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Essays on delegated asset management in illiquid markets
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Essays on delegated asset management in illiquid markets
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ESSAYS ON DELEGATED ASSET MANAGEMENT IN ILLIQUID MARKETS by Luis Goncalves-Pinto 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 2011 Copyright 2011 Luis Goncalves-Pinto To my parents, for their example of perseverance and commitment, and for the unyielding vision of the person they inspired me to be. ii Acknowledgments I would like to thank the following people who made this possible. First and foremost, I would like to thank my advisor, and dissertation chair, Fer- nando Zapatero, for his support, his patience and unwavering encouragement through- out my graduate work. It has been a real privilege to know and work with Fernando these past few years. I would also like to thank the other members of my dissertation committee, Pedro Matos, Andreas Stathopoulos, and Jianfeng Zhang, for investing their time and eort, and providing their wisdom and guidance during this process. A special thanks to Pedro Matos who has been not only a mentor but also a friend, and who has shown me by example what it means to rigorously pursue scientic understanding. I am also truly indebted to Elias Albagli, and cannot thank him enough, for the long and fruitful discussions we had, and for the invaluable support and guidance he oered me throughout my entire job market process. For other very helpful comments and discussions, I thank Gordon Alexander, Daniel Carvalho, Michael Gallmeyer, Igor Makarov, Alberto Manconi, Salvatore Migli- etta, Oguzhan Ozbas, Christopher Jones, Antonios Sangvinatsos, Breno Schmidt, Joshua Shemesh, Juan Sotes-Paladino, Giorgo Sertsios, Nico Singer, Denitsa Ste- fanova, Mark Westereld, and Costas Xiouros. I also appreciate the comments of participants at the Financial Management As- sociation 2010 Doctoral Student Consortium, the 6th Portuguese Finance Network Conference, the Western Finance Association 2010 Annual Meetings, the LBS 10th iii Annual Trans-Atlantic Doctoral Conference, the 7th Annual Paris Finance Inter- national Meeting, the IFID 2009 Conference on Retirement Income Analytics, the European Finance Association 2009 Annual Meetings, the LBS 9th Annual Trans- Atlantic Doctoral Conference, and seminar participants at Arizona State University, ESSEC, Hong Kong University of Science and Technology, IESE, Loyola University at Chicago, Nanyang Technological University, National University of Singapore (which I will be joining in June 2011), NOVA, Purdue University, Singapore Management University, University of Geneva, University of Hong Kong, University of Melbourne, University of New South Wales, University of Southern California, University of Utah, University of Western Ontario, UvA University of Amsterdam, and VU University Amsterdam. Last, and certainly not least, I have greatly beneted from the encouragement and regular humor of my ocemate, colleague, and very good friend, Joshua Shemesh, without whom it would have been a lot harder to complete this journey. The nancial support of doctoral fellowships from the Calouste Gulbenkian Foun- dation, from FLAD/Fulbright, and from the Funda c~ ao para a Ci^ encia e a Tecnologia (FCT), is also gratefully acknowledged. iv Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abstract viii Chapter 1: How Does Illiquidity Aect Delegated Portfolio Choice? 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 The nancial market . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 The fund manager's liquidity-unconstrained problem . . . . 10 1.2.3 The fund manager's liquidity-constrained problem . . . . . 14 1.3 Numerical results and discussion . . . . . . . . . . . . . . . . . . . 19 1.3.1 Analysis of illiquidity and explicit incentives . . . . . . . . . 20 1.3.2 Analysis of illiquidity and implicit incentives . . . . . . . . 31 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 2: The Value of Cross-Trading to Mutual Fund Families in Illiquid Markets: A Portfolio Choice Approach 44 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 The economy . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.2 The fund family's problem . . . . . . . . . . . . . . . . . . . 56 2.3 Numerical analysis and discussion . . . . . . . . . . . . . . . . . . . 60 2.3.1 Cross-trading and family portfolio strategies . . . . . . . . . 63 2.3.2 Portfolio delegation and utility implications of cross-trading 81 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 3: Co-Insurance in Mutual Fund Families 89 Bibliography 99 v List of Tables Table 1.1: Optimal buy-and-hold policies without incentives. . . . . . . . . 21 Table 1.2: Optimal policies without implicit incentives and limited trading of asset 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Table 1.3: Shadow costs of explicit incentives. . . . . . . . . . . . . . . . . . 30 Table 1.4: Optimal policies and costs of time-varying illiquidity. . . . . . . . 32 Table 1.5: Optimal policies and illiquidity discounts. . . . . . . . . . . . . . 33 Table 1.6: Illiquidity discounts and implicit incentives. . . . . . . . . . . . . 36 Table 1.7: Shadow costs of implicit incentives and buy-and-hold policies. . . 38 Table 1.8: Shadow costs of implicit incentives and limited trading of asset 2. 40 Table 2.1: Expected terminal portfolio weights on asset C by fund 2. . . . . 66 Table 2.2: Risk-shifting and return-to-risk ratios. . . . . . . . . . . . . . . . 79 Table 2.3: Utility implications of portfolio delegation to family-aliated and standalone funds and net eects of cross-trading to the family. . 83 vi List of Figures Figure 1.1: Shadow costs of explicit incentives. . . . . . . . . . . . . . . . . 28 Figure 1.2: Derived utility of terminal wealth function. . . . . . . . . . . . . 35 Figure 1.3: Shadow costs of explicit and implicit incentives due to portfolio delegation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 2.1: CIO's utility from cross-trading the illiquid asset C between funds 2 and 1 at t = 0, for dierent correlation coecients between the liquid asset returns. . . . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 2.2: CIO's utility from cross-trading the illiquid asset C between funds 2 and 1 at t = 0, for dierent correlation coecients between the illiquid and the liquid asset returns. . . . . . . . . . . . . . . . . 65 Figure 2.3: Distribution of investment period returns for fund 1. . . . . . . 68 Figure 2.4: Distribution of investment period returns for fund 2. . . . . . . 69 Figure 2.5: Correlation between funds 1 and 2 investment period returns, for dierent correlation coecients between the liquid asset returns. 73 Figure 2.6: Correlation between funds 1 and 2 investment period returns, for dierent correlation coecients between the illiquid and the liquid asset returns. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 2.7: Relative performance and risk-taking for fund 1. . . . . . . . . . 76 Figure 2.8: Relative performance and risk-taking for fund 2. . . . . . . . . . 77 Figure 2.9: Utility implications of portfolio delegation to family-aliated funds and net eects of cross-trading to the family for changing market liquidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 vii Abstract This dissertation consists of three chapters of interrelated work in which I investigate the implications of money management incentives to delegated asset allocation and to asset pricing in the context of illiquid markets. In the rst chapter, using a continuous-time dynamic portfolio choice framework, I study the problem of an investor who exogenously decides to delegate the administra- tion of her savings to a risk-averse money manager who trades multiple risky assets in a thin market. I consider a manager who is rewarded for increasing the value of assets under management, which is the product of both the manager's portfolio allocation decisions, taken over the investment period, and the money ows into and out of the fund, as a result of the portfolio performance relative to an exogenous benchmark. The model proposed here shows that, whenever the manager can substitute between more illiquid and less illiquid risky assets, she is likely to choose to hold an initial portfolio that is skewed toward more illiquid assets, and to gradually shift toward less illiquid assets over the investment period. The model further shows that several misalignments of objectives between the investor and the manager can lead to large utility costs on the part of the investor, and that these costs decrease with asset illiq- uidity. Solving for the shadow costs of illiquidity, the model indicates that delegated rather than direct investing is likely to lead to larger price discounts. The second chapter is joint work with Juan Sotes-Paladino. It builds upon the rst chapter and extends the theoretical framework to analyze a liquidity-constrained dynamic asset allocation problem in which investors delegate their portfolios to mutual viii funds that operate under a family organization. The funds are allowed to cross their trades of illiquid common holdings in response to the interests of the family as a whole, whereas investors' ows are assumed to respond asymmetrically to funds' performance, rewarding good performers disproportionately more than they penalize bad ones. We then study a previously unexplored channel through which fund families can play favorites among their aliated funds: to have some funds avoid the costs of illiquidity by making others adopt suboptimal investment decisions. 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 their standalone counterparts. We show that the additional costs of agency that investors incur under such a money management structure are likely to increase with asset liquidity. However, by imposing position limits on their funds' portfolios, we show that investors can improve their welfare outcome. We further nd that families' optimal strategies can induce a negative correlation between their aliated funds' after- ow returns, creating diversication benets to the family's overall portfolio. Nevertheless, the correlation between portfolios' returns can still be higher for aliated funds than for comparable standalones, due to the overlap in holdings. Finally, the model in this chapter suggests that, as market liquidity increases, fund families are likely to favor more correlated asset holdings within each member fund's portfolio. The third chapter is joint work with Breno Schmidt. In this third chapter we propose to test some of the empirical implications derived from the results of the second chapter. In particular, we propose to test the hypothesis that mutual funds aliated with large families tend to coordinate trades so as to mitigate the damaging eects that liquidity shortfalls can have on funds' performance and consequent share- holder redemptions. From this co-insurance hypothesis, we intend to derive important implications for the price reaction to asset re sales and for managerial risk-taking incentives. First, we use a bootstrap-based approach to show that osetting trades ix among funds aliated with larger families are more likely to be the result of coordi- nated trade strategies. Second, consistent with internal coordination, our preliminary results indicate that there is weak or no price pressure on traded securities mostly held in common by distressed funds aliated with large families. Finally, we make use of the known result that, due to strategic complementarities among investors' share redemptions, the generally convex relationship between ows and past performance does not appear to hold true for funds invested in less liquid assets. Consistent with our working hypothesis, our preliminary results indicate that this is true only for funds aliated with small families. As a result, by improving the convexity of their implicit payo structures, co-insurance strategies at the family level can potentially encourage individual fund managers to take extra risks. Keywords: Portfolio Delegation, Benchmarking, Liquidity, Mutual Fund Families, Internal Capital Markets, Co-Insurance, Cross-Trading, Portfolio Constraints, Asset Fire Sales, Risk-Taking Incentives. JEL Classication: C61, D60, D81, G11, G12, G23, G30, G32. x Chapter 1 How Does Illiquidity Aect Delegated Portfolio Choice? 1.1 Introduction In this chapter, I investigate how the inability to buy and sell assets as desired, aects the problem of an investor who decides not to invest her savings directly, but instead decides to hire a money manager to do so on a discretionary basis. As was recognized by Allen (2001), asset pricing theory cannot be indierent to the fact that portfolio delegation or institutional ownership have become increasingly dominant features of developed nancial markets. Indeed, even though direct ac- cessibility to nancial markets has been improving over the years, the tendency for individual investors to delegate the administration of their savings to professional money managers has not slowed down. According to data from the Federal Reserve Board, back in 1952, individual investors directly held over 90% of corporate equities, while by the end of 2008 this proportion was down to less than 25%. At the same time, the fraction of equities held by investment funds (including mutual funds, closed- end funds, and exchange-traded funds) rose from 2.9% to 28.5%. By hiring money managers to administer theirs savings, investors lose control over the composition of funds' portfolios and become subject to numerous misalignments of objectives that can be very costly. For instance, according to a survey by the Investment Company Institute (the national association of U.S. investment companies), by the end of 2007 over 90% of U.S. mutual fund-owning households indicated \saving-for-retirement" as their primary nancial goal. However, fund managers, who are concerned with their reputations and careers, and who may derive private benets from being in charge 1 of large funds, have incentives to boost short-term performance to increase the value of their funds' assets, actions that may be inconsistent with investors' risk tolerance and long-run investment goals. 1 It is therefore important to understand what factors aect delegated portfolio choice, and how they do so. To the best of my knowledge, existing models of delegated portfolio choice have assumed that investors and fund managers can trade assets continuously and with no frictions. Increasingly, however, both academics and practitioners recognize that, in addition to risk and return, liquidity (or the lack of it) is a critical component of the investment equation. In reality, when liquidity dries up, investors and money managers lose control over their portfolio allocations and may be forced to sit on their hands for long periods of time, unable to trade out of mistakes they may have made with less liquid assets. What's more, investing in illiquid assets presents an interesting set of challenges and risks. In particular, their true value is often unknown, their historical performance can often be misleading, and they typically cost more to transact. In addition, as if low liquidity wasn't already dicult enough to deal with, volatility also typically increases in down and more illiquid markets. In the face of such uncertainty, investors and money managers' optimal portfolio policies necessarily diverge from those they would choose in a less constrained context. Recall that the performance of mutual fund managers is usually measured relative to the performance of a benchmark, like the Wilshire 5000 or the S&P 500, which are usually simply paper-based calculations of stock prices as they are quoted on the exchange. However, as fund managers attempt to make a transaction, prices of illiquid assets can dier signicantly from those quoted on the exchange. The ows into and out of the managers' funds depend on the performance of their portfolios relative to these benchmarks, and the presence of illiquidity makes it more dicult for managers to track benchmarks and grow the value of their funds' assets. Given that illiquidity 1 Chevalier and Ellison (1999) analyze the incentives of mutual fund managers in terms of their career concerns, and nd that younger managers are both more likely to be red for poor performance and (as a result) take on less unsystematic risk than older managers 2 can result in a failure to transact, money managers' performance, not to mention their jobs, can be put at stake, and so the presence of illiquidity can ultimately aect managers' preferences and incentives. In this chapter, I relax the assumption of continuous and frictionless asset trading that has underlied existing models of delegated portfolio management to date and, using a familiar partial-equilibrium dynamic portfolio choice framework, I study the problem of an individual investor who exogenously delegates the administration of her savings to a fund manager who trades multiple risky assets in a market where prices are abnormally volatile and assets have limited marketability. I consider that the exogenous portfolio delegation decision is grounded on the assumption that, in comparison with the individual investor, the money manager is subject to lower trans- action costs, lower opportunity costs for engaging in active portfolio management, better information or ability, or better investing education. In addition, I consider a risk-averse manager whose compensation scheme is set exogenously and which is pro- portional to the value of her fund's assets under management at some terminal date. This fund's terminal value is the product of both the manager's portfolio allocation decisions, taken over the investment period, and the money ows into and out of the fund which are allowed to happen only once on the investment horizon and which depend on the manager's portfolio performance relative to a reference benchmark. I model illiquidity according to Longsta (2001), in which market participants are constrained to trading strategies that are of bounded variation, and investigate the extent to which the restricted ability to initiate or unwind portfolio positions aects the fund manager's optimal asset allocations and risk-shifting incentives created by benchmarking, and how that aects the welfare costs generated to a delegating in- vestor whose objectives may diverge from those of the fund manager. I study this problem in a Black-Scholes-Merton economy in which market participants have access to one riskless asset and two risky assets whose returns can be correlated. I consider liquidity restrictions to be asset-specic. Both of the risky assets can be traded at 3 the beginning of the investment period, but one of them cannot be traded again un- til after a \blackout" period, while the other is allowed to be traded, even if just in limited amounts, over the investment period. Market participants therefore have limited ability to rebalance their portfolio positions at any price, a characteristic that resembles the concept of depth in nancial markets. Examples of risky assets subject to these sorts of trading restrictions are stock of rms that are not publicly traded, restricted and unregistered stock in publicly traded companies, or a large holding of a particular stock that the market cannot absorb. My analysis shows that, compared to when there are no liquidity constraints, whenever a fund manager can choose between two risky assets that are identical in all respects except in terms of liquidity, she is likely to hedge her portfolio by lower- ing her total initial risk exposure, moving capital away from riskier investments and towards the safest one. This analysis conrms that in the framework of Longsta (2001) market illiquidity endogenously generates funding illiquidity. 