Delegation Uncertainty in the Era of Big Data

Transcription

1 Delegation Uncetainty in the Ea of Big Data Ye Li Chen Wang Octobe 10, 2018 Abstact Big data analysis equies expetise and ceates a division of labo and knowledge. We delegate tasks to expets fo thei skills in data collection and analysis, but delegation caies an intinsic fom of uncetainty its outcome depends on expets knowledge unknown to us. This pape studies delegation uncetainty in financial makets. The theoy offes explanation fo seveal puzzles on delegated potfolio management. It also poduces asset picing implications suppoted by ou empiical analysis: 1 CAPM alpha aises because investos patially delegate and hedge against delegation uncetainty; 2 the coss-section dispesion of alpha inceases in uncetainty; 3 manages bet on alpha, but alpha is immune to the ise of thei abitage capital delegation hedging is stonge when investos delegate moe. Finally, we offe a novel appoach to measue uncetainty fom asset etuns, delegation, and suvey expectations. We would like to thank Andew Ang, Patick Bolton, Jaoslav Boovi cka, Zhi Da discussant, Stefano Giglio, Las Pete Hansen, Gu Hubeman, Michael Johannes, Igo Makaov, Tano Santos, Thomas Sagent, José Scheinkman, Andei Shleife, and Zhenyu Wang fo helpful comments. We ae also gateful to confeence and semina paticipants at CEPR ESSFM Gezensee and Geneva Wokshop on Financial Stability in a New Ea. All eos ae ous. This pape was peviously ciculated unde the title Ambiguity and Delegated Potfolio Management, Columbia Business School Reseach Pape No The Ohio State Univesity. li.8935@osu.edu Yale School of Management. ch.wang@yale.edu

2 1 Intoduction Technological pogess is often accompanied by a division of labo. In the ea of big data, a division of knowledge emeges, and induces delegation: we hie data expets fo tasks that equie skills in data collection and analysis. 1 Howeve, delegation caies an intinsic fom of uncetainty, even in the absence of moal hazad. The delegation outcome depends on the expets supeio infomation that is unknown to us. This pape studies the implications of delegation uncetainty on delegated potfolio management and asset picing. The asset management industy is being evolutionized by exploding data souces and inceasingly sophisticated analytical techniques that help money manages bette estimate the pobability distibution of asset etuns. In contast, investos face the difficulty in gauging pobabilities. We model two types of agents: manages who know the etun distibution, and investos who face model uncetainty ambiguity given by a set of pobability distibutions models. Investos pay manages to allocate pat of thei wealth, and allocate the etained wealth unde ambiguity. 2 We abstact away fom moal hazad, which is studied by Miao and Rivea 2016 in a simila setup of heteogeneous belief. Hee manages use thei pobability knowledge to allocate the delegated wealth dutifully on the efficient fontie. Delegation impoves investos welfae by educing thei exposue to ambiguity in the etuns of individual assets. As in Gennaioli, Shleife, and Vishny 2015, such welfae view esolves impotant puzzles in the asset management liteatue. Fo example, we chaacteize conditions unde which delegation happens even when manages undepefom the maket o delive zeo alpha by holding potfolios popotional to the maket. Howeve, delegation uncetainty emains even though manages delive the efficient potfolio, the efficient fontie vaies acoss pobability models. Investos hedge the delegation uncetainty when allocating the etained wealth. Thei potfolio tilts towads away fom assets whose etuns move against with the fontie acoss models. Delegation hedging geneates CAPM alpha, and the coss-section dispesion of alpha inceases in investos model uncetainty. Moeove, the alpha of hedging assets is immune to the ise of abitage capital, i.e., the wealth allocated by manages, because investos hedging against delegation uncetainty becomes stonge when they delegate moe. 1 The division of knowledge is a tem boowed fom Hayek 1945 who descibe the impotance of pice system in sustaining the economic ode in wold with a division of labo and equally divided knowledge. 2 The management fee may epesent the manages wages, agency cost, seach and sceening costs, elative bagaining powe, o othe inefficiencies not modeled in the pape but elevant to delegation in eality. 1

3 In ou model, pofessional asset manages and investos ae diffeent in thei knowledge of etun distibution. To highlight such division of knowledge, we assume that investos do not lean the pobability distibution by obseving manages allocation in asset makets, and that manages cannot infom investos the tue etun distibution, which is in line with the difficulty in eality to explain the ationale and techniques behind investment stategies. 3 We povide closed-fom solutions fo investos delegation and the coss section of expected asset etuns by solving a quadatic appoximation of investos utility unde ambiguity. 4 As a technical contibution, ou appoximation extends that of Maccheoni, Mainacci, and Ruffino 2013 into functional spaces. When delegation is unavailable, and investos ae ambiguity-neutal, ou appoximation becomes the classic Aow-Patt appoximation, which geneates the mean-vaiance potfolio of Makowitz 1959 and a CAPM equilibium. In ou setup, delegation offes investos model-contingent allocation of wealth. Asset manages can be viewed as potfolio fomation machines with the knowledge of tue etun distibution as input and the coesponding efficient potfolio as output. In investos mind, the oveall stuctue of uncetainty is a two-step lottey: fist, a pobability model is dawn, obseved by manages who then allocates the delegated wealth on the fontie; second, a state of the wold is dawn accoding to the pobability model. Thus, the delegated potfolio is model-contingent, and though it, the etun fom delegation is both state- and modelcontingent. In contast, investos allocation of thei etained wealth is only state-contingent, because they cannot condition thei potfolio choice on the tue pobability model. Such model-contingency induced by delegation has two consequences that have implications espectively on delegation and the coss section of asset etuns. Fist, model-contingent allocation impoves investos welfae by allowing them to access the efficient potfolio unde each pobability model whicheve model is tue, manages know it and allocate efficiently. Investos optimal level of delegation depends on thei model uncetainty, the coss-model vaiation of fontie, management fee, and pefeence paametes, such as isk and ambiguity avesion. Appendix IV illustates how ou famewok can be used as a nomative model to 3 Leaning unde model uncetainty ambiguity has been studied by Epstein and Schneide 2007, and in the asset picing liteatue, Leippold, Tojani, and Vanini 2008, Ju and Miao 2012 and Choi Mele and Sangiogi 2015 study agents infomation acquisition unde Knightian uncetainty. Pásto and Stambaugh 2012 study how investos Bayesian leaning affects thei delegation decision. 4 We assume smooth ambiguity avesion utility function poposed by Klibanoff, Mainacci, and Mukeji 2005 and Nau 2006 and discussed by Epstein 2010 and Klibanoff, Mainacci, and Mukeji Ghiadato, Maccheoni, and Mainacci 2004 take an axiomatic appoach to study the sepaation between ambiguity and agents attitude towads ambiguity. 2

4 guide the delegation choice of investos unde model uncetainty. This new pespective on delegated asset management explains seveal puzzling findings in the liteatue, such as delegation in spite of undepefomance elative to the maket index. Investos cannot evaluate fund pefomances unde ational expectation, so econometicians measuements of pefomances ae based upon an infomation set diffeent fom investos. How delegation impoves welfae depends on the type of model uncetainty that investos face. We chaacteize conditions unde which delegation aises even though manages may undepefom the maket, delive negative alpha, o simply hold potfolios popotional to the maket potfolio Fama and Fench 2010; Lewellen Ou focus on subjective welfae echoes that of Gennaioli, Shleife, and Vishny The second consequence of model-contingency is the esulting delegation uncetainty. Investos ae avese to the coss-model comovement between assets and the efficient fontie i.e., the delegated potfolio. Hedging against delegation uncetainty induces a two-facto stuctue in the expected asset etuns: a typical CAPM isk pemium, and a model uncetainty pemium alpha that inceases in the level of delegation and model uncetainty. We would expect the alpha to convege to zeo if the economy appoaches full delegation e.g., diven by declining management fees, because manages who hold the mean-vaiance potfolio almost dominate the asset makets. Howeve, this is not the case. The moe investos delegate, the stonge they hedge against delegation uncetainty pe dolla of etained wealth. The inceasing hedging motive counte-balances the deceasing shae of wealth managed by investos themselves, which sustains the CAPM alpha. Theefoe, ou model sheds light on why cetain investment stategies still delive alpha in spite of the gowing abitage capital, i.e., the money managed by pofessionals who know those anomalies. Ou model delives othe asset-picing implications. The maket isk pemium declines in the level of delegation, which suggests that as the asset management secto gows, the secuity maket line will be inceasingly flat. Following Bewley 2011, we simplify investos model uncetainty by elating it to the statistical eos in paamete estimation. The oveall level of ambiguity, and investos sentiment, which is diectly mapped to suvey expectations, emege as the key deteminants of the coss-section vaiation of asset etuns. 5 is late used to extact ambiguity fom asset etun and suvey data. This setup 5 Ou model does not have limits to abitage, but due to delegation uncetainty, investos sentiment suvives in the expected asset etuns even if the level of wealth pofessionally managed appoaches 100%. 3

5 We test the model assumptions and asset picing implications using the U.S. equity factos that ae well studied in the liteatue of empiical asset picing. We use factos athe than individual stocks because a pasimonious facto stuctue spans stock etuns. 6 Fist, we find manages pefom facto timing, which suggests that they have a bette knowledge of etun distibution. Evey quate, we sot factos by thei institutional fund owneship. Factos with high fund owneship consistently outpefom those with low fund owneship. Paametic tests based on facto etun pediction confim this finding. A one standad deviation incease of fund owneship adds 1.76% annualized to a facto s futue etun, which tanslates to a 53% incease ove the aveage facto etun in ou sample. Second, factos with highe fund owneship have lage Shape atio. Finally, the CAPM alpha of factos with high fund owneship is consistently above zeo in olling windows, in spite of the ising delegation level in the U.S. stock maket. This is consistent with the model pediction that investos hedging against delegation uncetainty sustains alpha. In ou model, assets alpha is fom investos hedging against delegation uncetainty. A paticula implication is that when the level of model uncetainty inceases, the hedging becomes stonge, geneating a lage coss-section dispesion of alpha. This is confimed in data. Vaious measues of uncetainty positively pedict the coss-section dispesion of factos CAPM esiduals and aw etuns, with dispesion measued by the diffeence between maximum and minimum o by the coss-section standad deviation. The pedictive coefficient is sizable. Fo example, an 1% one standad-deviation incease of U CSA t uncetainty measue fom Juado, Ludvigson, and Ng 2015 pedicts a 0.46% annualized to 5.52% incease of the coss-section standad deviation of factos CAPM esiduals. Finally, we calculate investos model uncetainty by fitting the equilibium conditions of asset makets diectly to the data of facto etuns, delegation, and suveys of investos expectations Geenwood and Shleife The model-implied uncetainty exhibits cyclical dynamics and peaks aound maket tumoils, such as the dot-com bubble and the Geat Recession. Ou measue contains infomation distinct fom altenative uncetainty measues in the liteatue, but exhibits comovement. The coelation anges fom 0.15 to Among a lage set of fim chaacteistics that have been poposed to pedict etuns in the coss section, Hou, Xue, and Zhang 2015 show that a fou-facto model summaizes the coss section of aveage etuns, Feybege, Neuhiel, and Webe 2017 identify a small subset that povide distinct infomation, and Kozak, Nagel, and Santosh 2017 find the pincipal components appoximate the stochastic discount facto well. an 4

