University of Southern California Dissertations and Theses

Essays on financial markets

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Content ESSAYS ON FINANCIAL MARKETS by Min Seon Kim 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) August 2010 Copyright 2010 Min Seon Kim Acknowledgments My deepest gratitude is to my advisor, Prof. Wayne Ferson. I have been fortunate to have an advisor who teaches how to approach research questions and express ideas. I am also thankful to him for the long discussions and margin comments that helped me improve my dissertation. He carefully read and commented on countless revisions of all chapters in this manuscript. Also, his mentorship and guidance was paramount. I hope that one day I will do the same thing for my students as Prof. Ferson has done for me. I would like to sincerely thank Prof. William Zame, who helped me get interested in theoretical research through his classes and encouraged applications of game theory to the nance area. Although Prof. Zame is a professor at University of California, Los Angeles, hehasalwaystakenhistimetogivecommentsonmydissertation. Hisintuitionandinsights about economics also helped this manuscript go in interesting directions. Prof. Christopher Jones has been always there to listen to my questions and problems andsharehisthoughts. Iamalsodeeplygratefultohimforhisencouragementandguidance throughout my ve years at University of Southern California. I am pleased to thank Prof. Fernando Zapatero for his help and support during my graduate studies at University of Southern California. I am also indebted to him for the opportunity to work together on a research project. Prof. Mark Westereld also provided invaluable support to improve this manuscript. I am truly grateful to him for his insightful comments and constructive criticisms to help me focus my ideas. ii I would like to acknowledge Ricardo Alonso, Sangmok Lee, Anthony Marino, John Matsusaka, Kevin Murphy, Oguzhan Ozbas, and Heikki Rantakari for their comments on the rst chapter of my dissertation. I also thank Stephen Brown, Daniel Carvalho, George Cashman,HarryDeAngelo,DianaKnyazeva,JohnLong,TimLoughran,PedroMatos,Rosa Liliana Matzkin, Kevin Murphy, Oguzhan Ozbas, Raghavendra Rau, Antoinette Schoar, Clemens Sialm, David Solomon, Kumar Venkataraman, and Jerold Warner for their com- ments on the second chapter of my dissertation. IamalsogratefultoDr. LucyLeeforherinvaluablehelptoimprovemycommunication skillsandtoHelenPittsforhervariousformsofadministrativesupportandadvice. Finally, noneofthiswouldhavebeenpossiblewithouttheloveandsupportofmyfamilyandfriends in Seoul. iii Table of Contents Acknowledgments ii ListofTables vi ListofFigures viii Abstract x Chapter 1. Information Asymmetry and Incentives for Active Management 1.1. Introduction 1 1.2. Related Literature 7 1.3. A statement of the problem 9 1.4. E¢ cient bargaining solution 17 1.4.1. Special case (no performance fee) 20 1.4.2. Performance fee 23 1.5. Conclusion 35 Chapter 2. Changes in Mutual Fund Flows and Managerial Incentives 2.1. Introduction 36 2.2. Flow-performance relationship 42 2.2.1. Data and variable description 42 2.2.2. Kernel regression methodology 50 2.2.3. Linear regression methodology 51 2.2.4. Changes in the ow-performance relationship 52 2.2.5. Determinantsofsensitivityoftheow-performancerelationship 60 2.3. Managerial incentives 72 2.3.1. Data and variable description 73 2.3.2. Methodology 75 2.3.3. Changes in managersrisk-shifting behavior 79 2.4. Robustness check 83 2.4.1. Dummy variables for market conditions 83 2.4.2. Piecewise regression 84 2.4.3. Performance ranking 91 2.4.4. Flow-performance relationship for index funds 98 2.5. Conclusion 106 Chapter 3. Time Variation in Expected Returns and Aggregate Asset Growth 3.1. Introduction 108 3.2. Relatedliterature 113 3.3. Expected returns and consumption-wealth ratio 116 3.4. Predictiveregressions 119 3.4.1. Regressionmethodology 119 3.4.2. DataandsummaryStatistics 121 3.4.3. Results 131 3.5. Comparisons of out-of-sample predictive ability 135 3.6. Robustnesschecks 144 iv 3.6.1. Unbiased estimates of changes in returns 144 3.6.2. Does asset growth predict changes in stock returns? 148 3.7. Predictive regressions for the 25 Fama-French portfolios 149 3.8. Conclusion 170 References 172 AppendixA1. 181 A1.1. ProofforProposition2 181 A1.2. ProofforProposition3 184 A1.3. ProofforProposition4 192 A1.4. ProofforProposition5 196 A1.5. ProofforCorollary2 198 A1.6. ProofforCorollary3 199 A1.7. ProofforCorollary4 200 A1.8. Conditionforseparatingequilibrium 201 AppendixA2 207 A2.1. Kernelregression 207 A2.2. The e¤ect of the expense ratio on fund ows 208 AppendixA3 214 A3.1. Consumption-wealthratio 214 A3.2. Stationarycomponentofhumancapital 215 A3.3. Rsquared 217 v List of Tables Table1.1: Comparativestaticsinane¢ cientpoolingequilibrium 23 Table 1.2: Comparative statics in an e¢ cient separating equilibrium 33 Table1.3: Comparativestaticsfortheskillthreshold 35 Table2.1: Descriptivestatistics 46 Table 2.2: Estimates for control variables in kernel regression 54 Table2.3: OLSregressionofnetows 56 Table2.4: Determinantsofow-performancesensitivity 68 Table2.5: Descriptivestatisticsforriskshiftmeasure 76 Table2.6: Linearregressionforriskshiftmeasures 81 Table 2.7: Determinants of ow-performance sensitivity with dummy variables 84 Table 2.8: Determinants of ow-performance sensitivity using piecewise OLS 86 Table2.9: Determinantsofow-performance(ranking)sensitivity 95 Table 2.10: Linear regression for risk shift measures using performance ranks 96 Table2.11: Descriptivestatisticsforindexfunds 99 Table 2.12: OLS regression for net ows for index funds 102 Table 2.13: Determinants of ow-performance sensitivity for index funds 104 Table3.1: Descriptivestatisticsandcorrelations 123 Table3.2: Autocorrelationsandstandarderrors 130 Table3.3: OLSregressionforrealquarterlyreturns 132 Table3.4: OLSregressionforrealannualreturns 133 Table 3.5: Comparison of predictive ability for out-of-sample periods 139 vi Table 3.6: Regressions of changes on changes in returns 146 Table 3.7: Regressions using decomposed asset growth 149 Table 3.8: OLS regression of one-period ahead quarterly returns 152 Table 3.9: OLS regression of one-period ahead changes in quarterly returns 159 Table A2.1: Determinants of the coe¢ cients on the expense ratio 210 vii List of Figures Figure 1.1: Likelihood ratio and optimal performance fees 26 Figure 1.2: Expected utility for active and passive managers 31 Figure 1.3: Changes in expected utility for active and passive managers 33 Figure 2.1: Flow-performance relationship and 90% condence interval 38 Figure2.2: Samplecomposition 48 Figure 2.3: Flow-performance relationship by kernel regression 53 Figure 2.4: Flow-performance relationship by OLS regressions 58 Figure 2.5: Market volatility and performance dispersion 63 Figure2.6: Flow-performancesensitivityconditionalonmarketvolatility 71 Figure 2.7: Flow-performance relationship by piecewise OLS regressions 90 Figure2.8: TNA-weightednetows 92 Figure 2.9: Flow-performance (ranking) relationship by kernel regression 94 Figure2.10: Flow-performancerelationshipforindexfundsbykernelregression 101 Figure 3.1: Aggregate asset holdings and the CRSP VW index 124 Figure 3.2: Dividend-price ratio and estimated dividend-price ratio 128 Figure 3.3: Cayand recursive estimates for the cointegration parameters 129 Figure3.4: Out-of-samplepredictionsforrealquarterlyreturns 141 Figure 3.5: Recursive slope estimates of predictive regressions 147 Figure3.6: Actualandttedvaluesofsmallportfolios 166 Figure 3.7: Actual and tted values of large portfolios 168 Figure A2.1: Coe¢ cient on the expense ratio 209 viii Figure A2.2: Expense ratio and 12b-1 proportion 213 ix Abstract This dissertation consists of two chapters that examine agency issues in delegated portfolio management and one chapter that studies forecasts of stock market returns. Therstchapterproposesthattherecentgrowthinpassivemanagementinthemutual fund industry (e.g., closet-indexing) is caused by a weaker ow-performance relationship amid low skills, noisy performance, and more outside competition. Using a model that accommodates both moral hazard and adverse selection, I show that when skills decrease, when performance becomes noisier, and when competition from other investment vehicles grows, active management does not add much value relative to passive management. As a result, performance compensation also decreases, which discourages active management, causing even skilled managers to track market indexes. Given that the ow-performance relationship can serve as performance compensation, this model predicts that a weaker ow-performance relationship leads to more indexing. Thesecondchaptershowsthattheow-performancerelationshipindeedbecomesweaker, as the rst chapter argues. The relationship was convex prior to 2000, but is no longer con- vex. Variations in marginal ow contribute to the shape change and suggest that the relationship can serve as a dynamic incentive contract. The marginal ow to high perform- ing funds is lower when markets are highly volatile and when performance dispersion across fundsislower. Moreover, Ishowthatunderperformingmanagersengageinlessrisk-shifting after 2000, which can be also explained by performance dispersion and market volatility. These results are consistent with the view that the convex ow-performance relationship provides managers with risk-shifting incentives. x The third chapter proposes that under certain circumstances such as when expected returns are highly persistent, predicting changes in returns and then adding current returns results in a better performance than predicting returns directly. Regressions of changes in stock returns on changes in household net worth (aggregate asset growth) provide stable coe¢ cient estimates over time, which improve the out of sample predictive ability in terms ofmeansquaredpredictiveerror,whencomparedwiththedividend-priceratio,cayanda yieldspread. Theseresultssuggestthataggregateassetgrowthcapturesshockstoexpected returns, a main component of changes in returns. xi Chapter 1. Information Asymmetry and Incentives for Active Management 1.1. Introduction Mutual fund prospectuses disclose whether funds aim to outperform certain market indexes or simply follow them. Consumers who invest in active funds would like to ensure that their managers have skills and actually engage in active management. At the same time, investorswouldlikemanagerswhocanonlytrackmarketindexestopassivelymanage funds. Yet, given that investors cannot directly observe managersskills and actions, we may not expect such a separating equilibrium to arise. Recent growth in passive management in the U.S. mutual fund industry seems to indi- cate that managers have few incentives for active management. More assets in active funds are in fact passively managed, and the funds still charge high fees (i.e., closet-indexing). Cremers and Petajisto (2007) argue that 70% of assets in the mutual funds whose goal is to outperform the S&P500 index were passively managed in 2003. Provided that those funds have $700 billion of assets and the fee di¤erence between active funds and passive funds is 1%, investors overpay managers by around $5 billion a year. Passive funds also have dramatically increased. The number of exchange-traded funds increased from 19 in 1997 to 353 in 2006, and the fundsnet assets increased approximately 60 times to $422 billion. Index funds show comparable growth (2007 Investment Company Institute Fact Book). This chapter presents a model that explains the growth in passive management. Using this model, I show that the condition for the separating equilibrium is summarized by com- paring a value of active management to a threshold. When the value exceeds the threshold, 1 performancecompensationislargeenoughtoprovidehighskillmanagerswithincentivesfor activemanagement; atthesametime, lowskillmanagerscannotmimichighskillmanagers, and so they passively manage funds. When active management is less valuable than the threshold, low compensation discourages active management, and a pooling equilibrium in which even skilled managers track market indexes arises. In the model, the value of active management decreases when skills are less hetero- geneous across managers (e.g., a deterioration in skills due to a brain drain to the hedge fund industry; see Kostovetsky (2007) and Massa et. al. (2009)). Alternatively, more noise in performance decreases the value of active management. On the other hand, an increase in the threshold can also make active management less valuable than the required level. Several factors cause this increase. For example, when investors have other, better investments available to them, the separating equilibrium requires a higher value. Thus, competition from other investment vehicles, such as hedge funds, could contribute to the growth of passive funds in the mutual fund industry. Given that the ow-performance re- lationship can induce performance compensation under the xed fee structure of mutual funds (fees are proportional to assets under management), my model predicts that the re- centgrowthinindexingiscausedbyaweakerow-performancerelationshipamidlowskills, noisy performance, and more outside competition. These results are driven by two main elements that distinguish my model from other models for delegated portfolio management. First, my model considers both the issues of moral hazard and adverse selection. Fund managers withhold information about their own skills and management e¤orts. Investors can only observe performance, which depends on skill and e¤ort but also includes noise. To lead high skill managers to engage in active 2 management, compensation for active managers must be large compared to passive man- agers. However, this can bring the consequence that low skill managers pretend to be (or actually become) active to receive larger compensation. In particular, low skill managers can easily do so when high skill managers can add less value in active management. To prevent that consequence, compensation for passive management must increase (bribing theunskilledmanagers). Skilledmanagers,then,receiveevenmoretoengageinactiveman- agement. Therefore, when active management is insu¢ ciently valuable, sorting managers is too costly and a pooling equilibrium arises. AsLazear(2000)argues,variablepayisimportantbecauseofitssortingroleforhetero- geneous agents with private information. He shows that xed compensation induces e¤orts that are too low for high ability agents and too high for low ability agents. In my model, withtwolevelsofe¤ortactiveorpassivexedcompensationleadstoadistortioninhigh skillmanagerse¤orts(leadingtoindexing). Performancefeescanidentifymanagersofhigh ability and induce their e¢ cient e¤ort levels. Yet, I argue that when active management does not add much value, the distortion in e¤ort is preferred. The second major element of the model is that equilibrium is an e¢ cient bargaining outcomebetweeninvestorsandfundmanagers(Myerson(1979)). Inotherwords,themodel assumesthatthemutualfundindustryisneitherperfectcompetitionnormonopoly. Inpar- ticular, contrary to the assumption in the principal-agent literature that investors have full bargaining power, solving such a bargaining problem provides an interesting implication: Competition from outside the mutual fund industry cause indexing in the industry. When surpluses are allocated to both managers and investors, managerscompensation is nega- tively related to investorsoutside options, e.g., their payo¤s from investing in hedge funds. 3 Thus, more competition from other investment vehicles decreases mutual fund managers payo¤s, which can discourage costly active management. In contrast, when maximizing in- vestorssurplus, their reservation utility does not a¤ect equilibrium outcomes. Given that investors already have full bargaining power, better outside options make no di¤erences. 1 The bargaining approach has other advantages by maximizing a joint surplus of in- vestors and managers. When separating managers is preferred for improving the joint sur- plus, investorspayo¤s are also maximized, but not vice versa. (Appendix A1.8. illustrates this). In other words, there can be some cases in which separating managers is optimal for maximizing investorspayo¤s, but the joint surplus is in fact higher when all managers passively manage funds. Thus, given competitions both among investors and among fund managers, the bargaining approach can better describe separating equilibrium, as opposed to maximizing only investorspayo¤s. Another advantage is that I can examine how man- agerscompensation changes according to parameters in the model. This is comparable to the approach that maximizes investorspayo¤s. In this case, some types of managers receive only their reservation utilities; therefore, their compensation does not vary. My model also suggests optimal compensation for managers when both the skills and actions of managers are unobservable. I argue that the shape of the optimal incentive fee is in general nonlinear. This rationalizes the empirically observed nonlinear relationship between money ows and fund performance. 2 In particular, my model predictsconsistent with ndings in Chapter 2that the relationship can be concave, depending on the likeli- hood that the manager has skills and actively manages the fund, relative to the likelihood 1 Mathematically, maximizing their surplus is equivalent to maximizing their payo¤s without subtracting reservation utility. 2 Earlier studies that document convexity in the ow-performnace relationship include Ippolito (1992), Patel et. al. (1994), Goetzmann and Peles (1996), Chevalier and Ellison (1997), and Siri and Tufano (1998). 4 that she tracks a market index (likelihood ratio). Given that optimal performance com- pensation is not symmetric, one may argue that allowing an option-like incentive fee could help the industry encourage active management. Anecdotal evidence suggests that vir- tually no incentive fees are used in practice due to legal restrictions on the mutual fund industry: Performance-based compensation must be symmetric, not option-like (the 1970 Amendments to the Investment Company Act). My model implies that the compensa- tion increases with performance only when the monotone likelihood ratio property (MLRP) holds. An option-like fee is optimal only under a stronger condition. My model assumes that investors lack information about funds and fund managers. Although in practice mutual fund managers disclose information, such as investment ob- jectives (active or passive), benchmark indexes, and past performance, investors still lack important information. For example, performance is a noisy measure of managerial ability even net of a benchmark since the state of the world also plays a large role. Some studies also argue that performance may be manipulated, for instance, due to product di¤erenti- ation among funds (Massa (2003)), favoritism among funds in a fund family (Gaspar et. al. (2007)), or by high-frequency trading strategies (Goetzmann et. al. (2007)). Moreover, managersdisclosuresabouttheirfundscanbemisleading. Thecloset-indexingasdiscussed aboveisanexampleoffundmanagersprovidinginvestorswithinaccurateinformationabout their funds. Another example is studied by Sensoy (2009), who argues that the majority of stated benchmarks do not match fundsactual investment styles and that some funds attracted inows of investorsmoney based on misleading benchmarks. Given fund managers with private information, I use mechanism design theory, de- veloped by Hurwicz (1960, 1972), Maskin (1977) and Myerson (1979, 1981), to suggest 5 optimal compensation schemes and conditions for separating equilibrium. Previous studies in nance also have used the approach to suggest optimal contracts. For example, Dar- raough and Stoughton (1989) apply the theory to propose an optimal prot sharing rule for joint ventures under information asymmetry about member rmscosts. Harris and Raviv (1998) suggest an optimal capital budgeting rule when division managers have pri- vate information about the productivity of their projects. Marino and Matsusaka (2005) identify optimal capital allocations and discuss how to implement them through delegation or approval. Harris and Raviv (1981) identify conditions under which sellers, who do not knowbuyersvalues, canmaximizetheirexpectedrevenuesincommonlyobservedauctions. Under the Securities and Exchange Commission rules, mutual fund managers report someinformationinfundprospectuses. Iftheaccuracyofaprospectusishardtoverify,and a penalty for inaccurate information is unlikely, managers may not report truthfully. I view outcomes in the mutual fund industry as a response-plan equilibrium. This is a collection of response plans that dene each agents strategy for reporting information, given his true type, within which no agent can benet by unilaterally deviating from his reporting plan. Given a response-plan equilibrium, I focus on incentive-compatible direct mechanisms to examine the equilibrium according to the revelation principle. I describe managerstypes in terms of their skills (high or low) and fund types (active or passive). In other words, I view moral hazard on e¤ort as managersmisreporting of fund types (e.g., closet-indexing). The problem is equivalent to a generalized principal-agent problem in the spirit of Myerson (1982). To determine e¢ ciency, I assume that managers know their own types (interim- welfare criterion). I nd an equilibrium allocation by maximizing surpluses of investors and managers. This allocation is durable (renegotiation-proof) since only managers have 6 private information and the equilibrium results also hold for one manager (see Holmstrom and Myerson (1983)). I summarize two main contributions of this chapter to the literature. First, my model considers both adverse selection and moral hazard problems. 3 Unlike many models that focus on moral hazard, Allen (2001) argues that adverse selection may be more salient than moral hazard in the agency context for delegated portfolio management. Considering in- complete information about both ability and e¤ort is critical for studying a self-selection mechanism in the industry. Second, my model proposes conditions for separating equilib- rium. Many studies assume that a positive level of e¤ort is optimal 4 and examine which performance feesfor instance, an option-like or a symmetric feeare better for induc- ing e¤orts. This excludes pooling equilibrium in which managers exert no e¤ort. I show that such an equilibrium is optimal when active management is not su¢ ciently valuable. Separating equilibrium arises only when skills can add a large value in active management. 1.2. Related Literature This chapter is related to the theoretical literature on compensation for delegated portfolio management. Heckerman (1975) shows that an optimal compensation contract can induce managers with private information about stocks to align their interests with those of investors. The literature expanded greatly following Bhattacharya and Peiderer (1985). The main strand of the literature designs optimal compensation contracts for ac- tivelymanagedfunds, wheninvestorscannotobservefundmanagersactions suchase¤orts 3 Zame (2007) considers interactions between rms (and agents within rms) and the market. His model incorporates moral hazard, adverse selection, screening, and idiosyncratic risk, and shows that miscoordina- tion may happen, which can lead to Pareto ranked equilibria. 4 An exception is Dow and Gortons (1997) model, in which talented managers do noise trading although activelydoing nothing is optimal because investors cannot distinguish it from simplydoing nothing. 7 forcollectinginformationortheirrisk-takingbehavior. Stoughton(1993)andLiandTiwari (2008) suggest that an option-like incentive fee is optimal for the moral hazard on e¤ort. HeinkelandStoughton(1994)proposeadynamiccontractthathasoption-likeperformance compensation. On the other hand, Grinblatt and Titman (1989) show that an asymmetric performance fee may encourage fund managers to take excessive risk. Their results sup- port Starkss (1987) model, wherein a symmetric performance fee is optimal and preferred to an asymmetric performance fee, given moral hazard on risk-taking. 5 Carpenter (2000), Elton, Gruber and Blake (2003), Ross (2004) and Palomino and Prat (2003) argue that an option-like fee does not necessarily distort managersrisk-taking incentives. In Panageas and Westerelds (2009) model, risk-seeking behavior arises from an option-like incentive fee, only in nite horizons. Das and Sundaram (2002) incorporate risk sharing and adverse selection and conclude that an option-like contract is optimal for investorswelfare. Another strand of the literature studies optimal compensation in the presence of in- formation asymmetry about managersskills. Bhattacharya and Peiderer (1985) present a model with information asymmetry about managersforecasting ability and propose an optimal compensation contract that screens out managers with inferior ability. But their model only considers actively managed portfolios and, thus, no information asymmetry about managerse¤orts. Ippolito (1992) suggests a simple model for actively managed funds, in which managers are either good (skilled and diligent) or bad (unskilled or fraudu- lent)byanexogenouslygivenprobability. Themodelshowsthatmarketscanmaintaingood funds (actively managed funds by good managers) when investors chase good performers. 5 Admati and Peiderer (1997) show that a choice of performance benchmark is critical for risk-taking incentives. Also see Ou-Yang (2003), Agarwal et. al. (2007), Basak et. al. (2008) for discussions about benchmarking. In Brennans (1993) model, a choice of benchmark portfolios a¤ects asset returns. Murphy (2000) examines the role of performance standards for corporate executivescompensation. 8 However, the model does not discuss the equilibrium choices of e¤ort by the managers. Dy- bvig, Farnsworth and Carpenter(2004)propose an optimalcompensation contractthat can a¤ect managersinvestment strategies in the presence of private information about stock prices. In their model, managerial skills are common knowledge and only actively managed funds are considered. Berk and Green (2004) argue that the fund ow-performance relationship is a result of investorsrationalresponses,andthatthelackofperformancepersistenceisduetoinvestors competition. In their model, investors chase good performance because it is the evidence of superior managerial skills. Nevertheless, investors cannot earn excess prots by investing in good performers because managersskills have decreasing returns to scale and investors compete to invest their money in those funds until the excess returns they earn decrease to zero. Sinceinvestorsupdatetheirexpectationsaboutskillsbasedonperformance,managers compensation depends on how much skills contribute to performance. Therefore, their model, like mine, predicts a weaker ow-performance relationship when the heterogeneity of skills is low and when the noise in performance is large. Yet, their model does not incorporate the issues of moral hazard and adverse selection that I consider in this chapter. 1.3. A statement of the problem I consider an investor who makes contracts with a fund manager. 6 To simplify the analysis, I assume that the investors utility is additively separable in investment income and transaction cost. 7 I assume the following. 6 I also considerd two fund managers and obtains the same conclusions 7 Without the separability assumption, i.e., U 0 = u(;d); dene u(;0) u(;d) and rewrite U 0 = u(;0) where is interpreted as disutility from compensationg the manager by d: When solving the problem, working with instead of d does not change the results of the model. 9 Assumption 1 The investor s utility is given by U 0 =u()d where u() is a strictly increasing and concave function of utility from investment in funds and d is compensation for the fund manager. I assume the disutility from compensating the manager is linear, in particular, equal to the compensation. On the other hand, the fund managers utility is additively separable in money and e¤ort. It is increasing and concave in compensation and decreasing in management e¤ort. I assumethatdisutilityfrome¤ortisequaltoe¤ort,butonecanassumeanyformofdisutility (e.g., convex). Assumption 2 Fund manager s utility is given by U =v()e where v() is a continuous, di¤erentiable, strictly increasing and concave function of utility from compensation (normalization of v(0) = 0): I assume the disutility from management e¤ort is equal to the e¤ort e without loss of generality. Asaresultoftheseseparabilityassumptions,Icanrestrictmyattentiontothecontract aboutfeesbetweentheinvestorandthefundmanager. Inotherwords,theproblemistond a compensation contract that minimizes the investors transaction cost while maximizing the fund managers utility. 10 I assume that the investor is of only one type but that the fund manager can have high (H)skillwithprobabilityorlow(L)skillwithprobability1:Afterthemanagerrealizes her skill type, she can choose to manage an active (A) or a passive (P) fund. Choosing a fund type corresponds to choosing between a high e¤ort (active management) and a low e¤ort level (passive management). The probability of choosing a fund type is endogenous and depends on the skill type. I denote the probability that the high skill manager and the low skill manager choose an active fund by H and L respectively. Thus, the probability that a passive fund is chosen is equal to 1 H and 1 L respectively. I denote the probability distribution over the four types by , a four by one vector: Denition 1 We denote the type of manager, skill and fund type, by t2T and the set of four possible types by T =f(H;A);(H;P);(L;A);(L;P)g. Active management (A fund) takes more e¤ort than tracking an index (P fund), which I normalize to zero without loss of generality. I also assume that the e¤ort for active management is equal or larger for the L skill managerthanfortheH skillmanager. Usingthenotationse H ande L respectively,Iassume e H;A =e H e L;A =e L , and e H;P =e L;P = 0: Assumption 4 The return distribution of the P fund does not depend on the manager s skill but that of the A fund does. The return distribution of the A fund managed by the H skill manager rst-order stochastically dominates the return distribution when managed by the L skill manager. Similarly, the return distribution of the P fund rst-order stochastically dominates the return distribution of the A fund managed by the L skill manager. I denote the stochastic dominance by F H;A (e r) F L;A (e r) and 11 F P (e r) F L;A (e r) where F() is a cumulative distribution function that depends on skills and fund types as represented in the subscript, and e r is the return on the funds. It is worth discussing an equilibrium implication of the stochastic dominance assump- tion. Only two equilibria are plausible: pooling equilibrium with passive management and separating equilibrium in which the H skill manager chooses the A fund and the L skill manager chooses the P fund (excluding the case in which the H skill manager is indi¤erent between active and passive management). When the return distribution of active manage- ment by the L skill type are stochastically dominated by that of passive management; the L skill manager will not choose the A fund in equilibrium. By the revelation principle, the type (L;P) manager should earn as much as she would earn by pretending to actively manage the fund (type (L;A)). Since the return distribution of passive management is su- perior than that of active management by theL skill manager, the payo¤s when the passive manager pretends to be of the type (L;A) must exceed the payo¤s when the manager is actually of the type (L;A): As a result, the type (L;P) manager must obtain larger payo¤s than the type (L;A) manager in equilibrium. This leads the L skill manager to always choose the P fund in equilibrium. In addition, the assumptions 1 and 4 imply that the investor strictly prefers the A fund managed by the H skill manager or the P fund to the A fund managed by the L skill manager. IassumethattheinvestordoesnothaveapreferencebetweentheAfundmanaged by the H skill manager and the P fund, but that his utility from investment is larger in separatingequilibrium(e.g., diversication)thaninpoolingequilibriumwithpassivefunds. I denote this by u S > u P where u S (u P ) is utility from investment in separating (pooling) equilibrium: 12 When only the fund manager, as opposed to investors, knows the critical informa- tion for the investment contractsmanagement skill and fund type, an equilibrium with transactions between them may not be guaranteed. Suppose, therefore, that the investor can ask an arbitrator to mediate the contract with the fund manager. The arbitrator re- ceives condential information about the fund managers skill and the fund type. Then the arbitrator suggests compensation for each manager type (skill and fund type). If both the investor and the fund manager agree on the suggestions, the investor makes the investment andpaysthefeesaccordingtotheadvice. Otherwise, thereisnotransactionbetweenthem: no investment and no fund management. The arbitrator, thus, must select a compensation rule (mechanism) for the fund manager, given the managers report of her own skill and fund type. I consider two kinds of compensation: xed fees that do not depend on returns of the fund and performance fees that are contingent on them. Denition The set of alternatives for the investor and the fund manager consists of the elements (m;;) where m is xed compensation (in dollar); and (e r) is the perfor- mance compensation scheme (in dollar) as a function of net return on the fund. I assume that both fees are (uniformly) bounded. Finally, is the probability of com- pensating the manager with 2 [0;1]: Choosing the compensation probability corresponds to a mechanism that chooses a probability distribution over the manager types that receive compensation, in the case of more than one managers. In other words, a mechanism fully species the probabilities that each manager type obtains compensation, for every report of types (see Myerson (1979)). 13 Yet, for one manager, specifying only the compensation probability for her reported type is su¢ cient and simplies the model. It is straightforward to show that xed compensation for the passive manager (i.e., (H;P) or (L;P) type) is always preferred to performance compensation since the manager is risk-averse (Assumption 2). Moreover, allowing performance compensation for the (L;A) type manager does not change the results of the model. First, there is no (L;A) type in equilibrium under Assumption 4. In addition, when the manager reveals her type (which is guaranteed by the revelation principle), the low type manager should receive the same expected payo¤s from compensation, whether the compensation is xed or contingent on performance. Thus, I restrict my attention to compensation schemes that provide only the (H;A) type manager with performance compensation without loss of generality. Denition 3 The Bayesian collective choice problem is summarized by (, T, U 0 ; U, ) where is the set of alternatives, T the set of the manager s types, U 0 the investor s utility, U the manager s utility, and the probability distribution over T. Denition 4 A response set S is the set of all possible responses of the fund manager to the arbitrator, given her true type t 2 T. A standard response set is S = T, when responses are restricted to reports of types. I assume that each response of the managers to the arbitrator is condential and noncooperative. The fund managers are expected to be truthful as long as they have no incentive to lie. For a standard response set, a truthful response is the identity map. Denition 5 A choice mechanism is choosing an element in given the information reported to the arbitrator by the manager, s2S or t2T: 14 Aresponseplanequilibriumisaresponseplanofthefundmanagersuchthatshecannot bebettero¤bychangingtoanotherresponseplan,givenamechanism. Therevelationprin- ciple states that we can generate any response plan equilibrium by an incentive-compatible direct mechanism. Hence, I restrict my analysis to a mechanism by which the manager report her true type. Denition 6 A Bayesian incentive compatible mechanism is the one that does not give any type of the manager an incentive to lie: Z(;tjt)Z(;sjt) for all s6=t2T for every t2T where Z(;sjt) =(s)v(m(s);(e r(s)))e t : (1) In words, Z(;sjt) is the expected payo¤when the fund manager of type t reports her type as s, given a choice mechanism : A Bayesian incentive compatible mechanism leads to a response plan equilibrium in which responses are standard and the response plans are the identity map. An allocation of conditionally-expected payo¤s when the manager is honest is denoted by H() fH(jt)g t2T where H(jt) = Z(;tjt) given : In essence, this allocation of payo¤siswhatwecanachievewith truthful responseswhentheresponsesetis standard (re- porttypes). Equivalently, wecanachievethisallocationbyaBayesianincentivecompatible mechanism. This leads to the following denition of an incentive-feasible set. 15 Denition 7 An incentive-feasible set of allocations of conditionally-expected payo¤s is F =fH(): is is a Bayesian incentive compatible mechanism}. The allocation of conditionally-expected payo¤s when all managers are honest is pos- sible only through a mechanism which ensures that the fund manager reveals her type, that is, a Bayesian incentive compatible mechanism. Therefore, an incentive-feasible set is restricted to the set of allocations that can be achieved by a Bayesian incentive compatible mechanism. Denition 8 The conict outcome occurs when the investor and the fund manager fail to agree. It is no-transaction between them. They receive their reservation utilities in the conict outcome. Denition 9 A reference point is the payo¤s from the conict outcome. For simplicity, I assume that the fund managers reservation utility is zero for every type (relaxing the assumption does not change the results). The reservation utility of the investor is expected payo¤s from other investment alternatives. I denote the investors reservation utility by w 0 and will discuss how equilibrium outcomes change as w 0 changes, e.g., as other money management industries such as hedge funds grow. TheconictoutcomecanalwaysbeachievedbyaBayesian-incentivecompatiblemech- anismsincenofundmanagerhasanincentivetoliewhenthemechanismchoosestheconict outcomenomatterwhichtypeisreported. Thus,thepayo¤sfromtheconictoutcome(ref- erence point) are in F . 