2 In other words, because the fund manager's value of assets under management is a process bounded below by zero, and given that under liquidity constraints, returns of each risky asset cannot be replicated by those of the remaining assets in the portfolio, endogenously the manager cannot borrow to invest more in risky assets, because the prices of the illiquid risky assets can quickly fall, making it impossible to sell enough of those assets to guarantee positive terminal wealth. In addition, due to diversication reasons, and to incentives for the hedging of portfolio weight uncertainty, initial portfolio alloca- tions on the two risky assets appear to be generally tilted towards the more illiquid risky asset, and are likely to subsequently shift toward the less illiquid risky asset over the investment period. 3 According to the model, portfolio weights in the less 2 See Brunnermeier and Pedersen (2008) for a notable discussion on the mutually reinforcing eects of market liquidity and funding liquidity. 3 Note that, under liquidity constraints, portfolio weights become random variables, and the manager now needs to care about not only the mean and variance of the risky assets, but also the mean and variance of the portfolio weights, which are out of her control. 4 illiquid risky asset are also likely to tilt away from those in the benchmark portfolio signicantly more than are portfolio weights of the more illiquid risky asset, which suggests that a liquidity-constrained fund manager, whose portfolio allocations are no longer under her complete control, is likely to shift portfolio risk by trading on the less illiquid asset. Furthermore, fund managers have implicit incentives to distort portfolio allocations so as to increase the likelihood of future fund in ows, and explicit incentives to administer investors' savings according to their own attitudes towards risk, their (usually shorter) investment horizons, and their (eventually more favorable) participation and transaction costs. My analysis indicates that the misalignment of objectives between the investor and the fund manager can be very costly, and that these costs appear to decrease with asset illiquidity. Lastly, after comparing utilities derived under liquidity constraints, with those derived in a context without such con- straints, the model proposed here indicates that the shadow costs of illiquidity appear to be larger for the delegated portfolio problem, compared to the direct investing one. For ease of exposition, I conne attention to a nominal economy, a constant in- vestment opportunity set, and a constant relative risk aversion (CRRA, hereafter) preference structure. I also focus my analysis on the simple fund ow-to-performance function used in Browne (1999) and Binsbergen, Brandt, and Koijen (2008), where ows into and out of a fund are unbounded and do not depend on the performance of the portfolio chosen by the manager, but solely on the performance of the exogenous benchmark. I recognize that an alternative and more interesting specication for this ow-to-performance function would exhibit a local convexity and would take the size of the ows into and out of the fund to depend on the fund's own performance rel- ative to the benchmark, like the one studied in Basak, Pavlova, and Shapiro (2007), which is capped and calibrated according to the empirical estimations of Chevalier and Ellison (1997). Nevertheless, most of the main results presented in this chapter hold using either of these specications. 5 This work is closely related to the strand of research on risk-shifting incentives in delegated portfolio management, and the implications of benchmarking, represented on empirical grounds by Chevalier and Ellison (1997), Chevalier and Ellison (1999), and Sirri and Tufano (1998), and represented on theoretical grounds by Arora and Ou-Yang (2001), Browne (1999), Basak, Pavlova, and Shapiro (2007), Basak, Pavlova, and Shapiro (2008), and Basak and Makarov (2008). These studies describe a positive relationship between past performance relative to a given benchmark or peer group, and subsequent ows into and out of mutual funds, giving money managers, whose compensation is directly linked to the value of the assets they manage, an implicit incentive to distort portfolio allocations in order to increase the likelihood of nishing ahead of a given performance benchmark with the purpose of increasing future fund in ows. 4 This study is also related to the long literature on asset liquidity and its implications for asset pricing and portfolio choice, examples of which are Longsta (1995), Longsta (2001), Liu and Loewenstein (2002), Liu (2004), Jang, Koo, Liu, and Loewenstein (2007), and Isaenko (2008). Also relevant for this study is the empirical evidence of mutual funds' preference for large liquid stocks (Falkenstein (1996)), evidence of liquidity timing by mutual fund managers (Cao, Simin, and Wang (2007)), evidence that portfolio liquidity is actively managed and chosen as a function of the multiple liquidity needs of a fund (Massa and Phalippou (2005)), and evidence that fund managers tilt their holdings more heavily towards liquid stocks when the market is expected to be more volatile (Huang (2008)). The objective of the present work is then to integrate these blocks of literature and to provide a formal analysis of the implications of asset illiquidity in a delegated portfolio context. The chapter is organized as follows. In Section 1.2, I describe the theoretical model setup, which includes the continuous-time economic setting in which investors and money managers decide their optimal dynamic portfolio policies. The 4 Huang, Sialm, and Zhang (2008) show that funds that shift risk end up performing worse, which is consistent with risk-shifting being driven by money managers' opportunistic behavior, rather than their active portfolio management ability. 6 setup includes the money managers' unconstrained problem, with and without im- plicit incentives, and the constrained problem, with constant and stochastic liquidity constraints. In Section 1.3, I solve the model using numerical methods, and discuss its main results. Conclusions and implications for further research are presented in Section 1.4. 1.2 Model setup 1.2.1 The nancial market Consider two market participants. Let one be a household investor who exogenously decides to directly access the nancial market and manage her savings by herself or who, alternatively, decides to hire a money manager to whom she delegates the administration of all her savings. 5 Assume the money manager's fund belongs to a peer group consisting of a large number of competing funds, such that there are no incentives for strategic interactions among fund managers. 6 Moreover, let these market participants have constant investment opportunities, and nite investment time horizons, T i 2 [0;1), for i2fI;Mg. Let these participants take asset prices as given. Consider a nominal economy in which continuous and unlimited trading and short- selling are possible whenever liquidity restrictions are left out. In this economy, partic- ipants can invest inm + 1 assets, with prices denoted byS j (t), forj2f0; 1; 2;:::;mg. The rst asset is a non-redundant nominal riskless money market account, which price dynamics follow the process: dS 0 (t) S 0 (t) =rdt (1.1) 5 In a more general model, the investor could be allowed to dynamically choose how much of her portfolio to hold directly in a money market account and how much to hold indirectly on risky assets through the managed portfolios of mutual funds, pension funds, and the like. 6 See Basak and Makarov (2008) for an analysis of the dynamic portfolio choice implications of strategic interaction among money managers. 7 where r 0 is a constant, continuously compounded interest rate. The remaining m assets are non-redundant risky assets with nominal prices evolving according to the following equation: dS(t) S(t) = r + > s dt + > s dZ(t) (1.2) where is anm 1 vector of ones, denotes ad 1 vector of constant prices of risk, and s is adm matrix of constant loadings on the source of uncertainty generated by a d-dimensional standard geometric Brownian motion Z(t). Let N 0 (t) denote the number of units of the riskless money market account, and likewise let N(t) denote the m 1 vector of units of the risky securities held by a market participant at time t, for 0 t T . This market participant's wealth is therefore given by W (t) = N 0 (t)S 0 (t) +N(t) > S(t), which evolves according to the following equation: dW (t) =N 0 (t)rS 0 (t)dt +N(t) > diag[S(t)] h r + > s dt + > s dZ(t) i (1.3) where S(t) is an m 1 vector, and diag[S(t)] puts S(t) on the main diagonal of an mm matrix. Now, take this market participant's holdings in them+1 assets at each and every time t, to be self-nancing and to be constrained to lie in the closed solvency region of: S = n (S 0 (t);S(t))2R m+1 :N 0 (t)S 0 (t) +N(t) > S(t)> 0 o (1.4) for all t, and 0tT . Take the fund manager's compensation to be proportional to the value of the assets under her administration, and to be due at the investment horizon. Consider a fund manager whose performance is measured relative to the value of a self-designated benchmark,Y (t). 7 Assume this benchmark is a reference portfolio of risky and riskless 7 The benchmark portfolio Y (t) could be interpreted as a constraint in the contract decided between the investor and the fund manager at the beginning of the investment period. 8 assets which can be replicated by the fund manager. Let the benchmark evolve according to the process: dY (t) =Y (t) h r +(t) > > s dt +(t) > > s dZ(t) i (1.5) where the m 1 vector (t) given by (t) = Y (t) 1 diag[S(t)]M(t), denotes the vector of weights of the risky assets on the benchmark portfolio such that the weight on the riskless money market account is given by 1 > (t). Them 1 vectorM(t) denotes the number of units of risky assets that make up the benchmark at timet, for 0tT . I consider both a continuously rebalanced (active) benchmark, where (t) is set to be constant, and is determined at time t = 0, and a buy-and-hold (passive) benchmark portfolio, where M(t) is set to be constant, and where (t) becomes rather random. Note that by \self-designated benchmark" I do not mean that vector (t) should be included as a control in the manager's problem from which optimal performance benchmarks are derived (e.g. Binsbergen, Brandt, and Koijen (2008)). Note that this work is also not concerned about how the incentive to improve fund in ows could drive the fund manager to strategically mismatch her fund benchmark, as empirically illustrated in Sensoy (2008). Instead, here my solo focus is on how a fund manager allocates and manages resources to achieve investment objectives, given the manager's chosen performance benchmark. The benchmark is assumed to satisfy the manager's participation constraint. Note that this chapter focuses on the description of the continuous-time optimal control problem of the fund manager, and only occasionally refers to the investor's problem, which is taken as a special case of the manager's problem. For simplicity of notation, subscripts I andM are used to indicate variables or parameters pertaining to the investor and the fund manager, respectively, only when necessary for clarity of exposition. 9 1.2.2 The fund manager's liquidity-unconstrained problem As a point of reference, consider rst the problem of a fund manager who is not subject to liquidity restrictions, and who derives utility from the nominal value of the lump-sum cumulative amount of assets under management at the end of his or her investment horizon, t = T . Since admissible allocations require W (t) > 0, portfolio weights on risky assets are well dened and are given by the m 1 vector !(t) = W (t) 1 diag[S(t)]N(t), such that the remainder, 1 > !(t), denotes the portfolio weight on the money market account at time t, for 0 t T . Plugging !(t) into Equation (1.3) we get to the following functional form representing the dynamics of the value of assets under management: dW (t) =W (t) h r +!(t) > > s dt +!(t) > > s dZ(t) i (1.6) Subject to Equation (1.6), a fund manager guided by CRRA preferences, dynamically allocates the fund's assets valued initially at W (0) among a money market account and m risky assets, by choosing a vector of controls !(t), so as to solve: J(W;t) = sup !(t) E t [W (T )(T )] 1 1 (1.7) where > 0, and 6= 1, denotes the manager's coecient of relative risk aversion, and (T ) denotes the rate at which funds ow into ((T )> 1) or out ((T )< 1) of the fund at the terminal date, depending on the fund's performance relative to a given benchmark. 8 I assume that the fund manager's investment horizon,T , coincides with the date of fund ows, and that fund ows are nontradeable at that date. 9 8 In Section 1.3, for comparison purposes, I also present results for the simpler case of logarithmic preferences (CRRA utility with = 1). 9 In a more general model, the investment horizon would not coincide with the fund ows date (e.g. Basak and Makarov (2008)), in which case fund ows would be tradeable after the ow date, and(t<T ) would then enter the problem through the budget constraint, and not directly through the utility function. 