6 Liteatue. Ou pape contibutes to the ambiguity liteatue Hansen and Sagent Ambiguity also called Knightian uncetainty is the lack of knowledge of pobability distibution, and can be about uncetainty in pobability model o specific paametes Knight Ellsbeg paadox is an example of ambiguity-avese behavio. 7 Widely cited as a challenge to the expected utility theoy Dow and Welang 1992, ambiguity avesion has been intoduced in vaious fields in economics, such as asset picing e.g., Boyachenko 2012, Cao, Wang, and Zhang 2005, Chen and Epstein 2002, Epstein and Wang 1994, Galappi, Uppal, and Wang 2007, Hovath 2016, Maenhout 2004, Illeditsch 2011, Ilut 2012, Ju and Miao 2012, eal option Miao and Wang 2011, copoate govenance Izhakian and Yemack 2017, maket micostuctue Condie and Ganguli 2011; Easley and O Haa 2010; Ozsoylev and Wene 2011; Vitale 2018, secuitization Andeson 2015, and policy intevention in cises Caballeo and Kishnamuthy Epstein 2010 and Guidolin and Rinaldi 2010 eview the liteatue. Ou setup is a special case of the multi-agent envionments discussed by Hansen and Sagent Hee one type of agents, investos, face model uncetainty, while the othe type, manages, do not. Closely elated, Miao and Rivea 2016 study optimal contacting between a pincipal, who faces model uncetainty, and an agent, who does not. The model delives ich implications on copoate finance. We diffe by abstacting away moal hazad, and focus instead on the delegation uncetainty intinsic to the division of knowledge and its asset picing implications. Delegation unde ambiguity, and the associated incentive poblems, ae also studied by Fabetti, Hezel, and Ç. Pına 2014 and Rantakai Related to ou setup, Hishleife, Huang, and Teoh 2017 study whethe investos maket paticipation can be impoved by intoducing funds whose allocation is contingent upon ambiguous asset supply. Ou pape diffes by focusing on investos hedging against delegation uncetainty, and its implications on the coss-section vaiation of expected asset etuns. This pape contibutes to the asset picing liteatue by chaacteizing a hedging demand that aises fom the heteogeneity of pobability knowledge. The insight of ambiguity hedging is shaed with Dechsle 2013, who show that investos pay a pemium fo index options to hedge model misspecification. We also offe an altenative decomposition of expected etun, and show that the pices of model uncetainty and maket isk both depend on the endogenous level of delegation. We show that ou esults on asset picing hold even 7 A vesion of it was noted by John Maynad Keynes in his book A Teatise on Pobability

7 when investos ae not ambiguity-avese but face ambiguity, which is in stak contast to the existing liteatue e.g., Benne and Izhakian 2017, Epstein and Schneide 2008, Kogan and Wang 2003, Tojani and Vanini 2004, Ui Though delegation, investos etun on wealth is both state- and model-contingent, so even ambiguity-neutal investos cannot aveage out model uncetainty fo each state of the wold, acting as expected-utility agents. Instead, they ae foced to face the joint uncetainty in both the state and model space. We ae the fist to show that delegation aises fom ambiguity, and at the same time, fundamentally changes the ole of ambiguity in agents decision making. 8 Moeove, we identify assets whose CAPM alpha is obust to the gowth of abitage capital. Guided by the model, we find institutional owneship positively foecasts facto etuns. Nagel 2005 find the unconditional facto pemia ae most ponounced among stocks with low institutional owneship. We focus on the conditional facto pemia. Moeove, we use the model s asset-maket equilibium conditions to back out ambiguity fom investo suvey. Ou appoach is elated to Bhandai, Boovička, and Ho 2016 who use macoeconomic models to extact ambiguity shocks fom suvey data on households expectations about inflation and unemployment. Based on the theoy of Izhakian 2014, Benne and Izhakian 2017 measue ambiguity using intaday data of stock pices. Since Jensen 1968, a lage liteatue has documented that active potfolio manages fail to outpefom passive benchmaks o to delive alpha to investos. 9 Fama and Fench 2010 find that the aggegate potfolio of actively managed U.S. equity mutual funds is close to the maket potfolio see also Lewellen 2011, and vey few funds poduce sufficient benchmak-adjusted etuns to cove thei costs. Nevetheless, the asset management secto has been gowing damatically. As in Gennaioli, Shleife, and Vishny 2015, we popose an altenative pespective based on subjective welfae, but we poceed to chaacteize the conditions unde which manages undepefom the maket, delive negative alpha afte fees, and hold potfolios popotional to maket. Ou model is complementay to the existing liteatue on delegated asset management We show that delegation tansfoms the ambiguity at individual-asset level to that of efficient fontie, and its implications on asset picing. Uppal and Wang 2003 also emphasize diffeent types of ambiguity the oveall level and ambiguity on a subset of assets, but focus on the implications on unde-divesification. 9 See, e.g., Baas, Scaillet, and Wemes 2010, Cahat 1997, Del Guecio and Reute 2014, Fama and Fench 2010, Gube 1996, Malkiel 1995, Wemes See, e.g., Basak and Pavlova 2013, Bek and Geen 2004, Chevalie and Ellison 1999, Galeanu and Pedesen 2017,Gueiei and Kondo 2012, He and Xiong 2013, Kaniel and Kondo 2013, Kacpeczyk, Van Nieuwebugh, and Veldkamp 2016, Pásto and Stambaugh 2012, Pásto, Stambaugh, and Taylo 6

8 2 Model 2.1 Model setup Conside a economy whee agents make decisions on the fist day, and asset etuns ae ealized on the second and final day. Thee ae N isky assets, whose etuns ae stacked in a vecto = { i } N i=1, and one isk-fee asset that delives a isk-fee etun f. Define Ω as the set of states of the wold on the final day, so the vecto of asset etuns is a mapping fom the state space to eal numbes, : Ω R N. Thee ae a unit mass of homogeneous investos, and a unit mass of homogeneous fund manages. Fo simplicity, we assume that each investo is matched with one fund manage. Model uncetainty and pefeence. A epesentative investo is endowed with one unit of wealth. She chooses δ, which is the faction of wealth invested in the fund. We specify the delegation etun late afte laying out the investo s infomation set and pefeence. The investo needs to allocate 1 δ etained wealth, and chooses w o supescipt o fo own allocation, which is a column vecto of potfolio weights on the N isky assets. The investo does not know the etun distibution, so she foms he own potfolio unde model uncetainty o ambiguity. Hee ambiguity and model uncetainty ae used intechangeably. Model uncetainty is given by, a non-singleton set of candidate pobability distibutions of models. Fo a pobability measue, the investo assigns a pio π, which is the subjective pobability that is the tue etun distibution. The investo s pefeence is epesented by the smooth ambiguity-avese utility function in Klibanoff, Mainacci, and Mukeji 2005 KMM. The pupose of using this specification is to sepaate ambiguity itself and the avesion to ambiguity. 11 Utility is defined ove the teminal wealth, δ,w o,wd, whose subscipts show the dependence on the delegation level δ, the investo s own potfolio w o, and the delegation potfolio chosen by the manage w d supescipt d fo delegation that we intoduce shotly. The utility function is V δ,w o,w d = φ u δ,w o,w d ω dπ 1 d Ω Epstein 2010 has dawn the attention to the fact that KMM famewok may imply counteintuitive behavios, but Klibanoff, Mainacci, and Mukeji 2012 have eplied that those Ellsbeg-style thought expeiments do not pose difficulty fo the smooth ambiguity model. 7

9 φ and u ae stictly inceasing functions and twice continuously diffeentiable. Concavity of u and φ epesent isk and ambiguity avesion espectively. Delegation as model-contingent allocation. Fund manages pefeence is not modeled. A epesentative manage does not make any decision othe than constucting an efficient potfolio unde his knowledge of P, the tue pobability distibution of. We may think of a fund manage as a potfolio fomation machine that ceates a vecto of potfolio weights w d that achieves the efficient fontie moe details late on the definition of efficient potfolio. To access this machine, the investo pays an exogenous popotional fee ψ. In a iche setting, ψ can be detemined by the competition between fund manages, a manage s effot cost and asset management technology, agency cost, and bagaining powe. What can a fund manage offe? Fom the investo s pespective, fo any candidate model, if it is the tue model, the manage knows it and constucts the coesponding efficient potfolio w d. Theefoe, delegation makes investos wealth model-contingent. This is shown clealy once we wite out the total etun on the investo s wealth, [ ] [ ] δ,w o,w = 1 δ d f + f 1 T w o + δ f + f 1 T w d = f + f 1 T [ 1 δ w o + δw d ],. 2 The investo s own potfolio is a N-dimensional vecto, w o R N. In contast, the delegated potfolio, w d, is a mapping fom the model space to eal numbes, : R N, because if any is the tue model, the manage constucts the coesponding efficient potfolio w d. Though delegation, the total etun is a mapping fom the state space and the model space to eal numbes, δ,w o,w d : Ω R. If δ = 0, the potfolio etun is f + f 1 T w o, which just a mapping fom the state space Ω to R. Delegation impoves welfae though model-contingent allocation. As in Segal 1990, let us conside an imaginay economy with two stages: 1 investos choose w o and δ but cannot bet on which pobability model is tue the fist-stage state ; 2 the model is dawn and known by manages who allocate the delegated wealth. Hee, model uncetainty tanslates into a fom of maket incompleteness that can be educed by delegation. 12 Late we show that this welfae benefit is key to econcile the sizable delegation and medioce fund 12 This discussion is in line with Maenhout 2004 and Stzalecki 2013 who show an intinsic link between ambiguity avesion and the pefeence fo ealy esolution of isk e.g., Epstein and Zin

10 pefomances in data. Delegation fundamentally changes the natue of ambiguity and how it entes into investos potfolio choice. The delegated potfolio, w d, vaies acoss pobability models. This delegation uncetainty gives ise to a hedging motive the coss-model comovement between w d and an asset s etun distibution becomes a key consideation in investos potfolio decision. Without delegation, the etun on investos wealth does not vay with the pobability model and this hedging motive disappeas. In Section 2.4, we show that investos coss-model hedging motive in w o, induced by delegation, geneates a two-facto stuctue of asset etuns in equilibium. This motive becomes stonge when the delegation level is highe, so the equilibium neve conveges to CAPM a single-facto stuctue even if δ appoaches 100% and only manages tade assets. We will show that this hedging motive even appeas in the potfolio choice of ambiguityneutal investos with linea φ, so the two-facto stuctue of asset maket equilibium does not equie ambiguity avesion, which stands in contast with existing asset picing models with ambiguity e.g., Kogan and Wang In othe wods, once model uncetainty manifests into delegation uncetainty, it mattes fo asset picing even without ambiguity avesion. Note that without delegation, ambiguity-neutal investos simply pefom model-aveaging because the etun on wealth is only state-dependent, instead of stateand model-dependent. They calculate π-weighted aveage of pobabilities of any event, A = A dπ, fo any A Ω. 3 Unde this aveage model, ambiguity-neutal investos fom a potfolio, behaving as typical expected-utility agents, and do not hedge model uncetainty when delegation is unavailable. Befoe the fomal analysis, seveal obsevations ae in ode. Fist, vey impotantly in ou setting, manages do not diectly infom thei investos which model is tue. Othewise, the delegation uncetainty disappeas. This eflects the ealistic difficulty of communication between pofessional manages and investos. Paticulaly, big data and sophisticated techniques equip fund manages with inceasingly advanced tools to undestand etun distibution, but at the same time, ceate a division of knowledge. It is inceasingly difficult fo investos to undestand the infomation set and techniques of pofessional asset manages. Ou setup nests typical models in the liteatue of delegated potfolio management as 9

