with him in the wrong direction. This is, if the hired manager’s private signal suggests that he should buy the risky asset, but the self employed manager’s less reliable information suggests to sell the asset, the former may change his decision from buying to selling, thus herding with the less informed manager in the wrong direction. Finally, the authors show that even if the investor can design the contracts to make his manager trade in a market di¤erent form the self-employed manager and thus making bigger pro…ts from his monopolistic private information, in some cases he will prefer the manager to herd or trade in the same market as the other manager, since this substantially

219 reduces the risk premium that must be paid thanks to the use of a relative performance contract. Gümbel (2005) A good part of the literature treats herding as an undesirable phenomenon mainly because it induces intermediaries to disregard private information useful to make investment decisions. The work by Gümbel extends the work by Maug and Naik (1995) exploring the possibility that herding is a useful tool for investors, who may deliberately induce intermediaries to herd , i.e. trade in similar assets, in order to make relative performance contracts feasible. In this model there are two investors, two risk averse intermediaries with CARA utility functions, two market makers and noise traders. All agents live for one period. Each investor is assigned one intermediary. There are two risky assets in the economy and investors assign their intermediary to trade in one of these assets. This decision becomes public knowledge. Upon making the assignment investors o¤er intermediaries a linear contract that features a …xed payment, plus a percentage of the intermediary’s trading pro…ts minus a percentage of the other intermediary’s trading pro…ts. Once the intermediaries accept their contracts they have the option of making e¤ort to receive a private signal that is informative to predict the risky assets’ return. Based upon the information acquired intermediaries submit their trading orders to market makers, who setup prices and execute orders. There is one market maker for each asset and they are risk neutral and subject to Bertrand competition which means that they set the assets’ prices equal to their expected value conditional on the orders received. The presence of noise traders who make

220 random investment decisions guarantees that it is pro…table for intermediaries to acquire information, i.e. assets’prices won’t perfectly reveal their costly information. In equilibrium each intermediary will make portfolio decisions in order to maximize their expected utility; given a price function used by market makers; given his contract and his rival’s contract; and given the investors’ choice about in which asset they will trade. Investors in turn make the asset choice and design contracts taking into account the intermediaries’participation and incentive compatibility constraints. The author de…nes herding as a situation in which both investors assign their intermediary to trade in the same asset. If investors assign their intermediary to trade in di¤erent assets then each intermediary will act as a monopolist and will have better chance to exploit their private information. However, the existence of moral hazard means that contracts will have to satisfy the intermediaries’incentive compatibility constraints. Since intermediaries are risk averse increasing the percentage fee of pro…ts they gain will induce intermediaries to take more conservative portfolio decisions. This is related to the irrelevance result studied by Stoughton (1993) and Admati and P‡eiderer (1997). In these two works increasing the contract’s percentage fee has no e¤ect on intermediaries behavior. However, in Graham’s model the irrelevance is not complete since the intermediary’s portfolio choice will have an e¤ect on assets’ prices. This means that intermediaries will face a trade o¤ between hedging the risk implied by their contract and exploiting their private information. In order to avoid this type of behavior from intermediaries managers have the option of making some of the formers’pro…ts a function of their rival’s performance. The author assumes that if two intermediaries trade in the same asset the signal received is

221 correlated. This means that by making remuneration depend in relative as opposed to individual performance, intermediaries can be hedged against the risk of receiving useful although not perfect information. For example, if the intermediary’s information suggests buying the asset and the decision turns out to be wrong, the investor can observe if the other intermediary also made a bad decision, thus restraining from punishing his manager. Of course, in order to use relative compensation both intermediaries must trade in the same asset, i.e. there must be herding. There is a downside in making the intermediaries herd, since this will induce competition between intermediaries and principals. In particular, it is possible that one investor sets a relative performance contract strategically to induce aggressive trading by his intermediary, if this makes the other intermediary to reduce his trading intensity. Whether the investors choose to induce herding or not will depend on a tradeo¤ between alleviating the risk aversion e¤ects that absolute compensation causes and the competition e¤ect which makes the intermediaries’private information less pro…table. The form of the optimal contract under herding is not derived but the author uses numerical examples to show the situations in which inducing herding is desirable. This would be the case if intermediaries’ risk aversion is high , if their information’s quality is poor and if the cost of acquiring information is high. In the …rst case the ine¢ ciencies of using compensation based on absolute performance are exacerbated and herding is desirable. In the second case something similar happens but this time due to the fact that intermediaries are subject to greater risk resulting from their information lack of accuracy. In the last case since acquiring information is costlier the intermediaries would have to receive a greater

