Optimal Decentralized Investment Management

Transcription

1 THE JOURNAL OF FINANCE VOL. LXIII, NO. 4 AUGUST 2008 Optmal Decentralzed Investment Management JULES H. VAN BINSBERGEN, MICHAEL W. BRANDT, and RALPH S. J. KOIJEN ABSTRACT We study an nsttutonal nvestment problem n whch a centralzed decson maker, the Chef Investment Offcer CIO), for example, employs multple asset managers to mplement nvestment strateges n separate asset classes. The CIO allocates captal to the managers who, n turn, allocate these funds to the assets n ther asset class. Ths two-step nvestment process causes several msalgnments of objectves between the CIO and hs managers and can lead to large utlty costs for the CIO. We focus on 1) loss of dversfcaton, 2) unobservable manageral appette for rsk, and 3) dfferent nvestment horzons. We derve an optmal uncondtonal lnear performance benchmark and show that ths benchmark can be used to better algn ncentves wthn the frm. We fnd that the CIO s uncertanty about the managers rsk appettes ncreases both the costs of decentralzed nvestment management and the value of an optmally desgned benchmark. THE INVESTMENT MANAGEMENT DIVISIONS OF BANKS, mutual funds, and penson funds are predomnantly structured around asset classes such as equtes, fxed ncome, and alternatve nvestments. To acheve superor returns, ether through asset selecton or market tmng, gatherng nformaton about specfc assets and captalzng on the acqured nformatonal advantage requres a hgh level of specalzaton. Ths nduces the centralzed decson maker of the frm, the Chef Investment Offcer CIO), for example, to pck asset managers who are specalzed n a sngle asset class and to delegate portfolo decsons to these specalsts. As a consequence, asset allocaton decsons are made n at least two stages. In the frst stage, the CIO allocates captal to the dfferent asset classes, each managed by a dfferent asset manager. In the second stage, each manager decdes how to allocate the funds made avalable to hm, that s, to the assets wthn hs class. Ths two-stage process can generate several msalgnments of ncentves that may lead to large utlty costs on the part of the CIO. We show that desgnng approprate return benchmarks can substantally reduce these costs. Bnsbergen s at the Graduate School of Busness, of Stanford Unversty; Brandt s at the Fuqua School of Busness, of Duke Unversty, and Kojen s at Tlburg Unversty. Brandt s also assocated wth the NBER, and Kojen s also assocated wth Netspar. We thank Suleyman Basak, Phl Dybvg, Smon Gervas, Cam Harvey, Frank de Jong, Ron Kanel, Theo Njman, Anna Pavlova, Antonos Sangvnatsos, Hans Schumacher, Rob Stambaugh the edtor), Stjn Van Neuwerburgh, Dmtr Vayanos, Sunl Wahal, Bas Werker, an anonymous referee, and semnar partcpants at ABP Investments, Duke Unversty, Tlburg Unversty, the 2006 EFA meetngs, and the 2007 AFA meetngs for helpful comments and suggestons. Jules van Bnsbergen thanks the Prns Bernhard Cultuurfonds for generous fnancal support. 1849