11 special cases, whee manages obtain pedictive signals, i.e., bette knowledge of the fist moment of etun distibution. Hee we study the most geneal fom of skills distibution knowledge. Busse 1999 finds volatility-timing ability of mutual fund manages Chen and Liang 2007 fo hedge funds. 13 Jondeau and Rockinge 2012 study the economic value added by foecasting up to the fouth moments of etuns distibution timing. As the asset management industy inceasingly leveages on big data and nonlinea data pocessing techniques, such as machine leaning, it is impotant to model asset management unde this geneic specification of skills. As will be shown late, the model sheds light on many issues on delegated potfolio management and asset picing. 2.2 A quadatic appoximation To solve the investo s delegation and potfolio allocation in closed foms, we appoximate the utility function in a quadatic fashion by extending the esults of Maccheoni, Mainacci, and Ruffino 2013 MMR into functional spaces. MMR does not allow agents wealth to be model-contingent. Model-contingent allocation though delegation is the key in ou model. In this pape, we adopt thei technical egulaity conditions and the appoximation conditions. We will show that ou appoximation nests MMR s as a special case. Fist, we define the cetainty equivalent. Definition 1 A epesentative investo s cetainty equivalent is defined by C δ,w o,w = υ 1 d φ whee υ is a composite function υ = φ u. u δ,w o,w d ω dπ, 4 d Ω Accodingly, we wite the investo s delegation and potfolio poblem as follows: { } max C δ,w w o,δ o,w ψδ d 5 whee the etun on wealth, δ,w o,wd, is both state- and model-contingent Equation 2, and investos pay a popotional asset management fee ψ. 13 In line with the evidence, Feson and Mo 2016 povide a famewok to evaluate potfolio pefomance in both maket timing and volatility timing. 10

12 The quadatic fom is simila to the mean-vaiance pefeence but incopoates both isk and ambiguity. We define two paametes of isk avesion and ambiguity avesion espectively in a small neighbohood of the etun on wealth aound isk-fee ate f. Definition 2 At isk fee etun f, the local absolute isk avesion γ is defined as γ = u f u f 6 and maginal-utility-adjusted local ambiguity avesion θ is defined as θ = u f φ u f φ u f 7 Befoe the quadatic epesentation of investos pefeence, we intoduce notations: Define q as the Radon-Nikodym deivative of w..t., i.e., q ω = dω fo ω Ω. dω q and ae used intechangeably to epesent a candidate pobability model in. Let R w = f 1 T w denote the excess etun of any potfolio w. [ ] Let R w = E f 1 T w denote the expectation of excess etun of w unde. Given, let E X and σ 2 X denote the expectation and vaiance of any andom vaiable X espectively, and µ X and ΣX the matix of covaiance of any andom vecto espectively. denote the vecto of expectation and Given, the covaiance of two andom vaiables X and Y is denoted by cov X, Y. uadatic Pefeence. Using the Taylo expansion in the functional space, we appoximate the cetainty equivalent as in Poposition 1. The poof uses the genealized Féchet deivatives in the Banach spaces. Details ae povided in the Appendix. Poposition 1 uadatic pefeence The smooth ambiguity-avese pefeence ove the state- and model-contingent etun, δ,w o,wd, i.e. mappings fom Ω to R, can be epe- 11

13 sented by the cetainty equivalent, which has the following expansion: 1 δ2 γσ 2 2 [γe δ2 π σ 2 R wd ] + θσ 2 π R wd 2 θ + γ 1 δ δcov π R wo, R wd + R w o, w d, C δ,w o,w d =f + 1 δ 2 R wo δe π R wd R w o + θσ 2 π R w o + 8 whee R w o, w d is a high-ode tem that satisfies lim w o,w d 0 Rw o,w d w o,w d 2 = 0. Following MMR, we use the same appoximation condition if potfolio is sufficiently divesified such that its matix nom is close to zeo, the esidual tem can be ignoed. 14 In the following, we use this second-ode appoximation in investos objective function. The local quadatic appoximation allows us to intuitively undestand the investo s pefeence. As peviously defined, R wo is the expected excess etun to he own potfolio wo unde the aveage model. An inceases in R wo leads to highe utility, but the sensitivity, 1 δ2, deceases in the level of delegation δ. σ 2 R w o is the vaiance of excess etun to the own potfolio unde the aveage model. As a measue of isk, it deceases utility. The sensitivity to isk inceases in γ, the paamete of isk avesion. σ 2 π R w o measues model uncetainty. It is the coss-model vaiation of the expected excess etun, as R wo the expected etun on the investo s etained wealth unde a paticula model. denotes sensitivity to ambiguity inceases in θ, the paamete of ambiguity avesion. As δ inceases, and thus, the etained wealth deceases, both sensitivities to isk and ambiguity decline. The delegation etun entes into the utility in an intuitive manne. E π R wd is the expected excess etun of the delegated potfolio, aveaged ove models unde pio π, E π R wd = ] E [ f 1 T w d dπ, whee R wd is the expected excess etun of delegated potfolio if is the tue model. Utility inceases in the coss-model aveage of expected etun to delegation. σ 2 π measues the ambiguity in delegation etun. It is a coss-model vaiance of expected excess etun fom 14 Ou convegence citeion diffes fom that of Galappi and Skoulakis 2011, who study how the accuacy of polynominal appoximations to expected utility depends on the included Taylo expansion tems. Hlawitschka 1994 show i when a Taylo seies of expected utility diveges, then tuncated Taylo seies paticulaly second-ode expansions may povide excellent appoximations fo the pupose of potfolio selection; ii when a Taylo seies does convege, adding moe tems may wosen the appoximation. 12 R wd The

14 delegation, so it educes utility, and its sensitivity inceases in the level of delegation δ and ambiguity avesion θ. E π σ 2 R wd measues the isk in delegation etun aveaged ove models, as σ 2 R wd is the vaiance of delegation etun unde a paticula. Intuitively, the sensitivity to delegation isk inceases in isk avesion γ. The tems discussed so fa can be summaized into two categoies. Fist, aveaging ove models, what ae the expected etuns and etun vaiances isk. Second, the coss-model mean and vaiance of the expected etuns unde pio π ove the model space ambiguity. The quadatic appoximation shows how the these statistics ente into utility, and how the utility sensitivities to these statistics depend on isk avesion, ambiguity avesion, and the level of delegation. The last tem in the quadatic fom deseves moe attention. It is the coss-model covaiance between the expected delegation etun and the expected etun on etained wealth. Investos do not teat the delegation etun and thei own investment oppotunity set sepaately, but instead, they want to hedge the coss-model uncetainty. Specifically, if an asset tends to delive a highe expected etun unde models whee the expected delegation etun is low, then investos would like to invest moe in this assets. As long as δ < 100%, the investo has to deal with the coss-model uncetainty fom delegation when allocating etained wealth. cov π R wo, Rwd pecisely captues such coss-model hedging motive. This hedging tem has a utility sensitivity that inceases in both isk avesion γ and ambiguity avesion θ. Given γ and θ, the sensitivity is maximized at δ = 1. Intuitively, the 2 investo caes the most about the comovement between the delegation pefomance and the etun on he etained wealth, when she divides wealth 50/50. As will be shown late, this hedging motive has citical implications on the equilibium expected etuns of isky assets. Ou quadatic appoximation nests MMR s solution when δ = 0, i.e., no delegation and the standad mean-vaiance pefeence when δ = 0 and θ = 0, i.e., no delegation and no ambiguity avesion as special cases. Coollay 1 Without delegation, i.e., δ = 0, the appoximation degeneates to the quadatic appoximation of smooth ambiguity utility by Maccheoni, Mainacci, and Ruffino 2013: C f + f 1 T [ 1 δ w o + δw d ] f + R wo γ 2 σ2 R w θ o 2 σ2 π R wo `. 9 If δ = 0 and θ = 0, the quadatic fom degeneates to the standad mean-vaiance utility 13

15 unde the aveage model : f + R wo γ 2 σ2 R w o. 10 Late, we show that the investo s optimal potfolio choice w o nests MMR s solution of optimal potfolio and the mean-vaiance potfolio of Makowitz 1959 as special cases. Delegation potfolio. To deive the solution to the investo s poblem and equilibium asset picing implications, we need to specify the delegation potfolio. In line with Coollay 1, the investo infoms he isk avesion to the fund manage, and the manage foms the mean-vaiance efficient potfolio given his knowledge of the tue distibution of. Theefoe, in the investo s mind, fo any, the manages solves whee, as peviously defined, µ and Σ { µ max f 1 T w d γ w d T } Σ w d 2 w d ae the mean vecto and covaiance matix of unde pobability measue. The delegated potfolio is model-contingent, w d : R N : w d = γσ 1 µ f Unde Gaussian asset etuns and CARA u with absolute isk avesion γ, w d is the exact maximize of u fo any given. Even without ambiguity avesion i.e., unde linea φ, as long as φ > 0, the investo always achieves highe utility by delegating asset allocation to a fund manage who efficiently allocates wealth fo each candidate model. 2.3 Investo optimization Investo potfolio choice. We solve the optimal level of delegation δ and potfolio w o by maximizing the quadatic appoximation given by Equation 8. Poposition 2 gives the investo s choice of own potfolio of isky assets, w o. Details ae povided in the Appendix. Poposition 2 Investo potfolio unde ambiguity & delegation Given the optimal 14

16 level of delegation δ, the investo s own potfolio of isky assets is given by wδ o = γσ + θσµ π 1 µ f1 δ θ + γ cov π µ, R wd } 1 δ {{ } uncetainty hedging demand. 12 If the investo could not delegate δ = 0, he potfolio would be w o 0 = 1 γσ + θσµ π µ f1, whee the subscipt 0 epesent zeo delegation. This is also MMR s solution of ambiguity investo s potfolio poblem. 15 Σ measues isk, the covaiance matix of asset etuns unde the aveage model. It entes into the optimal potfolio scaled by γ, the paamete of isk avesion. In contast, Σ µ π is the coss-model covaiance matix of expected asset etun vecto µ. It measues ambiguity. The optimal potfolio s sensitivity to Σµ π depends on θ, the paamete of ambiguity avesion. If θ = 0, the optimal potfolio becomes the standad 1 fomula by Makowitz 1959 unde the aveage model, i.e. γσ µ f1. Without delegation, ambiguity-neutal investos use Bayesian model aveaging. Given δ > 0, the potfolio exhibits a hedging demand fom cov π µ, Rwd, the cossmodel comovement between the expected excess etuns of assets, µ, and the expected excess etun fom delegation, R wd. The investo knows that whicheve model is tue, the fund manage must know it and constuct the efficient potfolio accodingly, but the tue model is still unknown. Theefoe, the investo must design he own potfolio in a way that is obust to such ambiguity. The highe the ambiguity avesion is, the moe sensitive the investo s potfolio choice to this covaiance tem. Even if we shut down ambiguity avesion θ = 0, we still have the hedging demand, which is γ δ covπ µ 1 δ, Rwd, depending on the isk avesion paamete. Fund manages select the mean-vaiance efficient potfolio fo investos fo each model, but the investos still have allocate the etained wealth. To do that, they must conside all the pobability models and make thei own potfolio obust to the coss-model vaiation in investment oppotunity set and delegated etun. This coss-model hedging motive moves the investo s 15 Galappi, Uppal, and Wang 2007 deive a simila potfolio by incopoating estimation eos in expected etuns a maxmin appoach in the spiit of Gilboa and Schmeidle

17 total potfolio away fom the efficient fontie within each paticula model, so highe isk avesion makes investos moe cautious to the coss-model covaiance between asset etuns and delegation etun. Let cov π µ i, Rwd denote the i-th element of cov π µ, Rwd. It epesents the covaiance between asset i s expected etun and the delegation etun. When the expected delegation etun comoves with asset i s expected etun, i.e. cov π µ i, Rwd > 0, the investo educes investment in asset i. When asset i s expected etun moves against the expected delegation etun, i.e. cov π µ i, Rwd < 0, the investo demands moe of asset i as if buying an insuance against delegation uncetainty. This hedging motive will have citical implications on the equilibium coss-section of expected asset etuns. Optimal delegation. The optimal faction of wealth delegated to fund manages depends on the stuctue of investos ambiguity and delegation fee ψ. Poposition 3 Optimal delegation given w o Given the optimal potfolio w o, the investo s optimal delegation level δ is given by the fist ode condition: δ = E π R wd R wo θ + γ cov π R w o θ + γ cov π E π R w d R wo R w o, Rwd ψ + θσ 2 π R w d, Rwd. 13 The solution is vey intuitive. If the investo can achieve a high etun on he own, co- i.e. high R wo, delegation deceases. If the expected etun on etained wealth Rwo moves closely with the expected etun on delegated wealth R wd cov π R wo, Rwd acoss models i.e. high, delegation also deceases. The investo ae avese to the coss-model comovement, as eflected in the choice of w o. Delegation will incease if the delegation etun is expected to be high acoss models i.e. high E π R wd, and if it does not fluctuate much acoss pobability models i.e. low σ 2 π. Note that the investo s own potfolio w o de- R wd pends on δ, so Equation 13 only implicitly defines δ. The next coollay solves δ explicitly as a function of the investo s ambiguity stuctue and management fee. Coollay 2 Optimal delegation The investo s optimal delegation level δ is given by δ = E π R w d E π R wd + θσ 2 π R w d θ + γ B C ψ θ + γ 2 A 2 θ + γ B C, 14 16

