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Essays on delegated portfolio management under market imperfections
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Essays on delegated portfolio management under market imperfections
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ESSAYS ON DELEGATED PORTFOLIO MANAGEMENT UNDER MARKET IMPERFECTIONS by Juan Mart n Sotes-Paladino A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) May 2012 Copyright 2012 Juan Mart n Sotes-Paladino To my parents Mimi and Toty, my grandmother Cholita and my brothers Jose, Fede, Guchi and Juanqui, for their limitless support and patience. ii Acknowledgments I would like to thank my advisor and dissertation chair Fernando Zapatero for his constant guidance and encouragement. He has been a role model for the exceptional, yet humble scholar I aspire to be. I would also like to thank the other members of my dissertation committee: Elias Albagli, Pedro Matos, and Jin Ma, for sharing their wisdom and expertise with me. I am particularly indebted to Elias Albagli, whose insightful comments and arguments have constantly pushed me to a deeper level of understanding. His help during my last two years in the program has been invaluable. My special thanks to Luis Goncalves-Pinto; his willingness to lend me a hand over and over again has helped me make this journey easier and, certainly, much shorter. For very helpful suggestions and discussion, I thank Wayne Ferson, Sam Hartz- mark, Chris Jones, Haitao Mo, Marco Navone, Oguzhan Ozbas, Aris Protopapadakis, Marcel Rindisbacher, Antonios Sangvinatsos, David Solomon, Andreas Stathopoulos, Selale Tuzel and Tong Wang. I also appreciate the comments of participants at the FMA European Conference, World Finance Conference, 6th Annual Conference of the FIRS, 6th Portuguese Fi- nance Network Conference, LBS 10th Trans-Atlantic Doctoral Conference, and sem- inar participants at Boston University, EESP-FGV, IE Business School, HEC Lau- sanne, Indiana University, University of Melbourne, University of Miami, University of New South Wales, University of Southern California, and Virginia Tech. Finally, I cannot thank enough to Ana Pagani for her unconditional support and understanding. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abstract viii Chapter 1: Should We Expect Superior Managers to Be Stars? (Dis)Incentive Eects of Fund Flows in Money Management 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Investment Companies and Fund Flows . . . . . . . . . . . 10 1.3 Optimal Investment Strategies . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Peer Group's Performance: Uninformed Managers' Equilib- rium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Informed Managers' Trading and Payo Prole . . . . . . . 18 1.4 Average Risk-Taking and Performance . . . . . . . . . . . . . . . . 29 1.4.1 Managerial Risk-Taking . . . . . . . . . . . . . . . . . . . . 30 1.4.2 Measured Performance . . . . . . . . . . . . . . . . . . . . . 33 1.4.3 Fund Flows and Herding: Testable Implications . . . . . . . 36 1.5 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.6 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 2: The Value of Cross-Trading to Mutual Fund Families: A Port- folio Choice Approach 55 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.2.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.2.2 The Fund Family's Problem . . . . . . . . . . . . . . . . . . 67 2.3 Numerical Analysis and Discussion . . . . . . . . . . . . . . . . . . 71 2.3.1 Cross-Trading and Family Portfolio Strategies . . . . . . . . 74 iv 2.3.2 Utility Implications . . . . . . . . . . . . . . . . . . . . . . 92 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Chapter 3: A Rationale for Benchmarking in (Informed) Money Management 101 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.1 Financial Markets and Information Structure . . . . . . . . 105 3.2.2 Managerial Contract . . . . . . . . . . . . . . . . . . . . . . 107 3.3 Portfolio Choice Problem . . . . . . . . . . . . . . . . . . . . . . . 109 3.4 Optimal Contracting Problem . . . . . . . . . . . . . . . . . . . . . 113 3.4.1 Identical Risk Preferences . . . . . . . . . . . . . . . . . . . 116 3.4.2 General Risk Aversion . . . . . . . . . . . . . . . . . . . . . 117 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Bibliography 124 Appendices 130 Appendix A: Technical details of Chapter 1 . . . . . . . . . . . . . . . . 130 Appendix B: Technical details of Chapter 3 . . . . . . . . . . . . . . . . 143 v List of Tables Table 1.1: Summary statistics. . . . . . . . . . . . . . . . . . . . . . . . . . 40 Table 1.2: Summary statistics for top performers. . . . . . . . . . . . . . . . 43 Table 1.3: Flow-performance relationships, 5-piece specication. . . . . . . . 46 Table 1.4: Flow-performance relationships, 3-piece specication. . . . . . . . 47 Table 1.5: Flow-performance relationships: robustness checks. . . . . . . . . 51 Table 2.1: Initial and Expected Terminal Portfolio Weights on Illiquid Asset C by Fund 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Table 2.2: Risk-Shifting and Return-to-Risk Ratios. . . . . . . . . . . . . . 90 Table 2.3: Utility Implications of Portfolio Delegation to Family-Aliated and Standalone Funds and Net Eects of Cross-Trading to the Family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Table 3.1: Optimal Managerial Contract and Households' Derived Utility. . 119 vi List of Figures Figure 1.1: Managers' interim risk exposure . . . . . . . . . . . . . . . . . . 24 Figure 1.2: Optimal Payo Proles . . . . . . . . . . . . . . . . . . . . . . . 27 Figure 1.3: Average excess tracking error and portfolio volatility (% annual) 31 Figure 1.4: Informed managers' performance in excess of uninformed peers' 35 Figure 1.5: Flow-performance relationships . . . . . . . . . . . . . . . . . . 49 Figure 2.1: CIO's Utility from Cross-Trading for Dierent Correlation Coef- cients between the Liquid Asset Returns. . . . . . . . . . . . . 75 Figure 2.2: CIO's Utility from Cross-Trading for Dierent Correlation Coef- cients between the Illiquid and the Liquid Asset Returns. . . . 76 Figure 2.3: Distribution of Investment Period Returns for Fund 1. . . . . . 79 Figure 2.4: Distribution of Investment Period Returns for Fund 2. . . . . . 80 Figure 2.5: Correlation between Funds 1 and 2 Investment Period Returns, for Dierent Correlation Coecients between the Liquid Asset Returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 2.6: Correlation between Funds 1 and 2 Investment Period Returns, for Dierent Correlation Coecients between the Illiquid and the Liquid Asset Returns. . . . . . . . . . . . . . . . . . . . . . . . . 85 Figure 2.7: Relative Performance and Risk-Taking for Fund 1. . . . . . . . 87 Figure 2.8: Relative Performance and Risk-Taking for Fund 2. . . . . . . . 88 Figure 2.9: Utility Implications of Portfolio Delegation to Family-Aliated Funds and Net Eects of Cross-Trading to the Family. . . . . . 97 Figure 3.1: Excess Certainty Equivalent Return from Delegation. . . . . . . 120 vii Abstract This dissertation consists of three chapters of interrelated work in the area of delegated portfolio management. The common topic across the three chapters is the interaction of agency con icts in the relation between a delegating household and a portfolio manager on the one hand, and additional market frictions such as asymmetric and incomplete information or illiquidity on the other hand. The approach is primarily analytical and based on structural models of dynamic portfolio choice. Since the focus is ultimately applied, this dissertation contains both theoretical and empirical results. In the rst chapter I characterize analytically the investment policy of money managers with superior investment ability when fund ows are a convex function of end-of-period performance relative to peers. I show that skilled managers adopt contrarian strategies with respect to peers early in the period until a desired out- performance margin is achieved. Thereafter, they hedge against relative underper- formance by investing like the herd. Convex fund ows may then induce excessive risk-taking in certain circumstances but excessive conservatism on average, leading the best managers to look like average performers. More generally, small dierences in the ow-performance relationship can result in substantially dierent average risk-taking (herd vs. contrarian) behavior and return proles by identically skilled managers. I argue that traditional measures may fail to adjust fund performance for the type of risks these managers take. Using a sample of top performers in the US mutual fund industry I present evidence supporting the model-implied relation between funds' herd/contrarian behavior and their ow-performance relationships. viii The second chapter is joint work with Luis Goncalves-Pinto. It looks into the eects on portfolio delegation of the additional incentive misalignments between del- egating households and mutual fund managers that may appear when funds belong to a family organization like Vanguard or Fidelity. We focus on the possibility that family-aliated funds cross their trades of illiquid common holdings in response to the interests of the family as a whole, and investors' ows are a convex function of rela- tive returns. In such situations, we argue that fund families can play favorites among their members by having some funds reduce the costs of illiquidity while making others adopt suboptimal investment decisions. This eect makes the asset allocation decision of family-aliated funds substantially dierent from that of standalone coun- terparts, a feature that has been neglected by the literature on portfolio delegation so far. By including this feature in our framework, we nd that families' ability to cross-trade among member funds allows them to save on transaction costs but at the same time elicits higher risk-taking by aliated fund managers compared to identical standalone funds. We further nd that the additional costs of agency that investors incur under a fund family arrangement are likely to increase with asset liquidity. Our study has potential normative implications, as we show that investors can be better o in general when imposing position limits on their funds' portfolios. In the third chapter, I study the optimal design of compensation fees for a money manager when both she and delegating households face incomplete information about asset returns, but the manager has access to private information. This informational asymmetry is a potential source of value from delegation, and households' objective it to induce the manager to use her superior information in their best interest. The analysis focuses on the typical contracts observed in the industry, which include a pro- portional asset-based fee and benchmark-linked incentive fees of two types: (i) a linear (\fulcrum") fee, and (ii) a convex (\option-like") fee. I provide closed-form solution for the manager's optimal dynamic asset allocation over a xed investment horizon when she learns asset fundamentals over time, and analyze numerically the contract ix that maximizes households' utility from delegation. In contrast to prior literature, I show that simple benchmarking rules are highly valuable in allowing households to exploit manager's superior information in their favor. When the manager's risk- tolerance diers from households', the optimal contract|within this class|is linear and always includes a benchmark-linked fulcrum fee. In order to oset insucient or excessive risk-taking by the manager, the optimal benchmark has either a much higher or a much lower risk exposure than the unconditionally ecient portfolio the uninformed households would choose under self-management. I further show that option-like fees dominate pure proportional asset-based fees when the manager is more risk-averse than households. The optimality of benchmarked-linked fulcrum fees is robust to dierent precisions in manager's information and to dierent invest- ment horizons. Keywords: Portfolio Delegation, Mutual Funds, Incomplete Information, Fund Flows, Herding, Performance Evaluation, Liquidity, Mutual Fund Families, Internal Capital Markets, Cross-Trading, Portfolio Constraints, Risk-Taking Incentives, Benchmark- ing, Management Fees, Optimal Contracting. JEL Classication: C61, C63, D60, D81, D82, D83, G11, G12, G23, G30, G32. x Chapter 1 Should We Expect Superior Managers to Be Stars? (Dis)Incentive Eects of Fund Flows in Money Management 1.1 Introduction A well-established conclusion in the money management literature is that, induced by the empirically documented convex relationship between fund ows and relative performance, managers may unnecessarily shift fund risk in response to changes in their relative standing. 1 By shifting risk, managers maximize their chances to achieve outstanding returns and enjoy large money in ows in the future, but have delegating investors bear too much risk as well. In this chapter, I argue that convex fund ows may induce excessive risk-taking in certain circumstances but result in excessive conservatism on average, leading high-skill managers to attain just above-average returns. The argument takes into account that a manager may indeed gamble when her interim relative standing is very low but plays safe instead when this is suciently high, and hinges on recognizing that such relative standing is not exogenous to the manager but a function of her investment skills relative to the ability of all others. Within a standard portfolio choice framework, I derive the optimal trading strat- egy of managers with superior ability as they compete for investors' money ows against a large pool of peers. By determining peer performance endogenously, I char- acterize the skilled managers' relative performance as a function of the economy's 1 See, among others, Brown, Harlow, and Starks (1996), Chevalier and Ellison (1997), Taylor (2003), Kempf and Ruenzi (2008), Chen and Pennacchi (2009), and Huang, Sialm, and Zhang (2011). 1 driving states. I then use the probability distribution over the dierent states to assess, on a ex-ante (unconditional) basis, skilled managers' expected risk-taking, end-of-period fund returns, and households' utility from delegation, as a function of the ow-performance relationship they face. I explore the implications of skilled man- agers' strategies for performance measurement. I further derive predictions about the relation between managerial behavior and fund ows and test them over a sample of actively managed US mutual funds. I assume a simple nancial market consisting of one risky asset (a stock) and a risk-less bond. The stock mean return is a random variable realized at the beginning of the investment period, while its volatility and the risk-free rate are constant and known. The asset management industry in the model has a large number of managers, a relatively small fraction of which are \informed" (skilled) managers that observe the realization of the stock mean return. The rest are \uninformed" (unskilled) managers that only know the mean return prior distribution and learn its actual value from the observation of prices over time in a Bayesian fashion. 2 Whereas the emphasis in this chapter is on the strategies of the informed, uninformed managers' equilibrium poli- cies aect the peer group against which each fund's performance is assessed. Both types of managers have identical constant relative risk aversion (CRRA) preferences and dynamically allocate their funds' wealth between the two assets during a xed investment period to maximize utility over end-of-period compensation. This com- pensation is proportional to the terminal value of assets under management, which in turn depends on a fund's relative performance with respect to other funds through a ow-performance relationship. To capture dierent ow sensitivities at dierent lev- els of past returns according to the empirical evidence (Chevalier and Ellison (1997), Sirri and Tufano (1998), Huang, Wei, and Yan (2007) and Ivkovich and Weisbenner (2009) for mutual funds, Agarwal, Daniel, and Naik (2004) and Ding, Getmansky, 2 The qualitative results in this chapter do not hinge on the particular nature of the informational advantage, as long as the informed managers have the ability to deliver higher ex-ante expected risk-adjusted returns. 2 Liang, and Wermers (2008) for hedge funds), I assume this relationship is at as a function of relative performance up to a certain threshold, but increasing above it at a higher or lower rate depending on a given ow-performance elasticity. I derive the equilibrium strategies of all managers and characterize their dynamic trading and performance in closed-form. Uninformed managers' performance de- termines the peer group of the informed managers in the model. In a symmetric equilibrium in which managers of the same type choose identical strategies, a rst implication of the model is that ow-performance relationships have no eects on the strategies of the uninformed managers. Absent an informational advantage or, more broadly, absent heterogeneity in managers' preferences and incentives, individual at- tempts to outperform peers are too risky for these managers, who then invest as if they were trading for their own accounts. Uninformed managers' inference about the stock mean return drives their fund's risk exposure, boosting it when markets boom and cutting it down when markets plummet as a result of Bayesian learning. The consequent \trend chasing" behavior leads these managers to end the period with either a high risk exposure in bull markets or with a conservative risk exposure in bear markets. Informed managers' optimal portfolio includes a \contrarian" component that takes an aggressive opposite stance against peers' risk exposure (overweighting the stock when peers underweight it and vice versa when these overweight it), but also a \herding" component that partially replicates the peers' portfolio. Since investors evaluate relative performance at end-of-period, informed managers' dynamic trading strategy attaches a higher weight in the contrarian position in the rst part of the pe- riod, until the outperformance margin necessary to rank as high performer is achieved. The contrarian position can increase the absolute risk of their portfolio in some situa- tions, but also decrease it signicantly in others. Once on the upward-sloping portion of the ow-performance relationship, managers become highly risk-averse and hedge against the risk of underperforming to peers by increasing the weight in the herding 3 component. For empirically plausible ow-performance relationships and economic settings, I show that informed managers are likely to secure a sucient outperfor- mance margin early in the period and herd thereafter, resulting in very conservative average behavior. Moreover, herd behavior by informed managers can result in ex- pected returns in excess of less informed peers' as low as one third of what they could achieve if they were trading for their own accounts. More generally, informed managers' trading strategies under slightly dierent ow- performance relationships can result in substantially dierent end-of-period perfor- mance proles. These proles are highly non-linear functions of uninformed managers' returns in all cases. Since uninformed managers' trend-chasing strategy delivers a high performance in both extraordinarily good and bad markets, informed managers out- perform only in average markets in which asset returns do not deviate much from fundamentals. The consequent performance proles resemble the payos of an op- tion strategy that delivers gains when uninformed managers' returns do not uctuate much but large losses in volatile times. From the point of view of delegating households, herd managers invest too conser- vatively whereas contrarian managers take excessive risk. The agency problem is gen- erally signicant only in the second case: the informational advantage of skilled man- agers with a tendency to herd more than osets the costs of their conservative policies, but the opposite occurs|households incur large agency costs from delegation|for managers adopting contrarian strategies. For instance, under a typical model param- eterization a skilled manager without ow concerns delivers an average annual return of 34 basis points, on a certainty equivalent basis, in excess of unskilled peers. For ow-performance relationships that induce aggressive contrarian trading on average, the same manager can turn these benets into net annual costs as high as 100 ba- sis points. Furthermore, simulations of the model show that standard performance measures most likely fail to adjust for the type of risks these managers take due to 4 the non-linearities in the relation between their performance and the economy's driv- ing factors, as well as the resulting non-normality of their fund returns. Specically, Sharpe ratios and Jensen's alpha computed over time series of fund returns can un- derstate risk-adjusted performance for some ow relationships and overstate it for others. This bias is particularly pronounced for managers facing ows with high sen- sitivity to \stellar" returns, a feature present in the relationship estimated by several authors. 3 The model yields testable predictions relating skilled managers' average herding or contrarian behavior to the specic ow-performance relationship they face: (i) when ows are sensitive to medium relative performance, informed managers have a tendency to herd, and average herding is increasing in ow convexity; by contrast, (ii) when ows are insensitive to medium relative performance informed managers favor contrarian strategies, the more so the more convex is the ow-performance relation- ship. I take these implications to a sample of actively managed U.S. equity mutual funds during the period 1981-2010. I identify \informed" funds in the sample with a group of top performers during this period. Based on dierent measures of average herding, I select a group of \herding" and another group of \contrarian" funds out of the sample of informed mutual funds. In the absence of a conclusive theory re- lating fund ows to fund characteristics, I test whether any discrepancy between the fund- ow relationships faced by herding and contrarian informed funds in the data agrees with the dierences that could be expected from predictions (i) and (ii). I approximate the ow-performance relationships for these two groups by estimating piecewise linear regressions. In agreement with the model's predictions, I nd that ows to herding funds are more sensitive to middle-range performance, inducing sig- nicant relative concerns, while ows to contrarian funds display higher sensitivity on the top, consistent with a more convex ow-performance relationship. These ndings 3 This is the case, for instance, of the mutual funds in Ivkovich and Weisbenner (2009), and of the high-participation-cost funds in Huang, Wei, and Yan (2007). 5 suggest that the identied eects of dierent ow-incentives on managerial behavior and performance evaluation are of practical relevance in the mutual fund industry. The model developed in this chapter can help explain two phenomena that have drawn the attention of the literature recently: institutional investors' holdings of \bubble" stocks during the late 1990s, and \closet-indexing" by active mutual funds. 4 Bayesian learning by the uninformed managers in the model leads them to increase signicantly their exposure to stocks that have experienced sustained abnormally high returns (relative to their true but unobservable mean rate) in the past, very much like we observe during bubbles in asset prices. However, informed managers that are fully aware of this bubble in the model may still choose to \ride" it if their ows are sensitive enough to middle-range performance or show low convexity. Similarly, if the average (unskilled) fund manager within an objective category follows a benchmark closely (i.e. is a closet indexer) and money ows in this category are very sensitive to medium relative performance, the model predicts that a skilled manager in the same category may choose to optimally herd with the average fund and may appear to be a closet indexer in the data. This chapter is closely related to the literature on risk-taking behavior of mutual funds in a tournament setting. Basak, Pavlova, and Shapiro (2007, 2008) nd that managers subject to convex ow-performance relationships gamble to nish ahead of their benchmark but only over a nite-range of interim performance. 5 Chen and Pen- nacchi (2009) argue that portfolio managers have no incentives to increase the overall volatility of their portfolios but only the variance of the tracking error with respect to the benchmark portfolio. Cuoco and Kaniel (2011) investigate the equilibrium asset pricing implications of commonly used management compensation contracts. Basak 4 See e.g. Brunnermeier and Nagel (2004) and Dass, Massa, and Patgiri (2008) on institutional investors and bubbles, and Cremers and Petajisto (2009) on closet-indexing. 5 Although not in a tournament setting, Carpenter (2000) studies the incentives provided by option- like compensation within a dynamic portfolio choice approach and shows that option compensation does not strictly lead to greater risk seeking. 6 and Makarov (2011) analyze the strategic interactions that emerge from tournaments among managers for investors ows and nd an equilibrium in which managers' poli- cies are driven by chasing and contrarian behaviors in some situations, and by gam- bling behavior in others. Since the focus is on the risk-seeking incentives of portfolio managers, this literature assumes symmetric information both among managers and between managers and delegating households. Risk-shifting is then equivalent to pure gambling, and in addition households always derive negative returns from delegation. In order to reconcile the negative results in these models with the vast size of the asset management industry in practice, an implicit assumption that delegation brings about large savings on transaction costs to households is needed. By incorporating asym- metric information among managers and between these and investors, risk shifting in this chapter can be interpreted as contrarian herding based on superior information and can in principle render delegation valuable despite the incentive misalignments between managers and investors. 6 Furthermore, I show that industry equilibrium considerations may oset the risk-seeking incentives within a large pool of similar managers. Herd behavior by informed portfolio managers with less informed agents in multi- period settings has been analyzed in models featuring career concerns. Scharfstein and Stein (1990) and Zwiebel (1995) show that if managers worry about the markets perception about their ability, they could ignore useful private information in their portfolio choice and imitate the rest of managers. Froot, Scharfstein, and Stein (1992) argue that, since managers have short evaluation periods, they cannot aord to wait until their (eventually correct) private information is revealed and incorporated in asset prices. Short-term managers may then herd by acquiring the same information. Absent concerns with respect to investors' money ows, the threat of dismissal after 6 In a two-stage model with constant absolute risk aversion (CARA) preferences, Maug and Naik (2011) show that performance concerns relative to peers in managerial contracts may induce in- formed managers to \go with the ow" and disregard their own superior information, a situation that nevertheless may be in the interest of delegating agents. 7 poor performance in reputation concerns models make managers objective a more concave function of relative performance and herding results as an optimal response to increased risk aversion. The model in this chapter shows that herding behavior by skilled managers can arise even when managers' compensation is a convex function of relative performance. 7 The rest of the chapter proceeds as follows. Section 1.2 sets up the nancial markets, preferences and the investment management industry structure. I derive the equilibrium strategies and performance in Section 1.3. I analyze implications for managerial risk-taking and expected performance, along with testable predictions in Section 1.4. I present empirical support for the model's predictions in Section 1.5, and suggest other potential applications of the model's results in Section 1.6. Section 1.7 concludes, while the technical details are summarized in Appendix A.1 through A.3. 1.2 Model Setup I consider an economy in which households delegate their nancial wealth to portfolio management companies over a certain horizon denoted by [0;T ]. I refer to this com- panies as mutual funds, although the analysis in this chapter is equally applicable to hedge funds and other professional money managers whose fund ows are sensitive to past relative performance. Delegation occurs at t = 0 and no additional fund share purchases or redemptions take place untilt =T . All agents have constant relative risk 7 See Arora and Ou-Yang (2001) and Hu, Kale, Pagani, and Subramanian (2011) for models featuring both ow and career concern in a multi-period portfolio management setting. 8 aversion (CRRA) preferences, with coecient > 1 for fund managers and h > 1 for households: 8 u(w) = 8 < : w 1~ 1~ if w 0; 1 if w< 0, (1.1) for ~ 2f ; h g. Although the delegation decision is exogenous to the model, it is grounded on the assumption that some fund managers have private information about asset re- turns and all of them observe asset prices throughout the investment period [0;T ], so households may nd delegation valuable. 9 1.2.1 Financial Markets Mutual funds have access to nancial markets consisting of one risk-free and one risky assets, with prices andS respectively. The risk-less asset can be a short-term bond or a bank account, whereas the risky asset can be a stock or any portfolio of risky assets (e.g. the market portfolio or other traded benchmark). Each mutual fund is an atomistic participant in the asset markets and takes asset price dynamics as exogenously given. The bond has initial price 0 = 1 and pays a constant interest rate r per unit time, such that its price dynamics are d t = r t dt. The stock has initial price S 0 =s, and dynamics given by the following SDE: dS t =S t (dt +dB t ); (1.2) whereB is a standard Brownian motion process dened on a ltered probability space ( ;F;P;fFg tT ), on the interval 0tT . All parameters are constant. 8 The assumption that managers' risk-tolerance is the same for informed and uninformed managers is for notational simplicity only. All qualitative results hold with dierent type of managers featuring dierent risk aversion. 9 In a more general model, investors would be allowed to dynamically choose how much of their portfolio to hold directly and how much to hold indirectly through investment companies. For a model in which investors allocate money to mutual funds dynamically over time, see Hugonnier and Kaniel (2010). 9 The risk-free rate r 0 and volatility > 0 are observable by all mutual fund managers (and households), as are security prices. The stock mean return, however, is the unobservable realization att = 0 of a random variable with normal distribution N(r +m; 2 v 0 ), for some given constants m and v 0 > 0. Equivalently, the \market price of risk" (r)= is an unobservable draw from a normal distribution N(m;v 0 ) at t = 0. 1.2.2 Investment Companies and Fund Flows I assume an investment company industry consisting of a large pool of mutual funds that, except for their managers' ability, are otherwise identical. These mutual funds can be seen as a relatively homogeneous group following the same objective category (e.g. \Growth") and regarded as close substitutes by delegating households who, in determining a fund share purchases or redemptions at t = T , compare this fund performance to that of all others in the same group. To isolate the eects of relative performance concerns on managers' asset allocation decisions, I assume that this group of funds is suciently small relative to the overall asset markets and has no signicant impact on prices. Fund managers can be skilled or unskilled. I model superior skill/investment ability as access to private information about the realization of the market price of risk at t = 0. Therefore, all fund managers belong to either one of two sets: a set I of informed managers (type \I" or \I-managers") that observe (equivalently, the stock mean return) att = 0, and a setU of uninformed managers (type \U" or \U- managers") that do not observe but have to infer its value from the information they have available. Since v 0 > 0, uninformed managers face parameter uncertainty. This informational structure could be interpreted as stock-picking ability by the skilled managers in a multi-asset framework, and provides a tractable shortcut to modeling higher ability for some managers while at the same time allowing for the possibility 10 that the rest of them reduce their informational/ability disadvantage over time. 10 All I-managers are identical to each other, as are all U-managers. Furthermore, I assume thatU has unit measure (m(U) = 1) whereasI has zero measure (m(I) = 0). This assumption is meant to re ect an industry structure where superior managers are only a small fraction of the overall population of managers following the same objective category. In particular, it will imply that I-managers' portfolio choices are conditioned by relative performance concerns with respect mostly to uninformed peers. 11 Each manageri of typeJ2fI;Ug dynamically chooses an investment policy J t (i) representing the fraction of the fund wealth W J t (i), or assets under management, to be allocated to the risky asset at time t (fundi's risk exposure). Given initial wealth W J 0 (i) =w J , managers' portfolio value processes follow: dW J t (i) =W J t (i) r + J t (i) dt +W J t (i) J t (i)dB t ; (1.3) for J2fI;Ug. I assume that each manager's portfolio decisions are unobservable by other man- agers, as are fund returns during [0;T ). 12 Thus, uninformed manager's available 10 Proprietary research may lead fund managers to nd protable trading strategies that would characterize such managers as \informed". As this trading strategies becomes public (e.g. the \momentum" strategy) their informational advantage will shrink over time. Similarly, a reduction in the ability wedge between skilled and unskilled managers would occur naturally as the competitive forces in the industry drive the least skilled managers out of the market, only to be replaced by higher ability contestants. 11 For this reason, I shall use the terms \uninformed managers", \uninformed peers", or simply \peers" interchangeably henceforth. 