222 percentage of trading pro…ts which, given that they are risk averse, will lead to suboptimal decision making. Gümbel shows that if investors could, they would choose not to make their intermediaries’performance information available to the rest of agents in order to avoid the use of relative performance compensation and the increased competition that it brings about. This could be avoided by regulatory requirements such as those existing in the mutual funds market.

A.8 A.8.1

Other Topics Mutual Fund Performance and Persistence Berk and Green (2004) Berk and Green provide a rationalization for several stylized facts observed in the

delegated portfolio management market, such as asymmetric ‡ow-performance relationships and lack of persistence in returns. In order to explain this regularities the authors model an economy where all participants are symmetrically informed. There are investors who delegate wealth to managers, who are heterogeneous in terms of the ability they posses to generate positive excess expected returns. This ability is initially unknown to investors and managers. Instead, they both update their beliefs regarding this treat as time goes by and each manager builds a record of returns earned by their portfolios. A key aspect of this model is that skilled managers are able to generate abnormal returns through their stock picking skills. However, as the size of assets under management increases, managers’ ability to deliver high return to investors will decrease. This is an

223 exogenous assumption that captures the notion that as portfolios get bigger a manager will be forced to spread his stock-selecting activities too thin and also larger trades will be associated with a larger unfavorable price impact and higher execution costs. This means that, even though managers are skilled, it is possible that the …nal return paid to investors is lower as assets under management are bigger. Additionally, Berk and Green assume that investors are willing to supply capital with in…nite elasticity to funds that have positive excess expected return. This means that if a manager has a good investment record then all agents will assign a higher probability to the manager being skilled, and thus capable of generating positive excess returns. Therefore investors will supply new capital to this manager, who will be in charge of a bigger portfolio. This, however decreases his ability to deliver high excess return. In fact, in equilibrium all investors will end up obtaining the same expected excess return across managers, and this will be equal to zero. This means that there will be low persistence in performance, as perceived by investors (this is consistent with the evidence provided by Gruber, 1996). Also, note that this means that a good performance will lead to in‡ows for the managers, while a bad performance results in out‡ows. Moreover, the authors show that in their model this relationship is stronger for younger funds than for mature ones, which is also consistent with the stylized facts (see e.g. Chevalier and Ellison, 1997). The model also allows the authors to study the life cycle of funds. In fact, it is shown that as funds survive and grow the manager will invest an increasingly larger portion of his portfolio passively. This will lead to the fund presenting less idiosyncratic volatility and lower attrition rates. Performing calibration exercises Berk and Green’s model

224 predicts that the survival rate for fund decreases with age, which is a feature corroborated by the data. However, for older funds the model predicts much higher survival rates than those observed in their mutual funds sample. The authors suggest that this could be due to managerial turnover within mutual funds, such as good managers being promoted or defecting to other …rms. As a result, the low survival rates shown by the data could re‡ect a renewal in the learning process about managers’abilities. Finally, the authors also …nd evidence of a large percent of managers (about 80%) in their sample who show some degree of ability to generate positive excess return. Of course, the …nal return perceived by investors is much lower since as they compete to hire skilled managers they dissipate away such returns.