18 whee T 1 A =cov π µ, R wd γσ + θσµ π covπ µ, R wd, 15 T 1 B =cov π µ, R wd γσ + θσµ π µ f1, 16 T 1 C = µ f1 γσ + θσµ π µ f1. 17 The solution in Equation 14 depends on the complicated stuctue of the investo s model uncetainty that involves the coss-model mean and vaiance of expected delegation etun and the coss-model comovement of delegation etun and asset etuns. 16 In Appendix IV, we show how to calibate ou model with eal data and calculate the model-implied delegation. Compaative statics unde simplified model uncetainty. We deive compaative statics and exploe moe economic intuitions unde a simplified stuctue of model uncetainty. We make the following assumptions that coespond to typical settings whee delegated potfolio management has been studied pofessional asset manages obtain etun signals, i.e., supeio knowledge on the fist moment of asset etuns. Accodingly, investos face uncetainty in thei expectation of futue asset etuns. Assumption 1 The investo knows the tue covaiance matix: fo any, Σ = Σ P. Unde this assumption and the quadatic appoximation of investo pefeence, the model uncetainty is only about the expected etuns, which is captued by the subjective covaiance matix of expected etuns, Σ µ π, given pio π ove candidate models. 17 If the investo s model uncetainty is fom estimation eos, the diagonal of Σ µ π ecods the squaed standad eos of the expected etun estimato, which natually depends on the volatility 16 To solve δ, we substitute the investo s optimal potfolio into Equation 13, so the fomula is solved unde the assumption of an inteio solution, i.e., δ < 1. When δ = 1 and the investo does not etain any wealth to manage on he own, the investo s optimal potfolio given by Equation 12 is not well defined. This explains why even if delegation is fee i.e., ψ = 0, Equation 14 does not give 100% delegation. Intuitively, since the manage foms the efficient potfolio unde each pobability model, the investo with quadatic utility should fully delegate when ψ = 0. Theefoe, the complete solution of delegation should be 100% if ψ = 0, and the inteio value given by Equation 14 if ψ > Boyle, Galappi, Uppal, and Wang 2012, and Ilut and Schneide 2014 also intoduce ambiguity though uncetainty in the mean. 17

19 and covaiance of etuns unde the tue model i.e., data geneating pocess. Theefoe, we add the following assumption on π that links model uncetainty to volatility. Assumption 2 The investo s subjective belief of expected etun is given by a nomal distibution, whose covaiance is popotional to the tue etun vaiance: Since µ N µ, υσ P µ N µ, υσ P. 18, υ that paameteizes the level of model uncetainty. This setup can be easily undestood as paamete uncetainty o estimation eo when the investo ties to estimate the expected excess etuns, which echoes the intepetation of ambiguity by Bewley technical convenience. The nomality assumption of the pio ove µ also bings As shown in Appendix C, we can apply the Isselis theoem to damatically simplify investos optimal delegation and potfolio choice. N µ, υσ P is the popula conjugate pio. υ can be undestood as the invese of the size of estimation sample. If the investo has T obsevations of and she assumes the independence acoss obsevations, the method-of-moment estimato of the expected etun is 1 T ΣT t=1 and its covaiance is 1 T Σ P. This case diectly applies to Σµ π = υσ P with υ = 1 T. Lage υ means smalle sample and lage estimation eo o ambiguity. It is natual to assume that υ < 1 unde this intepetation, because 1 T Assumption 3 υ < 1. < 1 fo non-singleton sample. These assumptions highlight the link between volatility and ambiguity. When assuming the covaiance of asset etuns ae known to investos, lage volatility means the expected etuns ae hade to estimate highe paamete uncetainty. This model suggests that delegation should also elate to the potentially time-vaying uncetainty induced by the evolution of asset etun volatility. The case of known covaiance and unknown expected etuns echoes the obsevation by Meton Kogan and Wang 2003 also conside this case in thei discussion of potfolio selection unde ambiguity. Using these assumptions, we solve explicitly the optimal delegation as a function of the exogenous paametes, and simplifies the fomula of optimal potfolio choice. We povide 18 Bewley 2011 oiginally fomulated the agument that confidence intevals can be intepeted also as a measue of the level of ambiguity associated with the estimated paametes. Fo anothe ecent pape that uses Bewleys chaacteization of Knightian uncetainty, see Easley and OHaa

20 deivation details in the Appendix, but the main intuition can be simply undestood by noticing that fo any model, the expected delegation etun can be decomposed as follows, R wd = µ f 1 T w d = µ f 1 T γσ P 1 µ f 1 T T = µ µ γσ P 1 µ µ + µ µ γσ P 1 µ f1 + T T µ f1 γσ P 1 µ µ + µ f1 γσ P 1 µ f1, T T = µ µ γσ P 1 µ µ + 2 µ f1 γσ P 1 µ µ }{{}}{{}}{{} Chi-squaed distibution constant vecto Nomal distibution + R wd, }{{} constant whee the distibutional popeties labeled below each tem ae obtained unde the assumption that investos pio π is Gaussian, i.e., µ µ N, υσ P. Using Isselis theoem and the popeties of Chi-squaed and nomal distibutions, we solve the summay statistics A, B, C, E π R wd, and σ 2 π in Coollay 2 fo the optimal level of delegation, and the R wd key covaiance component in investos optimal potfolio, cov π µ, R wd = 2υ γ µ f1. 19 The solution is summaized in the following poposition fo compaative statics. Poposition 4 Compaative Statics Unde the thee assumptions, the investo s potfolio is given by w o = [ 1 δ 2υ Σ P 1 µ γ + θυ f1 γ + θ µ 1 δ γ f1 ]. 20 The optimal delegation decision is δ = θυ γ 2υθ+γ [ υ N ψ + 1 γ γ γ+υθ [ υ N θυ γ γ γ γ+υθ ] + 1 γ 2υθ+γ + 1 γ R wd 2 ] R wd, 21 whee the expected etun to the delegated potfolio unde the aveage model is R µ T wd = f1 γσ P 1 µ f A simple calculation shows that the fomula poduces easonable level of delegation. δ equals 49% unde 19

21 We have the following esults of compaative statics: 1 The optimal level of delegation δ inceases in N, the numbe of isky asset, and γ, the isk avesion: δ N > 0, δ γ > 0. 2 The optimal level of delegation δ deceases in θ, the ambiguity avesion, υ, the level of ambiguity, and ψ, the management fee: δ υ < 0, δ θ < 0, δ ψ < 0. 3 Given the delegation level δ, w o deceases in θ, the ambiguity avesion, υ, the level of ambiguity, and γ, the isk avesion: wo υ [ γ 4 When, N < 1 υ + θ + γ 2υ ψ + θ γ R w d wo < 0, θ < 0, wo γ < 0, given δ. ], w o 0 if and only if µ f1. Afte applying the Isselis theoem to simplify δ and w o details in the Appendix, a new summay statistic N, the numbe of assets, shows up. Because acoss models, the expected delegation etun, R wd = µ f1 T w d = µ f1 T γσ P 1 µ f1, follows Chi-squaed distibution because investos pio π is Gaussian, so N appeas because it is the degee of feedom in the mean and vaiance fomula of Chi-squaed distibutions. We povide detailed deivation in the Appendix. Intuitively, as the numbe of isky assets inceases, the fund manage s ability to constuct efficient potfolios of a lage set of assets is moe valuable, so the delegation level inceases. Highe isk avesion inceases the wealth delegated to manages who constuct efficient potfolios, because when isk avesion is high, being away fom the fonties significantly deceases the investo utility. Note that we can intepet N as the numbe of isk factos instead of pimitive isky assets. Suppose thee ae infinite numbe of assets, whose etuns ae spanned by N isk factos and thei own idiosyncatic shocks. By law of lage numbes, the investo can always divesify away idiosyncatic shocks at zeo cost no matte which pobability model is tue, as long as candidate pobability measues ae not point-mass. Effectively, the investo deals with N isk factos. Moe souces of isk motivates the investo to delegate moe. Holding constant N, delegation deceases in ambiguity avesion θ and the level of ambiguity υ, because the need to hedge against delegation uncetainty is stonge. The the following calibation: N = 10, γ = 5, θ = 1, R wd when N inceases to = 0.04, ψ = 0.01 and υ = δ inceases to 99%, 20

22 benefit of delegation is that the δ faction of wealth is allocated efficiently, but the moe the investo delegates, the stonge the coss-model hedging motive, which which educes benefits of delegation. A moe uncetain envionment tends to educe delegation. The compaative statics on investo s potfolio choice ae deived given the optimal delegation. The investo becomes moe consevative in holding isky assets, when facing moe ambiguity, o unde highe ambiguity avesion o isk avesion. 20 In eality, most investos hold long positions. In the model, investos takes all long positions, if unde thei aveage model, the expected excess etuns ae non-negative µ f 1. This esult equies N to be lowe than an uppe bound. As histoic data accumulates, υ, the estimation eo, aising the uppe bound of N. The uppe bound of N is equal to 272 unde the following calibation: γ = 5, θ = 1, R wd = 0.04, ψ = 0.01 and υ = This numbe is likely to be lage than the numbe of systematic isk factos. 2.4 Coss-section asset picing We chaacteize the coss section of expected asset etuns and thei CAPM alpha. Fist, we show that when delegation is unavailable, ou model poduces esults that nest key theoetical findings in the liteatue of asset picing unde ambiguity. Next, we show that delegation significantly changes the esults. In contast to the existing liteatue, the CAPM alpha the ambiguity pemium, does not disappea even when investos ae not ambiguity-avese. Also, if we conside a sequence of economies with ising levels of delegation appoaching 100%, the asset maket equilibium does not convege to CAPM. The esults aboved ae deived unde the geneal fom of model uncetainty. Then we simplify the stuctue of investos model uncetainty to obtain two esults that can be diectly tested in Section 3: 1 the coss-section dispesion of CAPM alpha inceases in the level of model uncetainty; 2 we can back out investos model uncetainty fom the elation between assets CAPM alpha and investos expectation unde the aveage model. What dives all the esults is investos hedging against delegation uncetainty. Equilibium without delegation. To chaacteize the equilibium expected etun, we define the maket potfolio m, which is equal to the exogenous supply of isky assets so that 20 Gollie 2011 investigated the compaative statics of moe ambiguity avesion in a static two-asset potfolio poblem. 21