12 This assumption prevents uninformed managers to learn about from the observation of in- formed managers' investment policies throughout the period. Such restriction seems plausible for many institutions like hedge funds that are largely secretive about their trading strategies. While it may seem less realistic for mutual funds given the ready availability of their net asset value (NAV) gures, it becomes more plausible once we account for the many fees and expenses involved in NAV's determination, usually available with some delay (e.g. one month/quarter) and on a less frequent basis than NAV data. 11 informationF S t at each moment of time t 2 [0;T ] is given by the market prices observed up to t:F S t fS u ; 0utg. Fund managers' compensation is due at horizon T and set in proportion to their funds' collected fees, which in turn consist of a fraction of their assets under man- agement. 13 Depending on a fund performance relative to its peers, households are assumed to purchase or redeem additional fund shares according to a ow-to-relative performance relationship that is exogenous to the model. Letting Y represent the average performance of all mutual funds, Y t R I W I t (i)di + R U W U t (i)di fort2 [0;T ], managers receive households' money ows at the rate f T at the end of the period following the functional form in Basak and Makarov (2011): f T =k1 fR J T (i)<R Y T g +k R J T (i) R Y T 1 fR J T (i)R Y T g ; (1.4) where k; > 0, R J T (i) W J T (i)=W J 0 (i), and R Y T Y T =Y 0 , J 2fI;Ug. Note that f T is always positive, with f T > 1 denoting additional in ows and f T < 1 denoting out ows. This relation resembles the convex payo of a call option: for managers underperforming a fraction Q 1 of the industry average return R Y T , fund ows are insensitive to relative performance and arrive (leave) at the constant rate k; for those outperforming the same fraction of the industry average return, fund ows are increas- ing in performance relative to all other mutual funds (peers) at a rate k R J T (i) R Y T . Since k is a scaling parameter that does not modify incentives in the margin, the two key parameters controlling the shape of the ow performance relationship are the performance threshold and the ow elasticity . The performance threshold is the minimum relative performance that has an eect on end-of-period fund ows, whereas the ow elasticity controls the rate of growth of fund ows in the top performance region (i.e. relative performance above the threshold ). 13 This compensation re ects the prevailing fee structure in the mutual fund industry. According to Elton, Gruber, and Blake (2003), 98.3% of all U.S. bond and equity mutual funds and around 89.5% of the assets under management had xed fees as of 1999. 12 Specication (1.4) is exible enough to capture the heterogeneity in the ow- performance relationships documented empirically by e.g. Chevalier and Ellison (1997), Sirri and Tufano (1998) and Ivkovich and Weisbenner (2009) for mutual funds, Agarwal, Daniel, and Naik (2004) and Ding, Getmansky, Liang, and Wermers (2008) for hedge funds, while at the same time capturing their main common features. These authors observe that investors withdraw money from poor performers at an approxi- mately at rate, but supply additional capital to good performers at an increasingly higher rate. In general, their estimations imply a value of k less than one and a ow elasticity greater than one, with the relative performance threshold taking values both above and below one depending on dierent fund characteristics. In all cases, this literature hints at a convex fund- ow relationship that is approximately insensitive to poor past relative performance. 14 Without loss of generality, I normalize the initial wealth of both types of funds to be equal: w I = w U = w, so Y 0 = w. Dening +( 1), (1.1) and (1.4) imply that manager i of type J2fI;Ug optimally chooses her fund's risk exposure f J t (i) : 0tTg to maximize expected utility of nal wealth: u f T W J T (i) = 8 < : k 1 1 W J T (i) 1 if W J T (i)<Y T (underperformance), k 1 1 W J T (i) 1 (Y T ) if W J T (i)Y T (outperformance), (1.5) subject to initial wealth w and the self-nancing constraint (1.3). The objective function (1.5) makes it clear how, whenever W J T (i)6=Y T , the ow-performance rela- tionship (1.4) aects manager i's incentives: in those states of the economy in which anI-manager outperforms the industry average (W I T Y T ), (i) her eective relative 14 The convexity in the ow-to-relative performance relationship may be the result of rational investors learning managerial ability from past performance, along with managers' implicit option to abandon poorly performing strategies (Lynch and Musto (2003)), decreasing returns to scale in money management (Berk and Green (2004)), or costs of participating in mutual funds (Huang, Wei, and Yan (2007)). 13 risk aversion increases to > , and (ii) the increase is larger for more convex (higher ) relationships in the outperformance region. 15 1.3 Optimal Investment Strategies When managers choose their investment policies to solve (1.5), each one considers her peers' optimal policies as determinants of the industry average performance Y . Managers' rationality then requires nding the industry equilibrium (provided this equilibrium exists) in which manager i's strategy (i2I[U) is an optimal response to all other managers' optimal policies. Whereas such equilibrium (equilibria) may be hard to characterize in a more general setting, the assumed small number of informed managers (m(I) = 0) simplies the analysis considerably. 16 In particular, if a symmetric equilibrium exists in which allJ-managers adopt the same investment policy ^ J t , identical initial assets under management w imply that W J (i) = W J for J2fI;Ug. Then: Y t = Z I W I t (i)di + Z U W U t (i)di =W I t m(I) +W U t m(U) =W U t ; (1.6) i.e. each fund's average peer performance in this economy is given by the performance of the uninformed managers. Equation (1.6) implies that U-managers' problem can 15 The increase in risk aversion occurs because changes in actual wealth are augmented by a ow rate k R I T R U T > k in the outperformance region, but only by a ow rate k outside of it. The ow rate for a top performer is not only increasing in , but also itself increasing in wealth. Therefore, eective wealth uctuates more in response to the same change in actual wealth in this region than in the underperformance region, raising manager's eective risk aversion. 16 Equally important in order to nd an equilibrium in this setup is the assumption that unin- formed managers do not observe informed managers' portfolio choice (or assets under management). This assumption rules out strategic interdependence, beyond that induced by the average industry performance Y , between the two types of managers. 14 be decoupled from that of I-managers' and their equilibrium strategies studied in isolation. If such equilibrium exists, the industry average Y in informed manager's problem is completely determined outside this problem and we can look for their optimal policies within the standard portfolio choice approach. The next two sections characterize an equilibrium in uninformed and informed managers' strategies. All proofs are given in Appendix A.1. 1.3.1 Peer Group's Performance: Uninformed Managers' Equilib- rium Uninformed managers have only partial information about the market price of risk but can learn about it by observing realized stock returns over time. The information structure I assume is the same as that considered by, among others, Brennan (1998) and Cvitanic, Lazrak, Martellini, and Zapatero (2006), and represents a particular case of the incomplete information case in continuous-time studied by Detemple (1986, 1991), Dothan and Feldman (1986) and Gennotte (1986). Given prior N(m;v 0 ) and ow of informationF S t as of time t2 (0;T ], uninformed managers seek to extract information about by solving the following ltering problem: 17 8 < : dR t = dSt St = (r +)dt +dB t ; R 0 = 0; (observation) d = 0; N(m;v 0 ): (state) (1.7) An application of the Kalman-Bucy lter (see e.g. Liptser and Shirayayev (2001)) to (1.7) allows us to characterize the distribution of conditional onF S t as Gaussian, with conditional mean ~ t E jF S t and variance v t E ( ~ t ) 2 jF S t satisfying: 8 < : d~ t =v t d ~ B t ; dv t =v 2 t dt; (1.8) 17 R is the stock \return" process, dened by Rt R t 0 (Su) 1 dSu. 15 with ~ 0 = m and v 0 as initial values for ~ and v, respectively. ~ B t is a standard Brownian motion with respect toF S t , known as the innovation process: 18 d ~ B t = 1 [dR t (r +~ t )dt] =dB t + ( ~ t )dt: (1.9) Bayesian updating of their prior leads U-managers to revise up their estimation ~ of after positive shocks to the stock market (d ~ B t > 0), and to adjust it down after negative shocks (d ~ B t < 0). Since uncertainty v t decays deterministically over time, their informational disadvantage with respect to informed managers shrinks as time goes by. Under parameter uncertainty, markets are not complete with respect to the true states of the world. However, they are complete with respect to the observable states of the economy (a single risky assetS driven by a single Brownian motion ~ B). Indeed, we can rewrite each U-manager's optimization problem (1.5) in terms of observables as: max ( U t (i)) 0tT ~ E u f T W J T (i) (1.10) s:t: dW U t (i) =W U t (i) r + U t (i)~ t dt +W U t (i) U t (i)d ~ B t ; (1.11) and initial wealth w. ~ E(:) denotes the expectation with respect to an equivalent probability ~ P under which ~ B is a standard Brownian motion. Problem (1.10) can now be addressed in a full-information framework. Absent arbitrage opportunities, uninformed managers see nancial markets as driven by a unique state-price de ator ~ with dynamicsd~ t =r~ t dt ~ t ~ t d ~ B t . The dynamic budget constraint (1.11) can be restated (see e.g. Karatzas and Shreve (1998)) as: ~ E ~ T W U T (i) =w (1.12) 18 The innovation process can be interpreted as the \surprise" component of observed returns, measured by the normalized deviation of returns from their conditional mean. 16 and, using the martingale/duality approach of Cox and Huang (1989) and Karatzas, Lehoczky, and Shreve (1987), the dynamic optimization problem (1.10) can be solved as a static problem over nal payos W U T (i). Let B Q t = ~ B t + R t 0 ~ s ds = B t +t denote the risk-neutral Brownian motion, and Tt the time remaining until the end of the period. Proposition 1 guarantees the existence of a symmetric equilibrium in the uninformed managers' strategies and characterizes the resulting optimal investment policies and wealth dynamics: Proposition 1. Under the economy of Section 1.2, there exists a symmetric equi- librium in managers' investment policies in which, for i2U and t2 [0;T ], optimal fund value ^ W U t (i) and risk exposure ^ U t (i) are given by: ^ W U t (i) = ^ W U t = ( U ~ t ) 1 e 1 1 r g 1 1 ;t; ~ t ;T ; (1.13) ^ U t (i) = ^ U t = 1 + ( 1)v t ~ t ; (1.14) where the Lagrange multiplier U is given in Appendix A.1 and the conditional mean ~ and variance v of the market price of risk are given by: 8 > < > : ~ t =v t B Q t + m v 0 ; v t = v 0 1+v 0 t ; (1.15) for g 1 (;t;x;T ) q (1+vt(Tt)) 1 1+(1)vt(Tt) exp n 2 (1)(Tt) 1+(1)vt(Tt) x 2 o . Moreover, the optimal fund values (1.13) and portfolio policies (1.14) characterize a Nash equilibrium among the uninformed managers for (1 +) 1 < 1 or > (1 +) 1+ > 1. A comparison of equations (1.14) and (1.15) with Theorem 1 in Cvitanic, Lazrak, Martellini, and Zapatero (2006) reveals that, in a symmetric equilibrium, uninformed managers invest as if the ow-performance relationship were at (f T = 1) and no risk-shifting incentives existed. Absent an informational advantage or, more broadly, 17 absent heterogeneity in managers' preferences and incentives, individual attempts to outperform peers are too risky for a risk-averse manager. The optimal policy is then the same a direct investor with the same information would follow. 19 Proposition 1 then highlights an important equilibrium implication of managers' policies that can be overlooked from an individual portfolio choice perspective: convex payos do not necessarily induce risk-shifting behavior when managers and their peers have similar risk-attitude and ability. 20 U-managers inference ~ t drives their fund's risk exposure, boosting it when mar- kets boom (high values of B Q t ) and cutting it down when markets plummet (low values of B Q t ). Bayesian learning then leads to rational \trend-chasing" behavior by uninformed managers. In both unexpectedly good and bad market years in which ob- served stock returns deviate signicantly from fundamentals|the stock mean return |byT , trend-chasing uninformed managers end the period with either an aggressive risk exposure in \bull" markets or with a very conservative risk exposure in \bear" markets. 1.3.2 Informed Managers' Trading and Payo Prole Informed managers have access to all the information their uniformed peers have, so even without observing their asset allocation at every point in time they can infer ^ U (and ^ W U ) perfectly. In addition, informed managers face complete information about asset returns: they form expectationE(:) with respect to the actual (objective) prob- ability measure P . They also observe past realizations of B and thus face complete 19 Note that ^ U t can be rewritten as: ~ t ( 1)v t +( 1)v t ~ t . The rst component is the standard (conditional) mean-variance ecient allocation to the risky asset scaled by U-managers' coecient of relative risk aversion , whereas the second component represents the manager's hedging demand against unexpected changes in her derived investment opportunity set (future reassessments of the market price of risk ). The hedging demand goes to zero as the end of the period approaches (t!T ), or as uncertainty vt vanishes. 20 Note that this implies that the range of past performance over which managers shift risk may not only be nite (Basak, Pavlova, and Shapiro (2007)) but even null if peers have similar preferences and information. 18 markets with respect to the actual states of the economy, with a unique state price de ator following dynamics d t =r t dt t t dB t . Once more, this allows us to re-express I-managers' dynamic budget constraint (1.3) as a static one: E T W I T (i) =w; (1.16) and transform the dynamic optimization program (1.5) into a static problem over nal payos W I T (i), subject to U-managers' optimal wealth process (1.13) at t =T . Absent implicit incentives (f T 1), problem (1.5) subject to (1.16) is identical to that solved by Merton (1971) and results in a constant weight in the risky asset ^ J (i) as the optimal investment policy: ^ I t (i) = M r 2 = : (1.17) Henceforth I refer to M as the Merton policy. It represents the normal risk expo- sure an informed manager would take if she were trading for her own account. The higher-than-normal (lower-than-normal) risk exposure associated with overweighting (underweighting) the risky asset relative to the Merton policy then has the interpreta- tion of risk over-exposure (under-exposure). I use this interpretation of both informed and uninformed managers' investment policies throughout. In the presence of fund ow incentives (f T 6= 1) informed managers will hedge against uctuations in fund ows and their portfolios will generally dier from the Merton policy. Let < 1 represent the (scaled) increase in managers' relative risk aversion as a result of ow concerns. Proposition 2 characterizes I-managers' optimal fund value and investment strategy during [0;T ] as a function of the state- price de ator and U-managers' inferred market price of risk ~ : 19 Proposition 2. Let(t;T ) 1+vt(Tt) 1+(1)vt(Tt) . Under the symmetric equilibrium forU- managers of Proposition 1, the optimal fund value of manager i2I at time t2 [0;T ] is given by: ^ W I t (i) = ^ W I t = ( I t ) 1 Z( ;) [N (d 1;t ) + 1N (d 2;t )] + ( I t ) 1 Z( ;)g 1 (;t; ~ t ;T )g 2 (t; ~ t ) exp 1 + (1)v t ~ t + 2 v t [N (d 3;t )N (d 4;t )]; (1.18) and her optimal risk exposure is given by: ^ I t (i) = ^ I t =! t + 1 p N 0 (d 2;t )N 0 (d 1;t ) N (d 1;t ) + 1N (d 2;t ) + (1! t ) (t;T ) + 1 + (1)v t ~ t + 1 r (t;T ) N 0 (d 4;t )N 0 (d 3;t ) N (d 3;t )N (d 4;t ) ! ; (1.19) where the Lagrange multiplier I solves E h T ^ W I T i =w,N (:) is the standard normal cumulative distribution function, ! t ( I t ) 1 Z( ;) (N (d 1;t ) + 1N (d 2;t )) ^ W I t ; 0! t 1; (1.20) and g 2 (t;x;T )A 1 p (1 +v 0 t) e rT + 2 x 2 v t m 2 v 0 ; Z(;t)e 1 r+ 2 2 t ; d 1;t ~ t v t + 1 '( I ) p ; d 2;t d 1;t + 2 '( I ) p ; d 3;t (t;T)~ t v t + (t;T) +'( I ) p (t;T ) ; d 4;t d 3;t 2 '( I ) p (t;T ) ; '(x) 1 p v T q (m) 2 v 0 + 2 ln x 1 A 0 p 1 +v 0 T ; for the positive constants A 0 and A 1 as given in the proof. 20 We can interpret informed managers' strategy in terms of \herding", \no-herding" and \contrarian" behavior with respect to uninformed managers' strategies. In the present setup, I say that I-managers herd with U-managers whenever their weight in the risky asset is between the normal (Merton) risk exposure and U-managers' risk exposure, overweighting the risky asset (relative to normal) when U-managers are over-exposed to risk and vice versa when U-managers are under-exposed. Con- versely, I say that I-managers follow a contrarian behavior if they are underexposed to risk when U-managers overweight the risky asset and overexposed to risk when U-managers underweight this asset. No-herding refers to the remaining situation in whichI-managers just follow the Merton policy without concerns aboutU-managers' risk exposure. Informed managers' optimal risk exposure (1.19) combines a mean-variance portfolio ! t + (1 ! t ) (t;T ) + 1+(1)vt ~ t and risk-shifting components !t p Tt N 0 (d 2;t )N 0 (d 1;t ) N (d 1;t )+1N (d 2;t ) and 1!t q 1 (t) Tt N 0 (d 4;t )N 0 (d 3;t ) N (d 3;t )N (d 4;t ) . The mean-variance portfolio assigns a time- and state-dependent weight ! t in the Merton (no-herding) policy , and a weight 1! t in a herding composite(t;T ) + 1+(1)vt ~ t . The weight! t is increasing in I-managers' conditional probability of trailing behind their uninformed peers by the end of the period,N (d 1;t ) + 1N (d 2;t ). As underperforming becomes likely, I-managers tilt their portfolio towards the Merton policy they would choose if they were trading for their own account. As outperforming becomes likely, 1! t rises and they tilt their portfolio towards the sum of the two mean-variance portfolios (t;T ) and 1+(1)vt ~ t that make up the herding composite. The rst of these portfolios corresponds to the risk exposure of a fully informed investor with time- varying coecient of relative risk aversion =(t;T ) and is aected by uninformed managers' uncertainty v t ; the second corresponds to the (conditional) mean-variance allocation to the risky asset chosen by an uninformed manager with coecient of rel- ative risk aversion > . A positive weight in the herding portfolio then has the interpretation that, to a greater or lesser extent, informed managers herd with their 21 uninformed peers and disregard their own superior information. Since this compo- nent is maximized when underperforming is an unlikely event (! t = 0), the herding portfolio represents informed managers' hedge against low fund ows triggered by un- derperformance, when they are already interim \winners". From managers' objective (1.5) we see that herding is generated by managers' increased risk aversion due to relative concerns in the outperformance region. Since relative concerns are increas- ing in ow elasticity , the higher the convexity of the ow-performance relationship the closer informed managers will stick to their peer group in order to lock-in their outperformance margin in this region. 21 The risk-shifting components depend on uninformed managers' current and ini- tial estimation error ~ t and m through the stochastic coecients d 1;t to d 4;t . In general, the sum of these components is signicantly positive when M > ^ U and strongly negative in the opposite case. This sum can then be interpreted as a contrar- ian portfolio representing a large long or short position in the risky asset depending on whether the uninformed managers under- or overweight this asset, respectively, relative to the Merton policy. Informed managers shift risk not by pure gambling but by taking large bets in the direction suggested by their private information, re ected in M . Moreover, the size of the contrarian portfolio is maximized for a moderate probability of underperforming (! t 0:5) but tends to zero as this probability goes to either zero or one. In this sense, the contrarian portfolio represents informed managers' hedge against low fund ows, when they are already interim (moderate) \losers". Figure 1.1 illustrates the risky asset allocation of managers as a function of unin- formed peers' estimation error ~ t . Results are in terms of interim excess (relative to the Merton policy) risk exposure as of the rst month (t = 1=12T ) and after the 21 As ! 1, ! and a fraction of the informed manager's portfolio perfectly replicates uninformed peers' position. 22 third quarter (t = 3=4T ). The solid lines correspond to informed managers' poli- cies when they are subject to typical ow-performance relationships as estimated by the literature, whereas the dash-and-dot line represents uninformed peers' policy; the dashed line illustrates the probability of the dierent states ~ t . Comparing each solid line across the two panels, we see that informed managers take contrarian po- sitions with higher probability after the rst month (Panel (a)) than after the third quarter (Panel (b)). Equivalently, they herd with higher probability as the investment horizon approaches. This is in general the dynamic strategy of informed managers: take aggressive contrarian positions against less informed peers early in the year until a desired outperformance margin is achieved, and herd with them thereafter as a means to lock in a high relative standing until the period end. Informed managers can take either a conservative herding or a risky contrar- ian stance in response to the same circumstances depending on the particular ow- performance relationship they face. For instance, an informed manager subject to a ow-performance relationship as estimated by Chevalier and Ellison (1997) herds with a 65% probability after the rst month (Panel (a) of Figure 1.1), whereas the same informed manager herds with almost 0% probability under the ow-performance re- lationship in Sirri and Tufano (1998). Even after three quarters, herding occurs with 90% probability in the rst case, but still with less than 45% probability in the sec- ond case, in which contrarian trading is still the prevailing strategy. The latter case suggests, as argued by the literature so far, that convex ow relationships may induce excessive managerial risk-taking on average as re ected by the aggressive contrarian trading of informed managers. By contrast, the rst case highlights an implication that has been overlooked by the existing literature: informed managers may take insucient risk on average as a result of herding behavior. The opposite average behavior of skilled managers depending on the fund- ow relation they face is a fea- ture that persists across dierent realizations of the market price of risk . In Section 1.4 I show that small dierences in the ow-performance relationship can give rise to 23 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 −0.2 0 0.2 0.4 Peers’ estimation error ˜η t −η ˆ φ t −φ M −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0 0.02 0.04 0.06 I−manager (Chev−Elli 97: δ=0.94, α=2.5) I−manager (Sirri−Tufa 98: δ=0.985, α=1.7) Peers (a) t = 1=12 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Peers’ estimation error ˜η t −η ˆ φ t −φ M −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 I−manager (Sirri−Tufa 98: δ=0.985, α=1.7) I−manager (Chev−Elli 97: δ=0.94, α=2.5) Peers (b) t = 3=4 Figure 1.1: Managers' interim risk exposure Panels (a) and (b) illustrate informed managers' and uninformed peers' risk exposure ^ I and ^ U in excess of the normal risk exposure M as of dierent times t. For the informed manager, risk exposures are shown under two dierent parameterizations for the ow-performance relationship: (i) =:985 and = 1:7 (approximately matching the relationship estimated by Sirri and Tufano (1998)), and (ii) = :94 and = 2:5 (approximately matching the relationship estimated by Chevalier and Ellison (1997) for \old" mutual funds). The dashed line represents the probability of the dierent states ~ t. Results correspond to a realized market price of risk = 0:557 att = 0. The rest of the parameters follow the baseline parameterization in Appendix A.2: T = 1;r = 3%; = :0158;m = 0:513;v0 = 0:192 2 ; = 5. substantially dierent expected risk-taking by skilled managers, an observation that motivates the empirical analysis of Section 1.5. 24 As noted by previous authors, risk-shifting need not increase overall portfolio risk: contrarian strategies can either increase the weight in the risky asset with respect to the Merton policy ( ^ I > M ) or decrease it ( ^ I < M ) depending on the risk exposure of peers. For instance, the informed manager's policy illustrated by the dark solid line in Panel (b) of Figure 1.1 increases the portfolio weight invested in the stock by almost 0.7 relative to the Merton policy when peers are underexposed to risk by 0.18 (for ~ t =0:1), but decrease it by the same magnitude when uninformed managers are overexposed to risk by 0.18 instead (for ~ t = 0:08). Equation (1.18) shows that informed managers' interim performance is a non- linear function of the state variable ~ . Corollary 1 relates these managers' end-of- period performance to uninformed managers' (normalized) error correction from the initial estimation error (~ T ) 2 v T (m) 2 v 0 : Corollary 1. Under the symmetric equilibrium for U-managers of Proposition 1, I-managers' optimal terminal fund value ^ W I T is given by: ^ W I T = 8 < : ( I T ) 1 ^ W U T ; if (~ T ) 2 v T (m) 2 v 0 > , ( I T ) 1 g 2 (T; ~ T ;T ) ^ W U T ; if (~ T ) 2 v T (m) 2 v 0 , (1.21) where 2 ln 1 I A 0 p 1 +v 0 T , (1 + ) 1+ < , and (1 +) 1 >. Whether I-managers opt to beat or to lose to U-managers depends more on the extent of learning by the latter as re ected on the size of their error correction and less so on the direction of the overall market. Indeed, informed managers outperform their peers whenever the uninformed managers' corrected error (~ T ) 2 v T in excess of their initial estimation error (m) 2 v 0 is small enough (less than ). 22 Otherwise they end up losing to their peers. Note that underperformance happens in both good and 22 In all numerical examples considered in this chapter, the outperformance regions is non-empty because I is such that (m) 2 v 0 + > 0. 25 bad states, as long as the stock return during the period is extreme (good or bad) enough, relative to the true distribution, to deviate U-managers inference of far away from the actual value. 23 The argument is illustrated in Panel (a) of Figure 1.2, displaying the informed managers' terminal assets under management in excess of uninformed peers' that result from following the trading policies of Figure 1.1. The informed manager that leans toward contrarian strategies delivers high excess end-of-period returns (dark solid line) in many situations (j~ T j < 0:08), even with respect to the Merton policy, as can be expected from average risk-shifting behavior. By contrast, the informed manager that has a tendency to herd delivers very small excess returns (light solid line) in the same situations, resulting in lower performance than could be accomplished by following the Merton policy. If she traded for her own account (Merton policy), this manager would deliver an average 31 basis points (bps) in excess of uninformed managers for this particular realization of the market price of risk ; the average excess return would rise to 158 bps under a ow-performance relationship like that in Sirri and Tufano (1998) but would reduce to only 15 bps under a ow relation like that in Chevalier and Ellison (1997). In this sense, skilled managers are not necessarily star performers. Since managers' skill (informational advantage) is the same across the depicted payo proles, this gure suggests that end-of-year performance may be a poor indicator of managers' true ability. If we tried to infer this ability based on excess returns without accounting for managers' incentives we would over-estimate it for some fund- ow relations and under-estimate it for others. This bias seems unlikely to be tackled by risk-adjustment based on linear factor models (e.g. CAPM) of funds' excess returns since informed managers' strategy is highly non-linear in the economy's driving state variable ~ . Indeed, whereas informed 23 This eect is similar to that found in the literature on limits of arbitrage (see, e.g., Shleifer and Vishny (1997)), where shocks pushing prices further away from fundamentals (rendering arbitrage opportunities even more protable) may end up hurting an arbitrageur's performance during a nite investment horizon. 26 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.4 −0.3 −0.2 −0.1 0 0.1 Peers estimation error ˜η T −η ˆ W T − ˆ W U T −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0 0.01 0.02 0.03 0.04 0.05 I−manager (Sirri−Tufa 98: δ=0.985, α=1.7) I−manager (Chev−Elli 97: δ=0.94, α=2.5) Merton Policy (a) Final payos as a function of uninformed peers' end-of-period inference error 0.8 1 1.2 1.4 1.6 1.8 2 −0.4 −0.3 −0.2 −0.1 0 0.1 ˆW U T ˆW T − ˆW U T 0.8 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 I−manager (Sirri−Tufa 98: δ=0.985, α=1.7) I−manager (Chev−Elli 97: δ=0.94, α=2.5) Merton Policy (b) Final payos as a function of uninformed peers' end-of-period fund value Figure 1.2: Optimal Payo Proles Panels (a) and (b) illustrate informed managers' end-of-period excess payos ^ W I T ^ W U T over unin- formed peers'. For the informed manager, nal payos are shown under two dierent parameteriza- tions for the ow-performance relationship: (i) =:985 and = 1:7 (approximately matching the re- lationship estimated by Sirri and Tufano (1998)), and (ii) =:94 and = 2:5 (approximately match- ing the relationship estimated by Chevalier and Ellison (1997) for \old" mutual funds). The dashed line represents the probability of the dierent states ~ T (Panel (a)) and ^ W U T (Panel (b)). Results correspond to a realized market price of risk = 0:557 att = 0. The rest of the parameters follow the baseline parameterization in Appendix A.2: T = 1;r = 3%; =:0158;m = 0:513;v0 = 0:192 2 ; = 5. 27 managers' conditional expected excess returns are a linear function of the market excess return: ~ E t d ^ W I t ^ W I t rdt = I t h ~ E t dSt St rdt i , the conditional beta I t ~ cov t d ^ W I t ^ W I t ; dSt St = ~ var t dSt St = ^ I t is highly non-linear in ~ t (see equation (1.19)). Standard approaches to performance evaluation usually estimate I t over time series of fund returns assuming stability of this parameter over time. Since I t is highly time-varying according to the model, such time series estimates can be substantially biased even when computed at the highest available frequency (daily returns in the case of US mutual funds), as argued in Section 1.4.2. The non-linearity of payos as a function of the economy's states is the result of the particular option strategy that informed managers' trading policy (1.19) replicates, as Panel (b) of Figure 1.2 shows. When the same payo proles (and density function over states) of Panel (a) are plotted as a function of uninformed managers' termi- nal fund wealth ^ W U T it becomes clear that investing in an informed manager's fund looks like simultaneously selling an out-of-the-money put and an out-of-the-money call (digital) options with maturityT and ^ W U T as underlying. 24 A delegating investor collects positive excess returns over delegation to uninformed managers when these managers' returns do not uctuate much, but suers large negative excess returns in volatile times. Exactly how high excess returns are and how much volatility can turn relative prots into losses depends on the particular ow-performance relationship the informed manager faces: for the light solid line of Figure 1.2 the strategy delivers frequent small excess returns (resembling a \nickel-picking" strategy) and rare large losses, whereas for the dark solid line the strategy delivers less frequent though larger excess returns, along with more frequent but smaller losses. 24 In the nancial jargon, such a strategy is known as a (short) \strangle" or \top vertical combi- nation". 28 1.4 Average Risk-Taking and Performance The examples in Section 1.3.2 are conditional on interim market performance, pre- sented in terms of a particular realization of the market price of risk, and calibrated to specic fund ow relations. In this section I simulate informed and uninformed managers' optimal investment strategies for a broader range of ow-performance spec- ications, under dierent economic settings. The analysis is in terms of expected behavior, averaging outcomes over the joint distribution of the random variables in- volved according to the numerical procedure detailed in Appendix A.3. This analysis is meaningful in the current setup because, by contrast with most of the prior liter- ature, skilled managers' and peers' strategies (pinning down skilled managers' rela- tive performance) are determined endogenously. Reported results correspond to the baseline and alternative parameterizations described in Appendix A.2, with market, preference and fund- ow parameters set to match typical values during the period 1980-2006 and to be in line with parameterizations used by previous authors. The baseline parameterization implies a potential advantage of informed over uninformed managers, if the former followed the Merton policy, of 34 basis points (bps) in terms of households' certainty equivalent returns and of 70 bps in terms of (log) returns over the period, or equivalently a 14.5% increase in Sharpe ratios. Section 1.4.1 examines the risk-taking behavior by informed managers in terms of both deviation from peers and of fund portfolio volatility. Section 1.4.2 looks into the relation between fund ows and informed manager's performance, with emphasis on the ability of standard performance measures to adjust for the risks informed managers take. Section 1.4.3 examines herding or contrarian behavior by informed managers as a function of the performance threshold and the ow elasticity and derives the model's testable implications. 