16 1.4. E¢ cient bargaining solution To achieve e¢ ciency, the arbitrator must nd a Bayesian incentive compatible mech- anism that is Pareto optimal. No other incentive compatible mechanism can make some better o¤without making others worse o¤. I use the interim welfare criterion for e¢ ciency, which considers the expected payo¤ for the fund manager conditional on her type. 8 In other words, the fund managers private information about her true type is considered in her expected payo¤s when we determine whether the manager would be better or worse o¤ by an alternative allocation. The interim-e¢ cient allocation in my model is durablein a sense that the investor and the fund manager will not unanimously approve a change to any other allocationsbecause only the fund manager has private information (Holmstrom and Myerson (1983)). Thearbitratorsproblemisa bargaining problemsincethefeasiblesetofexpectedallo- cation of payo¤sF includes a reference point when the investor and fund managers receive their reservation utilities, as discussed in Section 2. Moreover, in practice, both investment amount by investors and fee ratios charged by fund managers determine compensation for fundmanagers. Thus,amechanismthatgivesallsurplusestoeithertheinvestororthefund manager may not represent the industry. Rather, I consider mechanisms that provide both parties with positive surpluses. In particular, Harsanyi and Selton (1972), extending Nash (1950), identifyafeasiblesolutiontoabargainingproblemforN agentsasavectorfx t i g N i=0 that maximizes a generalized Nash Product. The generalized Nash Product is given by N Q i=1 ( Q t i 2T i (x t i w t i ) p(t i ) ) ; (2) 8 See Holmstrom and Myerson (1983) for detailed discussions about welfare criteria. 17 where x t i is the (expected) utility and w t i the reference point of the agent i of type t i 2T i ; and p(t i ) is the probability that the agent i is of type t i : In my model with one investor and one manager, the generalized Nash Product is simplied to (x 0 w 0 ) Q t2T x p(t) t where x 0 and w 0 are the investors expected utility and reservation utility and x t is the type t managers expected utility: Consequently, the arbitrator maximizes the generalized Nash Product over the set of feasibleallocationsofconditionally-expectedpayo¤sdenotedbyF (Denition7):However, no one will participate in the contract if the expected payo¤by a mechanism suggested by the arbitrator is less than the reference point (Denition 9). Therefore, at a minimum, an implementable mechanism design should provide reservation utility to the investor and the fund managers. Thus, an e¢ cient solution for N agents is the one that maximizes the generalized Nash Product over the set F + given by F + =F \fx :x t i w t i for all i and all t i 2T i g: Assumption 5 The allocation of conict outcome is strictly dominated. Since the choice set is compact (Denition 2), there exists a unique incentive-feasible bargaining solution under Assumption 5 (Myerson (1979)). In addition, Assumption 5 implies that the solution that maximizes the generalized Nash Product should also strictly dominate the conict outcome. Thus, I can maximize the log of the generalized Nash Product given by (2) when N = 2 (one investor and one manager), log(x 0 w 0 )+ P t2T p(t)logx t : 18 Therefore, the arbitrator must nd an e¢ cient Bayesian incentive compatible mecha- nism,f(t); m(t); (e r(t))g t2T ; which solves the following problem: max f(t);m(t);(e r(t))g t2T log(x 0 w 0 )+ P t2T p(t)logx t (3) subject to Z(;tjt)Z(;sjt) for all s6=t for every t2T (4) Z(;tjt) 0 for every t2T (5) (t) 2 [0;1] for every t2T; (6) where x 0 is the expected utility of the investor and x t =Z(;tjt) is the expected utility of the manager of type t when she truthfully reports her type. According to Assumption 1, the expected utility of the investor is given by x 0 = P t2T p(t)fu(t)(t)d(t)g = u P t2T p(t)(t)d(t); (7) where u P t2T p(t)u(t) is the expected utility from investment. 9 By Assumption 2, the expected utility of the manager of type t, Z(;tjt); is given by x t Z(;tjt) =(t)E t [v(m(t)(11 ft=(H;A)g )+(e r(t))1 ft=(H;A)g )]e t ; (8) 9 The value depends on p(t): In separating equilibrium, p(H;P) = p(L;A) = 0, and I denote the value by u S : In pooling equilibrium, p(H;A) = p(L;A) = 0, and I denote the value by u P : 19 where 1 ft=(H;A)g is the indicator function for the type (H;A) since the performance fees are only given to that type. E t [] is the expectation under the distribution of returns that depends on the type t. Nowwesolvethearbitratorsproblemgivenby(3)to(6)fortwocases: noperformance fee, and performance fee for the type (H;A) manager. I examine the rst case, considering that few mutual funds actually charge performance fees. In other words, the rst case assumes that only xed fees exist. On the other hand, even without explicit performance fees, investment ows can compensate performance by responding to performance when compensation is proportional to assets under management (industry practice). Thus, given this ow-performance relationship, the second case has no restrictions on fee structures. 1.4.1. Special case (no performance fee) Suppose that the manager receives only a management fee. This is a special case in which (e r(t)) =m(t) (independent ofe r). Then the equation (8), the expected utility of the manager of type t; becomes x t =(t)v(m(t))e t : (9) Proposition 1 The incentive compatibility constraints imply that the manager receives the same expected utility from the management fee irrespective of her skill and fund type. In essence, the set of feasible allocations F consists of those allocations that provide every type of manager with the same expected utility. 20 Proof. When the fund manager of type t truthfully reports her type, she receives the expected utility given by (9). However, if she reports a type s6=t; she receives Z(;sjt) =(s)v(m(s))e t : She will not lie if and only if (t)v(m(t))e t (s)v(m(s))e t for all s6=t: (10) The inequality (10) should hold for every t2T; which implies (t)v(m(t)) =(s)v(m(s))b y for all t;s2T. (11) In essence, the expected utility from the management fee does not depend on fund types and skills. I denote this expected utility for all types byb y: Corollary 1 Without performance fees, all managers choose to manage the P fund rather than the A fund (i.e. a pooling equilibrium arises). Proof. By Proposition 1, the expected utility from fee income is the same for both fund types. While the P fund requires no cost of e¤ort, managing the A fund decreases expected utility by the disutility from managing the A fund. Therefore, the manager is better o¤by choosing the P fund. Proposition 2 Without performance fees, the expected utility for the passive manager b y 21 decreases as (1) the investors utility from investing in passive funds u P decreases, or (2) the investors reservation utility from other investments w 0 increases. Proof. Let me dene v t = v(m(t)) and Q(v t ) = m(t) where Q = v 1 : That is, v t is the utility of the fund manager of type t when she receives a xed fee. Since v is strictly concave, Q is strictly convex. In equilibrium, as Appendix A shows, the probability to pay the fee is one, and the expected utility from passive managementb y is given by the following equation: 1 b y |{z} marginal benet = Q 0 (b y) u P Q(b y)w 0 : | {z } marginal cost The LHS is the marginal increase in social welfare (Equation (3)) when increasing one unit of utility of the manager. The RHS is its marginal cost since the investor should incur compensation cost. After rearrangement, I obtain b yQ 0 (b y)+Q(b y) =u P w 0 : (12) Note that the LHS of Equation (12) is increasing inb y since Q is strictly convex. Thus, it is straightforward to show @b y @u P > 0; @b y @w 0 < 0: Using Q(), the optimal xed fees, denoted by b m; are given by b m =Q(b y): Propositions 1 and 2 highlight the main results of the model when the managers com- pensation does not depend on performance. In essence, the H skill managers e¤ort is distorted provided that an optimal level would be active management without the restric- tion on the fee structure. The investor does not know who is skilled and for which funds 22 managerial skills matter. Yet, the investor must compensate the fund manager before he observes the fund performance. As a result, the investor ends up paying the same manage- ment fee on average to every type of the manager, and all types of the manager receive the same expected utility from fee income. This, in turn, leads every type of the manager to passively manage a fund. In this case, it is socially optimal that the investor pays fewer management fees when he has better outside investment opportunities like hedge funds. Table 1.1. Comparative statics in an e¢ cient pooling equilibrium u w 0 y + Note: y (a managers expected utility from the management fee), u (investorsutility from investment in mutual funds), w 0 (investorsreservation utility from other investments). Conversely,onecaneasilyverifythatoptimalcompensationinthepoolingequilibrium managerstrackindexesshouldbeaxedfee. Whenthereisnoneedtoseparatemanagers of di¤ering ability, xed compensation is e¢ cient since managers are risk-averse (Assump- tion 2). In this case, the restriction of (e r(t)) =m(t) does not a¤ect equilibrium outcomes. As a result, expected payo¤s to the passive manager in the pooling equilibrium should also satisfy Equation (12). Therefore, the above solution is also the optimal compensation in the pooling equilibrium. 1.4.2. Performance fee I solve a more general problem without restricting (e r(t)) to a constant, i.e., the type (H;A) manager receives a performance fee (plus a management fee), which is contingent 23 on e r. The expected utility of the manager of type t is given by the equation (8). I rst derive the optimal performance fee for the active manager and the optimal xed fee for the passive manager. Then I show that with a performance fee, we can achieve separating equilibrium provided that alphavalue added by the active management by the H skill managerexceeds some threshold. I restrict my analysis to the set of parameters so that the threshold is less than one (100%). Proposition 3 In separating equilibrium, optimal performance compensation (e r) is non- linear. It is positively related to the likelihood ratio of returnlikelihood that the fund is actively managed by the H skill manager compared to the likelihood that it is passively managed. As a result, it is increasing with e r; only when the monotone likelihood ratio property holds. Proof. Appendix A shows that the managers utility from performance compensation v(e r) satises Q 0 (v(e r))E HA [Q 0 (v(e r))]_ (1 1 LR(e r) ); (13) (_ indicates that they are equal up to a constant) where LR(e r) is the likelihood ratio, LR(e r) = f HA (e r) f P (e r) : Recall that Q 0 () is increasing by convexity of Q() under the assumption that the manager is risk-averse (Q =v 1 ). In words, Equation (13) implies that the investorsmarginal cost of providing the manager with additional unit of utility is increasing with performance only when the likelihood ratio is increasing. In particular, when the likelihood ratio is equal to 24 one,themarginalcostisthesameasitsaverage. Onlywhenthelikelihoodthatthemanager is of the (H;A) type is larger than the the likelihood that she passively manage the fund (the ratio exceeds one), the marginal cost is larger than the average. Since the marginal costisincreasing, thisimpliesthatthemanagerreceivesmoreutilitythantheaveragewhen she is more likely to be the H skill manager and engage in active management. Then, the incentive fee schedule (e r) is given by (e r) =Q(v(e r)): SinceQisanincreasing,convexfunction,performancecompensationalsoincreasesprovided that the monotone likelihood ratio property holds. Figure 1.1 illustrates how the optimal incentives fees (e r) depend on the likelihood ratio. Only when the ratio is increasing, the incentive fees also increase with performance (top). Otherwise, the fees are not increasing (bottom). Note that the solution v(e r) that satises Equation (13) only depends on the likelihood ratio. Since other parameters do not determine the optimal compensation, I denote the solution by v(e r): Using this solution, I dene alpha,value added by the active management by the H skill manager (compared to passive management), by the following: = E H;A [v(e r)]E P [v(e r)] E P [v(e r)] : (14) Only the likelihood ratio ( f HA (e r) f P (e r) ) determines : In other words, given the performance distributions, is also given. 25 Figure 1.1. Likelihood ration and optimal performance fees The optimal performance fees (red line) vary only according to the likelihood ratio f HA (e r) f P (e r) (dash line) that the manager has the H skill and actively manages the fund, compared to the likelihood that she tracks a market index. Performance of the active fund by theH skill manager is distributed by normal with mean 5% and the standard deviation 10% (15%) for the top (bottom) gure. Performance of the passive fund follows a normal distribution with mean 0% and standard deviation 10%. I assume a power function for the managers utility from income v(): 0.2 0.1 0 0.1 0.2 0 1 2 3 likelihood ratio performance 0.2 0.1 0 0.1 0.2 0 1 2 3 likelihood ratio performance 0.2 0.1 0 0.1 0.2 0 2 4 6 8 incentive fees performance 0.2 0.1 0 0.1 0.2 0 0.5 1 incentive fees performance 0.2 0.1 0 0.1 0.2 0 1 2 3 likelihood ratio performance 0.2 0.1 0 0.1 0.2 0 1 2 3 likelihood ratio performance 0.2 0.1 0 0.1 0.2 0 2 4 6 8 incentive fees performance 0.2 0.1 0 0.1 0.2 0 0.5 1 incentive fees performance Proposition 4 In separating equilibrium, the optimal xed fee for the passive manager and her expected utility increase with for small but decreases with it for large : Proof. As Appendix A shows the optimal probability to pay a xed fee to the passive manager,(L;P);isequaltoone, andtheexpectedpayo¤forthepassivemanager, denoted 26 by y; is a solution to the equation, yq()(u w 0 (1)Q(y)+e H )e H = 0; (15) where q() is a quadratic function as given by q() = 2 +((1+)+ )+ ; (16) E HA [Q(v(e r))] E P [v(e r)] and E HA [Q(v(e r))] E P [v(e r)] E HA [Q 0 (v(e r))] ( > 0 and > 0): Let me dene the left-hand side of (15) as F(y;): By the implicit function theorem, @y @ = @F(y;) @ @F(y;) @y : (17) First, @F(y;) @y =q()+(1)Q 0 (y)> 0 since Q() is increasing. Also, @F(y;) @ = yq 0 ()(u w 0 (1)Q(y)+e H ) = y(q 0 () q() )+ e H ; (18) where the last equality (18) uses Equation (15). Note that q 0 () q() = 1 isincreasingwith. Specically, itisnegativeforsmallandpositiveforlargealpha. Asa result, for someb > 0; we have @F(y;) @ 0 for b and @F(y;) @ 0 for b : Therefore, 27 by Equation (17), we obtain @y @ 0 for b ; and @y @ 0 for b : Since Q() = v 1 (); the optimal xed fee, which I denote by m, is given by m = Q(y): By Equation (15), the optimal compensation for the passive manager should satisfy v(m)q()(u w 0 (1)m+e H )e H = 0: Since Q() is increasing, m also increases for small ( b ) but decreases for large (b ): Proposition 5 In separating equilibrium, the expected utility for the active manager in- creases with : Proof. I dene the expected utility premium as the di¤erence of expected utility of the H skill manager between managing the A fund and the P fund: In other words, expected utility for the active manager is equal to y+; which is given by y+ = (+1)ye H : (see Appendix A). Using Equation (19) and (17), @(y+) @ =y+(+1) @y @ =y(+1) @F(y;) @ @F(y;) @y : 28 As Appendix A shows, we have y @F(y;) @y > (+1) @F(y;) @ ; (19) and, therefore, @(y+) @ > 0: Note that q(0) = by Equation (16): Thus, when = 0, we have y = e H since Q(0) = 0 (Assumption 2 and Q = v 1 ): Then by Equation (19), y + = 0. In addition, givenProposition3and4,y isincreasingandthendecreasingwithrespecttoalphawhereas y + is always increasing. Thus, one can easily see that y and y + should cross at one point, which I call : When > ; expected payo¤s of the active manager exceeds those of the passive manager (y+>y): In this case, the H skill manager is better o¤by activelymanagingthefund(separatingequilibrium). Otherwise,poolingequilibriumarises. Intuitively, when value of active management by the H skill manager is low, indexing is optimal. I formally show this result in Corollary 1. Corollary 1 The separating equilibrium is possible when value of active management by the H skill manager is high relative to a threshold, > ; (20) where is some threshold. Otherwise, we have a pooling equilibrium in which the manager always passively manages the fund. 29 Proof. By Equation (19), the utility premium earned by the active manager is given by =ye H ; (21) which increases with (propositions 3 and 4). Let me dene as the solution for = 0: Since is increasing, we have > 0 when > ; which is the condition for separating equilibrium. By Equations (22), the expected utility forthepassivemanagerisgivenbyy = e H :UsingEquation(15), isgivenbythefollowing equation: e H (1+) (u w 0 (1)Q( e H ))+ e H = 0: (22) A performance fee can screen managers of di¤ering ability when it is enough to com- pensatetheH skillmanagerbutnotenoughtoenticetheLskillmanager. Yet,performance compensation cannot be large when active management does not add much value. This is the case in which skills for active management are not su¢ ciently superior or noise in per- formance is large. The intuition is that when the two skill levels, H and L, are similar (or performance is too noisy to discriminate di¤erent skills), the low skill manager can easily mimic the high skill manager. Thus, separating them requires more bribes(information rents) to the low skill manager and less performance compensation for the active manager. When the decreased incentive fee cannot cover the high skill managers cost of e¤ort for 30 active management, the manager is better o¤by passively managing a fund. This leads to the pooling equilibrium in which both high and low skill managers passively manage funds. Figure 2 illustrates the separating equilibrium condition, assuming that Q() is a quadraticfunction. Ishowexpectedutilityofthemanagerasafunctionof withotherpa- rameters xed. As discussed, when = 0; expected utility of the passive manager, y (dash line), is equal to e H while that of the active manger, y+ (red line), is zero. The passive manager receives more as increases for small but less for large : On the other hand, the active managers expected payo¤s increase with : The two expected utilities cross at some value, which I call : In the region right to , we have the separating equilibrium while the pooling equilibrium arises in the region left to the threshold. Figure 2. Expected utility for active and passive managers The dash line is the expected utility for the passive manager y() and the red line is the expected utility for the active manager (y+)(): The two functions cross at the threshold for separating equilibrium. I assume a power function for the managers utility from income v(): 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 α* pooling separating ← cost α expected payoffs (net of cost) passive manager active manager 31 As a special case, consider e H = 0: When the H skill manager does not need to exert any e¤ort for active management (or have no disutility from management e¤ort for active management), the threshold becomes zero (use Equation (22) with e H = 0), and the condition for separating equilibrium, Equation (21), always holds since is nonnegative. Thus, separating the high skill and the low skill managers is always e¢ cient when the high skill manager is so e¢ cient that her disutility from active management is zero (see Figure 1.3 (B)). The following corollaries show how expected payo¤s to the passive manager and the active manager depend on other parameters in the model. Corollary 2 In the separating equilibrium, expected utility for the passive manager de- creases as (1) skilled managers are more e¢ cient for active management (their cost of e¤ort for active management e H decreases), (2) the fraction of skilled managers increases, (3) the investors utility from investment in the funds u decreases, or (4) the investors reservation utility from other investments w 0 increases. Proof. See Appendix A. Corollary 3 In the separating equilibrium, the expected utility for the active manager decreases as (1) skilled managers are less e¢ cient for active management (their cost of e¤ort for active management e H increases), (2) the fraction of skilled managers increases, (3) the investors utility from investment in the funds u decreases, or (5) the investors reservation utility from other investments w 0 increases. Proof. See Appendix A. 32 Table 1.2. Comparative statics in an e¢ cient separating equilibrium e H u w 0 y + + + y+ + Note: y (expected utility for the passive manager), (expected utility premium for the active manager), e H (H skill managers cost of e¤ort for active management), (fraction of H skill managers), u S (investorsutility from both fund types), w 0 (investorsreservation utility). Figure 1.3. Changes in expected utility for active and passive managers The dash line is the expected utility for the passive manager y() and the red line is the expected utility for the active manager (y+)(): The two functions cross at the threshold . Figures show how two functions change when (B) cost of e¤ort for active management decreases to zero; (C) investorsreservation utilityw 0 increases; (D) the fraction of skilled managers increases. (A) is the same as Figure 1.2. I assume a power function for the managers utility from incomev() 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H (A) expected utility 0 0.5 1 0 0.05 0.1 0.15 0.2 α* → separating (B) e H decrease to 0 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H expected utility (C) w 0 increase 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H (D) λ increase passive active 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H (A) expected utility 0 0.5 1 0 0.05 0.1 0.15 0.2 α* → separating (B) e H decrease to 0 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H expected utility (C) w 0 increase 0 0.5 1 0 0.05 0.1 0.15 0.2 α* 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H (A) expected utility 0 0.5 1 0 0.05 0.1 0.15 0.2 α* → separating (B) e H decrease to 0 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H expected utility (C) w 0 increase 0 0.5 1 0 0.05 0.1 0.15 0.2 α* pooling← → separating ← e H (D) λ increase passive active 33 The main results in the corollaries 2 and 3 are summarized in Table 1.2 and Figure 1.3. Wheninvestorshavebetteroutsideinvestmentopportunities(Figure1.3(C))andwhen theirutilityfrominvestmentinmutualfundsdecreases,theypayfewerperformancefeesand managementfees. Finally,whenskilledmanagersaccountforalargerfraction(increases), the investor also pays fewer performance fees and management fees. Intuitively, since a manger is more likely to be skilled, investors provide fewer bribes to passive managers. The active manager also receives less as increases since expected utility premium for active managers depends only on alpha and their cost of e¤ort. Proposition3suggeststhattheseparatingequilibriumrequiresasu¢ cientlevelofskills for active management. In particular, the separating equilibrium is unlikely when alpha is low relative to the threshold. This can happen due to a lower skills or a higher threshold. Corollary 4 suggests the conditions that increase the threshold (Table 1.3 shows how the threshold changes). For instance, as managers are less e¢ cient for active management, the required level of skills increases. Moreover, as there are more skilled managers in the industry, the benchmark level of skills becomes higher. On the other hand, if investors receive fewer payo¤s from investment in mutual funds or more from other investments like hedge funds, the standard also increases. When the managers skills are inferior compared to the higher standard, we have the pooling equilibrium in which managers do not actively manage funds. Corollary 4 A separating equilibrium is not likely if (1) skilled managers are ine¢ cient for active management (their cost of e¤ort for active management e H is high), (2) the fraction of skilled managers is large, (3) the investors utility u S is low, or (4) the investors reservation utility from other investments w 0 is high. 34 Proof. See Appendix A. Table 1.3. Comparative statics for the skill threshold e H u S w 0 + + + Note: (threshold of active management value), e H (cost of e¤ort for active management), (fraction of skilled managers), u S (investorsutility from investment in the separating equi- librium), w 0 (investorsreservation utility). 1.5. Conclusion When investors have only noisy information about managersskills and fund types (active or passive), fund managers can behave strategically. Yet, even under information asymmetry, if active managerscompensation depends on performance, and if performance issu¢ cientlyheterogeneousacrossskills,themarketcanscreenmanagersofdi¤eringability and achieve a separating equilibrium in which high skill managers actively manage funds whilelowskillmanagerstrackindexes. Imodeltheconditionsunderwhichsuchaseparating equilibrium obtains. If these conditions fail, the market fails to reward skills and active management. This, in turn, makes it optimal for managers to passively manage funds regardless of their skills and creates a pooling equilibrium. The recent growth in passive management (e.g., closet-indexing) suggests that skilled managers have recently faced reduced incentives for active management. I show that a lack of incentives for active management or lower performance compensation arises when there is a deterioration of skills in the pool of potential managers, more noise in performance, or an improvement of investorsinvestment alternatives (e.g., hedge funds). These results rationalize recent trends in the mutual fund industry. 35 Chapter 2. Changes in Mutual Fund Flows and Managerial Incentives 2.1. Introduction An important issue in the agency literature is the incentive e¤ects of compensation structures on agentsreal behavior. In the mutual fund industry, given that fees are pro- portional to assets under management, the relationship between money ows and past performance can induce implicit performance compensation. Brown, Harlow, and Starks (1996) and Chevalier and Ellison (1997) argue that convexity in the relationship provides managers who are behind the markets (or their peers) with incentives to take more risk toward the end of the year in an attempt to improve performance and thereby increase inows in the following year. 10 Underperforming managers may engage in this risk-shifting even at the expense of investorsinterests because of this option-like payo¤. This chapter examines whether managers indeed respond to the incentives provided by this implicit compensation scheme. To this end, I look at how their risk-shifting varies according to the shape of the relationship between net ows and past performance (e.g., benchmark-adjusted returns in the prior year). I show that contrary to the common view in the literature that the relationship is convex, its shape depends on conditioning vari- ables, particularly market volatility and performance dispersion across funds. I then study 10 Earlier studies that document the convex ow-performance relationship include Ippolito (1992), Goet- zmann and Peles (1996), Gruber (1996), Chevalier and Ellison (1997), and Sirri and Tufano (1998). In the models of Starks (1987) and Panageas and Westereld (2009), option-like incentive fees can lead to man- agersrisk-seeking behavior. Koski and Ponti¤(1999) argue that fund managers use derivatives to manage unexpected cash ows rather than to take more risk in an attempt to increase expected ows. Busse (2001) and Elton et. al. (2009) nd di¤erent results than those in Brown, Harlow, and Starks, when using daily return and monthly holding data respectively. Bergstesser and Poterba (2002), ONeal (2004), Johnson (2007), and Ivkovic and Weisbenner (2009) document that outows are unrelated to past performance and (net) ow-performance relationship arises from the response of inows to past performance. 36 managers risk shifting as it relates to time variation in the shape of the ow-performance relationship. I rst show that the ow-performance relationship varies over time. Consistent with earlier studies, it is convex from 1983 to 1999, but it is not convex in the 2000s. Intuitively, investors become less responsive to high performance because performance appears noisier in the 2000s. I nd that the expected marginal ow to high-performing funds is lower given high-volatility markets and low performance dispersion. Controlling for market volatility and performance dispersion, we cannot reject the hypothesis that ows are linearly related to performance. Managersrisk-shifting is also di¤erent depending on the two conditioning variables. Indthatmanagersperformingworsethanthemarketsreduceshiftingriskafter 2000. The level of risk-shifting can be also explained by performance dispersion and market volatility. When the expected shape of the ow-performance relationship conditional on thosevariablesisnotconvex(i.e., whenperformanceislessdispersedandwhenmarketsare more volatile), underperformers do not engage in risk-shifting. My results are consistent with the view that managersrisk-shifting responds to the nancial incentives provided by the ow-performance relationship. Using kernel regressions for the ow-performance relationship, I nd substantial de- creases in the expected marginal ow to high-performing funds over the period from 2000 to 2008, after controlling for the e¤ects of fund characteristics and money ows to the mu- tual fund industry as a whole (Figure 2.1). As a result, a fund outperforming the market value-weighted index by 20% attracted annual net ows of 30% prior to 2000 on average but only 10% in the subsequent decade. Given the average fund sizes of $1 billion and $1.4 billion during those periods, respectively, this type of fund had annual net inows of 37 $300 million prior to 2000, but only $140 million after that. Ordinary least square (OLS) regressions also conrm these changes. The coe¢ cient on squared performance decreases from 0.6 to -0.3 after 2000, which suggests a change from a convex to a concave relationship (when the relationship is increasing, the predominant di¤erence in marginal ow for high performance can be captured by convexity or concavity). I also nd a similar change in the shape from convexity to concavity using a piecewise OLS regression. Figure 2.1. Flow-performance relationship and 90% condence interval The y-axes represent expected annual net ows into and out of nonindex funds from 1983 to 1999 (before 2000) and from 2000 to 2008 (after 2000). Performance is annual returns minus the CRSP value weighted index (CRSP). The expected annual net ows are estimated using kernel regressions suggested by Robinson (1988) after controlling for contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, style ow and lagged ow. Log age is the natural logarithm (log) of the months since the inception date of a fund (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index fundsbutexcludingsectorfundsandinternationalfunds). Fundsareaggregatedacrossshareclasses, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 lagged return over the market valueweighted index expected annual net flows before 2000 after 2000 38 Ishowthattime-variationintheow-performancesensitivity(expectedmarginalow) contributes to the concave shape in the 2000s. Throughout the period from 1983 to 2008, the ow-performance relationship is concave in highly volatile markets, whereas it is convex in low-volatility markets. I also nd that the shape is more convex when performance is more dispersed across funds (after controlling market volatility). For example, 10% of performance dispersion reduces the concavity in highly volatile markets by around half. 11 TheseresultsareconsistentwiththepredictionsinBerkandGreen(2004)andChapter 1 that the marginal ow to high-performing funds is low when performance is less attribut- able to skills. In both models, investors use past performancerepresented as skills plus noiseas an inference of managersabilities (and e¤orts). When performance reects more luck relative to skill, the sensitivity of ows to superior performance is low. The contribu- tion of skills to performance increases with cross-sectional variation in skills but decreases with variation in noise. As a result, high-performing funds receive less on the margin when skills are less heterogeneous across managers and when performance is noisier. 12 Changes in risk-shifting behavior are consistent with the view that managers respond to the incentives provided by the implicit performance compensation. I examine variations in the relationship between performance and risk-shifting. Low-performing funds typically take more risk in the fourth quarter, but this risk-shifting is signicantly reduced after 2000, decreasing by about 70%. The same variables that determine the shape of the ow- 11 Usingrankingofrawreturnsasaperformancemeasure,Ialsondasubstantialdecreaseinmarginalow to top-ranked funds in the post-2000 period. Performance dispersion increases convexity of the relationship between net ows and ranking in the prior year, but market volatility appears statistically insignicant for the relationship. See Section 2.4.3 for the details. 12 In the model presented in Chapter 1which accomodates both adverse selection and moral hazard optimal incentive fees depend on the likelihood that the manager has high skills and actively manages the fund relative to the likelihood that she follows a passive index strategy. Thus, there is no intrinsic shape in the ow-performance relationship: it can be convex or concave. Del Guercio and Tkac (2002) nd a symmetric relationship for pension funds, and Kaplan and Schoar (2005) nd a concave relationship for private equity funds. 39 performance relationship also explain these changes. After conditioning the relationship between performance and risk-shifting, I nd that managers who are behind the markets tend to increase risk in the fourth quarter when performance in the middle of the year is more dispersed across funds. In periods when performance dispersion is low, it is the high- performing managers who engage in such risk-shifting. I also show that low-performing funds tend to take more systematic risk in the rest of the year when market volatility up to the third quarter is low. In addition to proposing a conditional relationship between ows and performance 13 and documenting changes in managersrisk-shifting behavior according to the relationship, this chapter makes several contributions. First, my ndings provide an explanation for the recent growth in passive management in the mutual fund industry, such as closet- indexing. Managers face fewer incentives to engage in active management because of lower marginal ow to high-performing funds. As suggested in Chapter 1, managers track market indexes when performance compensation is low, which occurs when active management is less valuable, for instance, due to less heterogeneous skills, noisier performance, and more competition from other investment vehicles. Second, while investors are less attracted to superior performance in the 2000s, fund ows are negatively related to expense ratios. I nd that funds with high expense ratios had more inows before 2000 (see also Barber, Odean, Zheng (2005) and Huang, Wei, and Yan (2005)). Yet, after 2000, my results show that a 1% increase in the ratios led to a 3-5% decrease in net ows. This is consistent with recent anecdotal evidence that past 13 Olivier and Tay (2009) and Wang (2009) show that contemporaneous GDP growth a¤ects the ow- performance relationship. My paper uses lagged variables to form a conditional expectation about the shape of the relationship. 40 performanceisnolongerthemostimportantfactor, whereasfeeshavebecomecriticalwhen investorschoosemutualfunds. AccordingtoasurveybytheInvestmentCompanyInstitute in 2006, more investors consider fees rather than past performance (see Investors Flock to Low-Cost Funds,J. Clements, 2007, The Wall Street Journal). As discussed in Appendix, the negative e¤ect of expense ratios after 2000 seems to be associated with a decrease in 12b-1 fees, most of which are used to compensate nancial advisers. Third, my results are consistent with investors attempting to distinguish skills associ- atedwithactivemanagementfromapassiveindexstrategy. Indthattheow-performance relationshipisinsignicantforindexfundsinmostcases(seealsoElton, Gruber, andBusse (2004)). Nonetheless, there is a strong relationship between index fund ows and perfor- mance compared to the funds with the same value and size characteristics. Given that outperforming those funds requires more than passive management, the results are con- sistent with the view that investors chase good performance because they perceive that it represents skills (e.g., Gruber (1996) and Zheng (1999)). Also, the changes in the ow- performance relationship according to market volatility and performance dispersion that I document seem less supportive of a story where investors irrationally chase recent winner funds (e.g., Sapp and Tiwari (2004)). 14 14 I also nd that in the areas where hedge funds are concentrated (New York and Boston), marginal ow to high performing funds decreases more compared to other areas after 2000. The market share of the funds (whose managers are) in those areas decreases from 50% in 1999 to 30% in 2008. These results can support a view that investors become less sensitive to high-performing funds because skilled managers leave the industry (e.g., a brain drain to hedge funds; see Kostovetsky (2007) and Massa, Reuter and Zitzewitz (2009)). 41 2.2. Flow-performance relationship 2.2.1. Data and variable description I obtain market indexes and mutual fund data from Morningstar, including returns, total net assets (TNA), 9 style categories (value and size), index fund ags, and funds benchmark indexes as disclosed in fund prospectuses. I aggregate across share classes based on their TNA. 15 Market returns are proxied by the Center for Research in Security Prices value-weighted (CRSP VW) index. My sample covers all U.S. equity mutual funds, excluding index funds (according to fund prospectuses), sector funds, specialized funds and international funds, from 1983 to 2008 annually (data starts in 1980 to obtain lagged variables). To compare my results with Chevalier and Ellison (1997), I follow their sampling criteria. I remove small funds (assets less than $10 million) and young funds (less than 3 years old) as of the beginning of the periodoverwhichfundowsaremeasured. 16 Ialsoexcludethefundsthatareclosedtonew or all investors in their closing years and afterwards (See Bris et. al. (2007) for a detailed discussion about fund closures), institutional share classes, funds that acquire other funds in their merger years, and the funds that are liquidated within 6 months before the date that fund ows are measured. Fund ows are measured annually at the end of December and lagged performance over the preceding calendar year. I dene fund ows as a percentage change in TNA, net of capital gains and dividends from investments. I measure net ows of a fund i at year t by 15 I obtained similar results using the CRSP data. Using only Morningstar increases the sample size (no cross-identication between two data sets) and better aggregates across share classes. 