23 asset makets clea. The maket potfolio is the sum of investos and manages potfolios: m = δw d P + 1 δ w o. 23 We fist study the case without delegation. Recall that w o 0, the zeo-delegation potfolio, is investo s potfolio when delegation is unavailable, w o 0 = 1 γσ + θσµ π µ f1 24 When δ = 0, substituting the maket cleaing condition, m = w0, o into Equation 24 and multiplying both sides by γσ + θσµ π, we have µ f1 = γσ + θσµ π m. 25 Note that Σ m is simply the vecto of covaiance unde between asset etuns and the maket etun, and Σ µ π m ecods the covaiance unde π between expected asset etuns and the expected maket etun. If investos aveage model is tue, i.e., = P, the lefthand side is the assets expected excess etuns unde the tue pobability measue, and the ight-hand side is decomposed into two covaiance tems. Poposition 5 Ambiguity pemium without delegation When delegation is unavailable δ = 0, the equilibium expected excess etuns of isky assets ae µ P f 1 =λ m β P,m + λ w o 0 β π µ,m, 26 if investos aveage model is the tue model, i.e., = P, whee we define maket pice of isk, λ m = γσ 2 P Rm, the isk beta, β P,m = cov P,R m σ 2 P Rm, maket pice of ambiguity, λ w o 0 = θσ 2 π R m, the ambiguity beta, β π µ,m = covπµ,rm. σ 2 πr m Equation 26 decompose the expected excess etun into two components. When investos ae the only maket paticipants, the expected excess etuns compensate them fo both thei isk exposue and ambiguity exposue. The fist component λ m β P,m is exactly 22

24 the standad CAPM beta multiplied by the maket pice of isk. The second tem λ w o 0 β π µ,m is the poduct of the ambiguity beta and pice of ambiguity. The ambiguity beta measues the coss-model comovement between the expected asset etuns and the expected maket etun i.e. the etun of zeo-delegation potfolio. If asset i s expected etun comoves with the expected maket etun acoss models i.e. β π µ i > 0,,m the asset must delive a highe aveage etun though λ w o 0 β π µ i > 0 in equilibium. If asset,m i s expected etun moves against the expected maket etun i.e. < 0, then it β π µ i,m seves as hedge against model uncetainty fom investo s pespective, and thus, it affods a discount in the aveage etun via λ w o 0 β π µ i < 0.,m The assumption of = P is impotant. Investos face model uncetainty, so they cannot evaluate the expected etuns of isky assets unde the tue model P. Instead, they examine the expected etuns by aveaging ove candidates models, i.e., µ, and accodingly, expected etuns unde eflect investos demand fo isk and ambiguity compensation. Only unde the assumption that = P, do investos expected etuns µ coincide with the expected etuns unde the tue model µ P, which ae obseved by econometicians, and thus, can we solve µ P using the potfolio optimality condition substituting out wo with m. Ambiguity geneates CAPM alpha as in Maccheoni, Mainacci, and Ruffino They analyze a special case of two assets whee one asset is pue isk whose distibution is known while the othe asset s etun is ambiguous. Using the constained-obust appoach, Kogan and Wang 2003 deive the simila two-facto stuctue of equilibium expected etuns. In those models and hee, if we shut down ambiguity avesion θ = 0, the pice of ambiguity, λ w o 0 = θσ 2 π, is zeo, and the model degeneates to CAPM. R wo 0 Coollay 3 CAPM without delegation When delegation is unavailable δ = 0, if investos ae ambiguity-neutal θ = 0, the equilibium excess etuns of isky assets ae µ P f 1 = λ m β P,m, 27 if investos aveage model is the tue model, i.e., = P. V ω = If the investo is ambiguity-neutal, the investo s utility function can be witten as ω Ω u ω d ω dπ = ω Ω [ ] u ω d ω dπ = u ω d ω ω Ω 23

25 which is simply the expected utility given the aveage pobability model. Ou quadatic appoximation becomes the standad mean-vaiance utility as shown in Coollay 1, so if = P, we ediscove CAPM. It is citical that u can be taken out of the integal opeato, because u, o equivalently, only depends on the state ω, but not on the model. This is in tun because delegation is unavailable, so investos wealth is not modelcontingent. Next, we show that when delegation is available, the equilibium deviates fom CAPM even when investos ae ambiguity-neutal. Equilibium with delegation. When delegation is available, the maket potfolio is equal to a mixtue of manages potfolio and investos potfolio, i.e., m = δw d P + 1 δ w o. We aange the fund manage s potfolio, w d P = γσ P 1 µ P f1, unde the tue pobability distibution P, and aive at the following expession of expected excess etuns: µ P f 1 = γσ P w d P Substituting the eaanged maket cleaing condition, w d P = 1m 1 δ δ δ w o, into the equation above, we have µ P f 1 = 1 1 δ δ γσ P m γσ P w o. 29 }{{} δ }{{} β P,m λ δ α Using the definition of maket beta in Poposition 5, we may ewite the fist tem on the ight-hand side as the poduct of assets maket beta and the pice of maket isk, λ δ. In contast to the equilibium without delegation, the pice of maket isk, λ δ = γ δ σ2 P Rm, deceases in the level of delegation δ. This popety is in line with the concuence of a gowing asset management industy and a declining equity pemium in the U.S. maket documented by Jagannathan, McGattan, and Schebina 2001 and Lettau, Ludvigson, and Wachte 2008 among othes. A declining maket pice of isk in esponse to a ising level of delegation suggests that the secuity maket line becomes flatte as moe money is poued into the pofessional asset management industy. The deviation fom CAPM, α, is due to the second tem on the ight-hand side, which 21 Note that because µ P aleady shows up in manages potfolio, we do not need to assume = P to solve the equilibium expected etuns as we did fo the case without delegation. 24

26 depends on the investos potfolio solved in Equation 2 and epoduced below, w o = γσ + θσµ π 1 µ f1 }{{} sentiment δ θ + γ cov π µ, R wd } 1 δ {{ } delegation hedging demand, that combines sentiment the expected asset etuns unde the aveage model, o aveage belief and the uncetainty hedging demand that depends on the coss-model covaiance between the expected asset etuns and the expected delegation etun, and is scaled by both isk Σ and model uncetainty Σµ π. Substituting this expession of w o into the α component of Equation 29, we have α = γσ P γσ + θσµ π 1 1 δ µ δ f1 θ + γ cov π µ, R wd }{{}}{{} fom aveage belief fom delegation hedging The sentiment component of α captues investos belief aveaged ove models. It eventually disappeas if the level of delegation appoaches 100% i.e., δ This is consistent with Coollay 3 because this component is exactly the full α when delegation is unavailable. Such deviation fom CAPM shinks as the maket is inceasingly dominated by ational-expectation manages who know the tue pobability model and allocate wealth on the efficient fontie. The othe component α, which is fom investos uncetainty hedging demand, is immune to the ise of delegation level, and in paticula, it does not disappea even when delegation appoaches 100%. This hedging demand aises fom delegation. In fact, as δ appoaches 100%, the hedging demand becomes inceasingly significant in magnitude as shown by the multiplie δ 1 δ in font of covπ µ, Rwd in investos potfolio. Theefoe, even if investos manage less wealth when δ inceases, the hedging incentive is stonge pe unit of etained wealth. Intuitively, the moe wealth is delegated to manages, the moe investos want to hedge against the coss-model vaiation of delegation etun. When investos potfolio ente into the expession of expected asset etuns in Equation 29, this multiplie exactly offsets 1 δ δ, the atio of etained-to-delegated wealth, sustaining the α so that the economy does not convege to CAPM even if δ appoaches 100%. 25

27 Inteestingly, we may set θ, the ambiguity avesion paamete, equal to zeo, but the component of α fom delegation hedging still exists and appoaches γcov π µ, Rwd as δ appoaches 100%. This popety distinguishes ou model fom existing models of asset picing with ambiguity, in which α disappeas if investos ae no longe ambiguity-avese e.g., Kogan and Wang The key is delegation, as we have shown in Coollay 3 that when delegation is unavailable, α disappeas when θ = 0. We can undestand the intuition behind ou esult by inspecting an ambiguity-neutal investo s utility function as we did fo the case without delegation, but now notice that the etun on wealth is both state-dependent and, though delegation, model-dependent. V ω, = u ω, d ω dπ, ω Ω whee a distibution of states of the wold, ω, can be viewed as a distibution conditional on. Due to delegation, an ambiguity-neutal investo cannot pefom Bayesian model aveaging and opeates unde the aveage pobability, but instead, has to deal with the joint uncetainty of state and model. Theefoe, the coss-model covaiance between the expected asset etuns and the expected delegation etun still appeas in investos potfolio choice, and the equilibium expected asset etuns, even if θ = 0. Poposition 6 summaizes ou esults so fa. Poposition 6 Ambiguity pemium with delegation The equilibium expected excess etuns of isky assets ae given by µ P f 1 = λ δ β P,m + α. 31 The maket betas, β P,m, ae defined as in Poposition 5. The pice of maket isk, λ δ = γ δ σ2 P Rm, suggesting that the secuity maket line become flatte as delegation inceases. The CAPM α is given by Equation 30, which depends investos sentiment, the aveage belief, and cov π µ, Rwd, the coss-model covaiance between the assets expected etuns and the expected delegation etun. γσ P µ f1 When δ appoaches 100% fo example because the management fee declines o the numbe of assets inceases as in Poposition 4, α does not convege to zeo. Its limit is 1 γσ + θσµ π θ + γ covπ µ, Rwd, a linea tansfomation of cov π µ, Rwd. 26

28 Even if investos ae not ambiguity-avese θ = 0, α still exists, and when δ appoaches 100%, it appoaches γcov π µ, Rwd. CAPM, Note that when δ is pecisely equal to 100%, we have m = w d P, and ediscove whee λ m = R wd P µ P f 1 = β P,mλ m, 32 = RP m. Howeve, as long as δ < 100%, investos need to allocate thei etained wealth unde ambiguity. The moe they delegate, the stonge they hedge against delegation uncetainty in thei potfolio choice Equation 12. As shown in Poposition 6, what geneates alpha is this hedging demand. Theefoe, even if the total amount of etained wealth declines as δ inceases, the hedging demand inceases pe unit of etained wealth, and thus, sustains the alpha. Coollay 4 Equilibium discontinuity with delegation When δ appoaches 100% fo example because the management fee declines o the numbe of assets inceases as in Poposition 4, the model equilibium does not convege to the CAPM equilibium. Howeve, when δ = 100%, the model poduces the CAPM equilibium. equilibium discontinuity in the limit. Theefoe, thee exists an In the past few decades, asset management industy has gown damatically, especially in the aea of quantitative investment that often tagets alpha aleady identified in the academic liteatue. Many have agued that investment stategies alpha shinks as abitage capital inceases. Yet many stategies suvive, and togethe, they constitute a ich set of anomalies in asset picing. In ou model, as shown in Poposition 4, the asset management industy gows i.e, δ inceases fo seveal easons. The numbe of systematic isk factos, N, may have inceased due to technological changes o globalization. The management fee, ψ, may have deceased thanks to inceasing competition and moe efficient data pocessing. As δ becomes highe, an inceasing shae of the asset maket is taken by manages who hold the fontie potfolio. Will the equilibium convege to CAPM? The answe is no. Empiically, we can obseve the gowth of pofessional asset management, but CAPM alpha neve disappeas fo cetain assets o investment stategies. It is difficult fo asset picing models to ationalize such anomalies. It is even moe difficult to econcile the obust alpha of seveal anomalies in a peiod of inceasing abitage capital. Ou model offes a new pespective to undestand such phenomena. 27