29 1.4.1 Managerial Risk-Taking According to the discussion in Section 1.3.2 contrarian managers shift risk by un- dertaking large long or short positions in the risky asset whenever their uninformed peers are under- or over-exposed to risk, respectively, relative to the Merton policy. As a result, end-of period returns exceed or fall short of uninformed managers' returns by a large margin for contrarian managers but by a very small margin for herding managers. This observation points to the distance between informed managers' and uninformed managers' returns as a candidate measure of herding behavior. Moti- vated by the empirical analysis of the next section, I look at tracking error volatility StdDev(log(R I T )log(R U T )) (hereafter just \tracking error"), in excess of the tracking error that would result from following the Merton policy, as such distance. A negative value of tracking error is then indicative of herding behavior, while a positive value is suggestive of contrarian behavior. Panels (a), (c) and (d) of Figure 1.3 illustrate the eects of and on the extent of herding by the informed manager, for the dierent values of the initial informational advantage v 0 (uninformed managers' prior uncertainty) of Appendix A.2. It is clear from all three panels that, for some fund ow relations, skilled managers can take aggressive contrarian stances on average (e.g. for = 3 and = 1:06 in Panel (a), an informed manager's annual excess tracking error is 6.3%), but also be conservative herd-like investors (e.g. for = 3 and = 0:9 in Panel (a), an informed manager's annual excess tracking error is -2.0%). Therefore, dierences in the sensitivity of ows to relative performance (as parameterized by in this example) have rst-order eects on the expected risk-taking behavior of skilled managers. In general, these panels show a non-monotonic relationship between tracking error and either or : herd managers turn contrarian and contrarian behavior intensi- es as both parameters go up until a maximum is reached, after which contrarian managers take less aggressive positions on average. This hump-shaped pattern is ex- plained by the fact that contrarian behavior should be highest when outperforming 30 0.5 1 1.5 2 2.5 3 0.9 0.95 1 1.05 1.1 −4 −2 0 2 4 6 8 α δ (a) Tracking Error (v0 = 0:192 2 ) 0.5 1 1.5 2 2.5 3 0.9 1 1.1 −1 0 1 2 3 α δ (b) Portfolio Volatility (v0 = 0:192 2 ) 0.5 1 1.5 2 2.5 3 0.9 1 1.1 −2 0 2 4 6 8 α δ (c) Tracking Error (v0 = 0:063 2 ) 0.5 1 1.5 2 2.5 3 0.9 1 1.1 −4 −2 0 2 4 6 α δ (d) Tracking Error (v0 = 0:317 2 ) Figure 1.3: Average excess tracking error and portfolio volatility (% annual) Panels (a) through (d) plot informed managers' tracking error with respect to peersStdDev(log(R I T ) log(R U T )), and portfolio volatilityStdDev(log(R I T )),inexcessof the \normal" tracking error/portfolio volatility resulting from following the Merton policy. Thus, negative values of the excess tracking er- ror correspond to the herd (contrarian) behavior according to the denition in Section 1.3.2. Results are shown for dierent ow-performance relationships as given by their ow elasticity and perfor- mance threshold . Standard deviations are computed over the actual (objective) joint probability distribution of the market price of risk and the driving Brownian motionB following the numerical procedure in Appendix A.3. The dierent panels correspond to dierent economic setup as deter- mined by uninformed managers' prior uncertaintyv0 in the baseline and alternative parameterizations of Appendix A.2 (T = 1, r = 0:03 and = 0:158 and = 5 across all panels). is a likely event but only towards the end of the period. For xed ow elasticity, only for moderately high values of the threshold (1 < < 1:05 in the baseline pa- rameterization) being a top performer is possible but requires informed managers to trade aggressively on their private information throughout the whole period in order to achieve the necessary outperformance margin. Medium to high values of then are 31 consistent with managers leaning towards contrarian strategies. Since a more convex compensation structure induces already contrarian traders to take higher risks, the intensity of this behavior is increasing in the ow elasticity for medium to high thresholds . Figure 1.3 further shows that skilled managers subject to low thresholds (less than 0.95) will herd on average, as re ected by a negative excess tracking error. More- over, herding is also increasing in , implying that more convex ow-performance re- lationships will actually induce more conservative strategies in this parameter region: investors rewarding large in ows to stars can actually lead informed managers not to dierentiate much from the herd despite having the informational advantage to rank high. We conclude that higher ow elasticity intensies both herd and contrarian behaviors. 25 From Panel (b) we see that, in general, whether skilled managers follow herd or contrarian strategies on average is not informative about the absolute risk of their portfolios: for all ow elasticities , contrarian managers may take less absolute risk than the Merton policy, and even less risk than herding managers. For instance, for = 1:5 and = :96 an informed manager's excess tracking error is 3.3% in Panel (a) but her excess portfolio volatility is -0.3% in Panel (b), whereas for = 2:5 and =:92 tracking error and portfolio volatility are -1.4% and -0.1% respectively. 25 Non-tabulated results for = 2; 8 show that this pattern remains qualitatively the same under dierent assumptions about managerial risk aversion. The ambiguous eect of on the extent of herding by informed managers is due to the interplay of two opposite eects. On the one hand, a higher convexity of the ow-performance relationship increases the incentives of informed but underperforming managers to deviate from the pack in order to be a star and enjoy large in ows. This is seen from Corollary 1 in Section 1.3.2: how much managers are willing to deviate is controlled by the ratio = = = ( =(1 +)), which is independent of but increasing in . On the other hand, once managers reach the upward-sloping part of the fund- ow relationship they become highly risk-averse and seek to lock in their outperformance by sticking close to their uninformed peers. As a result, whether average herding is increasing or decreasing in alpha depends on how likely it is to outperform and thus on the threshold . 32 1.4.2 Measured Performance Given that informed managers may either behave too conservatively or take exces- sive risks, it seems no clear a priori whether delegating to informed managers is in households' best interest despite their informational advantage. In the present setup, this question can be addressed by examining households' expected derived utility or, likewise, certainty equivalent returns (CER) from delegation. Panel (a) of Figure 1.4 plots the average excess CER that households would attain by delegating their wealth to an informed instead of to an uninformed manager for the model's baseline parameterization. 26 All herding managers (low values of ), as well as contrarian managers subject to fund ow relations with low convexity (< 0:7 for all values of ), deliver positive excess CER to households. Moreover, both low herding and con- trarian managers (< 0:7 and < 0:95 or > 1:08) attain the highest risk-adjusted performance, equivalent to 90% of the excess CER under the Merton policy (34 bps). By contrast, aggressive contrarian informed managers ( > 0:7 and 0:95) have households bear too much risk, in icting costs that more than oset the benets from their superior information. Net costs are increasing in the ow elasticity for these managers and can be as high as 100 bps (negative excess CER, for = 3 and = 1:025), indicating that delegation to uniformed managers would be preferred in this case. Since skilled managers taking aggressive contrarian strategies deliver higher average end-of-period excess returns over uninformed peers than herding managers (e.g. 75 bps for = 3 and = 1:025 versus 22 bps for = 3 and = 0:9 in non-reported simulations) but lower or even negative excess CER (e.g. -100 bps for = 3 and = 1:025 versus 18 bps for = 3 and = 0:9), these results highlight the importance of adjusting performance for the risks informed managers take. Sharpe ratios and Jensen's alpha computed over time series of fund returns are widely used measures to evaluate the performance of asset managers on a risk-adjusted 26 See Proposition 4 in Appendix A.1 and the procedure in Appendix A.3 for details on the com- putation of the excess CER in the simulations. 33 basis. Panels (b) and (c) of Figure 1.4 show that these measures do a poor job on informed managers subject to convex fund ows. 27 First, both fail to re ect the relative risk-adjusted performance of equally informed managers subject to dierent fund- ow relations: for each level of ow elasticity , Sharpe ratios and Jensen's al- pha are highest for informed managers taking the most extreme contrarian policies (middle-range values of), precisely where households suer the highest costs of dele- gation as measured by excess CER. Jensen's alpha is even increasing in the convexity whereas households' CER are decreasing instead for these managers. Second, both Sharpe ratios and Jensen's alpha may fail to provide an answer to the more fundamental question of whether delegating to informed managers is desirable or not. Indeed, delegation to funds with performance thresholds in a neighborhood of 0.97 is most appealing according to the positive and high value of both measures, but against households' interest according to the corresponding negative excess CER. The converse problem is also true in the case of Jensen's alpha: the negative values of this measure for high performance thresholds and moderate ow elasticity indicate that households do better by delegating to uninformed rather than to informed managers. However, households would actually be better o by delegating to informed managers according to the corresponding positive excess CER. This result implies that Jensen's alpha may not only under-estimate the risks taken by informed managers under cer- tain ow-performance relationships, but also over-estimate these risks under other ow-performance specications. Summing up, fund ows may distort measured performance using standard ap- proaches to the extreme that just better-than-average performers|according to such approaches|are actually most benecial to delegating investors and, conversely, star performers are in fact detrimental to these investors' interests. 28 The reason lies in 27 See Appendix A.3 for details on the computations of these measures in the simulations. 28 In non-tabulated results, I conrm that this bias remains signicant across dierent relative risk aversion misalignment between managers and delegating households ( h = 2; 8) and for dierent levels of initial uncertainty (v0 = 0:063 2 ; 0:317 2 ). 34 0.5 1 1.5 2 2.5 3 0.9 0.95 1 1.05 1.1 1.15 −150 −100 −50 0 50 δ α (a) Certainty equivalent returns (bps) 0.5 1 1.5 2 2.5 3 0.9 0.95 1 1.05 1.1 1.15 0 0.02 0.04 0.06 0.08 0.1 δ α (b) Sharpe ratios 0.5 1 1.5 2 2.5 3 0.9 0.95 1 1.05 1.1 1.15 −2 0 2 4 6 δ α (c) Jensen's alpha 1 1.5 2 2.5 3 0.94 0.96 0.98 1 1.02 1.04 1.06 20 40 60 80 100 α δ (d) Prob(R I T >R U T ) (%) Figure 1.4: Informed managers' performance in excess of uninformed peers' Panels (a) through (c) plot simulated average risk-adjusted performance of informed managers in excess of uninformed peers as a function of the ow elasticity and performance threshold in their ow-performance relationships, according to three measures: delegating households' certainty equivalent returns, Sharpe ratios and Jensen's alpha. Panel (d) plots the average probability (in %) that an informed manager's fund end-of-period return R I T is higher than that of an uninformed manager,R U T , as a function of the same parameters and. Averages are computed over the actual (objective) joint probability distribution of the market price of risk and the driving Brownian motion B following the numerical procedure in Appendix A.3. Results correspond to the baseline parameterization in Appendix A.2: T = 1;r = 3%; =:0158;m = 0:513;v0 = 0:192 2 ; = 5 = h . the specic option-like payos replicated by informed managers' strategy, as argued in Section 1.3.2. In consequence, assessing higher-order moments in the return dis- tribution of managed portfolios may be key in adjusting for risk, as suggested by e.g. Fung and Hsieh (2001) in the context of hedge funds. More generally, the results in this section point to the importance for performance evaluation of jointly estimat- ing managers' incentives, risk-preferences and skills within a structural approach, in the spirit of Becker, Ferson, Myers, and Schill (1999) or Koijen (2010). The empiri- cal analysis of Section 1.5 suggests that the concerns about traditional performance measures are of practical relevance in the mutual fund industry given the shape the ow-performance relationships skilled managers face. 35 1.4.3 Fund Flows and Herding: Testable Implications The characterization of skilled managers in terms of herd or contrarian behavior as a function of the performance threshold and the ow elasticity of Section 1.4.1 provides the model's testable implications. Specically, low values of the performance threshold induce herd behavior by skilled managers, whereas medium to high values of foster contrarian behavior. Moreover, the strength of both herd and contrarian behavior is increasing in fund ow convexity . Given that approximates the sensitivity of the ow relationship to dierent past relative returns (with low values of consistent with high sensitivity to medium relative performance and high values indicating low sensitivity), we can state the model's empirical prediction as follows: (i) when ows are sensitive to medium relative performance, informed managers have a tendency to herd, and average herding is increasing in ow convexity; by contrast, (ii) when ows are insensitive to medium relative performance informed managers favor contrarian strategies, the more so the more convex is the ow-performance relationship. I provide empirical support for these predictions in the next section. 1.5 Empirical Analysis Based on the model's empirical predictions, in this section I test the hypothesis that observed dierences in the risk-taking behavior of managers with superior informa- tion within a sample of actively managed U.S. equity mutual funds are consistent with dierent ow-performance relationships. A direct approach would require regressing managerial risk-taking, e.g. the extent of herding, on the funds' characteristics that determine the shape of their ow-performance relationships. In the absence of a conclusive theory relating fund ows to fund characteristics, I follow an indirect ap- proach instead and test whether any discrepancy between the fund- ow relationships faced by herding and contrarian informed funds agrees with the dierences that could 36 be expected from predictions (i) and (ii) in Section 1.4.3. Underlying this indirect methodology is the assumption that investors' money ows and managers' strategies are jointly determined in equilibrium. 29 In implementing this approach, I follow three steps: (1) identify potentially informed mutual funds in the sample; (2) rank informed funds according to estimated extent of herd/contrarian behavior during the sample period; and (3) test whether the sensitivity of fund ows to middle-range returns is higher for herding than for contrarian funds, consistent with the implications in Section 1.4.3. I obtain information about mutual fund returns, total net assets, net asset values and characteristics from the Center for Research in Security Prices (CRSP) Survivor Bias Free Mutual Fund Database. I also obtain market excess returns and the 30-day Treasury bill rate (r f ) from CRSP's Fama-French, Momentum and Liquidity dataset. The sample consists of monthly data over the period January 1981 through Decem- ber 2010. Since my focus is on open-ended, actively managed domestic equity funds, I apply the following screens. First, I exclude all funds classied as \index-based" or \index" funds as of 2008. Second, I exclude all funds with (Lipper) objective categories dierent from growth, growth and income, equity income and income, to facilitate comparison with the prior literature. Third, I omit all fund-month observa- tions with total net assets less than $5 million. The resulting total sample consists of 7,656 mutual fund share classes, 30 although data availability reduces this number to 3,060 mutual funds for steps 1 and 2 above. I next describe the approach to each of steps 1 to 3: 29 Therefore, the empirical analysis below has no pretension of establishing causality between risk- taking behavior and fund ows. 30 Dierent share classes re ecting diering fee structures for the same mutual fund are treated as stand-alone mutual funds by CRSP. Since my goal in this section is the estimation of the ow- performance relationships, I follow Huang, Wei, and Yan (2007) and Huang, Wei, and Yan (2011) and perform the analysis at the fund-share level in order to capture the reaction of investors with possibly dissimilar tastes to similar performance. In any case, I control for the fee structures of dierent share classes in the ow regressions. I refer to fund-shares as mutual funds henceforth. 37 1. Sample of \informed" managers: One of the main implications of the model is that some informed managers may choose to perform just better than average. Therefore, identifying privately informed managers only with the top mutual funds in a performance ranking in any given period (with or without factor-based risk- adjustment) is likely to miss truly informed funds from the sample. In order to circumvent this diculty, I use the model's predictions regarding informed managers' probability of outperforming in any given period. Panel (d) of Figure 1.4 plots the average probability that informed managers' total period returns (R I T ) are higher than those of their uninformed peers (R U T ), for the dierent ow-performance specications in Section 1.4. We see that, for almost all plausible fund- ow relationships, informed managers' performance is relatively homogeneous in this dimension: the probability that they perform better than average is greater than 0.5. 31 This result suggests identifying informed mutual funds in my sample with those that consistently rank in the top half (irrespective of their exact position) of the annual performance ranking for their respective objective category. More precisely, for each year and objective category I rank all funds according to their annual raw return and assign them a continuous rank (Rank) ranging from 0 (worst) to 1 (best). Funds with 0:45 Rank 0:55 are considered \median performers" whereas funds with Rank > 0:55 are considered \outperformers", for that year-objective class. 32 For all funds ranked during at least 5 years in the sample, I then compute the proportion of years in their life that they were outperformers and sort all funds in ascending order based on this proportion. Finally, I pick the top 30% as the \informed" mutual funds in the sample. I chose this threshold in an attempt to keep a fair balance between the sample size 31 The pattern in Panel (d) of Figure 1.4 is robust to dierent value of the initial informational advantagev0, as longv0 > 0: the probability of outperforming falls asv0 shrinks, but remain greater than 0.5 in most cases. Informed managers' probability of outperforming is smaller than 0.5 only for those subject to ow performance relationships with very high thresholds ( 1:05) or with high thresholds but low ow elasticity ( 1:025, < 1). 32 I use this selection criterion in order to construct a meaningful portfolio of average/median performers for each year-objective category. The returns on this portfolio are used to compute herding measures in step 2 below. 38 for the estimation of the ow-performance relationships on the one hand, and the fraction of \uninformed" managers expected to be incorrectly selected as \informed" on the other. A lower threshold favors a more accurate estimation, but turns the sample of informed managers noisier, and vice versa. Table 1.1 reports summary statistics for the overall sample and for the top 30% mutual funds according to the criterion above. Top performers are on average larger and achieve higher returns (both in terms of raw returns and in terms of Jensen's alpha with respect to the median performer in the respective objective class) without a signicant increase in volatility. However, none of these dierences are statistically signicant. In particular, top performers are not above-average systematic risk-takers (as measured by average beta), so their higher proportion of outperformance years seems not attributable to a higher load on risk factors. Moreover, the average mutual fund in the overall sample outperforms with a 46% probability, whereas the average top 30% mutual fund outperforms with 65% probability. 33 Since the average fund in the latter group was ranked in 11.2 years, the percentage of average but \lucky" mu- tual funds that are expected to be selected as top 30% performers is 12%. 34 Whereas this fraction suggests that the potential in uence of non-informed managers on the results in Step 3 may be non-negligible, the robustness of these results to a stricter selection criterion (including only the top 20% performers) in the robustness checks at the end of this section suggests that these results are not driven by pure chance. 33 From Panel (d) of Figure 1.4, informed managers subject to performance thresholds lower than 1.02 and all values of ow elasticity will in general outperform with 65% or higher probability. This parameter subset gives rise to enough heterogeneity in expected herding/contrarian behavior according to Panels (a)-(d) of Figure 1.3, so there is no reason to expect either herding or contrarian mutual funds to dominate the top 30% performers sample a priori. 34 This approximate proportion can be computed as f(7; 11; 0:46), where f(k;n;p) is the binomial probability of k successes in n trials when the probability of success is p. 39 Table 1.1: Summary statistics. The table reports summary statistics over the period 1981-2010 for mutual funds ranked in at least 5 years. Annual rankings for each objective category are computed over raw re- turns and normalized to be between 0 and 1. Jensen's alpha and beta are computed over monthly returns with respect to a portfolio of median performers comprised of mutual funds with performance rank between 0.45 and 0.55. A fund is considered to outperform in a year if its performance rank that year is greater than 0.55. For each mutual fund, the propor- tion of outperforming years is the ratio of number of years outperformed to the number of years ranked. Based on the proportion of outperforming years, all funds are sorted in as- cending order. The top 30% are selected as the \informed" mutual funds in the analysis. All MFs (N=3060) Top 30% Mean Std. dev. Mean Std. dev. TNA ($ mill) 922 3,876 1,584 5,542 Age 17.6 14.0 19.8 15.6 Expense Ratio (%) 1.22 0.58 1.11 0.56 Annual Return (%) 7.65 20.32 9.21 20.77 Std. dev. of Returns (% monthly) 4.18 2.03 4.27 2.01 Jensen's alpha (% monthly) 0.02 0.18 0.15 0.17 jbeta1j 0.01 0.16 0.04 0.18 Position in Return Ranking 0.51 0.29 0.60 0.28 Proportion of Outperforming Years 0.46 0.17 0.65 0.09 Number of years ranked 11.4 5.0 11.2 5.4 2. Herding measures: I compute three alternative herding measures follow- ing Chevalier and Ellison (1999) and Arora and Ou-Yang (2001). The rst, HerdTrackErr i , is based on fund i's tracking error of monthly returns relative to the median performer within i's objective category for each year: TrackErrVol i = v u u t 1 T i T i X t=1 r i;t r med i;t 2 ; (1.22) wherer i;t is the monthly return of fund i in montht,r med i;t is the monthly return of a portfolio of median performers withini's objective category during the same year (all funds in i's objective style with 0:45Rank 0:55), and T i is the total number of monthly observations for fund i. The second measure, HerdBeta i , is based on fund i's beta relative to the median performer ini's objective class: med i j i 1j, where 40 i is the usual regression-based measure of systematic risk-taking relative to r med i , over the entire sample: r i;t r f t = i + i r med i;t r f t + i : (1.23) The third measure, HerdCorr i , is based on the correlation coecient between fund i's monthly returns and those of the median performer in i's objective category over the entire sample: i;med corr(r i ;r med i ). The herding measures HerdTrackErr i , HerdBeta i and HerdCorr i then result from subtracting the median of, respectively, TrackErrVol i , med i and i;med for all funds in i's objective class from each of these variables. These measures proxy for managers' boldness in the sense of departing from the typical portfolio (HerdTrackErr andHerdCorr), or in the sense of taking above- or below-average systematic risk (HerdBeta). One should expect more herd- ing corresponding to lower HerdTrackErr and HerdBeta on the one hand, and to higher HerdCorr on the other hand. 35 Since the availability of monthly returns in CRSP database improves signicantly starting from 1990, I compute the herding mea- sures over the sample covering January 1990 through December 2010 so as to avoid noisy measurement of the portfolio of median performers for each year-objective style prior to 1990. I then rank all informed mutual funds as selected in Step 1 according to measured herding and identify the bottom and top 33% herding groups as the contrarian and herding mutual funds, respectively. The dummy variable Contrarian equals 1 for the group displaying, alternatively, the highest values forHerdTrackErr or BetaCorr, or the lowest values for HerdCorr, and 0 otherwise. I refer to \con- trarian" and \herding" groups those with Contrarian = 1 and Contrarian = 0, respectively. 35 The use of these three returns-based measures over portfolio holding measures will be bene- cial whenever managers actively trade within quarters (the frequency at which most mutual funds' portfolio holdings data are available) and whenever managers take osetting positions in assets other than equities or bonds (e.g. long or short positions in derivative to hedge their risk exposure). 41 Table 1.2 presents summary statistics for the contrarian and herding top funds. Even though dierences are not statistically signicant, contrarians seem to be smaller mutual funds with higher expense ratios, achieving higher Sharpe ratios and Jensen's alpha (with respect to the median portfolio in their respective styles), without a corresponding fall in the probability of outperforming. We see that herding funds according to HerdTrackErr are also lower HerdBeta and higher HerdCorr funds, signaling herding behavior in both cases, and similarly for the herding mutual funds according to the latter two measures. Herding funds then appear to take both lower systematic and unsystematic risk, delivering a high comovement with the median performers in their objective styles. 42 Table 1.2: Summary statistics for top performers. This table shows summary statistics for the top 30% performers over the period 1981-2010. HerdTrackErr,HerdBeta andHerdCorr are the return- based measures of herding described in Section 1.5, computed over the sample 1990-2010. Top performer mutual funds are ranked in ascending order according to the value of each of these measures. The bottom and top 33% are, respectively, the \Herding" and \Contrarian" funds forHerdTrackErr andHerdBeta, and conversely the \Contrarian" and \Herding" funds according toHerdCorr. The rest of the variables reported are as in Table 1.1. Herding measured by: HerdTrackErr HerdBeta HerdCorr Contrarian Group Herding Group Contrarian Group Herding Group Contrarian Group Herding Group Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev. TNA ($ mill) 821 2,431 1,866 7,670 1,214 4,230 1,621 7,670 1,288 4,325 1,760 7,670 Age 18.0 11.6 19.0 18.7 17.7 12.1 22.6 18.7 21.1 14.3 16.2 18.7 Expense Ratio (%) 1.32 0.52 0.81 0.50 1.27 0.57 1.01 0.50 1.31 0.48 0.85 0.50 Annual Return (%) 10.03 23.02 8.27 18.54 9.58 22.43 8.71 18.54 10.17 21.41 8.00 18.54 Std. dev. of Returns (% monthly) 4.77 2.24 3.93 1.79 4.55 2.24 4.11 1.79 4.41 2.20 4.05 1.79 Position in Return Ranking 0.62 0.33 0.59 0.22 0.61 0.32 0.59 0.22 0.61 0.32 0.59 0.22 Proportion of Outperforming Years 0.66 0.10 0.65 0.08 0.66 0.09 0.64 0.08 0.65 0.09 0.65 0.08 Jensen's alpha (% monthly) 0.23 0.23 0.09 0.08 0.19 0.22 0.12 0.08 0.24 0.22 0.09 0.08 HerdTrackErr (% monthly) 1.25 0.74 -0.46 0.23 0.89 0.88 -0.07 0.23 1.15 0.82 -0.40 0.23 HerdBeta 0.12 0.17 -0.03 0.06 0.15 0.14 -0.06 0.06 0.07 0.13 -0.01 0.06 HerdCorr -0.09 0.06 0.01 0.02 -0.06 0.06 -0.02 0.02 -0.10 0.05 0.01 0.02 43 3. Flow-performance relationships: I estimate informed mutual funds' ow- performance relationships following closely the methodology in Sirri and Tufano (1998). Their approach consists in regressing a mutual fund ow rate on its pre- vious year performance allowing for a piecewise linear relation between the two, and on a set of non-performance-related control variables. I dene the annual ow rate into a fund as: Flow i;t = TNA i;t TNA i;t1 (1 +R i;t ) TNA i;t1 (1 +R i;t ) ; (1.24) where R i;t is the return of fund i during year t and TNA i;t is fund i's total net asset value at the end of yeart. (1.24) re ects the growth rate of assets under management after adjusting for appreciation of the mutual fund assets, following the denition of f T in Section 1.2.2. 36 To avoid the impact of mutual funds mergers and stock splits leading to extreme values of ows, I exclude all funds merging with other funds during the sample and I lter out the top and bottom 2.5% of the net ow data. The regression equation is: Flow i;t =a +b 1 Size i;t1 +b 2 CategoryFlow i;t +b 3 Volatility i;t1 +b 4 Age i;t +b 5 ExpenseRatio i;t +b 6 Low i;t1 +b 7 4thPerfQuint i;t1 +b 8 3rdPerfQuint i;t1 +b 9 2ndPerfQuint i;t1 +b 10 Top i;t1 +b 11 Contrarian i +b 12 Contrarian i Low i;t1 +::: +b 16 Contrarian i Top i;t1 + i;t ; (1.25) where Size i is the natural log of fund i's TNA, CategoryFlow i is the aggregate ow rate into fund i's category, Volatility i;t is the standard deviation of fund i's monthly returns in year t, Age i is the natural log of 1 + fund i's age, ExpenseRatio i is fund i's expense ratio, and Contrarian i is i's \contrarian" dummy variable of Step 2. 36 This denition of fund ows is the same used by Huang, Sialm, and Zhang (2011) and Huang, Wei, and Yan (2011) 44 Performance quintiles are constructed as follows: Low i;t1 min(Rank i;t1 ; 0:2), 4thPerfQuint i;t1 min(Rank i;t1 Low i;t1 ; 0:2), and so forth, up to Top = Rank i;t1 Low i;t1 :::2ndPerfQuint i;t1 . Following the literature I also estimate an alternative 3-piece specication that combines the three middle quintiles in one: Mid i;t1 min(Rank i;t1 Low i;t1 ; 0:6). According to the testable implications of Section 1.4.3, we would expect funds in the herding group (Contrarian = 0) to be more sensitive to the medium relative performance pieces (quintiles 4thPerfQuint to 2ndPerfQuint, or Mid piece), or to be less sensitive on the top (Top fractional rank), consistent with a less convex relationship. The nal sample period for the estimation is 1998-2010. 37 Each year, I run cross-sectional regressions to estimate equation (1.25) and compute the means and t-statistics from the time series of coecient estimates following Fama and Mac- Beth (1973). In order to control for the high autocorrelation of mutual fund ows (see DelGuercio and Tkac (2002)), reported Fama-MacBetht-statistics are calculated using Newey and West (1987) autocorrelation and heteroskedasticity consistent stan- dard errors. Results are reported in Tables 1.3 and 1.4 for the quintile and 3-piece specications, respectively. 37 The length of the sample period is constrained by the lack of availability of fee data prior to 1998. 45 Table 1.3: Flow-performance relationships, 5-piece specication. This table examines the sensitivity of fund ows to relative performance for the top performance mutual funds in Table 1.2 (column (A)) and for only the \Herding" and \Contrarian" groups in this subsample (remaining columns) over the period 1998-2010. All funds in each objective category are annually ranked according to their annual raw returns and assigned a continuous rank (Rank) ranging from 0 (worst) to 1 (best). Performance quintiles are computed as follows: Lowi;t1 min(Ranki;t1; 0:2), 4thPerfQuinti;t1 min(Ranki;t1Lowi;t1; 0:2), and so forth, up to Top =Ranki;t1Lowi;t1::: 2ndPerfQuinti;t1. Each year a piecewise linear regression is performed by regressing net fund ows on funds' performance quintiles following the specication (1.25). Control variables include: Sizei;t1 (natural log of fundi's TNA),CategoryFlowi;t (aggre- gate ow rate into fundi's category),Volatilityi;t1 (standard deviation of fundi's monthly returns in yeart 1),Agei;t (natural log of 1 + fundi's age),ExpenseRatioi;t1 (fundi's expense ratio), andContrariani (equals 1 or 0 depending on whetheri is a \Contrarian" or \Herding" fund accord- ing to the herding measure in the corresponding column). Time-series average coecients and the Fama and MacBeth (1973)t-statistics (in parenthesis) calculated with Newey-West robust standard errors are reported. *, ** and *** denote signicance at the 10%, 5% and 1% level, respectively. \Contrarian" dummy based on: (A) HerdTrackErr HerdBeta HerdCorr Low 0.411*** 0.308 0.245*** 0.348 (5.12) (0.82) (3.49) (0.99) 4thPerfQuint 0.087 0.467** 0.173 0.467** (1.49) (2.86) (1.36) (2.94) 3rdPerfQuint 0.321** 0.303*** 0.213 0.251*** (2.79) (4.33) (1.28) (3.23) 2ndPerfQuint 0.235 0.380* 0.547*** 0.576* (1.75) (1.94) (3.39) (2.08) Top 1.218*** 1.426*** 0.945*** 0.862 (4.88) (3.49) (4.49) (1.59) Contrarian 0.037 -0.038 0.062 (0.73) (-1.14) (1.14) LowContrarian 0.008 0.151 -0.058 (0.02) (0.65) (-0.14) 4thPerfQuintContrarian -0.278 0.083 -0.333 (-0.95) (0.36) (-1.17) 3rdPerfQuintContrarian 0.137 -0.048 0.226 (0.61) (-0.19) (1.04) 2ndPerfQuintContrarian -0.704** -0.588*** -0.814* (-2.21) (-3.70) (-2.03) TopContrarian 0.306 0.443** 0.686 (0.55) (2.66) (1.29) Adjusted-R 2 0.177 0.183 0.190 0.191 Number of observations 6506 4298 4413 4357 46 Table 1.4: Flow-performance relationships, 3-piece specication. This table examines the sensitivity of fund ows to relative performance for the top perfor- mance mutual funds in Table 1.2 (column (A)) and for only the \Herding" and \Contrarian" groups in this subsample (remaining columns) over the period 1998-2010. All funds in each objective category are annually ranked according to their annual raw returns and assigned a continuous rank (Rank) ranging from 0 (worst) to 1 (best). Performance segments are com- puted as follows: Lowi;t1 min(Ranki;t1; 0:2), Midi;t1 min(Ranki;t1Lowi;t1; 0:6), and Top = Ranki;t1Lowi;t1Midi;t1. Each year a piecewise linear regression is per- formed by regressing net fund ows on funds' performance segments. Control variables include: Sizei;t1 (natural log of fund i's TNA), CategoryFlowi;t (aggregate ow rate into fund i's cat- egory), Volatilityi;t1 (standard deviation of fund i's monthly returns in year t 1), Agei;t (natural log of 1 + fund i's age), ExpenseRatioi;t1 (fund i's expense ratio), and Contrariani (equals 1 or 0 depending on whether i is a \Contrarian" or \Herding" fund according to the herding measure in the corresponding column). Time-series average coecients and the Fama and MacBeth (1973) t-statistics (in parenthesis) calculated with Newey-West robust standard er- rors are reported. *, ** and *** denote signicance at the 10%, 5% and 1% level, respectively. \Contrarian" dummy based on: (A) HerdTrackErr HerdBeta HerdCorr Low 0.315*** 0.485* 0.069 0.372 (4.43) (2.17) (0.51) (1.07) Mid 0.236*** 0.346*** 0.311*** 0.382*** (5.95) (5.14) (4.86) (4.36) Top 1.233*** 1.653*** 1.135*** 1.284** (5.13) (5.86) (5.41) (2.85) Contrarian 0.05 -0.061** -0.134 (1.61) (-2.43) (-1.21) LowContrarian -0.088 0.466** -0.03 (-0.33) (2.60) (-0.08) MidContrarian -0.191* -0.174** -0.200* (-2.10) (-2.35) (-2.10) TopContrarian -0.186 0.101 -0.017 (-0.56) (0.50) (-0.04) Adjusted-R 2 0.175 0.182 0.184 0.188 Number of observations 6506 4298 4413 4357 Column (A) in both tables, displaying the estimated regressions for all top mutual funds (i.e. including top funds not classied as \herding" nor \contrarian"), shows that the convex relationship between ows and relative performance found by the prior literature is also present in this sample. 