A.8.2

Multiple Agency Layers Gervais, Lynch and Musto (2005) So far the discussed works have focused on an agency problem between investors

and …nancial intermediaries. In practice, however, investors usually make a contract with a fund management …rm, which in turn hires some manager to make the portfolio decisions. This implies the existence of two delegation layers. Gervais, Lynch and Musto model this kind of situation in a setup where portfolio managers have the option of associating himself with a fund family. The authors assume that initially neither the managers themselves know their true types. As time goes by investors, managers and fund families gain knowledge about managers’abilities. However, investors take more time to learn than fund families. Therefore, the existence of families is useful if they adopt an appropriate …ring policy because they could increase the credibility of the managers they hire. For instance if the

225 family employs to managers and commits ex ante to …re one of them, investors know that the family will be better o¤ keeping the more skilled managers, since family’s revenue are positive function of the managers it hires. This reduces the problems caused by the existence of adverse selection.

A.9

Summary As we have seen the literature on the delegated portfolio management problem is

vast. There are several works (e.g. Bhattacharya and P‡eiderer, 1985, Stoughton, 1993, Carpenter et al, 2001, Ou-Yang, 2003) that attempt to derive closed-form solutions for optimal contracts between investors and …nancial intermediaries. It is clear from the above discussion that such contracts are derived under special conditions, such as particular utility functions. Hence the results lack generality. Moreover, sometimes the predictions made are not robust. In particular, the prediction that contracts should use symmetric performancebased fees is made by the above mentioned authors, such as Bhattacharya and P‡eiderer (1985) and Stoughton (1993). However, the works by Carpenter (2000), Ross (2004) and Panageas and Wester…eld (2007) highlight the fact that asymmetric contracts do not necessarily lead to ine¢ cient portfolio decisions by …nancial intermediaries. Also, a number of works such as Ou-Yang (2003) suggest that using benchmarks and relative compensation may be bene…cial for investors. On this point the stylized facts suggest that this type of performance-based contracts are not widely used, at least in the mutual fund industry (see Golec, 1992, Blake et al, 2003, and Cuoco and Kaniel, 2007). Also, the benchmarks are typically simple indexes such as the S&P 500 while the theoretical models suggest the use

226 of more sophisticated benchmarks that are functions of the intermediaries’promised return (Bhattacharya and P‡eiderer, 1985, and Stoughton, 1993), depend on the intermediary’s degree of risk aversion (Admati and P‡eiderer, 1997) or are used to punish those intermediaries with performance above the benchmark (Bhattacharya, 1999). An exception on this line is Gómez and Sharma (2005), who do suggest that simple benchmarks could be useful to align incentives if the intermediaries’trading strategies are restricted. Nevertheless the theoretical works have been successful in rationalizing stylized facts such as high transaction volume by institutional investors (Dow and Gorton, 1997). There is also substantial e¤ort being done to understand how the existence of delegated portfolio management may a¤ect assets’ prices (Cuoco and Kaniel, 2007, Goldman and Slezak, 2003). The existing literature also suggests several reasons for intermediaries to imitate the investment decisions of rivals, such as infering private information (Banerjee, 1992, Bikhchandani et al, 1992) and payo¤ externalities when the intermediary is evaluated against some benchmark (Maug and Naik, 1995). Also, there are several works showing that herding is a robust phenomenon and would prevail even in more realistic setups with endogenous prices (Avery and Zemsky, 1998, Beaudry and González, 2003, and Cipriani and Guarino, 2003) or endogenous ordering (Chari and Kehoe, 2000, and Zhang, 2006). While herding is often associated with ine¢ ciencies since it implies that intermediaries make no use of their private information, it may be the case that this phenomenon is bene…cial, as suggested by Gümbel (2005), since it may make rewarding intermediaries less costly using relative performance remuneration schemes.