29 Alpha unde the simplified uncetainty. The CAPM alpha of assets ae geneated by investos delegation hedging, so when the model uncetainty inceases, the hedging demand becomes stonge and the coss-section of alpha widens. To illustate this intuition, we simplify the stuctue of model uncetainty using the thee assumptions in Poposition 4, and deive expessions of α that we estimate diectly in Section 3. Unde the thee assumptions, investos know the tue etun covaiance matix, and uncetainty is popotional to the covaiance, i.e., Σ µ π = υσ P, so Equation 30 becomes γ α = 1 δ γ + θυ µ δ f1 θ + γ cov π µ, R wd. 33 }{{}}{{} fom aveage belief fom delegation hedging Next, we substitute cov π µ, Rwd = 2υ γ µ f1 into α. This simple solution of the coss-model covaiance uses the Isselis theoem and popeties of nomal and Chi-squaed distibutions as explained when it was fist intoduced in Equation 19. So, we have γ 1 δ α = µ γ + θυ δ f1 θ + γ 2υ µ }{{} γ f1, 34 }{{} fom aveage belief fom delegation hedging which can be futhe simplified to the following equation 2 θ/γ δ α = µ θ/γ + 1/υ 1 + υθ/γ δ f1. 35 }{{}}{{} fom delegation hedging zeo-delegation component Assets alpha ae popotional to the expected excess etuns unde investos aveage model. The coss-sectional vaiation of α is fom µ f1, the vecto of expected excess etuns unde investos aveage model. Theefoe, α dispesion becomes lage if its coefficient inceases, fo example when thee is moe model uncetainty highe υ. In Section 3, we find evidence suppoting this pediction using vaious measues of uncetainty. As δ appoaches 100%, the component of α fom investos zeo-delegation potfolio shinks to zeo, while the component fom delegation hedging emains. As peviously discussed, delegation hedging becomes stonge when the delegation level ises, offsetting the 28

30 decline of wealth managed by investos unde ambiguity. What is inteesting hee is that unde the simplified ambiguity, investos aveage belief eplaces the coss-model covaiance between assets and the efficient fontie, and theeby, suvives in α though delegation hedging when δ appoaches 100%. Any belief bias aveaged acoss models is captued by µ f1. The suvival of belief bias is not due to limits to abitage as in taditional models of behavioal finance Babeis and Thale 2003, but athe fom delegation uncetainty. Investos may well choose to delegate moe, but meanwhile, they hedge moe. 22 The following poposition summaizes ou esults. Poposition 7 Uncetainty and the coss-section dispesion of alpha Conditional on the level of delegation, δ, the coss-section dispesion of alpha is inceasing in model uncetainty, υ, as shown by Equation 35. As δ appoaches 100% fo example, due to a declining management fees o an inceasing numbe of assets as in Poposition 4, α conveges to α = 2 θ/γ + 1 µ θ/γ + 1/υ f1. 36 Anothe application of ou theoy is to extact investos model uncetainty fom data. Measuing model uncetainty is a vey challenging task, because by natue, ambiguity is subjective. Howeve, as show by Equation 35, if investos model uncetainty, υ, vaies ove time, the elation between alpha and investos expectation vaies. Theefoe, if we could obtain a measue of investos expectation, i.e., µ f1, we can back out the dynamics of investos model uncetainty by pojecting assets CAPM esiduals on investos expectation while contolling fo the delegation level in olling windows. In Section 3, we measue investos model uncetainty by using suveys on investos expectation of futue stock-maket pefomance as poxy fo µ f1. Poposition 8 Model-implied uncetainty measue Conditional on the level of delegation, δ, the elation between assets CAPM alpha and investos expectation unde the aveage model eveals the level of model uncetainty, as shown by Equation Ou model is static, so it does not speak to the evolution of wealth distibution between ational and iational agents ove time. 29

31 2.5 Delegation and fund pefomance So fa, we have been focusing on the model s asset picing implications, which guides us to identify new pattens in the etuns of stock-maket factos Section 3. Next, we show that the model also econciles the existing and puzzling evidence on delegated asset management. As eviewed by Fench 2008, the evidence on the medioce fund pefomance suggests that investos ae bette off not delegating and holding indices instead. This poses a challenge to undestand the gowth of pofessional asset management in the past few decades. In this pape, we shift the focus fom ex post pefomance to ex ante welfae. Pefomance measuement assumes lage sample and investos have ational expectation i.e., econometicians belief. In eality, investos face model uncetainty. In ou model, manages allocate the delegated wealth efficiently fo each model, but though delegation, investos wealth becomes model-contingent. When choosing the optimal level of delegation, the tade-off is now between within-model allocation efficiency and coss-model delegation uncetainty. Fund undepefoming the maket. Let us conside investing in the maket index, and compae the expected delegation etun and the maket etun unde the simplified stuctue of ambiguity. Substituting the investo s potfolio equation 20 into the expected maket excess etun, we solve the expected maket etun unde the tue pobability distibution: R m P = δr wd P + 1 δ R wo P = µ P f 1 T γσ P 1 [µ P f 1 δ + µ f1 The expected excess etun of the fund manage s potfolio is 1 δ γ δ2υ θ + γ γ + υθ ]. R wd P = µ P f 1 T γσ P 1 µ P f 1 The diffeence between the expected fund etun and the expected maket etun, R wd P Rm P, is equal to 1 δ µ P f 1 T γσ P [µ 1 P f 1 µ γ ] δ 1 δ 2υ θ + γ f1, 37 γ + υθ which is also the aveage pefomance diffeence in a lage sample. 30

32 Poposition 9 Delegation and undepefomance Unde the thee simplification assumptions, fund manages undepefom the maket if N i=1 w d i P µ i P f < κ N i=1 wi d P µ i f1, whee w d i P is the fund manages potfolio weight on asset i unde the tue pobability P, and which inceases in θ and υ and deceases in γ. κ = γ δ 1 δ 2υ θ + γ, 38 γ + υθ Whethe the fund manages undepefom o outpefom the maket depends on the compaison between the weighted-aveage of assets expected etuns unde tue model and the weighted aveage of assets expected etuns unde the investos aveage model scaled by κ. Because investos also paticipate in the maket, fund manages pefomance depends on thei elative aggession in isk- and ambiguity-taking. Fo example, if investos have in mind a high-etun maket i.e. high µ, then they can be moe aggessive and ean a highe expected etun than fund manages by taking on moe exposue to isk and ambiguity. Theefoe, in ou model, delegation can aise in spite of manages undepefomance elative to the maket. Investos do not know the tue pobability distibution, so they cannot evaluate fund pefomance unde ational expectation and choose between funds o the maket index. Note that we do not impose any estiction on investos potfolio choice, so holding the maket potfolio is cetainly within investos oppotunity set. But Delegation without alpha. Anothe commonly used pefomance metic is alpha Jensen It is defined as the esidual aveage fom egessing fund etun on the maket etun. Let us assume that µ = µ P. In this case, investos potfolio is popotional to fund manages potfolio and the maket potfolio unde the simplified ambiguity see Equation 20 and below: w o = 1 γ + θ δ 1 δ 2υ γ γ + θυ Σ P 1 µ P f Theefoe, CAPM holds. A egession of manages etun on the maket etun shows 31

33 exactly zeo alpha in a lage sample, so afte fees, investos eceive negative alpha fom delegation. Moeove, fund manages hold the maket potfolio up to a scaling facto, as some have documented in the empiical liteatue e.g., Lewellen Poposition 10 Delegation and negative alpha Unde the simplified model uncetainty and given µ = µ P, the delegated potfolio delives zeo goss alpha and negative alpha afte fees, and it is popotional to the maket potfolio. Why investos invest a significant shae of wealth in actively managed funds, in spite of thei undepefomance and negative alpha afte fees. This pape agues that unde ambiguity, they choose to delegate in ode to impove ex ante welfae. When choosing the optimal level of delegation, the tade-off is between within-model allocation efficiency and coss-model delegation uncetainty. Such focus on ex ante welfae echoes the obsevation by Gennaioli, Shleife, and Vishny Anothe inteesting implication is that even if fund manage possesses supeio knowledge and knows the tue model, this may not help them to geneate maket isk-adjusted etun. This esult challenges the taditional appoach of fund pefomance measuement: an asset management fim could be active in acquiing the knowledge of tue etun distibution, but this effot is not likely to be compensated if we only look at alpha. 3 Evidence We povide suppoting evidence fo ou model assumptions and esults using data fom the U.S. stock maket. As discussed peviously, idiosyncatic isks can be divesified away, so they should not affect investos evaluation of isk and model uncetainty. Theefoe, we use well-studied factos instead of individual stocks as the asset univese. Hee we offe a peview of ou findings. Fist, asset manages tilt thei potfolios to factos with highe futue etuns and Shape atio, which suggests that manages possess supeio infomation on asset etun distibution. Factos favoed by manages also exhibit obust CAPM alpha despite the ising level of delegation in the last few decades, consistent with Poposition 6. Moeove, vaious uncetainty measues positively pedict the cosssection dispesion of factos CAPM esiduals, as implied by Poposition 7. Finally, guided by Poposition 8, we calculate investos model uncetainty and its coelation with othe measues of uncetainty in the existing liteatue. 32

34 3.1 Data souces and vaiable constuction Asset space: factos. We conside the most well-studied stock-maket factos in the empiical asset picing liteatue. The factos can be divided into two categoies: accounting-based and etun-based. Accounting-based factos include value HML, accuals ACR, investment CMA, pofitability RMW, and net issuance NI. Retun-based factos include momentum MOM, shot-tem evesal STR, long-tem evesal LTR, betting-against-beta BAB, idiosyncatic volatility IVOL, and total volatility TVOL. To constuct each facto, we use monthly and daily etuns data of stocks listed on NYSE, AMEX, and Nasdaq fom the Cente fo Reseach in Secuities Pices CRSP. We include odinay common shaes shae codes 10 and 11 and adjust delisting by using CRSP delisting etuns. We obtain accounting data fom annual COMPUSTAT files to constuct fim chaacteistics. We follow the standad convention and lag accounting infomation by six months Fama and Fench If a fim s fiscal yea ends in Decembe in yea t, we assume that this infomation is available to investos at the end of June in yea t + 1. We constuct each facto in the typical HML-like fashion by independently soting stocks into six potfolios by size ME and the facto chaacteistic. We use standad NYSE beakpoints median fo size, and 30th and 70th pecentiles fo the facto chaacteistic. We compute value-weighted etuns and othe statistics of the six potfolios. A facto s etun is the value-weighted aveage etun of the two high-chaacteistic potfolios minus that of the two low-chaacteistic potfolios. We ebalance accounting-based factos annually at the end of each June and ebalance the etun-based factos monthly. Fund owneship: δ in data. We use quately institutional owneship data fom Thompson Financial CDA/Spectum database fom to Mutual fund chaacteistics e.g., investment objectives ae obtained fom the CRSP suvivoship-bias-fee mutual fund database. We apply standad filtes to holdings data following the liteatue: 1 we pick the fist vintage date FDATE fo each fund-epot date FUNDNO-RDATE pai to avoid stale infomation; 2 we adjust shaes held by a fund fo stock splits to account fo copoate events that happen between epot date RDATE and vintage date FDATE. We select funds focusing on the U.S. stock maket by excluding those with investment objective codes IOC of Intenational, Municipal Bonds, Bond & Pefeed, and Balanced. Fo the main esults, we map institutional investos to manages in ou model. As a 33

35 obustness check, we futhe naow down the definition of institutional investos to active domestic equity funds by utilizing investment objective codes fom CRSP, Lippe, Stategic Insight, and Wiesenbege. The esults using this naowe definition of manages ae vey simila to ou main esults available upon equest. We calculate manages owneship by summing up the stock holdings of institutional investos fo each stock in each quate. Stocks that ae listed in CRSP, but without any epoted institutional holdings, ae assumed to have zeo fund owneship. Table 1 epots summay statistics of monthly etuns and quately fund owneship fo all factos. Fund owneship at facto level. Ideally, we would like to teat each facto as an asset and compute the weight fo each facto as the faction of the total dolla amount invested by funds. Howeve, factos ae compised of numeous stocks and diffeent factos have ovelapping stock compositions. Fo example, stock A could be in the long leg of value and the shot leg of momentum. The exact dolla amount of stock A attibuted to each facto cannot be exactly identified, which complicates ou potfolio weight calculation. Instead of calculating the exact weights of factos in the fund potfolio, we calculate the elative ove/undeweight of each facto. Specifically, we measue the pofessional asset manages allocation to each facto by the spead of institutional owneship IN ST between the long leg and shot leg: INST i,t = INST long i,t INST shot i,t 40 whee INST j i,t, j = {long, shot} is the value-weighted aveage of the institutional owneship of all constituent stocks in long/shot leg of facto i. The intuition is simple. If manages have supeio knowledge of the tue etun distibution, when they oveweight cetain factos, the subsequent pefomance of these factos shall be stonge on aveage. Theefoe, in the following, we will use INST i,t to foecast facto i s futue etun. 3.2 Facto timing and obust alpha Using asset manages allocation to factos IN ST, we test whethe they have supeio knowledge of etun distibutions. Specifically, we estimate the following pedictive eges- 34