38 The remaining columns show that the estimated ow-performance relationships for the herding and contrarian groups dier, as re ected by the statistically signicant coecients for the interaction terms Contrarian2ndPerfQuint andContrarianMid across the regressions for all three 38 Specically, the estimated coecient for Top is signicantly higher than that of 2ndPerfQuint in Table 1.3, and similarly with Top and Mid in Table 1.4. 47 herding measures. Consistent with the model's predictions, the contrarian group is less sensitive to the second performance quintile (even at the 1% signicance level in the case of HerdBeta regression) and weakly more sensitive to the top quintile (sig- nicant at the 5% level forHerdBeta) in Table 1.3, and less sensitive to middle-range performance in Table 1.4. The dierences are economically signicant: according to Table 1.4, for instance, an increase from the 45th to the 55th ranking percentile will lead herding mutual funds to an expected increase in fund ows next year of 3.5%, 3.1% or 3.8% depending on the herding measure considered, whereas otherwise iden- tical contrarian fund ows will rise by only 1.5%, 1.4% and 1.8%. These estimates also suggest a more convex ow-performance relationship for the contrarian group. Panels (a) and (b) in Figure 1.5 illustrate this point, by depicting the multivariate 5- and 3-piece ow-performance relationships of herding and contrarian informed mu- tual funds according to the tracking error measure HerdTrackErr. 39 The graphs suggest: (i) fund ows for herding mutual funds are roughly as sensitive to very poor as to mid-range performance, delivering an approximately linear positive relationship between ows and performance in these regions, and (ii) contrarian mutual funds face a more convex relation between ows and performance. As robustness checks, I re-estimate equation (1.25) and its 3-piece counterpart for dierent alternatives to steps 1 to 3 above: (i) herding measures of step 3 computed over returns gross of expense ratios instead of over net of expense ratios as reported by CRSP; (ii) \informed" mutual funds dened as those belonging to the top quintile in the outperformance probability ranking of step 1. 40 Alternative (i) is meant to capture true managerial ability more closely, following Cohen, Coval, and Pastor 39 The pictures show the relation between expected ow rate and lagged relative performance for the average fund in each herding category, i.e. substituting average values for all included control variables into the estimated equations. 40 In non-tabulated results, I also checked the robustness of these empirical ndings to dierent denitions of the portfolio of average performers (0:49 Rank 0:51 and 0:4 Rank 0:6), as well as dierent thresholds for selecting contrarian and herding mutual funds (top and bottom 20% in the herding ranking). 48 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Performance Rank Net Fund Flow (%) Herding Contrarian (a) 5-piece specication 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Performance Rank Net Fund Flow (%) Herding Contrarian (b) 3-piece specication Figure 1.5: Flow-performance relationships Panels (a) and (b) illustrate the ow-performance relationships estimated in Tables 1.3 and 1.4 for the top 30% mutual funds classied as \Herding" or \Contrarian" according to their values of HerdTrackErr (second column in each table). The graphs show the relation between expected ow rate and lagged relative performance for the average fund in each herding category, i.e. substituting average values for all included control variables into the estimated equations. (2005). 41 Alternative (ii) imposes a stricter requirement for a mutual fund to be selected as \informed", reducing the importance of \uninformed but lucky" mutual funds in the sample. Estimates for the quintile specications with herding groups 41 To obtain gross returns, I add the annual expense ratio divided by 12 to the monthly returns in that year. Fee data is sparse prior to 1998 in CRSP Mutual Funds database, which is why the main results are presented in terms of net returns despite the arguments in favor of using gross returns instead. 49 classied by HerdTrackErr are reported in Table 1.5. In general, we see that the same dierences between the fund- ow relationships of herding and contrarian groups survive the dierent specications: herding funds face higher sensitivity to middle- range performance but lower sensitivity on the top, leading to a less convex ow performance relationship that is consistent with their more conservative trading. 42 42 Estimates for the 3-piece specication and for the herding classication of informed funds ac- cording to HerdBeta and HerdCorr are available from the author upon request. In all cases, the results follow the same patterns as in Tables 1.3 through 1.5. 50 Table 1.5: Flow-performance relationships: robustness checks. This table examines the sensitivity of fund ows to relative performance, over the period 1998-2010, for top mutual funds classied as \Herding" or \Contrarian" according to whether they are in the bottom or top 33% of a ranking based on the values of HerdTrackErr. In the second column, the sample includes the same top 30% performers as in Tables 1.3 and 1.4, but HerdTrackErr is computed over returns gross of expense ratios. In the third column, HerdTrackErr is com- puted over fund returns net of expense ratios as in Tables 1.3 and 1.4, but the sample includes only top 20% performers according to their proportion of outperforming years. All funds in each objective category are annually ranked according to their annual raw returns and assigned a con- tinuous rank (Rank) ranging from 0 (worst) to 1 (best). Performance quintiles are computed as follows: Lowi;t1 min(Ranki;t1; 0:2), 4thPerfQuinti;t1 min(Ranki;t1Lowi;t1; 0:2), and so forth, up toTop =Ranki;t1Lowi;t1::: 2ndPerfQuinti;t1. Each year a piecewise linear regression is performed by regressing net fund ows on funds' performance quintiles follow- ing the specication (1.25). Control variables include: Sizei;t1 (natural log of fund i's TNA), CategoryFlowi;t (aggregate ow rate into fundi's category),Volatilityi;t1 (standard deviation of fundi's monthly returns in yeart 1),Agei;t (natural log of 1 + fundi's age),ExpenseRatioi;t1 (fundi's expense ratio), andContrariani (equals 1 or 0 depending on whetheri is a \Contrarian" or \Herding" fund according to the value ofHerdTrackErr). Time-series average coecients and the Fama and MacBeth (1973)t-statistics (in parenthesis) calculated with Newey-West robust standard errors are reported. *, ** and *** denote signicance at the 10%, 5% and 1% level, respectively. Herding computed Top 20% performers over gross returns Low 0.719** 0.37 (2.43) (0.90) 4thPerfQuint 0.407** 0.562*** (2.63) (4.13) 3rdPerfQuint 0.285*** 0.149 (5.17) (1.17) 2ndPerfQuint 0.440** 0.480** (2.62) (3.04) Top 1.488*** 1.534** (3.22) (2.98) Contrarian 0.115** 0.003 (2.27) (0.06) LowContrarian -0.336 0.116 (-0.85) (0.23) 4thPerfQuintContrarian -0.285 -0.833* (-1.08) (-2.01) 3rdPerfQuintContrarian 0.116 0.599* (0.53) (2.02) 2ndPerfQuintContrarian -0.724* -0.736** (-1.91) (-3.02) TopContrarian 0.262 0.29 (0.47) (0.61) Adjusted-R 2 0.187 0.183 Number of observations 4241 2963 51 1.6 Other Applications Institutional investors and \bubbles". The optimal investment policy (1.19) implies that a manager with superior information but relative concerns with respect to less informed peers will not always adopt a contrarian stance against peer's suboptimal strategies (\lean against the wind"). This is true in the case of bubble-like asset price dynamics as well, which in the current setup would manifest as a sustained increase in the stock price S throughout the period, high above the appreciation that could be expected from its true mean return . Such a situation would correspond, e.g. to an estimation error ~ t = 0:07 by the uninformed managers in Panel (b) of Figure 1.1, resulting in risk over-exposure by these managers. We see that whereas some informed managers (dark solid line) would trade aggressively against uninformed funds and under-expose their portfolios to market risk, others (gray solid line) would herd with their uninformed peers and over-expose their portfolios to market risk. The model's implications are relevant in light of the recent empirical evidence on institutional investors' holdings of \bubble" stocks during the apparent technology bubble of the late 1990s 43 and suggests an alternative channel, i.e. the incentive eects of fund ows, behind some institutions' over-exposure to \bubble" stocks during this period. Active management and closet indexing Cremers and Petajisto (2009) nd that many actively managed U.S. equity mutual funds have holdings that are similar to those of their benchmarks, and point to the importance of distinguishing between funds that are truly active from those that are \closet index funds". They also nd that funds whose holdings are most dierent from their benchmarks have higher Jensen's alpha than their benchmarks. The results in Section 1.4.3 suggest that, if the average (unskilled) fund manager within an objective category follows a benchmark 43 See e.g. Brunnermeier and Nagel (2004) and Dass, Massa, and Patgiri (2008). 52 closely (i.e. is a closet indexer) and in addition money ows in this category are very sensitive to medium relative past performance, a skilled manager in the same category may choose to optimally herd with the average fund. Such a manager will look like an average manager and generate a too small Jensen's alpha according to the results in Section 1.4.2 in spite of its superior ability and of potentially delivering positive net benets to delegating investors. By contrast, the true ability of active and informed mutual funds (high return tracking error) responding to highly convex fund ows may be overstated by performance measures that fail to capture the non-linear risks inherent in their option-like strategies. 1.7 Concluding Remarks I study dynamic risk-taking and performance by money managers subject to convex fund ows in a model that allows for dierential ability across managers and in which relative performance is determined endogenously. I argue that fund ows provide skilled managers with complex incentives, inducing \limited liability" on the bottom relative performance region and concerns relative to less informed peers on the top. I show how the interplay of these eects can lead informed managers to take positions contrarian to their peers in some situations, but to herd with them and partially disregard their private information in others. I argue that for many plausible ow- performance relationships skilled managers' performance will not dierentiate much from their unskilled peers' despite facing highly convex incentives. Such policies re- sult not in excessive risk-taking but in excessive conservatism instead, leading high quality managers to look like average performers. I show that, in all cases, standard performance measures may fail to adjust for the risks informed managers take, result- ing in performance under- or over-statement in many situations. Using a sample of U.S. mutual funds I provide evidence supporting the model-implied relation between herding behavior and the shape of the ow-performance relationship. I suggest that 53 the developed framework can help explain the trading strategies of sophisticated in- stitutional investors during bubble-like asset price dynamics and closet-indexing by actively managed mutual funds. The assumed informational advantage by skilled managers in the model is arguably simplistic. In the real world, managers may have private information not only about assets mean returns but also about their volatility, about their correlation structure in a multi-asset framework, or about the relationship between asset returns and the economy's state variables when managers are market-timers. 44 However, the qualita- tive results in the model do not depend on the assumed informational advantage, and we could expect more informed managers to herd and perform like less informed peers in many situations even for more sophisticated information structures or distribution of skills among managers. The results in this chapter suggest that herding by skilled managers can make the inference problem of learning ability from performance very dicult. An im- portant extension of this model would then involve the equilibrium determination of both delegating investors' ows and managers' optimal trading in a unied frame- work. Another natural direction for future research are the asset pricing implications of informed and uninformed managers' strategies in response to convex fund ows. Although the task looks challenging, any progress in this area could prove fruitful in addressing the deeper question of how managers' implicit incentives aect the infor- mative role of asset prices and the overall eciency of capital markets. 44 See Detemple and Rindisbacher (2011) for a structural approach addressing performance mea- surement of fund managers with private information under several information structures and skills. 54 Chapter 2 The Value of Cross-Trading to Mutual Fund Families: A Portfolio Choice Approach 2.1 Introduction A fund manager does not usually work directly for investors but rather for a family like Vanguard or Fidelity. This organizational form creates an extra layer of agency that can lead to additional misalignments of incentives. Nevertheless, most of the existing analytical literature on portfolio delegation has neglected the eects of family- aliation on funds' asset allocation decisions and on investors' utility. The purpose of this chapter is to investigate, in a dynamic portfolio choice frame- work, the costs and benets that can potentially make the decision of investors to delegate their portfolios to family-aliated funds dierent from that of delegating to standalone funds. We characterize the optimal trading strategies of aliated funds that are allowed to coordinate trades and internally reallocate their illiquid asset hold- ings in response to the interests of the family as a whole. We examine how strategic cross-trading can distort portfolio allocations and incentives within fund families, as well as investors' utility implications of portfolio delegation in such an institutional arrangement. We attempt to capture several salient features of the mutual fund industry in our analysis. First, mutual fund families can constitute very large internal markets. Indeed, there exists a vast myriad of funds split into numerous categories and run by relatively few managing companies. Over 90% of all funds belong to multi-fund families, and the top 50 fund families concentrate over 80% of equity assets under 55 management (Gaspar, Massa, and Matos (2006)). Moreover, the average family has about 7 funds under its umbrella managing more than $4 billion in net assets, and some of those families group over 85 dierent funds. Second, there is substantial overlapping in asset holdings among funds belonging to the same family. Elton, Gruber, and Green (2007) nd that as much as 34% of total net assets of funds with the same objective, and as much as 17% of funds with dierent objectives, consist of stocks held in common within the family, compared to 8% outside the family. Third, managers of funds belonging to families are exposed to a number of in- centives that go beyond those of their standalone counterparts. Besides the implicit convex relation between the past performance of both family and standalone funds and the investors' new money into the funds, as documented in Chevalier and Ellison (1997) and Sirri and Tufano (1998), among others, aliated funds are subject to the star-performer phenomenon and spillover eects documented by Nanda, Wang, and Zheng (2004). These eects result in higher expected in ows to families having a top and a poor performer than to families having just two average performers. These features suggest that mutual fund families oer investors both benets and costs. Due to the prevalence and size of fund families we propose that, in the presence of constrained market liquidity for some assets, signicant benets can be derived from cross-trading assets within the family 1 as opposed to trading in external markets. 2 Mutual funds are permitted to cross-trade under SEC Rule 17(a)-7 of the Investment 1 A survey study conducted by the Bank for International Settlement (BIS (2003)) points to savings on transaction arising from \crossing of trades" as one of the main factors behind the trend towards consolidation among investment managers in the asset management industry. One way in which families can cross their trades among aliated funds is by using `crossing networks'. See Hendershott and Mendelson (1999) for an economic analysis of this alternative form of exchange. 2 Chalmers, Edelen, and Kadlec (2000) estimate that the annual trading costs for equity funds are of rst order relevance, averaging 0.78% of fund assets. 56 Act of 1940. 3 Hence, as long as interfund cross-trading is mutually benecial for the family members involved, and savings on transaction costs translate into higher net (of fees) expected returns for the funds, investors benet from their funds' aliation with a family. 4 However, inherent to the family organization is also some degree of centralization in the decision-making process, giving rise to strategic (interdependent) decisions that are absent for standalone funds. Depending on the family's managerial structure, this centralization is re ected to a greater or lesser extent in day-to-day asset allocation decisions. While duciary duty requires that managers execute transactions in the most favorable way to each fund's shareholders, the distorted incentives to which funds are subject, combined with a compensation scheme that rewards the family manager for the total value of the family's assets, imply an additional misalignment of objectives between investors and their agents. As a result, the family manager may be willing to cross trades between funds in order to benet one in detriment of the other when by doing so she can achieve a stellar performance for the former. 5 To the extent that cross-trading in the presence of illiquidity implies a suboptimal investment decision for at least one of the funds, investors in this fund pay an additional agency cost stemming only from their funds' aliation with the family. 3 Rule 17a-7 is an exemptive rule under the Investment Company Act of 1940 that permits pur- chase and sale transactions among aliated investment companies, or between an investment com- pany and a person that is aliated solely by reason of having a common (or aliated) investment adviser, common directors, and/or common ocers. 4 There are limits to cross-trading, though. SEC Rule 35(d)-1 of the Investment Act of 1940, for instance, requires a registered mutual fund with a name suggesting that it focuses on a particular type of investment to invest at least 80% of its assets in the type of investment suggested by its name. The Investment Company Act of 1940 also prevents mutual funds from short selling and buying securities on margin. 5 Positive transaction costs are essential for this argument because if markets are perfectly liquid there would be no incentives to cross-trade and all transactions would take place in public markets. Given that portfolio-distorting incentives can also be signicant in perfectly liquid markets, a fund family may still engage in star-creating strategies but not necessarily involving cross-trading in this situation. 57 We use a dynamic portfolio choice model to investigate how the interaction of these benets and costs can potentially make the asset allocation decision of ali- ated funds dierent from that of standalones. We consider a fund family consisting of two funds that follow dierent investment styles as re ected in the benchmarks with respect to which investors evaluate each fund's performance. 6 Investors' money withdrawals from and infusions to the funds are a non-linear function of their rela- tive (to the benchmark) past performance, with convexities in the top performance region. 7 Because managers are rewarded for increasing the value of assets under their management, distorted incentives to boost the performance of at least one of the funds in the family is a likely result. Additionally, we allow for some overlap in funds' asset holdings, consisting of two types of risky assets: a liquid asset specic to each investment style, and a relatively illiquid asset held in common by both funds. This overlap allows a centralized family manager to cross-trade the illiquid asset between the two funds in order to maximize the overall wealth of the family. Funds pay transaction costs indirectly when buying or selling the illiquid asset. We follow Longsta (2001)'s thin trading approach to illiquidity, in which market participants are constrained to trading strategies that are of bounded variation, i.e. the number of shares that can be bought or sold at any moment is bounded. 8 In these circumstances, one way to boost the performance of one of the funds in the family is to have it avoid the costs of illiquidity by buying or selling assets from the other fund. Of course, creating a top performer by cross-trading with another fund in the family is usually not in the best interest of the latter, implying a de facto 6 By investment style we mean the fund's stated investment objective (e.g. value, growth, market- oriented, small capitalization, etc.) 7 See Chevalier and Ellison (1997) and Sirri and Tufano (1998) for an empirical examination of investors' ow-performance relations, and Basak, Pavlova, and Shapiro (2007) for an analysis of their implications to managers incentives. 8 See Goncalves-Pinto (2010) for a detailed examination of an investor's delegated portfolio prob- lem to actively-managed and passive standalone funds in the context of illiquid markets, making use of a similar methodology to the one used in this chapter. 58 cross-subsidization between the two funds. A rst insight of the model is that fund families under illiquidity exacerbate the asset substitution problem induced by the convex shape of the ow-performance relation: two funds operating under the family umbrella more than duplicate the agency costs of delegation to identical standalone funds. This is because in some situations two independent funds will not be able to risk-shift due to their limited ability to trade in illiquid markets, but the same funds operating under a family arrangement will be able to circumvent illiquidity through cross-trading and thus engage in otherwise unfeasible risk-shifting strategies. We derive several additional implications from the solution of our model. First, families' ability to cross-trade between member funds may create benets to the family in the form of savings on transaction costs, but more likely even higher gains by the possibility of exploiting the convexities of funds' ow-performance relations. Second, the ability to cross-trade is likely to elicit higher risk-taking by family-aliated fund managers, compared to their standalone counterparts, entailing further utility losses on their investors. Fund-family aliation may even change the risk prole of its members by turning a relatively conservative standalone fund's return distribution more volatile, skewed and leptokurtic, and vice versa for a riskier fund. We show how investors can use position limits as contracting features that curb these risk- shifting incentives and improve the utility results from delegating their portfolios to family-aliated funds. Third, families' optimal strategies can induce a negative correlation between their aliated funds' after- ow returns, creating diversication benets on the family's overall portfolio. Still, these strategies can result in a higher correlation between member funds' portfolio returns than that between comparable standalones, due to overlap in holdings. Finally, we nd that families in our model favor highly diversied portfolios at the fund level when liquidity is low but correlated asset holdings under more liquid conditions. Overall, we hope that our work draws attention to a potential misspecication of the traditional approach to portfolio delegation. Given the importance of illiquidity 59 for many asset classes and the pervasiveness of the family organization of mutual funds, focusing on the asset allocation decisions of standalone funds can signicantly misestimate the agency costs of delegation. Related Literature The empirical literature on mutual funds families is vast and growing. This chapter is inspired in part by the analysis of Gaspar, Massa, and Matos (2006), who nd that mutual fund families transfer performance across member funds to favor those funds with an expected higher contribution to family prots. However, the kind of cross- subsidization the authors analyze comes through interfund transactions at below or above market prices to favor one fund over the others. Such transactions go against SEC Rule 17a-7 and are thus illegal. What our framework captures is a more subtle way of cross-subsidization that can take place even when interfund transactions are executed at fair market prices: the performance of one of the funds is increased above what would be achievable if it was an independent fund, at the cost of making the other fund in the family adopt a suboptimal investment policy. This is not so clearly against Rule 17a-7. However, the costs imposed on the fund out-of-favor do represent a clear breach of duciary duty. 9 In contrast, the theoretical body of research on the investment decisions of mutual fund families is less abundant. Closest to ours is the paper by Binsbergen, Brandt, and Koijen (2008), who study the costs of under-diversication and of misalignment of objectives to an investor that delegates her portfolio to a fund family in which a CIO employs multiple asset managers to invest in dierent asset classes. The authors obtain closed-form solutions for their problem and derive a performance benchmark 9 This breach of contract is arguably far more dicult to detect. The SEC sta provided clarica- tion on Rule 17a-7 later on, in the form of a no-action letter (Federated Municipal Funds (November 20, 2006)), with respect to an investment advisers duciary duties in connection with Rule 17a-7 transactions: an adviser must determine that the Rule 17a-7 transaction is in the best interests of both the selling and buying funds, thus prohibiting any transaction that is in the best interest of one fund but is otherwise neutral to the other fund. 60 that mitigates these costs. However, the problem they analyze is dierent from ours in important respects. Asset classes are mutually exclusive in Binsbergen, Brandt, and Koijen (2008) and thus cross-trading is of no interest in their framework. In the presence of nancial frictions (e.g. illiquidity) and risk-shifting incentives, we identify a motive for families to allow some overlap in asset holdings across its dierent in- vestment styles, in accordance with the evidence in Elton, Gruber, and Green (2007). Also related to ours is the work by Taylor (2003), who considers tournaments be- tween aliated funds' managers. If there is no strategic interaction between them, his paper shows that midyear losers (i.e. players with a below-average rank after the rst part of the year) should increase risk more than midyear winners. However, in the presence of strategic interactions, winners should instead increase risk more than losers do. Finally, though not in the context of fund families, the analysis in this chapter draws on the characterization of mutual funds' risk-shifting incentives induced by non-linear ow-to-performance relations studied by Basak, Pavlova, and Shapiro (2007). Our work can be seen as an application of their framework to funds aliated to fund families in the presence of illiquidity in nancial markets. The rest of the chapter proceeds as follows. In Section 2.2 we set up our analytical framework, including a description of the economic setting and the problem to be solved by the fund family. In Section 1.4 we solve the model using numerical methods, and discuss its main results. Conclusions and implications for further research are presented in Section 2.4. 61 2.2 Model Setup 2.2.1 The Economy We consider an economy in which investors (households) delegate the administration of their savings to mutual fund families over a certain investment horizon. 10 In general, this investment horizon extends over a sequence of periods [(h 1)T;hT ], h2f1; 2;:::;Hg and H <1, but we focus our analysis on one of such periods (e.g. one calendar year), which we denote by [0;T ]. Mutual fund companies have access to nancial markets consisting of three assets, with prices denoted by S i (t), for i2f1; 2;Cg. The rst two assets are risky assets that trade in perfectly liquid markets, but asset C is a risky asset that trades in a thin market, which makes us depart from the Black-Scholes-Merton complete nancial market structure. Following Longsta (2001), a market is thin when its participants can only adopt trading strategies that are of bounded variation, i.e. the number of shares of asset C that can be bought or sold per unit of time is limited. We make this notion of illiquidity more explicit below. The prices of these three assets evolve according to: dS i (t) = i S i (t)dt + i S i (t)dZ i (t); (2.1) where i and i are constant, and uncertainty is governed by the standard Brownian motion processesZ i (t), fori2f1; 2;Cg, with constant correlation coecients among 10 We consider investors' exogenous portfolio delegation decision to be grounded on the assumption that mutual fund managers are subject to lower transaction costs, lower opportunity costs for engaging in active portfolio management, better information or ability, and/or better investing education. In a more general model, investors would be allowed to dynamically choose how much of their portfolio to hold directly and how much to hold indirectly through the managed portfolios of mutual funds, pension funds, and the like. 62 the assets 0 kl < 1, for k;l2f1; 2;Cg, and k6=l (i.e. E[dZ k (t)dZ l (t)] = kl dt). 11 Asset prices are assumed to start the investment period at the value S i (0) =s i . We assume that investors are heterogeneous with respect to their appetites for risk and consequently allocate their savings to funds with dierent risk-return proles. In order to agree with the common denomination used in the nancial industry, we call the dierent risk-return proles as `investment styles' (e.g. value, growth, market- oriented, small capitalization). Funds with dierent investment styles allocate their resources to dierent (though not necessarily mutually exclusive) sets of assets. We consider two of such investment styles, style A and style B. Style A corresponds to a portfolio of assets 1 and C, with a higher weight on the liquid asset 1. Style B is characterized by a portfolio of assets 2 andC, with predominance of assetC. We can think of style A as focusing its strategy on liquid risky assets, like well-known publicly traded large-cap stocks, while style B can be thought of as investing predominantly in illiquid risky assets, like real estate or small-cap stocks from emerging economies. Mutual funds in our economy are organized in families, re ecting the prevalent organizational form in the U.S. mutual fund industry. 12 We consider a fund family consisting of two mutual funds j 2f1; 2g. Each of these funds is managed by a dierent portfolio manager, while a Chief Investment Ocer (CIO) decides on the extent of cross-trading between the two funds. 13 Fund 1 follows investment style A 11 The denition of illiquidity of our assetC parallels that in Longsta (1995) in the sense that the restriction on liquidity is mainly investor-specic. This makes clear the partial equilibrium sense of the analysis in this chapter given that our investors and managers may face trading restrictions even though the assetC is traded continuously in the market by other investors that are less constrained, leading asset C's price to evolve according to the dynamics expressed in Equation (2.1). 12 See Gaspar, Massa, and Matos (2006). We abstract from looking into the reasons behind the emergence of families as an organizational form in the rst place, or the factors determining families' optimal size. Both of these are interesting questions on their own right, but we take them as given. 13 In the nancial investment industry, Chief Investment Ocers are board-level managers for their investment companies. For most mutual fund families, CIOs have the responsibility for the investments and strategy of the overall group and oversee the team of investment professionals in charge of the individual funds' investments. The CIO in our setting cares about the overall value of the family but we greatly simplify his functions to that of deciding interfund transactions at the beginning of the investment period. 63 whereas fund 2 is a style B fund. We letN j C (t) be the number of shares of the illiquid asset C that fund j2f1; 2g holds as of time t, with dynamics: dN j C (t) =' j (t)dt; (2.2) where1 < ' j (t) < +1, and > 0. 14 Then the value of fund j's self-nancing portfolio, F j (t), has dynamics given by: dF j (t) = h F j (t) j +N j C (t)S C (t)( C j ) i dt +F j (t) j dZ j (t)+ +N j C (t)S C (t) [ C dZ C (t) j dZ j (t)] (2.3) and initial value F j (0) =f j , forj2f1; 2g. We constrain these portfolios to lie in the closed solvency region: S = n (S j (t);S C (t))2R 2 :N j (t)S j (t) +N j C (t)S C (t) 0 o (2.4) for all t2 [0;T ], where N j (t) denotes the number of shares of the liquid asset j that fund j2f1; 2g holds as of time t. In line with standard practice in the mutual fund industry, fund managers' com- pensation is set proportional to the value of the assets under their administration and is due at the investment horizon t = T . 15 There are no external cash in ows to or out ows from the funds during the investment period t2 [0;T ). Investors' fund share purchases and redemptions occur only at time t = T and depend on the fund's performance relative to the average performance of all the funds following the same investment style, in a way that we make clear in what follows. Let Y j (t) be the 14 This illiquidity parameter can be thought of as being investor-specic and/or asset-specic. It can also capture temporary price impact as in Isaenko (2009). 15 This compensation structure is also justied by some theoretical arguments holding that the size of a fund can be used as a proxy for managerial skill (see Berk and Green (2004)). 