16 Young funds may go through an incubation process. Chevalier and Ellison (1997) include funds between two and three years old. I exclude them since I also use lagged ows as a control variable. Results are similar if I include them. 42 net flow i;t = TNA i;t TNA i;t12 TNA i;t12 r i;t (23) where r i;t is the funds return over the period from t1 to t. 17 Following Chevalier and Ellison (1997), I measure performance as excess returns over CRSP VW returns. Since I focus on investorsreactions to performance, I also use simple performance measures that may be readily available to investors. In particular, relative returns compared to benchmark indexes or to the S&P500 index are available on fund com- panieswebsitesornancialwebsitessuchasYahoo! Finance. DelGuercioandTkac(2002) show that excess returns over market indexes, such as the S&P500 index, are important determinants of mutual fund ows. They also provide evidence that mutual fund ows are related to factor-adjusted performance measures as the sophisticated measures are corre- lated with readily available measures. When a funds benchmark index is missing, I use the most frequently used benchmark by other funds with the same styles (Sirri and Tufano (1998) use relative returns compared to the funds with the same investment objectives). Finally, I also compute average returns on equity funds in the same style categories (peer funds), which I call style returns. Relative returns compared to style returns is likely to be important if investors select funds based on value and size characteristics and make comparisons among funds within the same style category. To summarize, I use four perfor- mance measures depending on benchmark returns: CRSP VW, S&P500 index, the funds benchmark index, and style returns. Othervariablesusedintheregressionsincludefundsize,age,expenseratiosandvolatil- 17 To avoid e¤ects due to measurement errors or extreme observations, I winsorize fund ows at 1% and 99% levels following Barber, Odean and Zhang (2005). The conclusions of the paper do not depend on those outliers. 43 ity. Many studies nd that a small fund and a young fund grow faster (e.g., Chevalier and Ellison (1997), Sirri and Tufano (1998), Del Guercio and Tkac (2002), and Barber, Odean and Zhang (2005)). I measure size as the natural logarithm of ratio of a funds TNA to the average TNA of all equity funds in the sample at the beginning of each year (using the level could make the variable nonstationary). I use log age, which is the natural logarithm of the number of months since the inception dates. I also include expense ratios in the regression. Expense ratios do not include load fees (Morningstar does not provide historical load fees; using the CRSP data, I add one-seventh of load fees, as Sirri and Tufano (1998) suggest, and obtain similar results). Some studies nd that fund ows are negatively related to volatility of past returns. I measure volatility as the standard deviation of monthly returns over the prior two years (see Barber, Odean and Zheng (2005)). I also control for net ows to the industry (all equity mutual funds in the CRSP data- base) as a whole since they could inuence ows to individual funds. Industry ow can also handle xed time e¤ects if any. Cooper, Gulen, and Rau (2005) nd that investors chase styles and funds could attract more inows after changing their names to reect popular style characteristicseven without actual changes in investment stylesover 1994 to 2001. Thus, I also include style ows (net ows to funds with specic size and value character- istics according to the Morningstar categories) in the regressions. I add contemporaneous performance, and the second lag of performance to examine whether investors consider less recent performance. To control for fund xed e¤ects, I use lagged ows. My sample includes 17,679 observations (fund and year) for 2,264 funds from 1983 to 2008. Theyaccountfor83%ofnetassetsofnonindexfundsonaverage. Iconductaseparate analysis before and after 2000, around which the markets and the money management 44 industries seem to have changed. In early 2000s, the dot-com bubble burst and markets were highly volatile. Also, hedge funds experienced sharp growth. According to Hennesse Group LLC, total net assets of hedge funds increased by 50%, from $221 billion in January 1999to$324billioninJanuary2000. From1998to1999, thegrowthratewasonly6%. The observations are 6,771 and 10,908 for the pre-2000 and the post-2000 periods respectively. 18 Table 2.1. presents descriptive statistics. Over 1983 to 2008, nonindex funds have annual inows of around 10% on average. Before 2000, the average fund ows are 13%, which decrease to 9% after 2000. Yet, the decrease is not statistically signicant (the t- statistics for the equal means are adjusted for correlations among funds and across years). The average total net assets are about $1 billion before 2000 and grow to $1.4 billion after 2000. Duetotheintroductionofnewfundsinrecentyears,theaveragefundisyoungerafter 2000. The average expense ratios are slightly higher by 0.05% after 2000. The standard deviations of monthly returns are similar in the subperiods, around 4.3%. Industry ows and style ows are fewer in the 2000s. The mutual fund industry had net inows of 9% over 1983 to 1999, which decreased to 4%, after 2000 (the di¤erence is signicant). The average net ows to styles are around 10% before 1999 but only 4.7% after 2000. 18 In addition, there is a dramatic change in the time-series of the ow-performance relationship around 2000 (see Section 2.2..5. and Figure 5 (a)). I also conducted a separate analysis around 1999 and 2001 and obtained similar results. 45 Table 2.1. Descriptive statistics SampleisU.S.mutualfundswhoseobjectivesareoutperformingmarketindexesasstatedintheirprospectuses(nonindexfunds)andwhich meet sampling criteria described in the paper (e.g., excluding sector funds, international funds, funds closed to investors and small funds). Net owischangesintotalnetassets(TNA)excludingcapitalgainsanddividends, dividedbyTNAatthebeginningoftheperiod. Performanceof a fund is its annual return minus benchmark return. The benchmark return is return on the CRSP value weighted index (CRSP VW), return on the S&P500 index (SP500), return on the benchmark index designated by the fund (benchmark index), and the average return of the funds in the same style category as dened by the Morningstar (style return). Style return includes returns on index funds but exclude sector funds and international funds. All returns are net of expense ratios but before load fees. Log age is the natural logarithm (log) of the months since the inception date of fund (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly return of fund over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). CRSP VW returns and volatility is the mean and the standard deviation of daily returns on the CRSP VW respectively (annualized). All variables are measured over a year at the end of December except expense ratios, which are over fundsscal years. Funds are aggregated across share classes, excluding institutional shares. Di¤ represents the t-statistics for equal means between the two periods, adjusted for correlations among funds and autocorrelations over time. 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean ow (t) 0.105 -0.029 0.539 0.129 -0.009 0.539 0.090 -0.040 0.538 (-1.30) performance (t-1) return over CRSPVW 0.001 -0.015 0.134 -0.013 -0.018 0.111 0.010 -0.014 0.145 (0.95) return over SP500 0.004 -0.012 0.141 -0.032 -0.037 0.119 0.027 0.001 0.149 (1.98) 46 (Table 2.1 continued) 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean performance (t-1) return over benchmark 0.001 -0.011 0.120 -0.013 -0.020 0.107 0.010 -0.007 0.127 (1.07) return over style return 0.005 0.000 0.106 -0.002 -0.002 0.083 0.010 0.001 0.118 (1.97) log age (t-1) 4.859 4.762 0.766 4.966 4.875 0.849 4.793 4.710 0.702 (-3.01) age in years (t-1) 14.831 9.750 13.958 17.192 10.917 15.482 13.366 9.250 12.704 (-4.55) size (t-1) 0.958 0.914 1.630 1.023 0.964 1.520 0.918 0.879 1.694 (-1.04) TNA in billions (t) 1.285 0.274 4.619 1.026 0.242 3.035 1.445 0.298 5.366 (3.88) expense ratio (t-1) 0.012 0.012 0.004 0.012 0.011 0.005 0.012 0.012 0.004 (1.82) volatility (t-1) 0.043 0.038 0.022 0.044 0.040 0.020 0.043 0.037 0.023 (-0.11) industry ow (t) 0.070 0.060 0.076 0.087 0.082 0.086 0.037 0.040 0.041 (-2.00) style ow (t) 0.081 0.061 0.112 0.098 0.097 0.128 0.047 0.022 0.066 (-1.51) CRSPVW returns (t-1) 0.132 0.155 0.138 0.166 0.198 0.110 0.067 0.084 0.167 (-1.63) CRSPVW volatility (t-1) 0.144 0.127 0.055 0.130 0.121 0.050 0.169 0.160 0.057 (1.75) observations (funds) 17679 (2264) 6771 (1011) 10908 (2048) 47 The sample of funds does not have dramatic changes in age, size and style around 2000. Figure 2.2 shows the time-series composition of the sample. Given the median age (10 years) and size ($0.3 billion) of funds over 1983 to 2008, I dene young funds and small funds as those funds below the median values respectively. Panel (A) illustrates that the proportion of large funds increased dramatically in the 1990s. The proportion of young and old funds has been stable since early 1990s (Panel (B)). Moreover, the composition of fund styles is also similar throughout the years. 19 Figure 2.2. Sample composition Sample is U.S. nonindex mutual funds that meet sampling criteria described in the paper. Panel (A) shows the proportion of small (less than the median fund size of $0.3 billion) and large funds, and (B) shows the proportion of young (younger than the median fund age of 10 years) and old funds. Panel (C) shows the composition of fund styles (size and value characteristics according to Morningstar). (A) Fund size Large Small (< $0.3B) 0 10 20 30 40 50 60 70 80 90 100 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 % 19 I also use a sample of funds with nonmissing observations from 1992 to 2007 (220 funds) and obtain similar results (not reported but available upon request). 48 (Figure 2.2 continued) (B) Fund age Old Young (< 10yrs) 0 10 20 30 40 50 60 70 80 90 100 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 % (C) Fund style LargeBalance LargeGrowth LargeValue MidBalance MidGrowith MidValue SmallBalance SmallGrowth SmallValue 0 10 20 30 40 50 60 70 80 90 100 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 % 49 2.2.2. Kernel regression methodology To examine fund-ow relationships, I use the semi-nonparametric estimation sug- gested by Robinson (1988). Chevalier and Ellison (1997) use this procedure for the ow- performancerelationshipbutalsoexaminethesensitivitydi¤erencesacrossfundagegroups. They show that ows to old funds (5 years old or more) are less sensitive to performance. Yet, the di¤erences do not appear statistically signicant. Thus, I estimate the ow- performance relationship for funds of any age, controlling age e¤ects by an age variable in the regressions. I use a panel of funds and year and estimate net flow i;t = g(performance i;t1 )+ 3 performance i;t + 4 performance i;t2 + 5 age i;t1 + 6 size i;t1 + 7 expense i;t1 + 8 volatility i;t1 + 9 industry i;t + 10 style i;t + 11 net flow i;t1 +" i;t ; (24) where the variables are described in the preceding section. The error term is orthogonal to performance as E[" i;t jperformance i;t1 ] = 0. I estimate the function g(performance i;t1 ) usingkernelregressions. Ichooseoptimalbandwidthsbythecross-validationmethod,which improves e¢ ciency despite its computational costs (see Appendix). Previous studies do not use the cross-validation method (e.g., Chevalier and Ellison (1997) and Sensoy (2009)). As Robinson (1988) proves, we can obtain p n-consistent and unbiased estimates for 3 to 11 in the Equation (24) by the following steps: (a) Run each kernel regression of net ows and the control variables against lagged performance to obtain their expected values conditional on lagged performance; (b) Run OLS regressions of residual net ows on the residualcontrolvariables, denedasactualvaluesminusexpectedvaluesobtainedfrom(a), 50 to estimate 3 to 11 (Frisch-Waugh-Lovell theorem). Using the estimates b 3 to b 11 ; I can obtain net flow i;t by net flow i;t = net flow i;t ( b 3 performance i;t + b 4 performance i;t2 + b 5 age i;t1 + b 6 size i;t1 + b 7 expense i;t1 + b 8 volatility i;t1 + b 9 industry i;t + b 10 style i;t + b 11 net flow i;t1 ): (25) To estimate g(), I run kernel regressions of net flow i;t against performance i;t1 : This functionhastheinterpretation,E[netflow i;t jperformance i;t1 ] =g(performance i;t1 ):In words, I estimate the ow-performance relationship as expected fund ows conditional on past performance, controlling other factors, such as industry growth, size, and age. One limitation of this method is its inability to identify because the regression Equation (24) is (observationally) equivalent to net flow i;t = +g(performance i;t1 )+ ::: Therefore, I normalize g(performance i;t1 ) so that we have g(0) = 0: To account for correlations among funds and autocorrelations over time, I report stan- dard errors after clustering observations by year and fund. 2.2.3. Linear regression methodology I run OLS regressions of net ows on each performance measure and the control vari- ables,afterassumingthatg()inEquation(24)isquadraticinlaggedperformance. Previous studiesalsousethesquareoflaggedperformancetocapturenonlinearityofow-performance 51 relationships(e.g.,Barber,OdeanandZheng(2005)andSensoy(2009)). Irunthefollowing regressions using panel data: net flow i;t = + 1 performance i;t1 + 2 performance 2 i;t1 + 3 performance i;t + 4 performance i;t2 + 5 age i;t1 + 6 size i;t1 + 7 expense i;t1 + 8 volatility i;t1 + 9 industry i;t + 10 style i;t + 11 net flow i;t1 +" i;t ; (26) where the variables are the same as in the Equation (24). Similar to kernel regressions, I report standard errors after clustering by year and fund. 2.2.4. Changes in the ow-performance relationship I rst present the kernel regression results for the two subperiods. Figure 2.3 shows the estimates of the ow-performance relationship, i.e., the function g(performance t1 ) in Equation (24); along with their 90% condence intervals in each period. For all four perfor- mance measures, the striking changes are in shapes: convexity before 2000 and linearity or concavityafter2000. Inparticular, expectedinowsafter good performancearemuch fewer after 2000 than before that year, and the decreases in expected inows are larger for higher performance. For example, expected inows after outperforming the CRSP VW by 10% are 14.3% before 2000, but they decrease to 6.6% after 2000. Net inows to funds with 20% outperformance decreased from 27.8% to 9.4%. On the other hand, the ow-performance relationship for poor performance are similar between the two periods. Table 2.2. presents the coe¢ cients on the control variables (the linear part in Equation (24)). 52 Figure 2.3. Flow-performance relationship by kernel regression Estimates of the ow-performance relationships using kernel regressions and their 90% con- dence intervals. The y-axes represent expected annual net ows into and out of nonindex funds from 1983 to 1999 (before 2000) and from 2000 to 2008 (after 2000). Performance is annual returns minus benchmark returns. The benchmark returns are the CRSP value weighted index (CRSP), the S&P500 index (SP500), returns on the benchmark indexes designated by the funds (benchmark), andtheaveragereturnsofthefundsinthesamestylecategoryasdenedbytheMorningstar(style). Style returns include returns on index funds but exclude sector funds and international funds. The expected annual net ows are estimated after controlling contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, style ow and lagged ow. Log age is the natural logarithm (log) of the months since the inception date of a fund (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). Funds are aggregated across share classes, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over CRSP annual flows before 2000 after 2000 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over SP500 annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over benchmark annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over style annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over CRSP annual flows before 2000 after 2000 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over SP500 annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over benchmark annual flows 0.2 0.1 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over CRSP annual flows before 2000 after 2000 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over SP500 annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over benchmark annual flows 0.2 0.1 0 0.1 0.2 0.2 0 0.2 0.4 lagged return over style annual flows 53 Table 2.2. Estimations for control variables in kernel regressions The dependent variable of the semi-nonparametric regressions is annual net ows in the year t and the independent variables are listed in the rst column. Numbers for each independent variable are estimates and standard errors, clustered by year and fund. Regressions are di¤erent depending on performance measures, as described in the rst row. Style returns are the average returns of the funds in the same style categories as dened by the Morningstar. Log age is the natural logarithm (log) of the months since the inception date of the fund. Log size is the log of TNA of the fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly return of the fund over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). All variables are measured over a year at the end of December except expense ratios, which are over the fundscal year. The numbers of observations are 6,771 and 10,908 from 1983 to 2008 and from 2000 to 2008 respectively. Funds are aggregated across share classes, excluding institutional share classes. performance: return over CRSP VW return over style return 1983-1999 2000-2008 1983-1999 2000-2008 performance (t) 0.800 1.041 0.835 1.265 (0.138) (0.119) (0.191) (0.170) performance (t-2) 0.456 0.287 0.581 0.346 (0.133) (0.106) (0.139) (0.117) log age (t-1) -0.032 -0.027 -0.017 -0.032 (0.009) (0.008) (0.009) (0.009) log size (t-1) -0.026 -0.039 -0.029 -0.040 (0.005) (0.006) (0.005) (0.005) expense ratio (t-1) 4.532 -2.747 3.404 -1.980 (1.499) (1.338) (1.562) (1.372) volatility (t-1) -2.163 0.460 -0.701 -0.118 (0.535) (0.745) (0.544) (0.448) industry ow (t) 0.389 -0.028 0.146 0.129 (0.199) (0.477) (0.174) (0.358) style ow (t) 0.278 0.049 0.597 0.889 (0.079) (0.144) (0.063) (0.063) ow (t-1) 0.301 0.230 0.282 0.209 (0.030) (0.038) (0.027) (0.035) The most dramatic changes are for performance relative to peer funds with the same styles. As shown in Figure 2.3, the sensitivity of the relationship (i.e., the slope of the function g(performance t1 )) is sharply increasing with the performance before 2000. Yet, 54 after 2000, the sensitivity declines. For instance, for 10% outperformance relative to peer funds, the sensitivity is around 3.6 before 2000 but only 1.1 after that. The shape of the relationship between fund ows and excess returns over styles is odd for the performance region less than -20% or more than 20% prior to 2000. This is because there are not many data points at those extremes. The 5 percentile and 95 percentile of excess returns over style returns are around -13% and 12% respectively (other performance measures are almost -20% and 16% respectively). The results from the OLS regressions are similar. I only report the results for perfor- mance relative to the CRSP VW index and for performance relative to style returns since using the S&P500 index and fundsbenchmark indexes lead to almost the same results as using the CRSP VW index and style returns respectively. Table 2.3 shows the estimates followed by standard errors (clustered by year and fund) and p-values for each independent variable. Thedependentvariableisnetows. Oneofthestrikingdi¤erencesisfortheslope estimates on square of lagged performance. They are positive in the pre-2000 period but decrease to around -0.3 -0.4. The di¤erences of those coe¢ cients are signicant at 1% signicancelevel(thetestiswhethertheinteractiontermbetweenthesquaredperformance and the second period dummy is zero). The estimates on lagged performance are positive in both periods but the magnitudes are a bit smaller after 2000. I plot the estimated ow-performance relationship (the quadratic function) in Figure 2.4 (A). 55 Table 2.3. OLS regression of net ows The dependent variable of the ordinary least square regressions is annual net fund ows in the year t and the independent variables are listed in the rst column. Numbers for each independent variable are estimates, standard errors, and p-values respectively. The standard errors are clustered by year and fund. Regressions are di¤erent depending on the performance measures, as described in the rst row. Style returns are the average returns of the funds in the same style categories as dened by the Morningstar. Squared performance is square of performance. Log age is the natural logarithm (log) of the months since the inception date of the fund. Log size is the log of TNA of the fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly return of the fund over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). All variables are measured over a year at the end of December except expense ratios, which are over the funds scal year. The numbers of observations are 6,771 and 10,908 from 1983 to 2008 and from 2000 to 2008 respectively. Funds are aggregated across share classes, excluding institutional share classes. The columns Di¤show the test results for equal coe¢ cientsontheindividualvariables. Thevaluesrepresentthecoe¢ cientsontheinteractionterms between the variables and the second period dummy, their standard errors and p-values respectively (standard errors are clustered by year and fund). P-values are the Chow-test p-values for joint tests of equal coe¢ cients on the variables. return over CRSP VW return over style returns 1983-1999 2000-2008 Di¤ 1983-1999 2000-2008 Di¤ intercept 0.489 0.511 -0.060 0.281 0.484 -0.013 (0.089) (0.106) (0.029) (0.082) (0.100) (0.019) (0.000) (0.000) (0.036) (0.001) (0.000) (0.501) performance (t-1) 1.507 0.933 -0.664 1.798 1.214 -0.678 (0.186) (0.209) (0.232) (0.193) (0.239) (0.310) (0.000) (0.000) (0.004) (0.000) (0.000) (0.029) squared performance 0.662 -0.323 -1.469 2.264 -0.409 -3.037 (t-1) (0.643) (0.088) (0.314) (0.919) (0.124) (0.757) (0.303) (0.000) (0.000) (0.014) (0.001) (0.000) performance (t) 0.731 0.978 0.244 0.750 1.262 0.511 (0.123) (0.133) (0.190) (0.174) (0.177) (0.247) (0.000) (0.000) (0.199) (0.000) (0.000) (0.039) performance (t-2) 0.905 0.494 -0.460 1.194 0.618 -0.660 (0.143) (0.135) (0.208) (0.135) (0.160) (0.198) (0.000) (0.000) (0.028) (0.000) (0.000) (0.001) log age (t-1) -0.080 -0.075 0.005 -0.058 -0.076 -0.017 (0.013) (0.016) (0.020) (0.012) (0.016) (0.021) (0.000) (0.000) (0.813) (0.000) (0.000) (0.424) log size (t-1) -0.008 -0.032 -0.014 -0.014 -0.033 -0.018 (0.005) (0.005) (0.008) (0.004) (0.005) (0.007) (0.152) (0.000) (0.075) (0.001) (0.000) (0.019) 56 (Table 2.3 continued) return over CRSP VW return over style returns 1983-1999 2000-2008 Di¤ 1983-1999 2000-2008 Di¤ expense ratio (t-1) 8.094 -4.073 -9.305 6.017 -2.844 -6.547 (2.021) (1.864) (2.653) (1.942) (1.570) (2.404) (0.000) (0.029) (0.000) (0.002) (0.070) (0.007) volatility (t-1) -2.028 -0.103 0.830 -0.462 -0.135 -0.515 (0.532) (0.897) (1.080) (0.479) (0.700) (0.813) (0.000) (0.909) (0.442) (0.335) (0.847) (0.527) industry ow (t) 0.441 0.232 0.009 0.236 0.185 -0.045 (0.206) (0.502) (0.525) (0.153) (0.307) (0.323) (0.032) (0.644) (0.987) (0.122) (0.547) (0.888) style ow (t) 0.489 0.014 -0.459 0.709 0.982 0.246 (0.089) (0.163) (0.241) (0.093) (0.065) (0.116) (0.000) (0.930) (0.057) (0.000) (0.000) (0.034) Adjusted R 2 (p-values) 0.214 0.126 (0.000) 0.228 0.150 (0.000) Since the sensitivity of ow-performance relationship is given by the rst derivative of the Equation (26) with respect to performance, 1 +2 2 performance t1 ; (27) theestimatessuggestthatthesensitivitychanges,forexample,from1:5+1:3performance t1 to 0:9 0:6performance t1 when performance is excess returns over CRSP VW. Figure 2.4 (B) illustrates these changes in sensitivity by plotting the sensitivity as a function of past performance (i.e., Equation (27)) before 2000 and after 2000. For all four performance measures, the sensitivity increases with past performance before 2000 (convexity), but it decreaseswithpastperformanceafterthat(concavity). Yet, themostdramaticchangesare for performance relative to peer funds with the same styles, which are consistent with the kernel regression results. The sensitivity for 10% outperformance is around 2.5 on average from 1983 to 1999 but decreases to around 1 in the post-2000 period. 57 Figure 2.4. Flow-performance relationship and its sensitivity by OLS regressions (A) Flow-performance relationship and (B) its sensitivity from 1983 to 1999 (before 2000) and from 2000 to 2008 (after 2000) for each performance measure. The sensitivity is the rst derivative of the ow-performance relationship with respect to lagged performance, i.e., 1 +2 2 performance in Equation (27). The estimates used for the plots are presented in Table 2.3. The relationship is estimatedafterregressingannualnetfundowsonlaggedperformance,itssquare,contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, and style ow. Performance is annual returns minus benchmark returns. The benchmark returns are the CRSP value weighted index (CRSP), the S&P500 index (SP500), returns on the benchmark indexes designated by the funds (benchmark), and the average returns of the funds in the same style category as dened by the Morningstar (style). Style returns include returns on index funds but exclude sector funds and international funds. Log age is the natural logarithm (log) of the months since the inception dates of funds (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). Funds are aggregated across share classes, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. (A) Flow-performance relationship using OLS regression 0.2 0.1 0 0.1 0.2 0.5 0 0.5 lagged return over CRSP annual flows before 2000 after 2000 0.2 0.1 0 0.1 0.2 0.5 0 0.5 lagged return over SP500 annual flows 0.2 0.1 0 0.1 0.2 0.5 0 0.5 lagged return over benchmark annual flows 0.2 0.1 0 0.1 0.2 0.5 0 0.5 lagged return over style annual flows 58 (Figure 2.4 continued) (B) Sensitivity of the ow-performance relationship using OLS regression 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over CRSP sensitivity before 2000 after 2000 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over SP500 sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over benchmark sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over style sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over CRSP sensitivity before 2000 after 2000 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over SP500 sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over benchmark sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over style sensitivity Another di¤erence between the two periods is that net ows are positively related to expenseratiosoffundsbefore2000butnegativelyrelatedtothataftertheyear. Bothkernel regressions (Table 2.2.) and OLS regressions (Table 2.3) show these contrasting impacts of expenseratios. Forexample,asshowninTable2.3,whenmeasuringperformancecompared to the CRSP VW index, a 1% increase in expense ratios leads to a 8% increase in net ows inthefollowingyearonaveragefrom1983to1999. After2000,expenseratioshavenegative impact on fund ows. A 1% increase in expense ratios loses around 3% of net ows in the following year. Barber, Odean, Zheng (2005) and Huang, Wei, and Yan (2005) also nd a positive e¤ect of (operating) expense ratiosnot including front-end loadson fund ows for sample periods before 2000. Barber, Odean, and Zheng argue that investors pay less 59 attention to ongoing operating expenses that are embedded in fund returns than front-end load fees. Rather, they are even attracted to funds with higher marketing expenses, leading to a positive relationship between expense ratios and fund ows. I discuss variations in the e¤ect of the expense ratio on fund ows in Appendix. Coe¢ cients estimates for other variables have the same signs in both periods (Table 2.2. and Table 2.3; results for other performance measures are available upon request). For example, smaller funds or younger funds grow more, consistent with earlier studies. A fund with less volatile returns over the last two years attracts more inows. Funds with more popular style characteristics received more in the pre-2000 period. However, style ows appear insignicant in the post-2000 period when performance is measured relative to CRSP VW (or S&P500 index). In essence, investors seem to chase funds that outperform the markets irrespective of their styles in the 2000s. 2.2.5. Determinants of sensitivity of the ow-performance relationship I showed that the ow-performance relationship changes around 2000. In particu- lar, substantial changes are marginal ow with respect to performance (i.e., the slope of g(performance t1 ) or Equation (27)). For example, Figure 2.4 (B) shows that marginal ow increases with performance before 2000 but decreases after 2000. Thus, I investigate what impacts the sensitivity of the ow-performance relationship. I focus on market- or industry-widefactors,ratherthanfund-specicones,toexaminevariationsinthesensitivity through time. To this end, I look at how 2 in Equation (27) changes, more specically, how it varies 60 depending on market- or industry-wide conditions. Assuming linearity, I estimate 1 +2 2;t performance t1 = 1 +2( + 0 Z t )performance t1 ; (28) where Z t is a vector of conditioning variables that are known at time t. Since the Equation (28) is the rst derivative of the regression Equation (26), I run the following regression over the whole sample period, including the interaction terms between those variables and squared performance net flow i;t = + 1 performance i;t1 + performance 2 i;t1 + 0 Z t performance 2 i;t1 + 3 performance i;t +::::::+ 11 net flow i;t1 +" i;t : (29) Tomotivatemarketvolatilityasamarket-widevariable, IpresentinFigure2.5(A)the cross-sectional estimates on the square of excess returns relative to the CRSP VW (i.e., 2 in Equation (26)) along with their 10% condence intervals, and lagged market volatility (annualizedstandarddeviationofthedailyCRSPVWreturns). Theestimateonthesquare term dramatically decreases in 1998 and becomes negative in 2000, suggesting convexity in the ow-performance relationship. The estimate and the lagged market volatility have almost the opposite patterns through time. While the estimates in the rst period (1983- 1999) and in the second period (2000-2008) are positive and negative respectively, there have been some periods with negative coe¢ cients before 2000 and positive coe¢ cients after 2000, which seemtodepend on market volatility. Forinstance, afterthe 1987market crash, the estimate on squared performance signicantly drops in 1988. Following the less volatile markets in 2005 and 2006, the slope estimates increase in 2006 and 2007. 61 These results are consistent with the predictions that marginal ow to superior perfor- mance is low when performance is noisy (Berk and Green (2004) and Chapter 1). Figure 2.5 (C) shows performance distributions. In low-volatility markets (less than 10%), funds underperform markets on average, and outperformance is rare: The mean and the skewness of excess returns relative to the CRSP VW are -3% and -0.03 respectively. When markets are highly volatile (more than 16%), funds outperform the markets on average (3%) and the performance is skewed to the right (skewness 0.7). Thus, performance tends to improve and appears noisier in highly volatile markets. I also look at how the sensitivity of the ow-performance relationship depends on heterogeneityinskills. Toobtainaproxyforthat,Irstregressthecross-sectionalstandard deviation of performance on the market return and the market volatility (the mean and the standard deviation of daily returns on the CRSP VW index respectively). Given that performance is represented as skills plus noise, its variance is the sum of the variance of skills and the variance of noise. Provided that the variance of skills is uncorrelated with market conditions, only the variance of noise depends on market conditions. Thus, I use the regression residual, which I call performance dispersion, as a conditioning variable. In addition to market volatility and performance dispersion, I use industry ow as conditioning variables. Industry ow is the same variable used in the previous regressions. Sinceindicatorvariablesareeasytointerpret(e.g., high-andlow-volatilitymarkets), Irank market return, market volatility and industry ow into three groups respectively and assign values of -1 (low), 0 (medium), and +1 (high). Approximately, the medium volatility is between 10% and 16%; the medium return between 2% and 20%; and the medium industry ow between 3% and 9% (the results do not change if I use their levels, or if I use dummies 62 for high- and low-volatility markets). High volatility and low industry ow are prevalent in the 2000s. Figure 2.5 (B)shows the marketvolatilityindicatorand performance dispersion. Figure 2.5. Market volatility, performance dispersion, and performance distribution (A) The blue line represents the estimates on square of lagged excess returns relative to the CRSP VW index after running cross-sectional regressions of annual net ows on the lagged excess return, its square, and other control variables in each year. The control variables are described in Figure 2. The dash lines are their 90% condence intervals. The sample is U.S. mutual funds whose objectives are outperforming market indexes as stated in their prospectuses and which meet sampling criteria described in the paper. The red dash line is lagged market volatility as measured by annualized standard deviation of daily return on the CRSP VW index in the prior year. (B) Performance dispersion is the residual obtained from the regressions of the cross-sectional standard deviation of performance on the mean and the volatility of the daily CRSP VW returns. Perfor- mance is excess returns over the CRSP VW index. Market volatility indicator is -1, 0, and +1 if market volatility is low, medium and high respectively based on its ranking. (C) The histogram represents the distribution of excess returns relative to the CRSP VW index when the volatility of returns on the CRSP VW index is low (under 10%) during the period from 1982 to 2007. (D) The histogram represents the distribution of excess returns relative to the CRSP VW index when the volatility of returns on the CRSP VW index is high (over 16%) during the period from 1982 to 2007. The volatility is annualized standard deviation of daily returns and ranked into three groups (low/mid/high) over 1982 to 2007. 63 (Figure 2.5. continued) 64 (Figure 2.5 continued) 65 The results are provided in Table 2.4 (the results for performance relative to S&P500 indexaresimilartoPanel(A),andthoseforperformancerelativetothebenchmarkindexare similartoPanel(B)).Beforediscussingthemainregressions,Irstpresentsomebenchmark cases. Therstregressionpoolsallyears. Theslopeestimatesonsquaredperformancefrom 1983 to 2008 are negative and signicant for all four performance measures, suggesting the concave ow-performance relationship on average over the whole period. The concavity e¤ect in the second period dominates because there are more observations after 2000. The e¤ect of expense ratio appears negligible since its contrasting impacts in two subperiods are cancelled out each other over the entire sample period. When adding lagged ows (the second regression), the adjusted R-squared increases by around 7-8% while most estimates change little, except expense ratios. I include lagged ows in the rest of regressions. Consistent with the results in Table 2.2., using the second period dummy variable changes the estimates on squared performance and expense ratios dramatically (the third regression). The ow-performance relationship before 2000 appears convex but become concave in the second period. The expense ratios are positively related to fund ows over 1983 to 1999 but have a negative impact in the 2000s. These results are similar for all performance measures. Inowdiscussthemainregressionsthatexaminethedeterminantsoftheow-performance sensitivity. When including the interaction terms between squared performance and lagged stock market indicators (the regression (4)), the coe¢ cients on those terms are negative (marketreturnsappearinsignicantforthesensitivity). Notethattheestimatesonsquared performance are statistically insignicant. We cannot reject that in the medium volatility markets, the ow-performance relationship is linear. In periods following highly volatile 66 markets, the coe¢ cient on the squared excess returns over the CRSP VW decreases by around -0.9, which indicates concavity in the ow-performance relationship. In low volatil- ity markets, the coe¢ cient increases by +0.9, which implies convexity in the relationship (see Figure 2.6). 20 I interpret these results as supporting evidence that investors become less responsive to superior performance in highly volatile markets because they perceive performance arising primarily from luck. After adding performance dispersion (uncorrelated with market volatility and returns) as a conditioning variable (regression (5)), I nd that a 1% increase in the cross-sectional standard deviation in performance increases the coe¢ cient on squared performance by around 7%. Thus, investors appear more responsive to superior performance when perfor- manceismoredispersed. Providedthatperformancedispersioncanproxyforheterogeneity in skills, high dispersion indicates that performance is more attributable to skills. Industry ow appears to increase the sensitivity of the owperformance relationship (regression (6)). This can suggest that more money in mutual funds actually goes to high- performing funds. Provided that fund performance exhibits decreasing returns to scale (Chen et. al. (2004)), investors invest in funds with higher expected returns when they investmore. Thelastregressionsshowtheresultsincludingallconditioningvariables. Mar- ket volatility and performance dispersion remain signicant when measuring performance relativetostylereturns. Yet,whentheCRSPVWisusedasthebenchmark,marketvolatil- ity becomes insignicant. This could be due to the signicant correlation between market volatility and the second period dummy. 21 20 When using dummy variables for market volatility, I nd consistent results that convexity decreases in highly volatile markets (see Section 2.4.1 for details). 