10 Optimal portfolio policies without benchmarking incentives Absent benchmarking considerations, in which case (t) = 1, and under regularity conditions on the value function, the Hamilton-Jacobi-Bellman Partial Dierential Equation (HJB PDE, hereafter) representing the problem described above, suppress- ing time indicators, is given by: J t +rWJ W + sup ! WJ W ! > > s + 1 2 W 2 J WW h ! > > s s ! i = 0 (1.8) with terminal condition J(W;T ) = [1=(1 )]W (T ) 1 . The corresponding optimal portfolio allocations on risky assets are given by the m 1 vector: ! (t) = 1 > s s 1 > s (1.9) with the remainder, 1 > ! (t), invested in the money market account. In this uncon- strained liquidity setting, if < 0 m1 , then! (t)< 0, the money manager chooses to hold a short position in risky assets. Likewise, if = 0 m1 , then! (t) = 0, and if > > s 1 > s s , then ! (t)> 1, and the money manager chooses to hold a lever- aged position in risky assets. Moreover, this is so in a complete market setting, where s is invertible, and where > s s 1 = 1 s > s 1 = 1 s 1 s > is possible. Given that s is assumed to be constant, these optimal investment strategies are indepen- dent of the investment horizon, as shown in Merton (1969). Such myopic allocations require continuous trading, and clearly N (t) = 1 W (t)diag[S(t)] 1 > s s 1 > s is of unbounded variation. Hence, after plugging (1.9) into (1.6), to solve the resul- tant stochastic dierential equation, and plugging the solution then into (1.7), and 11 after linearizing the term involving the Wiener process before placing the expecta- tion operator in front of it, the utility derived by the fund manager, as a result of implementing these unconstrained optimal policies, is given by: J(W;t) = W (t) 1 1 exp r + 1 2 A 1 (1 ) (1.10) whereA 1 = > s > s s 1 > s , and =Tt. This complete solution to the man- ager's liquidity-constrained problem coincides with that of the investor whenever any explicit incentives are left out. The fund manager has explicit incentives to administer the investor's savings according to her own attitude towards risk, her (usually shorter) investment horizon, and her (eventually more favorable) participation and transaction costs. The implications of these explicit incentives are considered in Section 1.3. Optimal portfolio policies with benchmarking incentives In the presence of benchmarking incentives, a fund manager experiences money ows into and out of her fund at a rate(T ), depending on the manager's portfolio perfor- mance, at time T , relative to a benchmark chosen (exogenously) at time t = 0. For ease of exposition, in this chapter I focus the analysis on the results from a fund ow- to-performance function that is in line with Browne (1999), and Binsbergen, Brandt, and Koijen (2008), and according to which fund ows do not depend on the manager's own portfolio performance, but solely on the absolute performance of an exogenous benchmark. In this case, the ow-performance function is given by (T ) = 1=Y (T ), for aY (t) that evolves according to Equation (1.5). The convenience of this congura- tion is that it allows the derivation of a closed form solution for the optimal portfolio policies, whenever trading restrictions are left out, which helps with the intuition of 12 the results I present in Section 1.3.2. If we letX(T ) =W (T )(T ), and we apply Ito's rule, we arrive at the dynamics of X(t), for (t) = 1=Y (t), as follows: dX(t) =X(t)[!(t)(t)] > h > s > s s (t) dt + > s dZ(t) i (1.11) and, under regularity conditions on the value function, suppressing time indicators, and given a continuously rebalanced benchmark, the HJB PDE for this problem is given by: sup ! XJ X ! > A 2 + 1 2 X 2 J XX h ! > > s s (! 2) i XJ X > A 2 + 1 2 X 2 J XX h > > s s i +J t = 0 (1.12) where A 2 = > s > s s , and with terminal condition J(X;T ) = [1=(1 )]X(T ) 1 . Hence, when the performance of the fund manager is measured rela- tive to an exogenous benchmark, the manager's optimal portfolio is given by: ! # (t) = 1 > s s 1 > s + 1 1 (t) (1.13) which is independent of the investment horizon when our money manager chooses to be compared to a continuously rebalanced benchmark, in which case (t) is set to be constant. When the money manager chooses to have her performance be measured against a (passive) buy-and-hold benchmark, her optimal portfolio weights, as give by Equation (1.13), become rather time-varying, because the vector of weights on the benchmark portfolio becomes a moving target, for a money manager who chooses ac- tive investing. This optimal portfolio policy contains an actively managed component and a herd component, the latter mimicking the benchmark against which the man- ager's performance is measured. Note that, in fact, the manager will tend to track the benchmark more and more closely as her risk aversion increases. Clearly, when > 1, if (t) > ! (t), then the manager has an incentive to increase risk exposure. 13 On the contrary, if (t)<! (t), the manager has an incentive to herd and decrease risk exposure. In either case, the dynamics of the optimal number of units held by the fund manager on risky assets, N (t), is that of a process of unbounded variation, where both unlimited leveraged and short positions are admissible. This assumption of continuously and unlimited trading, which has been common in delegated portfolio management literature, is going to be relaxed in Section 1.2.3. The solution to the derived utility of the terminal value of assets under manage- ment when the fund manager chooses a continuously rebalanced exogenous benchmark against which to measure performance, is given by: J(X;t) = X(t) 1 1 exp 1 2 h A 1 A 3 (t) > A 2 i (1 ) (1.14) where is constant, A 3 = > s (t), and =Tt. No such explicit solution exists for the case in which the fund manager chooses to have his performance measured relative to (value-weighted) buy-and-hold benchmark, because in that case (t) is a rather stochastic variable which distribution is unknown. Alternative specications are investigated in Basak, Pavlova, and Shapiro (2007), in which ow-to-performance functions exhibit a local convexity and take the size of the ows into and out of the manager's fund to depend on the fund's own performance relative to the benchmark. However, given that the main results of this chapter hold for either of these specications, I focus my analysis on the simpler and more intuitive case where (T ) = 1=Y (T ). 1.2.3 The fund manager's liquidity-constrained problem In practice, fund managers face multiple constraints. Sudden liquidity dry-ups, like the one that accompanied the 2007-08 meltdown in sub-prime lending, result in fund managers nding themselves forced to sit on their hands for long periods of time, 14 unable to deal in any size in shares of even the more liquid large-cap companies. 10 As a result, managers loose control over their portfolio allocations, which potentially puts their short-term performance records (not to mention their jobs) in jeopardy. Thus, let the fund manager choose, at time t = 0, anm 1 portfolio vector!(0), which she will want to revise later on if she chooses to actively invest. However, for t> 0, take the size of the money manager's trades each period, for a given cost, to be out of her complete control, and restricted to lie in a bounded interval. Specically, and in parallel with Longsta (2001), assume that the dynamics of the number of units of risky assets that the money manager can hold each time, is given by: dN(t) =(t)dt (1.15) where (t) is an m 1 vector,1 < (t) (t) (t) <1 and (t) 0. This specication captures the aspect of depth in nancial markets, which I allow to be asset-specic. 11 I also allow it to be either constant or time-varying. In this context, the dynamics of the value of a fund's assets under management is given by the expression: dW (t) =rW (t)dt +N(t) > diag[S(t)][ > s dt + > s dZ(t)] (1.16) which will coincide with the budget constraint of an individual investor, in case we leave out any explicit and implicit incentives provided to the fund manager by her compensation scheme, risk appetite, or investment horizon. Thus, when liquidity is constrained, the manager can nd herself in a situation where portfolio allocations 10 See Brunnermeier (2008) and Allen and Carletti (2008) for excellent accounts of the sequence of events that have mapped out the 2007-08 nancial crisis, focusing on a wide range of factors, among which the typical fragility of market liquidity. 11 It may also be the case that the fund manager has access to the nancial markets in more favorable terms than the investor. In a more general setting, this situation could be captured by allowing liquidity constraints to be investor-specic and more relaxed to the fund manager. 15 are no longer under her control, and it can take a long time for her to \trade out of mistakes" in less liquid assets, which may very well lead to bankruptcy. Therefore, in order to guarantee the solvency of the fund, the manager has to abstain from taking leveraged positions or short extensions in the available risky assets. Accordingly, as in Longsta (2001), in order for portfolio policies to be admissible, optimal controls !(0) and (t) need to be such that 0 !(t) 1, and 0 1 > !(t) 1, for all 0tT . Constant liquidity constraints Consider rst the case in which the liquidity constraint, as measured by the value of the parameter (t), is set to be constant, (t) = , for all 0 < t T . As a result, and because Equation (1.16) is now a function of W (t), N(t), and S(t), the problem of our CRRA fund manager, who decides on an initial allocation of capital among m risky assets, !(0), and a money market account, 1 > !(0), as well as on the subsequent revisions of that initial portfolio, as denoted by the m 1 vector of continuous controls (t), is stated as: J(W;N;S;t) = sup !(0);(t) E t [W (T )(T )] 1 1 (1.17) subject to the budget constraint (1.16). Under regularity conditions on the value function, absent benchmarking incentives,(T ) = 1, and suppressing time indicators, the HJB PDE for this problem is given by: J t +J W rW +J W h N > diag[S] > s i + 1 2 J WW h N > A 4 N i + +J S h diag[S] r + > s i + 1 2 tr [J SS A 4 ] +J WS [A 4 N] + sup n > J N o = 0 (1.18) with terminal condition J(W;N;S;T ) = [1=(1 )][W (T )(T )] 1 , where J S and J WS are 1m vectors, J N is an m 1 vector, J SS is an mm matrix, and A 4 = diag[S] > s s diag[S]. In this case, because (t) is constrained, the HJB PDE is 16 optimized by choosing (t) so as to maximize the term > J N for a given initial portfolio vector!(0). Therefore, the constrained money manager follows a bang-bang control policy, according to which she chooses either = if J N > 0, or = if J N < 0, or = 0 ifJ N = 0, as long as it is guaranteed thatW (t)> 0, and the trading strategies are admissible, meaning N 0 (t) 0, N(t) 0, and N 0 (t) + > N(t) > 0, for all 0 t T . Absent benchmarking incentives, the formal solution to the fund manager's derived utility is given by: J(W;N;S;t;!(0)) = W (t) 1 1 E t exp Z T t A 5 (u)du + Z T t A 6 (u)dZ(u) (1.19) whereA 5 (t) =r(1 ) +A 6 (t) ( (1=2) s !(t)), andA 6 (t) = (1 )!(t) > > s . Note that the portfolio weight vector !(t) enters the derived utility function both linearly and quadratically, which means that liquidity restrictions introduce a second layer of mean-variance analysis into the manager's problem. This problem coincides with the problem of a liquidity-constrained investor, if we also leave out the fund manager's explicit incentives. In Section 1.3, I use numerical techniques to solve this optimization problem, as well as the problem that accounts for the manager's benchmarking incentives, as discussed in Section 1.2.2. In particular, I use the methodology suggested in Longsta (2001). It consists of an application of the Least-Squares Monte Carlo algorithm, proposed by Longsta and Schwartz (2001). Succinctly, it involves replacing the conditional expectation function in (1.19) by its orthogonal projection on the space generated by a nite set of basis functions of the values of the state variables that are part of the manager's problem. Next, from that explicit functional approximation, we can then solve for the optimal control variable (t), as dened above, for any given !(0). Portfolio weights held in risky assets are then easily retrieved, for each time t, for 0tT , from the relation: ! j (t) =! j (0) + Z t 0 S j (u) W (u) j (u)du (1.20) 17 where j (0) = 0, and j2f1; 2;:::;mg. Time-varying liquidity constraints Existing literature provides evidence of liquidity timing by mutual fund managers (Cao, Simin, and Wang (2007)), evidence that portfolio liquidity is actively managed and is chosen as a function of the multiple liquidity needs of a fund (Massa and Phalippou (2005)), and also evidence that fund managers tilt their holdings more heavily towards liquid stocks when the market is expected to be more volatile (Huang (2008)). Moreover, according to Acharya and Pedersen (2005), liquid funds are likely to overperform in illiquid periods, and to underperform during liquid periods. What's more, stock market downturns have been showing that liquidity has a way of suddenly drying up when it is needed the most, has commonality across assets (Chordia, Roll, and Subrahmanyam (2000)), is related to volatility, and co-moves with the market. In order to capture some of these aspects of market liquidity, including the \ ight- to-quality" incentive induced by the liquidity uncertainty in the risky assets' market, consider the following mean-reversion process: d(t) =K((t))dt + > dZ(t) (1.21) where (t) and , are m 1 vectors, K is an m m diagonal matrix of speed of reversion parameters, and is a dm matrix of loadings on the sources of uncertainty, generated by the same d-dimensional geometric Brownian motion used in previous sections, Z(t). I then allow the fund manager to be able to hedge against liquidity risk. Note, however, that we need the stochastic liquidity parameter (t), as dened above, to assume only positive values, and to have a long-run equilibrium 18 level denoted by . In this case, an adequate mean-reverting model for (t) can be represented by: (t) = exp (t) 1 2 1 2 expf2KtgK 1 D (1.22) where (t) is an m 1 vector, K 1 is the inverse matrix of K, and D is an m 1 vector that contains the diagonal of the matrix > . The long-run equilibrium level for (t) is related to the equilibrium level for (t), and is given by the relation = expfg. Here, like in Section 1.2.3, the money manager's optimal portfolio strategy is to trade as much as possible, whenever possible. 1.3 Numerical results and discussion Let the investor and the fund manager trade in two risky assets (m = 2), which may have distinct degrees of liquidity. We could think of one of those risky assets to be a liquid well-known publicly traded large-cap stock, while the other would be, for instance, real estate or a small-cap stock from an emerging economy. I addition, let uncertainty be generated by a two-dimensional Brownian motion (d = 2), such that the price dynamics for these two risky assets is to include the following 2 2 matrix of loadings on the sources of risk: s = 0 @ 11 21 12 22 1 A (1.23) where the term jk denotes the loading that assetj puts on the source of riskk. From (1.23) we get that the variance of assetj is given by 2 j = 2 j1 + 2 j2 and the correlation between the two assets is given by = ( 11 21 + 12 22 )=( 1 2 ). In addition, let the 21 vector of expected returns for this pair of risky assets be given by =r+ > s . The numerical results in this section are based on 100,000 runs and 20 time steps per 19 year. Investment horizons T i , for i2fI;Mg, are expressed in years. Initial prices of assets (risky and riskless) are set to unity, such that S j (0) = 1, where j2f0; 1; 2g. Furthermore, let the riskless money market account earn a constant riskless interest rater = 0:02. Bear in mind that, when looking at the results presented in the sections that follow, we should focus on their comparative statics and predictions, not on the reality of their assumptions. 1.3.1 Analysis of illiquidity and explicit incentives The fund manager has an explicit incentive to administer the investor's savings ac- cording to her own attitude towards risk, her (usually shorter) investment horizon, and her (eventually more favorable) participation and transaction costs. In this sec- tion, I assess the economic signicance of these explicit incentives, in particular the case in which the appetites for risk diverge between the investor and the manager, and to what extent the presence of liquidity constraints aects its likely outcome. The case of symmetric asset liquidity constraints First, consider the case where the two risky assets are identical in every respect, in- cluding their liquidity characteristics. Table 1.1 reports the optimal initial portfolio weights for the unconstrained, and the constrained liquidity cases, ! u j (0), and ! c j (0), respectively, for dierent values of T i , , j , and i , when in the absence of bench- marking incentives ((T ) = 1). Take the two risky assets to be identical in every respect, so we can isolate the eects of the misalignment of incentives between the investor and the fund manager. Therefore, I set both risky assets to have the same expected return, j = 0:10, and to be non-tradeable ( j = 0), for t> 0, throughout the periodT i . In terms of comparative statics, these results show that, under liquidity constraints, there is \ ight-to-quality" through which either the investor or the fund manager choose to hold fewer of the riskier assets and more of the safest one. 20 Table 1.1: Optimal buy-and-hold policies without incentives. Optimal buy-and-hold (j = 0) investment policies and illiquidity discounts, with no in u- ence of either explicit or implicit incentives. Illiquidity discount (ID) is dened as the num- ber of basis points the price of the identical risky assets would have to be reduced in order to make the investor/manager indierent between holding the liquidity-constrained (! c j (0)) and the liquidity-unconstrained (! u j (0)) portfolios. CSVj denotes the cross-sectional volatility of the simulated constrained portfolio weights, at the terminal date, ! c j (T ), for j 2f1; 2g. Parame- ters j , , and RA( i), denote the return volatility of the risky assets, the correlation coe- cient between the risky assets' returns, and the coecient of relative risk aversion, respectively. Panel A: Constrained liquidity, j = 0, Ti=1 j = 0:3 j = 0:5 RA( i) ! u j (0) ! c j (0) CSVj ID(bp) ! u j (0) ! c j (0) CSVj ID(bp) -0.5 1 1.778 0.500 0.1197 704.03 0.640 0.500 0.1838 83.99 2 0.889 0.500 0.1197 149.17 0.320 0.268 0.1111 44.48 5 0.356 0.328 0.0833 20.29 0.128 0.103 0.0499 19.34 10 0.178 0.165 0.0464 10.71 0.064 0.050 0.0261 9.99 0 1 0.889 0.500 0.0996 136.55 0.320 0.305 0.1075 12.82 2 0.444 0.440 0.0891 5.73 0.160 0.153 0.0649 8.93 5 0.178 0.175 0.0444 3.54 0.064 0.060 0.0298 4.39 10 0.089 