227 One of the reasons for intermediaries to herd that has received particular attention in the literature is that of reputational concerns. The works by Scharfstein and Stein (1990), Avery and Chevalier (1999), Graham (1999) and Ottaviani and Sørensen (2006) show how intermediaries worried about their reputation may herd instead of using their private information. In fact, Dornbusch et al (2000) suggest that this is one of the contagion mechanisms that operated during the Asian crisis. This view contrasts with that of Heinkel and Stoughton (1994), Chemmanur and Fulghieri (1994) and Farnsworth (2003), who argue that the presence of implicit incentives provided by reputation may alleviate the ine¢ ciencies caused by informational asymmetries even without the use of bonus of performance based fees. Also, the predictions about the relation between reputation and incentives to herd are mixed; Avery and Chevalier (1999) predict a negative relation while Graham (1999) predicts that this relation is positive. There is also mixed evidence with Chevalier and Ellison (1999) and Hong et al (2000) validating the prediction by Avery and Chevalier (1999) and Graham (1999) presenting evidence supporting his own predictions. Given the existent lack of consensus regarding the e¤ects of the possibility of investing in reputation in a delegated portfolio management context we make a contribution by studying the relation between reputation and herding in a delegated portfolio management context recognizing that investing in reputation is a slow process that takes place over several periods and that absent some source of permanent uncertainty about the intermediaries’characteristics steady state reputational equilibria cannot exist. We thus follow the methodology developed by Mailath and Samuelson (1998), (2001) and Vial (2008), which hasn’t been applied before in a delegated portfolio management context with herding.

228

Appendix B

MATLAB Code

%This program solves de skilled …nancial intermediary’s decision problem %between investing in information or herding. %The program asks for an initial guess for critical reputation value below %(above) which intermediaries choose to herd (invest in information). On %output the program gives the correct critical value as well as information %regarding the economy’s risk free return, the reputation for which %function omega equals zero (Mu hat); as well as the lowest and highest %values reputation can take (Mu min and Mu max respectively). The program %also plots the value function, price function, dynamic system that %describes the evolution of beliefs, and functions wmu and vmu. clear; %Setting parameter values. Values in parentheses correspond to baseline %case global k W p PH y rh rl q mucrit Eta Ri Rh Lambda Theta c Ph yh rh rl q Lambda = 0.15; Theta = 0.5; Eta = 0.45;

%Replacement probability (0.15) %Skilled FI mass (0.5) %Successful imitation probability (0.5)

229 PH = 0.85;

%Good signal probability given r=rH if FI invests (0.85)

y = 1-PH; p = 0.5;

%r=rH unconditional probability (0.5)

rh = 2.2;

%Risky asset pay in good state (2.1)

rl = 1;

%Risky asset pay in low state (1.2)

q = 1;

%Risky asset price (1.1)

W=600;

%Wealth under management (600)

c = 3;

%Information investment cost (1.25)

Ph = 0.5*(1+Eta*(2*PH-1));

%d=g probability given FI herds

yh = 1-Ph; Delta = 0.75;

%Discount factor (0.75)

Phi = Delta*(1-Lambda);

%Discount factor adjusted by replacement prob.

R = p*(rh/q)+(1-p)*(rl/q);

%Ex ante expected return (1.55)

Ri = R+ p*(1-p)*(2*PH-1)*((rh-rl)/q); Rh = R+ Eta*p*(1-p)*(2*PH-1)*((rh-rl)/q); k = Rh;

%Expected return given FI invests %Expected return given FI herds

%Price function constant

Mumax = Theta;

%(0.925)

%Discretizing Reputation points = 1000; Mu = linspace(0,1,points)’; %Initial reputation values N = size(Mu); %Reward matrix

230 f = zeros(points,2); %Transition probability matrix if FI invests in information Mi = zeros(points,points); %Transition probability matrix if FI herds Mh = zeros(points,points); %Initial Mu* guess mucrit = input(’Mu* Guess: ’) %Obtaining reputation value one period ahead with "mub" and "mug" functions for i = 1:N(1,1) Mugraw(i,1) = mug(Mu(i,1)); %d=g Mubraw(i,1) = mub(Mu(i,1)); %d=b end %Nearing next period’s reputation to closest value in grid for i = 1:N(1,1) [Mugindex(i,1), Mugindex(i,2)] = min(abs(Mu-Mugraw(i,1))); Mug(i,1) = Mu(Mugindex(i,2)); [Mubindex(i,1), Mubindex(i,2)] = min(abs(Mu-Mubraw(i,1))); Mub(i,1) = Mu(Mubindex(i,2)); end %Filling Mi matrix for i = 1:N(1,1) if Mugindex(i,2) == Mubindex(i,2)