21 I also obtained similar results using the sample of funds with at least 8 annual observations both before and after 2000 respectively (from 1992 to 2007, total 220 funds). The results are not reported and available upon request. 67 Table 2.4. Determinants of ow-performance sensitivity The dependent variable is annualnetfundowsintheyeartandtheindependentvariablesare listedintherstcolumn. Numbersforeachindependentvariableareestimates,standarderrors,and p-values respectively. The standard errors are clustered by year and fund. Regressions are di¤erent depending on performance measures. Style returns are the average returns of the funds in the same style category as dened by the Morningstar. Squared performance is square of performance. This variable has interaction terms with ve conditioning variables. Market ret state is -1, 0, and +1 when the mean of daily returns on the CRSP VW index over the year is in the low, middle, and high group respectively. Similarly, market vol state is -1 (low volatility), 0 (medium), and +1 (high) according to the standard deviation of daily returns; and industry ow state is equal to -1 (low), 0 (medium), and +1 (high demand), depending on net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Performance dispersion is the residual obtained from the regressions of the cross-sectional standard deviation of performance on the mean and the volatility of the daily CRSP VW returns. The second period is one if the year is between 2000 and 2008 and zero otherwise. The expense ratio does not include load fees. The variables not presentedinthetableincludeperformance(t); performance(t-2); logage(t-1; thelogofthemonths since the inception date of fund); log size (t-1; the log of TNA of a fund divided by the average TNA of sample funds, including index funds); volatility (t-1; the standard deviation of monthly return of fund over the last two years); industry ow (t); and style ow (t; net ows to the funds in each of 9 styles, including index funds). All variables are measured over a year at the end of December except expense ratios, which are over fundsscal years. The number of observations is 17,679 over 1983 to 2008. Fund variables are aggregated across share classes, excluding institutional share classes. 68 (Table 2.4. continued) (A) performance: excess return over CRSP VW (1) (2) (3) (4) (5) (6) (7) performance (t-1) 1.064 0.822 0.829 0.844 0.897 0.920 0.907 (0.165) (0.151) (0.149) (0.151) (0.140) (0.133) (0.131) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) squared performance (t-1) -0.370 -0.384 0.647 0.761 0.697 0.628 0.966 (0.079) (0.066) (0.312) (0.620) (0.638) (0.566) (0.572) (0.000) (0.000) (0.038) (0.220) (0.274) (0.267) (0.092) *market ret state (t-1) -0.304 -0.717 -0.604 -0.728 (0.482) (0.532) (0.360) (0.416) (0.528) (0.178) (0.094) (0.081) *market vol state (t-1) -0.879 -1.564 -0.951 -0.375 (0.440) (0.564) (0.557) (0.662) (0.046) (0.006) (0.088) (0.571) *performance dispersion 6.536 8.773 9.370 (t-1) (2.573) (0.909) (0.896) (0.011) (0.000) (0.000) *industry ow state (t) 1.083 0.736 (0.363) (0.389) (0.003) (0.059) *second period (t) -1.056 -1.238 (0.307) (0.699) (0.001) (0.077) expense ratio (t-1) 0.032 -0.324 1.583 1.915 2.119 2.094 1.827 (2.034) (1.567) (2.153) (2.197) (2.196) (2.252) (2.268) (0.987) (0.836) (0.463) (0.384) (0.335) (0.353) (0.420) *second period (t) -3.017 -3.883 -3.854 -3.709 -2.982 (2.160) (2.181) (2.053) (2.176) (2.144) (0.163) (0.075) (0.061) (0.089) (0.165) ow (t-1) 0.260 0.257 0.257 0.259 0.259 0.259 (0.028) (0.028) (0.028) (0.028) (0.028) (0.028) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Adjusted R squared 0.147 0.225 0.228 0.228 0.230 0.232 0.232 69 (Table 2.4 continued) (B) performance: excess return over style return (1) (2) (3) (4) (5) (6) (7) performance (t-1) 1.401 1.119 1.118 1.127 1.142 1.158 1.158 (0.193) (0.159) (0.160) (0.168) (0.163) (0.160) (0.160) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) squared performance (t-1) -0.508 -0.470 1.631 2.878 2.834 2.755 2.995 (0.095) (0.086) (0.698) (0.810) (0.761) (0.769) (1.024) (0.000) (0.000) (0.020) (0.000) (0.000) (0.000) (0.004) *market ret state (t-1) -0.659 -1.171 -1.102 -1.184 (0.410) (0.525) (0.307) (0.360) (0.108) (0.026) (0.000) (0.001) *market vol state (t-1) -2.738 -3.451 -2.865 -2.571 (0.861) (0.881) (0.891) (0.887) (0.002) (0.000) (0.001) (0.004) *performance dispersion 7.657 11.615 12.119 (t-1) (3.889) (1.183) (1.092) (0.049) (0.000) (0.000) *industry ow state (t) 1.254 1.091 (0.462) (0.531) (0.007) (0.040) *second period (t) -2.125 -0.700 (0.704) (1.125) (0.003) (0.534) expense ratio (t-1) 0.388 -0.062 -0.528 -0.671 -0.546 -0.533 -0.617 (1.705) (1.325) (1.806) (1.854) (1.827) (1.849) (1.852) (0.820) (0.963) (0.770) (0.718) (0.765) (0.773) (0.739) *second period (t) 0.432 -0.165 -0.215 -0.155 0.052 (1.648) (1.584) (1.548) (1.597) (1.535) (0.794) (0.917) (0.889) (0.923) (0.973) ow (t-1) 0.249 0.246 0.244 0.246 0.247 0.247 (0.027) (0.027) (0.027) (0.027) (0.027) (0.027) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Adjusted R squared 0.169 0.240 0.242 0.245 0.246 0.247 0.247 70 Figure 2.6. Flow-performance sensitivity conditional on market volatility Market volatility is ranked into low, medium and high groups based on the standard deviation of daily return on the CRSP VW index. The sensitivity is the rst derivative of the relationship with respect to lagged performance (i.e., Equation (27) in the paper). The estimates used for the plots are presented in Table 4 (regression (4)). The relationship is estimated after regressing annualnetowsintoandoutofnonindexfundsonlaggedperformance,itssquare, contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, and style ow. Performance is annual returns minus benchmark returns. The benchmark returns are the CRSP value weighted index (CRSP), the S&P500 index (SP500), returns on the benchmark indexes designated by the funds (benchmark), and the average returns of the funds in the same style category as dened by the Morningstar (style). Style returns include returns on index funds but exclude sector funds and international funds. Log age is the natural logarithm (log) of the months since the inception dates of funds (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database(includingsectorfundsandinternationalfunds). Styleowisnetowstothefundsineach of 9 styles (including index funds but excluding sector funds and international funds). Funds are aggregated across share classes, excluding institutional shares. The total number of funds is 2,264 over 1983 to 2008. 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over CRSP sensitivity lowvolatility highvolatility 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over SP500 sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over benchmark sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over style sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over CRSP sensitivity lowvolatility highvolatility 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over SP500 sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over benchmark sensitivity 0.2 0.1 0 0.1 0.2 0 1 2 3 lagged return over style sensitivity 71 2.3. Managerial incentives I examine whether managers change their risk-shifting according to variations in the ow-performance relationship. Brown, Harlow and Starks (1996) and Chevalier and Ellison (1997)showthatsomemutualfundmanagersincreasetheriskinessoftheirfundstowardthe end of the year as a result of the incentives provided by the ow-performance relationship. They also argue that the increase in the riskiness is positively related to expected net ows in the following year. Given a decrease in fund ows to outperforming funds in the 2000s, managers should engage in less risk-shifting. In addition, variation in risk-shifting according to performance up to the third quarter should be also di¤erent in the 2000s. When the sensitivity of the ow-performance relationship is larger for high performance than for low performance, underperforming managers have proper incentives to increase the riskiness of their funds while good performers lock in their gains. Otherwise, managers who are behind the markets would have fewer incentives for such risk-shifting. Therefore, I also look at how performance up to the third quarter a¤ects risk-taking behavior in the fourth quarter depending on variations in the ow-performance relationship. I measure the riskiness of funds equity holdings rather than fund returns. Busse (2001) obtain di¤erent results than those in Brown, Harlow and Starks (1996) when using daily returns instead of monthly returns. Thus, I examine risk measures that use holding data, such as those suggested by Chevalier and Ellison (1997) and Huang, Sialm and Zhang (2009). Chevalier and Ellisons measure compares fund risk at the end of December and at the end of September. Huang, Sialm and Zhangs measure compares fund risk with a long term risk level, namely, over the prior 36 months. 72 2.3.1. Data and variable description I use the Thomson-Reuters Mutual Fund Holdings for equity holdings and the CRSP Survivor-Bias-Free US Mutual Fund for TNA and monthly returns (Morningstar does not providehistoricalholdingdata). Iincludeonlythefundswiththeobjectivecodesofaggres- sive growth, growth or growth and income, as provided by the Thomson-Reuters. I exclude index funds by fund name, small funds (less than $10 million at the end of September) and young funds (fewer than 2 years). The sample period is from 1983 to 2008. The rst risk measure uses volatility of excess returns over the CRSP VW. It decom- poses the volatility into two parts (Chevalier and Ellison (1997)): var(r i r m ) =var(r i i r m )+( i 1) 2 var(r m ); where r i is the fund is return, r m the market return proxied by the CRSP VW index, and i isthefundisCAPMbeta. Totalrisk, unsystematicriskandsystematicrisk, aredened as their square roots, STD(r i r m ); STD(r i i r m ); andj i 1j respectively: Risk-shiftsaretheriskdi¤erencebetweenSeptemberandDecember,denotedbyRISK Dec RISK Sep for total risk, unsystematic risk and systematic risk. To estimate RISK Sep and RISK Dec ,IconstructtwoportfoliosthatholdthestocksinthefundattheendofSeptember andDecemberrespectivelyandcalculatethevolatilityofthosehypotheticalportfoliosusing dailyreturndataintheprioryear. AsChevalierandEllisonpointout,RISK Dec RISK Sep does not depend on changes in market conditions but only on changes in equity holdings in the fourth quarter: To estimate total risk STD(r i r m ); I use the (annualized) stan- dard deviation of daily excess returns on the hypothetical portfolio over the CRSP VW 73 returns. Likewise, unsystematic risk, STD(r i i r m ); is the (annualized) standard devia- tion of r i i r m where i is the beta of the portfolio. One disadvantage of this measures is that betas of individual stocks should be estimated. I only consider the funds of which hypothetical portfolios have 80% of fundsTNA. Another measure is suggested by Huang, Sialm and Zhang (2009). It represents how much riskier the fund would have been if it had held the current assets over the prior 36 months. It measures the di¤erence between the standard deviation of returns on a hypothetical portfolio that holds the current assets of the fund and the standard deviation of actual returns on the fund over the prior 36 months: hypothetical risk actual risk. A positive risk shift of a fund implies that the fund has higher risk, compared to the prior three years. I focus on the risk-shift measure at the end of December. I only consider a fund if the value of its hypothetical portfolio is at least 80% of its TNA. I include some control variables in the regression. Chevalier and Ellison (1997) argue that small or young funds may have stronger incentives forincreasing riskiness and that the risk level at the end of September would be also relevant for risk-shifting. Thus, I use the log of equity value in million and the log of months since the inception date (or the rst monththatTNAdataisavailable)ascontrolvariables. Tocontroltherisklevelfromwhich managers deviate, I use total risk, idiosyncratic risk and systematic risk in September (for the other risk-shift measure, I use actual risk). The descriptive statistics are provided in Table 2.5. The sample size is small because the regressions require a match between the Thomson-Reuters holding database and the CRSP data. Moreover, I have some screening criteria as discussed above. The average fund has $1.2-1.5 billion of TNA and is 16-18 years old. A typical fund has 8.3% of total 74 risk and 7.5% of idiosyncratic risk (annualized) at the end of September. These values are larger by around 3% than in Chevalier and Ellison. Systematic risk, dened as j i 1j; is similar to Chevalier and Ellison. The magnitudes of risk shift are smaller in my sample than in Chevalier and Ellison, for instance, -0.05% versus 0.2% for total risk. The risk-shift measure by Huang, Sialm and Zhang is around 0.4% on average, which is larger than the average of -0.33% in their paper. I only look at equity holdings and restrict my sample to the funds of which hypothetical portfolios cover at least 80% of their TNA. Yet, Huang, Sialm and Zhang also consider bond and cash holdings by proxing them using the Lehman Brothers Aggregate Bond Index and the Treasury Bill rate, and do not have restrictions on the hypothetical portfolio value. Another di¤erence is that my sample includes only the end of December, while Huang, Sialm and Zhang also consider all months, including the months for which the quarterly holding data are unavailable (e.g., January and February) and use the most recent holding data for a given month. 2.3.2. Methodology The main goal is to examine whether the ow-performance relationship provides man- agers with risk-shifting incentives toward the end of the year. Given the dramatic changes in the relationship after 2000 and its variations according to market conditions and per- formance dispersion, I look at how managersrisk-shifting behavior in the fourth quarter changes after 2000, and how such changes are related to those variables that a¤ect the ow-performance relationship. 75 Table 2.5. Descriptive statistics for risk shift measures Descriptive statistics for four risk-shift measures: total risk (Dec-Sep), idiosyncratic risk (Dec-Sep), systematic risk (Dec-Sep), and Risk- shift. Total risk of a fund in a given month in year t is the annualized standard deviation of daily excess returns over the CRSP VW index of a portfolio that holds the same stocks in the fund at the end of the month. Daily returns are over the prior calendar year (i.e., year t-1). Total risk (Dec-Sep) is total risk in December minus total risk in September. Idiosyncratic risk of a fund in a given month is the annualized standard deviation of unexpected daily returns of a portfolio, dened as returns on the portfolio minus the portfolio beta times returns on the CRSP VW. The portfolio beta is weighted average of betas of the stocks, which are the slope estimates when regressing the stocks daily returns on the CRSP VW returns over the prior calendar year. Systematic risk (Dec-Sep) in year t is the di¤erence between the absolute value of beta in December minus one and the absolute value of beta in September minus one. Hypothetical volatility is the annualized standard deviation of monthly return on a portfolio over the prior 36 months that holds the same stocks at the end of the month. Actual volatility is the annualized standard deviation of the funds actual returns over the prior 36 months. The sample includes only the fund whose corresponding portfolios have at least 80n% of the fund TNAs. Age is the number of months since the inception date of a fund. The sample period is from 1983 to 2008. Di¤represents the t-statistics for equal means between the two periods, adjusted for correlations among funds and autocorrelations. 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean total risk (Dec - Sep) (%) -0.055 -0.012 0.807 -0.038 -0.003 0.950 -0.060 -0.014 0.759 (-0.18) idiosyncratic risk (Dec-Sep) (%) -0.039 -0.008 0.693 -0.048 -0.006 0.881 -0.036 -0.009 0.627 (0.12) systematic risk (Dec-Sep) (%) -0.339 -0.095 4.808 -0.360 0.055 5.647 -0.332 -0.118 4.529 (0.05) total risk (Sep) 0.083 0.071 0.050 0.087 0.077 0.041 0.082 0.069 0.052 (-0.38) idiosyncratic risk (Sep) 0.075 0.065 0.042 0.080 0.070 0.038 0.073 0.063 0.043 (-0.71) systematic risk (Sep) 0.194 0.141 0.177 0.224 0.183 0.178 0.185 0.128 0.175 (-1.44) equity in billion 1.249 0.257 4.000 0.543 0.127 1.738 1.459 0.323 4.433 (4.63) age in years 16.461 11.667 14.123 12.636 6.667 14.008 17.597 12.667 13.957 (4.95) return over CRSP VW (Sep) 0.005 -0.006 0.094 -0.003 -0.008 0.101 0.007 -0.006 0.092 (0.83) number of observations 9644 2207 7437 76 (Table 2.5 continued) 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean hypothetical - actual volatility (Dec) 0.004 0.004 0.038 0.020 0.016 0.033 0.001 0.002 0.038 (-2.24) hypothetical volatility (Dec) 0.221 0.213 0.085 0.244 0.242 0.079 0.217 0.207 0.085 (-0.48) actual volatility (Dec) 0.217 0.209 0.086 0.224 0.222 0.072 0.216 0.206 0.089 (-0.08) equity in billion 1.486 0.315 4.686 0.855 0.155 3.336 1.580 0.350 4.848 (2.87) age in years 18.169 13.667 14.451 15.789 9.667 15.405 18.525 13.667 14.269 (2.15) return over CRSP VW (Sep) -0.002 -0.010 0.084 -0.008 -0.008 0.094 -0.001 -0.010 0.082 (0.98) number of observations 6723 952 5771 77 Provided that managers take more risk as an attempt to increase inows in the fol- lowing year, risk-shifting should also depend on performance according to the shape in the ow-performance relationship. For example, when the relationship is convex, it would be underperformingmanagerswhohavestrongincentivestogamblebytakingmorerisk. Thus, I also examine how the relationship between risk-shifting and performance varies according to the variables that determine the ow-performance sensitivity. To this end, I include the interactiontermsbetweenperformanceandthoseconditioningvariablesasregressors. Irun the following regression: risk shift i;t;Dec = + 1 period t + 0 2 controls i;t + 3 ret i;t;SEP + 4 ret i;t;SEP period t + 0 5 ret i;t;SEP Z t;SEP +" i;t ; (30) where risk shift i;t;Dec is RISK i;tDec RISK i;tSep when using total, unsystematic, and systematic risk, and is hypothetical risk i;t;Dec actual risk i;t;Dec by Huang, Sialm and Zhang. The variable period t is one when the year t is in the second period (from 2000 to 2008). The conditioning variables Z t;SEP include market return (demeaned), market volatility (demeaned), and performance dispersion (residual) as explained in Section 1. 22 Given that I look at managersrisk-shifting behavior in the fourth quarter, I measure these variables as of the third quarter. In essence, these variables can be used to form expectations aboutthe ow-performance relationship in the followingyear. The below time line illustrates the case of RISK i;tDec RISK i;tSep : 22 I exclude industry ow at time t+1 since it is not known when managers engage in risk-shifting. 78 Jan (t-1) Dec (t-1) Jan (t) Sep (t) Dec (t) risk Sep(t) risk Dec(t) | {z } equity returns for risk Sep(t) and risk Dec(t) | {z } Z t, SEP I also include year dummy and report standard errors that are clustered by fund (I do not include lagged risk shift since its slope estimates are close to zero and insignicant). 2.3.3. Changes in managersrisk-shifting behavior Table 2.6 presents the linear regression results for managersrisk-shifting behavior. The rst table (A) provides the results for total risk, unsystematic risk and systematic risk. First, the coe¢ cient on performance (excess returns over the CRSP VW) at the end of September is negative and signicant for total risk, suggesting that managers who are behind the markets increase risk in the fourth quarter prior to 2000. Yet, this risk-shifting lessensbyaround70%in the2000s, asthenegativecoe¢ cientonthesecond perioddummy shows. This seems consistent with the weaker ow-performance relationship after 2000. Provided that convexity in the relationship leads underperforming managers to increase risk, they should engage in less risk-shifting when the shape is not convex. On the other hand, when decomposing risk, performance seems to have little impact on risk-shifting for the systematic part. The regression model (2) shows how these changes in risk-shifting are related to the variables that a¤ect the ow-performance relationship. For all three risk-shifting measures, performance dispersion across funds plays a signicant role. The coe¢ cients on its interac- 79 tion term with fund performance are negative. In essence, when performance is dispersed, those managers who are behind the markets (CRSP VW index) increase riskboth unsys- tematic and systematic riskin the fourth quarter. On the other hand, market volatility appears important for systematic risk. After including those conditioning variables, I nd that underperforming managers also increase systematic risk when market volatility is low and when performance is more dispersed. The nal regression (3) also includes the second period dummy and shows that those conditioning variables can explain changes in man- agersrisk-shifting behavior after 2000. Given that the ow-performance relationship is more likely to be convex following large performance dispersion and low market volatility, my results support the view that managers respond to the ow-performance relationship by changing the riskiness of their funds toward the end of the year. Table 2.7 provides the results for risk-shifting from the risk level over the prior 36 months. The positive, signicant intercept in the regression model (1) suggests that prior to2000,fundsriskattheendofDecemberwashigherbyaround0.03comparedtothelong- termlevel. Afterthatyear,themagnitudedecreasesbyaround30%,ascanbeseenfromthe negative coe¢ cient on the second period dummy. Yet, this change appears to be unrelated to changes in the ow-performance relationship. The risk-shift measure is positively related toperformancethroughoutthe sampleperiodfrom 1983 to2008. Good performersincrease risk compared to the long-term risk level. Moreover, the relationship between performance and a deviation from the long-term risk level does not depend on performance dispersion, which determines the expected sensitivity of the ow-performance relationship. Rather, such relationship increases with market volatility, suggesting that underperformers take risk when market volatility is low while outperformers take even more risk when it is high. 80 Table 2.6. Linear regressions for risk shift measures by Chevalier and Ellison (1997) The dependent variables are risk-shift measures as listed in the column header and described in the below. The independent variables are listed in the rst column. The numbers for each independent variable are estimates, standard errors, and p-values respectively. The regressions include year dummy, and standard errors are clustered by fund. Second period is a dummy variable that takes one if the year is between 2000 and 2008 and zero if the year is between 1983 and 1999. Risk (Sep) is total risk, idiosyncratic risk and systematic risk at the end of September respectively. Market return (Sep) and market volatility (Sep) are the mean and the standard deviation of daily returns on the CRSP VW index from January to September (annualized) respectively. Log size is the natural logarithm (log) of the value of the hypothetical portfolio constructed as described in the below. Log age is the log of months since the inception date. Performance (Sep) is excess returns on funds over the CRSP value-weighted index from January to September. Performance dispersion (Sep) is the residual of the cross-sectional standard deviation of performance of sample funds from January to September after regressing it on market return (Sep) and market volatility (Sep). The dependent variables are as follows. Total risk of a fund in a given month in year t is the annualized standard deviation of daily excess returns over the CRSP value-weighted index (CRSP VW) of the hypothetical portfolio that holds the same stocks in the fund at the end of that month. Daily returns are over the prior calendar year (i.e., year t-1). Idiosyncratic risk of a fund in a given month is the annualized standard deviation of unexpected daily returns of the hypothetical portfolio, dened as returns on the portfolio minus the portfolio beta times returns on the CRSP VW. The portfolio beta is weighted average of betas of the stocks, which are the slope estimates when regressing the stocks daily returns on the CRSP VW returns over the prior calendar year. Systematic risk (Dec-Sep) in year t is the di¤erence between the absolute value of beta in December minus one and the absolute value of beta in September minus one. The sample period is from 1983 to 2008 and the numbers of observations is 9,644. total risk idiosyncratic risk systematic risk (1) (2) (3) (1) (2) (3) (1) (2) (3) intercept -0.001 -0.001 -0.001 0.001 0.001 0.001 -0.020 -0.019 -0.019 (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.016) (0.015) (0.015) (0.576) (0.659) (0.658) (0.683) (0.598) (0.605) (0.211) (0.205) (0.206) second period 0.001 0.001 0.000 0.000 0.026 0.025 (0.001) (0.001) (0.002) (0.002) (0.015) (0.015) (0.451) (0.549) (0.895) (0.816) (0.082) (0.088) 81 (Table 2.6 continued) total risk idiosyncratic risk systematic risk (1) (2) (3) (1) (2) (3) (1) (2) (3) risk (Sep) -0.015 -0.015 -0.015 -0.018 -0.018 -0.018 -0.044 -0.043 -0.043 (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) (0.004) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) log size (Sep) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.701) (0.756) (0.755) (0.985) (0.915) (0.924) (0.875) (0.983) (0.974) log age (Sep) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.001) (0.001) (0.541) (0.476) (0.476) (0.829) (0.821) (0.820) (0.607) (0.505) (0.506) performance (Sep) -0.011 -0.004 -0.004 -0.010 -0.003 -0.004 -0.010 -0.005 0.001 (0.003) (0.001) (0.003) (0.002) (0.001) (0.003) (0.014) (0.008) (0.016) (0.000) (0.013) (0.165) (0.000) (0.043) (0.156) (0.474) (0.520) (0.941) *market return (Sep) 0.002 0.003 -0.025 -0.022 0.255 0.240 (0.013) (0.014) (0.012) (0.013) (0.071) (0.074) (0.895) (0.852) (0.032) (0.074) (0.000) (0.001) *market volatility (Sep) 0.024 0.024 -0.037 -0.040 0.895 0.907 (0.036) (0.036) (0.034) (0.033) (0.203) (0.201) (0.500) (0.510) (0.266) (0.234) (0.000) (0.000) *performance dispersion (Sep) -0.310 -0.305 -0.252 -0.235 -1.037 -1.125 (0.053) (0.060) (0.046) (0.051) (0.273) (0.326) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) *second period 0.007 0.000 0.008 0.002 0.015 -0.009 (0.003) (0.004) (0.003) (0.003) (0.016) (0.020) (0.011) (0.889) (0.001) (0.573) (0.345) (0.634) adjusted R squared 0.060 0.067 0.067 0.054 0.058 0.058 0.056 0.062 0.062 82 The results that performance dispersion is insignicant for risk-shifting from the level over the prior 3 years do not seem to support the argument that the ow-performance relationship provides managers with an incentive to deviate from a long-term risk level. Thus, I interpret that to increase inows during the following year, managers increase risk relative to the risk level at the end September, but not relative to a long-term risk level. 2.4. Robustness check 2.4.1. Dummy variables for market conditions Instead of using indicator variables for market returns and market volatility as in Section 2.2.5, I use dummy variables for those market conditions as conditioning variables for the ow-performance relationship. I dene market volatility state-High and market volatility state-Low as the dummy variables that have a value of one if market volatility is high (more than 16%) and low (below 10%) respectively. Similarly, I dene market return state-High and market return state-Low. Table 2.7 presents the regressions using those dummy variables (other independent variables not reported). The ndings are consistent with the results obtained from the indicator variables (i.e., Table 2.4). Market returns do not appear to have a signicant impactontheshapeoftheow-performancerelationship,butmarketvolatilitydecreasesits convexity. In highly-volatile markets, the relationship is concave. In low-volatility markets, the shape seems more convex, but we cannot reject the hypothesis that it is linear. On the other hand, performance dispersion and industry ow have positive e¤ects on convexity of the ow-performance relationship. 83 Table 2.7. Determinants of ow-performance sensitivity when dummy variables for market conditions are used Market ret state and market vol state in Table 2.4 are replaced with the dummy variables, market ret state-High/-Low, and market vol state-High/-Low respectively. Market ret state-High is 1 when the mean of daily returns on the CRSP VW index over the year is in the high group (i.e., when market ret state is +1) and 0 otherwise. Similarly, other dummy variables are dened. See Table 2.4 for details. (The complete results are available upon request). (1) (2) (3) (4) (5) (6) performance (t-1) 0.822 0.829 0.840 0.887 0.884 0.889 (0.151) (0.149) (0.156) (0.140) (0.137) (0.136) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) squared performance (t-1) -0.384 0.647 0.250 0.788 -0.058 -0.318 (0.066) (0.312) (0.977) (1.026) (1.393) (0.901) (0.000) (0.038) (0.798) (0.442) (0.967) (0.724) *market ret state-High 0.315 -0.596 -0.506 -0.039 (t-1) (0.996) (1.103) (1.430) (0.784) (0.752) (0.589) (0.724) (0.961) *market ret state-Low 1.019 0.907 0.978 1.703 (t-1) (1.438) (1.489) (1.659) (1.230) (0.479) (0.543) (0.556) (0.167) *market vol state-High -0.982 -1.790 -1.285 0.335 (t-1) (0.310) (0.458) (0.341) (0.541) (0.002) (0.000) (0.000) (0.535) *market vol state-Low 0.842 0.595 0.450 1.128 (t-1) (2.035) (2.058) (2.175) (2.082) (0.679) (0.772) (0.836) (0.588) *performance dispersion 6.650 9.920 11.233 (t-1) (2.579) (2.438) (1.433) (0.010) (0.000) (0.000) *industry ow (t) 20.420 17.812 (6.899) (7.475) (0.003) (0.017) *second period (t) -1.056 -2.116 (0.307) (0.572) (0.001) (0.000) Adjusted R squared 0.225 0.228 0.228 0.230 0.232 0.233 2.4.2. Piecewise regression I also estimate the ow-performance relationship for seven intervals of performance, lower than 0:2, between 0:2 and 0:1, between 0:1 and 0; and so forth. More speci- 84 cally, I run piecewise OLS regression: net flow i;t = + A 1 performance A i;t1 + B 1 performance B i;t1 +:::+ G 1 performance G i;t1 + 3 performance i;t + 4 performance i;t2 + 5 age i;t1 + 6 size i;t1 + 7 expense i;t1 + 8 volatility i;t1 + 9 industry i;t + 10 style i;t + 11 net flow i;t1 +" i;t ; whereperformanceintervalsaredenedasperformance A i;t1 = min[performance i;t1 ;0:2]; performance B i;t1 = min[performance i;t1 performance A i;t1 ;0:1]; and so on. Ialsointeractallthoseperformancevariableswiththeconditioningvariables(e.g.,mar- ketvolatility,performancedispersion,andsecond-perioddummy)toexaminetimevariation in marginal ow (the coe¢ cients from A 1 to G 1 ): Table 2.8 presents the piecewise OLS regression results for the market excess returns. All regressions include the control variables as shown in the above regression equation, such as fund characteristics, industry ow, and lagged ow (these coe¢ cients are not reported). The results conrm that the relationship is not convex on average over the whole sample periodfrom1983to2008(regression(1))andtheshapechangesfromconvexitytoconcavity around the year 2000. Using the coe¢ cient estimates in the regression (2), I plot the ow- performance relationship in Figure 2.7 (A). The regressions from (3) to (6) show the conditional version of the ow-performance relationship. Similar to the OLS results, market volatility and performance dispersion a¤ect marginal ow. In particular, market volatility appears important for net ows to outperformance below 10%. Figure 2.7 (B) shows the relationship conditional on market 85 volatility (regression (5)) with other conditioning variables xed. Investors appear less sensitivetofundsoutperformingthemarketby10%orlesswhenmarketsarehighlyvolatile. Iinterpretthisresultasconsistentwiththeviewthatinvestorsperceivesuchoutperformance arising primarily from luck in highly volatile markets. On the other hand, performance dispersion and industry ow are more salient for net ows to outperformance of more than 20%. When performance is more dispersed and when the mutual fund industry receives more investment money, funds that are way ahead of the market attract more inows. Table2.8. Determinantsofow-performancesensitivityusingpiecewiseOLSregression The dependent variable is annual net fund ows in the year t: Numbers for each independent variable are estimates, standard errors (clustered by year and fund), and p-values respectively. Performance is annual excess returns over the CRSP value-weighted index returns. Performance [ , - 0.2]isequaltomin(performance,-0.2)andperformance[-0.2,-0.1]tomin(performance-performance [ , -0.2], 0.1) and so on. Market ret state is -1, 0, and +1 when the mean of daily returns on the CRSP VW index over the year is in the low, middle, and high group respectively. Similarly, market volstateis-1(low), 0(medium), and+1(high)accordingtothestandarddeviationofdailyreturns; and industry ow state is equal to -1 (low), 0 (medium), and +1 (high demand), depending on net owstoallequitymutualfunds. Performancedispersionistheresidualobtainedfromtheregressions of the cross-sectional standard deviation of performance on the mean and the volatility of the daily CRSPVWreturns. Thesecondperiodisoneiftheyearisbetween2000and2008andzerootherwise. See Table 2.4 for descriptions of variables not presented in the table. The number of observations is 17,679 over 1983 to 2008. (The complete results are available upon request). 86 (Table 2.8. continued) (1) (2) (3) (4) (5) (6) performance (t-1) [ , -0.2] -0.082 0.028 0.007 -0.037 -0.053 -0.010 (0.350) (0.283) (0.358) (0.329) (0.331) (0.343) (0.815) (0.920) (0.984) (0.909) (0.873) (0.977) *market ret state (t-1) 0.210 0.120 0.007 0.059 (0.183) (0.207) (0.195) (0.247) (0.251) (0.561) (0.971) (0.810) *market vol state (t-1) -0.008 -0.144 -0.225 -0.271 (0.250) (0.208) (0.209) (0.206) (0.976) (0.489) (0.282) (0.189) *performance dispersion 3.135 5.395 0.067 (t-1) (1.508) (2.769) (0.361) (0.038) (0.052) (0.852) *industry ow (t-1) 0.296 0.320 (0.319) (0.322) (0.354) (0.321) *second period (t) -0.229 4.834 (0.394) (3.009) (0.561) (0.109) performance (t-1) [-0.2, -0.1] 0.556 0.271 0.670 0.605 0.539 0.334 (0.392) (0.509) (0.404) (0.381) (0.388) (0.582) (0.156) (0.594) (0.098) (0.113) (0.166) (0.566) *market ret state (t-1) 0.194 -0.038 -0.292 -0.146 (0.435) (0.424) (0.431) (0.497) (0.656) (0.928) (0.499) (0.769) *market vol state (t-1) 0.123 -0.210 -0.311 -0.407 (0.545) (0.512) (0.494) (0.469) (0.821) (0.682) (0.529) (0.386) *performance dispersion (t-1) 8.848 13.178 0.490 (3.331) (7.169) (0.747) (0.008) (0.066) (0.512) *industry ow (t-1) 0.367 0.385 (0.564) (0.586) (0.515) (0.511) *second period (t) 0.462 11.687 (0.730) (7.675) (0.528) (0.128) 87 (Table 2.8 continued) (1) (2) (3) (4) (5) (6) performance (t-1) [-0.1, -0] 1.295 1.598 1.171 1.192 1.193 1.116 (0.280) (0.331) (0.173) (0.179) (0.178) (0.303) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) *market ret state (t-1) 0.176 0.242 0.237 0.236 (0.222) (0.240) (0.249) (0.272) (0.430) (0.313) (0.342) (0.386) *market vol state (t-1) -0.839 -0.775 -0.811 -0.825 (0.186) (0.222) (0.214) (0.193) (0.000) (0.001) (0.000) (0.000) *performance dispersion -1.765 -2.071 0.096 (t-1) (4.227) (4.286) (0.395) (0.676) (0.629) (0.809) *industry ow (t-1) -0.091 -0.031 (0.267) (0.257) (0.733) (0.905) *second period (t) -0.524 -1.455 (0.552) (4.235) (0.343) (0.731) performance (t-1) [0, 0.1] 1.065 1.477 1.882 1.880 1.866 1.845 (0.354) (0.466) (0.329) (0.338) (0.333) (0.348) (0.003) (0.002) (0.000) (0.000) (0.000) (0.000) *market ret state (t-1) 0.023 -0.054 -0.067 -0.094 (0.340) (0.348) (0.335) (0.327) (0.946) (0.877) (0.842) (0.773) *market vol state (t-1) -0.865 -0.917 -0.892 -0.945 (0.454) (0.444) (0.436) (0.466) (0.057) (0.039) (0.041) (0.043) *performance dispersion 2.666 3.087 0.136 (t-1) (4.972) (4.868) (0.556) (0.592) (0.526) (0.806) *industry ow (t-1) 0.157 0.156 (0.272) (0.285) (0.565) (0.584) *second period (t) -0.695 2.209 (0.653) (5.365) (0.288) (0.681) 88 (Table 2.8 continued) (1) (2) (3) (4) (5) (6) performance (t-1) [0.1, 0.2] 0.472 1.502 1.904 1.818 1.837 2.291 (0.535) (0.607) (0.752) (0.802) (0.844) (0.982) (0.378) (0.014) (0.011) (0.024) (0.030) (0.020) *market ret state (t-1) -0.473 -0.254 -0.227 -0.338 (0.384) (0.333) (0.228) (0.211) (0.219) (0.445) (0.320) (0.109) *market vol state (t-1) -0.880 -0.641 -0.478 0.028 (0.868) (0.887) (0.931) (0.877) (0.311) (0.470) (0.608) (0.974) *performance dispersion -1.309 0.285 -1.384 (t-1) (3.902) (3.371) (0.814) (0.737) (0.933) (0.089) *industry ow (t-1) 0.493 0.242 (0.488) (0.400) (0.313) (0.545) *second period (t) -1.234 2.255 (0.832) (3.249) (0.138) (0.488) performance (t-1) [0.2, ] 0.236 0.842 0.710 0.792 0.508 0.419 (0.122) (0.191) (0.637) (0.645) (0.468) (0.501) (0.054) (0.000) (0.265) (0.219) (0.277) (0.404) *market ret state (t-1) 0.036 -0.297 -0.074 -0.106 (0.540) (0.562) (0.368) (0.382) (0.946) (0.597) (0.840) (0.781) *market vol state (t-1) -0.476 -1.053 0.024 0.850 (0.300) (0.305) (0.484) (0.820) (0.113) (0.001) (0.961) (0.300) *performance dispersion 4.819 5.630 -0.912 (t-1) (0.989) (0.686) (0.942) (0.000) (0.000) (0.333) *industry ow (t-1) 1.156 0.963 (0.452) (0.591) (0.011) (0.104) *second period (t) -0.575 5.683 (0.222) (0.661) (0.010) (0.000) Adjusted R squared 0.227 0.233 0.242 0.244 0.246 0.247 89 Figure 2.7. Flow-performance relationship using piecewise OLS regression The y-axes represent expected annual net ows into and out of nonindex funds. Performance is annual excess returns over the CRSP value weighted index (CRSP VW). The expected annual net ows are estimated over seven intervals of performance: lower than 0:2, between 0:2 and 0:1, between0:1 and 0; between 0 and 0:1; between 0:1 and 0:2; higher than 0:2; after control- ling contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, style ow and lagged ow. Log age is the natural logarithm (log) of the months since the inception date of a fund (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). Funds are aggregated across share classes, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. (A) Flow-performance relationship from 1983 to 1999 (before 2000) and from 2000 to 2008 (after 2000) (B) The ow-performance relationship in the low volatility market and in the high volatility market using piecewise OLS regression. Market volatility is ranked into low, medium and high groups based on the standard deviation of daily return on the CRSP VW index. (A) Flow-performance relationship before and after 2000 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.2 0.1 0 0.1 0.2 0.3 lagged return over CRSP annual flows before 2000 after 2000 90 (Figure 2.7 continued) (B) Flow-performance relationship conditional on market volatility 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 lagged return over CRSP annual flows low volatility high volatility 2.4.3. Performance ranking Sofar,Ilookatbenchmark-adjustedreturnsasperformance(e.g.,ChevalierandEllison (1997), Barber, Odean, and Zheng (2005), Sensoy (2009)). Goetzmann and Peles (1996) and Sirri and Tufano (1998) among others use performance ranking and nd that the ow-performance relationship is signicant only for top-ranked funds, leading to a convex ow-performance relationship. To examine whether my results are robust to performance measures, I rank raw returns in the ascending order and divide the rank by the number of funds (i.e., the fund with highest return has the rank of one) and estimate the relationship between net ows and that performance ranking. I also estimate the conditional version of the relationship and examine how performance ranking is related to managersrisk-shifting in the fourth quarter. 