64 (exogenously given) benchmark capturing the style of fund j's industry average per- formance, for j2f1; 2g. 16 Y 1 (t) and Y 2 (t) are the value processes for (self-nancing) reference portfolios, holding B 1 C (t) and B 2 C (t) shares of asset C (with the remaining invested in the asset specic to the style of fund j, for j2f1; 2g), respectively, and dynamics given by: dY j (t) = h Y j (t) j +B j C (t)S C (t)( C j ) i dt +Y j (t) j dZ j (t)+ +B j C (t)S C (t) [ C dZ C (t) j dZ j (t)]; (2.5) and Y j (0) = y j , j2f1; 2g. These benchmark portfolios represent passive buy-and- hold strategies that keep the number of shares of the illiquid asset as of time t = 0 constant over the investment period. A continuously-rebalanced benchmark would instead hold the relative weights of the assets in their portfolios constant over this period. 17 We then set B j C (t) =y j j C (0)=s j , where j C (0) is the initial weight of asset C on benchmark j, for j2f1; 2g. According to the assumed investment styles, we set 0 1 C (0)< 2 C (0) 1. We let R F j (T ) = ln (F j (T )=F j (0)) and R Y j (T ) = ln (Y j (T )=Y j (0)) denote the continuously-compounded return of fund j's portfolio and of benchmark j's port- folio, forj2f1; 2g, respectively, over the time periodt2 [0;T ]. We setY j (0) =F j (0) without loss of generality. Investors' in ows and out ows of money at time t = T depend on a fund's performance relative to its style benchmark, which can be mea- sured by the dierence in their returns: R F j (T )R Y j (T ). In particular, we draw on the estimations by Sirri and Tufano (1998) nding that investors reward good fund 16 These benchmark portfolios could also be interpreted as constraints in the contract decided between the investor and the fund managers at the beginning of the investment period. For theoretical arguments justifying the use of benchmarks in the principal-agent contract between investors and fund management companies see, for instance, Maug and Naik (1996), Basak, Pavlova, and Shapiro (2008), Li and Tiwari (2009) and Dybvig, Farnsworth, and Carpenter (2010). 17 Since asset C is illiquid in our framework, it can be innitely costly for a manager to keep the relative weight of this asset in her portfolio constant. Keeping the number of shares constant (a buy-and-hold strategy on assetC) is then a more natural specication for these benchmark portfolios. 65 performers with increasingly higher cash in ows but punish very poor performers as little as they punish slightly bad performers. We choose this ow-performance re- lation over Chevalier and Ellison (1997)'s empirical specication in order to capture the greater cash in ows to star performers in a fund family as found empirically by Nanda, Wang, and Zheng (2004). 18 We approximate this type of fund-performance relation by a linear-convex function, 19 one per investment style: j (T ) = 8 > < > : L j if R F j (T )R Y j (T )< j L j + j h e R F j (T )R Y j (T ) e j i if R F j (T )R Y j (T ) j ; (2.6) for L j and j positive, and j 2R, forj2f1; 2g. Note that j (T ), the rate at which money ows into ( j (T ) > 1) or out of ( j (T ) < 1) fund j at the terminal date, depends on that fund's performance relative to its benchmark. Agents in this economy derive utility from the value of their wealth at the terminal date T . We assume standard constant relative risk aversion (CRRA) preferences for the CIO and the funds' managers. In order to capture investors' heterogeneous preferences for dierent investment styles, we introduce a relative concern with respect to their corresponding benchmarks in their utility functions as follows: U j (F j (T );Y j (T )) = [F j (T )=Y j (T )] 1 i 1 i ; (2.7) for investor in fund j2f1; 2g, and investor's risk aversion parameter given by i . 20 18 However, we are not considering the \spillover" eect to other funds in the same family as the star performer, also documented by Nanda, Wang, and Zheng (2004), as this would introduce strategic interactions between the portfolio decision problems of the fund managers, rendering the problem signicantly more complicated to solve. 19 We borrow this functional specication from Basak, Pavlova, and Shapiro (2007). 20 Note that this specication of preferences is similar to the used in the \external habit formation" literature (see, e.g. Chan and Kogan (2002)) but with benchmarkYj (T ) instead of the external habit for investor j. 66 2.2.2 The Fund Family's Problem As described before, the family consists of funds 1 and 2 following investment styles A and B, respectively. What makes a family dierent from just a portfolio of two independent funds (standalones) in our specication is the possibility of avoiding public markets in some circumstances by cross-trading assets between the member funds. 21 In the presence of an illiquid market for the asset that both funds hold in common in their portfolios (designated as assetC), there may exist situations in which both funds are willing to place opposite orders on this asset in the market. 22 In such circumstances it would be mutually benecial for both funds to cross trades within the family and save on the transaction costs imposed by illiquid external markets. 23 If these mutual funds were to be run by managers that are completely independent of each other, cross-trading would take place if and only if both funds' wealth was to increase by doing so. 24 As long as the benets of cross-trading translate into higher net (of fees) expected returns, investors in aliated funds should then be better o than investors in comparable standalones. However, to the extent that funds belonging to families are not completely independent but are subject to some degree of centralization in the asset allocation decisions, a potential misalignment of 21 SEC Rule 17(a)-7 of the Investment Act of 1940 allows interfund cross-trading as long as the transaction is eected at the \independent current market price of the securities", among other conditions. 22 For instance, a fund manager may nd it necessary to reduce her holding of a particular asset in order to keep her account in compliance with the mandated asset allocation requirements, due to changes in the market value of a portfolio holding, or due to share withdrawals by fund investors. At the same time, another fund manager within the same family may nd that her portfolio is under- weighted in that same asset, or that the asset being sold is a good addition to her portfolio. If each manager places opposite orders on the same asset in the open market, both funds will have to pay commissions and other transaction costs on their respective transactions. They can reduce these costs by \crossing" their trades, especially when the asset being traded is illiquid. The market value of the asset is readily obtainable so the transaction price is unquestionably fair to both fund managers. 23 This is similar to the distinction between internal and external capital markets in the corporate nance literature. 24 We assume that managers do not compete with each other within the fund family. See Tay- lor (2003) for an analysis of the risk-shifting incentives induced by the strategic interaction among managers of the same family when their compensation is based on tournaments. 67 objectives between the family's goals and those of the investors may bring costs to the latter when the family is allowed to cross-trade among its funds. We assume a \minimal" degree of centralization in the funds' portfolio choice decisions. A CIO of the fund family company decides, at the beginning of the in- vestment period, the extent of cross-trading in asset C between funds 1 and 2. From that initial moment on, until the end of the investment period, each fund is managed independently according to its own style. We acknowledge that, in reality, the ability of the CIO to coordinate trades internally may very well be exercised more frequently than just at the beginning of the investment period, but that framework revealed to be intractable. Therefore, the results of our analysis can be seen as a sort of `lower bound' on the actual eects of a family's ability to cross-trade. Rather than weaken- ing our conclusions, this leads us to believe that many of the predictions of our simple model may be even stronger in the real world. The family's problem is then solved in two stages, in a backward fashion. Let n j C 0 be the initial number of shares of assetC in fundj's portfolio, forj2f1; 2g. 25 Given the extent of cross-trading X decided by the CIO att = 0, in the second stage t2 (0;T ] fund j's manager solves the following problem: V j (F j ;Y j ;N j C ;S C ;t) = sup ' j (t) E t [ j (T )F j (T )] 1 j 1 j ; (2.8) where j > 1 is the manager-specic coecient of relative risk aversion. Note that each manager is rewarded exclusively in proportion to the terminal value of its own assets under management, and that this is the result of each manager's portfolio allocation decisions taken over the investment period, as well as the value of investors' ows into and out of the fund at the terminal date. The fund manager's investment 25 We can think of n j C as the number of shares of the illiquid asset that fund j2f1; 2g `inherits' from the previous investment period. 68 horizon coincides with the date of fund ows, and fund ows are nontradable at that date. 26 In solving (2.8), manager j is subject to the price process for asset C, given by Equation (2.1), the dynamics of the assets under management (2.3), the benchmark process (2.5), and the following initial conditions: N 1 C (0) =n 1 C X; (2.9) N 2 C (0) =n 2 C +X: (2.10) Note that a positive value of X represents a purchase of asset C by fund 2 from fund 1, whereas a negative value represents the opposite transaction. We impose the condition thatn 2 C Xn 1 C (no `internal' short-selling). In the rst stage of the problem, which happens at timet = 0, the CIO solves the following static problem: V c (f 1 ;f 2 ;y 1 ;y 2 ;n 1 C ;n 2 C ;s C ) = sup X E 0 W (T ) 1 c 1 c (2.11) where W (t) = 1 (t)F 1 (t) + 2 (t)F 2 (t), and c > 1 is the CIO's coecient of relative risk aversion. The CIO solves (2.11) subject to the optimal trading strategies ' j (t) derived from (2.8), which she expects fund managers will adopt over the period t2 (0;T ], for any given N j C (0). Note that the CIO's compensation is proportional to the total value of the family (joint value of funds 1 and 2) at the terminal date, which we denote byW (T ). Given the convexities in the ow-performance relations of the individual funds, the objective of the CIO may induce her to pursue star-creating 26 In a more general model, the investment horizon need not coincide with the fund ows date (e.g. Basak and Makarov (2008)), in which case fund ows would be tradable after the ow date and j (t<T ) would then enter the problem through the budget constraint, and not directly through the utility function. See also Hugonnier and Kaniel (2008) for an analysis of the implications of dynamic ows on a mutual fund manager's portfolio decisions. 69 strategies, a phenomenon that is empirically documented in Nanda, Wang, and Zheng (2004). 27 We allow our setup to incorporate the eects of SEC Rule 35(d)-1 of the Invest- ment Act of 1940. This Rule requires that an investment company whose name sug- gests that the company focuses on a particular type of investment, or on investments in a particular industry, invests at least 80% of its assets in the type of investment suggested by its name. We capture this institutional feature by including position limits at time t = 0 on asset C for funds 1 and 2, 28 respectively, as follows: 0 N 1 C (0)s C f 1 b 1 ; (2.12) b 2 N 2 C (0)s C f 2 1; (2.13) whereb 1 < 1 andb 2 > 0 are given. According to the investment styles dened above, fund 1 (2) can hold at most (at least)b 1 (b 2 ) of its resources in the illiquid assetC in order to comply with the regulation on position limits at timet = 0 . When we impose position limits, CIO's problem (2.11) is constrained by the following two simultaneous conditions, which already integrate the no internal short-selling constraints above: 29 n 1 C b 1 f 1 s C Xn 1 C (2.14) 27 The convexities in the high-performance region of the ow-performance relation represent an implicit high-incentive contract for the fund managers. Massa and Patgiri (2009) nd evidence that high-incentive contracts induce managers to take more risk (reducing the funds' probability of survival), but at the same time deliver higher risk-adjusted return, with the superior performance remaining persistent. 28 In the continuous-time setting that we use in this chapter, working with position limits and asset illiquidity at the same time throughout the whole investment period may result in the non-existence of a solution to the investment problem in some circumstances. Intuitively, because of the portfolio weight uncertainty created by the limited trading in assetC, there may exist no portfolio policy that satises both the illiquidity and position limits constraints simultaneously. 29 Even under no position limits, we impose these conditions (evaluated at b1 = 1 and b2 = 0) to enforce no internal short-selling when we study the optimal amount of cross-trading for funds with dierent initial values of assets under management in Section 1.4. 70 b 2 f 2 s C n 2 C X f 2 s C n 2 C (2.15) The fund managers' optimal investment strategies are as follows: if @V j =@N j C > 0 then ' j (t) = , and if @V j =@N j C < 0 then ' j (t) =, whenever such amount of trading is admissible, otherwise ' j (t) = 0, for j2f1; 2g. The following section describes our approach to approximating these conditions as well as the solution to the CIO's problem (2.11). 2.3 Numerical Analysis and Discussion In this Section, we solve for the optimization problems of the investor, the fund man- agers, and the fund family's CIO, as described in Section 1.2. 30 The introduction, in our setup, of the many institutional features that we believe are key to understanding a fund family's asset allocation decisions (linear-convex ow-performance relations, position limits, liquidity constraints) made us unable to nd closed-form solutions to our model and led us to rely on numerical techniques instead. In particular, we use the methodology adopted in Longsta (2001) and applied in a portfolio delegation setting by Goncalves-Pinto (2010). It consists of an application of the Least-Squares Monte Carlo algorithm, proposed in Longsta and Schwartz (2001). Succinctly, it involves replacing the conditional expectation function in (2.8) by its orthogonal projection on the space generated by a nite set of basis functions of the values of the state vari- ables that are part of the managers' problem. 31 Next, from that explicit functional approximation, we can solve for the optimal control variable ' j (t) for t2 (0;T ] and for any starting value of N j C (0). Given the optimal trading strategies ' j (t) the CIO 30 The investor's problem can be taken as a particular case of the managers' problem once we shut-o both the cross-trading (i.e. X = 0) and the implicit incentives induced by the convexities of the funds' ow-performance functions (i.e. j (T ) = 1=Yj (T ), for j2f1; 2g). 31 We used up to the third order power polynomials of all the state variables (accounting for their interactions) and the rst three powers of the utility function as basis functions. 71 then choosesX that solves (2.11). Portfolio weights held in the risky illiquid asset C by fund j can then be easily retrieved, for each time t2 (0;T ], from the relation: ! j C (t) =! j C (0) + Z t 0 S j (u) F j (u) ' j (u)du (2.16) where ' j (0) = 0 and ! j C (0) = N j C (0)s C =f j , and the remainder (1! j C (t)) being invested in the liquid asset j that is specic to the investment style of fund j 2 f1; 2g. Note that fund manager j starts o her investment period with the portfolio weights she `inherits' from a previous period: n j C s C =f j in the illiquid risky asset C and 1n j C s C =f j in the liquid risky asset j. These will in general dier from ! j C (0) and 1! j C (0), respectively, by the amount of cross-tradingX. We assume throughout that the inherited weights replicate those of the benchmark for investment style j, i.e. n j C s C =f j = j C (0), implying N j C (0) =B j C (0) only when X = 0. The numerical results presented in this Section are based on 100 time steps|the discretization period is 0.01 years|and 80,000 simulated paths for the state variables. We use the following set of initial values, unless otherwise noted: asset prices, the values of the funds' assets under management and of their corresponding benchmarks are all normalized to unity at t = 0, i.e. s C =s j =f j =y j = 1, for j2f1; 2g. The benchmark for investment style A (fund 1) is a portfolio invested 30% in illiquid asset C and the remaining in the liquid risky asset 1. The benchmark for style B (fund 2) is a portfolio invested 70% in illiquid asset C, with the remainder being invested in liquid asset 2. This implies ! 1 C (0) = 1 C (0) = 0:30 and ! 2 C (0) = 2 C (0) = 0:70 if X = 0. Whenever we impose position limits on the amount of cross-trading at t = 0 we allow fund 1 to have at most 40% and fund 2 to have at least 60% of their respective resources invested in the illiquid asset C, meaning that b 1 = 0:40 and b 2 = 0:60. 72 Regarding the ow-performance relationship of fund j2f1; 2g, we follow Basak, Pavlova, and Shapiro (2007) in setting L j = 0:97, j = 1:6, and j =0:05 in order to match the estimated relation documented in Sirri and Tufano (1998). Lastly, we consider a baseline case and dierent alternative congurations for the preference and asset price parameters. The baseline coecient of relative risk aversion for the investors (i) is i = 5, and those of the managers (m1 and m2) and the CIO are set to match the baseline conguration in Binsbergen, Brandt, and Koijen (2008): i = m1 = m2 = 5, c = 10. 32 The baseline expected returns and return volatilities are 1 = 2 = 0:11, C = 0:18, 1 = 2 = 0:20, and C = 0:37. These values roughly match the average annual return and volatility, over the period 1927-2009, of the high (for the liquid assets 1 and 2) and low (for the illiquid asset C) quintile of the size-sorted (value-weighted) Fama-French portfolios. The baseline correlations of returns between pairs of risky assets are 12 = 1C = 2C = 0, but we examine both positive and negative correlations in the alternative cases. We set the baseline illiquidity parameter to = 0:50. Because can be seen as a fund portfolio's maximum turnover rate over the investment period, we chose this value so that our resulting portfolio turnover roughly matches, on average, the turnover rate of equity mutual funds over the period 1974-2008. 33 The alternative parameter congurations are specied in more detail in each of the subsections below. 32 Bar, Kempf, and Ruenzi (2005) show that team-managed funds exhibit lower (unsystematic) risk than single-managed funds. In our model, allowing the family's CIO to take part of the funds' asset allocation decisions can be thought of as having the aliated funds to be managed by a team, though only partially at time t = 0. Thus, drawing on the results of Bar, Kempf, and Ruenzi (2005), we can consider the CIO in our model as exhibiting a weaker appetite for risk ( c = 10) than individual funds' managers ( 1 = 2 = 5). In addition, we consider the investor to have the same risk aversion as the fund managers so that we can evaluate the costs investors bear as a result exclusively of the incentives implicit in the funds' ow-performance relation and the internal asset reallocation, and not as a result of fund managers' explicit incentives to administer the investor's savings according to their own (and dierent) appetites for risk. 33 See the 2009 Investment Company Fact Book published by the Investment Company Institute (ICI). 73 The rest of this Section proceeds as follows. In Subsection 2.3.1 we determine the family's optimal portfolio allocation and examine its implications in terms of funds cross-subsidization, funds return correlations and risk-shifting incentives for managers and the CIO. Subsection 2.3.2 looks at the utility implications for investors of family- aliated versus standalone funds and analyze how these change as asset illiquidity changes. 2.3.1 Cross-Trading and Family Portfolio Strategies Overview We present in Figures 2.1 and 2.2 the utility derived by the CIO from choosing dierent amounts of cross-trading (X) between fund 2 and fund 1 at t = 0. We show results for dierent correlation coecients between the liquid assets i = 1; 2 (Figure 2.1), and between the liquid assets and the illiquid asset C (Figure 2.1). A rst observation is that, by actively cross-trading between the family funds the CIO can derive a higher utility in all cases with respect to the situation in which no cross-trading occurs (NC dotted line). Moreover, when no position limits constrain funds' portfolios (lines OC 0 , OC and OC + in Figure 2.1, lines OC n , OC p and the middle dashed line in Figure 2.2) the CIO maximizes her utility by having fund 1 buy a signicant number of shares from fund 2, ranging from 0:34 (almost 50% of fund 2's holding of asset C) when 12 =0:5 to a maximum of 0:70 when 12 = +0:5 (100% of fund 2's holding of asset C). However, such amounts of cross-trading would most likely violate the SEC Rule 35(d)-1 of the Investment Act of 1940|regarding the relation between a fund's port- folio composition and its stated investment style|and position limits should come into play. LP and UP indicate the lower and the upper bounds of the position limits to which the CIO is now subject when cross-trading. In this case, she is not able to choose OC anymore but needs to choose the amount X that maximizes her utility 74 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 x 10 −4 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] CIO’s derived utility NC LP UP OC 0 OC − OC + ρ12 =0 ρ12 =−0.5 ρ12 =+0.5 Figure 2.1: CIO's Utility from Cross-Trading for Dierent Correlation Coecients between the Liquid Asset Returns. A negative (positive) value ofX means that fund 2 sells (buys)X (X) shares of the illiquid assetC to (from) fund 1 att = 0. The dotted line NC indicates the no-crossing point at which the two funds start o their investment periods with the portfolio weights that the managers `inherit' from the previous investment period. These are assumed to replicate those of their respective benchmarks at time t = 0: 1 C (0) = 0:30, 2 C (0) = 0:70. OC 0 , OC , OC + indicate the amount of cross-trading that maximizes the CIO's derived utility (with no position limits) for 12 = 0;0:5; +0:5, respectively. LP and UP indicate the lower and the upper bounds of the position limits to which the CIO may be subject to when cross-trading at time t = 0. Fund 2 is restricted to hold at least 60% of its TNA in the illiquid assetC, while fund 1 can hold at most 40% in the illiquid asset. The remaining parameter values are the same as in the baseline case of Table 2.1. in the interval between LP and UP. We have a corner solution at LP in all cases, in which the CIO has fund 2 sell 0:10 shares of the illiquid assetC to fund 1. The CIO's utility falls with respect to OC but the individual managers' utility may improve, eventually increasing the overall utility created inside the family. We look into this trade-o in more detail in Section 2.3.2. Panel A of Table 2.1 (columns OC and PL) presents the initial and expected terminal weights of fund 2 in asset C after the optimal cross-trading for the baseline and alternatives cases in Figures 2.1 and 2.2 (rst ve rows), as well as for additional 75 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −6 −5 −4 −3 −2 −1 0 x 10 −4 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] CIO’s derived utility NC LP UP OC n OC p ρiC =0 ρiC =−0.5 ρiC =+0.5 Figure 2.2: CIO's Utility from Cross- Trading for Dierent Correlation Coecients between the Illiquid and the Liquid Asset Returns. A negative (positive) value of X means that fund 2 sells (buys)X (X) shares of the illiquid asset C to (from) fund 1 at t = 0. The dotted line NC indicates the no-crossing point at which the two funds start o their investment periods with the portfolio weights that the managers `inherit' from the previous investment period. These are assumed to replicate those of their respective benchmarks at time t = 0: 1 C (0) = 0:30, 2 C (0) = 0:70. OCn, OCp indicate the amount of cross-trading that maximizes the CIO's derived utility (with no position limits) for iC =0:5; +0:5, respectively, for i = 1; 2. LP and UP indicate the lower and the upper bounds of the position limits to which the CIO may be subject to when cross-trading at time t = 0. Fund 2 is restricted to hold at least 60% of its TNA in the illiquid asset C, while fund 1 can hold at most 40% in the illiquid asset. The remaining parameter values are the same as in the baseline case of Table 2.1. cases. We see that the results described above are robust to dierent parameters congurations. Under no position limits the sale of share of asset C from fund 2 to fund 1 att = 0 is particularly high when the CIO is relatively risk-tolerant ( C = 5), and when fund 1 is twice as large as fund 2 (F 1 (0) = 2 row). We also see that, on average, fund 2's manager does not re-weight her holdings of asset C closer to that of her benchmark towards the end of the period, except when the CIO sells all her holdings of this asset at the beginning of the period. 76 Table 2.1: Initial and Expected Terminal Portfolio Weights on Illiquid Asset C by Fund 2. Fund 2's initial (! 2 C (0)) and expected terminal (E[! 2 C (T )]) portfolio weights on asset C, after cross-trading on asset C between the two funds by the CIO. Column OC shows the results when the CIO's maximizes her own utility, while columns OM1 and OM2 display the results when the CIO maximizes the utility of fund 1's and fund 2's manager's, respectively. The column PL denotes the allocations that result after imposing position limits on cross-trading. The baseline analysis (rst row) shows the results under the following parameter values: expected returns 1 = 2 = 0:11, and C = 0:18, return volatilities 1 = 2 = 0:20, and C = 0:37, coecient of relative risk aversion for the investors (i), the managers (m1 and m2), and the CIO of i = m1 = m2 = 5, c = 10, illiquidity parameter = 0:50, and correlations of returns among the risky assets 12 = 1C = 2C = 0. The comparative statics analysis (second to eleventh rows) shows the results of changing the baseline parameters one at a time, as indicated. Panel A considers the joint eects of cross-trading and the convexity in the ow-performance relation a la Sirri and Tufano (1998). Panel B shuts-o the ow-performance convexity by making it linear. Panel A: Linear-convex ow-performance relation a la Sirri and Tufano (1998) OC OM1 OM2 PL ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] Baseline 0.31 0.27 0.69 0.50 0.42 0.32 0.60 0.42 iC = +0:50 0.32 0.27 0.84 0.78 0.27 0.26 0.60 0.42 iC =0:50 0.30 0.28 0.62 0.43 0.50 0.36 0.60 0.42 12 = +0:50 0.00 0.12 0.69 0.50 0.42 0.31 0.60 0.42 12 =0:50 0.36 0.29 0.69 0.49 0.40 0.30 0.60 0.41 c = 5 0.00 0.16 0.69 0.50 0.42 0.32 0.60 0.42 1&2 = 0:08 0.31 0.28 0.66 0.48 0.44 0.33 0.60 0.43 F1(0) = 2 0.01 0.16 0.68 0.49 0.42 0.32 0.60 0.42 F2(0) = 2 0.415 0.38 0.695 0.60 0.40 0.34 0.65 0.53 Panel B: Non-convex ow-performance relation (j (T ) = 1=Yj (T )) OC OM1 OM2 PL ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] ! 2 C (0) E[! 2 C (T )] Baseline 0.73 0.55 0.60 0.42 0.76 0.59 0.73 0.55 iC = +0:50 0.74 0.57 0.57 0.40 0.77 0.62 0.74 0.57 iC =0:50 0.22 0.26 0.61 0.42 0.76 0.59 0.60 0.42 12 = +0:50 0.14 0.19 0.60 0.42 0.76 0.60 0.60 0.42 12 =0:50 0.75 0.57 0.60 0.41 0.77 0.60 0.75 0.57 c = 5 0.16 0.21 0.60 0.42 0.76 0.59 0.60 0.42 1&2 = 0:08 0.79 0.65 0.56 0.40 0.79 0.65 0.79 0.65 F1(0) = 2 0.76 0.59 0.52 0.36 0.76 0.59 0.76 0.59 F2(0) = 2 0.575 0.48 0.65 0.55 0.725 0.64 0.65 0.55 Columns OM1 and OM2 in Table 2.1 indicate the optimal initial and expected terminal weights of fund 2's portfolio in asset C that would maximize the utilities of manager 1 and manager 2, respectively. We include these results to illustrate how the misalignment between the CIO's objective and the objectives of its aliated funds' managers determines portfolio allocations for the individual funds that may 77 dier substantially from those they would optimally choose as standalones: manager 1 would buy only 0:10 and manager 2 would sell only 0:28 shares of asset C if they were to start new funds and operate them as standalones. What do these strategies imply for the distribution of family-aliated versus stan- dalone fund returns? Panel A (Panel B) of Figures 2.3 and 2.4 plot fund 1's and fund 2's distribution of before- ow (after- ow) returns. The rst column of graphs (NCT) corresponds to the standalone case while the remaining two correspond to the fam- ily case. 34 Under the standalone organization, fund 2's average before- ow return is higher than fund 1's though at the cost of higher risk, as expected from its more aggressive investment style. However, funds' risk-return proles may change dramat- ically when the same two funds operate under the family umbrella. The CIO seeks to increase each fund's after- ow returns with respect to the standalones counterpart in this case. If she is not constrained by position limits, she achieves this by heavily loading fund 1's initial holding of asset C. In so doing, however, she turns fund 1 the higher return (and risk) fund in the family on a before- ow basis. This strategy on the part of the CIO seems to be more consistent with a risk-shifting motive than with a transaction costs saving reason for cross-trading. The preceding analysis suggests that, although circumventing the costs of illiq- uidity may be an important motive for allowing cross-trading among family-aliated funds it seems to be not the only, or even the most important, one. The next section explores other potential though more subtle purposes behind this type of interfund transactions. 34 The histograms in Figures 2.3 and 2.4 approximate the distribution of log-returns, and so the average in these gures are average log-returns (E[log(1+r)]). The equivalent holding-period average returns E[r] can be approximated by E[r]E[log(1 +r)] + 1 2 (Vol) 2 (e.g., fund 1's average before- ow return at NCT is:113 + (:5)(:1804) 2 = 12:9%). These are the average returns we (implicitely or explicitely) use throughout the dierent numerical exercises in this chapter. 78 −1 −0.5 0 0.5 1 1.5 0 500 1000 1500 2000 2500 3000 3500 PANEL A1: Histogram of R F 1 (T) at NCT (before flows) Aver = 0.11321; Vol = 0.1804 Skew = 0.045229; Kurt = 3.2427 −2 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL A2: Histogram of R F 1 (T) at OC (before flows) Aver = 0.12259; Vol = 0.23972 Skew = 0.19896; Kurt = 3.6507 −1 −0.5 0 0.5 1 1.5 0 500 1000 1500 2000 2500 3000 3500 PANEL A3: Histogram of R F 1 (T) at LPL (before flows) Aver = 0.1167; Vol = 0.18793 Skew = 0.086215; Kurt = 3.4098 −1 −0.5 0 0.5 1 1.5 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL B1: Histogram of R F 1 (T) at NCT (after flows) Aver = 0.15918; Vol = 0.1871 Skew = 0.11509; Kurt = 3.5623 −2 −1 0 1 2 3 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL B2: Histogram of R F 1 (T) at OC (after flows) Aver = 0.21518; Vol = 0.34172 Skew = 0.70395; Kurt = 4.2944 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL B3: Histogram of R F 1 (T) at LPL (after flows) Aver = 0.16825; Vol = 0.207 Skew = 0.28053; Kurt = 4.261 Figure 2.3: Distribution of Investment Period Returns for Fund 1. Results are shown for the no cross-trading case (NCT) in which funds operate as standalones, for the optimal cross-trading without position limits case (OC), and for the optimal cross-trading with position limits (LPL). The parameter values are the baseline parameters of Table 2.1. Cross-Fund Subsidization Fund families may also use the ability to cross-trade between family members for strategic purposes. Nanda, Wang, and Zheng (2004) observe that fund families that possess at least one fund with an exceptional performance record benet from ow spillovers among their aliated funds. Guedj and Papastaikoudi (2005) and Gaspar, Massa, and Matos (2006) nd that this eect, along with the convexity of funds' 79 −2 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL A1: Histogram of R F 2 (T) at NCT (before flows) Aver = 0.12292; Vol = 0.2436 Skew = 0.16886; Kurt = 3.4662 −1 −0.5 0 0.5 1 1.5 0 500 1000 1500 2000 2500 3000 3500 PANEL A2: Histogram of R F 2 (T) at OC (before flows) Aver = 0.11361; Vol = 0.18107 Skew = 0.0018545; Kurt = 3.1605 −2 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL A3: Histogram of R F 2 (T) at LPL (before flows) Aver = 0.12177; Vol = 0.2175 Skew = 0.14591; Kurt = 3.4883 −2 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 PANEL B1: Histogram of R F 2 (T) at NCT (after flows) Aver = 0.17755; Vol = 0.23387 Skew = 0.096893; Kurt = 3.8528 −1 −0.5 0 0.5 1 1.5 0 500 1000 1500 2000 2500 3000 PANEL B2: Histogram of R F 2 (T) at OC (after flows) Aver = 0.22095; Vol = 0.2362 Skew = 0.33803; Kurt = 2.9765 −2 −1 0 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 PANEL B3: Histogram of R F 2 (T) at LPL (after flows) Aver = 0.1904; Vol = 0.20537 Skew = 0.060191; Kurt = 3.354 Figure 2.4: Distribution of Investment Period Returns for Fund 2. Results are shown for the no cross-trading case (NCT) in which funds operate as standalones, for the optimal cross-trading without position limits case (OC), and for the optimal cross-trading with position limits (LPL). The parameter values are the baseline parameters of Table 2.1. ow-to-performance sensitivities, encourage some fund families to `play favorites' among aliated funds in order to maximize the family's total amount of assets un- der management. In particular, the latter authors conjecture that fund families may cross-subsidize some member funds over others within the family through interfund transactions at below or above market prices. Whenever this is the case, though, 80 cross-trading at non-market prices goes against SEC Rule 17a-7 and should thus be illegal. 35 We suggest that, in the presence of illiquidity, cross-subsidization may take place in a more subtle way|and without going so clearly against SEC Rule 17a-7: the CIO may enhance the performance of one of the funds above what would be achievable if it were an independent fund through cross-trading the illiquid asset with another fund in the family. By doing so, the CIO enables the favored fund to avoid the costs of illiquidity. But it also makes the other fund in the family adopt a suboptimal investment policy, and thus pay a cost of aliation. Therefore, cross-subsidization within fund families under illiquidity can take place even when interfund transactions are executed at fair market prices. Moreover, both the benets and costs of the family's ability to cross-trade are inextricably related under our interpretation of the motives for cross-trading. The presence of cross-subsidization in our setting can be detected by comparing Panels A and B in Table 2.1. Panel B shows the same quantities as Panel A (described in Subsection 2.3.1) but in the absence of convexities in the ow-performance relation. When no risk-shifting incentives exist, the CIO actively cross-trades only if it allows her to increase the family's wealth by saving on transaction costs|the shadow costs of illiquidity, in our setting. The extent of cross-trading changes dramatically in this case: except for the cases 12 = +0:50 and C = 5, the CIO has fund 2 buy shares of the illiquid asset C from fund 1 at the beginning (even for 12 = +0:50 and C = 5, the CIO cross-trades much less in the absence of risk-shifting incentives), with fund 2's manager eventually re-balancing her portfolio towards a more liquid position by the end of the period. Therefore, trading in assetC by the CIO att = 0 re ects a lot of risk-shifting and much less of saving on transaction costs. Moreover, if performance 35 Note that the non-market prices are key for this hypothesis to remain valid in perfectly liquid markets, since at the fair market prices each fund would be just indierent between cross-trading with another fund in the same family and trading in the public markets. 