0.085 0.0243 2.15 0.032 0.030 0.0158 2.40 0.5 1 0.593 0.500 0.0718 11.70 0.213 0.210 0.0713 5.76 2 0.296 0.293 0.0527 1.61 0.107 0.100 0.0430 4.92 5 0.119 0.115 0.0291 1.92 0.043 0.040 0.0199 2.61 10 0.059 0.058 0.0164 1.23 0.021 0.020 0.0105 1.41 Panel B: Constrained liquidity, j = 0, Ti=2 j = 0:3 j = 0:5 RA( i) ! u j (0) ! c j (0) CSVj ID(bp) ! u j (0) ! c j (0) CSVj ID(bp) -0.5 1 1.778 0.500 0.1620 1,380.80 0.640 0.473 0.2258 285.84 2 0.889 0.500 0.1620 336.82 0.320 0.230 0.1351 158.06 5 0.356 0.313 0.1124 76.34 0.128 0.088 0.0645 68.84 10 0.178 0.155 0.0645 40.49 0.064 0.043 0.0347 35.52 0 1 0.889 0.500 0.1364 286.80 0.320 0.293 0.1419 63.84 2 0.444 0.430 0.1209 29.70 0.160 0.140 0.0873 41.65 5 0.178 0.170 0.0634 16.78 0.064 0.053 0.0405 19.67 10 0.089 0.083 0.0356 9.80 0.032 0.025 0.0210 10.43 0.5 1 0.593 0.500 0.0999 29.83 0.213 0.205 0.0984 28.23 2 0.296 0.298 0.0754 9.41 0.107 0.095 0.0605 21.92 5 0.119 0.115 0.0431 8.38 0.043 0.038 0.0292 10.96 10 0.059 0.058 0.0250 5.24 0.021 0.018 0.0149 5.86 This lower total initial risk exposure is due to hedging demands against the port- folio weights uncertainty measured by CSV j , at t = T i , which is the cross-sectional variation of the simulated portfolio weights for each asset j and represents the extent to which portfolio weights are out of the control of market participants when liquidity 21 is constrained. As a result, under liquidity constraints, market participants need to care about not only the mean and variance of the risky assets, but also the mean and variance of the portfolio weights, which are out of their complete control. For the set of input parameters I consider, note that CSV j increases with j for indepen- dent and negatively correlated risky assets, while it decreases for positively correlated risky assets. Not unexpectedly, constrained portfolio weights' variation increases with the investment horizon. These simulated variations range from .0105 (for = 0:5, i = 10, j = 0:5, and T i = 1), to .2258 (for = 0:5, i = 1, j = 0:5, and T i = 1), while they are all null for the unconstrained portfolio weights, by construction. To trade-o the expected value of ! c j (t) with its variation is also part of the problem of a liquidity constrained market participant. Note also that, when the unconstrained investor holds a leveraged position, the constrained one restricts her portfolio weights to lie in the interval 0 ! c 1 (0) +! c 2 (0) 1, to prevent against insolvency. Because the wealth of the investor or the value of the fund's assets are processes bounded below by zero, and given that, under liquidity constraints, returns of each risky asset cannot be replicated by those of the remaining assets in the portfolio, endogenously the market participants cannot borrow to invest more in risky assets, because the price of the illiquid assets can quickly fall and it would not be possible to sell enough of these assets to guarantee positive terminal wealth. This one dimension of liquidity is generally referred to as funding liquidity, which has to do with the availability of credit or the ease with which the investor and the manager an borrow or take on leverage. Note that this funding illiquidity was generated endogenously in this model by the market illiquidity, which is the ease with which market participants can trans- act, or the ability of markets to absorb large purchases or sales without much eect on prices. Lastly, Table 1.1 also reports the number of basis points we would have to discount the prices of the identical risky assets so as to make the investor or the fund manager indierent between holding the constrained portfolio, and the constrained one. These 22 illiquidity discounts can also be thought of as the extra premiums that one would require for holding an illiquid asset instead of a perfectly liquid one. For this set of parameters, illiquidity discounts (ID) increase with T i , and decrease with i , , and j . They range from 1.23 to 1,380.80 basis points. In particular, forT i = 2, j = 0:3, = 0, and i = 1, the price of the identical risky assets would have to be discounted by 2.87% so as to compensate the investor for holding a buy-and-hold portfolio, instead of one he or she could rebalance without restrictions. Not surprisingly, the largest discounts occur when endogenous constraints on leverage are binding. Note also that, when the risky assets' volatility is decreased from j = 0:5 to j = 0:3, two opposite forces in uence the value of the discounts for the lack of liquidity. On the one hand, when assets' volatility decreases, unconstrained initial portfolio weights increase and borrowing constraints bind more quickly, which leads to increases in discounts in order to compensate for funding illiquidity. On the other hand, a decrease in assets' return volatility also makes it less likely that constrained portfolios will deviate from the unconstrained ones, which can lead to decreases in pricing discounts for illiquidity. The case of asymmetric asset liquidity constraints Consider now the case in which we relax the trading constraint on one of the assets. Let the maximum number of shares of asset 2, that can be traded per year, be 2 = 0:10, while keeping 1 = 0. Table 1.2 reports the optimal initial portfolio weights for the unconstrained, and the constrained liquidity cases,! u j (0), and! c j (0), respectively, forT i = 1, j = 0:5, and dierent values of, and i , where benchmarking incentives are still left out ((T ) = 1). Because the risky assets are not identical in terms of tradability anymore, in this table I show the optimal portfolio allocations, with and without constraints, separately for each of the risky assets. The main result of this table is that, when using two identical risky assets with dierent liquidity constraints, the investor and the manager's initial portfolios are likely to be skewed towards the more illiquid asset, which is to say! c 1 (0)>! c 2 (0), and that over the investment period, 23 these portfolio allocations are likely to shift weights towards the less illiquid asset, as we can see from the values of E[! c j (T )] in Table 1.2. Table 1.2: Optimal policies without implicit incentives and limited trading of asset 2. Optimal investment policies and costs of constant illiquidity, with no in uence of implicit in- centives (Y (t) = 1), when asset 1 is non-tradeable (1 = 0), and asset 2 has limited trading per year (2 = 0:1), j = 0:5, and Ti = 1. Illiquidity cost (IC) is dened as the amount of initial wealth (in basis points) that we would have to give the investor/manager in order to make her indierent between holding the liquidity constrained and the liquidity unconstrained portfolios. CSVW denotes the cross-sectional volatility of the simulated value of assets under management, under liquidity constraints, at the terminal date, W(T). The variable E[! c j (T )] de- notes the expected value of the constrained portfolio weight for asset j, at the terminal date. RA( i) IC(bp) CSVW Asset j ! u j (0) ! c j (0) E[! c j (T )] CSVj 1 84.70 0.3195 1 0.640 0.500 0.500 0.1838 2 0.640 0.500 0.500 0.1838 2 52.27 0.1660 1 0.320 0.276 0.279 0.1100 -0.5 2 0.320 0.184 0.287 0.1230 5 28.61 0.0671 1 0.128 0.105 0.110 0.0505 2 0.128 0.045 0.147 0.0693 10 23.63 0.0389 1 0.064 0.054 0.057 0.0278 2 0.064 0.006 0.106 0.0493 1 16.18 0.2583 1 0.320 0.285 0.284 0.1039 2 0.320 0.285 0.376 0.1271 2 12.45 0.1266 1 0.160 0.150 0.154 0.0641 0 2 0.160 0.100 0.200 0.0837 5 11.58 0.0508 1 0.064 0.059 0.062 0.0291 2 0.064 0.007 0.107 0.0501 10 17.70 0.0368 1 0.032 0.030 0.032 0.0157 2 0.032 0 0.097 0.0435 1 7.82 0.2132 1 0.213 0.222 0.222 0.0745 2 0.213 0.148 0.244 0.0842 2 8.30 0.1027 1 0.107 0.105 0.108 0.0450 0.5 2 0.107 0.045 0.145 0.0606 5 9.49 0.0479 1 0.043 0.040 0.042 0.0199 2 0.043 0 0.097 0.0440 10 19.88 0.0352 1 0.021 0.010 0.011 0.0053 2 0.021 0 0.098 0.0430 We know that if these market participants were able to trade as much of these risky assets as desired, they would choose to hold the unconstrained portfolio weights, as denoted by! u j (0), which are their most ecient allocations. However, under liquidity 24 constraints, they need to hedge against the possibility that, beyond time zero, they may not be able to hold those ecient portfolio weights and at the same time they will not want to deviate too much from them. One would expect that, after relaxing the liquidity constraint on asset 2 its initial portfolio weight could now be placed closer to the unconstrained weight, but that is not the case. There are hedging and diversication forces simultaneously at play here in order to create this apparently counter-intuitive eect. if these market participants were to be ale to trade only on one illiquid risky asset and one riskless asset, and the liquidity constraint on that one illiquid asset was to be partially relaxed, then we would nd its portfolio weight to get closer to its unconstrained weight. It is important to note, however, that in the one risky asset case, market participants have no diversication concerns. In the two risky assets case presented in this chapter diversication does matter. Asset 1 is not allowed to trade beyond time zero, while asset 2 can be traded after time zero, even if in limited amounts. The price of both assets has a positive drift, which moves the mass of probability to the upside of the price movement. Therefore, it becomes likely that after time zero the market participants would want to increase their holdings of both assets, but given that they are restricted from doing that for asset 1, it makes sense to hold more of asset 1 than of asset 2 to begin with, and to hold a portfolio weight on asset 1 that is as close as possible to that they would choose in an unconstrained context. Moreover, combining the facts that asset 2 is allowed to trade beyond time zero, the market participants are likely to want to increase their holdings on asset 2 given its positive drift and, at the same time, they need to diversify their portfolios in order to reduce their idiosyncratic risks as much as possible, then they optimally choose to allocate relatively smaller fractions of capital on asset 2 to start with, and to subsequently buildup those positions over the investment period, this way expecting to minimize the costs that could arise from poor diversication strategies. Another important aspect of this problem is that, under liquidity constraints, market participants have concerns about portfolio weights' uncertainty. When there are two 25 risky assets with dierent liquidity constraints, one should expect portfolio weight uncertainty to be higher for the more illiquid asset 1 than for asset 2, not only because of the price volatilities of assets 1 and 2, but also because every time one trades asset 2, that aects the portfolio weight of asset 1. So, one would expect the market participants to take advantage of their ability to trade asset 2 to somehow decrease the uncertainty of the portfolio weight on asset 1, and this way increase their overall derived utilities. Note that the ratio of the more liquid asset initial portfolio weight, ! c 2 (0), to the one for the less liquid asset, ! c 1 (0), decreases monotonically with i , meaning that, a more risk averse investor or money manager optimally chooses to hold relatively more of the less liquid asset. The ratio! c 2 (0)=! c 1 (0) can be thought of as a measure of the level of portfolio liquidity, and Table 1.2 shows that the optimal initial portfolio liquidity decreases, in general, with the correlation of assets' returns. It also shows that these initial optimal portfolios shift liquidity levels over the investment period. The variableE[! c j (T )] denotes the expected value of the constrained portfolio weights at time t =T i . Note that E[! c 2 (T )] is generally larger than ! c 2 (0) by, approximately, 2 = 0:10, the maximum number of shares of asset 2 that can be traded per year. Note also that the cross-sectional variation of ! c 2 (T ), given by CSV 2 , is generally larger than the one we discussed above for Table 1.1, where asset 2 could not be traded, while CSV 1 remains of the same kind as in Table 1.1. Thus, the constrained money manager, in trading o the expected value of! c 2 (T ) against its variation, does so by taking much smaller initial positions and by trading asset 2, so as to keep the non-tradeable asset 1's portfolio weights' expected value, and variation, under control. However, the optimal utility levels that an investor/manager attains when actively trading asset 2, turn out to be lower than those she would obtain under a passive buy- and-hold strategy as the one shown in Table 1.1. Table 1.2 shows the variable CSV W , which denotes the cross-sectional variation of the value of assets under management 26 at t =T i . Generally, these variations of W (T ) are larger when trading for asset 2 is allowed than when it is not. I cannot directly assign to each of these two assets the responsibility for the total cost of the illiquidity eect. Therefore, in Table 1.2 I report instead the total illiquidity cost (IC), which denotes the percent (measured in basis points) of the investor's (or, likewise, the manager's) initial wealth, that one would have to give her in order to compensate her for holding a liquidity constrained portfolio, instead of a portfolio that she could revise with no restrictions. Note that, as expected, IC increases when short-selling constraints bind. For brevity, I do not tabulate the simulation results for the case where, leaving everything else constant, I let T i = 2, or alternatively, I let j = 0:3. Succinctly, for longer investment horizons, initial portfolio weights decrease substantially, but illiquidity costs, and portfolio value variations, increase drastically. For instance, when = 0, and i = 10, then IC= 2:022%, CSV 2 = 0:1254. If, alternatively, I let j = 0:3, then illiquidity costs increase dramatically for i = 1, while they decrease for i > 1, which is the result of having endogenous borrowing constraints to bind. Dierently, when I keep all the same parameters used in Table 1.2 except that I also allow asset 1 to trade during the investment period, with 1 = 0:10, then the assets return to identical, as in Table 1.1, but optimal initial portfolio risk exposure decreases, while illiquidity discounts rise. The case of divergence in appetites for risk We observe the SEC regularly advising investors to carefully read fund prospectus and shareholder reports, to learn about its investment strategy and the risks it takes to achieve its returns, so that these risks can be factored in and be tested for consistency with the investor's nancial goals and risk tolerance. In fact, signicant misalignments of incentives can be derived from the extensive dierences in appetites for risk between investor and fund manager. Figure 1.1 reveals the shape of the shadow costs associated 27 with this particular explicit incentive, as a function of both the investor's and the money manager's risk appetites. 1 2 3 4 5 1 2 3 4 5 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 Investor risk aversion Manager risk aversion Shadow cost of portfolio delegation Figure 1.1: Shadow costs of explicit incentives. Shadow cost derived from explicit incentives only, due to dierences in risk appetites between the investor and the fund manager. These costs are measured as factors we would have to multiply the investor's initial wealth with, in order to compensate her for the eect of suboptimal policies derived from portfolio delegation. This is also the case of identical, independent, and non-tradeable (j = 0) risky assets, where r = 0:02, j = 0:10, and j = 0:5, for j2f1; 2g, and Ti = 1. Note that it is asymmetric, that the costs of delegation are the lowest when the manager's risk aversion parameter is equal to that of the investor, and the highest when the manager exhibits lower risk aversion than the investor. Table 1.3 reports these costs for investment horizons T i = 1, and where input parameter values are as those in Table 1.1. These costs are expressed as the percent of wealth the suboptimal 28 investor would be willing to give away in return for being allowed to follow the optimal strategy. Panel A reports the shadow costs for the unconstrained liquidity case. Panel B reveals the results for the case where liquidity is totally constrained ( j = 0). Not unexpectedly, when investor and manager have the same attitude towards risk, given implicit incentives are left out, the money manager is acting in the best interest of the investor and, as a result, no losses are derived from this delegated portfolio relationship. However, in case of divergence in risk attitudes, utility costs can become very signicant. These costs range from 0.08% to 250.33% of the investor's initial wealth, for the unconstrained liquidity case, while they range from 0% to 32.66% in the constrained liquidity case. Largest costs occur for higher values of I , and lower values of M , which is what we should nd, professional money managers to be much less risk averse than fund investors. What is interesting to see in these results is that utility costs become signicantly lower when in the presence of liquidity constraints. This has to do with the endogenous leveraging constraints, and hedging demands, which restrain portfolio weights to fall into a closed limited set of values, in order to prevent for bankruptcy. These costs signicantly increase with the investment horizon T i . When the investment horizon increases to 2 years, utility losses derived from the dierence in appetites for risk between the investor and the money manager, for the same parameter values as in Table 1.3, range from 0.17% (for I = 5, M = 10, = 0:5, and j = 0:5) to 709.27% (for I = 10, M = 1, =0:5, and j = 0:3), in the unconstrained liquidity case, while they range from 0% (for I = 1, M = 2, =0:5, and j = 0:3) to 59.59% (for I = 10, M = 1, =0:5, and j = 0:5), in the constrained liquidity case. For brevity, I do not tabulate either the results forT i = 2, or the results for 1 = 0 and 2 = 0:10. When I let the maximum number of shares of asset 2, that can be traded per year, be 2 = 0:10, then utility losses for a constrained investor whose risk appetite may not be consistent with that of the fund manager to whom she delegates the management of all her savings, are generally reduced. 