91 Using ranking as a performance measure does not change the conclusion that marginal ow to high performing funds is lower in the post-2000 period than before 2000. I rst plot TNA-weighted net ows of funds in each bin from 1 to 10 according to deciles of the performance ranking as described above (i.e., a ranking of 0.15 is in the 2nd bin, and a ranking of 0.95 is in the 10th bin)in Figure 2.8. The average netows to high-ranked funds decrease dramatically after 2000. Figure 2.8. TNA-weighted net ows TNA-weightednetowsineachbinfrom1to10before2000andafter2000. Fundsaregrouped into 10 based on lagged ranks in the ascending order (ranks divided by the total number of funds in a given year) based on annual returns minus the CRSP value weighted returns. The groups are assigned the bin numbers from 1 to 10. Funds are aggregated across share classes, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. TheresultsofkernelregressionandOLSregressionareconsistentwiththendingthat marginal ow to high-ranked funds is much lower in the post-2000 period. Figure 2.9 shows 92 the ow-performance relationship estimated using kernel regression, after controlling other factors such as fund characteristics and industry ow. For example, those funds ranked at the90percentilereceived30%ofinowsonaveragebefore2000,whichdecreaseto10%after 2000. This di¤erence is comparable to the results obtained using the absolute performance measures (e.g., excess returns over the CRSP VW returns). Outperforming the market by 0.2 leads to 30% of inows before 2000 but only 10% after that year. The shape of the ow-performance relationship is still convex even after 2000. Yet, convexity decreases dramatically. Before 2000, the relationship seems linear up to around the 70 percentile, after which the sensitivity increases sharply. After 2000, the linear rela- tionship holds up to the 90 percentile and it becomes convex after that. The slope after the 90 percentile in the post-2000 period is also less steep than the slope after the 70 percentile before 2000. The OLS regression results are also consistent. The estimate on squared ranks ispositiveinbothperiods,butthemagnitudedecreasesbyabout70%after2000(regression (3) in Table 2.9). Ialsoexaminewhetherconvexityintherelationshipbetweennetowsandperformance ranking depends on the conditioning variables. I nd that performance dispersion increases convexity of the relationship between net ows and performance ranking. As regression models (6) and (7) in Table 2.9 show, when fund returns are more dispersed, investors are much more sensitive to high-ranked funds. Yet, the e¤ect of market volatility on fund ows responding to performance ranking seems noisy and insignicant in the OLS regression. Given a right-skewed distribution of fund returns in highly-volatile markets (see Figure 2.5 (C) and (D)), fundsabsolute performance can improve in such markets, but relative performance may not. 93 Figure 2.9. Flow-performance (ranking) relationship by kernel regression Estimatesoftheow-performancerelationshipsusingkernelregressionssuggestedbyRobinson (1988) and their 90% condence intervals. The y-axes represent expected annual net ows into and out of nonindex funds from 1983 to 1999 (before 2000) and from 2000 to 2008 (after 2000). Performance is rankin the ascendingorderbasedon annualreturnsminusthe CRSPvalueweighted index (CRSP), divided by the total number of funds in a given year. The expected annual net ows are estimated after controlling contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, style ow and lagged ow. See Figure 7 for variable descriptions. Funds are aggregated across share classes, excluding institutional share classes. The total number of funds is 2,264 over 1983 to 2008 and the numbers of observations are 6,771 and 10,908 before 2000 and after 2000 respectively. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 lagged performance rank expected annual net flows before 2000 after 2000 Finally, I also conrm changes in managersrisk-shifting behavior after 2000 when performance is measured as ranking. Similar to low-performing funds (based on excess returns over the market), low-ranked funds tend to increase the riskiness of their fund in the fourth quarter, but this risk-shifting lessens in the post-2000 period. This change in managersrisk-shifting is also explained by performance dispersion, similar to the results when using excess returns over the CRSP VW index as performance measure (Table 2.10). 94 Table 2.9. Determinants of ow-performance (ranking) sensitivity See Table 4 for details. The only di¤erence is performance, which is ranking in the ascending order based on annual returns minus the CRSP value weighted index (CRSP), divided by the total number of funds in a given year. (The complete results are available upon request). (1) (2) (3) (4) (5) (6) (7) performance (t-1) 0.445 0.360 0.361 0.363 0.364 0.363 0.363 (0.046) (0.043) (0.044) (0.044) (0.044) (0.044) (0.044) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) squared performance (t-1) 0.517 0.352 0.610 0.365 0.363 0.365 0.654 (0.123) (0.119) (0.131) (0.122) (0.124) (0.109) (0.152) (0.000) (0.003) (0.000) (0.003) (0.003) (0.001) (0.000) *market ret state (t-1) -0.138 -0.167 -0.166 -0.233 (0.143) (0.143) (0.126) (0.133) (0.336) (0.243) (0.185) (0.078) *market vol state (t-1) -0.103 -0.133 -0.125 -0.072 (0.141) (0.135) (0.145) (0.131) (0.465) (0.325) (0.387) (0.583) *performance dispersion 2.429 2.937 3.280 (t-1) (1.622) (1.420) (1.514) (0.134) (0.039) (0.030) *industry ow state (t) 0.216 0.124 (0.143) (0.126) (0.131) (0.323) *second period (t) -0.410 -0.479 (0.187) (0.170) (0.028) (0.005) expense ratio (t-1) 0.772 0.300 -0.173 1.400 1.538 1.435 0.170 (2.055) (1.558) (2.149) (1.972) (1.960) (1.981) (2.080) (0.707) (0.847) (0.936) (0.478) (0.433) (0.469) (0.935) *second period (t) 1.026 -1.716 -1.930 -1.685 0.642 (2.014) (1.946) (1.808) (1.851) (1.995) (0.610) (0.378) (0.286) (0.363) (0.748) ow (t-1) 0.251 0.252 0.252 0.251 0.252 0.252 (0.029) (0.029) (0.029) (0.029) (0.029) (0.029) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Adjusted R squared 0.161 0.233 0.234 0.234 0.234 0.235 0.236 95 Table 2.10. Linear regressions for risk shift measures using performance ranks The dependent variables are risk-shift measures as listed in the column header and described in the below. The independent variables are listed in the rst column. The numbers for each independent variable are estimates, standard errors, and p-values respectively. The regressions include year dummy, and standard errors are clustered by fund. Second period is a dummy variable that takes one if the year is between 2000 and 2008 and zero if the year is between 1983 and 1999. Risk (Sep) is total risk, idiosyncratic risk and systematic risk at the end of September respectively. Market return (Sep) and market volatility (Sep) are the mean and the standard deviation of daily returns on the CRSP VW index from January to September (annualized) respectively. Log size is the natural logarithm (log) of the value of the hypothetical portfolio constructed as described in the below. Log age is the log of months since the inception date. Performance (Sep) is rank in the ascending order based on excess returns on funds over the CRSP value-weighted index from January to September, divided by the total number of funds in a given year. Performance dispersion (Sep) is the residual of the cross-sectional standard deviation of excess returns over the CRSP value-weighted index of sample funds from January to September after regressing it on market return (Sep) and market volatility (Sep). The dependent variables are as follows. Total risk of a fund in a given month in year t is the annualized standard deviation of daily excess returns over the CRSP value-weighted index (CRSP VW) of the hypothetical portfolio that holds the same stocks in the fund at the end of that month. Daily returns are over the prior calendar year (i.e., year t-1). Idiosyncratic risk of a fund in a given month is the annualized standard deviation of unexpected daily returns of the hypothetical portfolio, dened as returns on the portfolio minus the portfolio beta times returns on the CRSP VW. The portfolio beta is weighted average of betas of the stocks, which are the slope estimates when regressing the stocks daily returns on the CRSP VW returns over the prior calendar year. Systematic risk (Dec-Sep) in year t is the di¤erence between the absolute value of beta in December minus one and the absolute value of beta in September minus one. The sample period is from 1983 to 2008 and the numbers of observations are 9,644. total risk idiosyncratic risk systematic risk (1) (2) (3) (1) (2) (3) (1) (2) (3) intercept 0.000 0.000 0.000 0.002 0.002 0.002 -0.021 -0.028 -0.026 (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.016) (0.015) (0.015) (0.843) (0.817) (0.867) (0.237) (0.264) (0.149) (0.186) (0.060) (0.080) second period 0.000 0.000 -0.001 -0.003 0.025 0.037 (0.002) (0.002) (0.002) (0.002) (0.015) (0.015) (0.933) (0.888) (0.385) (0.059) (0.097) (0.014) 96 (Table 2.10 continued) total risk idiosyncratic risk systematic risk (1) (2) (3) (1) (2) (3) (1) (2) (3) risk (Sep) -0.018 -0.017 -0.018 -0.020 -0.019 -0.019 -0.045 -0.047 -0.047 (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) (0.004) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) log size (Sep) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.901) (0.926) (0.979) (0.833) (0.653) (0.747) (0.785) (0.867) (0.918) log age (Sep) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.001) (0.001) (0.453) (0.431) (0.426) (0.729) (0.765) (0.755) (0.543) (0.453) (0.447) performance (Sep) -0.002 -0.001 -0.003 -0.002 -0.001 -0.003 0.002 0.003 -0.002 (0.001) (0.000) (0.001) (0.001) (0.000) (0.001) (0.004) (0.002) (0.004) (0.005) (0.011) (0.000) (0.001) (0.002) (0.000) (0.563) (0.071) (0.670) *market return (Sep) 0.009 0.013 0.002 0.005 0.073 0.083 (0.004) (0.004) (0.003) (0.003) (0.021) (0.022) (0.014) (0.001) (0.604) (0.136) (0.001) (0.000) *market volatility (Sep) 0.022 0.023 0.012 0.013 0.191 0.194 (0.009) (0.009) (0.008) (0.008) (0.051) (0.051) (0.014) (0.008) (0.141) (0.102) (0.000) (0.000) *performance dispersion (Sep) -0.018 -0.017 -0.033 -0.033 0.180 0.180 (0.006) (0.006) (0.005) (0.005) (0.036) (0.036) (0.002) (0.003) (0.000) (0.000) (0.000) (0.000) *second period 0.002 0.003 0.002 0.002 0.002 0.007 (0.001) (0.001) (0.001) (0.001) (0.005) (0.005) (0.027) (0.003) (0.003) (0.004) (0.741) (0.146) adjsuted R squared 0.057 0.058 0.059 0.051 0.055 0.056 0.057 0.063 0.063 97 2.4.4. Flow-performance relationship for index funds I discuss the results for index funds to compare with nonindex funds. The descriptive statistics are provided in Table 2.11. Index funds received large inows in the 1990s. Index funds are younger, but the average size is almost double compared to nonindex funds, reectinga growth in passive management. The averageexpenseratios arelowerbyaround 0.08% than nonindex funds. After 2000, the ratios increase by 0.1% (not signicant). Table 2.12 shows that the ow-performance relationship for index funds are barely signicant. Yet, there seems to be a shift from concave relationship to convex relationship after 2000. In particular, investors appear to be more sensitive to index fundsinferior performance relative to their peer funds before 2000. On the other hand, similar to the results for nonindex funds, style ows do not have a signicant, positive impact on index fund ows after controlling for performance relative to the CRSP VW index (results are similar for performance relative to the S&P500 index) after 2000. Investors prefer funds thatbeatthemarketsirrespectiveoftheirinvestmentstylesafter2000. Theexpenseratiois the most important determinant for index fund ows throughout the years. An 1% increase in expense ratios leads to almost 20% decrease in fund ows in the following year. In Figure 2.10, I present kernel regression results for index funds from 2000 to 2008. Duetoasmallsamplesize,theestimationisimpossiblefortherstperiod(1983-1999), and the condence intervals of the estimates are large even in the second period. Nevertheless, exceptionally, investors appear to respond to the performance of index funds compared to their peer funds (same styles), consistent with the linear regression results. 98 Table 2.11. Descriptive statistics for index funds The sample is U.S. mutual funds whose objectives are tracking market indexes as stated in their prospectuses (index funds) and which meet sampling criteria described in the paper (e.g., excluding sector funds, international funds, funds closed to investors and small funds). Net owischangesintotalnetassets(TNA)excludingcapitalgainsanddividends, dividedbyTNAatthebeginningoftheperiod. Performanceof a fund is its annual return minus benchmark return. The benchmark return is return on the CRSP value weighted index (CRSP VW), return on the S&P500 index (SP500), return on the benchmark index designated by the fund (benchmark index), and the average return of the funds inthesamestylecategoryasdenedbytheMorningstar(stylereturn). Stylereturnincludereturnsonnonindexfundsbutexcludesectorfunds and international funds. All returns are net of expense ratios but before load fees. Log age is the natural logarithm (log) of the months since the inception date of the fund. Log size is the log of TNA of the fund divided by the average TNA of sample funds (including nonindex funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly return of the fund over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Style ow is net ows to the funds in each of 9 styles (including nonindex funds but excluding sector funds and international funds). CRSP VW returns and volatility is the mean and the standard deviation of daily returns on the CRSP VW respectively (annualized). All variables are measured over a year at the end of December except expense ratios, which are over fundsscal years. Fund variables are aggregated across share classes, excluding institutional share classes, for each fund. Di¤represents the t-statistics for equal means between the two periods, adjusted for correlations among funds and autocorrelations over time. 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean ow (t) 0.125 0.033 0.449 0.298 0.202 0.430 0.077 0.009 0.443 (-4.03) performance (t-1) return over CRSPVW -0.001 -0.017 0.089 0.013 0.013 0.080 -0.005 -0.026 0.091 (-0.67) return over SP500 0.008 -0.006 0.097 -0.012 -0.002 0.082 0.013 -0.009 0.100 (0.81) return over benchmark 0.001 -0.006 0.074 0.006 0.001 0.056 0.000 -0.014 0.078 (-0.23) return over style return 0.004 0.002 0.057 0.023 0.018 0.046 -0.001 0.000 0.059 (-1.78) 99 (Table 2.11 continued.) 1983-2008 1983-1999 2000-2008 Di¤ variable mean median std mean median std mean median std mean log age (t-1) 4.528 4.477 0.546 4.399 4.277 0.576 4.563 4.543 0.532 (1.42) age in years (t-1) 9.074 7.333 6.067 8.317 6.000 6.888 9.281 7.833 5.811 (0.73) size (t-1) 1.533 1.553 1.770 1.592 1.583 1.593 1.517 1.532 1.816 (-0.27) TNA in billions (t) 3.451 0.590 11.815 3.010 0.636 9.594 3.571 0.569 12.353 (2.83) expense ratio (t-1) 0.005 0.004 0.004 0.004 0.004 0.003 0.005 0.004 0.004 (1.19) volatility (t-1) 0.038 0.033 0.018 0.040 0.036 0.018 0.037 0.033 0.018 (-0.35) industry ow (t) 0.070 0.060 0.076 0.087 0.082 0.086 0.037 0.040 0.041 (-2.00) style ow (t) 0.081 0.061 0.112 0.098 0.097 0.128 0.047 0.022 0.066 (-1.51) CRSPVW returns (t-1) 0.132 0.155 0.138 0.166 0.198 0.110 0.067 0.084 0.167 (-1.63) CRSPVW volatility (t-1) 0.144 0.127 0.055 0.130 0.121 0.050 0.169 0.160 0.057 (1.75) observations (funds) 1172 (185) 251 (52) 921 (181) 100 Figure 2.10. Flow-performance relationship for index funds by kernel regression Estimates of ow-performance relationships for index funds using kernel regressions suggested by Robinson (1988) and 90% condence intervals (dotted lines). The y-axes represent expected annual net ows into and out of index funds from 2000 to 2008. Performance is annual returns minus benchmark returns. The benchmark returns are the CRSP value weighted index (CRSP), the S&P500 index (SP500), returns on the benchmark indexes designated by the funds (benchmark), andtheaveragereturnsofthefundsinthesamestylecategoryasdenedbytheMorningstar(style). Style returns include returns on index funds but exclude sector funds and international funds. The expected annual net ows are estimated after controlling contemporaneous performance, the second lag of performance, log age, log size, expense ratio, volatility, industry ow, style ow and lagged ow. Log age is the natural logarithm (log) of the months since the inception dates of funds (age). Log size is the log of TNA of a fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly returns over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database (including sector funds and internationalfunds). Styleowisnetowstothefundsineachof9styles(includingindexfundsbut excludingsectorfundsandinternationalfunds). Fundsareaggregatedacrossshareclasses,excluding institutional share classes. The total number of funds is 181 and the numbers of observations is 921. 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over CRSP annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over SP500 annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over benchmark annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over style annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over CRSP annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over SP500 annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over benchmark annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over CRSP annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over SP500 annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over benchmark annual flows 0.1 0.05 0 0.05 0.1 0.2 0.1 0 0.1 0.2 lagged return over style annual flows 101 Table 2.12. OLS regression for net ows for index funds The dependent variable of ordinary least square regressions is annual net ows for index funds in the year t: Numbers for each independent variable are estimates, standard errors (clustered by year and fund), and p-values respectively. Regressions are di¤erent depending on the performance measures, as described in the rst row. Style returns are the average returns of the funds in the same style categories as dened by the Morningstar. Squared performance is square of performance. Log age is the natural logarithm (log) of the months since the inception date of the fund. Log size is the log of TNA of the fund divided by the average TNA of sample funds (including index funds but excluding sector funds and international funds). Expense ratio does not include load fees. Volatility is the standard deviation of monthly return of the fund over the last two years. Industry ow represents net ows to all equity mutual funds in the CRSP database. Style ow is net ows to the funds in each of 9 styles (including index funds but excluding sector funds and international funds). All variables are measured over a year at the end of December except expense ratios, which are over the funds scal year. The numbers of observations are 251 and 921 from 1983 to 2008 and from 2000 to 2008 respectively. Fund variables are aggregated across share classes, excluding institutional share classes, for each fund. The columns Di¤show the test results for equal coe¢ cientsontheindividualvariables. Thevaluesrepresentthecoe¢ cientsontheinteractionterms between the variables and the second period dummy, their standard errors and p-values respectively (standard errors are clustered by year and fund). Chow-test p-values are the p-values for joint tests of equal coe¢ cients on the variables. returns over CRSP VW returns over style returns 1983-1999 2000-2008 Di¤ 1983-1999 2000-2008 Di¤ intercept 0.548 0.471 -0.146 0.444 0.429 -0.102 (0.294) (0.123) (0.051) (0.254) (0.135) (0.048) (0.062) (0.000) (0.004) (0.080) (0.001) (0.034) performance (t-1) 0.242 0.018 -0.244 0.801 0.322 -0.772 (0.291) (0.258) (0.310) (0.435) (0.219) (0.711) (0.405) (0.945) (0.433) (0.066) (0.141) (0.277) squared performance -0.763 0.422 4.163 -4.355 0.088 10.560 (t-1) (2.366) (0.220) (1.861) (7.346) (0.245) (6.982) (0.747) (0.055) (0.025) (0.553) (0.719) (0.131) performance (t) 1.169 0.329 -0.865 1.019 0.306 -0.723 (0.483) (0.204) (0.479) (0.625) (0.219) (0.741) (0.016) (0.106) (0.072) (0.103) (0.162) (0.329) performance (t-2) 1.317 -0.238 -1.620 1.741 -0.383 -2.274 (0.520) (0.097) (0.545) (0.551) (0.119) (0.795) (0.011) (0.015) (0.003) (0.002) (0.001) (0.004) log age (t-1) 0.003 -0.053 -0.043 0.003 -0.054 -0.038 (0.066) (0.032) (0.050) (0.062) (0.033) (0.051) (0.966) (0.097) (0.382) (0.956) (0.100) (0.456) log size (t-1) -0.038 -0.041 -0.037 -0.037 -0.040 -0.036 (0.021) (0.016) (0.022) (0.023) (0.015) (0.020) (0.078) (0.010) (0.095) (0.108) (0.009) (0.077) 102 (Table 2.12 continued.) returns over CRSP VW returns over style returns 1983-1999 2000-2008 Di¤ 1983-1999 2000-2008 Di¤ expense ratio (t-1) -25.222 -19.489 16.242 -25.841 -19.805 15.341 (11.694) (6.884) (11.185) (11.324) (7.464) (11.180) (0.031) (0.005) (0.147) (0.023) (0.008) (0.170) volatility (t-1) -3.141 0.711 4.120 -3.373 1.438 5.209 (1.764) (1.145) (1.649) (1.975) (1.263) (1.688) (0.075) (0.535) (0.013) (0.088) (0.255) (0.002) industry ow (t) -0.402 -0.760 -1.283 0.087 -0.367 -1.050 (0.503) (0.268) (0.634) (0.640) (0.214) (0.603) (0.425) (0.005) (0.043) (0.891) (0.087) (0.082) style ow (t) 0.548 0.548 -0.243 0.646 0.733 0.123 (0.294) (0.450) (0.327) (0.120) (0.215) (0.302) (0.062) (0.223) (0.457) (0.000) (0.001) (0.685) Adjusted R 2 and Chow-test p-values 0.136 0.057 (0.000) 0.117 0.057 (0.000) The results after pooling all years are presented in Table 2.13 (complete results are available upon request). As opposed to its negative e¤ect for ow-performance sensitivity for nonindex funds, market volatility has a positive impact on index funds ows when per- formanceismeasuredcomparedtothemarkets. Investorsaremoresensitivetoperformance intheperiodsfollowingvolatilemarkets. Thispositiveimpactonthesensitivitycanexplain the positive coe¢ cients on squared performance after 2000 as shown in Table 2.11. Finally, unlike nonindex funds, lagged ows do not have explanatory power for ows into and out of index funds. This suggests that there are few fund xed e¤ects on index fund ows. Moreover, performance and the control variables have less explanatory power for index funds. While the adjusted R-squared is 22-25% for nonindex funds, it is less than 10% for index funds. Thus, there could be other market- or industry-wide factors that inuence fund ows into and out of index funds. 103 Table 2.13. Determinants of ow-performance sensitivity for index funds Thedependentvariableofordinaryleastsquareregressionsisannualnetowsforindexfundsin theyeartandtheindependentvariablesarelistedintherstcolumn. Numbersforeachindependent variable are estimates, standard errors (clustered by year and fund), and p-values respectively. Regressions are di¤erent depending on performance measures. Style returns are the average returns of the funds in the same style category as dened by the Morningstar. Squared performance is square of performance. This variable has interaction terms with ve conditioning variables. Market ret state is -1, 0, and +1 when the mean of daily returns on the CRSP VW index over the year is in the low, middle, and high group respectively. Similarly, market vol state is -1 (low volatility), 0 (medium), and +1 (high) according to the standard deviation of daily returns; and industry ow state is equal to -1 (low), 0 (medium), and +1 (high demand), depending on net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Performance dispersion is the residual obtained from the regressions of the cross-sectional standard deviation of performance on the mean and the volatility of the daily CRSP VW returns. The second period is one if the year is between 2000 and 2008 and zero otherwise. Expense ratio does not include load fees. The variables notpresented areperformance(t); performance(t-2); logage(t-1); logsize(t-1); volatility(t-1); industryow(t); andstyleow(t). Allvariablesaremeasuredoverayearattheend of December except expense ratios, which are over fundsscal years. The number of observations is 1,172 over 1983 to 2008. Funds are aggregated across share classes, excluding institutional share classes,foreachfund. (Thecompleteresults,includingforotherperformancemeasures,areavailable upon request.) 104 (Table 2.13. continued) (A) performance: excess return over CRSP VW (1) (2) (3) (4) (5) (6) (7) performance (t-1) 0.149 0.094 0.065 0.070 0.045 0.021 -0.002 (0.163) (0.154) (0.140) (0.138) (0.139) (0.146) (0.153) (0.363) (0.542) (0.640) (0.615) (0.746) (0.886) (0.992) squared performance (t-1) 0.380 0.289 -2.195 -3.516 -2.894 -2.999 -0.276 (0.000) (0.128) (2.193) (2.161) (2.623) (2.633) (3.692) (0.000) (0.024) (0.317) (0.104) (0.270) (0.255) (0.940) *market ret state (t-1) -0.443 0.248 0.398 0.574 (0.309) (0.798) (0.792) (0.804) (0.152) (0.756) (0.615) (0.475) *market vol state (t-1) 4.373 4.537 4.149 5.872 (2.235) (2.434) (2.507) (2.785) (0.051) (0.063) (0.098) (0.035) *performance dispersion -36.739 -54.999 -95.602 (t-1) (30.460) (34.576) (43.990) (0.228) (0.112) (0.030) *industry ow state (t) -1.129 -3.244 (0.750) (1.615) (0.133) (0.045) *second period (t) 2.655 -5.158 (2.253) (3.986) (0.239) (0.196) expense ratio (t-1) -19.940 -19.277 -12.251 -11.722 -13.653 -13.662 -14.845 (6.004) (5.716) (7.779) (8.010) (8.267) (8.305) (7.807) (0.001) (0.001) (0.116) (0.144) (0.099) (0.100) (0.058) *second period (t) -9.010 -9.224 -7.933 -8.226 -6.350 (7.743) (8.033) (7.834) (7.883) (7.645) (0.245) (0.251) (0.311) (0.297) (0.406) ow (t-1) 0.051 0.051 0.053 0.052 0.051 0.048 (0.052) (0.052) (0.052) (0.052) (0.052) (0.052) (0.329) (0.326) (0.312) (0.316) (0.321) (0.351) Adjusted R squared 0.090 0.092 0.093 0.094 0.094 0.094 0.094 105 (Table 2.13 continued.) (B) performance: excess return over style return (1) (2) (3) (4) (5) (6) (7) performance (t-1) 0.462 0.419 0.442 0.709 0.652 0.543 0.620 (0.183) (0.169) (0.161) (0.240) (0.230) (0.233) (0.257) (0.012) (0.013) (0.006) (0.003) (0.005) (0.020) (0.016) squared performance (t-1) 0.202 0.092 -6.000 -1.384 0.161 0.138 -2.799 (0.000) (0.000) (7.318) (4.275) (4.535) (4.521) (7.148) (0.000) (0.000) (0.412) (0.746) (0.972) (0.976) (0.695) *market ret state (t-1) -1.549 -0.019 0.533 0.410 (0.787) (1.278) (1.226) (1.159) (0.049) (0.988) (0.664) (0.724) *market vol state (t-1) 2.411 3.735 3.169 1.616 (4.156) (4.096) (4.089) (4.590) (0.562) (0.362) (0.438) (0.725) *performance dispersion -103.642 -181.112 -161.312 (t-1) (62.091) (77.728) (81.924) (0.095) (0.020) (0.049) *industry ow state (t) -3.470 -2.354 (1.600) (2.516) (0.030) (0.350) *second period (t) 6.167 4.805 (7.308) (8.410) (0.399) (0.568) expense ratio (t-1) -19.843 -19.288 -15.830 -15.919 -17.200 -17.783 -17.367 (6.278) (6.071) (7.378) (7.352) (7.355) (7.219) (7.231) (0.002) (0.002) (0.032) (0.031) (0.020) (0.014) (0.016) *second period (t) -4.596 -4.031 -2.854 -2.325 -3.190 (7.308) (6.971) (6.659) (6.790) (7.348) (0.530) (0.563) (0.668) (0.732) (0.664) ow (t-1) 0.048 0.047 0.047 0.046 0.046 0.045 (0.050) (0.050) (0.050) (0.049) (0.049) (0.049) (0.336) (0.343) (0.345) (0.350) (0.354) (0.356) Adjusted R squared 0.093 0.095 0.094 0.095 0.095 0.096 0.095 2.5. Conclusion I show that the ow-performance relationship for U.S. mutual funds, which was convex prior to 2000, is no longer convex. In particular, the marginal ow to high-performing 106 funds signicantly decreases in the 2000s. According to my ndings, when markets are highly volatile and when performance is less dispersed, the sensitivity of ows to superior performance is low. These variations contribute to the change in the shape in the 2000s. Fundmanagersappeartorespondtochangesintheow-performancesensitivity. Their risk-shifting toward the end of the year lessens after 2000, and this change can be also explained by the same variables that determine the sensitivity of the ow-performance relationship. In particular, underperforming managers engage in less risk-shifting when performance across funds is less dispersed. They also take less systematic risk in the fourth quarter when market volatility is high. I argue that the ow-performance relationship can serve as a dynamic incentive contract and managers react to the incentives provided by the implicit performance compensation scheme. My results are consistent with the view that investors use performance to infer about skills. Investors chase good performance because they interpret it as superior ability, not because they are irrationally guided by the desire to invest in recent winner funds. Changes in the ow-performance relationship according to market volatility and performance dis- persion and an exceptionally strong relationship between index fund ows and performance compared to their peer funds support this view. 107 Chapter 3. Time Variation in Expected Returns and Aggregate Asset Growth 3.1. Introduction Many studies suggest that expected returns are highly persistent (e.g., Conrad and Kaul (1989), Huberman and Kandel (1991), and Ferson, Sarkissian and Simin (2008)). Van Binsbergen and Koijen (2009) estimate that the autocorrelation of annual expected stock returns is around 0.96 (assuming dividends are reinvested in the market). Given a model of stock price with both random walk and stationary components (e.g., Fama and French (1988)),stockreturnshavenegativeserialcorrelationduetothemean-reversioncomponent. Yet, when the stationary component, equivalently expected returns, are highly persistent, the autocovariance of stock returns over a short-term horizon is virtually zero. 23 In this case, changes in returns tomorrow are roughly the sum of three terms: shocks to expected returns today, unexpected returns tomorrow, minus unexpected returns today. This chapter suggests a new approach for predicting stock returns. Given a linear relationship between nonstationary variables (e.g., consumption and dividends) and future stock returns, we can estimate the model after rst di¤erencing the variables and stock returns. The di¤erenced model suggests that the growth rates of nonstationary variables conveyinformationaboutexpectedchangesinreturns,ofwhichamaincomponentisshocks toexpectedreturns. Hence,usingthegrowthratesofthosevariables,wecanpredictchanges in returns. By adding current returns to the predicted changes, we can forecast returns. 23 Fama and French (1988) provide an example. Given a stationary component z of stock prices with the autocorrelation ;the covariance of stock returns over time T is(1 T ) 2 2 (z)where 2 (z)is the variance. When 1, stock returns appear uncorrelated for small T (e.g., T = 1). From 1952 to 2006, quarterly returns on the U.S. stock markets have virtually zero serial correlation (insignicant). For annual horizons, the correlation coe¢ cient is -0.06 (insignicant), consistent with Fama and French. 108 Using the di¤erence approach, I nd that among macroeconomic variables, aggregate asset growththe rst di¤erences of the logarithm of household net worthhas powerful andsimplepredictiveabilityforstockreturnsoverquarterlyandannualhorizons. Ordinary least square (OLS) regressions of changes in stock returns on asset growth lead to stable slopecoe¢ cientestimatesovertime,whichimproveout-of-samplepredictabilityforreturns. In terms of a mean squared predictive error out of sample, aggregate asset growth performs well when compared with other predictors, such as cay,the dividend-price ratio, and a yieldspread. Whenusedtopredictreturnsdirectly,assetgrowthperformsworse,consistent withexistingndingsthatmacroeconomicvariablesarenotusefultopredictreturnsdirectly. An extension of the model by Campbell and Mankiw (1989) developed here suggests that changes in the components of the consumption-wealth ratiothe growth rates of con- sumption, asset holdings and labor incomeconvey information about expected changes in the market returns. Provided that consumption growth and labor income growth are less volatilethanassetgrowth, themaincomponentdrivingchangesintheratioisassetgrowth, providing information about shocks to discount rates. When investors expect future stock returnstoincrease,higherdiscountratesdecreasethecurrentvalueofassetholdings. When expected future stock returns are lowgiven a negative shock to discount rateswe have a positive asset growth (e.g., Campbell and Shiller (1988) and Campbell (1991)). In-sample empirical results are consistent with the implication of the model. The regressionspredictingchangesinreturnsoverquarterlyandannualhorizonsarestatistically signicant, and the magnitudes are economically signicant. A 1% increase in aggregate asset holdings predicts that returns on the market value-weighted index will decrease by around 3.6% in the following quarter. This coe¢ cient estimate is insensitive to sample 109 periods, consistent with the model that the slope is equal to the reciprocal of the steady- state investment ratio (the implied ratio is 0.28). Variations in asset growth explain around 39% and 46% of the variation in quarterly and annual changes in returns respectively. Adding other variables, such as consumption growth and labor income growth, does not improvethepredictivepowerofassetgrowth. Testsshowthatusingaggregateassetgrowth provides unbiased estimates for expected changes in stock returns. The stable estimates on asset growth for changes in stock returns improve out-of- sample predictability for the returns. I compare the out-of-sample predictive ability of various variablessuch as the dividend-price ratio, the estimated dividend-price ratio (the cointegrating relationship of dividend and price), cay(Lettau and Ludvigson (2001)), a yield spread, and a risk-free ratein terms of a mean squared predictive error suggested by Clark and West (2007). The comparison indicates that from the rst quarter of 1980 to the fourth quarter of 2006, the variable caywhen estimated using the whole sample periodperformsbetterthanthenullmodelthatexpectedreturnsareconstant. However,if recursively estimated, caydoes not outperform the null model. In contrast, asset growth has better predictive ability for stock returns than the null model, when the di¤erence approach is used (predicted returns are the predicted changes plus current returns). Its meansquaredpredictiveerrorforstockreturnsissmallerthanthehistoricalaveragemodel, especially in expansions. Given that cash ow shocks can be more salient than discount rate shocks in recessions, better predictability of asset growth in expansions conrms that asset growth is related to expected changes in returns through shocks to the discount rate. I also nd that asset gowthwhen used for the di¤erence approachhas better pre- dictive abilityforgrowth stocks than forvalue stocks. When assetgrowth is used to predict 110 returns directly or other variables, such as cay,are used, there are no signicant di¤er- ences. Giventhatgrowthstocksaremoresensitivetodiscountrateshocks(e.g., Lettauand Wachter (2008), and Campbell, Polk, and Vuolteenaho (2010)), these results are consistent with the model that asste growth conveys information about discount rate shocks. When estimating a relationship among nonstationary variables, di¤erencing each vari- able in the regression relationship can solve the statistical problems documented by Yule (1926)andGrangerandNewbold(1974). Giventhatequityreturnsareastationaryprocess, regressing changes in returns on lagged asset growth might be an overdi¤erenced model. Overdi¤erencing causes the errors to be autocorrelated. Plosser and Schwert (1977, 1978) argue that di¤erencing makes little di¤erencesince the least square estimator is still con- sistent and unbiased. Yet, the estimator can be ine¢ cient (e.g., Maeshiro and Vail (1988)) because e¢ ciency decreases with serial correlation of the regressor. When the regressor is not serially correlated, the estimator is also e¢ cient (Plosser and Schwert (1978)), which is the case of asset growth (serial correlation of 0.06, insignicant). As a result, regressing changes in returns on lagged asset growth should not cause a signicant loss of estimation e¢ ciency. Thus,thedi¤erenceapproachforpredictingreturnsworkswellwithassetgrowth. Also, given that the growth rates of the nonstationary variables are stationary, we do not need an assumption of cointegrating processes when using the di¤erence approach to predict stock returns. Models for time variation in expected stock returns often provide re- lationships between future discount rates and nonstationary variables, such as consumption anddividends. Somestudiesusethemodelsbyassumingcointegratingprocessesofthevari- ables. For example, Lettau and Ludvigson (2001) suggest that a cointegrating relationship of consumption, assets, and labor income (the variable cay) has information about future 111 expected returns. This cointegrating relationship requires the assumption that assets and human capitaltwo components of wealthare also cointegrated. Campbell and Shiller (1989)assumethatlogdividends andpricesare cointegratedprocessesandshowthat the (log) dividend-price ratio can predict future discount rates. Some evidence calls into question the assumption of cointegrating processes, given that cointegration tests, such as Johanson test, have low power and detecting a unit root may be challenging (e.g., Schwert (1989), Eitrheim (1992), and Cheung and Lain (1993)). Avramov (2002) and Goyal and Welch (2008) document that once recursively estimated, the out of sample predictive ability of cayis not as good as when estimated over the whole sample period (see also Brennan and Xia (2005)). Campbell and Yogo (2002) show that the dividend-price ratio is highly persistent and its predictive ability is limited to the annualfrequencywhenpersistenceis considered intheinference. Lettau and Nieuwerburgh (2007) nd changes in steady-state means of the dividend-price ratio. Finally, another advantage of using predicted changes in returns to predict returns is that autocorrelation of the changes increases with the discount rate e¤ect, i.e., the negative covariance between unexpected returns and shocks to expected returns. As a result, lagged changes in returns or variables that are correlated with those lagged changes can be used to predict changes in returns. On the other hand, when the discount rate e¤ect is sizable, stock returns tend to have little serial correlation. In this case, predicting returns can be more challenging because lagged returns are not useful. 