81 is measured by funds' risk-adjusted returns (the ratio of average return to volatility), as many empirical studies do, this strategy would look like cross-subsidization of fund 2 by fund 1: while fund 1's risk-adjusted return falls from 0:72 in the standalone case to 0:64 in the fund family case when no position limits are imposed, fund 2's risk-adjusted return increases from 0:63 to 0:72. 36 A closer look at the higher moments of the return distributions in Figures 2.3 and 2.4 may help to understand the CIO's strategy and the consequent performance of the fund family. Because the CIO cares about after- ow returns, and because these are a convex function of before- ow returns in the high performance region, the distinctive feature of CIO's strategy is that she favors a highly volatile, right-skewed and leptokurtic distribution of after- ow returns for at least one of the funds in the family. In so doing, she induces an also highly volatile and right-skewed distribution of before- ow returns for the relatively more conservative fund 1, contrary to what is mandated by its investment style. This strategy is optimal from CIO's perspective because the more conservative fund is evaluated with respect to a correspondingly conservative benchmark featuring a low expected volatility but also a low expected return. When penalties for poor fund returns are low, as induced by a linear-convex ow-performance relation, increasing the risk of the ex-ante more conservative fund in the family is a more eective means to outperform the benchmark by a large amount and thus get large investors' cash infusions at the end of the period. It is at this point that imposing position limits becomes key for investors: since the risk-adjusted return remains almost invariant for the investors in fund 1 (around 0:72 at both NCT and LPL) and increases for investors in fund 2 (from 0:63 at NCT to 0:68 at LPL), by introducing position limits investors conserve the aliated funds' return-risk prole corresponding to their respective styles and may even (weakly) enhance their 36 These values are displayed in the baseline case of Table 2.2, under the RRR column. 82 performance with respect to that of comparable standalones. 37 Even in this case, however, we see that the family organization increases the skewness and kurtosis of fund 1's returns without equivalently reducing those of fund 2's returns, relative to the standalone case. The following subsection analyzes an additional driving force underlying the fam- ily's portfolio strategy, namely the search for diversication eects. Family Strategies and Funds' Return Correlations The analysis in the previous section focuses on the eects of cross-trading on family funds' performance both at the individual level and relative to standalone funds. In this section we change the focus to the within-family eects of interfund transactions by looking at the induced correlation, between family members, of funds' returns. This issue is of particular interest in light of the evidence presented by Elton, Gruber, and Green (2007) showing that family-aliated funds impose an under- diversication cost on family investors by inducing a higher correlation of their port- folio returns compared to their standalone counterparts. Figures 2.5 and 2.6 plot the correlation between fund 1's and 2's returns before and after ows, respectively. Figure 2.5 shows the induced return correlations for dierent degrees of correlation between the liquid assets 1 and 2, 12 . This correlation is of no relevance to each fund manager because each of them either holds asset 1 or asset 2, but not both. However, this correlation is key to the CIO's strategy: indeed, Panel B shows that if the CIO is not constrained by position limits, the amount of cross-trading she optimally chooses induces a negative correlation between the funds' after- ow returns. When assets are themselves negatively correlated ( 12 =0:5), this negative correlation is attained by a relatively small amount of cross-trading (OC ), but when it is positive ( 12 = +0:5) 37 We expand on this last point in Section 2.3.2. See Almazan, Brown, Carlson, and Chapman (2004) for empirical evidence documenting investors' utility-improving eects of investment policy constraints. 83 the CIO cross-trades as much as she can in order to achieve the maximum diversi- cation eects between funds' returns (OC + ). CIO's pursuit of diversication benets is also clear in Panel B of Figure 2.6, which displays funds' after- ow return correla- tions for dierent correlation coecients iC between the liquid assetsi = 1; 2 and the illiquid asset C. In all cases, family-aliated funds exhibit a substantially lower cor- relation of after- ow returns than that of otherwise identical standalone funds (NC) provided the CIO freely chooses the extent of cross-trading. Thus, an (unconstrained) family organization allows the CIO to attain greater diversication benets on the family's overall portfolio, thus decreasing the family's risk. −0.6 −0.4 −0.2 0 0.2 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] Corr[R F 1 (T),R 2 F (T)] PANEL A: Correlation between fund 1 and fund 2 investment period returns BEFORE fund flows at t = T NC LP UP OC 0 OC − OC + ρ12 =0 ρ12 =−0.5 ρ12 =+0.5 −0.6 −0.4 −0.2 0 0.2 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] Corr[R F 1 (T),R 2 F (T)] PANEL B: Correlation between fund 1 and fund 2 investment period returns, AFTER fund flows at t = T NC LP UP OC 0 OC − OC + ρ12 =0 ρ12 =−0.5 ρ12 =+0.5 Figure 2.5: Correlation between Funds 1 and 2 In- vestment Period Returns, for Dierent Correlation Coecients between the Liquid Asset Returns. The meanings of the vertical dotted lines are as in Figure 2.1. The parameter values are the baseline parameters of Table 2.1. Panel A in Figures 2.5 and 2.6 indicates that this strategy translates into a positive correlation (0:50) between before- ow returns in the OC cases, but this correlation is still lower than that of standalones (NC) in all cases. Qualitatively, this outcome 84 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] Corr[R F 1 (T),R 2 F (T)] PANEL A: Correlation between fund 1 and fund 2 investment period returns BEFORE fund flows at t = T NC LP UP OC n OC p ρiC =0 ρiC =−0.5 ρiC =+0.5 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Amount X of cross−trading of illiquid asset C between fund 2 and fund 1 at time t = 0 [fund 2 sells to fund 1 when negative and purchases from fund 1 when positive] Corr[R F 1 (T),R 2 F (T)] PANEL B: Correlation between fund 1 and fund 2 investment period returns AFTER fund flows at t = T NC LP UP OC n OC p ρiC =0 ρiC =−0.5 ρiC =+0.5 Figure 2.6: Correlation between Funds 1 and 2 Investment Period Returns, for Dierent Correlation Coecients between the Illiquid and the Liquid Asset Returns. The meanings of the vertical dotted lines are as in Figure 2.1. The parameter values are the baseline parameters of Table 2.1. of our model seems to go against the evidence in Elton, Gruber, and Green (2007), in which results are in terms of before- ow returns. However, we also see in these graphs that when the CIO is subject to position limits (LP)|as CIOs and managers usually are in the real world|the aliated funds' returns show a higher before- ow correlation than that for standalones in all cases, the dierence being as high as 0:13 in some cases. Our model with position limits is then consistent with the qualitative results of Elton, Gruber, and Green (2007). The following subsection examines in more detail the dynamic risk-taking behavior of the individual funds' managers and compare those in family-aliated funds against their standalone peers. 85 Risk-Shifting Incentives The results presented so far suggest that the ability to cross-trade is likely to elicit higher risk-taking by the CIO. However, we have said little about whether the indi- vidual funds' managers attempt to counterbalance the higher risk-taking favored by the CIO (by attempting to track the benchmark more closely aftert = 0) or if, on the contrary, they reinforce it. We examine this question in Figures 2.7 and 2.8, which plot fund 1 and fund 2 managers' risk exposure, respectively, as a function of their year-to-date performance relative to the benchmark. These graphs show the amount of unsystematic risk the funds end up taking at t = T as a function of their end of the third quarter (year-to-date) relative performance. A fund is not taking any unsystematic risk if the weights it invests in each of the assets equal those of their respective benchmarks. Thus, we compute the dierence between the funds' position in the asset that corresponds the least to the fund's style (the illiquid asset C for fund 1 and the liquid asset 2 for fund 2) and the benchmarks' weights on those same assets, 38 and plot those dierences against the funds' relative performance as of the end of the third quarter (3=4T ). A rst observation is that family-aliated funds (Panel B in both gures, cor- responding to the case of cross-trading subject to position limits) take more unsys- tematic risk than comparable standalones (Panel A in both gures), even though the risk-taking patterns dier substantially by investment styles. This leads us to a sec- ond observation: while fund 2 overweights the liquid asset in a neighborhood of the 0 interim excess return and underweights it in the very poor and very good relative performance regions, fund 1 does exactly the opposite, i.e. underweights the illiquid asset around the 0 past excess return region and overweights it for very good or poor interim performance. This implies that, after three quarters of the investment period 38 Note that these weights are time-varying given that the benchmarks are passive buy-and-hold portfolios 86 −0.1 −0.05 0 0.05 0.1 −0.2 −0.1 0 0.1 0.2 0.3 PANEL A: Unsystematic risk−taking by fund 1 on illiquid asset C as a function of relative performance as of 3T/4 for X = 0 and E[ω C 1 (T) − β C 1 (T)] = −0.05 R F 1 (3T/4) − R Y 1 (3T/4) ω C 1 (T) − β C 1 (T) −0.1 −0.05 0 0.05 0.1 0.15 −0.2 −0.1 0 0.1 0.2 0.3 0.4 PANEL B: Unsystematic risk−taking by fund 1 on illiquid asset C as a function of relative performance as of 3T/4 for X = −0.10 and E[ω C 1 (T) − β C 1 (T)] = −0.01 R F 1 (3T/4) − R Y 1 (3T/4) ω C 1 (T) − β C 1 (T) Figure 2.7: Relative Performance and Risk-Taking for Fund 1. Amount of unsystematic risk fund 1 takes (post cross-trading) as a function of its end of the 3rd quarter (year-to-date) relative performance. Unsystematic risk-taking by fund 1 is the dierence between its position and benchmark 1's position in asset C. Panel A shows results for the case of no cross-trading (NCT) at t = 0 (standalone fund). Panel B shows results for the case in which cross-trading is subject to position limits (LPL). The meanings of the vertical dotted lines are as in Figure 2.1. The parameter values are as in the baseline case of Table 2.1. (assumed to be one year), incentives to risk-shift for fund 2 (fund 1) are lower (higher) the less likely beating the benchmark becomes or the most exceptionally good relative performance is. 39 39 Although not displayed in the text, results are even more pronounced when cross-trading is not subject to position limits. 87 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 PANEL A: Unsystematic risk−taking by fund 2 on liquid asset 2 as a function of relative performance as of 3T/4 for X = 0 and E[ω 2 (T) − β 2 (T)] = 0.19 R F 2 (3T/4) − R Y 2 (3T/4) ω 2 (T) − β 2 (T) −0.2 −0.1 0 0.1 0.2 0.3 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 PANEL B: Unsystematic risk−taking by fund 2 on liquid asset 2 as a function of relative performance as of 3T/4 for X = −0.10 and E[ω 2 (T) − β 2 (T)] = 0.28 R F 2 (3T/4) − R Y 2 (3T/4) ω 2 (T) − β 2 (T) Figure 2.8: Relative Performance and Risk-Taking for Fund 2. Amount of unsystematic risk fund 2 takes (post cross-trading) as a function of its end of the 3rd quarter (year-to-date) relative performance. Unsystematic risk-taking by fund 2 is the dierence between its position and benchmark 2's position in asset 2. Panel A shows results for the case of no cross-trading (NCT) at t = 0 (standalone fund). Panel B shows results for the case in which cross-trading is subject to position limits (LPL). The meanings of the vertical dotted lines are as in Figure 2.1. The parameter values are as in the baseline case of Table 2.1. An important implication of these results is that the family-aliated fund whose expected return falls compared to the standalone situation (fund 2) is the one that takes the higher unsystematic risk in the mediocre performance region. In doing so, however, this fund chooses a more conservative policy with respect to the overall variance of its portfolio than in the absence of cross-trading. This implies that when funds are aliated to families, it might be dicult to detect eective risk-shifting 88 empirically by measuring portfolio volatility. The use of the tracking error variance as an empirical measure of risk-shifting seems more appropriate instead, as advocated by Basak, Pavlova, and Shapiro (2007). Panel A in Figure 2.8 makes it clear that standalone funds may also take excessive unsystematic risk, although in a much less aggressive way than their family-aliated counterparts. Table 2.2 complements our analysis of managers' risk-shifting incentives and their impact on the return-to-risk ratios of funds' portfolios for the baseline case and dif- ferent alternative parameter congurations. As in Table 2.1, Panel A presents results in the presence of risk-shifting incentives induced by convex ow-performance rela- tions, while Panel B displays results in the absence of such incentives. We denote by RS the average of the deviations of funds' portfolio weights on the assets that cor- respond the least to their investment styles from the corresponding weights in their benchmarks, as oft =T . Negative values correspond to average under-weights in the funds' portfolios compared to the weights in their respective benchmarks. Similarly, RRR denotes the reward-to-variability ratio and measures how much funds' returns compensate investors for the risk they bear. 89 Table 2.2: Risk-Shifting and Return-to-Risk Ratios. Risk-shifting (RS) measures how much unsystematic risk each fund takes as measured by the dierence (in percentage points) between the fund's position in the asset that corresponds the least to its style (the illiquid asset C for fund 1, and the liquid asset 2 for fund 2) and that of its benchmark at t = T . Return-to-Risk ratio (RRR) measures a fund's expected return per unit of average fund's return volatility (in dec- imal points), over the investment period. Results are shown for the cases in which cross-trading is subject to position limits (CT Poslim), position limits are not imposed (CT NoPoslim), and there is no cross-trading (No CT, funds operate as standalones). The baseline analy- sis (rst row) uses the same parameters as the baseline case in Table 2.1. The comparative statics analysis (second to eleventh rows) shows the results of changing the baseline parameters one at a time, as indicated. Panel A considers the joint eects of cross-trading and the con- vexity in the ow-performance relation a la Sirri and Tufano (1998). Panel B shuts-o the ow-performance convexity by making it linear. Panel A: Linear-convex ow-performance relation a la Sirri and Tufano (1998) CT Poslim CT NoPoslim No CT Fund 1 Fund 2 Fund 1 Fund 2 Fund 1 Fund 2 RS RRR RS RRR RS RRR RS RRR RS RRR RS RRR Baseline 0:76 0:72 27:97 0:68 17:41 0:64 42:29 0:72 4:91 0:72 18:56 0:63 iC = +0:50 0:70 0:60 28:04 0:58 16:56 0:56 42:55 0:60 4:35 0:60 18:71 0:56 iC =0:50 0:58 0:97 27:67 0:86 19:56 0:75 41:23 0:95 3:22 0:96 18:59 0:75 12 = +0:50 0:76 0:72 27:94 0:68 67:68 0:49 57:50 0:61 4:91 0:72 18:75 0:64 12 =0:50 0:76 0:72 28:61 0:68 13:14 0:66 41:13 0:72 4:91 0:72 19:78 0:64 c = 5 0:76 0:72 27:97 0:68 67:68 0:49 54:14 0:63 4:91 0:72 18:56 0:63 1&2 = 0:08 0:83 0:62 27:83 0:61 17:34 0:59 42:27 0:60 5:03 0:60 18:43 0:59 F1(0) = 2 0:39 0:73 27:97 0:68 22:59 0:64 53:81 0:63 3:43 0:73 18:56 0:63 F2(0) = 2 0:76 0:72 15:08 0:64 27:17 0:60 31:31 0:71 4:91 0:72 9:43 0:62 90 Table 2.2, Continued Panel B: Non-convex ow-performance relation (j (T ) = 1=Yj (T ) for j2f1; 2g) CT Poslim CT NoPoslim No CT Fund 1 Fund 2 Fund 1 Fund 2 Fund 1 Fund 2 RS RRR RS RRR RS RRR RS RRR RS RRR RS RRR Baseline 5:91 0:72 15:01 0:62 5:91 0:72 15:01 0:62 4:61 0:72 18:77 0:63 iC = +0:50 5:71 0:60 12:75 0:55 5:71 0:60 12:75 0:55 4:18 0:60 17:88 0:56 iC =0:50 0:70 0:97 27:97 0:86 31:55 0:66 43:91 0:90 3:23 0:96 19:17 0:75 12 = +0:50 0:26 0:72 27:34 0:68 43:78 0:55 50:47 0:68 4:61 0:72 17:97 0:63 12 =0:50 6:79 0:71 13:02 0:61 6:79 0:71 13:02 0:61 4:61 0:72 19:51 0:63 c = 5 0:26 0:72 28:19 0:68 40:14 0:56 48:57 0:69 4:61 0:72 18:77 0:63 1&2 = 0:08 8:69 0:57 5:71 0:55 8:69 0:57 5:71 0:55 4:67 0:60 18:72 0:58 F1(0) = 2 5:22 0:72 10:59 0:60 5:22 0:72 10:59 0:60 3:34 0:73 18:77 0:63 F2(0) = 2 0:26 0:72 14:67 0:64 7:89 0:69 22:06 0:67 4:61 0:72 9:12 0:62 91 Several results regarding risk-shifting are robust across the dierent scenarios. First, RS is in general smaller (indicating less over-weighting or more under-weighting) in the absence of incentives to risk-shift. Second, the convexities in the ow- performance relation aect the behavior of family-aliated funds (CT Poslim and CT NoPoslim) much more than that of standalones (NoCT): the values of RS and RRR for CT Poslim and CT NoPoslim dier in general between Panels A and B, while these remain almost unaltered for NoCT. Third, both under convex and under linear incentives the extent of unsystematic risk-taking is higher for unconstrained family-aliated funds (CT NoPoslim) than for family funds subject to position limits (CT Poslim). Finally, fund 2's RRR generally improve upon imposing position limits on cross-trading without hurting (or even improving in some cases) fund 1's RRR. This nal observation implies that an investor who was to decide between investing in an equally-weighted portfolio of aliated funds or in an equally-weighted portfolio of standalones would enjoy a better return-risk prole by investing in the family as long as position limits were imposed. We next look in more detail at the implications of cross-trading and of portfolio delegation to aliated versus standalone funds on investors' utility. 2.3.2 Utility Implications One of the advantages of taking a portfolio choice approach to studying the invest- ment decision problem of fund families is that it allows us to approximate the utility implications of portfolio delegation under this arrangement. An investor delegating her portfolio to a standalone fund instead of actively managing her savings herself bears an agency cost stemming from the misalignment between her objectives and the manager's. When instead the same investor decides to delegate her portfolio to a family-aliated fund, not only she pays this agency cost but also an additional cost due to the interference of centralized (family-level) decisions on each fund's invest- ment policy. That is, because family concerns are not necessarily perfectly aligned 92 with those of its individual member funds, an extra layer of con ict of interests is added between the investor and institutional management. 40 However, the possibil- ity of cross-trading among aliated funds so as to circumvent the costs of illiquidity may in principle oset part or all of this additional agency costs to the investor. A quantitative assessment of the resulting net benets (costs) of delegation under fund family aliation is the purpose of this Section. We express these net utility benets in monetary terms by computing the certainty equivalent rate of return. For an investment policy , this is the risk-free rate of returnCE() that makes an agent indierent between following the policy over the investment horizon T and earning this risk-free rate on the same initial investment over the same period. For an agent with CRRA coecient and initial wealth z, CE() solves: [CE()z] 1 1 =E (Z (T )) 1 1 ; (2.17) where the superscript `' denotes that the nal wealth Z(T ) is attained under the optimal investment policy. 41 We can also compute the net benets NB( 1 ; 2 ) of switching from investment policies 1 to 2 as NB( 1 ; 2 ) = CE( 2 )=CE( 1 ) 1. Note that a positive value of NB( 1 ; 2 ) represents a utility gain of moving from policy 1 to policy 2 , while a negative value has the opposite meaning. Using these measures, Table 2.3 reports two main net gains/costs of delegation: the total benet of delegation (TBD) and the benets derived from family aliation (BA). 42 40 An example of this extra layer of agency con ict is provided by Bhattacharya, Lee, and Pool (2010), who nd evidence that some funds within families may coordinate actions to increase the value of the family as a whole, by providing liquidity to their siblings in distress. 41 For investor in fund j, Z(t) =Fj (t)=Yj (t), j2f1; 2g. 42 In all cases, investor's derived utility under delegation is obtained by plugging her managed fund's terminal wealth (as well as the corresponding benchmark's) in her utility function (2.7). 93 Table 2.3: Utility Implications of Portfolio Delega- tion to Family-Aliated and Standalone Funds and Net Eects of Cross-Trading to the Family. The net eects of cross-trading (NBF) to the family as a whole are given by the ratio of the sum of the certainty equivalents for the CIO and the two fund managers under optimal cross- trading (with and without position limits) and the sum of their same certainty equivalents but under no cross-trading. The total benets of delegation (TBD) measures investors' net utility gain from delegating to funds that belong to families instead of managing the funds' portfolios themselves. TBD is computed by dividing the sum of the certainty equivalents that the investors obtain from delegating their savings to family-aliated funds, by the sum of the certainty equiv- alents the investors would get by managing the funds' portfolios themselves (starting o with the benchmark portfolios). The benets of aliation (BA) measures investors' net utility gain from delegating their portfolios to aliated funds compared to delegating to standalone funds. Results are shown for the cases in which cross-trading is subject to position limits (CT Poslim) and not subject to position limits (CT NoPoslim). No cross-trading (No CT) denotes investors' net utility gain from delegating to standalone funds instead of investing directly. All numbers are expressed in net returns (i.e. subtracting one from the computed ratios) and in percent- ages (%). The baseline case (rst row) uses the same parameters values as the baseline case in Table 2.1. The comparative statics analysis (second to eleventh rows) shows the results of changing the baseline parameters one at a time, as indicated (i = 0 denotes the situation in which the investor holds the benchmark portfolio instead of investing directly). Panel A con- siders the joint eects of cross-trading and the convexity in the ow-performance relation a la Sirri and Tufano (1998). Panel B shuts-o the ow-performance convexity by making it linear. Panel A: Linear-convex ow-performance relation a la Sirri and Tufano (1998) CT Poslim CT NoPoslim NBF TBD BA NBF TBD BA NoCT Baseline 2:01 0:43 0:41 2:33 4:12 4:10 0:02 iC = +0:50 2:13 0:32 0:28 3:53 2:43 2:40 0:04 iC =0:50 1:93 0:64 0:64 1:66 5:80 5:80 0:00 12 = +0:50 1:67 0:46 0:42 2:80 16:01 15:98 0:04 12 =0:50 2:02 0:44 0:42 2:45 3:30 3:29 0:02 c = 5 1:56 0:43 0:41 1:78 15:48 15:47 0:02 1&2 = 0:08 1:91 0:47 0:45 2:20 4:09 4:06 0:03 F1(0) = 2 1:42 0:46 0:45 1:10 7:93 7:91 0:02 F2(0) = 2 1:62 0:02 0:00 3:00 3:55 3:53 0:02 Panel B: Non-convex ow-performance relation (j (T ) = 1=Yj (T ) for j2f1; 2g) CT Poslim CT NoPoslim NBF TBD BA NBF TBD BA NoCT Baseline 0:00 0:00 0:00 0:24 0:42 0:42 0:00 iC = +0:50 0:01 0:00 0:00 0:20 0:27 0:27 0:00 iC =0:50 0:39 0:65 0:65 4:22 8:78 8:78 0:00 12 = +0:50 0:21 0:41 0:41 4:46 9:61 9:61 0:00 12 =0:50 0:00 0:02 0:02 0:00 0:02 0:02 0:00 c = 5 0:16 0:42 0:42 3:95 8:48 8:48 0:00 1&2 = 0:08 0:07 0:18 0:18 0:07 0:18 0:18 0:00 F1(0) = 2 0:04 0:02 0:02 0:04 0:02 0:02 0:00 F2(0) = 2 0:08 0:00 0:00 0:22 0:70 0:70 0:00 94 TBD denotes investors' net benets of switching from direct investment (managing the funds' portfolios according to their own appetites for risk, no cross-trading, and no payo convexities) to delegating their savings to family-aliated funds. The eect of funds' aliation on investors' utility (BA) represents investors' net benets of switching from investing in a standalone fund to delegating their portfolios to a family- aliated fund in the same investment style. 43 Table 2.3 also provides the net gains of cross-trading (NBF) to the family as a whole, which we compute as the net benets accruing to the CIO and the two fund managers (as one unique entity) of allowing cross-trading within the family. Some observations are robust to the dierent values of the model parameters. First, a fund's aliation to a family is very costly for its investors when there are no position limits (CT NoPoslim), as shown by the signicant negative values under the BA columns of Panel A. As a result, the total costs of delegation (negative of TBD) can be as high as 4:1% in the baseline case (and even 16:0% when 12 = +0:50). However, investors on average can be better o by imposing position limits on their funds' holdings (CT Poslim), rendering these costs less than 50 basis points per year in almost all cases. Second, the fund family as a whole generally benets from its ability to cross-trade between member funds, with net gains being higher when the CIO is not restricted by position limits. Benets to the family amount to 200 (233) basis points per year when position limits are imposed (no position limits are imposed) in the baseline case. Intuitively, we would expect cross-trading to be particularly protable when iC is high, because in this case each fund's return can deviate from its benchmark's (and thus exploit the convexity in the ow-performance relation) only by trading large amounts of the illiquid asset. This is indeed the case in our model: when the liquid 43 Note that these computations are based on the sum of the certainty equivalents of style A investors (those investing in fund 1) and style B investors (those investing in fund 2). Because we assume both funds start the investment period with equal wealth, the resulting net benets are a simple average of the net benets to each type of investor. 95 asset is highly correlated with the illiquid one ( iC = +0:5) the certainty equivalent returns for the family as whole can be as high as 353 basis points per year under no position limits. Third, convexities in the ow-performance relation are an important component of the cost of delegation under a fund family arrangement: the comparison of Panel A with Panel B shows that investors pay substantially lower costs of aliation (BA) absent this type of convexities. The same comparison also evidences that the benets of the family organization are much higher in the presence of convexities than in the absence of them. According to these results, the possibility of cross-subsidizing some funds in the family is a more powerful reason to gather funds under the family umbrella than is purely saving on transaction costs or avoiding the costs of illiquidity. It then seems clear that, under normal liquidity conditions|we have kept the liquidity parameter equal to 0:50 throughout the analysis: 44 (a) investors bear additional agency costs by investing in family-aliated funds over and above the costs of delegating to standalone funds; and (b) fund families benet by the possibility of cross-trading between aliated funds instead of keeping them as standalone funds. It is thus natural to wonder how these costs and benets change, or even whether costs may eventually turn into benets for investors and vice versa for the fund family, under dierent liquidity conditions. We look into this question in Figure 2.9. This plots the BA measure of investors' utility (Panel A) and the NBF measure of family's benets (Panel B) for the case when the CIO is subject to position limits (CT Poslim), 45 for dierent liquidity conditions . Results for the investors look dierent from those for the fund family. No matter how liquid assets are (high value of ), investors in our model always bear costs of delegating to family-aliated instead of to standalone funds. Moreover, the higher the 44 This value of approximates market conditions during the period 1974-2008, see Section 1.4. 45 As in previous sections, we favor this case as the most likely in the real world. Results for the no position limits case are available upon request. 96 liquidity the higher the additional costs of delegation: in very illiquid conditions, the benets of saving on transaction costs almost oset the costs of the exacerbated risk- shifting by aliated funds, whereas in more liquid conditions the benets to investors in terms of saving on transaction costs get negligible. Additionally, the lower the correlation between the liquid and the illiquid asset the higher the investors' costs of aliation, irrespective of liquidity conditions. This happens because each fund manager exacerbate more the risk-shifting induced by the CIO the more diversied her fund's portfolio is (the lower iC is). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 liquidity parameter (α) PANEL A: Investors’ utility effects (in percent points) from delegating to two affiliated funds instead of two standalones, with position limits ρiC =0 ρiC =+0.5 ρiC =−0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 liquidity parameter (α) PANEL B: Net utility effects (in percent points) derived by the family (the CIO and the two fund managers) from cross−trading, with position limits ρiC =0 ρiC =+0.5 ρiC =−0.5 Figure 2.9: Utility Implications of Portfolio Delegation to Family-Aliated Funds and Net Eects of Cross-Trading to the Family. Investors' utility eects (PANEL A) are the benets of aliation (BA) and Net utility eects derived by the family (PANEL B) are the net benets to the family (NBF) measures computed in Table 2.3. Results are shown for the baseline parameters of Table 2.1 and for negative (iC =0:5) and positive (iC = +0:5) correlation between the liquid and the illiquid asset returns. 97 Panel B suggests that benets never turn into costs for the fund family in our model under position limits. 46 But in contrast to the case of the investor, the family attains the maximum benets at intermediate values of liquidity|`interior' optimal liquidity obtains: when the illiquid and the liquid assets are negatively (positively) correlated, the maximum benets are attained in relatively illiquid (liquid) condi- tions, in which = 0:3 ( = 0:7). That is, at each fund level, families will favor highly diversied portfolios in dry markets (when 0 0:35 NBF is highest for iC = 0:5), but highly correlated portfolios when market are very liquid (when > 0:35 NBF is highest for iC = +0:5). This result is not surprising in light of our previous analysis: when the assets in her portfolio (either asset 1 or asset 2 and asset C) are negatively correlated, each manager can exploit the convexities in the ow-performance relation by trading a small amount of shares in external markets, and so illiquidity (low ) is less of a problem given the extent of cross-trading at t = 0. When the same assets are positively correlated, substantially more portfolio re-balancing (by trading in external markets) is needed for the same purpose, and the maximum family benets are achieved in relatively liquid markets. 2.4 Conclusion We examine institutional and market features that can make the aliation of mutual funds with families matter for the portfolio delegation decisions of investors. When mutual funds invest in at least one asset whose market is not perfectly liquid, mutual fund companies with a larger number of funds under their umbrella can avoid the costs of illiquidity by cross-trading between member funds, potentially increasing investors' utility. But the family organization will also enable these companies to exploit the convexities in the implicit ow-performance relation more fully by cross-subsidizing 46 Results can be quite dierent when the CIO is allowed to cross-trade as much as she desires st t = 0, though, since in this case she will impose too high costs of deviating from the benchmark on the individual managers, rendering the overall family benets into costs in some cases. 98 some funds in the family at the expense of others. By lowering the correlation be- tween the individual funds' after- ow returns, the family as a whole will additionally obtain diversication benets even though the resulting before- ow return correlation will usually be higher than that between comparable standalones. In implementing these strategies, family-aliated funds generally shift risk more than their standalone counterparts, imposing additional agency costs on investors. These costs of family- aliation increase as liquidity conditions improve and savings on transaction costs become relatively smaller than the costs of delegation. We illustrate how the inclusion of position limits on funds' asset holdings can curb these risk-shifting incentives and benet investors. Ultimately, our purpose in this chapter is to show that the costs and benets of portfolio delegation to family aliated funds can dier substantially from those in the independent fund paradigm in which most of the existing portfolio delegation literature is framed. We are aware that this study is limited in several important ways. For ease of exposition, we conne attention to constant investment opportunities in nancial markets, constant illiquidity, and totally passive investors. It would be useful in future research to explore a more complex investment opportunity environment including, for instance, predictability in assets returns, and its interaction with asset illiquidity and cross-trading. It would also be of interest to let the investor endogenously decide how much of her savings to hold in a money market account and how much to invest in the managed portfolios of family-aliated or standalone funds. This would allow the investor to trade around incentive misalignments and improve her utility results. Other interesting extensions could focus on deriving optimal performance bench- marks that better align incentives between investors and money managers in the context of illiquid markets. Future work could also study the sensitivity of the results presented here to alternative measures of asset liquidity, in particular transaction costs (as studied in Liu and Loewenstein (2002), Liu (2004), and Jang, Koo, Liu, and Loewenstein (2007), among others). 