29 Table 1.3: Shadow costs of explicit incentives. Shadow costs of explicit incentives derived from dierences in risk appetites between the investor and the money manager, where Ti = 1. Shadow cost is dened as the additional percentage of the investor's initial wealth that we would have to give her in order to make her indierent between delegating the administration of her savings to a money manager, and administering those savings herself. Using standard working practice, these utility losses are computed via taking the ratio of the annualized certainty equivalent rates of return, achieved under the in- vestor's portfolio delegated and centralized problems, after which I subtract one and multiply by 100 to express the losses in percent points per year. Parameter values are as in Table 1.1. Panel A: Unconstrained liquidity, Ti=1 I = 1 I = 2 I = 5 I = 10 j = 0:3 0:5 0:3 0:5 0:3 0:5 0:3 0:5 -0.5 0 0 7.78 2.72 64.60 18.40 250.33 55.13 M = 1 0 0 0 3.67 1.30 26.20 8.54 82.92 22.96 0.5 0 0 2.41 0.86 16.47 5.57 46.52 14.54 -0.5 3.51 1.22 0 0 6.79 2.41 27.12 8.71 M = 2 0 1.77 0.63 0 0 3.26 1.17 12.12 4.17 0.5 1.19 0.42 0 0 2.15 0.77 7.83 2.74 -0.5 9.38 3.24 2.55 0.89 0 0 1.46 0.54 M = 5 0 4.63 1.63 1.28 0.45 0 0 0.72 0.26 0.5 3.07 1.09 0.86 0.31 0 0 0.47 0.17 -0.5 12.05 4.14 4.61 1.61 0.70 0.24 0 0 M = 10 0 5.90 2.08 2.29 0.81 0.35 0.12 0 0 0.5 3.91 1.39 1.53 0.54 0.24 0.08 0 0 Panel B: Constrained liquidity, j = 0, Ti=1 I = 1 I = 2 I = 5 I = 10 j = 0:3 0:5 0:3 0:5 0:3 0:5 0:3 0:5 -0.5 0 0 0 1.62 0.69 11.70 5.50 32.66 M = 1 0 0 0 0.07 1.17 4.96 7.06 16.95 16.28 0.5 0 0 1.13 0.81 10.37 4.62 29.93 10.12 -0.5 0 1.00 0 0 0.69 1.80 5.50 5.82 M = 2 0 0.44 0.59 0 0 3.19 1.03 11.46 3.24 0.5 1.09 0.43 0 0 1.93 0.61 6.59 1.97 -0.5 2.04 2.71 1.35 0.74 0 0 1.21 0.35 M = 5 0 3.20 1.54 1.24 0.41 0 0 0.66 0.23 0.5 2.97 1.06 0.86 0.28 0 0 0.40 0.15 -0.5 4.31 3.46 3.19 1.34 0.04 0.20 0 0 M = 10 0 4.48 1.96 2.26 0.74 0.35 0.10 0 0 0.5 3.79 1.34 1.51 0.51 0.23 0.07 0 0 They range from 0:04% to 32:35%, when j = 0:5. For instance, when M = 10, I = 2, = 0, and j = 0:5, and T i = 1, the utility loss is equal to 0:63%, instead of 30 0:74%. These utility costs increase for longer investment horizons and for lower asset return volatilities, everything else constant. Implications of time-varying liquidity constraints Furthermore, I examine the implications of allowing marketability bounds to follow a stochastic process like the one described by expression (1.22), in Section 1.2.3. Table 1.4 reports optimal initial investment policies that the investor, and the fund manager alike, would choose, absent benchmarking incentives ((t) = 1), for the case where asset 2 is allowed to trade throughout the year, and the following parameter values: volatility of the marketability bound for asset 2, 2 = 0:2, speed of reversion 2 = 0:1 (where h denotes theh th element of the diagonal matrixK), initial value 2 (0) = 0:1 (annualized), long-run equilibrium levels 2 = 0:1 (annualized), in Panel A, 2 = 0:2, in Panel B, j = 0:5, andT i = 1. Succinctly, these results suggest that illiquidity costs slightly increase when we allow marketability to be stochastic, and increase more for larger long-run equilibrium levels of liquidity. In addition, cross-sectional variations for! c 2 (T ) rise, while they remain roughly level for! c 1 (T ). Furthermore, Table 1.4 also reports estimates for the parameter j2 , which denotes the simulated average time- series correlation coecients between ! c j (t) and the stochastic 2 (t), for j2f1; 2g. The values for these coecients suggest that swings in asset 2's marketability are directly accompanied by ! c j (t), progressively more closely the more risk tolerant is the investor/manager, and the larger the equilibrium level . Finally, on the whole, shadow costs of explicit incentives, under stochastic liquidity, generally decrease. 1.3.2 Analysis of illiquidity and implicit incentives In this section, I investigate the case in which the fund manager derives her utility not just from the value of assets under management, but the ratio of this to the value of an exogenous benchmark. Assume that both investor and fund manager guide their 31 portfolio allocations by similar risk appetites, investment horizons, and participation constraints. Table 1.4: Optimal policies and costs of time-varying illiquidity. Optimal investment policies and costs of time-varying illiquidity, with no in uence of implicit incentives (Y (t) = 1), for independent risky assets ( = 0), where asset 1 is non-tradeable (1(t) = 0), and asset 2 has limited time-varying trading per year, for a liquidity volatility parameter value of 2 = 0:2, speed of reversionK2 = 0:1, initial level 2(0) = 0:1 (annualized), long-run equilibrium level2 = 0:1 (annualized),j = 0:5, andTi = 1. The parameterj2 denotes the simulated average time-series correlation between ! c j (t) and the stochastic liquidity boundary 2(t), for j2f1; 2g. Panel A: Stochastic constrained liquidity, 2 = 0:1 RA( i) IC(bp) CSVW Asset j ! u j (0) ! c j (0) j2 E[! c j (T )] CSVj 1 17.00 0.2580 1 0.320 0.280 -0.3646 0.282 0.1033 2 0.320 0.280 0.8221 0.377 0.1340 2 13.59 0.1283 1 0.160 0.150 -0.1551 0.154 0.0641 2 0.160 0.100 0.6649 0.205 0.0930 5 13.46 0.0508 1 0.064 0.054 -0.0556 0.057 0.0271 2 0.064 0.006 0.2548 0.111 0.0603 10 20.65 0.0386 1 0.032 0.030 -0.0476 0.032 0.0157 2 0.032 0 0.2276 0.102 0.0525 Panel B: Stochastic constrained liquidity, 2 = 0:2 RA( i) IC(bp) CSVW Asset j ! u j (0) ! c j (0) j2 E[! c j (T )] CSVj 1 17.18 0.2586 1 0.320 0.280 -0.3452 0.282 0.1034 2 0.320 0.280 0.9035 0.380 0.1352 2 13.89 0.1289 1 0.160 0.150 -0.1554 0.154 0.0641 2 0.160 0.100 0.8528 0.208 0.0946 5 13.99 0.0516 1 0.064 0.054 -0.0645 0.057 0.0271 2 0.064 0.006 0.5012 0.115 0.0622 10 22.06 0.0394 1 0.032 0.030 -0.0572 0.032 0.0157 2 0.032 0 0.4760 0.105 0.0543 Implications of benchmarking for buy-and-hold policies Table 1.5 reports optimal initial portfolio weights a money manager would choose in case her fund's performance was to be measured against the performance of a continuously rebalanced benchmark portfolio, Y (t), for j (t) = 0:5, with a ow-to- performance function (T ) = 1=Y (T ) (e.g. Browne (1999), Binsbergen, Brandt, and 32 Koijen (2008)), where risky assets are set to be identical in all respects, independent ( = 0), and totally illiquid ( j = 0). I assume the benchmark portfolio weights are equal for both of the risky assets so that the manager has equal incentives to load her optimal allocations on the assets that are in the benchmark. One may ask whether or not it is reasonable to assume that a performance benchmark can be composed of illiquid assets. This assumption could be justied by the ex-ante fear of fund managers about market crashes. In other words, one can assume ex-ante that some of the assets in a benchmark can become illiquid, like nancials and real estate during the current subprime crisis, or the tech stocks in the internet crash. Panel A shows the results for the case where T M = 1, while Panel B shows the results for T M = 2. Figure 1.2 illustrates how the shape of the money manager's derived utility function looks like, for a particular set of parameters, and for (T ) = 1=Y (T ). Table 1.5: Optimal policies and illiquidity discounts. Optimal investment policies and illiquidity discounts, under the in uence of benchmarking in- centives ((T ) = 1=Y (T )), for a continuously rebalancing benchmark with j = 0:5, and independent ( = 0) non-tradeable (j = 0) risky assets. Illiquidity discounts are dened as the number of basis points the price of the identical risky assets would have to be re- duced in order to make the manager indierent between holding the liquidity constrained and the liquidity unconstrained portfolios. CSVj denotes the cross-sectional volatility of the simulated constrained portfolio weights, at the terminal date, ! c j (T ), for j 2 f1; 2g. Panel A: Rebalancing Benchmark with j = 0:5, j = 0, TM =1, = 0 (CSVY =0.404) RA( M ) ! u j (0) ! c j (0) CSVj TE ID(bp) CSVW TEY P[W<Y] 1 0.320 0.305 0.1075 19.71 12.82 0.2543 3.09 0.5173 2 0.410 0.395 0.1289 12.60 39.27 0.3294 2.03 0.5344 5 0.464 0.450 0.1424 10.07 108.95 0.3752 1.56 0.5864 Panel B: Rebalancing Benchmark with j = 0:5, j = 0, TM =2, = 0 (CSVY =0.655) RA( M ) ! u j (0) ! c j (0) CSVj TE ID(bp) CSVW TEY P[W<Y] 1 0.320 0.290 0.1412 22.01 63.84 0.4049 3.55 0.5264 2 0.410 0.385 0.1685 15.42 156.63 0.5375 2.54 0.5503 5 0.464 0.445 0.1862 13.40 360.01 0.6213 2.12 0.6099 Optimal initial portfolio weights are, therefore, those that attain the maximum of this function. When compared to the results in Table 1.1, Table 1.5 shows that 33 the money manager's optimal initial portfolio policies have now an actively man- aged component, and an herd component. It just conrms the result of Equation (1.13), in Section 1.2.2, for the case where (T ) = 1=Y (T ). If we take the dierence between the optimal initial portfolio weights of Table 1.5, and those in Table 1.1, and then divide these dierences by the latter values, we get a potential measure of herding demands. From these computations, it becomes clear that, generally, for (T ) = 1=Y (T ), herding incentives become slightly larger for the constrained liquid- ity case, when compared to the liquidity unconstrained one. These results show that a more and more risk averse money manager tends to choose portfolio allocations that converge more and more to the benchmark portfolio weights when (T ) = 1=Y (T ). For longer investment horizons, these results get amplied. Table 1.5 also shows that cross-sectional variations of the portfolio weights, CSV j , increase in the presence of benchmarking. It also shows the simulated cross-sectional variations for the value of assets under management, and the benchmark portfolio ( j (t) = 0:5), at the horizonT M , which are denoted by CSV W and CSV Y , respectively. These results show that, generally, CSV W < CSV Y , given that the benchmark is continuously rebalancing in order to preserve j (t) = 0:5, while the risky assets in the manager's portfolio are totally illiquid, which makes it a buy-and-hold portfolio. Nevertheless, the tracking error of Y , TE Y , is relatively small. As a tracking error measure, I use the square root of the non-central second moment of the deviations between the money manager's portfolio and benchmark returns, which is the measure that is frequently used in practice. The tracking error for j (t), denoted by TE , appears to be signicantly larger than TE Y and decreasing with M . As a result, illiquidity discounts (ID) for our identical illiquid risky assets, come out dramatically larger when compared to the results in Table 1.1, and which get amplied for longer investment horizons. Finally, Table 1.5 also reports the simulated probabilities that the money manager's optimal portfolio values end up below that of the benchmark, by the terminal date, T M , for a given optimal control ! c j (0), which are denoted by 34 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 −1.14 −1.12 −1.1 −1.08 −1.06 −1.04 −1.02 −1 −0.98 Initial portfolio risk exposure Initial holdings of risky asset 1 Derived utility Figure 1.2: Derived utility of terminal wealth function. Derived utility of terminal wealth function, for a CRRA manager, with M = 2, whose performance is measured relative to a benchmark, with j = 0:5, and where the ow-to-performance specication is given by: (T ) = 1=Y (T ). This manager chooses, at time t = 0, to hold !1(0) and !2(0), on risky assets 1 and 2, respectively, and these initial allocations cannot be revised for t> 0 (j = 0). The values on the axis for initial holdings of risky asset 1 are fractions of the initial total portfolio risky exposure (!1(0) +!2(0)). These are identical and independent risky assets, where r = 0:02, j = 0:10, and j = 0:5, for j2f1; 2g. P[W < Y ]. Bear in mind that we normalized W (0) = Y (0) = 1. For brevity, I do not tabulate the results I obtain for j (t) = 0:2, in which case the benchmark has a money market exposure of 0 = 0:6. Obviously, in this case, the cross-sectional variations of Y (T ) become signicantly lower, where CSV Y = 0:151 for T M = 1, 35 and CSV Y = 0:225 for T M = 2. Moreover, the probabilities of under-performing the benchmark rise, despite the reductions in TE Y , and TE . Table 1.6 shows the results for the case in which the benchmark is value-weighted or buy-and-hold. It shows that hedging demands for illiquidity, as measured by the dierence between ! u (0) and ! c (0), are smaller for the case in which the benchmark is (value-weighted) buy-and-hold, when compared to those in which the benchmark is (equally-weighted) continuously rebalanced. Table 1.6: Illiquidity discounts and implicit incentives. Optimal investment policies and illiquidity discounts, under the in uence of benchmark- ing incentives, where (T ) = 1=Y (T ), for a buy-and-hold benchmark with j = 0:5, and independent ( = 0) non-tradeable (j = 0) risky assets. Tracking errors (TE) are measured as the square root of the non-central second moment of the deviations be- tween the money manager's portfolio weights/returns and the benchmark weights/returns. The simulated probabilities that the money manager's optimal portfolio values end up below those of the benchmark, by the terminal date, are represented by P[W<Y]. Buy-and-Hold Benchmark with i = 0:5, i = 0, TM =1, = 0 (CSVY =0.4169) RA( M ) ! u i (0) ! c i (0) CSVu CSVc TE ID(bp) CSVW TEY P[W<Y] 1 0.320 0.305 0.1021 0.1075 19.27 2.24 0.2543 2.97 0.5147 2 0.410 0.405 0.1309 0.1313 9.50 2.05 0.3377 1.46 0.5147 5 0.464 0.465 0.1481 0.1464 3.54 1.13 0.3878 0.55 0.5146 10 0.482 0.480 0.1538 0.1505 2.03 0.63 0.4003 0.31 0.5146 Compared to Table 1.5, note that in Table 1.6 one extra column was added which includes information on CSV u , the cross-sectional variation of the optimal uncon- strained weights, which was null for a rebalanced benchmark, but is now stochastic when using a buy-and-hold benchmark. Note that, when using a buy-and-hold bench- mark, the tracking errors for j andY decrease signicantly, as well as the value of the discounts for the lack of liquidity, ID(bp), which almost vanish for the case of a very risk averse fund manager. Note that the likelihood that the manager's portfolio will under-perform, by the time horizon t =T , a buy-and-hold benchmark, is in general lower than that for the case in which the benchmark is continuously rebalanced. 36 Shadow costs of benchmarking and symmetric liquidity constraints Table 1.7 reports the results for the shadow costs of benchmarking incentives, for the case of a continuously rebalanced benchmark, measured in percent points of the investor's initial wealth, for the case of identical ( j = 0:5, j = 0:1), independent, and non-tradeable risky assets ( = 0, j = 0). Figure 1.3 reveals the shape of this shadow cost function, for a particular set of input parameters. Shadow costs range from 0% to 188.76% in the unconstrained liquidity case, while ranging from 0% to 136.21% for the constrained liquidity one. For i = 10, a liquidity constrained investor, with an investment horizon of 2 years, and j = 0:5, requires 136.21% extra initial wealth in order to be indierent between delegating the management of her savings to the professional money manager and directly accessing the nancial markets to do it herself. I also investigate whether a constrained professional money manager adjusts her portfolio riskiness through taking on unsystematic rather than systematic risk. Thus, I let 1 = r = 0:02, so that asset 1 does not command any risk premium, and set 12 = 0, so that asset 1 is solely driven by the source of risk Z 1 (t). Moreover, let 1 = 0 and 2 = 1, such that the benchmark portfolio is given by asset 2 alone, and is solely driven by the source of risk Z 2 (t), so that 21 = 0, which also implies that = 0. The results obtained under this setup show that, the money manager optimally chooses to hold asset 2 only, in her portfolio, which means that risk-taking happens only through systematic risk. This is the case either when both risky assets are non-tradeable ( j = 0), or when we let asset 2 trade during the year ( 1 = 0 and 2 = 0:20). 