24 24 Given rt+1 = t + "t+1 and t = t1 + t where t Et[rt+1]; the autocovariance of returns is cov(rt+1;rt) = var( t ) + cov(t;"t) where t and "t denote discount rate shocks and unexpected re- turns respectively. Thus, when the discount rate e¤ect is sizable, i.e., cov(t;"t) var( t ); stock returns have little serial correlation. On the other hand, the autocovariance of changes in stock returns is cov(rt+1;rt) =(1) 2 var( t )+(2)cov(vt;"t)var("t): 112 3.2. Related literature Although this chapter estimates expected changes in returns to predict returns, it is based on a view that many studies in the predictive regression literature adopt: Time variation in expected returns in e¢ cient markets can result in predictable components of stockreturns. Fama(1976)providesagraphicalillustration,andGibbonsandFerson(1985) notethattime-varyingconditionalexpectedreturnsarenotinconsistentwiththehypothesis that the unconditional mean is constant and the market is e¢ cient. Therearethreemainapproachestoinvestigatingtimevariationinexpectedreturnsand itsimplicationsforrealizedreturns. Oneapproachistotestforthepresenceofthestationary component in stock prices. This approach typically proposes a specic process for the expected returns and uses autocorrelations in stock returns (Poterba and Summers (1989) provide an excellent summary of test methods for the presence of stationary components of prices). Alternatively, many studies use nancial variables such as dividend-price ratios to estimate expected returns. The third approach is based on the fact that stock returns appears to vary with business cycles and uses macroeconomic variables. Fama and French (1988) and Poterba and Summers (1989) are among the rst re- searchers to examine a stationary component in stock prices. Their ndings show that the time-varying expected returns due to the stationary component of prices may account for a large fraction of return variation for more than a one-year holding period. In par- ticular, Fama and French argue that if prices have both a random walk and a stationary component following an autoregressive process of order one (AR (1)), we may observe a U-shaped pattern of the regression coe¢ cients of current returns on past returns as a func- tion of the investment horizon. In predicting returns over short horizons, the covariance 113 between changes in the stationary component of prices is close to zero, but for longer hori- zons, the covariance becomes negative. However, as the horizons get longer, the variance of changesintherandomwalkcomponentdominatesthevarianceofchangesinthestationary component. This results in zero slopes at very long horizons. Alternatively, other studies use nancial variables such as dividend-price ratios and earnings-price ratios to capture time variation in expected returns (e.g., Roze¤ (1984), Shiller (1984), Keim and Stambaugh (1986), Campbell and Shiller (1987) and Fama and French (1988)). Although those variables show economically signicant predictive power for stock returns, their statistical signicance is often marginal. Their forecasting ability is particularly weak for returns over short and intermediate horizons. Nevertheless, Cochrane (2006) provides indirect evidence for the predictive ability of dividend-price ratios. If the dividend-price ratio does not predict stock returns or dividend growth, we should observe a constant dividend-price ratio. Since dividend growth is not predictable by the dividend- priceratio,thevariationinthedividend-priceratio,therefore,suggestsvariationinexpected stock returns. A model for the implication of macroeconomic variables for expected stock returns is suggested by Campbell and Mankiw (1989). These researchers derive the relationship between the consumption-wealth ratio and expected returns on wealth, based on the bud- get constraint of an optimal consumption problem. Despite their theoretical appeal that the results hold for a general class of investorspreferences, macroeconomic variables have not been very successfully linked with stock returns. This is because most of the relevant variablesincluding consumption and labor incomeare not stationary and their growth rates are empirically shown to have little predictive power for stock returns. On the other 114 hand, Lettau and Ludvigson (2001) show that consumption, asset holdings and labor in- come share common trends, and that the variable caycan predict stock returns over intermediate horizons. Although all three approaches provide evidence that expected returns vary through time, few are very successful at predicting stock returns over intermediate horizons. Only inlonghorizonsandforthesampleperiodincludingpre-1940datadothetestsofFamaand French show signicant support for a stationary component in prices. This may be because a stationary but highly persistent component in prices results in autocorrelations of stock returns that are close to zero in intermediate-horizons, as Fama and French suggest. They also argue that after 1940, stock returns appear close to white noise. Ferson, Heuson, and Su (2005) nd no evidence of weak-formpredictability of monthly returns on individual stocks in recent sample periods. Financial variables, such as a dividend-price ratio, seemed successful at predicting re- turns over long horizons in early studies. However, many researchers call their predictive ability into question based on econometric issues. Stambaugh (1999) looks at a small sam- ple bias that arises when autocorrelated regressors are correlated with contemporaneous unexpected returns. He shows that with an adjustment for the bias, the nancial variables have much less predictive power. Ferson, Sarkissian, and Simin (2003) explore spurious regression and data mining when regressors are serially correlated. If the underlying ex- pected returns are persistent over time, highly autocorrelated independent variables lead to spurious regression bias. Boudoukh, Richardson, and Whitelaw (2008) show that sampling errors increase OLS coe¢ cient estimates and R squares for longer horizons when regressors are persistent. Goyal and Welch (2003, 2008) examine the out-of-sample performance of 115 nancial variables and argue that nancial variables have little predictive ability compared to historical averages. 25 Finally, the predictive power of the variable cayfor intermediate-horizon returns is sensitive to sample periods. Lettau and Ludvigson (2001) show that the variable caycan achieve an adjusted R- squared as high as 10% from the fourth quarter of 1952 to the third quarter of 1998. Yet, its predictability decreases in more recent sample periods. Pastor and Stambaugh (2006) show that the adjusted R-squared drops to 4% from the rst quarter of 1978 to the fourth quarter of 2003. 3.3. Expected returns and consumption-wealth ratio Assume that aggregate wealth at time t (W t ) comprises asset holdings (A t ) and human capital (H t ) and that both are tradable. Let me denote a variable in logarithm by a lower case letter. For example, r w;t is the logarithm of the gross return on wealth for one period from t 1 to t; and c t is the logarithm of consumption C t in the period t: As shown in Appendix, the logarithm of the consumption-wealth ratio; c t w t ; may be approximated as c t w t 1 X i=1 i w (r w;t+i c t+i ) (31) where w is the steady-state level of investment to aggregate wealth ( WC W ). First di¤erencing the equation (31) gives us the model in changes, c t w t 1 X i=1 i w (r w;t+i 2 c t+i ); 25 Two recent studies show that diversifying across individual forecasts (Rapach, Strauss and Zhou (2008)) and forecasting separately dividend yield, earnings growth, and price-earnings ratio growth (Ferreira and Santa-Clara (2008)) can outperform the historical mean model. 116 where 2 denotesthesecond-di¤erencing(i.e., 2 c t+i =c t+i 2c t+i1 +c t+i2 ):Giventhat consumption and wealth (assets plus human capital) are nonstationary, the above model in changes does not require cointegrating relationships between the nonstationary variables. Moreover, as discussed in the below, weaker assumptions are needed for deriving testable implications for variations in expected returns, compared to a model in levels. Since the above equation holds ex-post, we can take expectation conditional on infor- mation at time t, which is denoted by E t ; as c t w t E t [ 1 X i=1 i w (r w;t+i 2 c t+i )]: (32) The equation (32) suggests that the current change in the consumption-wealth ratio provides information about expected changes in returns in the future. Provided that the asset holdings to wealth ratio ( A t1 W t1 ) does not vary much from the steady-state level ( A W ) and the human capital to labor income ratio, Ht Yt Z t ; is a stationary process, 26 I can express changes in wealth as w t a t +(1)(y t +z t ) (33) where y t and z t are the logarithms of human capital Y t and the stationary process Z t respectively. Following Campbell (1996) and Jagannathan and Wang (1996), I regard labor income as the dividends of human capital. Then, the return on human capital is given by R h;t+1 = 26 This is equivalent to the assumption of Lettau and Ludvigson (2001) that human capital (ht) may be decomposed into some stationary process zt and a nonstationary component that is captured by labor income: 117 H t+1 +Y t+1 Ht ; and we have r w;t+i r a;t+i +(1)r h;t+i ; where r a;t is the return on assets and r h;t the return on human capital: This, along with (33), allows us to rewrite the equation (32) as c t (a t +(1)y t ) E t " 1 X i=1 i w (r a;t+i +(1)r h;t+i 2 c t+i ) # +(1)z t ; (34) which suggests that current changes in the consumption-wealth ratio convey information about expected changes in returns on assets and returns on human capital. For comparison, the equation (35) shows the model in Lettau and Ludvigson (2001), c t (a t +(1)y t )E t " 1 X i=1 i w (r a;t+i +(1)r h;t+i c t+i ) # +(1)z t ; (35) which suggests that the deviation from a cointegration relationship among consumption, asset holdings and labor income can predict future asset returns provided that expected future returns on human capital (r h;t+i ) and consumption growth (c t+i ) in the right-hand side (RHS) do not vary much. Yet, given the success of the long-run risk literature, as originated by Bansal and Yaron (2000), the assumption that expected consumption growth is negligible may not be valid. Moreover, Equation (35) requires the stronger assumption that asset holdings and human capital are cointegrated (log-linearizing around the steady-state ratio of A H yields to w t a t +(1)h t a t +(1)(y t +z t ) up to a constant). Given that human capital 118 is nonobservable, this assumption has few empirical evidence. An alternative without the cointegration assumption is using the model in changes. Note that if Equation (35) is a correct specication, the relationship (34) should also hold. To derive empirical implications for expected returns using Equation (34), I make weaker assumptions than under Equation (35). Suppose second di¤erences of consumption and changes in returns on human capital do not vary much. Then the expectations are constants or almost zero (the sample average of the second di¤erences of consumption is close to zero from the rst quarter of 1952 to the fourth quarter of 2006). Then we may ignore E t [ 2 c t+i ] and E t [r h;t+i ] in the equation (34). In addition, provided that labor incomecanberegardedasdividendsonhumancapital,E t [ 2 y t+i ]shouldbealsonegligible. This leads to z t 0 (see Appendix). As a result, I can further simplify the expression (34) as c t (a t +(1)y t ) 1 X i=1 i w E t [r a;t+i ]; (36) which suggests that expected changes in returns on the market portfolio can be predicted by current changes in the consumption-wealth ratio. 3.4. Predictive regressions 3.4.1. Regression methodology To test the relationships suggested by the equation (36), I regress changes in returns on assets against consumption, asset and labor income growths as r t+1 =+ 1 c t + 2 a t + 3 y t +u t+1 ; (37) 119 where u t+1 is an error term, uncorrelated with the regressors. Based on the equation (36), I expect that 1 1 w ; 2 1 w , and 3 1 w : Consumptiongrowthandlaborincomegrowtharemuchlessvolatilethanassetgrowth. Over the sample period, consumption growth has mean 0.0051 and standard deviation 0.0045. Labor income growth (proxy for return on human capital) has the sample mean 0.0056 and the standard deviation 0.009. These are much less volatile than asset growth with a mean of 0.0063 and a standard deviation of 0.02. As a result, a t may be the main component of changes in the consumption-wealth ratio that captures the variation in changes in returns. Thus, I also run the following regression using only asset growth as independent variable, r t+1 =+a t +u t+1 : (38) My hypothesis is that is equal to the reciprocal of the investment ratio, 1 w : Changes in the consumption-wealth ratio are also related to changes in returns more than one-period ahead as the equation (36) suggests. Yet, by w < 1 and < 1, many of the terms in the RHS of (36) must be small. 27 Therefore, I focus on the regression equation (38) but also present the results for predicting changes in returns two periods ahead. I also estimate regressions including lagged changes in returns, motivated by the observed autocorrelation of changes in returns. Using asset growth and changes in returns as regressors, I run the following regression: r t+1 =+ 4 a t1 + (r t r t1 )+u t+1 : (39) 27 For instance, using w = 0:3 and = 0:1; we have w = 0:03, 2 w = 0:009, 3 w = 0:00081 and so on: I take w = 0:3 based on the regression results that the investment ratio is around 0.28. On the other hand, = 0:1 is consistent with Jorgenson and Fraumeni (1989) and Lustig, Nieuweiburgh and Verdelhan (2008), who estimate the human capital to wealth ratio, 1; to be 0.9. 120 3.4.2. Data and summary Statistics I use quarterly and annual real returns on value-weighted CRSP index to proxy for asset returns (r a;t ): Although the CRSP VW index does not include all assets in the market portfolio but only public stocks, the index should be highly correlated with a broader market portfolio (e.g., Stambaugh (1982)). Many studies use the index to proxy for the unobservable market portfolio. For asset holdings (a t ); I use the household net worth data provided by the Board of Governors of the Federal Reserve System: The net worth of households is calculated as total assets including real estate, durable goods (at replacement cost) and nancial assets minus total liabilities such as home mortgages and consumer credit. Financial assets include deposits, Treasury securities, corporate equities, mutual fund shares and pension fund reserves. As of the fourth quarter of 2005, the ratio of (directly and indirectly held) equity to total assets and to net worth are 23.2% and 38.2% respectively (Table B.100.e of Flow of Funds provided by the Board of Governors of the Federal Reserve). The net worth usedinthetestsisin2000chain-weighteddollarswithaPersonalConsumptionExpenditure (PCE) deator and divided by population. 28 The PCE deator and population data are from National Income and Product Accounts (NIPAs). I also use quarterly consumption and labor income on Martin Lettaus website. All values are in logarithms. The period is from the rst quarter of 1952 to the fourth quarter of 2006 for quarterly returns and from 1952 to 2006 for annual returns. Table3.3.1summarizesdescriptivestatisticsforthequarterlygrowthofaggregateasset holdings and stock returns. Quarterly real equity returns are 1.8% on average and 2.9% 28 Lettau and Ludvigson (2001) also used a PCE deator (1992=100) as a common deator for consump- tion, asset holdings and labor income. The results provided here do not depend on which deator is used. 121 in median while changes in the returns are 0.01% on average. Changes in equity returns are highly volatile compared to the levels of returns: variance 0.013 vs. 0.007. Since the variance of the changes is given by var(r t+1 ) = 2(var(r t )cov(r t+1 ;r t )); those sample variances are consistent with the sample autocorrelation in returns cov(r t+1 ;r t ), which is around 0.0004. The autocorrelation in returns close to zero implies that cov(v t ;" t ) must be negative and sizable (see the footnote 2). A variable that captures the discount rate e¤ect, for example, the asset growth as discussed earlier, should be useful for predicting changes in returns. The sample mean and median of the logarithm of asset holdings are 11.26 and 11.21 respectively, each of which corresponds to around $75,000 and $70,000 in 2000 dollars. 29 They have grown at 0.6% in a quarter on average and show a sizable variation (standard deviation of 2%). Consumption and labor income grow at similar rates to that of asset holdings. Yet, their variations are much smaller. Figure 3.1 shows the time series plots of asset growth and equity returns. Since asset growth and changes in equity returns appear stationary (augmented Dickey-Fuller tests also reject unit roots), we can use least square estimation for the regression model for changes. Correlations among the variables are noteworthy as shown in Table 3.1 (panel B). Although equities are not a major part of net worth of households, contemporaneous asset growth and equity returns are highly correlated, consistent with Stambaugh (1982, 1983). Changesinequityreturnsarenegativelycorrelatedwithlaggedassetgrowth,whichsuggests thediscountratee¤ect. Figure3.1showsthetimeseriesoflaggedassetgrowthandchanges in equity returns for quarterly data (bottom left) and annual data (bottom right). 29 Asset holdings are nonstationary. The statistics for the sample period are for a reference only. 122 Table 3.1. Descriptive statistics and correlations Equity returns are the natural logarithm of quarterly real returns on the CRSP value-weighted (VW) index (NYSE/AMEX/NASDAQ). Changes in returns are the rst di¤erences in equity returns. Asset holdings are the natural logarithm of net worth of households (from the Flow of Funds data provided by the FRB Board of Governors) in 2000 chain dollars, divided by populations (quarterly). The statistics of this variable (nonstationary) is for reference only. Personal Consumption Expenditure deator and population data are from the National Income and Product Accounts. Asset growth is the rst di¤erences of asset holdings. Consumption growth and (labor) income growth are the rst di¤erences of the natural logarithms of real consumption per capita and real labor income per capita respectively (quarterly) as provided in Martin Lettaus website. The data period is from the rst quarter of 1952 to the fourth quarter of 2006. (A) Descriptive statistics equity return change in return asset holdings asset growth consumption growth labor income growth Mean 0.0175 0.0001 11.2565 0.0063 0.0051 0.0056 Median 0.0293 -0.0037 11.2104 0.0085 0.0051 0.0064 Maximum 0.2036 0.4024 11.995 0.0793 0.021 0.0455 Minimum -0.3137 -0.3115 10.616 -0.0587 -0.0124 -0.0268 Std. Dev. 0.0836 0.1151 - 0.0202 0.0045 0.009 Skewness -0.8501 0.6098 - -0.4243 -0.4416 0.1928 Kurtosis 4.5806 4.0989 - 4.7195 4.7158 5.5143 Observations 219 219 219 219 219 219 (B) Correlations change in return asset growth lagged asset growth lagged consumption growth lagged income growth equity return 0.6888 0.8807 0.0175 -0.0826 -0.1315 change in return 1 0.5931 -0.6263 -0.1691 -0.1473 asset growth 1 0.0607 0.0009 -0.0742 lagged asset growth 1 0.2291 0.1519 lagged consumption growth 1 0.4665 lagged labor income growth 1 123 Figure 3.1. Aggregate asset holdings and real returns on the CRSP VW index Asset holdings are the natural logarithm of net worth of households (from the Flow of Funds data provided by the FRB Board of Governors) in 2000 chain dollars, divided by populations (quarterly). Personal Consumption Expenditure deator and population data are from the National Income and Product Accounts. Asset growth is the rst di¤erences of asset holdings and lag asset growth is its lagged value. Equity returns are the natural logarithm of quarterly real returns on the CRSP value-weighted (VW) index (NYSE/AMEX/NASDAQ). Changes in returns are the rst di¤erences of equity returns. The data period is from the rst quarter of 1952 to the fourth quarter of 2006. The last gure uses annual data from 1952 to 2006. 10.8 11.2 11.6 .04 .00 .04 55 60 65 70 75 80 85 90 95 00 05 asset holdings per capita asset growth .4 .2 .0 .2 .4 .4 .2 .0 .2 .4 .6 55 60 65 70 75 80 85 90 95 00 05 equity return change in return 124 (Figure 3.1 continued) .08 .04 .00 .04 .08 .12 .4 .2 .0 .2 .4 .6 55 60 65 70 75 80 85 90 95 00 05 lag asset growth change in return .15 .10 .05 .00 .05 .10 .8 .4 .0 .4 .8 55 60 65 70 75 80 85 90 95 00 05 annual asset growth annual change in return 125 For comparisons tests, I use a dividend-price ratio, an estimated dividend-price ratio, yield spread, risk free rate and cayas predictive regressors. The dividend-price ratio of theCRSPVWindexistheratioofdividendsforthepreviousyeartothecurrentvalueofthe index. ThedividendsarecalculatedusingthemethodadoptedbyLettauandNieuwerburgh (2007). Specically, I get dividend yields by subtracting the quarterly returns on the index without dividends from the quarterly returns with dividends. Then, dividends are obtained by multiplying the corresponding index value at the end of the previous quarter. The one- year dividends are the sum of the previous four-quarter dividends. The dividend-price ratio is the logarithm of the one-year dividends minus the logarithm of the CRSP VW index without dividends. I also estimate the cointegrating relationship between the logarithm of dividends and thelogarithmofthepricewithoutdividends. Theestimatedcointegrationparameteris0.77 (without lags and leads). Augmented Dickey-Fuller tests for the estimated dividend-price ratio reject the null of a unit root at the 5% signicance level (the results are not sensitive to the lag choices) while they do not reject for the dividend-price ratio. 30 The estimated dividend-price ratio is less volatile (variance is 0.067 vs. 0.19) and its correlation with the stock returns is larger and signicant at the 10% signicance level, compared to the dividend-price ratio. Figure 3.2 (A) shows the time series of the dividend-price ratio and theestimatedratio. Whenthe cointegratingrelationshipisestimated, theratioseems to be movingaroundaxedmean. Foroutofsampleprediction, Ialsoestimatethecointegrating relationship using a recursive method (the rst estimation uses the sample from the rst quarter of 1926 to the fourth quarter of 1979 and then add one observation at a time). 30 For example, using 8 lags, the p-values of the tests are 0.04 and 0.62 for the estimted dividend-price ratio and the dividend-price ratio respectively. 126 The recursive estimates of the cointegration parameter is compared to the estimate using the whole sample period in Figure 3.2 (B). The substantial variation in the cointegration parameterssuggeststhatout-of-samplepredictiveabilitywouldbeworsewhenthedividend- price ratio is estimated recursively than when estimated over a whole sample period. For the variable cay, I use the quarterly data (from the rst quarter of 1952 to the fourth quarter of 2006) for consumption, asset holdings and labor income growth on Martin Lettaus website and estimate the cointegration relationships, using 8 lags and 8 leads. An estimation using the whole sample period leads to the same cointegration parameters as the ones on the website. I also obtain the variable cayby a recursive method similar to that for the estimated dividend-price ratio. Figure 3.3 shows the time-series of caywhen estimated over the whole sample period and the recursive estimates of the cointegration parameters. The cointegration parameters for cayare sensitive to estimation periods. In particular, the parameter for labor income is larger than the estimate from the whole sample (0.65) while that for asset growth is smaller than the value obtained for the whole period (0.24) in the 1980s. These variations in the cointegration relationship provide an explanation for Goyal and Welchs (2008) ndings that the out-of-sample predictive ability of caydecreases once recursively estimated. I also use a yield spread and a risk free rate to compare the out-of-sample forecasting ability. CampbellandYogo(2006)showthatthesevariablesarestationaryandndevidence that they predict returns. As used in their study and Fama and French (1989), the yield spread is the di¤erence between Moodys Aaa corporate bond yield and one-month T-bill rate, and the risk free rate is the three-month T-bill rate. 127 Figure 3.2. Dividend-price ratio and estimated dividend-price ratio dp ratio(dividend-price ratio) is the logarithm of the ratio of the previous one-year dividends for the CRSP value-weighted index to the index value without dividends at the end of each quarter. estimated dp ratiouses a cointegrating parameter, i.e., the logarithm of the ratio of the previous one-year dividends for the CRSP value-weighted index to the cointegrating parameter times the index value without dividends. The estimated cointegrating parameter is around 0.77 over the whole sample period. Figure (A) shows the time-series of dp and estimated dp ratios and their sample means in dashed lines. Augmented Dickey Fuller tests reject the null of unit roots for estimated dp ratio but do not reject for the dp ratio. Figure (B) shows the recursive estimates of the cointegrating parameter. I use the data from the rst quarter of 1926 to the fourth quarter of 1970 for the rst estimate and then add one observation at a time to recursively estimate the parameter up to the fourth quarter of 2006. The dashed line is the cointegrating parameter estimated over the whole sample (around 0.77). The data period is from the rst quarter of 1926 to the fourth quarter of 2006. 128 Figure 3.3. cayand recursive estimates of the cointegration parameters cayis the cointegrating relationship of consumption, asset holdings and labor income (from Martin Lettaus website) estimated over the whole same sample period from the rst quarter of 1952 to the fourth quater of 2006. Figure (A) shows its time series. Augmented Dickey Fuller tests reject the null of a unit root for cay. Figure (B) shows the recursive estimates of the cointegrating parameters: red plus marks for the cointegration coe¢ cient for labor income and blue circle marks for the cointegration coe¢ cient for asset holdings. I use the data from the rst quarter of 1952 to the fourth quarter of 1970 for the rst estimate and then add one observation at a time to recursively estimate the parameters up to the fourth quarter of 2006. The dashed lines are the cointegration coe¢ cients estimated over the whole sample, which are used for cay in (A). The data period is from the rst quarter of 1952 to the fourth quarter of 2006. 129 Table 3.2. Autocorrelations and standard errors Autocorrelation and its standard error in the below for lags from 1 to 8. The p-value for white noise is the p-value of the Chi-square statistics for the tests of no serial correlations up to the lags of 6. See Table 3.1 for the variable description. Dividend-price is the ratio of the previous year dividends for the CRSP value-weighted index to the index value without dividends at the end of each quarter. Estimated dividend-price is the ratio of the previous year dividends for the index to the cointegrating coe¢ cient times the index value without dividends. The estimated cointegrating coe¢ cient is around 0.77 over the whole sample period. cayis the cointegrating relationship of consumption, asset holdings and labor income (from Martin Lettaus website) estimated over the whole same sample period. All variables are in natural logarithm. Data periods for the variables are in parentheses. Lags 1st 2 3 4 5 6 7 8th p-value for white noise CRSP VW -0.062 0.011 0.140 -0.139 -0.004 0.037 -0.169 0.012 (0.020) (1926 Q1-2006 Q4) (0.056) (0.056) (0.056) (0.057) (0.058) (0.058) (0.058) (0.060) CRSP VW 0.052 -0.040 -0.022 -0.002 -0.033 -0.012 -0.125 0.024 (0.970) (1952 Q1-2006 Q4) (0.067) (0.068) (0.068) (0.068) (0.068) (0.068) (0.068) (0.069) asset holdings 0.984 0.969 0.953 0.936 0.920 0.904 0.888 0.872 (0.000) (1952 Q1-2006 Q4) (0.068) (0.116) (0.149) (0.174) (0.196) (0.215) (0.232) (0.247) asset growth 0.061 0.014 0.054 0.030 -0.069 0.047 -0.077 0.037 (0.770) (1952 Q1-2006 Q4) (0.068) (0.068) (0.068) (0.068) (0.068) (0.069) (0.069) (0.069) dividend-price 0.951 0.908 0.865 0.821 0.789 0.763 0.738 0.724 (0.000) (1926 Q1-2006 Q4) (0.056) (0.093) (0.117) (0.136) (0.150) (0.162) (0.173) (0.183) estimated dividend-price 0.878 0.783 0.693 0.615 0.560 0.514 0.472 0.450 (0.000) (1926 Q1-2006 Q4) (0.056) (0.089) (0.108) (0.121) (0.130) (0.137) (0.143) (0.148) cay 0.854 0.748 0.662 0.575 0.517 0.474 0.422 0.368 (0.000) (1952 Q1-2006 Q4) (0.068) (0.106) (0.128) (0.143) (0.153) (0.161) (0.167) (0.172) 130 Table 3.2. presents serial correlations of equity returns, asset growth, the dividend- price ratio, the estimated dividend-price ratio, and cay. The tests show that we cannot reject that aggregate asset growth and equity returns after 1952 have zero autocorrelations. When expected returns are persistent, expected changes in returns are serially uncorrelated (E t [r t+1 r t ] =(1) t1 +v t " t v t " t when 1: see footnote 2 for notations) since v t " t is uncorrelated over time under the assumption that v t and " t are independent white noise processes. The serial independence of asset growth is consistent with the view that asset growth conveys information about expecetd changes in returns. On the other hand,thedividend-priceratioishighlypersistentwhiletheautocorrelationoftheestimated dividend-price ratio is smaller. cayis also serially correlated. 3.4.3. Results Table 3.3 summarizes the OLS regression results for forecasting changes in returns one quarterahead. ThedependentvariableischangesinquarterlyrealreturnsontheCRSPVW index, and the independent variables include lagged consumption growth, asset growth and labor income growth, and changes in the returns. Rows for independent variables show the estimate of the coe¢ cient, Newey-West corrected standard errors and p-value respectively. I test the regression equation (37) by the regression model (9). The result shows that consumption growth and labor income growth do not play a role in predicting changes in equity returns. These variables are not statistically signicant and their magnitudes are not economically important. In contrast, asset growth shows signicant and important predictive power. Hence, I focus on the predictive ability of asset growth in the rest of the chapter. 131 Table 3.3. OLS regression of one-period ahead changes in quarterly real returns on the CRSP value-weighted index Dependent variables are changesin quarterly(log) real returns on the CRSP VW index. Asset growth isthe rstdi¤erencesof the natural logarithm of aggregate asset holdings. second lagged asset growthis two lags of asset growth. See table 1 for a detailed data description. The rst row of each independent variable is the parameter estimate and the other rows represent Newey-West corrected standard error (with lag length 4) and p-value respectively. The data period is from the rst quarter of 1952 to the fourth quarter of 2006. (1) (2) (3) (4) (5) (6) (7) (8) (9) Intercept 0.000 0.022 0.020 0.018 0.022 0.020 0.000 0.026 (0.004) (0.006) (0.007) (0.006) (0.007) (0.008) (0.005) (0.010) (0.945) (0.000) (0.003) (0.004) (0.002) (0.009) (0.960) (0.007) lagged asset growth -3.577 -3.160 -2.339 -2.164 -3.516 (0.314) (0.423) (0.453) (0.474) (0.327) (0.000) (0.000) (0.000) (0.000) (0.000) second lagged asset growth -2.897 -1.235 -1.054 (0.417) (0.417) (0.458) (0.000) (0.003) (0.023) lagged change in returns -0.451 -0.451 -0.123 -0.769 -0.344 -0.390 -0.600 (0.059) (0.058) (0.073) (0.069) (0.095) (0.100) (0.070) (0.000) (0.000) (0.091) (0.000) (0.000) (0.000) (0.000) second lagged change -0.105 -0.329 (0.065) (0.052) (0.104) (0.000) lagged consumption growth -0.114 (1.570) (0.942) lagged labor income growth -0.699 (0.898) (0.437) adjusted R squared 0.200 0.204 0.390 0.397 0.354 0.409 0.413 0.283 0.386 132 Table 3.4. OLS regression of one-period ahead changes in annual real returns on the CRSP value-weighted index Dependent variables are changes in annual (log) real returns on the CRSP VW index. Asset growth is the rst di¤erences of the natural logarithm of aggregate asset holdings. second lagged asset growthis two lags of asset growth. See table 1 for a detailed data description. The rst row of each independent variable is the parameter estimate and the other rows represent Newey-West corrected standard error (with lag length 4) and p-value respectively. The data period is from 1952 to 2006. (1) (2) (3) (4) (5) (6) (7) (8) (9) Intercept -0.001 0.094 0.089 0.107 0.114 0.098 0.001 0.026 (0.016) (0.025) (0.025) (0.027) (0.029) (0.025) (0.018) (0.010) (0.972) (0.000) (0.001) (0.000) (0.000) (0.000) (0.947) (0.007) lagged asset growth -3.788 -3.574 -1.864 -1.013 -3.516 (0.594) (0.758) (0.722) (0.639) (0.327) (0.000) (0.000) (0.013) (0.120) (0.000) second lagged asset growth -4.331 -2.668 -2.902 (1.017) (0.683) (0.725) (0.000) (0.000) (0.000) lagged change in returns -0.414 -0.414 -0.074 -0.953 -0.569 -0.763 -0.603 (0.088) (0.083) (0.126) (0.113) (0.094) (0.127) (0.120) (0.000) (0.000) (0.562) (0.000) (0.000) (0.000) (0.000) second lagged change -0.205 -0.458 (0.098) (0.097) (0.040) (0.000) lagged consumption growth -0.114 (1.570) (0.942) lagged labor growth -0.699 (0.898) (0.437) adjusted R squared 0.155 0.171 0.459 0.452 0.476 0.502 0.516 0.320 0.386 133 The model (3) in Table 3.3 tests Equation (38), whose result is summarized in r t+1 = +a t +u t+1 = 0:02243:5773a t : (40) Lagged asset growth shows predictive power at a signicance level of 1% with an adjusted R squared of around 39%. This R squared roughly corresponds to an R squared of around 3.5% when I regress the level of returns on the lagged asset growth (Appendix shows the relationship). Moreover, the slope of aggregate asset growth has a negative sign as predicted by the equation (38). Its value is -3.58, which implies that the steady-state level of the investment to wealth ratio ( w ) is around 0.28. Model (4) demonstrates regressions with an additional independent variable, lagged changes in returns. The result shows that it does not improve on the predictive power of aggregate asset growth to include lagged changes in returns. Given that lagged changes in returns have relevant information about unexpected returns, this result suggests that asset growth also has relevant information about such returns. 31 I test Equation (39) by the regression model (5). It shows that the second lagged asset growthandthelaggedchangesinreturnscanpredictchangesinreturnswithlesspredictive power. Although the signs are correct, the coe¢ cient level of the second lag of asset growth does not appear reasonable based on the models predictions. Theresultsforannualreturnsaresimilartothoseforquarterlyreturnsdescribedabove. 31 Change in returns is given by rt+1rt =(1) t1 +vt"t | {z } E t [r t+1 r t ] +"t+1, where vt is shocks to expected returns and "t is unexpected returns. Likewise, rtrt1 =(1) t2 +vt1"t1+"t: When expected returns are highly persistent ( 1); we have Et[rt+1rt] vt"t: 134 As shown in Table 3.4, annual aggregate asset growth has signicant predictive ability for annual returns at the 1% level with a coe¢ cient estimate of -3.79, which is very close to that for quarterly returns. The adjusted R squared is as high as 46%. Predicted changes in returns may be used to predict future returns by adding them to the current returns. In this case, the prediction errors for returns are the same as those for changes since actual lagged returns are added. 3.5. Comparisons of out-of-sample predictive ability I compare the out-of-sample predictive ability of various variablessuch as histori- cal means, dividend-price ratios and cayin terms of a mean squared predictive error (MSPE) as suggested by Clark and West (2007). Their statistics are used for an alternative model that nests a null model. The null model for predictive regressions is that equity returns are a white noise process with a non-zero mean (constant expected returns, i.e., historical average returns are the forecasted returns). For models in levels, I estimate the mean by running regressions on a constant, using a recursive estimate method. Similarly, for alternative models, I run regressions on a constant and predictive variables. Predictive variables are lagged asset growth, the dividend-price ratio, the estimated dividend-price ratio, yield spread, risk-free rate (three-month T-bill rate), and cay. On the other hand, I use models in changes in returns for asset growth and changes in the dividend-price ratio and then add current returns to tted changes in returns to predict returns one period ahead. In this case, the null model is equivalent to the hypothesis that one-period ahead change in returns is white noise with nonzero mean minus the current return. Thus, for the null model in changes, I regress changes in returns on a constant and 135 lagged returns (i.e., the predicted changes in returns are historical average returns minus currentreturns). Alternativemodelsuseaconstant,laggedreturnsandpredictivevariables. When using lagged asset growth and lagged returns together, their correlation (correlation coe¢ cient0.88,signicant)causesmulticollinearity. Inthiscase,Iuseapartoflaggedasset growth that is uncorrelated with lagged returns. Since labor income growth is uncorrelated with returns (correlation coe¢ cient 0.07, insignicant), regressing lagged asset growth on lagged labor income growth provides the tted value of lagged asset growth, which is used along with lagged returns for the regressions. Whencomparingtheout-of-samplepredictiveabilityofanullmodelandanalternative model, a standard t-test is inappropriate for testing equal MSPEs due to a size distortion. ClarkandWest(2007)showthatthestandardtestresultsinastatisticthatisnotnormally distributed but is centered around a negative value in nite samples. This implies that an alternative model has a larger MSPE than the null model on average, when the null is true. The intuition behind this distortion is that an additional variable in an alternative model is not only unhelpful under the null but also harmful by adding noise to the predictive process. Therefore, in nite samples, the MSPE of an alternative model is expected to be larger than the MSPE of a null model. A simulation in Clark and West shows that the size of a standard t-test is 3% for a 30-year quarterly period when a normal size is 10%. The size distortion becomes worse as we increase the out-of-sample periods; for example, it is 1% for a 60-year quarterly period. 