99 Lastly, we note that since the results we present are in the context of a partial equilibrium, they should be taken only as suggestive. An interesting|and presumably very complex|extension could pursue the asset pricing implications of fund families' strategies in illiquid markets. 47 47 On empirical grounds, Goncalves-Pinto and Schmidt (2010) investigate the asset pricing impli- cations of mutual fund families co-insurance strategies. In particular, they hypothesize that fund families have an incentive to coordinate trades across their aliated funds in order to avoid costly external nancing. Consistent with this hypothesis, they nd weak or no price pressure on traded securities mostly held in common by distressed funds aliated with large families. 100 Chapter 3 A Rationale for Benchmarking in (Informed) Money Management 3.1 Introduction The ubiquitous use of benchmarking for the compensation of money managers in the nancial industry remains largely a puzzle in light of the existing literature on delegated portfolio management. Indeed, the use by institutional investors of market indices such as those in the MSCI or S&P families to evaluate the performance of asset managers is a common practice in the pension fund industry (see e.g. BIS (2003)). Moreover, nearly 10% of the total assets in the mutual fund industry are under the management of funds that charge benchmark-linked incentive fees (see Elton, Gruber, and Blake (2003)). 1 Similarly, many hedge funds typically charge incentive fees over fund returns in excess of a risk-free rate (e.g. a Treasury bill rate), whenever such excess returns are positive. Yet, the theory of portfolio management has been mostly pessimistic about the usefulness of benchmarks in managers' compensation contracts. In a setup with asym- metric information between delegating households and managers, and constant abso- lute risk aversion (CARA) preferences, Stoughton (1993) and Admati and P eiderer (1997) argue that linear benchmarking is generally inconsistent with optimal risk- sharing or with the goal of obtaining the optimal portfolio for the investor, and is irrelevant either for inducing the manager to exert more eort or for screening out 1 The abundant evidence on the positive relation between a fund performance relative to peers or to a benchmark portfolio, and future investors ows, suggest that most mutual funds are also indirectly subject to relative performance evaluation. 101 bad managers. By broadening the class of contracts or considering logarithmic pref- erences instead, subsequent authors (Li and Tiwari (2009), Dybvig, Farnsworth, and Carpenter (2010)) have proved that the inclusion of a benchmark in the compensation contract can be optimal to mitigate the moral hazard (eort under-provision) prob- lem. However, the question of whether there exists any role for linear or non-linear benchmarking in aligning managers' and fund investors' interests, when managers have access to superior information, remains by and large unanswered. 2 In this chapter, I show that linear benchmarking can be highly valuable for dele- gating investors in inducing a manager to use her superior information in their best interest. For benchmarks to fulll this purpose in the setup I consider, two crucial conditions have to be met. First, investors' and the manager's appetites for risk have to dier, for preferences that are not of the CARA type. Second, managers must have access to superior information or investment skills that can potentially turn portfolio delegation valuable. Under these conditions, benchmark-linked incentives fees have independent value even in the absence of moral hazard problems: 3 they distort the risk exposure of the manager's portfolio in the direction preferred by the households, while at the same time enable higher performance by allowing the manager's superior information to be incorporated in her investment policy. I approach this problem in a familiar dynamic portfolio choice framework in continuous-time, in which I allow for non-linear compensation contracts in the del- egation relation and for incomplete information about market parameters for both 2 Under symmetric information between managers and investors, several authors have shown that benchmarking practices can improve (Starks (1987), Basak, Pavlova, and Shapiro (2008)) or be even optimal (Cadenillas, Cvitanic, and Zapatero (2007), Li and Zhou (2009)) for risk-sharing, and that the optimal contract in the principal-agent problem includes a benchmark-linked fee (Ou-Yang (2003)). 3 In some situations, the problem of aligning manager's and investors' interests may be argued to be more important than that of inducing the manager to exert costly eort since, presumably, the manager may have enough incentives to collect information to administrate an existing portfolio. For instance, the investment division of a large bank may administer not only individual investors' wealth but also some of the funds of its parent bank, in which case the incentives to collect valuable information for investment decision may be already in place before a delegation contract with the individual investor is signed. 102 parties. In this setup, I assume the manager has access to private information, cre- ating an informational asymmetry that can render delegation valuable. Households' optimal contract design problem consists in aligning the manager's incentives with their own while exploiting the manager's information to improve the performance of their portfolios. The analysis focuses on the typical contracts observed in the indus- try, which include a proportional asset-based fee and benchmark-linked incentive fees of two types: (i) a linear (\fulcrum") fee rewarding or penalizing fund performance relative to a benchmark, and (ii) a convex (\option-like") fee rewarding relative out- performance but without penalizing underperformance.. A rst contribution of this chapter is to provide a closed-form solution for the manager's optimal dynamic asset allocation over a xed investment horizon under this wide class of contracts, when she learns asset fundamentals over time. Due to the assumption of CRRA preferences, the manager's hedging demand against changes in her future investment opportunities represent an important component of the optimal portfolio. I decompose this hedging demand into several sub-portfolios and show how the compensation contract aects each of them, in a way that can be used by households to distort the manager's portfolio in their favor. A second contribution of this chapter is a numerical characterization of the con- tract that maximizes households' utility from delegation. In contrast to prior litera- ture, I show that simple benchmarking rules are highly valuable in allowing households to exploit manager's superior information. When the manager's risk-tolerance diers from households', the optimal contract|within the class considered|is linear and always includes a benchmark-linked fulcrum fee. In order to oset insucient or ex- cessive risk-taking by the manager, the optimal benchmark has either a much higher or a much lower risk exposure than the unconditionally ecient portfolio the unin- formed households would choose under self-management. Option-like fees are domi- nated by fulcrum fees but, in turn, dominate pure proportional asset-based fees when 103 the manager is more risk-averse than households. The optimality of benchmarked- linked fulcrum fees is robust to dierent precisions in manager's information and to dierent investment horizons. The analysis of this particular subset of contracts is highly relevant given the widespread use of all or some of its component in the asset management industry. The topic is of interest not only to practitioners but also to regulation authorities, in particular as regards the compensation arrangement between mutual fund managers and fund owners. Prior to 1971, both fulcrum and \bonus" (option-like) perfor- mance fees had been used in this industry. In 1971, the SEC ruled that if investment companies used performance-based compensation contracts, the contracts had to be symmetric: the use of bonus performance contracts was prohibited. Since then, the issue has been the object of a long-standing debate. The results in this chapter sug- gests that the prohibition may have entailed no welfare loss, and most likely might have been welfare-improving, for delegating investors. 4 The rest of the chapter is organized as follows. Section 3.2 introduces the eco- nomic environment, the nature of the delegation problem, and the class of managerial contracts considered. Section 3.3 presents the manager's portfolio choice problem and provides an analytical characterization of the optimal investment strategy for an arbitrary contract. I examine households contracting problem and solve numerically for the optimal compensation parameters in Section 3.4. Section 3.5 oers concluding remarks, whereas the technical details are summarized in Appendix B.1. 4 Under the appropriate changes in the setup and interpretations, the analysis here could be easily extended to more general corporate settings (i.e., beyond the asset management industry) in order to possibly shed light on the even longer-standing debate about the appropriate structure of CEOs' compensation. 104 3.2 Model Setup I consider an economy in which households (henceforth, also referred to as investors) can opt to delegate their nancial wealth w to investment companies (pension funds, mutual funds or hedge funds) over a certain investment horizon denoted by [0;T ], and in which no additional funds share purchases or redemptions take place. In case dele- gation occurs, each investment company manages investors' portfolio by dynamically allocating their wealth among the available nancial assets in return for a monetary sum according to a pre-specied (at t = 0) compensation contract. The rest of the chapter focus on the relationship between the fund manager in one of such companies, and her fund investors. I assume constant relative risk aversion (CRRA) preferences, with coecients of relative risk aversion ; h > 1 for the fund manager and households, respectively: u(w; ~ ) = 8 > < > : w 1~ 1~ if w 0; 1 if w< 0, (3.1) for ~ 2f ; h g. 3.2.1 Financial Markets and Information Structure Financial markets consist of one risk-free, with price , and one risky, with price S, assets. The riskless asset can be a short-term bond or a bank account, whereas the risky asset can be a stock or any portfolio of risky assets (such as the market portfolio or other traded benchmark). All agents are atomistic participants in the asset markets, so they take asset price dynamics as exogenously given. The bond has initial price 0 = 1 and pays a constant interest rate r per unit time, such that its 105 price dynamics are d t = r t dt. The stock has initial price S 0 = s, and dynamics given by the following SDE: dS t =S t (dt +dB t ); (3.2) whereB is a standard Brownian motion process dened on a ltered probability space ( ;F;P;fFg tT ), over the interval 0tT . All parameters are constant. The risk-free rate r and volatility > 0 are public information, as are security prices. The mean return , however, is the unobserved realization at t = 0 of a random variable with normal distribution N(r + m; 2 v 0 ), for given constants m and v 0 0. Equivalently, the \market price of risk" (r)= is the unobserved draw from a normal distribution N( m; v 0 ) at t = 0. 5 I introduce asymmetric information between households and the manager by as- suming that the latter has access to superior information about asset return fun- damentals. This informational advantage is the only potential source of value from delegation: both parties have access to nancial markets at no cost, but in addition the manager has access to a private (noisy) signal ~ =+ att = 0, with N(0; 2 ) and ? . In this setup, households may decide to delegate their portfolio to pro- fessional managers not because these face lower transactions costs on traded assets, or because households do not have the time for active management|both of which may represent additional benets from delegation|, but because the manager has superior investment skills. Arguably, the belief that money managers may have valuable information is a key reason behind the existence of active money management. This assumption 5 The structure of the uncertainty I assume is the same as that considered by, among others, Brennan (1998) and Cvitanic, Lazrak, Martellini, and Zapatero (2006), and represents a particular case of the \incomplete information" case in continuous-time studied by e.g. Detemple (1986, 1991), Dothan and Feldman (1986) and Gennotte (1986). 106 represents a departure from the literature on delegated portfolio management un- der symmetric information, in which delegation has no value unless households are exogenously prohibited from participating in the nancial markets, or are assumed to face higher|though unspecied|transaction cots relative to fund managers. As described in Section 3.4, the assumption in this chapter implies a more demanding participation constraint on households' delegation decision, and allows a quantita- tive analysis of this decision to be centered around the value of manager's private information. 3.2.2 Managerial Contract For a realization ~ of her signal at t = 0, the fund manager dynamically chooses an investment policy t representing the fraction of the fund's wealthW t allocated in the risky asset (the fund's risk exposure) at time t2 [0;T ] to maximize expected utility over end-of-period compensation X T : E " X 1 T 1 # ; (3.3) subject to the self-nancing constraint for assets under management: dW t =W t (r + t )dt +W t t dB t ; (3.4) and initial wealth W 0 =w. As is standard in the literature, I assume that contracts cannot be written on the manager's portfolio choicef t g t2[0;T ] but only on end-of-period fund returns. For tractability purposes, I restrict the set of contracts considered to the typical class observed in the asset management industry. More precisely, the manager's compensation X T is the product of a management fee rate f T , agreed upon with the delegating investors at the outset, and the fund's value as of the beginning of the 107 periodW 0 . In turn, the management fee rate is a function of the fund's absolute and relative performance, with respect to a benchmark Y , during the investment period [0;T ]. The benchmark is a value-weighted portfolio with a constant weight Y in the risky asset and the remaining 1 Y in the risk-free asset. I impose the condition 0 Y 1 to prevent the benchmark from representing a short position in either assets. Its value process satises the self-nancing dynamics: dY t =Y t r + Y dt +Y t Y dB t ; (3.5) where without loss of generality I normalize the initial value Y 0 = y to equal w. The fee rate ties manager's compensation to performance according to f T =f(R W T ;R Y T ; 1 ; 2 ; 3 ; ), with R W T W T =W 0 , R Y T Y T =Y 0 , 1 ; 2 ; 3 ; > 0, f(x;y;k 1 ;k 2 ;k 3 ;) =k 1 x +k 2 (xy) +k 3 (xy) + ; (3.6) and x + max(x; 0). The managerial fee is thus a random variable with three components: a propor- tional fee 1 R W T that rewards fund's absolute performance, a benchmark-linked linear or \fulcrum" incentive fee 2 (R W T R Y T ) that rewards or penalizes fund performance relative to the benchmark, and a benchmark-linked convex or \option-like" incentive fee 3 (R W T R Y T ) + that rewards fund's performance in excess to that of the bench- mark by a threshold . This contract specication is general enough to encompass existing management fees for dierent type of investment companies (pension, mutual and hedge funds), and either explicit or implicit (fund- ow to relative performance) contracts between a fund manager and the fund owners. 108 Letting W 0 =Y 0 , total managerial compensation can be expressed as: X T f T W 0 = 1 W T + 2 (W T Y T ) + 3 (W T Y T ) + =f (W T ;Y T ; 1 ; 2 ; 3 ;): (3.7) A management fee contractC species the dierent fee components 1 , 2 and 3 , the benchmark's risk exposure Y as well as the moneyness of the option-like fee. Households' problem then consists in designing a contract: C ( 1 ; 2 ; 3 ; Y ;) so as to induce the manager to use her private information in their best interest. The following additional parameters are used to simplify notation throughout the rest of this chapter: 3 2 + 3 , 4 2 + 3 . Both 3 and 4 represents the sum of the benchmark-linked incentives fees, scaled by the moneyness of the option-like fee in the case of 4 . The following section describes the manager's optimal portfolio choice for a given (arbitrary) compensation contractC . 3.3 Portfolio Choice Problem As long as <1 the private signal ~ provides the manager with a more accurate (relative to the common prior N( m; v 0 ) she shares with households) initial assessment of the realized market price of risk . Following the projection theorem for normal variables, the manager updates her prior about after the observation of ~ to a normally distributed variable with conditional mean: mE[j ~ ] = 2 m + v 0 ~ v 0 + 2 ; (3.8) and conditional variance v 0 var[j ~ ] = v 0 2 v 0 + 2 . Note that the noisier the signal (the higher ), the more the manager relies on the common prior m to estimate , and the lower the value of her private information relative to investors'. Naturally, this value also falls as v 0 declines, independently of the precision of the manager's signal, 109 since in this case there's a small overall uncertainty to be reduced by the private signal. In contrast, as the signal's noise falls ( ! 0) the correlation between ~ and increases, 6 approaching the perfect (complete) information case in which m = and v 0 = 0 in the limit. From then on, the manager updates her inference about at any t2 [0;T ] ac- cording to Bayes rule based on the information owF S t fS u ; 0utg, i.e. on the ltrationF S t generated by the stock price process. Given the updated prior N(m;v 0 ), the distribution of conditional onF S t and ~ is Gaussian, with conditional mean ~ t E h j ~ ;F S t i and variance v t E h ( ~ t ) 2 j ~ ;F S t i satisfying (see Chapter 1): 8 > < > : d~ t =v t d ~ B t ; dv t =v 2 t dt; (3.9) with ~ 0 =m and v 0 as initial values for ~ and v, respectively. Note that ~ B t is a standard Brownian motion with respect toF S t , with dynamics given by d ~ B t = 1 h dSt St (r +~ t )dt i = dB t + ( ~ t )dt. Bayesian updating of her prior leads the managers to revise up her estimation ~ of after positive shocks to the stock market (d ~ B t > 0), and to adjust it down after negative shocks (d ~ B t < 0). Since uncertaintyv t decays deterministically over time, so does the manager's sensitivity of the revisions to surprises ~ B t in stock returns according to (3.9). Under parameter uncertainty, markets are still complete with respect to the ob- servable states of the economy (a single risky asset S driven by a single Brownian motion ~ B). Absent arbitrage opportunities, the manager sees nancial markets as driven by a unique state-price de ator with dynamicsd t =r t dt t ~ t d ~ B t . Let B Q t = ~ B t + R t 0 ~ s ds =B t +t denote the risk-neutral Brownian motion, andTt the time remaining until the end of the period. For a given compensation contract, 6 Note that cov(; ~ ) = cov(; +) = v0 given the orthogonality assumption between and . Thus, corr(; ~ )(; ~ ) = v 0 p v 0 p v 0 + 2 = q v 0 v 0 + 2 , and (; ~ )! 1 as ! 0. 110 the manager's optimal fund value ^ W and investment strategy ^ are characterized in the following: Proposition 3. Let ~ (t) + ( 1)v t (Tt) and V (t) ~ (t) (1 +v t (Tt)). Dene the constants 1 ( 1 + 2 ) 1 1 and 2 ( 1 + 3 ) 1 1 , and let 3 2 1 + 2 and 4 4 1 + 3 be the relative weight of the fulcrum fee in a linear contract, and the relative weight of the benchmark-linked incentive fees in a general fee contract, respectively. For an arbitrary contractC ( 1 ; 2 ; 3 ; Y ;), the manager's optimal fund value at time t2 [0;T ] is given by: ^ W t =( t ) 1 e 1 1 r g 1 ;t; ~ t ;T [ 2 ( 2 1 ) (N (d 1;t )N (d 2;t ))] (3.10) +Y t [ 4 ( 4 3 ) (N (d 3;t )N (d 4;t ))] and her optimal risk exposure is given by: ^ t = ~ t ~ (t) +! t Y ~ t ~ (t) + t ; (3.11) where the conditional mean ~ and variance v of the market price of risk are given by: 8 > < > : ~ t =v t B Q t + m v 0 ; v t = v 0 1+v 0 t ; (3.12) the Lagrange multiplier solvesE h T ^ W T i =w,N (:) is the standard normal cumula- tive distribution function, g (;t;x;T ) q (1+vt(Tt)) 1 1+(1)vt(Tt) exp n 2 (1)(Tt) 1+(1)vt(Tt) x 2 o , ! t Y t [ 4 ( 4 3 ) (N (d 3;t )N (d 4;t ))] ^ W t 0; (3.13) 111 t 1! t r V (t) ( 2 1 ) (N 0 (d 1;t )N 0 (d 2;t )) 2 ( 2 1 ) (N (d 1;t )N (d 2;t )) (3.14) + ! t p ( 4 3 ) (N 0 (d 3;t )N 0 (d 4;t )) 4 ( 4 3 ) (N (d 3;t )N (d 4;t )) ; and d 1;t ~ (t) v t V (t) Y ~ t ~ (t) +'() p V (t) ; d 2;t d 1;t 2 '() p V (t) ; d 3;t ~ (t) v t Y ~ t ~ (t) +'() p ; d 4;t d 3;t 2 '() p ; for the function '(:) as given in the proof. Proof. See Appendix B.1. The importance of Proposition 3 lies in providing a closed-form solution (module the scaling constant ) for the manager's optimal dynamic portfolio choice problem, under a wide range of compensation contracts and degrees of a uncertainty about asset return fundamentals. In particular, expression (3.10), evaluated at t = T , identies the optimal fund value for a particular realization of the triple (;;B T ) that allows for the computation of investors' and the manager's certainty equivalent returns from delegation. These returns are a main input for the numerical optimization exercise over managerial contracts performed in Section 3.4. A closer inspection of (3.11) allows for an intuitive interpretation of the manager's investment policy. Indeed, this equation can be rewritten as: ^ t = ~ t |{z} A +! t Y ~ t | {z } B (1! t ) ( 1)v t + ( 1)v t ~ t | {z } C + t |{z} D : (3.15) The optimal portfolio has several components, denoted A to D in equation (3.15). Component A is a standard mean-variance portfolio scaled by manager's coecient of relative risk aversion , whereas portfolios B to D represent the manager's hedging demand against changes in her future investment opportunities. In particular, port- folio B is a hedge against future deviations from the benchmark, and implies that the 112 manager's portfolio partially mimics this benchmark|e.g. increasing her fund's risk exposure whenever the benchmark's is higher than the mean-variance portfolio's|in order to avoid being penalized for relative underperformance. Since! t is positive and increasing in the ratioY t = ^ W t , this mimicking eect intensies as relative performance falls. Portfolio C represents the manager's hedge against future errors in the estimation of , ( 1)vt +( 1)vt ~ t , scaled by the coecient (1! t ) 1. This implies that a benchmarked manager's (scaled) hedge against forecast errors is never greater than that of an otherwise identical direct investor ( 1)vt +( 1)vt ~ t and, depending on the benchmark's performance, can even adopt the opposite sign. Finally, component D is a risk-shifting portfolio that hedges against the possibility that the option-like fee goes out of the money. This portfolio represents a large long or short position in the risky asset depending on whether the benchmark under- or overweight this asset, respectively, relative to the mean-variance portfolio. Thus, the manager shifts risk by taking large bets in the direction suggested by her private information as re ected in ~ t . Summing up, under CRRA preferences households' choice of the contract param- eters aects the manager's asset allocation through her hedging demand, components B to D in equation (3.15). The next section analyses how these contract parameters can be chosen so as to have the manager use her private information in households' best interest. 3.4 Optimal Contracting Problem Whenever households decide to delegate their portfolio to a fund manager, a potential agency problem arises from the hidden nature of the latter's private information: 113 even if the manager's portfolio choice at each point in time were observable, the non- observability of her private signal ~ would still prevent investors from judging the optimality of the decision taken. As a benchmark case, it is useful to consider the portfolio strategy that would allow households to reap the total value, according to their own preferences, of the manager's information. This is the strategyf M t : 0tTg that households would implement themselves if they had access to the manager's private signal ~ and results from solving, for each realized pair (;), the following optimization problem: max (t) t2[0;T] E " W 1 h T 1 h # ; (3.16) subject to the self-nancing constraint (3.4), where the expectation is with respect to the distribution of B T . The solution to this problem is presented in the following: Lemma 1. When households observe the signal ~ at t = 0, their optimal portfolio strategy M t and wealth W M t for t2 [0;T ] are given by M t = 1 h + ( h 1)v t ~ t = ~ t h ( h 1)v t h + ( h 1)v t ~ t h ; (3.17) W M t = ( M t ) 1 h e 1 1 h r g 1 h ;t; ~ t ;T ; (3.18) where M = h g 1 h ; 0;m;T =w i h , and ~ t , v t and g(:) are given in Proposition 3. Proof. Problem (3.16) is identical to problem (3.3) for = h ; 1 = 1, and 2 = 0 = 3 (for any values of Y and ). For a given M , equations (3.17) and (3.18) then follow from plugging these parameter values in equations (3.11) and (3.10) of Proposition 3. The Lagrange multiplier follows from solving for M in (3.10) fort = 0, with W M 0 =w. 114 Before the signal ~ is realized, households' expected utility from following strategy (3.11) is given by U h ( M )E (W M T ) 1 h 1 h ; (3.19) where the expectation is to be taken with respect to the joint distribution of (;;B T ). Given the closed-form expression for W M T of equation (3.10), this expectation can be approximated with arbitrary precision using, for instance, Gaussian quadrature. When manager's private signal cannot be contracted upon, households' will not be able to derive a higher expected utility thanU h ( M ) and, in general, they will obtain a strictly smaller expected utility (after paying manager's compensation if they decide to delegate, or as a result of the lower quality of their information if they manage their funds themselves). For this reason, I refer to U h ( M ) and M f M t g t2[0;T ] as \households' maximum utility" and \maximum utility portfolio strategy", respec- tively. I assume that delegating households oer the privately informed manager a com- pensation contract before she receives her signal. Households' problem then consist in choosing the contract parameters, within the class considered in Section 3.2.2, that induce the manager to implement the investment policy resulting in the closest expected utility to U h ( M ) possible, subject to both parties participating, i.e.: max f 1 ; 2 ; 3 ; Y ;g E 2 6 4 ^ W T X T 1 h 1 h 3 7 5 (3.20) s:t: = 8 > > > > > < > > > > > : E X 1 T 1 U 0 ; (M's PC) E ( ^ W T X T ) 1 h 1 h E h (W h T ) 1 h 1 h i ; (HH's PC) 0 Y 1;> 0 where the expectations are to be taken with respect to the joint distribution of (;;B T ). Conditions (M's PC) and (HHs PC) represent the manager's and the 115 households' participation constraints, respectively, where U 0 is manager's reservation utility, and W h T denotes households' terminal wealth under the investment policy f h t : 0tTg that they would follow under self-management. In general, a solution to problem (3.20)|in case it exists|will depend on the specic value of the manager's reservation utility U 0 . In order to eliminate the de- pendence on this parameter, I assume throughout that competition among privately informed managers drives their reservation utility U 0 to a positive value arbitrarily close to zero. 7 This assumption allows 1 to be set to a positive but arbitrarily small value and focus the search for the optimal compensation contract on the remaining parameters ( 2 ; 3 ; Y ;) by expressing 2 and 3 as a multiple of 1 . Thus, the rest of the analysis in this section concentrates entirely on the value of delegation to households, at the cost of abstracting from risk-sharing considerations. The following two sections analyze the optimal compensation contracts for the cases of identical and diering risk aversion for households and the manager. 3.4.1 Identical Risk Preferences When = h there is no incentive misalignment between the manager and households. The optimal contract in this case is well-known to be a constant sharing rule that compensates the manager in proportion to the investors' payos (see e.g. Cadenillas, Cvitanic, and Zapatero (2007)). In the class of contracts considered in this chapter, this proportionality is achieved by oering the manager a pure proportional fee: Lemma 2. When households and the manager have identical risk-preferences ( h = ), a pure proportional fee contractf( 1 ; 0; 0; Y ;) : 0 Y 1;> 0g implements 7 This assumption seems reasonable in the present setup, in which managers are just endowed with private information at no cost. If the information quality of the private signal depended instead on costly eort by the manager, an analogous assumption would require that the certainty equivalent of U0 covers the costs of eort necessary to receive a signal with the specied information quality. 116 the maximum utility portfolio strategy M . Moreover, households can attain closest- to-maximum utility by setting 1 such that E X 1 h T 1 h =U 0 . Proof. When 2 = 0 = 3 , the manager's compensation is X T = 1 W T , so the benchmark's risk exposure Y and the moneyness of the option-like fee play no role and can be set to any value 0 Y 1; > 0. The manager's objective function is then E ( 1 W T ) 1 h 1 h = ( 1 ) 1 h E " W 1 h T 1 h # ; (3.21) which is a monotone increasing transformation of the households' objective function in (3.16). The resulting optimal portfolio ^ then equals M . Moreover, (3.21) is positive and continuous in 1 , so there exists a value 1 such that E X 1 h T 1 h =U 0 . SinceU 0 is arbitrarily close to 0 by assumption, so is the resultingX T and households obtain closest-to-maximum utility. The result in Lemma 2 implies that there is no role for benchmarking, either linear or convex, when the manager and households have identical risk-tolerances. The relevant question is then whether this result is robust to the more general case in which manager's and households' risk-tolerances are allowed to dier, a question I address in the next section. 3.4.2 General Risk Aversion A comparison of equation (3.15) with equation (3.17) shows that, for an arbitrary contractC , the optimal portfolio ^ chosen by the manager will dier from households' maximum utility portfolio strategy M when 6= h . Moreover, it can be shown in this case that no contract, within the class considered in this chapter, can induce the manager to choose this portfolio. The problem of the optimal contract design then consists in choosing the values ( 1 ; 2 ; 3 ; Y ;) that result in a portfolio close to M so that households' derived utility gets as close to U h ( M ) as possible. 117 Proposition 3 and equation (3.15) provide an intuition of how the linear and con- vex benchmark may aid to attain this goal. A fulcrum fee induces relative concerns with respect to benchmark on the manager and gives her incentives to partially mimic it in all states. On the one hand, this mimicking eect can be used by households to align manager's risk-tolerance with theirs, setting a risky benchmark for relatively more risk-averse managers and vice versa for relatively risk-tolerant ones (see com- ponent B in (3.15)). Moreover, under incomplete information the fulcrum fee has an additional risk-aversion alignment eect via the manager's hedge against inference errors due to its impact on ! t (see equation (3.13)), as is clear from component C in (3.15). On the other hand, when the manager has superior information, mimicking a benchmark that is constructed with information of lower quality (that of house- holds) leads her to partially disregard valuable private information. An option-like benchmarked-linked incentives fee attenuates the benchmark concern in many states and fosters more aggressive trading on manager's private information, while still at- taining some alignment in risk-attitude. The ip side of the coin is that this type of fee encourages excessive risk-taking in other situations as well. The optimal weight to the fulcrum and option-like fees should then result from a trade-o between the benets of manager's informational advantage on the one hand, and the costs of risk-aversion misalignment on the other. Table 3.1 shows the optimal contracts that results from solving problem (3.20) numerically under standard parameterizations of the model. Figure 3.1 illustrates households' derived utility from the optimal contract across the dierent basic types: proportional-only fees ( 1 > 0; 2 = 0 = 3 ), linear benchmarking ( 1 ; 2 > 0; 3 = 0), and option-like benchmarking ( 1 ; 3 > 0; 2 = 0). Derived utility under the dif- ferent contracts is expressed as a percentage of the excess certainty equivalent returns (CER), relative to investors' own portfolio management, of households' maximum utility U h ( M ). 118 Table 3.1: Optimal Managerial Contract and Households' Derived Utility. The table reports the optimal contract (1;2; 3; Y ;) and associated excess certainty equivalent returns (CER) from delegation for dierent time horizonsT , quality of manager's private informa- tion , and manager's relative risk aversion . Benchmark-linked fees 2 and 3 are expressed as a multiple of the proportional fee1. Excess CER are computed with respect to CER from house- holds' own portfolio management, and results are shown both in basis points (column `bps') and as percentage of the excess CER corresponding to households' maximum utility U h ( M ) (column `%'). The rest of the model parameters are: r = 3%; = :0158; m = 0:513; v0 = 0:037; h = 5. Optimal Contract Excess CER T 2 3 Y bps % 1 0.2 3 0.80 0.00 1.00 0.11 18.27 96.59 4 0.28 0.00 1.00 0.07 18.81 99.47 5 0.00 0.00 1.00 1.00 18.91 99.97 6 0.23 0.00 1.00 1.00 16.76 88.63 7 0.41 0.00 1.00 1.00 14.01 74.09 0.4 3 0.92 0.00 1.00 0.17 7.09 94.72 4 0.35 0.00 1.00 0.18 7.41 99.04 5 0.00 0.00 1.00 1.00 7.49 100.10 6 0.26 0.00 1.00 1.00 6.56 87.73 7 0.45 0.00 1.00 1.00 5.45 72.82 5 0.2 3 1.03 0.00 1.00 0.19 14.07 90.77 4 0.34 0.00 1.00 0.16 15.25 98.44 5 0.00 0.00 1.00 1.00 15.50 99.99 6 0.19 0.00 1.00 1.00 13.99 90.31 7 0.33 0.00 1.00 1.00 11.95 77.09 As expected from Lemma 2, the optimal contract is a \pure" total asset-based proportional fee ( 2 = 0 = 3 ) when manager's and households' relative risk-aversions coincide ( = 5 = h ). However, as long as there is the slightest misalignment between their attitudes towards risk ( h = 5, < 5 or > 5), benchmarking increases households' returns from delegation signicantly: the inclusion of benchmarked-linked fees ( 2 > 0 or 3 > 0) dominates proportional fee-only contracts. This is true for both linear and convex benchmarking, as shown by the dierence between the solid lines and the dash-and-dot line in Figure 3.1. Option-like fees improve substantially over pure total asset-based fees when the manager is more risk-averse than households. Moreover, sticking to a proportional fee-only contract in all situations can render delegation undesirable, as illustrated by the negative values adopted by the dash- and-dot line in Figure 3.1 for < 4. 