37 Table 1.7: Shadow costs of implicit incentives and buy-and-hold policies. Shadow costs to the investor of benchmarking incentives ((T ) = 1=Y (T )), when the benchmark is either continuously rebalanced (Panel B) or buy-and-hold (Panel C), the risky assets are indepen- dent ( = 0) and non-tradeable (j = 0). Shadow cost is dened as the additional percentage of the investor's initial wealth that we would have to give her in order to make her indierent between delegating the administration of her savings to a money manager, and administering those sav- ings herself. These shadows costs are measured for dierent investment horizons Ti, and dierent benchmark portfolio weightsj . Using standard working practice, these utility losses are computed via taking the ratio of the annualized certainty equivalent rates of return, achieved under the in- vestor's portfolio delegated and centralized problems, after which I subtract one and multiply by 100 to express the losses in percent points per year. Parameter values are as in Table 1.1. Panel A: Unconstrained liquidity j = 0:2 j = 0:5 RA( i) Ti=1 2 1 2 1 0 0 0 0 2 0:51 1:02 3:20 6:50 5 3:26 6:64 22:25 50:31 10 8:39 17:78 67:11 188:76 Panel B: Rebalanced bench., j = 0 j = 0:2 j = 0:5 RA( i) Ti=1 2 1 2 1 0 0 0 0 2 0:44 0:85 3:00 5:82 5 2:81 4:82 20:32 41:28 10 6:44 10:71 58:25 136:21 Panel C: Buy-and-hold bench., j = 0 j = 0:2 j = 0:5 RA( i) Ti=1 2 1 2 1 0 0 0 0 2 0:49 1:04 3:25 6:62 5 2:99 5:80 22:46 46:84 10 6:81 11:94 63:86 160:21 Implications of benchmarking and asymmetric liquidity constraints Consider now the case where we let the maximum number of shares of asset 2, that can be traded per year, be non-zero. In particular, let 2 = 0:20. Table 1.8 shows that, not unexpectedly, in this case the optimal initial portfolio risk exposure (the fraction of assets under management invested in risky assets) declines, when compared to the results in Table 1.5. As the risk aversion parameter of the manager increases, 38 1 2 3 4 5 1 2 3 4 5 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Investor risk aversion Manager risk aversion Shadow cost of portfolio delegation Figure 1.3: Shadow costs of explicit and implicit incentives due to portfolio delegation. Shadow cost (vertical axis) derived from explicit and benchmarking (implicit) incentives, together, due to portfolio delegation, as a function of the risk aversion parameters for the investor and the fund manager (horizontal axes). This is the case of a manager whose performance is measured relative to a benchmark Y (t), with j = 0:5, and where the fund ow-to-performance specication is given by: (T ) = 1=Y (T ). These costs are measured as factors we would have to multiply the investor's initial wealth with, in order to compensate her for the eect of suboptimal policies derived from portfolio delegation. This is also the case of identical, independent, and non-tradeable (j = 0) risky assets, where r = 0:02, j = 0:10, and j = 0:5, for j2f1; 2g, and Ti = 1. her incentives to herd and to track the exogenous benchmark also increase, and both constrained and unconstrained optimal portfolio allocations tend to get closer to the benchmark portfolio weights. One of the main results of this table is that, compared to the direct investing case, here the illiquidity costs are higher, and increase with M . because the closer the manager wants to track the benchmark, the more she is going to need to have less illiquid assets to do so. Therefore, the costs of not being 39 able to trade and track the benchmark necessarily increase. In other words, it is more costly to the manager to have illiquid assets when she needs more desperately to track a benchmark that includes those assets. The parameter CV W denotes the cross-sectional variation of the simulated values of the assets under management, which increase with the risk aversion parameter M because the manager's portfolio is tracking a benchmark that is riskier that the Merton myopic portfolio allocations. Note that now it becomes more likely that the money manager will underperform the benchmark, as given by the parameter P[W <Y ]. Table 1.8: Shadow costs of implicit incentives and limited trading of asset 2. Optimal investment policies and costs of constant illiquidity, under the in uence of benchmarking incentives ((T ) = 1=Y (T )), for a continuously rebalanced benchmark with j = 0:5, when asset 1 is non-tradeable (1 = 0), and asset 2 has limited trading per year (2 = 0:2), j = 0:5, and Ti = 1. Illiquidity cost (IC) is dened as the additional amount of initial wealth (in basis points) that we would have to give the fund manager in order to make her indierent between holding the liquidity constrained and the liquidity unconstrained portfolios. CSVY denotes the cross-sectional volatility of the simulated value of the rebalanced benchmark. These panels consider the case of independent risky assets ( = 0), which results in CSVY = 0:4040. Each panel also includes values for the parameters and , which denote the simulated time-series correlation coecients of the portfolio liquidity (ratio of ! c 2 (t) to ! c 1 (t)), and the portfolio risk exposure (! c 2 (t) +! c 1 (t)), respec- tively, with the ratio of the assets under management to the benchmark portfolio (W(t)/Y(t)). M IC(bp) CSVW TEY P[W<Y] j ! u j (0) ! c j (0) E[! c j (T )] CSVj TE 1 19.32 0.2572 3.22 0.5215 1 0.320 0.306 0.306 0.1073 19.63 2 0.320 0.204 0.395 0.1409 21.57 2 54.18 0.3318 2.21 0.5367 1 0.410 0.414 0.411 0.1293 11.29 2 0.410 0.276 0.465 0.1575 16.39 5 138.21 0.4061 1.51 0.6476 1 0.464 0.455 0.457 0.1514 10.28 2 0.464 0.455 0.539 0.1529 10.28 M = 1 2 5 10 -0.0660 -0.1923 0.1626 0.1376 -0.3804 -0.2620 -0.2571 -0.2507 This result seems to relate with the empirical evidence that actively managed funds have, on average, an inferior performance than that of index funds (e.g. Gruber (1996)). These probabilities increase with the risk aversion parameter, and that may be the case because the more risk averse the manager, the higher are her incentives to track the benchmark very closely and, because of illiquidity, the harder to do so. 40 Compared to Table 1.5, here in Table 1.8 the cross-sectional variations of ! c 2 (T ) slightly increase, while CSV W slightly decreases. Furthermore, illiquidity costs sig- nicantly increase when we allow asset 2 to trade, even for limited amounts, during the year. Another important result of this table is that denoted by parameter TE , which denotes the tracking error of the manager's portfolio weights, given by ! c j (t), with respect to the benchmark weights given by j . According to these results, the manager's portfolio weights on the less illiquid asset 2 appear to be likely to tilt away from the benchmark weights more than the portfolio weights of more illiquid assets. These tracking errors can be viewed as a measure of risk-shifting by the fund man- ager. Therefore, we can conclude from these results that the manager's risk-shifting is more likely to happen using less illiquid assets. I also include in this Table 1.8 estimates of the parameters and, which denote the simulated time-series correlation coecients of the portfolio liquidity (ratio of ! c 2 (t) to ! c 1 (t)), and the portfolio risk exposure (! c 2 (t) +! c 1 (t)), respectively, with respect to the ratio of the assets under management to the benchmark portfolio (W(t)/Y(t)). Note that switches from a negative gure to a positive one, as the money manager becomes more conservative. A negative means that, when the performance of the manager's portfolio deteriorates relative to the benchmark, then on average, contemporaneously, the manager optimally chooses to distort her portfolio composition towards the more liquid risky asset, which occurs for the more risk loving managers. These results would be consistent with risk-shifting behavior to take place in the more liquid asset class. Note, however, that these correlations don't appear specially strong. The negative signs of all over Table 1.8 suggest that increasing fund distress implies escalating risk exposure. Shadow costs of benchmarking incentives to a liquidity constrained investor, when asset 2 is tradeable, j = 0:20, range from 0% to 25.87% of her initial wealth. 41 1.4 Conclusion This study was limited in several ways. For ease of exposition, I conne attention to CRRA preferences, continuously rebalanced benchmarks, and totally passive in- vestors. Nevertheless, it suggests that asset illiquidity can signicantly aect money managers' risk-shifting incentives as well as the investors' utility costs of misaligned objectives. The main results of this chapter also suggest that the value of liquidity is likely to increase with the rise of nancial intermediation, and that we should ex- pect the investor's propensity to delegate portfolio decisions to a fund manager to be higher in the presence of illiquid assets and in more illiquid periods. It would be useful in future research to include a more complicated preference structure in this analysis, and allowing the investor to endogenously decide how much of her savings to hold in the money market account and how much to invest in the managed portfolio, which would allow the investor to trade around incentive misalign- ments and improve her welfare results. Other interesting extensions could focus on deriving optimal performance benchmarks that would account for asset illiquidity and could then be used to better align incentives between investors and money managers in more illiquid markets. Future work could also focus on studying the implications of time-varying investment opportunities and their interaction with asset illiquidity, as well as on the sensitivity of the results presented here to alternative measures of asset liquidity. Examples of those alternative measures of liquidity include the bid- ask spread (Amihud and Mendelson (1986)), the price impact of trade (Brennan and Subrahmanyam (1996)), turnover (Datar, Naik, and Radclie (1998)), trading vol- ume (Brennan, Chordia, and Subrahmanyam (1998)), and transaction costs (Liu and Loewenstein (2002), Liu (2004), Dai and Yi (2006), Jang, Koo, Liu, and Loewenstein (2007), and Dai, Jin, and Liu (2008)). Lastly, given that only partial equilibrium results are presented here, they should be taken only as suggestive. Future research could focus on assessing the asset pricing implications of liquidity restrictions in a 42 delegated portfolio general equilibrium setting. Examples of related recent literature on this topic are Longsta (2005) and Leippold and Rohner (2008). 43 Chapter 2 The Value of Cross-Trading to Mutual Fund Families in Illiquid Markets: 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 investors' welfare. 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' welfare 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 44 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 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 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. 45 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. 46 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 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 set of tools as the ones used in this chapter. 47 facto 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 a standalone fund. 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 fam- ily 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. On average, aliated funds exhibit a more volatile, skewed and leptokurtic distribution of returns. Moreover, these extra costs of delegation increase with asset liquidity. We show how investors can use position limits as contracting features that curb these risk-shifting incentives and improve the welfare results from delegating their portfolios to family-aliated funds. Third, families' optimal strate- gies 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. We also nd that fami- lies in our model favor highly diversied portfolios at the fund level when liquidity is low but correlated asset holdings under more liquid conditions. Finally, our analysis sheds light on fund families' incentives to play favorites and to build one star fund at other funds' expense. 48 Overall, we hope that our work draws attention to a potential misspecication of the traditional approach to portfolio delegation. Given the importance of illiquidity 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. Our work 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 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. 49 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 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 our work 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 2.3 we solve the model using numerical methods, and discuss its main results. Conclusions and implications for further research are presented in Section 2.4. 50 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. 51 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, so as to keep asset C's price 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 also 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. 52 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' 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)). 53 (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 performers with increasingly higher cash in ows but punish very poor performers as 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. 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. 54 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, 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 . 19 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 Note that this specication of preferences is similar to the used in the \External Habit Forma- tion" literature (see, e.g. Chan and Kogan (2002)) but with benchmarkYj (T ) instead of the external habit for investor j. 55 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. 20 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. 21 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. 22 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. 23 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 20 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. 21 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. 22 This is similar to the distinction between internal and external capital markets in the corporate nance literature. 23 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. 56 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. 24 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 24 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. 57 horizon coincides with the date of fund ows, and fund ows are nontradable at that date. 25 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 25 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. 58 strategies, a phenomenon that is empirically documented in Nanda, Wang, and Zheng (2004). 26 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, respectively, as follows: 27 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 asset C in order to comply with the regulation on position limits at time t = 0 . When we impose position limits, the CIO's problem (2.11) is constrained by the following two simultaneous conditions, which already integrate the `no internal short-selling' constraint dened above: 28 n 1 C b 1 f 1 s C Xn 1 C (2.14) 26 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. 27 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. 28 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 2.3. 59 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 the approach we use to approximate 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 2.2. 29 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. 30 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 29 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). 30 We used as basis functions 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. 60 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 assetC 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, depending on the amount of cross-trading X. 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) 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 the value 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. 61 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. 31 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. 32 The alternative parameter congurations are specied in more detail in each of the subsections below. 31 Bar, Kempf, and Ruenzi (2005) show that team-managed funds exhibit lower (unsystematic) risk than single manager 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 being 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. 32 See the 2009 Investment Company Fact Book published by the Investment Company Institute (ICI). 62 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 welfare 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 2.2). 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 63 −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 the illiquid asset C between funds 2 and 1 at t = 0, for dierent correlation coecients between 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. 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 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. 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 64 −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 the illiquid asset C between funds 2 and 1 at t = 0, 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. 