36 sion: fo facto i at time t, R i,t,t+3 = α + β INST i,t + γ X i,t + ε i,t,t+3 41 whee i = {HML, ACR, CMA, RMW, NI, MOM, ST R, LT R, BAB, IV OL, T V OL}, and R i,t,t+3 is the etun next quate i.e., month t to t + 3, and X i,t includes contol vaiables such as facto volatility that may also pedict facto etuns. We use the next-quate etun because institutional owneship data is available quately fo individual stocks. Note that IN ST at facto level vaies evey month due to the monthly ebalancing of value-weighted facto potfolios. Theefoe, ou estimation is at monthly level but with ovelapping lefthand side vaiables. Ou hypothesis is that a facto will delive highe etun in the futue if its manage owneship INST is highe now. To incease statistical powe, we pool all factos togethe to estimate a panel pedictive egession. In Table 2 Panel A, we epot the egession esults using pooled OLS and vaious fixed effect models. RV i,t is the ealized volatility of facto i estimated using pevious 36 months of facto etuns Moeia and Mui Standad eos ae double-clusteed by facto and quate. As typical in the liteatue of etun pedictability, we addess the concen ove biased standad eos due to ovelapping obsevations. Specifically, we follow the suggestion of Hodick 1992 and un the following evese egession to test the facto etun pedictability of IN ST at thee-month hoizon. 3 R i,t+1m = α + β INST i,t j + γ X i,t + ε i,t+1 42 j=0 On the left-hand side is R i,t+1m, the futue one-month etun multiplied by 3 so that it is compaable in magnitude with quately etuns. Results ae epoted in Table 2 Panel B. Ou key pediction is confimed in all specifications. In both panels, the pedictive coefficient of IN ST is positive and significant, obust to altenative standad eos and vaious fixed effects. The coefficients estimated using panel egessions and Hodick evese egessions ae vey close. Moeove, the pedictability we document is economically meaningful. Fo example, the coefficient 0.31 in the fist column of Panel B implies that, when the institutional owneship of one facto ises by one standad deviation, futue facto etun 35

37 incease by 44 bps in the following quate 1.76% annualized. Given the aveage annual facto etun of 3.31% in ou sample, a one standad-deviation change in IN ST is associated with 53% incease of expected facto etun. The evidence of facto timing by fund manages lends substantial suppot to ou model setup, the key assumption that asset manages possess supeio knowledge of etun distibution. Ou findings ae inteesting even independent fom the theoetical setup and add to the empiical liteatue on institutional owneship and asset etun pedictability. Fo example, Nagel 2005 find that the etun pedictive powe of fim chaacteistics is moe ponounced among stocks with low institutional owneship. Hee we focus on time-seies pedictability and find at the facto level institutional owneship positively pedicts futue etuns. We also implement a coss-sectional tading stategy that exploits the infomation advantage of asset manages. At the end of each quate, we ank all factos based on thei INST. We long the top 4 factos and shot the bottom 4 factos fo the next quate, weighing each facto equally. Fo compaison, we also fom an M potfolio by equally weighing the factos with medium IN ST. The potfolio is ebalanced quately. The pefomances of high IN ST factos, low IN ST factos, and that of long-shot facto potfolio ae plotted in Figue 1 cumulative etuns and Figue 2 olling aveage etuns. Factos with high fund owneship consistently outpefom factos with low fund owneship since The fact that this patten only stated to appea in the 1990s suggests that asset management industy may benefit fom the exploding eseach effots devoted to stock-maket factos in the academia, moe data souces, and the developments of data pocessing techniques especially financial econometics in the 1990s. So fa, we have only focused on the fist moment of facto etuns. In Table 3, we epot vaious moments and statistics of etuns of facto potfolios soted by fund owneship. Factos with high fund owneship exhibit highe mean etun, lowe volatility, and smalle skewness. These statistics all vay monotonically in fund owneship, suggesting that asset manages tend to invest in a set of factos with a desiable statistical pofile. Manages also tend to hold stocks with highe kutosis. This suggests that unde ambiguity, investos efain fom factos with moe exteme etuns, while asset manages ae moe willing to take on such exposue likely due to thei confidence in gauging etun distibution. Poposition 6 shows that even if the level of delegation appoaches 100%, the equilibium does not convege to CAPM. Thee exists a set of assets factos in ou empiical 36

38 context whose CAPM alpha is obust. In Figue 3, we plot the aggegate fund owneship ight Y-axis and the 60-month olling-window estimate of CAPM alpha of potfolio of high IN ST factos. We focus on high IN ST factos because given ou pevious esults, high INST factos ae moe likely to exhibit highe etuns and CAPM alpha, as they ae selected by asset manages with supeio infomation. Fund owneship exhibits a steady linea tend upwad, ising fom less than 4% in the 1980s to moe than 20% ecently. Duing this peiod, the alpha also tended up, fom negative 40bps monthly to positive 60 bps monthly with occasional decline. But oveall, thee is no evidence that a gowing asset management secto is associated with declining alpha o convegence to a CAPM economy. We also plot the 60-month olling CAPM alpha of the long-shot facto potfolio in Figue 4, and find simila pattens. 3.3 Uncetainty and alpha dispesion While model uncetainty is subjective by natue, thee exists an intinsic link between model uncetainty and time-vaying volatility. As in the case of simplified ambiguity, uncetainty in the mean aises fom estimation eos that in tun depend on the level of volatility see also Bewley Numeous studies have taken on the challenge to measue uncetainty and chaacteize its dynamics, eithe in the specific fom of time-vaying volatility o the geneal peception of uncetainty in the media o financial makets. In paticula, we conside the following uncetainty measues: the coss-sectional aveage Ut CSA and fist pincipal component Ut P CA of uncetainties estimated using a lage set of maco and financial vaiables fom Juado, Ludvigson, and Ng 2015; baseline Economic Policy Uncetainty EP U and news-based Economic Policy Uncetainty EP U news fom Bake, Bloom, and Davis 2016; CBOE stock maket volatility indexes V IX and V XO Williams Poposition 7 states that given δ, the coss-section dispesion of alpha inceases in the level of model uncetainty. Figue 5 plots fo each month, the coss-section dispesion the diffeence between maximum and minimum of factos CAPM esiduals against each of the six uncetainty measues in the pevious month. CAPM esiduals ae fom full-sample 23 Related, Diouchi et al extend the Black-Scholes option picing model by intoducing ambiguity though Choquet-Bownian motions. Using this model, they measue implied ambiguity using the minimum absolute eo between obseved index options pices and the suggested model s intinsic values. Andeouyz et al measue ambiguity using the dispesion of volume-weighted stike pices of the S&P Index options. Ulich 2013 uses entopy of inflation to measue ambiguity. 37

39 time-seies egession. The uncetainty measues ae lagged by a month because the model implies a elation between uncetainty and the expected dispesion of CAPM esiduals i.e., alpha dispesion athe than the ealized dispesion. A stong positive coelation emeges. We find simila pattens in Figue 6 that plots the dispesion of factos aw etuns. Table 4 epots the esults of paametic tests. We conside two measues of dispesion, the diffeence between maximum and minimum, and the coss-section standad deviation of factos CAPM esiduals. We foecast the dispesion with uncetainty measues Panel A. In Panel B and C, we contol fo the aw and detended fund owneship espectively, consistent with Poposition 7. Table 5 epots the esults fo facto etun dispesion. Acoss specifications, measues of uncetainty positively pedict the dispesion of factos CAPM esiduals and etuns. The economic magnitude is sizable. Fo example, an 1% one standad deviation incease of U CSA t of the coss-section standad deviation of factos CAPM esiduals. pedicts a 0.46% annualized to 5.52% incease 3.4 Model-implied uncetainty measue It is challenging to measue the uncetainty that investos face when making delegation and asset-allocation decisions, especially due to the subjective natue of model uncetainty. Poposition 8 shows how to extact investos model uncetainty fom assets CAPM alpha and investos expectations, contolling fo delegation. Empiically, we can deploy a two-step pocedue to estimate the dynamics of model-implied uncetainty, υ. Fist, given a 60-month olling window that stats in month t, we un a panel egession of factos excess etuns on the maket excess etun and investos expectations fom suvey data the maket foecasts fom Ameican Association of Individual Investos used in Geenwood and Shleife 2014: fo facto i in month s [t, t + 59], i,s f,s = a t + b i,t M,s f,s + c t suvey s 1 + ε i,s, 43 whee f,s is the isk-fee ate and the egession coefficients all have subscipt t, indicating the olling window. This egession is the empiical countepat of Equation 35. Howeve, the left-hand side of Equation 35 is CAPM alpha, while that of the egession is ealized facto etun. Theefoe, we contol fo the maket excess etun, i.e., the CAPM component Accodinglly, we allow diffeent factos to have diffeent maket beta. 38

40 Moeove, suvey is lagged because in the model, investos expectation is matched with ex ante alpha instead of ex post, ealized CAPM esiduals. Finally, note that ou suvey data is on investos expectation of futue maket etun instead of etuns of individual factos, i.e., µ f1. It is an impefect poxy, but eadily available and widely used. Next, we use the time seies of egession coefficient ĉ t to back out investos model uncetainty. In the model, ĉ t combines both the level of delegation δ and the model uncetainty υ. Theefoe, we egess ĉ t on δ t, and take the OLS esidual as ou empiical measue of model uncetainty, which we denote as ˆυ t. Figue 7 plots the time seies of ou estimated model uncetainty ˆυ t, togethe with othe measues of uncetainty in the existing liteatue which we also use in the pevious analysis. They ae the composite uncetainty measue U P CA of Juado, Ludvigson, and Ng 2015, the Economic Policy Uncetainty EP U of Bake, Bloom, and Davis 2016, and CBOE stock maket volatility index V IX. The estimated model uncetainty exhibits an economically meaningful dynamics, peaking aound majo episodes of maket tumoils such as the dotcom bubble and financial cisis. While the diffeent uncetainty measues ae not captuing the same object in theoy, they ae highly coelated. Specifically, ou measue ˆυ t has a coelation of 0.5 with the composite uncetainty measue of Juado, Ludvigson, and Ng 2015 who use infomation fom a lage set of maco and financial vaiables. 4 Conclusion Big data equies expetise. The esulting division of knowledge natually gives ise to delegation, while at the same time, geneates delegation uncetainty. The welfae impovement fom delegation helps esolve seveal puzzles in the liteatue of delegated potfolio management. In data, asset manages appea to pefom facto timing. Ou model also poduces asset picing implications suppoted by evidence. Investos hedging against delegation uncetainty ceates CAPM alpha that is obust to the ise of abitage capital wealth allocated by manages to ean alpha, and the coss-section dispesion of alpha inceases in uncetainty. We extend the Aow-Patt appoximation of ambiguity-avese utility by Maccheoni, Mainacci, and Ruffino 2013 to have closed-fom solutions and intuitive compaative statics. Delegation and delegation uncetainty aise wheeve a subset of agents ae endowed with o can invest in bette access to infomation. While we focus on the application in 39