32 Clark and West (2007) suggest an adjustment to make the distribution of the sta- tistic approximate the standard normal distribution. The adjustment is to subtract out 32 A recursive method is used to get the out-of-sample estimates. 136 the noise that makes the MSPE with additional variables larger than the MSPE without the variables under the null. Without the adjustment, MSPE of an alternative would be overestimated because more variables introduce more noise. With the adjustment, size is improvedfrom3%to9%andfrom1%to7%for30-yearand60-yearquarterlyout-of-sample periods respectively (the normal size is 10%). The distribution of t-statistics is closer to the standard normal distribution, centered around zero, after the adjustment. I, therefore, use their suggested t-statistics as described in Appendix but also report t-statistics without the adjustment. The comparison indicates that excluding recessions (16 out of 108 quarters out of sample)accordingtotheNationalBureauofEconomicResearch(NBER),thebestpredictor isaggregateassetgrowth,whenusedfortheregressioninchanges. Thevariablecay,when estimatedusingthewholesampleperiod,isthebestpredictorovertheout-of-sampleperiod from the rst quarter of 1980 to the fourth quarter of 2006. Yet, recursive estimates of the variable caydecreases its predictive ability and it underperforms the null model that uses only a constant as the regressor. Table 3.5 presents the comparison results. The rst column provides the regressors (lagged values). In the second column, the levelmodel represents the regression in levels: regressing returns on a constant and the regressors. The changemodel regresses changes in returns on a constant, lagged returns, and predictive variables. I include lagged returns to nest the null model for changes, which uses a constant and lagged returns as discussed earlier. The third column provides the mean and median squared errors of the alternative models while the fourth column is for the null model. In the following three columns, I compare their squared errors in terms of mean and median. t valueis the standard t- 137 statistics for equal MSPEs while CW t valueis the one after the adjustment suggested by Clark and West (2007). I also report CW t valuefor expansions (92 quarters) and recessions (16 quarters) separately. The comparison suggests that aggregate asset growth performs signicantly better than the null model during expansions (92 out of 108 quarters) when used for the change model. When it is used to predict returns directly, the performance is not as good as the change model (CW t-stat in expansions of 1.80 vs. 1.32). This conrms that asset growth provides information about changes in returns one period ahead, especially shocks to expected returns. Thus, it appears that asset growth is better used for regressions in changes, as the theoretical model suggests. On the other hand, the predictive ability of the other variables does not apear superior compared to historical average returns. The dividend-price ratio does not outperform the null model. Once we estimate the cointegration parameter for the dividend-price ratio, the predictive ability improves, but the the statistical signicance is still marginal (CW t-stat of 1.24). Moreover, when the cointegrating relationship is recursively estimated (feasible estimated dividend-price ratio), the ratio performs worse. Given that the dividend-price ratio is highly persistent (the rst-order autocorrelation of 0.95, signicant; see Table 3.2.), we can use the variable for the model in changes, which does not require the cointegrat- ing relationship of dividend and price. 33 Regressing changes in returns on changes in the dividend-price ratio results in better forecasts in recessions, compared to the null model (CW t-stat of 2.24). Yet, the sample size of recessions is small (16 periods). 33 Campbell and Shiller (1989) show that changes in the logarithm of the sum of price and dividend can be approximated by changes in the dividend to price ratio and the dividend growth. Then changes in the ratio are related to expected changes in returns and expected second di¤erences of dividends. 138 Table 3.5. Comparison of predictive ability for the out-of-sample period The regression models arer t =+ 0 X t1 +" t for the level and r t =+ 0 X t1 +" t for the change model. The intercept and the slope coe¢ cient are estimated using a recursive method (the rst estimates use the data from 1952 Q1 to 1979 Q4). The returns r t is the quarterly real return on the CRSP VW index and lagged returns are r t1 . The independent variables are in the rstcolumn. Assetgrowthistherstdi¤erencesofthenaturallogarithmofaggregateassetholdings (net worth of households in the Flow of Funds data provided by the FRB Board of Governors) in 2000 chained dollars divided by population. dpis the ratio of the previous year dividends for the index to the index value without dividends. estimated dpis the logarithm of the ratio of the previous year dividends for the index to the cointegrating coe¢ cient (0.77; estimated over 1926 Q1 to 2006 Q4) times the index value without dividends. The feasible estimated dpuses the cointegrating coe¢ cient estimated using a recursive method. Yield spread is the di¤erence between Aaa Moodys corporate bond yields and the one-month T-bill yield. rfis the three-month T-bill yield. cayuses the cointegrating parameters among consumption, asset and labor income from Martin Lettaus website while feasible cay uses a recursive estimation. Squared errors in (B) uses the null model that includes only a constant for the level model and a constant and lagged returns for the change model. All variable are in logarithm. The t-value is the standard t-statistic for equal meansquaredpredictionerrors. TheCWtvalueistheadjustedt-statisticassuggestedbyClarkand West (2007). Expansion and recession are according to the National Bureau of Economic Research. The out-of-sample period is from 1980 Q1 to 2006 Q4. 139 (Table 3.5 continued) lagged variable Model (A) (B) null (B-A)/B in % t value CW t value squared errors squared errors mean median mean median mean median all all expansion recession asset growth, returns Level 0.0074 0.0024 0.0072 0.0025 -2.19 3.66 -0.6 0.51 1.32 -0.66 asset growth, returns Change 0.0074 0.0024 0.0073 0.0026 -0.19 7.82 -0.07 0.77 1.8 -0.69 dp Level 0.0075 0.003 0.0072 0.0025 -2.53 -16.8 -0.92 0.41 -0.23 1.03 estimated dp Level 0.0071 0.0023 0.0072 0.0025 1.64 6.61 0.45 1.24 1.15 0.59 feasible estimated dp Level 0.0073 0.0027 0.0072 0.0025 -1.3 -7.67 -1.31 -0.9 -1.06 0.04 dp, returns Change 0.0077 0.0031 0.0073 0.0026 -4.26 -20.75 -0.77 0.56 -2.5 2.24 yield spread Level 0.0075 0.0027 0.0072 0.0025 -3.62 -9.71 -0.62 0.54 -0.02 1.03 rf Level 0.0077 0.0024 0.0072 0.0025 -6.37 4.11 -1.18 -0.12 -0.53 0.04 cay Level 0.0073 0.0025 0.0072 0.0025 -1.15 -0.12 -0.19 2.03 2.6 0.22 feasible cay Level 0.0075 0.0025 0.0072 0.0025 -4.51 -0.19 -0.64 1.41 1.15 0.59 asset growth, returns, dp Level 0.0073 0.0025 0.0072 0.0025 -1.59 0.49 -0.29 0.91 1.22 -0.01 asset growth, returns, dp Change 0.0074 0.0023 0.0073 0.0026 -1.36 10.91 -0.51 0.41 0.9 -0.43 asset growth, returns, spread Level 0.0076 0.0031 0.0072 0.0025 -5.44 -23.51 -0.86 0.92 0.77 0.5 asset growth, returns, rf Level 0.0077 0.0026 0.0072 0.0025 -7.2 -3.29 -1.19 0.26 1.18 -0.31 asset growth, returns, cay Level 0.0074 0.0023 0.0072 0.0025 -3.09 8.96 -0.46 1.69 2.33 -0.03 140 Figure 3.4. Out-of-sample predictions for real quarterly returns on the CRSP VW index The gures show predicted returns against actual returns for the out-of-sample period from the rst quarter of 1980 to the fourth quarter of 2006 using a recursive method. Returns are the natural logarithm of quarterly real returns on the CRSP value-weighted (VW) index. The rst two gures show the null models for regressions in levels and for regression in changes respectively. The null model for regressions in levels regresses returns on only a constant and obtains predicted returns (historical means). The null model for regressions in changes regresses changes in returns on a constant and lagged returns and adds lagged returns to the predicted changes to obtain predicted returns. The other gures show the results when additional regressors are used as independent variables. See Table 3.5 for detailed description of the independent variables. When asset growth and returns are used together as independent variables, a part of asset growth that is uncorrelated with returns is used. The uncorrelated part is obtained by regressing asset growth on labor income growth (correlation of labor income growth and stock returns is 0.07, insignicant). 141 (Figure 3.4 continued) 142 (Figure 3.4 continued) 143 The predictive ability of all other variables is not as good as the constant expected return model. Exceptionally, when estimating over the whole sample period, the variable cayperforms signicantly better than the historical mean model (CW t-stat of 2.03). However, the variable cayobtained by a recursive method (feasible cay) does not yield a better result than the null model. Finally, I plot predicted returns against actual returns in Figure 3.4. The rst two gures demonstrate that there are few variations in predicted returns in the null models (historical means and lagged return model). On the other hand, using asset growth results in an improvement and a substantial variation in predictions. The estimated dividend-price ratio and changes in the dividend-price ratio also has better predictive ability than the dividend-price ratio. However, once recursively estimated (feasibleestimated dp ratio), the estimated dividend-price ratio does not perform well. Similar conclusions can be made for the variable cay. 3.6. Robustness checks 3.6.1. Unbiased estimates of changes in returns I check if out-of-sample predicted changes in returns using asset growth are unbiased estimates for actual changes in returns, i.e., the estimates are expected changes in returns. To this end, I regress actual changes in returns on predicted changes in returns obtained usinganout-of-samplemethod(e.g.,recursiveorxedmethod). Iftheestimatedcoe¢ cient, b ; is close to one, r t+1 = b c E t [r t+1 ]+u t+1 c E t [r t+1 ]+u t+1 ; 144 then I may conclude that my estimates of changes in returns are unbiased forecasts of subsequent realized changes in returns. The results are presented in Table 3.6. For quarterly returns, a xed out-of-sample method provides us with unbiased estimates for changes in returns. I rst estimate the slope coe¢ cienton asset growth by runninga regression of changes in returns on a constant and asset growth from the rst quarter of 1952 to the fourth quarter of 1979. I obtain predicted changes in returns for the out-of sample period from the rst quarter of 1980 to the fourth quarter of 2006. Then I regress actual changes in returns on the predicted changes for the out-of-sample period. The slope estimate on predicted changes is close to one (1.08) with the p-value of almost zero. The intercept is not signicant. When using a recursive method, the results are similar (not tabulated). Figure 3.5 shows the slope estimate on asset growth when a recursive method is used (the rst estimate uses the data fromtherstquarterof1952tothefourthquarterof1979). Theestimatesarenotsensitive to data periods but stable over time. This can explain the out-of-sample predictive power of asset growth for changes in returns. On the other hand, when recursivelt estimated, the slope coe¢ cients on cayare very volatile, which an explain why it performs poor for perdicting returns in this case. Fortheannualsample, axedmethoddoesnotyieldunbiasedestimatesforchangesin returns. The slope estimate for predicted changes in returns is 0.57, as presented in Table 3.6. When using a recursive method, the slope coe¢ cient improves but is not close to one (0.89). Theinterceptisnotclosetozero(0.09)andsignicantat5%level. Thus, itappears that we may not obtain unbiased estimates for changes in annual returns due to the small sample size. 145 Table 3.6. Regressions of changes on estimated changes in returns for the out-of-sample period from 1980Q1 to 2006Q4 Regressions of actual changes in returns on estimated changes in expected returns using asset growth. The dependent variables are actual changes in quarterly real (log) returns on the CRSP VW index from the rst quarter of 1980 to the fourth quarter of 2006. The independent variables are estimated changes in expected returns. When the intercept is close to 0 and the slope of the estimated changes in expected returns is close to 1, the estimated changes in expected returns can be unbiased estimates of changes in expected returns. Panel (A) and (B) use a xed method to estimate changes in expected returns. After regressing changes in returns on a constant and asset growth from the third quarter of 1952 to the fourth quarter of 1979, the estimates of the intercept and the slope of asset growth are used to obtain estimated changes in expected returns from the rst quarter of 1980 to the fourth quarter of 2006. Asset growth is the rst di¤erences of the natural logarithm of aggregate asset holdings (net worth of households in the Flow of Funds data provided by the FRB Board of Governors) in 2000 chained dollars divided by population (provided by the National Income and Product Accounts). On the other hand, Panel (C) uses a recursive method. The estimates of the intercept and the slope of asset growth are recursively obtained, using the data up to the previous period. For example, the estimates of the intercept and the slope of asset growth used for estimating changes in expected returns are from the regression using the data up to the fourth quarter of 2003. (A) Quarterly returns: xed method (B) Annual returns: xed method (C) Annual returns: recursive method intercept slope adjusted intercept slope adjusted intercept slope adjusted R 2 R 2 R 2 estimate 0.007 1.078 0.432 0.021 0.570 0.305 0.088 0.893 0.324 standard error (0.009) (0.119) (0.035) (0.162) 0.042 0.244 t-value (0.731) (9.071) (0.609) (3.521) 2.125 3.666 p-value (0.467) (0.000) (0.548) (0.002) 0.044 0.001 146 Figure 3.5. Recursive slope estimates of predictive regressions The least square slope estimates for the out-of-sample period from the rst quarter of 1980 to the fourth quarter of 2006 by a recursive method. The rst estimates use the data from the rst quarter of 1952 to the fourth quarter of 1979. The gure (A) uses the model in changes that regresses changes in returns on a constant and lagged asset growth while the gure (B) uses the model in levels that regresses returns on a constant and lagged ntextquotedblleft cay.ntextquotedblrightn The dashed lines are the estimates over the whole sample period from the rst quarter of 1952 to the fourth quarter of 2006. Returns are the natural logarithm of quarterly real returns on the CRSP value-weighted (VW) index (NYSE/AMEX/NASDAQ). Asset growth is the rst di¤erences of the natural logarithm of net worth of households (from the Flow of Funds data provided by the FRB Board of Governors) in 2000 chain dollars, divided by populations (quarterly). cayis the cointegrating relationship of consumption, asset holdings and labor income (from Martin Lettaus website) estimated over the whole same sample period from the rst quarter of 1952 to the fourth quarter of 2006. 147 3.6.2. Does asset growth predict changes in stock returns? Asset growth does explain variation in future changes in stock returns. Although stock returns and asset growth are correlated as high as 88% during the sample period (Table 3.1), asset growth explains more than contemporaneous stock returns in one-period ahead changes in returns. As shown earlier, predictive ability of asset growth for stock returns is better than lagged stock returns. I further investigate the issue by decomposing asset growth into two parts: the part uncorrelated with stock returns and the correlated part. Since labor income growth is uncorrelated with (contemporaneous) stock returns (correlation coe¢ cient of 0.07 with the p value of 0.3), I regress asset growth on labor income growth. The tted values are, then, not correlated with stock returns while the residuals are. I regress one-period ahead stock returns on the decomposed asset growth or on tted asset growth and stock returns. Table 3.7 conrms that asset growth predicts stock returns or changes in returns. The tted asset growth explains one-period ahead returns and the estimated slope is similar to the one obtained using asset growth without a decomposition. The residual asset growth or stock returns cannot explain future stock returns. When using only stock returns and a constant as regressors, only the intercept is signicant and the adjusted R squared is negative (not tabulated). On the other hand, when explaining changes in returns, the residual and stock returns become signicant. The results show that asset growth predicts changes in returns by explaining both shocks to the discount rate and unexpected returns (v t and " t in the equation (13)): This is consistent with the implication of the the theoretical model that changes in asset holdings are associated with future changes in returns. 148 Table 3.7. Regressions using decomposed asset growth OLS results, coe¢ cient estimates and their standard errors in parentheses, using asset growth as two parts: one part uncorrelated with contemporaneous stock returns and the other part that is correlatedwithcontemporaneousstockreturns. Dependentvariablesarequarterlyreal(log)returns on the CRSP VW index (mkt), its lagged value (lag mkt) and their di¤erences (change). Asset growth is the rst di¤erences of the natural logarithm of aggregate asset holdings (net worth of households in the Flow of Funds data provided by the FRB Board of Governors) in 2000 chained dollars divided by population (provided by the National Income and Product Accounts). After regressing asset growth on labor income growth, I obtain tted value (tted lag asset growth) and residuals (residual of lag asset growth). Labor income growth is the rst di¤erences of the natural logarithm of real labor income per capita as provided in Martin Lettaus website. Since labor income growth cannot explain contemporaneous returns on the CRSP VW index (p-value of the slope estimate 0.30 and the adjusted R squared 0.05%), the tted value of asset growth is not correlated with contemporaneous returns on the CRSP VW index. The data period is from the rst quarter of 1952 to the fourth quarter of 2006. independent variables dependent variable returns returns changes in returns intercept 0.034 0.039 0.039 (0.013) (0.015) (0.015) tted lagged asset growth -3.595 -3.715 -3.715 (2.003) (2.025) (2.025) residual of lagged asset growth 0.160 (0.315) lagged mkt 0.062 -0.938 (0.067) (0.067) mean squared errors 0.0069 0.0068 0.0068 R squared 0.0187 0.0211 0.483 Adjusted R squared 0.0096 0.012 0.4781 3.7. Predictive regressions for the 25 Fama-French portfolios Iexaminepredictivepowerofaggregateassetgrowthforthe25Fama-Frenchportfolios constructed by size and the BE/ME ratios. Some researchers argue that growth stocks are particularlya¤ectedbychangesinthemarketdiscountratebecausetheircashowsoccurin the distant future (Lettau and Wachter (2007), Campbell, Polk, and Vuolteenaho (2010)). 149 Value stocks, on the other hand, are more sensitive to cash ow shockse.g., long-run macroeconomic risk in the cash ows (Hansen, Heaton, and Li (2008))rather than to discount rate shocks. Given that the theoretical model in Section 3.3. suggests that asset growthconveysinformationaboutshockstothemarketdiscountrate, growthstockscanbe more sensitive to variation of asset growth and better predicted by asset growth, compared with value stocks. I obtain the data for returns on the 25 Fama-French portfolios from Kenneth Frenchs website. Giventhatsomeassetsaremorevolatilethaothers(e.g., smallstockstendtohave large variances than large stocks), it is misleading to compare mean squared errors across di¤erent assets. Thus, I look at the average of standardized squared errors, which is the mean squared errors divided by the variance of returns on a given asset. This is in fact a test to compare predictive ability of a given regression model across various assets in terms of R squared by the following. The R squared of a given portfolio i can be written as Rsquared i = 1 T P t=1 (r it b r it ) 2 T P t=1 (r it r it ) 2 = 1 T P t=1 (r it b r it ) 2 T T P t=1 (r it r it ) 2 T ; where b r it is the predicted return and r it is the sample average. I rewrite the equation by dividing each element of the summation part in the numerator by the denominator, R squared i = 1 T P t=1 0 B B B @ (r it b r it ) 2 T P t=1 (r it r it ) 2 T 1 C C C A T : 150 In other words, the R squared is equal to one minus the sample mean of the standardized squared predictive errors (r it b r it ) 2 T P t=1 (r it r it ) 2 T : Thus, when comparing the R squared, I test whether thestandardizedvalueshaveequalmeansacrossdi¤erentportfolios(t-statisticsareadjusted for serial correlation by the Newey-West method with 4 lags). Table 3.8 shows the predictive regression results for the 25 portfolios, using the return model, i.e., the dependent variable is returns on the portfolios. Panel A presents the coef- cients on the independent variables (e.g., lagged asset growth and lagged cay) and the standard errors (adjusted by the Newey-West method with 4 lags). The results suggest that lagged returns and lagged asset growth do not have signicant predictive power, but onlylaggedcayappearsusefulforpredictingthereturns. PanelBshowsthatthevariable cayachievestheRsquaredofabout0.02-0.06. TheRsquaredisnotsignicantlydi¤erent between growth and value stocks, given the t-statistics of about 0.14 (Panel B). I do the same analysis using the change model, i.e., predicted returns are the predicted change in returns plus lagged actual returns (Table 3.9). The betas for asset growth are negative, suggesting that a 1% increase in aggregate asset holdings predicts that returns on those portfolios decrease by 0.02-0.04 in the following quarter (these results are comparable to the prediction that market returns decrease by 0.036 as shown in Section 3.4.3.). The magnitudes of the betas are larger for growth portfolios than for value portfolios (e.g., the small growth portfolio has a beta of -3.53 and the small value portfolio has a beta of -2.06). Other independent variables, including lagged cay,are also statistically di¤erent from zero at the 1% signicance level. Yet, asset growth has better predictive power for changes in returns on the 25 portfolios than other variables do as I discuss in the below. 151 Table3.8. OLSregressionofone-periodaheadquarterlyreturnsonthe25Fama-French portfolios Dependent variables are quarterly returns (t) on the Fama-French 25 portfolios from 1952 Q1 to2006Q4. The rst number representssmall(1), 2, 3, 4, andlarge(35)portfoliosbasedon thesize and the second number represents growth (1), 2, 3, 4, and value (5) portfolios based on the BE/ME ratio. Panel (A) reports the coe¢ cient estimates and standard errors, which are adjusted for serial correlation using Newey-West method (4 lags). Panel (B) reports the R-squared. The t-statistics in the last row is for equal R-squared between growth portfolios and value portfolios. Asset growth is the rst di¤erences of the natural logarithm of aggregate asset holdings (net worth of households in theFlowofFundsdataprovidedbytheFRBBoardofGovernors)in2000chaineddollarsdividedby population. Personal Consumption Expenditure deator and population data are from the National IncomeandProductAccounts. Cayisthecointegratingrelationshipofconsumption, assetholdings, and labor income as provided on Martin Lettaus webiste. 152 (Table 3.8. continued) (A) Coe¢ cient and standard error portfolio variable (1) (2) (3) (4) (5) small growth intercept 0.024 0.023 0.023 0.024 0.024 (0.010) (0.010) (0.010) (0.010) (0.010) return (t-1) 0.001 -0.026 0.015 (0.071) (0.108) (0.075) asset growth (t-1) 0.141 0.284 (0.572) (0.859) cay (t-1) 1.468 1.489 (0.884) (0.874) 12 intercept 0.041 0.040 0.040 0.040 0.040 (0.009) (0.009) (0.009) (0.009) (0.009) return (t-1) -0.016 -0.046 -0.004 (0.065) (0.101) (0.067) asset growth (t-1) 0.052 0.275 (0.487) (0.754) cay (t-1) 1.491 1.487 (0.727) (0.730) 13 intercept 0.041 0.039 0.040 0.041 0.041 (0.008) (0.008) (0.008) (0.008) (0.008) return (t-1) -0.010 -0.068 0.002 (0.071) (0.118) (0.072) asset growth (t-1) 0.221 0.488 (0.398) (0.667) cay (t-1) 1.273 1.275 (0.641) (0.642) 14 intercept 0.047 0.045 0.047 0.047 0.047 (0.007) (0.007) (0.007) (0.007) (0.007) return (t-1) 0.001 -0.080 0.012 (0.063) (0.101) (0.064) asset growth (t-1) 0.360 0.656 (0.419) (0.647) cay (t-1) 1.183 1.190 (0.595) (0.594) small value intercept 0.052 0.049 0.052 0.051 0.051 (0.008) (0.008) (0.008) (0.008) (0.008) return (t-1) -0.012 -0.105 -0.001 (0.061) (0.095) (0.063) asset growth (t-1) 0.438 0.853 (0.402) (0.621) cay (t-1) 1.256 1.255 (0.622) (0.619) 153 (Table 3.8 continued.) portfolio variable (1) (2) (3) (4) (5) 21 intercept 0.030 0.030 0.030 0.028 0.029 (0.009) (0.009) (0.009) (0.008) (0.009) return (t-1) -0.070 -0.058 -0.058 (0.070) (0.110) (0.072) asset growth (t-1) -0.410 -0.106 (0.494) (0.783) cay (t-1) 1.913 1.873 (0.688) (0.719) 22 intercept 0.040 0.039 0.040 0.036 0.039 (0.007) (0.007) (0.008) (0.007) (0.007) return (t-1) -0.090 -0.103 -0.080 (0.072) (0.110) (0.074) asset growth (t-1) -0.337 0.101 (0.432) (0.665) cay (t-1) 1.405 1.357 (0.573) (0.601) 23 intercept 0.044 0.043 0.044 0.043 0.044 (0.007) (0.007) (0.007) (0.006) (0.007) return (t-1) -0.031 -0.045 -0.021 (0.067) (0.115) (0.069) asset growth (t-1) -0.079 0.091 (0.366) (0.624) cay (t-1) 1.353 1.343 (0.531) (0.537) 24 intercept 0.046 0.044 0.046 0.044 0.045 (0.007) (0.007) (0.007) (0.006) (0.007) return (t-1) -0.031 -0.083 -0.025 (0.064) (0.088) (0.066) asset growth (t-1) 0.087 0.378 (0.383) (0.538) cay (t-1) 1.506 1.501 (0.528) (0.535) 25 intercept 0.049 0.047 0.049 0.048 0.049 (0.007) (0.007) (0.007) (0.007) (0.007) return (t-1) -0.034 -0.097 -0.024 (0.058) (0.084) (0.060) asset growth (t-1) 0.157 0.516 (0.385) (0.567) cay (t-1) 1.150 1.136 (0.551) (0.556) 154 (Table 3.8 continued.) portfolio variable (1) (2) (3) (4) (5) 31 intercept 0.032 0.033 0.033 0.030 0.032 (0.008) (0.008) (0.008) (0.007) (0.008) return (t-1) -0.066 -0.008 -0.055 (0.070) (0.124) (0.073) asset growth (t-1) -0.480 -0.441 (0.434) (0.769) cay (t-1) 1.907 1.880 (0.625) (0.648) 32 intercept 0.037 0.038 0.037 0.037 0.037 (0.007) (0.007) (0.007) (0.006) (0.007) return (t-1) 0.003 0.050 0.014 (0.068) (0.121) (0.068) asset growth (t-1) -0.109 -0.305 (0.363) (0.643) cay (t-1) 1.556 1.562 (0.554) (0.558) 33 intercept 0.039 0.038 0.039 0.038 0.039 (0.006) (0.006) (0.006) (0.006) (0.006) return (t-1) -0.025 -0.038 -0.016 (0.062) (0.101) (0.062) asset growth (t-1) -0.052 0.077 (0.312) (0.506) cay (t-1) 1.341 1.335 (0.477) (0.483) 34 intercept 0.039 0.041 0.039 0.042 0.039 (0.006) (0.006) (0.006) (0.006) (0.006) return (t-1) 0.062 0.102 0.069 (0.056) (0.080) (0.058) asset growth (t-1) 0.077 -0.257 (0.320) (0.439) cay (t-1) 1.451 1.467 (0.535) (0.522) 35 intercept 0.044 0.045 0.044 0.045 0.044 (0.007) (0.007) (0.007) (0.007) (0.007) return (t-1) 0.012 0.028 0.020 (0.058) (0.082) (0.059) asset growth (t-1) -0.026 -0.123 (0.331) (0.461) cay (t-1) 1.181 1.189 (0.566) (0.561) 155 (Table 3.8 continued.) portfolio variable (1) (2) (3) (4) (5) 41 intercept 0.032 0.034 0.034 0.032 0.032 (0.007) (0.008) (0.008) (0.007) (0.007) return (t-1) -0.022 0.098 -0.008 (0.073) (0.127) (0.077) asset growth (t-1) -0.351 -0.804 (0.413) (0.724) cay (t-1) 1.769 1.765 (0.639) (0.646) 42 intercept 0.034 0.034 0.034 0.033 0.033 (0.006) (0.006) (0.006) (0.006) (0.006) return (t-1) -0.033 0.004 -0.027 (0.070) (0.124) (0.069) asset growth (t-1) -0.210 -0.224 (0.310) (0.560) cay (t-1) 1.574 1.569 (0.517) (0.527) 43 intercept 0.038 0.039 0.038 0.039 0.037 (0.005) (0.006) (0.005) (0.006) (0.006) return (t-1) 0.042 0.059 0.052 (0.057) (0.099) (0.057) asset growth (t-1) 0.096 -0.097 (0.267) (0.453) cay (t-1) 1.307 1.323 (0.480) (0.477) 44 intercept 0.038 0.039 0.038 0.040 0.037 (0.006) (0.006) (0.006) (0.006) (0.006) return (t-1) 0.047 0.070 0.059 (0.058) (0.089) (0.058) asset growth (t-1) 0.090 -0.135 (0.293) (0.443) cay (t-1) 1.183 1.209 (0.522) (0.516) 45 intercept 0.042 0.041 0.042 0.042 0.042 (0.007) (0.007) (0.007) (0.006) (0.007) return (t-1) -0.018 -0.044 -0.010 (0.066) (0.095) (0.065) asset growth (t-1) 0.046 0.199 (0.351) (0.512) cay (t-1) 1.421 1.417 (0.524) (0.529) 156 (Table 3.8 continued.) portfolio variable (1) (2) (3) (4) (5) large growth intercept 0.027 0.029 0.028 0.029 0.027 (0.006) (0.007) (0.006) (0.006) (0.006) return (t-1) 0.064 0.211 0.075 (0.053) (0.119) (0.051) asset growth (t-1) -0.035 -0.794 (0.292) (0.592) cay (t-1) 1.671 1.693 (0.453) (0.439) 52 intercept 0.029 0.030 0.029 0.030 0.029 (0.005) (0.006) (0.005) (0.005) (0.005) return (t-1) 0.039 0.119 0.046 (0.058) (0.093) (0.054) asset growth (t-1) -0.018 -0.394 (0.254) (0.412) cay (t-1) 1.485 1.494 (0.378) (0.378) 53 intercept 0.031 0.032 0.031 0.032 0.031 (0.005) (0.005) (0.005) (0.005) (0.005) return (t-1) 0.029 0.074 0.029 (0.066) (0.100) (0.060) asset growth (t-1) -0.003 -0.207 (0.225) (0.329) cay (t-1) 1.360 1.360 (0.371) (0.368) 54 intercept 0.028 0.032 0.028 0.032 0.028 (0.005) (0.006) (0.005) (0.005) (0.005) return (t-1) 0.123 0.223 0.126 (0.056) (0.073) (0.053) asset growth (t-1) 0.078 -0.523 (0.248) (0.312) cay (t-1) 1.330 1.340 (0.405) (0.389) large value intercept 0.031 0.033 0.031 0.034 0.031 (0.007) (0.006) (0.007) (0.006) (0.007) return (t-1) 0.098 0.104 0.103 (0.065) (0.092) (0.063) asset growth (t-1) 0.268 -0.038 (0.305) (0.416) cay (t-1) 1.428 1.445 (0.436) (0.426) 157 (Table 3.8 continued.) (B) R squared portfolio (1) (2) (3) (4) (5) small growth 0.000 0.000 0.001 0.015 0.016 12 0.000 0.000 0.001 0.022 0.022 13 0.000 0.001 0.004 0.021 0.021 14 0.000 0.004 0.008 0.020 0.020 small value 0.000 0.005 0.011 0.019 0.019 21 0.005 0.004 0.005 0.034 0.037 22 0.008 0.003 0.008 0.026 0.032 23 0.001 0.000 0.001 0.031 0.031 24 0.001 0.000 0.004 0.039 0.039 25 0.001 0.001 0.006 0.019 0.020 31 0.004 0.006 0.006 0.041 0.044 32 0.000 0.000 0.001 0.041 0.041 33 0.001 0.000 0.001 0.036 0.036 34 0.004 0.000 0.005 0.042 0.047 35 0.000 0.000 0.000 0.022 0.023 41 0.001 0.004 0.007 0.043 0.043 42 0.001 0.002 0.002 0.048 0.049 43 0.002 0.000 0.002 0.038 0.041 44 0.002 0.000 0.003 0.031 0.034 45 0.000 0.000 0.001 0.033 0.034 large growth 0.004 0.000 0.015 0.062 0.067 52 0.001 0.000 0.005 0.060 0.062 53 0.001 0.000 0.002 0.061 0.062 54 0.015 0.000 0.025 0.053 0.069 large value 0.010 0.004 0.010 0.046 0.057 growth-value (t-stat) 0.006 0.010 0.012 0.141 0.143 158 Table 3.9. OLS regression of one-period ahead changes in quarterly returns on the Fama-French 25 portfolios Dependent variables are changes in quarterly returns (t) on the Fama-French 25 portfolios from 1952 Q1 to 2006 Q4. The rst number represents small (1), 2, 3, 4, and large (5) portfolios based on the size and the second number represents growth (1), 2, 3, 4, and value (5) portfolios based on the BE/ME ratio. Panel (A) reports the coe¢ cient estimates and standard errors, which are adjusted for serial correlation using Newey-West method (4 lags). Panel (B) reports the R- squared. The t-statistics in the last row is for equal R-squared between growth portfolios and value portfolios. Asset growth is the rst di¤erences of the natural logarithm of aggregate asset holdings (networthofhouseholdsintheFlowofFundsdataprovidedbytheFRBBoardofGovernors)in2000 chained dollars divided by population. Personal Consumption Expenditure deator and population data are from the National Income and Product Accounts. Cay is the cointegrating relationship of consumption, asset holdings, and labor income as provided on Martin Lettaus webiste. 159 (Table 3.9. continued) (A) Coe¢ cient and standard error portfolio variable (1) (2) (3) (4) (5) small growth intercept 0.001 0.034 0.022 0.000 0.000 (0.008) (0.009) (0.009) (0.008) (0.009) change (t-1) -0.497 -0.327 -0.497 (0.059) (0.074) (0.057) asset growth (t-1) -5.423 -3.531 (0.647) (0.699) cay (t-1) 2.845 2.811 (0.776) (0.924) 12 intercept 0.001 0.030 0.020 0.000 0.000 (0.007) (0.009) (0.008) (0.007) (0.007) change (t-1) -0.501 -0.319 -0.500 (0.058) (0.071) (0.056) asset growth (t-1) -4.780 -3.135 (0.502) (0.547) cay (t-1) 2.304 2.276 (0.656) (0.768) 13 intercept 0.000 0.023 0.014 0.000 0.000 (0.006) (0.007) (0.007) (0.006) (0.006) change (t-1) -0.494 -0.355 -0.492 (0.063) (0.081) (0.061) asset growth (t-1) -3.701 -2.192 (0.496) (0.550) cay (t-1) 1.989 1.895 (0.575) (0.661) 14 intercept 0.000 0.021 0.013 0.000 0.000 (0.005) (0.007) (0.007) (0.006) (0.006) change (t-1) -0.468 -0.334 -0.467 (0.058) (0.075) (0.057) asset growth (t-1) -3.346 -1.981 (0.526) (0.605) cay (t-1) 1.835 1.769 (0.540) (0.620) small value intercept 0.001 0.022 0.013 0.000 0.001 (0.006) (0.007) (0.007) (0.006) (0.006) change (t-1) -0.459 -0.338 -0.459 (0.055) (0.071) (0.053) asset growth (t-1) -3.527 -2.061 (0.499) (0.612) cay (t-1) 1.945 1.917 (0.599) (0.688) 160 (Table 3.9 continued.) portfolio variable (1) (2) (3) (4) (5) 21 intercept 0.000 0.035 0.025 0.000 0.000 (0.007) (0.009) (0.008) (0.007) (0.007) change (t-1) -0.520 -0.296 -0.519 (0.056) (0.066) (0.054) asset growth (t-1) -5.691 -4.065 (0.568) (0.564) cay (t-1) 2.602 2.539 (0.670) (0.761) 22 intercept 0.000 0.029 0.020 0.000 0.000 (0.006) (0.008) (0.007) (0.006) (0.006) change (t-1) -0.521 -0.324 -0.519 (0.064) (0.083) (0.062) asset growth (t-1) -4.572 -3.100 (0.524) (0.579) cay (t-1) 1.996 1.914 (0.555) (0.633) 23 intercept 0.000 0.024 0.017 0.000 0.000 (0.005) (0.007) (0.007) (0.005) (0.005) change (t-1) -0.495 -0.285 -0.494 (0.061) (0.082) (0.059) asset growth (t-1) -3.872 -2.704 (0.441) (0.534) cay (t-1) 1.813 1.793 (0.517) (0.587) 24 intercept 0.000 0.021 0.013 0.000 0.000 (0.005) (0.007) (0.006) (0.005) (0.005) change (t-1) -0.494 -0.338 -0.494 (0.063) (0.082) (0.060) asset growth (t-1) -3.403 -2.099 (0.505) (0.606) cay (t-1) 1.719 1.711 (0.523) (0.596) 25 intercept 0.000 0.022 0.014 0.000 0.000 (0.005) (0.007) (0.007) (0.005) (0.006) change (t-1) -0.458 -0.306 -0.457 (0.058) (0.076) (0.056) asset growth (t-1) -3.526 -2.257 (0.494) (0.597) cay (t-1) 1.711 1.681 (0.580) (0.659) 161 (Table 3.9 continued.) portfolio variable (1) (2) (3) (4) (5) 31 intercept 0.000 0.034 0.026 0.000 0.000 (0.006) (0.009) (0.008) (0.006) (0.007) change (t-1) -0.501 -0.247 -0.503 (0.060) (0.072) (0.058) asset growth (t-1) -5.391 -4.198 (0.522) (0.534) cay (t-1) 2.389 2.467 (0.596) (0.682) 32 intercept 0.000 0.025 0.020 0.000 0.000 (0.005) (0.007) (0.007) (0.005) (0.005) change (t-1) -0.464 -0.213 -0.463 (0.060) (0.078) (0.057) asset growth (t-1) -4.015 -3.169 (0.467) (0.581) cay (t-1) 1.964 1.929 (0.515) (0.584) 33 intercept 0.000 0.022 0.016 0.000 0.000 (0.004) (0.007) (0.007) (0.005) (0.005) change (t-1) -0.484 -0.276 -0.482 (0.054) (0.080) (0.052) asset growth (t-1) -3.490 -2.474 (0.489) (0.602) cay (t-1) 1.695 1.639 (0.484) (0.544) 34 intercept 0.000 0.020 0.016 0.000 0.000 (0.005) (0.006) (0.007) (0.005) (0.005) change (t-1) -0.426 -0.218 -0.427 (0.060) (0.084) (0.057) asset growth (t-1) -3.201 -2.465 (0.429) (0.567) cay (t-1) 1.684 1.719 (0.522) (0.601) 35 intercept 0.000 0.022 0.016 0.000 0.000 (0.005) (0.007) (0.007) (0.005) (0.006) change (t-1) -0.449 -0.281 -0.450 (0.059) (0.076) (0.056) asset growth (t-1) -3.465 -2.455 (0.475) (0.554) cay (t-1) 1.579 1.609 (0.527) (0.619) 162 (Table 3.9 continued.) portfolio variable (1) (2) (3) (4) (5) 41 intercept 0.000 0.031 0.026 0.000 0.000 (0.006) (0.008) (0.008) (0.006) (0.006) change (t-1) -0.482 -0.179 -0.484 (0.064) (0.078) (0.061) asset growth (t-1) -4.988 -4.162 (0.423) (0.511) cay (t-1) 2.371 2.422 (0.582) (0.673) 42 intercept 0.000 0.024 0.018 0.000 0.000 (0.004) (0.007) (0.007) (0.005) (0.005) change (t-1) -0.492 -0.254 -0.492 (0.059) (0.083) (0.055) asset growth (t-1) -3.831 -2.885 (0.475) (0.620) cay (t-1) 1.784 1.781 (0.498) (0.561) 43 intercept 0.000 0.020 0.016 0.000 0.000 (0.004) (0.006) (0.007) (0.005) (0.005) change (t-1) -0.417 -0.182 -0.417 (0.054) (0.083) (0.050) asset growth (t-1) -3.182 -2.573 (0.414) (0.574) cay (t-1) 1.622 1.617 (0.475) (0.538) 44 intercept 0.000 0.020 0.015 0.000 0.000 (0.004) (0.006) (0.006) (0.005) (0.005) change (t-1) -0.441 -0.230 -0.444 (0.054) (0.076) (0.051) asset growth (t-1) -3.138 -2.390 (0.407) (0.534) cay (t-1) 1.628 1.684 (0.468) (0.538) 45 intercept 0.000 0.021 0.014 0.000 0.000 (0.005) (0.007) (0.007) (0.005) (0.006) change (t-1) -0.476 -0.319 -0.477 (0.063) (0.077) (0.060) asset growth (t-1) -3.415 -2.234 (0.440) (0.481) cay (t-1) 1.731 1.734 (0.547) (0.613) 163 (Table 3.9 continued.) portfolio variable (1) (2) (3) (4) (5) large growth intercept 0.000 0.023 0.019 0.000 0.000 (0.004) (0.007) (0.007) (0.005) (0.005) change (t-1) -0.437 -0.158 -0.441 (0.053) (0.070) (0.050) asset growth (t-1) -3.630 -3.118 (0.334) (0.437) cay (t-1) 1.970 2.050 (0.439) (0.483) 52 intercept 0.000 0.020 0.016 0.000 0.000 (0.004) (0.006) (0.006) (0.004) (0.004) change (t-1) -0.451 -0.182 -0.450 (0.053) (0.067) (0.050) asset growth (t-1) -3.188 -2.629 (0.326) (0.399) cay (t-1) 1.675 1.666 (0.431) (0.449) 53 intercept 0.000 0.017 0.013 0.000 0.000 (0.004) (0.005) (0.005) (0.004) (0.004) change (t-1) -0.468 -0.239 -0.470 (0.055) (0.077) (0.052) asset growth (t-1) -2.779 -2.139 (0.278) (0.335) cay (t-1) 1.364 1.397 (0.364) (0.387) 54 intercept 0.000 0.016 0.013 0.000 0.000 (0.004) (0.005) (0.005) (0.004) (0.004) change (t-1) -0.407 -0.183 -0.413 (0.061) (0.085) (0.057) asset growth (t-1) -2.623 -2.146 (0.315) (0.427) cay (t-1) 1.411 1.498 (0.407) (0.467) large value intercept 0.000 0.017 0.012 0.000 0.000 (0.004) (0.006) (0.006) (0.004) (0.005) change (t-1) -0.438 -0.283 -0.437 (0.057) (0.074) (0.054) asset growth (t-1) -2.683 -1.887 (0.390) (0.423) cay (t-1) 1.596 1.588 (0.463) (0.526) 164 (Table 3.9 continued.) (B) R squared portfolio (1) (2) (3) (4) (5) small growth 0.247 0.245 0.322 0.029 0.275 12 0.251 0.256 0.328 0.025 0.275 13 0.244 0.205 0.297 0.025 0.267 14 0.219 0.185 0.266 0.024 0.241 small value 0.211 0.169 0.254 0.022 0.232 21 0.270 0.325 0.386 0.029 0.298 22 0.271 0.292 0.366 0.024 0.293 23 0.245 0.284 0.339 0.027 0.271 24 0.244 0.224 0.305 0.024 0.269 25 0.210 0.202 0.269 0.020 0.229 31 0.251 0.363 0.407 0.031 0.283 32 0.215 0.321 0.352 0.033 0.247 33 0.234 0.278 0.331 0.028 0.261 34 0.181 0.255 0.289 0.030 0.212 35 0.202 0.229 0.288 0.020 0.223 41 0.232 0.392 0.414 0.038 0.272 42 0.242 0.324 0.369 0.030 0.272 43 0.174 0.276 0.299 0.031 0.204 44 0.195 0.267 0.305 0.031 0.227 45 0.227 0.222 0.297 0.024 0.252 large growth 0.191 0.363 0.381 0.046 0.241 52 0.203 0.338 0.361 0.040 0.243 53 0.219 0.309 0.350 0.032 0.253 54 0.166 0.276 0.301 0.034 0.205 large value 0.192 0.210 0.272 0.032 0.224 growth-value (t-stat) 0.432 2.074 1.829 0.118 0.648 165 Figure 3.6. Actual and tted values of small portfolios The gures plot predicted quarterly returns and actual quarterly returns on small growth and small value portfolios among the 25 Fama-French 25 portfolios. Change model regresses quarterly chanagesinreturnsoncontant,laggedchangesinreturnsandlaggedassetgrowth. Predictedreturns arethettedchangesinreturnspluslaggedactualreturns. Returnmodelregressesquarterlyreturns on contant, lagged returns and lagged cay.See Table 3.6 or 7 for the data description. 