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) foriC =0:5; +0:5, respectively, fori = 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. Moreover, under no position limits, the sale of shares of asset C from fund 2 to fund 1 at t = 0 is particularly high when the CIO is relatively risk-tolerant ( C = 5), and even 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. 65 Table 2.1: Expected terminal portfolio weights on 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 66 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. 33 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 family- aliation case. 34 Under the standalone organization, fund 2's average before- ow return is higher than fund 1's but 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 compared to their standalone counterparts. 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. 33 Or, equivalently, if markets att = 0 for assetC were perfectly liquid so that the managers could `at no cost' rebalance their `inherited' portfolios to their optimizing initial allocations. 34 The histograms in Figures 2.3 and 2.4 approximate the distribution of log-returns, and so the averages 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 (implicitly or explicitly) use throughout the dierent numerical exercises in the chapter. 67 −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' 68 −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, 69 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 rebalancing 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. 70 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 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 distinc- tive feature of the 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 the CIO's perspec- tive 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 overperform the benchmark by a large amount and thus get large investor 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 36 These values are displayed in the baseline case of Table 2.3, under the RRR column. 71 corresponding to their respective styles and may even (weakly) enhance their perfor- mance 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 reducing those of fund 2's returns, relative to the standalone case. The following subsection uncovers an additional driving force underlying the fam- ily's portfolio strategy, namely the search for diversication benets. 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 the returns of funds 1 and 2, 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 op- timally chooses induces a negative correlation between the funds' after- ow returns. When assets are themselves negatively correlated ( 12 =0:5), this negative corre- lation is attained by a relatively small amount of cross-trading (OC ), but when it 37 We expand on this last point in Section 2.3.2. See Almazan, Brown, Carlson, and Chapman (2004) for empirical evidence documenting welfare-improving eects on investors derived from im- posing investment policy constraints. 72 is positive ( 12 = +0:5) the CIO cross-trades as much as she can in order to achieve the maximum diversication eects from funds' returns (OC + ). The CIO's pursuit of diversication benets is also clear in Panel B of Figure 2.6, which displays funds' after- ow return correlations for dierent correlation coecients iC between the liq- uid assetsi = 1; 2 and the illiquid assetC. In all cases, family-aliated funds exhibit a substantially lower correlation 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 diver- sication benets on the family's overall portfolio, which may decrease the family's overall portfolio 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 investment 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 73 −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. is still lower than that of standalones (NC) in all cases. Qualitatively, this outcome 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 compares them between family-aliated and their standalone peers. 74 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 the individual funds' managers attempt to counterbalance the higher risk-taking favored by the CIO (by attempting to track the benchmark more closely after t = 0) or if, on the contrary, they reinforce it. We examine this question in Figures 2.7 and 2.8, which plot the risk exposure of fund 1 and fund 2, 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 benchmark weights on those same assets, 38 and plot those dierences against the funds' relative performance as of the end of the third quarter (3T=4). 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, even though the risk-taking patterns dier substantially by investment styles. This leads us to a second observation: while fund 2 takes high unsystematic risk in a neighborhood of the 0 excess return and lower risk in the very poor and very good relative performance regions, fund 1 does exactly the opposite. This implies that, after three quarters of the investment period (assumed to be one year), incentives to risk-shift for fund 2 (fund 1) are lower (higher) the 38 Note that these weights are time-varying given that the benchmarks are passive buy-and-hold portfolios 75 −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 in which cross-trading att = 0 is subject to position limits. Panel B shows results for the case in which cross-trading is not subject to position limits. 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. less likely beating the benchmark becomes or the most exceptionally good relative performance is. 39 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 39 Although not displayed in the text, results are even more pronounced when cross-trading is not subject to position limits. 76 −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 at t = 0 (fund 2 as a standalone). Panel B shows results for the case in which cross-trading is not subject to position limits. 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. 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 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 77 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 dier- ent alternative parameter congurations. As in Table 2.1, Panel A presents results in the presence of risk-shifting incentives induced by convex ow-performance relations, while Panel B displays results in the absence of such incentives. We denote by RS the average of the deviations of the funds' portfolio weights on the assets that cor- respond the least to their investment styles from the corresponding weights on 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. 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. 78 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 79 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 80 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' welfare. 2.3.2 Portfolio delegation and utility implications of cross-trading One of the advantages of taking a portfolio choice approach to studying the investment decision problem of fund families is that it allows us to approximate the welfare 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 fund 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 investment policy. That is, because family concerns are not necessarily perfectly aligned 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 possibility of cross-trading among aliated funds so as to circumvent the costs of illiquidity may in principle oset part or all of these 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. A positive net cost|as is usually the case in portfolio delegation problems|is then easily interpreted as a measure of how much the investor values the manager's ability (stock selection and timing ability, 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. 81 information gathering, etc.) or of how costly she considers managing her savings herself. 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 policy 1 to investment policy 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 represents a utility loss. 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 TBD denotes investors' net benets of switching from direct investment (man- aging 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. 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. 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). 82 Table 2.3: Utility implications of portfolio delegation 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 No CT 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 No CT Baseline 0:00 0:00 0:00 0:00 0:00 0:00 0:00 iC = +0:50 0:01 0:00 0:00 0:01 0:00 0:00 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 83 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 Some observations are robust to the dierent values of the model parameters. First, a fund's aliation to a family is very costly to 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 improve their welfare by imposing position limits on their funds' holdings (CT Poslim), rendering these costs small 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. Moreover, cross-trading can be particularly protable when the liquid asset is highly correlated with the illiquid one ( iC = +0:5), with certainty equivalent returns for the family as whole 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 bene- ts 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 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. 84 umbrella than is purely saving on transaction costs or avoiding the costs of illiquid- ity. 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 ad- ditional 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 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 funds' aliation to the family, irrespective of liquidity conditions. This happens because each fund manager exacerbates more the risk-shifting induced by the CIO the more diversied her fund's portfolio is (the lower iC is). 44 This value of approximates market conditions during the period 1974-2008, see Section 2.3. 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. 85 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 for changing market liquidity. 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. 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) conditions, in which = 0:3 ( = 0:7). Finally, Panel B makes it clear that families will favor highly diversied portfolios in dry markets (NBF is highest for iC =0:5 when 0 0:35), but highly correlated portfolios when markets are more liquid (NBF is highest for iC = +0:5 when > 0:35). 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. 86 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' welfare. But the family organization will also enable these companies to exploit the convexities in the implicit ow-performance relation more fully by cross-subsidizing 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. We illustrate how the inclusion of position limits on funds' asset holdings can curb these risk-shifting incentives and benet investors. Ultimately, the purpose of this chapter is to highlight the extent to which the costs and benets of family aliation can dier substantially from those in an independent fund paradigm on which most of the existing analytical literature has been focusing to date. 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 87 in the managed portfolios of family-aliated or standalone funds. This would allow the investor to trade around incentive misalignments and improve her welfare 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). 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. 88 Chapter 3 Co-Insurance in Mutual Fund Families In this chapter, we propose to study how co-insurance can serve as a rationale for corporate diversication in the mutual fund industry. The corporate nance literature has long recognized that the ability to reduce conglomerate risk by co-insuring dierent segments is a potential benet of corporate diversication. Essentially, by internally reallocating resources across dierent divi- sions, companies can avoid costly external nancing. 1 Similarly, we postulate that fund families have an incentive to take advantage of their internal capital markets and to strategically coordinate actions across their individual member funds in order to smooth the impact of liquidity shortfalls, whenever that results in the creation of value for the fund complex as a whole. Consistent with our working hypothesis, our preliminary results indicate that open-end funds aliated with large families are likely to coordinate trades with their siblings in order to (i) mitigate the potential harmful eects of nancial pressure imposed by their investors' share purchases and redemptions, as well as to (ii) alleviate the ownership costs associated with the lack of liquidity of their portfolio holdings. Our preliminary results further indicate that the widespread ability that fund families have to cross-trade assets among member funds, and in this way circumvent the costs associated with having to deal with public markets, then results in price pressure to 1 Lewellen (1971) was among the rst to suggest that cash ow co-insurance can serve as a purely nancial rationale for corporate diversication by reducing the probability of nancial distress. 89 be less signicant in traded securities that are mostly held in common by distressed funds that belong to large families and which transactions cluster by time. However, we recognize that the ability to co-insure aliated funds does not nec- essarily imply that it is optimal for families to do so. The incentives of the fund family may be misaligned with those of their aliated funds. Our preliminary re- sults indicate that individual fund managers anticipate to be \bailed-out" by their co-insuring families against liquidity shortfalls, which improves the convexity of their implicit payo schemes, and ultimately encourages them to take extra risks. Thus, the purpose of this chapter is to propose a study of the importance of the bene- ts of co-insurance relative to its costs, and the conditions under which co-insurance strategies at the family level create or destroy value. One of the most glaring characteristics of the U.S. mutual-fund industry is the ex- istence of a vast myriad of funds, split into numerous categories, and run by relatively few managing companies. Virtually all funds belong to multi-fund families, and the size and number of funds oered by these families vary widely, even among families of equal market share. The top fund families concentrate most of equity assets under management, the average family has more than 10 funds under its umbrella, and some of the families group over 100 dierent funds. This suggests that mutual fund families can constitute large (and potentially very active) internal markets. Although these empirical regularities have been previously recognized, relatively modest attention has been given to their impact on the complex structure of incen- tives underlying the mutual fund industry. In particular, the extent to which both the number and organization of aliated funds aect the strategic behavior of their families and of individual managers is a question that remains largely unexplored. We intend to contribute to this area of research by investigating the conditions under which the organizational structure of certain families allows for their ability to co-insure aliated funds. In particular, we focus on the incentives that families have to co-insure their open-end funds against the possibility of being forced by heavy 90 investors' ows to engage in costly transactions, especially so when the degree of liquidity of their holdings is relatively low. Unlike their closed-end and hedge fund counterparts, open-end fund managers must deal with daily money ows into and out of their funds. In particular, fund managers often see signicant redemptions in periods when their funds have fared poorly, forcing them to sell into markets where fund securities have been declining in value. In ows, on the other hand, often occur after years of strong performance, when investment opportunities are presumably less attractive. 2 Moreover, when nancial distress in mutual funds clusters by time, the aggregate ow-induced transactions of overlapping holdings can create signicant price pressure, which generates perva- sive cascading eects that amplify the detrimental consequences of forced portfolio rebalancing. We postulate that one of the mechanisms that fund families can use to co-insure their funds against the harmful eects of ow-induced trading, as well as to mitigate the ownership costs of the lack of liquidity of their underlying holdings, is through cross-trading, which we assume is likely to be processed through the fund family's trading desks, or through crossing networks. 3 Cross-trades can represent a signicant source of savings in transaction costs and market maker spreads for mutual funds, given that no brokerage commission, fee or other remuneration is paid in connection with these transactions. The ability to engage in cross-trading also eliminates the ineciencies, administrative costs, and time involved in breaking up a block of assets in order for them to be absorbed by the market. More importantly, by crossing 2 Alexander, Cici, and Gibson (2007) show that the reason behind the decision to trade matters when assessing trade performance. They show how unanticipated investor ows force managers to rebalance their portfolios to control liquidity, which subsequently penalizes funds' returns. 