119 50 100 150 % of Maximum Utility CE Excess Return -100 -50 0 3 3.5 4 4.5 5 5.5 6 6.5 7 % of Maximum Utility CE Excess Return γ Figure 3.1: Excess Certainty Equivalent Return from Delegation. Households' excess certainty equivalent returns (CER) from delegation for the optimal proportional-only fee contract (dash-and-dot line), linear benchmarking (solid thick line), and convex benchmarking (solid thin line). Excess CER are computed with respect to CER from households' own portfolio management, and results are in percentage of the excess CER corresponding to households' maximum utility U h ( M ). The model parameters are: T = 1; = 0:2;r = 3%; =:0158; m = 0:513; v 0 = 0:037; h = 5. An important observation from Table 3.1 and Figure 3.1 is that, irrespective of the manager's risk aversion relative to households', the optimal contract is always linear (i.e. has 3 = 0). In particular, linear benchmarking dominates option-like fees, and by a high margin in most cases. Since linear benchmarking serves the purpose of risk- alignment better than convex benchmarking, we see that this margin is increasing in the misalignmentj h j. In striking contrast with prior literature, benchmarking is highly valuable for delegating investors: by including benchmark-linked fulcrum fees, households can collect almost the entire (from 74% to 99% for the range of included in Figure 3.1) excess returns made possible by the manager's informational advantage. As is clear from Table 3.1, these results are highly robust to dierent investment horizons T and precision of the manager's private information 1 . 120 How is the optimal fulcrum fee 2 determined? An examination of Table 3.1 reveals that this fee is increasing in the risk-aversion misalignmentj h j. The intuition here is the same as above: the higher the misalignment in incentives, the lower the relative value of the manager's private information to the investors, who then decides to give the manager higher incentives to take the appropriate risks (higher or lower, depending on whether > h or < h ). By contrast, an increase in the precision of manager's private information 1 (keepingj h j constant), reduces the optimal fulcrum fee in the contract. The optimal fulcrum fee also depends on the investment horizon T : 2 increases with T for < h but decreases with it for > h . Notably, the optimal benchmark in the fee contract is very dierent from the unconditionally ecient portfolio that has been suggested by priori literature (see e.g. Dybvig, Farnsworth, and Carpenter (2010)). This is the portfolio that less informed households would choose themselves at t = 0 (the moment the contract is oered) and, for the case of T = 1 in Table 3.1, assigns a weight of 1 h +( h 1) v 0 T m = 0:63 to the risky asset. 8 In contrast, the choice of the optimal benchmark follows a simple rule: choose a very risky (\all equity") benchmark Y = 1 if > h , and a very conservative (\almost risk-free") benchmark Y 0 if < h . The intuition is that, from households' perspective, a conservative manager ( > h ) underreacts to good news|increasing the weight in the risky asset in her portfolio less than households would prefer|and overreacts to bad news|lowering the weight in the risky asset in her portfolio more than households would prefer. Both under- and overreaction problems can be alleviated by setting a risky benchmark in the managerial contract, and conversely when the manager is more risk tolerant ( < h ). Note that this rule is robust to uncertainty about the precise value of manager's coecient of relative 8 This formula obtains from letting t = 0 and !1 (households have no private information) in the determination of m and v0, and then evaluating (3.17) for ~ 0 = m;v0 = v0. 121 risk aversion , and all the delegating investors needs to know is whether the manager is more or less conservative than themselves. 3.5 Conclusions In this chapter I examine the optimal design of fee contracts for the compensation of money managers when they have access to private but incomplete information about asset return fundamentals, and their preferences are allowed to dier from delegating households'. I show that, irrespective of the manager's risk aversion, the optimal contract is always linear and, in general, includes a benchmark-linked fulcrum fee. In contrast to prior literature, I show that benchmarking increases households' returns from delegation signicantly by enabling them to almost fully exploit the manager's superior information while aligning manager's incentives with theirs. Option-like fees improve substantially over pure total asset-based fees when the manager is more risk- averse than households. When the manager is very risk-tolerant, benchmarking is not only optimal but necessary for delegation to exist, since in this case a pure total asset- based fee can turn delegation undesirable despite the manager's valuable information. I explore how the optimal component fees and the benchmark's composition depend on the risk-aversion misalignment, the quality of the manager's information, and the investment horizon. The analysis in this chapter can be of interest not only to practitioners but also to regulation authorities. Prior to 1971, both fulcrum and option-like performance fees had been in place in the mutual fund industry. In 1971, the SEC ruled that if investment companies used performance-based compensation contracts, the contracts had to be symmetric: the use of option-like performance contracts was prohibited. The results in this chapter contribute to the debate that has emerged since then on this issue by suggesting that this ban may have entailed no welfare loss, and most likely might have improved welfare, on the delegating investors. 122 This chapter assumes frictionless markets and no constraints on managers' invest- ment policies. In practice, managers usually face legal or contractual restrictions on the risk exposure of their portfolios, the allowed amount of short-selling, their expo- sure to derivative instruments, among others. 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For 2R, 0tu, we have e 2 R u t ~ 2 s ds R u t ~ sdB Q s = q (1 +v t (ut)) e 2 ~ 2 t v t vu 2 (B Q u B Q t + ~ t v t ) 2 : (A.1) Proof. First, note that the dierential equation for v in (1.8) is of the Ricatti type, whose solution is given by (see e.g. Gennotte (1986), p.741): v t = v 0 1 +v 0 t : (A.2) We can now solve the dierential equation for ~ in (1.8). Since v t > 0 for all t when v 0 > 0, we can write: d~ t v t =d ~ B t =dB Q t ~ t dt Using (A.2) and integrating both sides between 0 and t gives: ~ t v t m v 0 Z t 0 ~ s ds =B Q t B Q 0 Z t 0 ~ s ds; i.e. ~ t =v t B Q t + m v 0 : (A.3) Next, following Cvitanic, Lazrak, Martellini, and Zapatero (2006) denote E t v0 2 B Q t 2 + mB Q t . An application of It^ o's Lemma gives: dE t = v 0 B Q t +m dB Q t + v 0 2 dt) v 0 B Q t +m dB Q t =dE t v 0 2 dt: (A.4) 130 We have: Z u t ~ s dB Q s = Z u t v s B Q s + m v 0 dB Q s = Z u t v s v 0 dE s 2 Z u t v s ds = v 0 Z u t v s dE s ln 1 +v 0 u 1 +v 0 t 2 ; (A.5) where the last equality follows from direct integration of the expression for v t in (1.15). Integrating by parts, Z u t v s dE s =v u v 0 2 B Q u 2 +mB Q u v t v 0 2 B Q t 2 +mB Q t + Z u t v 0 2 B Q s 2 +mB Q s v 2 s ds ) v 0 Z u t v s dE s = v u v 0 v 0 2 B Q u 2 +mB Q u v t v 0 v 0 2 B Q t 2 +mB Q t + 2 Z u t ~ 2 s ds 2 m v 0 2 Z u t v 2 s ds: (A.6) Direct integration gives R u t v 2 s ds = 2 m v0 2 (v u v t ). Plugging the result in (A.5), we get: 2 Z u t ~ 2 s ds Z u t ~ s dB Q s = ln 1 +v 0 u 1 +v 0 t 2 2 v u B Q u 2 + 2 m v 0 B Q u v t 2 B Q t 2 + 2 m v 0 B Q t )e 2 R u t ~ 2 s ds R u t ~ sdB Q s = s 1 +v 0 u 1 +v 0 t e 2 m v 0 2 (vuvt) 2 h vu (B Q u ) 2 +2 m v 0 B Q u v t 2 (B Q t ) 2 +2 m v 0 B Q t i (A.7) Using the expressions in (1.15) for ~ t andv t we nally get, after some algebraic manipulation, equation (A.1). Lemma 4. Let z N(0; 2 z ), and let ;c; z2R. We have: (i) E e z 1 fz zg =e 2 z 2 2 N z 2 z z , (ii) E h e (zc) 2 1 fz zg i = e c 2 1+2 2 z p 1+2 2 z N z 2 2 z 1+2 2 z c z= p 1+2 2 z ! , whereN (:) is the standard normal cumulative distribution function. Proof. Follows from direct integration against the normal density, using the change of vari- ables ~ z = z 2 z z for part (i) and z = z 2 2 z 1+2 2 z z= p 1+2 2 z , z l = z 2 2 z 1+2 2 z z= p 1+2 2 z for part (ii). 131 Proof of Proposition 1. The expressions for ~ and v in (1.15) are given in the proof of Lemma 3 above (equations (A.3) and (A.2)). To prove (1.13) and (1.14) we rst show that a symmetric equilibrium in nal payos ^ W U T exists and characterize it. We rst conjecture that such equilibrium exits. By (1.6), Y T = W U T . Each U-manager seeks to solve (1.10) subject to (1.12). Standard optimization techniques cannot be applied directly to this problem because the objective function is locally non-concave in a neighbor- hood of W U T (i) = Y T = W U T , 1 where we used (1.6) in the last equality. The rst step is then to construct the concavication ~ u(:) of u(:) (i.e. the smallest concave function ~ u(w) satisfying ~ u(w)u(w) for all w 0), restate and solve the original problem (1.10) in terms of ~ u(:), and then verify that the solutions of the non-concave problem can be derived from those of the concavied problem. Applying this approach gives each U-manager's optimal terminal wealth as (see Proposition 1 in Basak and Makarov (2011)): ^ W U T (i) = 8 > < > : ( U (i)~ T ) 1 if U (i)~ T >b ^ W U T ; (I) (1 +) 1 ( U (i)~ T ) 1 ^ W U T if U (i)~ T b ^ W U T ; (II) (A.8) where U (i) is the Lagrange multiplier attached to the i manager's time-T budget constraint (1.12) solvingw = ~ E h ~ T ^ W U T (i) i , andb(x) (1 +) 1+ x . In region (I) we have: ^ W U T (i)< (1 +) 1+ ^ W U T < ^ W U T ; (A.9) whereas in region (II): ^ W U T (i) (1 +) 1 ^ W U T > ^ W U T : (A.10) This implies that in any symmetric equilibrium in which ^ W U T (i) = ^ W U T for alli2U, whether allU-managers end up the period in region (I) or in region (II) depends on . More precisely, if 1 only ^ W U T ^ W U T can be true, so if an equilibrium exists it occurs in region (II), where all U-managers outperform in all economic states: ^ W U T (i) = (1 +) 1 ( U (i)~ T ) 1 ^ W U T ) ^ W U T (i) = U (i) ~ T 1 ; (A.11) where (1 +) . if> 1 only ^ W U T < ^ W U T can be true, so if an equilibrium exists it occurs in region (I), where all U-managers underperform in all economic states: ^ W U T (i) = ( U (i)~ T ) 1 : (A.12) 1 Moreover, it is non-dierentiable at this point. 132 For this to be a candidate equilibrium it has to be true that U (i) = U for all i2U. This is indeed the case when 1, since: w = ~ E h ~ T ^ W U T (i) i = ~ E " ~ T U (i) ~ T 1 # ) U (i)= = U = = ~ E ~ 1 1 T =w ; (A.13) and similarly (setting 1) for the case > 1. Moreover, this is the unique symmetric equilibrium. To nd the Nash equilibria we additionally require that, given: ^ W U T = 8 > > < > > : ( U =~ T ) 1 ; U = = ~ E ~ 1 1 T =w if 1 ( U ~ T ) 1 ; U = ~ E ~ 1 1 T =w if > 1, (A.14) noU-manager has incentives to deviate and adopt a dierent policy ~ W U T (i)6= ^ W U T . Consider rst the case 1. From (A.8) and (A.14): ^ W U T (i) = 8 > < > : ( U (i)~ T ) 1 if U (i)> (1 +) 1+ U ; (I') ( U (i)= U ) 1 ( U =~ T ) 1 if U (i) (1 +) 1+ U ; (II'). (A.15) Conditions in (I') and (II') are state-independent, so the manager ends up either in region (I') or in region (II') depending on the value of U (i). From the discussion above, condition (I') cannot be true in a symmetric equilibrium when 1. If condition (II') is true, w = ~ E h ~ T ^ W U T (i) i = ( U (i)= U ) 1 ~ E " ~ T U ~ T 1 # = ( U (i)= U ) 1 w , U (i) = U : Since (1+) 1+ = = (1+) = , condition (II') indeed veries if and only if (1 +) 1 < 1, in which case ^ W U T (i) = ^ W U T = ( U =~ T ) 1 . A similar argument shows that, when > 1, the optimal strategies ~ W U T (i) = ^ W U T = ( U ~ T ) 1 for all i2U characterize a Nash equilibrium if and only if > (1 +) 1+ > 1. Therefore, (A.14) represents indeed a symmetric Nash equilibrium in U-managers' policies when either (1 +) 1 or > (1 +) 1+ . We can nally prove equations (1.13) and (1.14). For all values of , (A.14) can be summarized as: ^ W U T = ( U ~ T ) 1 ; U = ~ E ~ 1 1 T =w : (A.16) 133 Under the risk-neutral measure the de ated wealth process is a martingale, so for i2U and t2 [0;T ] the optimal wealth is given by: ^ W U t (i) = ^ W U t =e r(Tt) E Q t h ^ W U T i =e r(Tt) ( U ~ t ) 1 E Q t " ~ T ~ t 1 # (A.17) where E Q t (:) denotes the expectation under the risk-neutral measure Q conditional onF S t , and the state-price de ator is: ~ t =e rt+ 1 2 R t 0 ~ 2 s ds R t 0 ~ sdB Q s : (A.18) Therefore, ^ W U t = ( U ~ t ) 1 e (1 1 )r(Tt) E Q t h e 1 2 R T t ~ 2 s ds+ 1 R T t ~ sdB Q s i = ( U ~ t ) 1 e (1 1 )r(Tt) q (1 +v t (Tt)) 1 E Q t e 1 2 ~ 2 t v t + v T 2 (B Q T B Q t + ~ t v t ) 2 ; (A.19) where the last equality follows from applying Lemma 3 with = 1 and u = T to the expectation on the RHS. Using part (ii) of Lemma 4 with z = B Q T B Q t , 2 z = Tt, = v T 2 , c = ~ t vt , and z!1 to compute this expectation: ^ W U t = ( U ~ t ) 1 e (1 1 )r(Tt) v u u t (1 +v t (Tt)) 1 1 1 + 1 1 v t (Tt) exp ( 1 2 (1 1 )(Tt) 1 + (1 1 )v t (Tt) ~ 2 t ) : (A.20) The Lagrange multiplier follows from solving for U in (A.20) for t = 0 with ^ W U 0 =w: U = g 1 1 ; 0;m;T =w : (A.21) In order to derive the investment policy (1.14) replicating the optimal portfolio value (1.13), note that this can be rewritten as ^ W U t =f(t; ~ t ;T ), withd~ t =v t ~ t +v t dB Q t andf2C 1;2 . Applying It^ o's Lemma the diusion term of d ^ W U t is: v t @d ^ W U t @~ t . Under the risk-neutral measure, d ^ W U t satises the self-nancing constraint: d ^ W U t = ^ W U t rdt + ^ W U t ^ U t dB Q t ; (A.22) Equating diusion terms: ^ U t = v t ^ W U t @d ^ W U t @~ t : (A.23) Substituting the derivative of (A.20) with respect to ~ t in (A.23) gives the optimal risk exposure (1.14). 134 Proof of Proposition 2. Following the same approach as in the derivation of (A.16) above, the optimal terminal wealth of manager i2I is: ^ W I T (i) = 8 > < > : ( I (i) T ) 1 if I (i) T >b ^ W U T ; (I) (1 +) 1 ( I (i) T ) 1 ^ W U T if I (i) T b ^ W U T ; (II) (A.24) where I (i) is the Lagrange multiplier attached to the i manager's time-T budget constraint (1.16) solvingw =E h T ^ W I T (i) i andb(x) is given in the proof of Proposition 1. It is immediate to see that, since initial wealth i is the same for all I-managers, we have I (i) = I and thus ^ W I T (i) = ^ W I T in (A.24) for alli2I. In order to express regions (I) and (II) in terms of the state variable ~ T and the model's parameters, note that T =e (r+ 2 2 )TB T =e ( 2 2 r)TB Q T , and that U-managers' optimal terminal wealth is given by (A.16) with: ~ T =e rT+ 1 2 R T 0 ~ 2 s ds R T 0 ~ sdB Q s = p 1 +v 0 Te rT+ m 2 2v 0 v T 2 B Q T + m v 0 2 ; (A.25) where the last equality follows from the application of Lemma 3 with = 1;t = 0;u = T . Therefore, region (I) can be rewritten (omitting the index i) as: I T >b ^ W U T , I e ( 2 r)TB Q T > (1 +) 1+ U p 1 +v 0 Te m 2 2v 0 v T 2 B Q T + m v 0 2 , I e ( 2 r)T ~ v T m v 0 > (1 +) 1+ U p 1 +v 0 Te m 2 2v 0 1 2 ~ 2 T v T , (~ T ) 2 v T (m) 2 v 0 > 2 ln h 1 A 0 p 1 +v 0 T i ; (A.26) for A 0 (1+) 1+ U > 0. Region (II) is just the relative complement in R of region (I). Note that by letting < 1, plugging (A.16) and (A.25) in the second part of (A.24) and rearranging we obtain the alternative expression for I-managers' wealth in region (II) of Corollary 1: (1 +) 1 ( I T ) 1 ^ W U T = ( I T ) 1 g 2 (T; ~ T ;T ); (A.27) where: g 2 (t;x;T )A 1 q (1 +v 0 t) exp rT + 2 x 2 v t m 2 v 0 ; (A.28) and A 1 (1 +) 1 U > 0. 135 We can now derive the interim performance (1.18). Under the risk-neutral measure the de ated wealth is a martingale, so using (A.24) the optimal wealth ^ W I t for all t2 [0;T ] is given by: ^ W I t =e r(Tt) E Q t h ^ W I T i =e r(Tt) 1 I E Q t 1 T 1 R I + (1 +) 1 1 I E Q t 1 T ^ W U T 1 R II (A.29) whereR I denotes the underperformance region in (A.26): (~ T ) 2 v T > (m) 2 v 0 + 2 ln h 1 A 0 p 1 +v 0 T i , ~ T p v T > s (m) 2 v 0 + 2 ln h 1 A 0 p 1 +v 0 T i = p v T '( I ) , ~ T v T ~ t v t < v T ~ t v t '( I ) [ ~ T v T ~ t v t > v T ~ t v t +'( I ) ; (A.30) andR II is the outperformance regionRnR I . The rst expectation on the RHS of (A.29) is: E Q t 1 T 1 R I = 1 t E Q t " T t 1 1 R I # = 1 t e 1 r 2 2 (Tt) E Q t h e (B Q T B Q t ) 1 R I i = 1 t e 1 (r 1 2 2 )(Tt) [N (d 1;t ) + 1N (d 2;t )]; (A.31) where the last equality follows from the applying part (i) of Lemma 4 to the expectation on the RHS, with: z =B Q T B Q t = ~ T v T ~ t vt , 2 z =Tt, = , z = v T ~ t vt '( I ), and: d 1;t v T ~ t vt (Tt)'( I ) p Tt = ~ t vt + 1 '( I ) p ; (A.32) d 2;t v T ~ t vt (Tt) +'( I ) p Tt =d 1;t + 2 '( I ) p : (A.33) The second expectation on the RHS of (A.29) is: E Q t 1 T ^ W U T 1 R II = U 1 t ~ t E Q t " T t 1 ~ T ~ t 1 R II # : (A.34) 136 We have: E Q t " T t 1 ~ T ~ t 1 R II # =e 1 r 2 2 (Tt) E Q t " e (B Q T B Q t ) ~ T ~ t 1 R II # = q (1 +v t (Tt)) e 1 r 2 2 (Tt) 2 ~ 2 t v t E Q t e (B Q T B Q t )+ v T 2 (B Q T B Q t + ~ t v t ) 2 1 R II = q (1 +v t (Tt)) e 1 r 2 2 (Tt) 2 ~ 2 t v t + v T 2 ~ 2 t v 2 t ~ t v t + v T 2 E Q t e v T 2 B Q T B Q t + ~ t v t + v T 2 1 R II (A.35) Using part (ii) of Lemma 4 for z =B Q T B Q t , 2 z =Tt, = v T 2 , c = ~ t vt + v T , and z = v T ~ t vt '( I ), the expectation on the RHS is: E Q t e v T 2 B Q T B Q t + ~ t v t + v T 2 1 R II = e 1 2 v T 1v T (Tt) ~ t v t + v T 2 p 1v T (Tt) [N (d 3;t ) + 1N (d 4;t )]; (A.36) where: d 3;t v T ~ t vt v T 1v T (Tt) ~ t vt + v T +'( I ) p (Tt)=(1v T (Tt)) = ~ t vt + 1 '( I ) p ; (A.37) d 4;t v T ~ t vt v T 1v T (Tt) ~ t vt + v T '( I ) p (Tt)=(1v T (Tt)) =d 3;t 2 '( I ) q 1v T (Tt) : (A.38) Using the formulas in (A.35) and (A.36): E Q t 1 T ^ W U T 1 R II = U 1 t ~ t q (1 +v t (Tt)) e 1 r 2 2 (Tt) 2 ~ 2 t v t + v T 2 ~ 2 t v 2 t ~ t v t + v T 2 e 1 2 v T 1v T (Tt) ~ t v t + v T 2 p 1v T (Tt) [N (d 3;t ) + 1N (d 4;t )]: (A.39) Plugging (A.31) and (A.39) in A.29 and collecting like-terms: ^ W I t = ( I t ) 1 Z( ;) [N (d 1;t ) + 1N (d 2;t )] + ( I t ) 1 Z( ;)g 1 (;t; ~ t ;T )g 2 (t; ~ t ;T ) exp 1 + (1)v t ~ t + 2 v t [N (d 3;t )N (d 4;t )]; (A.40) where g 1 (:) is given in Section 1.3.1 and Z(;t) exp n 1 r + 2 2 t o . 137 In order to derive the investment policy (1.19) replicating the optimal portfolio value (1.18), note that this can be rewritten as ^ W I t = f(t; ~ t ;T ), with d~ t =v t ~ t +v t dB Q t and f2C 1;2 . Applying It^ o's Lemma the diusion term of d ^ W I t is: v t @d ^ W I t @~ t . Under the risk-neutral measure, d ^ W I t satises the self-nancing constraint: d ^ W I t = ^ W I t rdt + ^ W I t ^ I t dB Q t ; (A.41) Equating diusion terms: ^ I t = v t ^ W I t @d ^ W I t @~ t : (A.42) Substituting the derivative of (A.40) with respect to ~ t in (A.42) gives the optimal risk exposure (1.19). Proof of Corollary 1. Equation (1.21) follows from equations (A.24) to (A.28) in the proof of Proposition 2. The minimum underperformance and outperformance margins and follow from Proposition 1 in Basak and Makarov (2011). The last result in this section concerns households' derived utility from delegation. For a realization at t = 0 of the market price of risk, households' expected utility u J () of delegating wealth w to a type J manager (J2fI;Ug) can be computed as the expectation, with respect to the true probability P , of households' utility over the nal wealth ^ W J T . 2 Proposition 4 summarizes the results for delegation to the uninformed and informed managers of sections 1.3.1 and 1.3.2: Proposition 4. Let ^ h 1 and h 1 . For a realized market price of risk , under the symmetric equilibrium for U-managers of Proposition 1 households' expected utility from delegating to U-managers is given by: u U ()E 2 6 4 ^ W U T 1 h 1 h 3 7 5 = ^ U e ^ rT 1 h s (1 +v 0 T ) 1+^ 1 + (1 + ^ )v 0 T exp ^ (1 + ^ ) 1 + (1 + ^ )v 0 T m 2 2 T ^ v 0 T 1 + (1 + ^ )v 0 T 2 T + m v 0 ; (A.43) 2 This expected utility is gross of management fees paid to the investment fund. Therefore, cer- tainty equivalent results in Section 1.4.2 should be interpreted as indicating the maximum fee house- holds would be willing to pay up front in order to be indierent between delegating or not. 138 and their expected utility from delegating to I-managers is given by: u I ()E 2 6 4 ^ W I T 1 h 1 h 3 7 5 = e ^ rT 1 h ^ I e ^ (1^ ) 2 2 T N (d 0 1 ) + 1N (d 0 2 ) + I A 1 h 1 s (1 +v 0 T ) ( h 1) 1 +( h 1)v T T exp (1 + v T T ) 2 2 T ( h 1)T 1 +v 0 T m m 2 2 + 2 2 ( h 1) 2 v 2 T T 1 +( h 1)v T T T + m v 0 + v T 2 ) N (d 0 3 )N (d 0 4 ) ) ; (A.44) where: d 0 1 v T + ( 1)T m v0 '( I ) p T ; d 0 2 d 0 1 + 2 '( I ) p T ; d 0 3 v T + ( 1)T m v 0 1+( h 1)v T T +'( I ) p T=(1 +( h 1)v T T ) ; d 0 4 d 0 3 2 '( I ) p T=(1 +( h 1)v T T ) : Proof. The Brownian motion B under the actual probability P is related to the risk-neutral Brownian motion B Q through: B t =B Q t t. Delegation to an uninformed manager results in nal wealth ^ W U T as given by (A.16) and (A.25), where ~ T can be rewritten as: ~ T = p 1 +v 0 Te rT+ m 2 2v 0 v T 2 Bt+t+ m v 0 2 : (A.45) Letting ^ = 1 h , households' expected utility from delegating to U-managers is then: E 2 6 4 ^ W U T 1 h 1 h 3 7 5 = ^ U 1 h E h ~ ^ U T i = ^ U e ^ rT 1 h s (1 +v 0 T ) 1+^ 1 + (1 + ^ )v 0 T exp 8 > < > : ^ 2 m 2 v 0 ^ v T 1 + ^ v T T + m v0 2 2 9 > = > ; ; (A.46) where the last equality follows from applying part (ii) of Lemma 4 for z =B T ; = ^ v T 2 ;c = (t + m v0 ) and z = +1. Rearranging, we get the formula in (A.43). 139 When households delegate their portfolio to an informed manager, utility over nal wealth depends crucially on whether the I-manager underperforms or outperforms the benchmark. As of t = 0, (A.30) tells us that the rst will be the case when: ~ T v T m v 0 < v T m v 0 '( I ) [ ~ T v T m v 0 > v T m v 0 +'( I ) , B Q T < v T m v 0 '( I ) [ B Q T > v T m v 0 +'( I ) , B T < v T m v 0 T '( I ) [ B T > v T m v 0 T +'( I ) R 0 I ; (A.47) with outperformance occurring inR 0 II RnR 0 II . Letting h 1 and noting that T = e (r+ 2 2 )TB T under P , households' derived utility can then be computed as: E 2 6 4 ^ W I T 1 h 1 h 3 7 5 = 1 1 h ( E h ( I T ) h 1 1 R I i +E " (1 +) 1 1 I 1 T ^ W U T 1 h 1 R 0 II #) = ^ U e ^ r+ 2 2 T 1 h E h e B T 1 R 0 I i + A 1 h 1 p (1 +v 0 T ) ( h 1) e r+ 2 2 T+( h 1) m 2 2v 0 1 h E e ( h 1)v T 2 Bt+t+ m v 0 2 B T 1 R 0 II (A.48) Finally, using exactly the same steps as in the proof of Proposition 2 to compute the expec- tations in (A.48) we arrive at the formulas in Proposition 4. The results in Proposition 4 allow for the computation of the certainty equivalent returns an investor may expect to collect from delegation, as detailed in Appendix A.3. A.2 Model Parameterization Except otherwise noted, the investment horizon is assumed to be T = 1 year. I identify the risk-less asset with the 3-month U.S. Treasury bill, and the stock S with a broad-based market portfolio. The baseline scenario is as follows. I assume the real risk-less interest rater is 3%. For the prior market price of risk m and market volatility , I use historical estimates during the sample January 1980-December 2006. This corresponds to a recent and relatively long period for which the hypothesis of normality of annual returns cannot be rejected. 3 Following Brennan and Xia (2001), I set the prior for the market excess return r equal to the sample mean return of the Fama and French (1996) market portfolio during the period, 8:1%. The corresponding standard deviation of the market portfolio, , equals 15:8%. I set the prior variance v 0 equal to (the square of) the standard error of the sample mean market price of risk. This standard error equals 0:192 for the period 1980-2006 and 3 The Jarque-Bera test of normality results in a p-value of 16:6% during this period. 140 corresponds to a standard error for the mean return of 3%, in line with baseline values used in the literature. As alternative scenarios, I also examine the cases v 0 = 0:063 2 (corresponding to a standard error for the mean return of 1%) andv 0 = 0:3167 2 (corresponding to a standard error for the mean return of 5%). I assume baseline coecients of relative risk aversion equal to 5 for both managers ( ) and households ( h ) in order to approximately match the median curvature parameter found by Kimball, Sahm, and Shapiro (2007) from the Health and Retirement Survey using hypothetical income gambles. This value is also in line with the empirical estimates in Koijen (2010) in a similar setup. This author highlights a high dispersion in managerial risk attitude, and in addition a strong correlation between ability and risk aversion. Because more skilled managers correspond to those with complete information in the present setup, I favor a moderately high coecient of risk aversion in the numerical analysis of Section 1.4, but consider both lower (minimum= 2) and higher (maximum= 8) values as alternative specications. Without loss of generality, I set funds' initial w equal to 1. For the ow-performance relationship, I allow the performance threshold to vary between 0.9 and 1.1. For the per- formance rankings of mutual funds in the sample of Section 1.5, the average minimum return relative to the median in deciles 9 (second lowest performance decile) is -8.8%, corresponding to 0:92, whereas the average minimum return for the top decile is 7.8%, corresponding to 1:078. In consequence, I expect the range considered for to be wide enough to include most plausible performance thresholds in empirically estimated fund ow-relations. With the same purpose, I allow the ow elasticity to vary between 0.5 and 3. A.3 Numerical Simulations The results in gures 1.3 and 1.4 are generated by simulating equations (1.13)-(1.15), (1.18)- (1.20) and (A.43)-(A.44) for all possible economic states according to the baseline and alterna- tive parameterizations in Appendix A.2, and averaging their values over their joint probability distribution. More precisely, I rst simulate a grid of possible realizations of the market price of risk att = 0 over the support of its prior distribution N(m;v 0 ). For each realization I compute the plotted variable as follows: Tracking error, Sharpe ratios and probability of outperformance: I rst simulate a grid of realizations of the Brownian motion processB as oft =T over the support of N(0;T ), and compute end-of-period log returns for the informed and uninformed managers using equations (1.13) and (1.18). Tracking error varianceTE() and probability of outperformance p O () are then the averages over N(0;T ) of r I T r U T 2 and 1 fr I T >r U T g , where r J T log(R J T ) for J 2fI;Ug. 4 Analogously, manager J's Sharpe ratio SR J () is the average of r J T r divided by the standard deviation of r J T , whereas the excess Sharpe ratio is the dierence SR I ()SR U (). 4 Averages are approximated by Gaussian quadrature of the corresponding mathematical expec- tation, similar to (A.51) below. 141 Certainty equivalent returns (CER): The certainty equivalent return CER(u J ()) for the derived utility u J (J2fI;Ug) of delegating to a J-manager as characterized by Propo- sition 4 is given by: CER(u J ()) = 1 w [(1 h )u J ()] 1 1 h : (A.49) I denote () the risk-less return dierential from delegating to an informed instead of to an uninformed manager: () =CER(u I ())=CER(u U ()) 1. Jensen's alpha: I approximate managers' realized Jensen's alpha during the period by Monte Carlo simulations. First, I simulateM = 10; 000 paths ofK = 52 periods (weeks) of the Brownian motionB. Letting tT=K, the market premiumr S t rt (r S t log(S t =S tt )) represents the only risk factor for the simple nancial market structure in this setup. I then estimate Jensen's alpha as the intercept in the CAPM regression: r J t rt = J () + J ()(r S t rt) +; (A.50) wherer J t log(R J t =R J tt ) andJ2fI;Ug, over the time-series fort = 1;:::;K for them-th path ofB. Informed managers' excess alpha for the realized market price of risk is then the average over the M paths of B of I () U (). 5 The nal step consists in averaging these variables over N(m;v 0 ) using Gaussian quadra- ture over the grid of . The reported values in gures 1.3 and 1.4 are computed as: xE [x] = Z 1 1 x() e (m) 2 2v 0 p 2v 0 d; (A.51) wherex equals, alternatively, tracking error, (), excess Sharpe ratio, probability of outper- formance or Jensen's alpha. 6 5 I check the reliability of the Monte Carlo estimates by increasing, alternatively, the number of pathsM to 50,000 and the number of periods K to 252. In all cases, the relative gain in accuracy is less than 1%. 6 For a ne enough grid of , the expectation in (A.51) can be approximated with arbitrary precision. 142 Appendix B Technical Details of Chapter 3 B.1 Proof of Proposition 3. The manager's inference problem is the same as that of the uninformed managers in Chapter 1, so equation (3.9) follows directly from Proposition 1. The dynamic self-nancing condition (3.4) can be re-expressed as a static one (see e.g. Karatzas and Shreve (1998)), so for a given contractC ( 1 ; 2 ; 3 ; Y ;) problem (3.3)-(3.4) becomes: max W T ~ E u 1 W T + 2 (W T Y T ) + 3 (W T Y T ) + ; ; (B.1) s:t: ~ E [ T W T ] =w: (B.2) ~ E(:) denotes the expectation with respect to the equivalent probability ~ P under which ~ B is a standard Brownian motion. For 3 > 0, u(:; ) is globally non-concave and non- dierentiable at W T = Y T . This is due to the presence of a kink in the compensation function f (:;y; 1 ; 2 ; 3 ;) at x = y. Such kink implies that the manager is risk-loving in the interval over whichu(:; ) is non-concave and prefers to take on gambles that result in val- ues ofW T in either extreme of this interval. In order to use standard optimization techniques, we rst need to construct the concavication ~ u(:) ofu(:; ) (i.e. the smallest concave function satisfying u). Following the approach in Basak and Makarov (2011), the concavied objective function can be obtained as: 1 ~ u(W T ) = 8 > < > : 1 1 [( 1 + 2 )W T 2 Y T ] 1 ; W T <W a +b(Y T )W T ; WW T W 1 1 [( 1 + 2 + 3 )W T ( 2 + 3 )Y T ] 1 ; WW T (B.3) whereW ( 1 + 2 ) 1 1 b(Y T ) 1 + 2 1+2 Y T ,W ( 1 + 2 + 3 ) 1 1 b(Y T ) 1 + 2+ 3 1+2+ 3 Y T andb(x) = ( 1 + 2 + 3 ) 1 n 1 1 1 h 2+ 3 1+2+ 3 2 1+2 i x o , for 1+2+ 3 1+2 1 1 > 1 and a = 1 1 [( 1 + 2 )W 2 Y T ] 1 b(Y T )W . Replacing u for ~ u in problem (B.1), the optimal terminal wealth ^ W T needs to satisfy the rst order condition: @~ u( ^ W T ) @W T = T for a Lagrange multiplier of the budget constraint (B.2). Using the constants dened in Section 3.2.2, a solution can be characterized as: ^ W T = ( ( 1 + 2 ) 1 1 ( T ) 1 + 2 1+2 Y T ; T >b(Y T ) (R I ) ( 1 + 3 ) 1 1 ( T ) 1 + 4 1+3 Y T ; T b(Y T ) (R II ) (B.4) 1 See also Cuoco and Kaniel (2011) and the references therein. 143 Using the denition of b(:) above, and the state-price de ator (see Appendix A.1): T =e rT+ 1 2 R T 0 ~ 2 s ds R T 0 ~ sdB Q s = p 1 +v 0 Te rT+ m 2 2v 0 v T 2 B Q T + m v 0 2 ; (B.5) regionR I is given by all values ~ T such that: Y v T '()< ~ T v T < Y v T +'(); (B.6) where '(x) 1 p v T r (m Y ) 2 v0 + 2( 1) h r + 2 ( Y ) 2 i T + 2 ln xb(y) p 1 +v 0 T . Region R II is just the relative complement: RnR I . Under the risk-neutral measure Q the de ated wealth e rt ^ W t is a martingale (as is the de ated benchmark value e rt Y t ), so using (B.4) the optimal wealth ^ W t , for all t2 [0;T ] is given by: ^ W t =e r(Tt) E Q t h ^ W T i =e r(Tt) ( 1 + 2 ) 1 1 1 E Q t 1 T 1 R I + 2 1 + 2 E Q t [Y T 1 R I ] (B.7) + ( 1 + 3 ) 1 1 1 E Q t 1 T 1 R II + 4 1 + 3 E Q t [Y T 1 R II ] =e r(Tt) ( ( 1 + 2 ) 1 1 ( t ) 1 E Q t " T t 1 1 R I # + 2 1 + 2 E Q t Y T Y t 1 R I + ( 1 + 3 ) 1 1 ( t ) 1 E Q t " T t 1 1 R II # + 4 1 + 3 E Q t Y T Y t 1 R II ) : The rst and second expectations on the RHS are: E Q t " T t 1 1 R I # = v u u t (1 +v t (Tt)) 1 1 1 + 1 1 v t (Tt) (B.8) e r (Tt) 1 2 (1 1 )(Tt) 1+(1 1 )v t (Tt) ~ 2 t [N (d 1;t )N (d 2;t )]; and E Q t Y T Y t 1 R I =e r(Tt) [N (d 3;t )N (d 4;t )]; (B.9) with d 1;t ~ (t) v t V (t)( Y ~ t ~ (t) )+'() p V (t) , d 2;t d 1;t 2 '() p V (t) , d 3;t ~ (t) v t ( Y ~ t ~ (t) )+'() p and d 4;t d 3;t 2 '() p , for ~ (t) and V (t) as dened in Proposition 3. These results follow from applying Lemma 4 in Appendix A.1 to: T t 1 =e r (Tt) 1 2 R T t ~ 2 s ds+ 1 R T t ~ sdB Q s (B.10) = q (1 +v t (Tt)) 1 e r (Tt) 1 2 ~ 2 t v t + 1 v T 2 (B Q T B Q t + ~ t v t ) 2 144 and to Y T =Y t = e h r 1 2 ( Y ) 2 i (Tt)+ Y (B Q T B Q t ) . Similar computations give the third and fourth expectations on the RHS as: E Q t " T t 1 1 R II # = v u u t (1 +v t (Tt)) 1 1 1 + 1 1 v t (Tt) (B.11) e r (Tt) 1 2 (1 1 )(Tt) 1+(1 1 )v t (Tt) ~ 2 t [1 +N (d 2;t )N (d 1;t )]; and E Q t Y T Y t 1 R II =e r(Tt) [1 +N (d 4;t )N (d 3;t )]: (B.12) Plugging the four expectations in (B.7), using constants 1 - 4 as dened in Proposition 3 and rearranging, we get equation (3.10). The Lagrange multiplier results from solving E h T ^ W T i =w for ^ W T as given by equation (3.10), for t evaluated at T . In order to derive the investment policy (3.11) replicating the optimal portfolio value (3.10), note that the latter can be rewritten as ^ W t =h(t; ~ t ;T ), with d~ t =v t ~ t +v t dB Q t , for some function h2C 1;2 . Applying It^ o's Lemma the diusion term of d ^ W t is: v t @d ^ Wt @~ t . Under the risk-neutral measure, d ^ W t satises the self-nancing constraint: d ^ W t = ^ W t rdt + ^ W t ^ t dB Q t ; (B.13) Equating diusion terms: ^ t = v t ^ W t @d ^ W t @~ t : (B.14) Substituting the derivative of (3.10) with respect to ~ t in (B.14) gives the optimal risk exposure (3.11). 145
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Creator
Sotes-Paladino, Juan Martín
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Core Title
Essays on delegated portfolio management under market imperfections
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Business Administration
Publication Date
04/27/2012
Defense Date
03/21/2012
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benchmarking,cross-trading,fund flows,herding,incomplete information,internal capital markets,liquidity,management fees,mutual fund families,mutual funds,OAI-PMH Harvest,optimal contracting.,performance evaluation,portfolio constraints,portfolio delegation,risk-taking incentives
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Zapatero, Fernando (
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Tags
benchmarking
cross-trading
fund flows
herding
incomplete information
internal capital markets
liquidity
management fees
mutual fund families
mutual funds
optimal contracting.
performance evaluation
portfolio constraints
portfolio delegation
risk-taking incentives