166 (Figure 3.6 continued.) 167 Figure 3.7. Actual and tted values of large portfolios The gures plot predicted quarterly returns and actual quarterly returns on large growth and large value portfolios among the 25 Fama-French portfolios. Change model regresses quarterly chanagesinreturnsoncontant,laggedchangesinreturnsandlaggedassetgrowth. Predictedreturns arethettedchangesinreturnspluslaggedactualreturns. Returnmodelregressesquarterlyreturns on contant, lagged returns and lagged cay.See Table 3.6 or 7 for the data description. 168 (Figure 3.7 continued.) 169 Panel B in Table 3.9 shows the R squared of the regression models in Panel A (the adjusted R squared is lower by 0.003-0.007). Comparing the predictive power across the 25 portfolios, only asset growth predicts growth stocks better than value stocks. Finally, I plot predicted returns and actual returns on small growth, small value , large growth, and large value portfolios in Figure 3.6 and 7, using the change model and asset growth. I also present similar gures, using the return model and cay.Consistent with results for regressions of the market returns, the predicted values by cayvary much less than the predicted values by asset growth. 3.8. Conclusion This chapter shows that changes in nonstationary variables are useful for predicting returns through predicting changes in equity returns. In particular, I nd changes in aggre- gate asset holdings (asset growth) is a univariate, powerful predictor of subsequent changes in equity returns. The changes in returns on stocks estimated by the asset growth explain a substantial portion of the variation in realized changes in returns. These empirical results areconsistentwiththenegativerelationshipbetweenchangesinvalueofassetsandchanges in expected future returns (shocks to discount rates). Therefore, asset growth allows us to learnabouttheagentsupdatesonexpectationsforfuturereturnsonstocks. Estimatingthe cointegrating relationship of dividends and stock prices can improve the predictive ability of the dividend yield. I show that asset growth predicts better out of sample in terms of a mean squared predictive error (Clark and West (2007)) than other predictors, such as the dividend-price ratio, cayand yield spread. 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Proof for Proposition 2 Proposition 1 and Corollary 1 reduce the arbitrators problem to max y;fm(t);(t)g t2T log(x 0 w 0 )+ P t2T p(t)log(b ye t ) (1-1) subject to (t)v(m(t)) =b y for every t2T; (1-2) where x 0 =u P P t2T p(t)(t)m(t): (1-3) Note that I drop the participation constraints because an optimal allocation gives the manager of type t more than her reservation utility zero (as assumed in Section 2) and, therefore, the constraints are not binding. Let me dene v t = v(m(t)) and Q(v t ) = m(t) where Q v 1 : That is, v t is utility of the manager of typet when she receives fee income: SinceV is strictly concave, Q is strictly convex. Then, the above problem can be rewritten as max b y;fvt;(t)g t2T log(u P P t2T p(t)(t)Q(v t )w 0 )+ P t2T p(t)log(b ye t ) (1-4) subject to (t)v t = b y for every t2T (1-5) (t) 2 [0;1] for every t2T: 181 The constraint (1-5) is the incentive compatibility by Proposition 1. I dene a Lagrangian ((t) 0 will not bind), L(b y;v t ;(t); t ) = log(u P P t2T p(t)(t)Q(v t )w 0 ) + P t2T p(t)log(b ye t )+ P t2T t f(t)v t b yg+ P t2T t f1(t)g: (1-6) Since my objective function is now strictly concave and the constraints are linear, the Kuhn-Tucker theorem applies. In other words, the rst-order conditions are also su¢ cient. FOC for (1-6) with respect to (t) gives us p(t) x 0 w 0 Q(v t ) v t = t t v t : (1-7) Since p(1) =p(3) = 0 by Corollary 1; we should have H;A = L;A = 0: On the other hand, FOC for (1-6) with respect to v t is given by p(t) x 0 w 0 Q 0 (v t ) = t : (1-8) FOC for (1-6) with respect tob y is P t2T p(t) b y = P t2T t : (1-9) Recall that Q =v 1 is strictly convex with Q(0) = 0 (Assumption 2), which implies Q 0 (v t )> Q(v t ) v t for all v t 2R (real values): 182 Then (1-7) and (1-8) imply that t > 0; so that (t) = 1 for t = (H;P);(L;P): (1-10) Thus, by (1-5) v t =b y for every t2T: (1-11) Finally, using the fact that P t2T p(t) = 1; (1-9) becomes 1 b y = P t2T t ; whose RHS is given by P t2T t = Q 0 (v t ) x 0 w 0 ; by (1-8). Since x 0 = u P Q(b y) by (1-3), (1-10) and (1-11), the above equation can be written as 1 b y |{z} marginal benet = Q 0 (v t ) u P Q(b y)w 0 | {z } marginal cost ; (1-12) wheretheLHSisthemarginalbenetofincreasingoneunitofthemanagersutilitywhereas the RHS is its marginal cost since the investor should incur compensation cost. A1.2. Proof for Proposition 3 The H skill manager who manages the A type fund receives a performance fee (ad- ditional xed fees can be regarded as included in performance fees). If the type (H;A) 183 manager receives compensation, she receives utility, v(e r)v((e r)) wheree r isthenetreturnon the fund. Note thatIdenoteitbyv(e r) withoutthetype (H;A) because only that type receives performance compensation (see Section 2). The manager of other type t6= (H;A) only receives the management fee (without loss of generality; see Section 2) as given by v t v(m(t)) for t6= (H;A): Similar to the case without performance fees, I use the inverse function of a mangers utility from income, Q =v 1 ; and write Q(v t ) = m(t) for t6= (H;A) Q(v(e r)) = (e r(H;A)): (2-1) The expected utility for the type t manager with the truthful report is x H;A =(H;A)E H;A [v(e r)]e H (2-2) x t = (t)v t for t = (H;P) and (L;P) (2-3) x L;A = (L;P)v L;A e L ; (2-4) 184 where E H;A [] is the expectation with respect to F H;A (e r), the distribution of net returns of the A fund when managed by an H skill manager. Now, I rst derive the incentive compatibility (IC) constraints for each type of the manager. The ICs for t = (H;P) (or (L;P)) not to report a type (L;P) (or (H;P)) or (L;A) yield (t)v t = y for t = (H;P) and (L;P) (2-5) y (L;A)v L;A : (2-6) If the manager of type (H;P) or (L;P) pretends to be of type (H;A), she receives (H;A)E P [v(e r)]; where E P [] is the expectation with respect to F P (e r), the distribution of net returns of the P fund. Therefore, I need another IC constraint, y(H;A)E P [v(e r)]: (2-7) A manager of type (L;A) will not pretend to be of type (H;P) or (L;P) if (L;P)v L;A e L ye L holds. Then, by (2-6), we should have (L;P)v L;A =y: (2-8) 185 Note that ye L is the expected utility of the type (L;A) manager from the truthful report, which is less that the type (L;P) manager receives since x L;P =y >ye L =x L;A : Therefore, it is optimal for the low skill manager to choose the P fund (become the type (L;P)) rather than the A fund (become the type (L;A)). So in equilibrium, the low skill manager always passively manages a fund. Ontheotherhand, ifthemanageroftype (L;A)liestobethetype (H;A), shereceives (H;A)E L;A [v(e r)]e L ; where E L;A [] is the expectation with respect to F L;A (e r), the distribution of net return on the A fund when managed by an L skill manager. Then the IC that the type (L;A) manager does not lie to be the type (H;A) manager is y(H;A)E L;A [v(e r)]: (2-9) For the type (H;A) manager, she will not lie to be of any other type if (H;A)E H;A [v(e r)]y: (2-10) 186 Let me dene the di¤erence of the H skill managers expected utility between active management and passive management as = ((H;A)E H;A [v(e r)]e H )y; which allows me to rewrite (2-10) as e H : (2-11) For future reference, let me derive the condition for the separating equilibrium. The H skill manger will choose the A type fund if (H;A)E H;A [v(e r)]e H >y; (2-12) which is equivalent to > 0: (2-13) The arbitrators problem (3)-(6) in Section 3 becomes max y;;v(e r);(H;A)fvt;(t)g t2T;t6=(H;A) log(x 0 w 0 )+ P t2T p(t)log(x t ) subject to (H;A)E H;A [v(e r)]e H =y+ for t = (H;A) 187 e H (IC 1) (t)v t = y for t = (H;P) and (L;P) (IC 2) (L;A)v L;A = y (IC 3) (H;A)E L;A [v(e r)] y (IC 4) (H;A)E P [v(e r)] y (IC 5) (t) 2 [0;1]: where x 0 =u S fp(H;A)(H;A)E H;A [Q(v(e r))]+ P j6=1 p(t)(t)Q(v t )g is the investorsexpected utility and the managers expected utility of the type t, x t ; is given by (2-2) to (2-4). (IC 1) states that the manager of type (H;A) does not misreport her type as derived in (2-11). (IC 2) is the condition that the L (H) skill fund manager who manages the P fund will not lie to have the H (L) skill, and this is derived in (2-5). (IC 3) is that the type (L;A) manager will not report to be of the type (H;P) or (L;P) as I derive in (2-8). (IC 4) ensures that the L skill manager who actively manages a fund (type (L;A)) will not pretend to have the H skill (type (H;A)), as derived in the condition (2-9). The condition that the passive manager (type (H;P) or (L;P)) will not lie to be of the type (H;A) to earn the performance fee is provided by (IC 5), which is derived in (2-7). Under the assumption that return on the P fund stochastically dominates the return on the A fund managed by the L skill manager, only (IC 5) can hold with equality and (IC 4) holds with strict inequality. Thus, I can ignore (IC 4). 188 To solve the problem, I consider a separating equilibrium, which requires the condition (2-13). Then the inequality constraint (IC 1) is not binding. Also, we have p(H;A) = ; p(L;P) = 1; and p(H;P) = p(L;A) = 0: As a result, we can also ignore (IC 2) since the Lagrangian multiplier must be zero. DeneaLagrangian(Iconjecturethat(t)2 [0;1]willnotbindexceptfor(L;P) 1), L(y;;v(e r);(H;A);v t ;(t); H;A ; L;P ; L;P ) = log(x 0 w 0 )+flog(y+w H;A )+(1)log(yw L;P )g + H;A f(H;A)E H;A [v(e r)]e H yg+ L;P f(L;P)v L;P yg + L;P fy(H;A)E P [v(e r)]g+ L;P f1(L;P)g: (2-14) where x 0 =u S (H;A)E H;A [Q(v(e r))](1)(L;P)Q(v L;P ): (2-15) Lemma 1 In the e¢ cient outcome, the managers utility from performance compensation is only determined by the ratio of the likelihood that she is H skill manager and actively manage the fund to the likelihood that she tracks a market index (likelihood ratio). Proof. We need to solve v(e r) and show that it depend only on the likelihood ratio, f HA (e r) f P (e r) ; where f H;A (e r) and f P (e r) are the density functions of net returns on the active fund managed by the H skill manager and on the passive fund respectively. FOC for (2-14) with 189 respect to (H;A) is 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] = H;A L;P E P [v(e r)] E H;A [v(e r)] (2-16) For simplicity, if I dene a H;A ;b 1 x 0 w 0 ; and c L;P ; (2-17) the above equation becomes b E H;A [Q(v(e r)] E H;A [v(e r)] =ac E P [v(e r)] E H;A [v(e r)] : (2-18) If I di¤erentiate (2-14) with respect to v(e r); the pointwise optimization yields (H;A)Q 0 (v(e r))F H;A (e r) x 0 w 0 = H;A (H;A)F H;A (e r) L;P (H;A)f P (e r) and by rearranging it, I obtain Q 0 (v(e r)) x 0 w 0 = H;A L;P f P (e r) f H;A (e r) : (2-19) If I multiply both sides of (2-19) by f H;A (e r) and then integrate, I get 1 x 0 w 0 E H;A [Q 0 (v(e r)] = H;A L;P : (2-20) 190 Using (2-21), I can rewrite (2-25) as a =c+bE H;A [Q 0 (v(e r)]: (2-21) If I use (2-17) and (2-21), the equation (2-19) becomes bfQ 0 (v(e r))E H;A [Q 0 (v(e r)]g =cf f H;A (e r)f P (e r) f H;A (e r) g: Then b = cf f H;A (e r)f P (e r) f H;A (e r) g Q 0 (v(e r))E H;A [Q 0 (v(e r)] ; (2-22) and by (2-21), a =cf1+ E H;A [Q 0 (v(e r)] Q 0 (v(e r))E H;A [Q 0 (v(e r)] f H;A (e r)f P (e r) f H;A (e r) g: Since we can express (2-18) as b E H;A [Q(v(e r)] E H;A [v(e r)] =ac E P [v(e r)] E H;A [v(e r)] ; we have cf f H;A (e r)f P (e r) f H;A (e r) g Q 0 (v(e r))E H;A [Q 0 (v(e r)] E H;A [Q(v(e r)] E H;A [v(e r)] (2-23) = cf1+ E H;A [Q 0 (v(e r)] Q 0 (v(e r))E H;A [Q 0 (v(e r)] f H;A (e r)f P (e r) f H;A (e r) E P [v(e r)] E H;A [v(e r)] g: 191 If I cancel c in (2-23) and rearrange it, I obtain f H;A (e r)f P (e r) f H;A (e r) = fQ 0 (v(e r))E H;A [Q 0 (v(e r)]gfE H;A [v(e r)]E P [v(e r)]g E H;A [Q(v(e r)]E H;A [Q 0 (v(e r)]E H;A [v(e r)] : (2-24) Thus, Q 0 (v(e r))E H;A [Q 0 (v(e r)]_ 1 1 LR(e r) ; gives us the solution, which I denote by v(e r): Notice that v(e r) does not depend on the parameters such as and e H but only on f H;A (e r); f P (e r) and the function Q =v 1 : A1.3. Proof for Proposition 4 As shown in (2-14), we have 10 unknowns: y, y+ (instead of ), (L;P);v L;P ; L;P ; L;P ; L;P ; H;A ; (H;A); v(e r): Let me rst derive their FOCs respectively in the listed order. y+ + 1 y = H;A + L;P L;P (3-1) y+ = H;A (3-2) 1 x 0 w 0 Q(v L;P ) v L;P = L;P 1 L;P (1)v L;P (3-3) 1 x 0 w 0 Q 0 (v L;P ) = L;P 1 (3-4) (L;P) 1 (3-5) (L;P)v L;P =y (3-6) (H;A)E P [v(e r)] =y (3-7) 192 (H;A)E H;A [v(e r)]e H =y+ (3-8) 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] = H;A L;P E P [v(e r)] E H;A [v(e r)] (3-9) Q 0 (v(e r)) x 0 w 0 = H;A L;P f P (e r) f H;A (e r) ; which becomes E H;A [Q 0 (v(e r))] x 0 w 0 = H;A L;P ; (3-10) after taking the expectation E H;A []: By (3-4) and (3-5), we require L;P > 0 sinceQ() is strictly convex. Then (3-5) should hold with equality, which gives us (L;P) = 1: (3-11) Then (3-11) and (3-6) suggest v L;P =y: (3-12) Using (3-12) and (3-4), we have L;P 1 = Q 0 (y) x 0 w 0 ; (3-13) which can be used for (3-3) along with (3-12) as in 1 x 0 w 0 Q(y) y = Q 0 (y) x 0 w 0 L;P (1)y : Thus, we obtain L;P (1) = Q 0 (y)yQ(y) x 0 w 0 : (3-14) 193 On the other hand, using (3-2), we have H;A = 1 y+ : (3-15) By (3-15), (3-9) can be written as 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] = 1 y+ L;P E P [v(e r)] E H;A [v(e r)] ; where E P [v(e r)] E H;A [v(e r)] = 1 +1 : Thus, we obtain L;P = 1 y+ 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] ! (+1): (3-16) Using (3-15) and (3-16), we can write (3-10) as E H;A [Q 0 (v(e r))] x 0 w 0 = 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] 1 y+ ! + 1 x 0 w 0 E H;A [Q(v(e r)] E H;A [v(e r)] ; which can be rearranged to E H;A [Q 0 (v(e r))] E H;A [Q(v(e r)] E H;A [v(e r)] = E H;A [Q(v(e r)] E H;A [v(e r)] x 0 w 0 y+ ! : (3-17) In (3-17), we can express both y+ and x 0 w 0 , using y: First, by (3-7), (H;A) = y E P [v(e r)] : (3-18) 194 Second, if we divide (3-8) by (3-7) +1 = y++e H y ; which gives us y+ = (+1)ye H : (3-19) By denition (2-15), Proposition 4, (3-11), and (3-12), x 0 w 0 =u S (H;A)E H;A [Q(v(e r))](1)Q(y); where we can replace (H;A) using (3-18). Then we arrive at x 0 w 0 = u S E H;A [Q(v(e r))] E P [v(e r)] y(1)Q(y) = u S y(1)Q(y) (3-20) where = E H;A [Q(v(e r))] E P [v(e r)] : Therefore, (3-17) can be rewritten as E H;A [Q 0 (v(e r))] E H;A [Q(v(e r)] E H;A [v(e r)] = E H;A [Q(v(e r)] E H;A [v(e r)] u S y(1)Q(y) (+1)ye H ! : (3-21) After rearrangement, (3-21) becomes yq()(u S w 0 (1)Q(y)+e H )e H = 0 (3-22) 195 where q() = 2 +((1+)+ )+ ; (3-23) and E HA [Q(v(e r))] E P [v(e r)] E HA [Q 0 (v(e r))]( > 0; see Lemma 3) as desired: A1.4. Proof for Proposition 5 Equation (19) is derived in (3-19) in the previous section. I need to show Equation (20): y @F(y;) @y > (+1) @F(y;) @ : (4-1) The LHS is given by yq()+(1)yQ 0 (y); which can be extended by (3-23): y(+1)+y+y(1) | {z } yq() +(1)yQ 0 (y): (4-2) On the other hand, by Equation (18), the RHS of (4-1) becomes +1 fy(q 0 ()q())+e H g = +1 fy( 2 )+e H g (4-3) where I use (3-23). 196 After rearranging (4-3), I obtain (+1)y +1 (ye H ) : (4-4) By Lemma 2, ye H > 0 and I restrict my attention to the set of parameters that > 0 by Lemma 3. Comparing (4-2) and (4-4) leads to (4-1) as desired. Lemma 3 E HA [Q(v(e r))] E P [v(e r)] E HA [Q 0 (v(e r))]> 0shouldholdforthesetofparameterssuch thatexpectedutilityforthepassivemanagerislargerthancostofactivemanagement e¤ort for the active manager (y >e H ) for some : Proof. Suppose not (it is negative) Then by Equation (18), @F(y;) @ =y (ye H )> 0; (4-5) which implies that @y @ < 0: In other words, expected utility for the passive manager is always less thane H sincey =e H when = 0: A1.5. Proof for Corollary 2 I derived the equation that y should satisfy in (3-22), which I call F; F =yq(;)(u S w 0 (1)Q(y)+e H )e H = 0; (5-1) 197 where q(;) = 2 +((1+) )+ : By the implicit function theorem, @y @ = @F @ @F @y ; where @F @y =q()+(1)Q 0 (y)> 0 (5-2) as shown in Proposition 4 in Section 3. Thus, the sign of @y @ is the opposite sign of @F @ ; which is given by @F @ = y @q(;) @ Q(y) = (yQ(y)): Note that as shown in (3-20), the investors expected utility is given by x 0 =u S y |{z} fees for active (1) Q(y) |{z} : fees for passive Thus, byx 0 =u S (yQ(y))Q(y);Icaninterpret yQ(y)asexpectedfeepremium for the active manager, which must be positive since the active manager receives more than the passive manager and costs of compensation increase with compensation. This proves that @F @ > 0; and thereby @y @ < 0: (5-3) 198 It is easy to see that @F @u S < 0; @F @w 0 > 0; so that @y @u S > 0; @y @w 0 < 0: (5-4) On the other hand, @F @e H = < 0; (5-5) and thus I show @y @e H > 0: (5-6) A1.6. Proof for Corollary 3 As shown (3-19), y+ = (+1)ye H ; which indicates that the comparative statics should have the same signs as those for y; except with respect to e H : In other words, by (5-2) and (5-3), @y+ @ < 0; @y+ @u S > 0; @y+ @w 0 < 0: In contrast, @y+ @e H = @y @e H 1 = @F @e H @F @y 1: 199 By (5-2) and (5-5), @y+ @e H = q()+(1)Q 0 (y) 1 = q()(1)Q 0 (y) q()+(1)Q 0 (y) = 2 (1+) (1)Q 0 (y) q()+(1)Q 0 (y) < 0: A1.7. Proof for Corollary 4 I already discussed that y = e H (since = 0 at and by Equations (22)), where y is expected utility for the passive manager when = : In Equation (23), I replace = e H y and e H =y to obtain (looking at y is convenient) (1+) = 1 y (u S w 0 (1)Q(y))y e H | {z } decreasing in y : .Notice that the RHS is decreasing in y; which indicates that it should be increasing in : When u S increases, the RHS increases but the LHS does not change. We should have @ @u S < 0: Similarly, one can show that @ @w 0 > 0; @ @ > 0; @ @e H > 0: 200 A1.8. Condition for separating equilibrium I compare conditions for separating equilibrium between two approaches: maximiz- ing investorssurplus and maximizing the sum of managersand investorssurpluses. For simplicity, I assume no asymmetric information. Investor s problem Pooling equilibrium maxx 0 w 0 subject to x 0 w 0 0; y H 0; y L 0; where x 0 is the investors expected utility, given by x 0 =u P Q(y H )(1)Q(y L ): Here, u P is the utility from investments when there is only passive funds. High (low) skill managers expected utility is denoted by y H (y L ): The investors cost for giving utility y H is equal to Q(y H ) by Q() =v 1 (): Obviously, the participation constraints are binding, and the solution is given by y H =y L = 0; 201 and the investor receives, by Q(0) = 0 (Assumption 2), u P w 0 : (6-1) Separating equilibrium max x 0 w 0 subject to x 0 w 0 0; y H 0; y L 0; where x 0 is the investors expected utility, given by x 0 =u S Q(y H +e H )(1)Q(y L ); where u S is the utility from investments when there are two types of funds, active and passive. High (low) skill managers expected utility is denoted by y H (y L ): The investors cost for giving utility y H is equal to Q(y H +e H ) since y H =v(m)e H and Q() =v 1 (): Similar to the pooling case, the constraints are binding, and thereby, y H =y L = 0: The investors expected utility is, by Q(0) = 0; u S Q(e H ): (6-2) 202 Condition for separating equilibrium Comparing (6-1) and (6-2), we have the condition u S u P Q(e H ): (6-3) Planner s problem Pooling equilibrium max x 0 w 0 +y H +(1)y L subject to x 0 w 0 0; y H 0; y L 0; where x 0 is the investors expected utility, given by x 0 =u P Q(y H )(1)Q(y L ); whereu P istheutilityfrominvestmentswhenthereisonlyonetypeoffunds, passive. High (low) skill managers expected utility is denoted by y H (y L ): The investors cost for giving utility y H is equal to Q(y H ) by Q() =v 1 (): Assuming Q 01 (1) 0; 203 we have interior solutions for y H and y L : Then the FOCs with respect to y H (y L ) are Q 0 (y H )+ = 0 (1)Q 0 (y L )+(1) = 0; which lead to Q 0 (y H ) = Q 0 (y L ) = 1 y H = y L =Q 01 (1): Denoting the solution for y H and y L by y ; the social welfare is equal to u P w 0 Q(y )+y : (6-4) Separating equilibrium max x 0 w 0 +y H +(1)y L subject to x 0 w 0 0; y H 0; y L 0; where x 0 is the investors expected utility, given by x 0 =u S Q(y H +e H )(1)Q(y L ); 204 where u S is the utility from investments when there are two types of funds, active and passive. High (low) skill managers expected utility is denoted by y H (y L ): The investors cost for giving utility y H is equal to Q(y H +e H ); since y H =v(m)e H and Q() =v 1 (). Assuming Q 01 (1)e H ; (6-5) we have interior solutions for y H (y L ). Their FOCs are Q 0 (y H +e H )+ = 0 (1)Q 0 (y L )+(1) = 0; which lead to Q 0 (y H +e H ) = Q 0 (y L ) = 1 (6-6) y H +e H = y L =Q 01 (1): Notice that the low skill manager still gets the same utility as in the pooling case whereas the high skill manager gets less. Denoting the solution for y L by y ; the social welfare is equal to u S w 0 Q(y )+y e H : (6-7) 205 Condition for separating equilibrium Comparing (6-4) and (6-7), we have the condition u S u P e H : (6-8) Comparison between the planner s and the investor s problems To compare the separating equilibrium conditions (6-3) and (6-8), I show that e H >Q(e H ) (6-9) by the following. Since Q() is increasing, (6-6) implies Q 0 (e H ) 1: (6-10) On the other hand, the mean-value theorem implies Q(e H ) e H =Q 0 (b e) (6-11) for some 0<b e<e H : Since Q 0 () is also increasing by convexity, Q 0 (b e)<Q 0 (e H ) 1; 206 where the last inequality follows by (6-10). Then (6-11) implies Q(e H ) e H < 1; so that (6-9) holds. Therefore, when Q(e H )<u S u P <e H ; the separating equilibrium is preferred by the investor, but the pooling equilibrium can increase the sum of surpluses of the investor and the manager. Appendix A2. A2.1. Kernel regression Idiscusskernelregressions. Givendataofpairsoffundowsandpastperformance,ker- nel regressions estimate the functiong(performance) as a weighted sum of fund ows. The weight for observed fund ow is large when the corresponding performance is close to the conditioning value ofperformance: More specically, the weight is proportional to the nor- mal density (kernel function) with the mean equal to the conditioning value performance and the standard deviation equal to a bandwidth. Asymptotically, the estimates do not depend on kernel functions used for weights under some conditions, and normal, uniform, and Epanechnikov kernels are often used. I select optimal bandwidths using the cross-validation method (minimize integrated 207 squarederrors). Whenweuseabandwidthhthatgoestozeroasthenumberofobservations n increases to innity but not as fast as n (nh!1), the estimated function converges to thetrueoneinprobability. Thecross-validationmethodofchoosingbandwidthisconsistent with these criteria and preferred by researchers despite its high computational costs. On the other hand, xed bandwidths (i.e., arbitrarily choosing bandwidths) or a rule of thumb method for bandwidths may not satisfy those criteria for consistency. Since the cross- validation method minimizes integrated squared errors, it also results in more e¢ cient estimates. Many studies use simple averages to estimate g(performance i;t1 ). For example, Sirri and Tufano (1998) and Huang, Wei and Yan (2007) rank past performance into 20 and 10 bins respectively and use equally-weighted averages of fund ows in each bin. This method may be understood as kernel regressions with the uniform kernel. In this case, bandwidths do not decrease as sample size increases but vary across bins. Take an example of 10 bins. Thenweestimateexpectedfundowsfor10valuesofperformancewhicharethemidpoints ofrangesofbins. Thebandwidthforthebinbishalfofthebinsrange. Sincetherangesdo not shrinkas the sample size increases, the bandwidth does not go to zero. Thus, estimated ow-performance relationships may not converge to the true function in probability. A2.2. The e¤ect of the expense ratio on fund ows Figure A2.1 shows the coe¢ cient on the expense ratio over time. It is hard to make an inference due to the wide condence intervals. Yet, the coe¢ cient is negative in most of the yearsinthepost-2000period. Thetime-seriespatterndoesnotlooktoberelatedtomarket volatility. In fact, the interaction term between the expense ratio and market volatility is 208 insignicant, as is the interaction term between the expense ratio and performance disper- sion(notreported). Theseresultssuggestthatmarketvolatilityandperformancedispersion cannot explain the variation in the e¤ect of the expense ratio on net ows. Rather, the negative e¤ect of the expense ratio on fund ows after 2000 seems to be associatedwithadecreaseinthefractionof12b-1feesafter1999. Ilookataveragefractions of 12-1 fees over time, using a sample of nonindex equity funds with nonmissing 12b-1 fees in the CRSP database (the data starts in 1992). 34 Figure A2.1. Coe¢ cient on the expense ratio The blue line represents the estimates on the expense ratio after running cross-sectional regres- sions of annual net ows for U.S. nonindex mutual funds on the lagged excess return, its square, expense ratio, and other control variables in each year. The dash lines are their 90% condence in- tervals. The red dash line is lagged market volatility (annualized standard deviation of daily return on the CRSP VW index). I allow the coe¢ cient on the expense ratio to vary according to the TNA-weighted proportion of 12b-1 fees in the prior year. I nd that expense ratios have a negative e¤ect 34 The CRSP database uses 0 (zero)when 12b-1 information is missing in the data years until 2000. 209 onfundows,butthenegativee¤ectdecreasesastheaveragefractionof12b-1feesincreases. When 12b-1 fees count for more than 30% of fund expenses on averageas in the pre-2000 periodexpense ratios have a positive e¤ect on fund ows (Table A2.1). Table A2.1. Determinants of the coe¢ cient on the expense ratio The dependent variable is annual net fund ows in the year t and the independent variables are listed in the rst column. Numbers for each independent variable are estimates, standard errors, andp-valuesrespectively. Thestandarderrorsareclusteredbyyearandfund. Performanceisexcess returns over the CRSP VW returns. Squared performance is square of performance. This variable has interaction terms with ve conditioning variables. Market ret state is -1, 0, and +1 when the mean of daily returns on the CRSP VW index over the year is in the low, middle, and high group respectively. Similarly, market vol state is -1 (low volatility), 0 (medium), and +1 (high) according to the standard deviation of daily returns; and industry ow state is equal to -1 (low), 0 (medium), and +1 (high demand), depending on net ows to all equity mutual funds in the CRSP database (including sector funds and international funds). Performance dispersion is the residual obtained from the regressions of the cross-sectional standard deviation of performance on the mean and the volatility of the daily CRSP VW returns. The second period is one if the year is between 2000 and 2008andzerootherwise. Theexpenseratiohasaninteractiontermwiththe12b-1proportion,which is the TNA-weighted average fraction of 12b-1 fees in the expense ratio. The variables not presented in the table include performance (t); performance (t-2); log age (t-1); log size (t-1); fund volatility (t-1); industry ow (t); style ow (t); and 12b-1 proportion (t-1). The number of observations is 17,679 over 1983 to 2008. Fund variables are aggregated across share classes, excluding institutional share classes (The complete results are available upon request). 210 (1) (2) (3) (4) (5) (6) (7) performance (t-1) 1.023 0.781 0.801 0.809 0.856 0.886 0.888 (0.167) (0.155) (0.164) (0.164) (0.148) (0.143) (0.145) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) squared performance (t-1) -0.368 -0.375 0.891 0.782 0.647 0.762 1.595 (0.085) (0.072) (0.803) (1.244) (1.303) (1.270) (1.844) (0.000) (0.000) (0.267) (0.530) (0.620) (0.549) (0.387) *market ret state (t-1) -0.157 -0.516 -0.390 -0.583 (0.453) (0.475) (0.311) (0.508) (0.728) (0.278) (0.211) (0.252) *market vol state (t-1) -1.034 -1.462 -1.059 -0.691 (1.192) (1.260) (1.283) (1.116) (0.386) (0.246) (0.409) (0.536) *performance dispersion (t-1) 5.953 8.271 8.568 (2.594) (1.070) (1.098) (0.022) (0.000) (0.000) *industry ow (t) 1.033 0.550 (0.396) (0.704) (0.009) (0.435) *second period (t) -1.292 -1.537 (0.820) (1.735) (0.115) (0.376) expense ratio (t-1) -85.259 -52.731 -51.189 -51.851 -43.434 -38.461 -36.610 (21.514) (18.662) (18.421) (16.823) (13.545) (13.149) (13.196) (0.000) (0.005) (0.006) (0.002) (0.001) (0.004) (0.006) *12b-1 proportion (t-1) 285.701 175.104 169.483 171.009 143.285 126.827 120.703 (72.279) (63.639) (62.703) (56.965) (46.659) (45.424) (45.261) (0.000) (0.006) (0.007) (0.003) (0.002) (0.005) (0.008) ow (t-1) 0.258 0.257 0.258 0.260 0.260 0.260 (0.031) (0.031) (0.031) (0.031) (0.031) (0.031) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Adjusted R squared 0.139 0.217 0.217 0.217 0.220 0.222 0.222 211 A survey by the Investment Company Institute (ICI) in 2004 nds that 92% of 12b-1 fees are used for paying nancial advisers, and only 2% for advertising and promotion. An ICI report (Fundamentals,14 (2), 2005) argues that mutual funds shifted the way of compensating nancial advisors from front-end load fees to 12b-1 fees. FigureA2.2. (A)showstheaveragefractionof12-1bfeesandtheaverageexpenseratio from 1983 to 2008 (both variables are lagged). When weighted by fund size (TNA), the lagged 12b-1 proportion is about 31% before 2000. Yet, it decreases to 29% after 2000. The ICI report argues that the total amount of 12b-1 fees increased from $7 billion in 1998 to $11billion after2000(Fundamentals,14 (2), February 2005, ICI). This dramatic increase in total amount of 12b-1 fees is not because of an increase in 12b-1 fees but because of an increase in fund assets (see Figure A2.2. (B)). Giventhatmost12b-1feesarepaidtonancialadvisers,thepositivecoe¢ cientsonthe interaction term between expense ratio and the fraction of 12b-1 fees suggest that nancial advisers help attract ows. As direct sales become more prevalent in the 2000s, the role of thoseintermediariesbecomeslessimportant. Asaresult, adecreaseinthefractionof12b-1 fees is associated with a negative impact of the expense ratio on fund ows after 2000. Ontheotherhand,thelaggedaverageexpenseratiosdecreasesince2005. Inparticular, thelaggedTNA-weightedexpenseratiois95basispointin2004,whichdecreasesto77basis point in 2008. Given a fund size of $3 billion, the di¤erence in expenses is about $5 million peryear. Finally, theTNA-weightedexpenseratioislowerthanthesimpleaverageexpense ratio, but the fraction of 12b-1 fees is higher when weighted by fund size. This suggests that large funds tend to charge lower expense ratios but higher 12b-1 fees. 212 Figure A2.2. Expense ratio and 12b-1 proportion The sample consists of equity nonindex funds with nonmissing data for 12b-1 fees in the CRSP database. (A) Equally weighted (average) or value-weighted (TNA-weighted) expense ratio and 12- 1 proportion in the expense ratio in the prior year. (B) Total assets under management in billion dollars and total 12b-1 fees in million dollars. 213 Appendix A3. A3.1. Consumption-wealth ratio I start from the budget constraint W t+1 = R w;t+1 (W t C t ) (7-1) W t+1 W t = R w;t+1 (1 C t W t ) (7-2) where W t is the aggregate wealth in period t, R w;t+1 is the gross return on aggregate wealth and C t is the consumption. I take the natural logarithm of Equation (7-2) and use lower case letters to denote the logs: w t+1 w t =r w;t+1 +log(1exp(c t w t )): (7-3) Assuming the log consumption-wealth ratio does not vary much, I can approximate the last element of RHS of (7-3) by a rst Taylor expansion around a stationary level cw = log( C W ), following Campbell (1993), as log(1exp(c t w t )) exp(cw) 1exp(cw) ((c t w t )(cw)): (7-4) Letmek representtheremainingconstanttermsinRHSof(4)and w thesteady-state investment to wealth ratio, WC W : This allows me to express as log(1exp(c t w t )) C W 1 C W (c t w t )+k = (1 1 w )(c t w t )+k: (7-5) 214 Then I can approximate (7-3) using (7-5) and then solve forward the rst di¤erential equation: w t+1 w t r w;t+1 +(1 1 w )(c t w t )+k w (w t+1 w t ) w r w;t+1 +( w 1)(c t w t )+ w k c t w t w r w;t+1 w (w t+1 w t )+ w (c t w t )+ w k c t w t w r w;t+1 + w (c t+1 w t+1 c t+1 +c t c t +w t )+ w (c t w t )+ w k c t w t w r w;t+1 + w (c t+1 w t+1 ) w (c t+1 c t )+ w k where c t+1 w t+1 w r w;t+2 + w (c t+2 w t+2 ) w (c t+2 c 2 )+ w k ::: c t w t 1 X i=1 i w (r w;t+i c t+i )+ lim n!1 n w (c t+n w t+n )+ 1 X i=1 i w k: (7-6) Finally, if I use lim n!1 n w (c t+n w t+n ) = 0 and omit the constant term for simplicity in the RHS of (7-6), I can get the equation (1) in Section 3: c t w t 1 X i=1 i w (r w;t+i c t+i ): A3.2. Stationary component of human capital If I consider labor income (Y) as dividend of human capital, R h;t+1 H t =H t+1 +Y t+1 : 215 where R h;t+1 is the gross return on human capital H t . Dividing both sides by Y t+1 and then taking log gives R h;t+1 H t Y t+1 = H t+1 Y t+1 +1 r h;+1 +h t y t+1 = log( H t+1 Y t+1 +1) = log(exp(h t+1 y t+1 )+1): Denoting Ht Yt =Z t ; we have h t =y t +z t and, thereby, r h;t+1 +y t y t+1 +z t = log(exp(h t+1 y t+1 )+1): (8-1) Assuming z t is stationary, I can use a Taylor approximation around its steady-state level, hy; to (8-1), r h;t+1 +y t y t+1 +z t log(exp(hy)+1)+ exp(hy) exp(hy)+1 (h t+1 y t+1 (hy)) b+ exp(hy) exp(hy)+1 (h t+1 y t+1 ) b+ exp(hy) exp(hy)+1 z t+1 where b is the remaining constant. Dene exp(hy) exp(hy)+1 = H Y H Y +1 = H H+Y = h : I have the 216 approximate expression for z t : z t (r h;t+1 y t+1 )+b+ h z t+1 (r h;t+1 y t+1 )+b+ h (k(r h;t+2 y t+2 )+b+ h z t+2 ) (r h;t+1 y t+1 )+b+ h ((r h;t+2 y t+2 )+b+ 2 h z t+2 )) 1 P j=0 j h (r h;t+1+j +y t+1+j )+v where v is the remaining constant, assuming lim n!1 n h = 0: Then the change in z t is: z t 1 P j=0 j h (r h;t+1+j + 2 y t+1+j ): A3.3. R squared If I regress changes in returns, r t+1 r t ; on a t ; I get the R squared, R 2 = Var(a t ) var(r t+1 r t ) = cov(r t+1 r t ;a t ) 2 var(r t+1 r t )var(a t ) : If r t is uncorrelated over time, the above R squared is approximately equal to R 2 = fcov(r t+1 ;a t )cov(r t ;a t )g 2 2var(r t+1 )var(a t ) = cov(r t+1 ;a t ) 2 2cov(r t+1 ;a t )cov(r t ;a t )+cov(r t ;a t ) 2 2var(r t+1 )var(a t ) = cov(r t+1 ;a t ) 2 2var(r t+1 )var(a t ) cov(r t+1 ;a t )cov(r t ;a t ) var(r t+1 )var(a t ) + cov(r t ;a t ) 2 2var(r t+1 )var(a t ) : 217 Nowdenethecorrelationcoe¢ cientbetweenthecontemporaneousequityreturnsand assetgrowth,andthatoftheequityreturnsandlagofassetgrowthby 0 and 1 respectively. Then I can express as cov(r t+1 ;a t )cov(r t ;a t ) var(r t+1 )var(a t ) = cov(r t+1 ;a t ) p var(r t+1 ) p var(a t ) cov(r t ;a t ) p var(r t ) p var(a t ) = 1 0 and cov(r t ;a t ) 2 var(r t+1 )var(a t ) = cov(r t ;a t ) p var(r t ) p var(a t ) ! 2 = 2 0 : These lead to R 2 = 1 2 R 2 1 0 + 1 2 2 0 ; where R 2 is the R squared when I regress r t+1 on a t : Thus, I have the R 2 as given by R 2 = 2R 2 ( 2 0 2 1 0 ): Usingb 0 = 0:8803 andb 1 = 0:0175 over the sample period from 1952 to 2006, I have R 2 = 2R 2 0:7441: A3.4. Clark and West (2006, 2007) adjustment Considerapredictionfory foranout-of-sampleperiodP fromTP+2toT+1using a rolling estimate method. Dene the out of sample mean squared prediction (MSPE) as b 2 = 1 P T P t=TP+1 e 2 t+1 ; 218 where e t+1 =y t+1 b y t+1 : I may test equal mean squared prediction error (MSPE) between two models (null and alternative) out of sample. Clark and West (2006, 2007), however, alarms for the use of a standard t-test for the di¤erence between two MSPE, b 2 0 b 2 1 = 1 P T P t=TP+1 e 2 0;t+1 1 P T P t=TP+1 e 2 1;t+1 : where e 2 0;t+1 is the squared error of a null model and e 2 0;t+1 is the squared error of an alternative model. Clark and West show that, in many cases, a t-test for this di¤erence results in statistics whose distribution is centered around a negative value. This distortion in nite sample increases as an out-of-sample period P increases. Therefore, Clark and West suggest to test if 1 P T P t=TP+1 2e 0;t+1 (e 0;t+1 e 1;t+1 ) is zero. This adjusted di¤erence of MSPE is based on the observation that, in nite sample, b 2 2 b 2 1 ;islikelytobenegativebecauseadditionalvariablesofanalternativemodelintroduce noise in a prediction process. Consider a simple example. A null model is that the variable y is white noise while an alternative model states that it can be predicted using predictors X. The di¤erence 219 between MSPE is, then, b 2 2 b 2 1 = 1 P T P t=TP+1 (y t+1 ) 2 1 P T P t=TP+1 y t+1 X 0 t b t 2 = 1 P T P t=TP+1 (y t+1 ) 2 1 P T P t=TP+1 (y t+1 ) 2 2y t+1 X 0 t b t + X 0 t b t 2 = 1 P T P t=TP+1 2y t+1 X 0 t b t 1 P T P t=TP+1 X 0 t b t 2 : (9-1) Under the null that y t+1 =" t+1 ; I expect E[y t+1 X 0 t b t ] = E[e t+1 X 0 t b t ] = 0 since e t+1 is notcorrelatedwithX:Thenthersttermintheequation(9-1)is 1 P T P t=TP+1 2y t+1 X 0 t b t 0; in nite sample by the law of large numbers. However, the second term may not be zero but negative even though the null is true due to noise in nite sample. Therefore, I may subtract the second term and then test if the following adjusted di¤erence is zero: b 2 2 b 2 1 adjusted = 1 P T P t=TP+1 2y t+1 X 0 t b t = 1 P T P t=TP+1 2y t+1 n y t+1 (y t+1 X 0 t b t ) o = 1 P T P t=TP+1 2e 0;t+1 (e 0;t+1 e 1;t+1 ): (9-2) When the null model is more general, Clark and West (2007) show that a similar argument also applies and suggest to do a t-test for the equation (9-2). 220

Abstract (if available)

Abstract This dissertation consists of two chapters that examine agency issues in delegated portfolio management and one chapter that studies forecasts of stock market returns.

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University of Southern California Dissertations and Theses

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Creator Kim, Min Seon (author)

Core Title Essays on financial markets

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 06/24/2010

Defense Date 04/13/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag delegated portfolio management,flow-performance relationship,incomplete information,mutual funds,OAI-PMH Harvest,predictive regression,risk-shifting

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ferson, Wayne (committee chair), Alexander, Kenneth S. (committee member), Jones, Christopher S. (committee member), Westerfield, Mark (committee member), Zame, William (committee member), Zapatero, Fernando (committee member)

Creator Email minkim2010@gmail.com,minskim@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3154

Unique identifier UC190646

Identifier etd-Kim-3740 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-356414 (legacy record id),usctheses-m3154 (legacy record id)

Legacy Identifier etd-Kim-3740.pdf

Dmrecord 356414

Document Type Dissertation

Rights Kim, Min Seon

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