231 Mi(i,Mugindex(i,2)) = PH + y; else Mi(i,Mugindex(i,2)) = PH; Mi(i,Mubindex(i,2)) = y; end end %Filling Mh matrix for i = 1:N(1,1) if Mugindex(i,2) == Mubindex(i,2) Mh(i,Mugindex(i,2)) = Ph + yh; else Mh(i,Mugindex(i,2)) = Ph; Mh(i,Mubindex(i,2)) = yh; end end %Obtaining transition probability matrix M = cat(1,Mi,Mh); %Filling reward matrix using "fee" function for i = 1:N(1,1) f(i,1) = (fee(Mu(i,1)))*Ri*W-c; f(i,2) = (fee(Mu(i,1)))*Rh*W; end

232 %Packing model structure model.reward = f; model.transprob = M; model.discount = Phi; %Solving model using Compecon Toolbox [v,a] = ddpsolve(model); %Obtaining function wmu for i = 1:N(1,1) wmu(i,1) = c-(fee(Mu(i,1)))*(Ri-Rh)*W; end %Obtaining function vmu for i = 1:N(1,1) VMug(i,1) = v(Mugindex(i,2)); VMub(i,1) = v(Mubindex(i,2)); end vmu = (VMug - VMub).*(Delta*(1-Lambda)*(PH-Ph)); %Obtaining Bayes rule for i = 1:N(1,1) Mudg(i,1)= mug(Mu(i,1)); Mudb(i,1)= mub(Mu(i,1)); end %Finding muhat (Reputation value for which wmu=0)

233 Muhat=((W*(Ri-Rh)*(k-Rh)+c*Rh))/((Ri-Rh)*((Ri-Rh)*W-c)); %Obtaining price function for i = 1:N(1,1) bmu(i,1) = fee(Mu(i,1))*10000; end %Plotting …gure (1); plot(Mu,v); xlabel(’Reputation’) ylabel(’$’) title(’Value Function’) axis([0, 1, -Inf, Inf]) …gure(2); plot(Mu,wmu,Mu,vmu,’–’); xlabel(’Reputation’) ylabel(’$’) title(’Functions w(mu) and v(mu)’) legend(’w(mu)’,’v(mu)’,1) axis([0, 1, -Inf, Inf]) …gure(3); plot(Mu,Mudg,Mu,Mudb,Mu,Mu); xlabel(’Reputation in t’)

234 ylabel(’Reputation in t+1’) title(’Bayes Rule’) legend(’Good decision’,’Bad decision’,’45o ’,4) axis([0, 1, 0, 1]) …gure(4); plot(Mu,bmu); xlabel(’Reputation’) ylabel(’Basis Points’) title(’Fee’) axis([0, 1, -Inf, Inf]) %Finding Mu* using the policy function j=1; while a(j,1) == 2; j=j+1; end MuCritic = Mu(j,1); %Finding Mumin difmumin=Mub-Mu; jjj=1; while difmumin(jjj,1)>0; jjj=jjj+1; end

235 Mumin = Mu(jjj,1); Table = [R MuCritic Muhat Mumax Mumin]; disp(’R Mu* Muhat Mumax Mumin’) disp(Table)

236

Appendix C

Additional MATLAB Functions

C.1

Mug Function function [mug] = mug(x) %This function updates initial reputation "x", conditional on a good %investment decision being made (d=b). %Speci…cally, we have x(t+1)=(A*x(t)+B)/(C*x(t)+D) %Setting parameter values global Lambda Theta mucrit Eta PH A = Lambda*Theta*(1-Eta)*(PH-0.5)+(1-Lambda)*PH; B = Lambda*Theta*(0.5+Eta*(PH-0.5)); C = (1-Eta)*(PH-0.5); D = (0.5+Eta*(PH-0.5)); %De…ning function if x

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