41 financial makets, the eseach questions can be ecast into othe economic settings. Fo example, when attention is scace, communication within an oganization is impefect Dessein, Galeotti, and Santos In such cases, distotion no longe happens though the deviation of the aggegate potfolio fom the efficient fontie, but instead, takes the fom of eal esouces that ae devoted to hedge delegation uncetainty by senio manages. As we show in this pape, uncetainty and specialization geneate delegation, while delegation fundamentally changes how uncetainty entes into agents decision making by inducing delegation hedging. Enomous attention has been devoted to uncetainty in macoeconomy e.g., Bloom How does the division of knowledge and labo among agents and the esulting delegation change the ole of uncetainty? Does agents hedging against delegation uncetainty amplify o dampen the impact of uncetainty shocks? Reseach in this diection deepens ou undestanding of the macoeconomic consequences of specialization. Ambiguity has pofound implications on macoeconomic dynamics Ilut and Schneide 2014; Bianchi, Ilut, and Schneide Delegation is ubiquitous, but often ignoed in macoeconomic models. It is an inteesting to undestand how delegation due to agents diffeent pobability knowledge affects the macoeconomy by tansfoming the ole of ambiguity in agents decision making. 40

42 Table 1 Summay Statistics of Facto Retuns and Institutional Owneship This table shows the mean, median, standad deviation, count, quintile values and autocoleation coefficient ρ of monthly etuns and quately elative institutional owneship fo each facto. The constuction of the longshot factos etuns and institutional owneship follows the Fama and Fench 1993 pocedue fo constucting HML and is descibed in detail in the text. Panel A summaizes monthly annualized facto etuns. Panel B summaizes quately facto institutional owneship IN ST in pecentage. The etuns data is avaiable fo : and the owenship data is available fo 19801: ACR HML BAB CMA IVOL LTR MOM NI RMW STR TVOL Panel A: monthly etun annualized count mean std % % % ρ Panel B: quately institutional owneship IN ST % count mean std % % % ρ

43 Table 2 Pedicting Futue Facto Retuns with Fund Owneship IN ST This table shows pedictive egessions of monthly long shot facto etuns on lagged values of the facto elative institutional owneship IN ST contolling fo othe facto etun pedictos such as ealized volatility RV. Panel A epots estimations fom pooled OLS and fixed effect panel egessions: R 3m i,t+1 = α + β INST i,t + γ X i,t + ε i,t+1 The left hand vaiable is monthly ovelapping 3-month etuns. Since owneship data is efeshed quately, standad eos ae double-clusteed at quate and facto levels. Panel B epots estimations using Hodick evese pedictive egessions 3 Ri,t+1 1m = α + β 1 2 INSTi,t j n + γ X i,t + ε n i,t+1 3 j=0 The left hand vaiable is monthly non-ovelapping etuns multiplied by a facto of 3 to be compaed with estimates fom Panel A. The sample peiod is : Standad eos ae in paentheses. *,**, and *** indicate 10%, 5% and 1% statistical significance espectively. Panel A: panel egessions R 3m i,t INST RV Constant uate FE Facto FE Obsevations 4,884 4,884 4,884 4,513 4,513 4,513 Adjusted R Residual Std. Eo Panel B: Hodick 1992 evese pedictive egessions Ri,t+1 1m j=0 INST n t j RV Constant uate FE Facto FE Obsevations 4,884 4,884 4,884 4,535 4,535 4,535 Adjusted R Residual Std. Eo

44 Table 3 Summay Statistics: Equal-weighted Potfolios of Factos by Fund Owneship This table shows the annualized mean, volatility, Shape atio, skewness, kutosis, best/wost month of the etuns of potfolios of factos soted on institutional owenship IN ST. At the end of each quate, we ank all factos based on thei institutional owneship INST. We long the top 4 factos H and shot the bottom 4 factos L fo the following 3 months, weighting each facto equally. We fom an M potfolio by weighting the emaining medium IN ST factos equally. The potfolio is ebalanced quately. The sample peiod is : H M L H-L Mean ann. 4.98% 2.68% 2.06% 2.91% Vol ann. 6.16% 7.34% 11.38% 10.93% Shape Skewness Kutosis Obsevations

45 Table 4 Dispesion in CAPM Residual and Uncetainty This table epots the elationship between ex-ante uncetainty measues and subsequent dispesions in facto etun s CAPM esiduals. Panel A egesses dispesions on vaious uncetainty measues; Panel B contols fo aw institutional owneship δ; Panel C contols fo detended institutional owneship δ detend. We measue dispesion in two ways. The left-hand side uses the coss-sectional diffeence between maximum and minimum, and the ight-hand side uses coss-sectional standad deviation. The uncetainty measues include the coss-sectional aveage U CSA and fist pincipal component U P CA of uncetainties estimated using a lage set of maco and financial vaiables fom Juado, Ludvigson, and Ng 2015; baseline Economic Policy Uncetainty EP U and news-based Economic Policy Uncetainty EP U news fom Bake, Bloom, and Davis 2016; CBOE stock maket volatility indexes V IX and V XO. Fo each specification, the sample size is detemined by the availability of the uncetainty measue and fund owneship. All uncetainty measues ae nomalized to have mean of 0 and standad deviation of 1%. *,**, and *** indicate 10%, 5% and 1% statistical significance espectively. Dispesion t = ɛ max i,t ɛ min i,t Dispesion t = σ t ɛ i,t U CSA U P CA EP U EP U news V IX V XO U CSA U P CA EP U EP U news V IX V XO 44 Panel A: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + ɛ t γ *** 1.34*** 0.80*** 1.15*** 2.54*** 2.29*** 0.46*** 0.36*** 0.20** 0.31*** 0.73*** 0.66*** N R Panel B: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + γ 2 δ t 1 + ɛ t γ *** 1.67*** 0.76*** 1.16*** 2.55*** 2.34*** 0.45*** 0.46*** 0.19** 0.31*** 0.73*** 0.67*** γ *** 0.21*** ** 0.04*** 0.07*** ** N R Panel C: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + γ 2 δ detend t 1 γ *** 1.49*** 0.80*** 1.20*** 2.56*** 2.31*** 0.58*** 0.40*** 0.20** 0.32*** 0.73*** 0.67*** γ ** ** N R ɛ t

46 Table 5 Dispesion in Retun and Uncetainty This table epots the elationship between ex-ante uncetainty measues and subsequent dispesions in facto etuns. Panel A egesses dispesions on vaious uncetainty measues; Panel B contols fo aw institutional owneship δ; Panel C contols fo detended institutional owneship δ detend. We measue dispesion in two ways. The left-hand side uses the coss-sectional diffeence between maximum and minimum, and the ight-hand side uses coss-sectional standad deviation. The uncetainty measues include the coss-sectional aveage U CSA and fist pincipal component U P CA of uncetainties estimated using a lage set of maco and financial vaiables fom Juado, Ludvigson, and Ng 2015; baseline Economic Policy Uncetainty EP U and news-based Economic Policy Uncetainty EP U news fom Bake, Bloom, and Davis 2016; CBOE stock maket volatility indexes V IX and V XO. Fo each specification, the sample size is detemined by the availability of the uncetainty measue and fund owneship. All uncetainty measues ae nomalized to have mean of 0 and standad deviation of 1%. *,**, and *** indicate 10%, 5% and 1% statistical significance espectively. Dispesion t = max i,t min i,t Dispesion t = σ t i,t U CSA U P CA EP U EP U news V IX V XO U CSA U P CA EP U EP U news V IX V XO 45 Panel A: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + ɛ t γ *** 1.34*** 1.10*** 1.56*** 3.21*** 2.94*** 0.55*** 0.40*** 0.32*** 0.47*** 1.01*** 0.92*** N R Panel B: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + γ 2 δ t 1 + ɛ t γ *** 1.67*** 0.76*** 1.16*** 2.55*** 2.34*** 0.45*** 0.46*** 0.19** 0.31*** 0.73*** 0.67*** γ *** 0.21*** ** 0.04*** 0.07*** ** N R Panel C: Dispesion t = γ 0 + γ 1 Uncetainty t 1 + γ 2 δ detend t 1 γ *** 1.49*** 0.80*** 1.20*** 2.56*** 2.31*** 0.58*** 0.40*** 0.20** 0.32*** 0.73*** 0.67*** γ ** ** N R ɛ t

47 Figue 1 Cumulative Retuns of Factos Soted by Fund Owneship Equal-Weighted Cum etuns of soted facto potfolios, equally weighted 6 Cumulative Retuns H L Date Cum etuns of long-shot factos potfolio, equally weighted Cumulative Retuns Date 46

48 Figue 2 Weighted 60-month Rolling Aveage of Facto Retuns Soted by Fund Owneship, Equal month olling etuns of soted facto potfolios, equally weighted H L Rolling Retuns Date 60-month olling etuns of long-shot factos potfolio, equally weighted Rolling Retuns Date 47

49 Figue 3 60-month Rolling CAPM Alpha of Factos with High Delegation Owneship Equalweighted and Aggegate Fund Owneship month olling alpha of high INST factos potfolio, equally weighted Co. = 0.49 H alpha EW inst ight DATE

50 Figue 4 60-month Rolling CAPM Alpha of Facto Long-Shot potfolio Equal-weighted and Aggegate Delegation Owneship Co. = month olling alpha of long-shot facto potfolio, equally weighted L-S alpha EW inst ight DATE

51 Figue 5 Uncetainty and Subsequent Coss-sectional Dispesions in Facto CAPM Residuals The dispesion is measued as the time t coss-sectional diffeence between maximum and minimum of facto s CAPM esiduals. The coelation between a uncetainty measue and dispesion is shown top-ight in each panel. The uncetainty measues include the coss-sectional aveage Ut CSA and fist pincipal component Ut P CA of uncetainties estimated using a lage set of maco and financial vaiables fom Juado, Ludvigson, and Ng 2015; baseline Economic Policy Uncetainty EP U and news-based Economic Policy Uncetainty EP U news fom Bake, Bloom, and Davis 2016; CBOE stock maket volatility indexes V IX and V XO. 0.4 Co = Co = ɛ max ɛ min i,t i,t 0.2 ɛ max ɛ min i,t i,t U CSA t Ut 1 P CA 0.4 Co = Co = ɛ max ɛ min i,t i,t 0.2 ɛ max ɛ min i,t i,t EP U t EP Ut 1 news 0.4 Co = Co = ɛ max ɛ min i,t i,t 0.2 ɛ max ɛ min i,t i,t V IX t V XO t 1

52 Figue 6 Uncetainty and Subsequent Coss-sectional Dispesions in Facto Retuns The dispesion is measued as the time t coss-sectional diffeence between maximum and minimum of facto s etuns. The coelation between each uncetainty measue and dispesion is shown top-ight in each panel. The uncetainty measues include the coss-sectional aveage Ut CSA and fist pincipal component Ut P CA of uncetainties estimated using a lage set of maco and financial vaiables fom Juado, Ludvigson, and Ng 2015; baseline Economic Policy Uncetainty EP U and news-based Economic Policy Uncetainty EP U news fom Bake, Bloom, and Davis 2016; CBOE stock maket volatility indexes V IX and V XO. 0.4 Co = Co = max min i,t max i,t min 0.2 max min i,t max i,t min U CSA t Ut 1 P CA 0.4 Co = Co = max min i,t max i,t min 0.2 max min i,t max i,t min EP U t EP Ut 1 news 0.4 Co = Co = 0.50 max min i,t max i,t min max min i,t max i,t min V IX t V XO t 1

53 Figue 7 Time Seies of the Model-implied and Existing Uncetainty Measues This figue plots the model-implied uncetainty level the left Y-axis ove time, which is estimated using the two-step egession pocedue details in the main text. It also plots thee uncetainty measues the ight Y-axis, including the fist pincipal component Ut P CA of uncetainties estimated fom a lage set of maco and financial vaiables by Juado, Ludvigson, and Ng 2015, the Economic Policy Uncetainty EP U of Bake, Bloom, and Davis 2016, and CBOE volatility index V IX. The coelations between the model-implied and othe uncetainty measues ae epoted in panel titles. The shaded aeas aeas ae the NBER ecession peiods. 52