3 A crossing network is an ATS (Alternative Trading Systems) that matches buy and sell orders electronically for execution without rst routing the order to an exchange or other displayed market. The advantage of the crossing network is the ability to execute a large block order without impact- ing the public quote. Examples of crossing networks would be Liquidnet, Pipeline, ITG's Posit or Goldman Sachs' SIGMA X. 91 buy and sell orders within their families' internal markets, mutual funds may have access to better pricing conditions for their transactions. For instance, a mutual fund experiencing heavy share redemptions is looking to sell a large block of shares of a security and makes a trade on the open market. The spike in volume might prompt a change in the price of the security, which in turn could deteriorate the outcome of selling such a block of shares. When thousands of shares are involved, even a small change in share price can translate into a signicant amount of money. If instead the non-distressed siblings of this selling fund coordinate actions in order to absorb such a large transaction, they could skip over market forces and get a price that's better suited. 4 Ultimately, we predict that the generalization of this sort of strategies at the family level can lead to the smoothing of the potentially negative impact that asset re sales often impose on a security's price. Flow-driven trading has been shown to trigger price impact. Specically, Coval and Staord (2007), Khan, Kogan, and Serafeim (2010), and Lou (2010) show that mutual funds typically scale up and down their existing portfolio positions in response to in ows and out ows from investors and that these \passive" trades create price impact. Our preliminary results are consistent with the prediction that ow-induced transactions generated mostly within large fund families, which are more likely to co-insure their aliated funds, do not result in signicant price reactions on the underlying securities, compared to those generated mostly within small families. Thus, in order for a family to be able to co-insure its aliated funds, it needs to possess enough latitude, i.e. to have a sucient number of funds among which it can more likely nd opportunities for osetting transactions. 5 We conjecture that funds 4 SEC Rule 17(a)-7 of the Investment Company Act of 1940 allows interfund cross-trading, under certain conditions. One of those conditions is that the transaction must be eected at the \indepen- dent current market price of the securities," which is usually taken to be the average of the highest current independent bid and lowest current independent oer. Such price calculations leave out the information on volume and liquidity level of the securities involved in the interfund transactions. 5 SEC Rule 35(d)-1 requires that an investment company with a name that suggests that it focuses its portfolio holdings in a particular type of investment, or in investments in a particular industry, 92 have more exibility to build-up or unwind their current positions than to add new positions to their portfolios, by reason of e.g. economies of research resources and the familiarity and superior knowledge about the stocks already covered by a fund manager. The combination of the above may explain the existence of a substantial overlap in asset holdings among funds oered by one same family. 6 In order to show that the existence of osetting trades within fund families is not a spurious result but the outcome of a coordinated strategy between aliated funds, we propose to use a bootstrap-based technique similar to that used to distinguish skill from luck in e.g. Fama and French (2010), or Kosowski, Timmermann, Wermers, and White (2006). We compare the actual degree of osetting trades of individual security sales within each large fund family with their simulated distributions generated from data of all the remaining families in our sample. In particular, we propose to replicate each one of our large families using matching funds from the remaining families and analyze how much of the aggregate sells of a particular security across the funds aliated with that family is oset by the aggregate buys of that same security across the remaining funds in the family. The funds would be matched rst by fund size and then by the characteristic-based classications of Daniel, Grinblatt, Titman, and Wermers (1997). Our preliminary tests consistently indicate that the large degree of absorption of aggregate stock selling within large families is very unlikely to arise solely due to sampling variability. should invest at least 80% of its assets in the type of investment suggested by its name. As a result, not only it should be more likely to nd osetting trades within large fund families, but it should also be easier to absorb large block trades within large families without having to deviate from compliance with the SEC Rule 35(d)-1. 6 Elton, Gruber, and Green (2007) nd evidence that mutual fund returns are more closely corre- lated within than between fund families, due primarily to common stock holdings. They show that, depending on the objective group being considered, as much as 34% of total net assets (TNA) consist of stocks held in common by funds in the same objective. For funds with dierent objectives, the median percent of the portfolio held in the same securities is 17% inside the family compared to 8% outside the family. 93 Next, our preliminary results are consistent with the idea that coordination of trades between funds and their distressed siblings is common practice among large families, and that such family strategies can have important asset pricing implications. In particular, we extend the methodology in Lou (2010) and create a measure of demand shocks to individual stocks from the projection of mutual fund ows onto the stocks they hold, controlling for multiple stock and fund characteristics, in line with Pollet and Wilson (2008). We document a signicant ow-induced price pressure eect on individual stock returns when most of the distressed funds involved in trading those stocks are funds aliated with small families, which we dene as having less than 10 aliated funds. However, consistent with our co-insurance hypothesis, no ow-induced price pressure is found for securities that are mostly traded by distressed funds that belong to large families. Finally, we recognize that co-insurance strategies may not produce the eects desired from the point of view of the fund family. That is most likely the case when there are reasons to believe that the incentives of the aliated fund managers are misaligned with those of the family as one entity. 7 Thus, we argue that insulating aliated funds from the risk of liquidity shortfalls may lead individual managers to prevent the liquidity shortfalls from happening in the rst place. Consistent with this argument, we propose to use a semi-parametric approach to estimate the shape of the sensitivity of investors' ows to past performance of illiquid funds aliated with large families, and our preliminary results indicate that it is relatively more convex than that of their small family counterparts. Following Chevalier and Ellison (1997), we then propose to show that the convexity of the shape of the ow-performance relationship then creates incentives for fund managers of illiquid funds aliated with large families to take more risks than their small family counterparts. 7 Kempf and Ruenzi (2008) provide evidence for this misalignment of incentives within fund families. They show that funds of a family should not be viewed as coordinated entities, but as individualities that also compete in a family tournament for investors' ows. 94 Existing research has documented that both fund families and individual managers possess strong incentives to maximize the total value of the net assets under man- agement (TNA). A fundamental component of it has to do with investors' response to fund performance. There is ample empirical evidence that the best performing mutual funds receive a disproportional share of total ows (e.g. Sirri and Tufano (1998)). Although a generally convex relationship between fund ows and past per- formance has been observed for the average individual fund, it may not hold for all types of funds. In general, investors are reluctant to redeem shares of losing funds, which contributes to the convexity of the ow-performance relationship observed for the average fund. This behavior is consistent with the existence of economic or psy- chological costs associated with the redemption of fund shares. Chen, Goldstein, and Jiang (2009) argue, however, that ow-performance sensitivities are not the same for all types of funds. In particular, conditional on poor past performance, funds that hold illiquid assets experience more out ows than other funds. They argue that, fol- lowing substantial out ows, funds need to adjust their portfolios and conduct costly and unprotable trades, which damage future returns. Since mutual funds conduct most of the resulting trades after the day of redemption, most of the costs are borne by the remaining investors. This leads the remaining investors to want to pull their money out of the funds creating a \fund-run" eect that greatly amplies the sub- sequent damage to the value of illiquid funds. This eect might be enough to oset the potential costs preventing investors from liquidating their positions in poorly per- forming illiquid funds, which then helps alleviate the convexity on the negative side of the ow-performance sensitivity of illiquid funds. As a result, the generally con- vex shape of the ow-performance relationship, as so documented by Chevalier and Ellison (1997), and Sirri and Tufano (1998), appears to hold only for liquid funds. However, our preliminary results are consistent with the idea that when funds aliated with large families commit to provide liquidity to one another, they are likely to preserve the convexity of the sensitivity of their ows to past performance. 95 They also suggest that the convexity in the ow-performance relationship creates incentives for managers of illiquid funds in large families to take more risks than their matched counterparts aliated with small families. These results are aimed to contribute to the relatively small body of research on how the structure of the industry aects mutual fund behavior, in particular how family aliation aects individual fund managers' incentives and investment decisions. For instance, Khorana and Servaes (1999) look at how economies of scale and scope aect the fund family decisions to open new funds. Gaspar, Massa, and Matos (2006) show that fund families selectively allocate performance across member funds in order to favor those that are more likely to generate higher fee income or future ows into the family. Pollet and Wilson (2008) show that funds with many siblings diversify less rapidly as they grow. Our results are aimed to complement those in Massa (2003) by showing how the organization of funds into families creates positive externalities across them. Our preliminary ndings also complement those in Nanda, Wang, and Zheng (2004), which show that \winner-picking" strategies can be used to create performance spill-overs within the families. We explore the \socialism" side of the mutual fund family organization and show that strategies aimed at smoothing out ows of funds that more sensitive to underperformance can be an important instrument to help families achieve the maximization of the value of their net assets under management. Our results are related to those in Massa and Patgiri (2009), which show that family aliation increases risk taking. What we nd is also consistent with the ndings in Chen, Hong, Huang, and Kubik (2004), who provide evidence that the performance of a fund is related to the family it belongs to. Guedj and Papastaikoudi (2005) show that performance persists at the family level. We believe that co-insurance strategies at the family level can contribute to the performance persistence of aliated funds. Our work is related to the extensive literature on mutual fund herding and on the price impact of institutional ows. Examples of work on this issue are Wermers 96 (1999), Coval and Staord (2007), Frazzini and Lamont (2008), and Lou (2010). Another important topic related to our work has to do with the eect of mutual fund ownership on stock price volatility. Recent papers by Anton and Polk (2010) and Greenwood and Thesmar (2010) show that the concentration of ownership of a nancial asset can create stock price fragility via the correlation of the trading needs of its owners. We believe that our results can shed some light on this issue. Our work is also related to the work by Almazan, Brown, Carlson, and Chapman (2004), which shows that fund managers in large families are less constrained by the investors than managers in small families, because the family is supposed to function as a delegated monitor. 8 Moreover, Huang, Wei, and Yan (2007) documents that fund investors are subject to lower participation costs when investing in funds that belong to large families, which may also reduce the transaction costs associated with switching from one fund to another. We believe that such evidence strengthens the signicance of our ndings. Specically, even though we should expect out ows to be more sensitive to underperformance, due the fact that it is easier for investors to substitute between funds within larger families, we still nd a signicantly more convex ow-performance relationship for illiquid funds aliated with large families, compared to their matched counterparts from small families. The research on risk-sharing strategies of mutual fund families is almost non- existent. Closest to ours is a very recent work by Bhattacharya, Lee, and Pool (2010), which investigates how aliated funds of mutual funds provide liquidity in the form of new share purchases from other aliated funds that may be experiencing heavy redemptions from their outside investors. They show that this action reduces the performance of the liquidity providers but improves the performance of the liquidity receivers, by allegedly preventing them from engaging in asset re sales. However, the problem they analyze is dierent from ours in important respects. They focus on 8 See also Gervais, Lynch, and Musto (2005). 97 the unilateral provision of liquidity from aliated funds of funds to their distressed investments, which seems somewhat hard to rationalize. We instead focus on the ability that families have to coordinated actions among aliated funds in order for them to provide liquidity to one another. The reciprocity associated with the co- insurance strategy that we postulate in seems easier to justify from the point of view of the liquidity providers. We argue that funds aliated with large families provide liquidity to their distressed siblings and may eventually receive liquidity from them, should their positions reverse later on. Moreover, we explore the asset pricing implications of internal liquidity provision as well as the potential for \moral hazard" behavior on the part of liquidity receivers, which we believe are important dimensions of this issue, and which are not pursued in Bhattacharya, Lee, and Pool (2010). We believe that our work is set to contribute to the literature on mutual funds in a number of dimensions. First, we emphasize the possibility of co-insurance within mutual fund families, which has been largely overlooked. In doing so, we highlight the fact that the characteristics and organizational form of the fund families can be fundamental determinants of their strategic behavior, and that their strategic behavior can have important implications for asset prices. 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Abstract (if available)
Abstract
This dissertation consists of three chapters of interrelated work in which I investigate the implications of money management incentives to delegated asset allocation and to asset pricing in the context of illiquid markets.
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University of Southern California Dissertations and Theses
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Creator
Goncalves-Pinto, Luis
(author)
Core Title
Essays on delegated asset management in illiquid markets
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Business Administration
Publication Date
04/12/2011
Defense Date
03/16/2011
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,portfolio delegation
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Zapatero, Fernando (
committee chair
), Matos, Pedro (
committee member
), Stathopoulos, Andreas (
committee member
), Zhang, Jianfeng (
committee member
)
Creator Email
lgoncalv@usc.edu,luis.goncalvespinto@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3739
Unique identifier
UC1175199
Identifier
etd-GoncalvesPinto-4510 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-445249 (legacy record id),usctheses-m3739 (legacy record id)
Legacy Identifier
etd-GoncalvesPinto-4510.pdf
Dmrecord
445249
Document Type
Dissertation
Rights
Goncalves-Pinto, Luis
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
portfolio delegation