2 1850 The Journal of Fnance We focus on the followng mportant, although not exhaustve, lst of msalgnments of ncentves. Frst, the two-stage process can lead to severe dversfcaton losses. The unconstraned sngle-step) soluton to the mean-varance MV) optmzaton problem s lkely dfferent from the optmal lnear combnaton of MV effcent portfolos n each asset class, as ponted out by Sharpe 1981) and Elton and Gruber 2004). Second, there may be consderable, but unobservable, dfferences n appettes for rsk between the CIO and each of the asset managers. When the CIO only knows the cross-sectonal dstrbuton of rsk appettes of nvestment managers, but does not know where n ths dstrbuton a gven manager falls, delegatng portfolo decsons to multple managers can be very costly. Thrd, the nvestment horzons of the asset managers and of the CIO may be dfferent. Snce the managers are usually compensated on an annual bass, ther nvestment horzon s generally relatvely short. The CIO, n contrast, may have a much longer nvestment horzon. In practce, the performance of each asset manager s measured aganst a benchmark comprsed of a large number of assets wthn hs class. In the lterature, the man purpose of these benchmarks has been to dsentangle the effort and achevements of the asset manager from the nvestment opportunty set avalable to hm. In ths paper we show that an optmally desgned uncondtonal benchmark can also serve to mprove the algnment of ncentves wthn the frm and to substantally mtgate the utlty costs of decentralzed nvestment management. Our results provde a dfferent perspectve on the use of performance benchmarks. Admat and Pflederer 1997) take a realstc benchmark as gven and show that when an nvestment manager uses the condtonal return dstrbuton n hs nvestment decsons, restrctng hm by an uncondtonal benchmark dstorts ncentves. 1 In ther framework, ths dstorton can only be prevented by settng the benchmark equal to the mnmum-varance portfolo. We show that the negatve aspect of uncondtonal benchmarks can be offset, at least n part, by the role of uncondtonal benchmarks n algnng other ncentves, such as dversfcaton, rsk preferences, and nvestment horzons. We use a stylzed representaton of an nvestment management frm to quantfy the costs of the msalgnments for both constant and tme-varyng nvestment opportuntes. We assume that the CIO acts n the best nterest of a large group of benefcares of the assets under management, whereas the nvestment managers only wsh to maxmze ther personal compensaton. Usng two asset classes bonds and stocks) and three assets per class government bonds, Baa-rated corporate bonds, and Aaa-rated corporate bonds n the fxed ncome class, and growth stocks, ntermedate, and value stocks n the equtes class) the utlty costs can range from 50 to 300 bass ponts per year. We therefore argue that decentralzaton has a frst-order effect on the performance of nvestment management frms. We demonstrate that when the nvestment opportunty set s constant and rsk atttudes are observable, the CIO can fully algn ncentves through an 1 See also Basak, Shapro, and Teplá 2006).

3 Optmal Decentralzed Investment Management 1851 uncondtonal benchmark consstng only of assets n each manager s asset class. In other words, cross-benchmarkng s not requred. Furthermore, we derve the perhaps counterntutve result that the rsk averson levels of the asset managers for whch the utlty costs of the CIO are mnmzed can substantally dffer from the rsk averson of the CIO. We then consder the case of tme-varyng nvestment opportuntes and show that an uncondtonal passve) benchmark can stll substantally, though not fully, mtgate the utlty costs of decentralzed nvestment management. Next we generalze our model by relaxng the assumpton that the CIO knows the asset managers rsk appette. Specfcally, we derve the optmal benchmark assumng that the CIO only knows the cross-sectonal dstrbuton of nvestment managers rsk averson levels, but does not know where n ths dstrbuton a gven manager falls. We fnd that the qualtatve results on the benefts of optmal benchmarkng derved for a known rsk averson level apply to ths more general case. In fact, we fnd that uncertanty about the managers rsk appettes ncreases both the costs of decentralzed nvestment management and the value of an optmally desgned benchmark. The negatve mpact of decentralzed nvestment management on dversfcaton was frst noted by Sharpe 1981), who shows that f the CIO has ratonal expectatons about the portfolo choces of the nvestment managers, he can choose hs nvestment weghts such that dversfcaton s at least partally restored. However, ths optmal lnear combnaton of MV effcent portfolos wthn each asset class usually stll dffers from the optmally dversfed portfolo over all assets. To restore dversfcaton further, Sharpe 1981) suggests that the CIO mposes nvestment rules on one or both of the nvestment managers to solve an optmzaton problem that ncludes the covarances between assets n dfferent asset classes. Elton and Gruber 2004) show that t s possble to overcome the loss of dversfcaton by provdng the asset managers wth nvestment rules that they are requred to mplement. The asset managers can then mplement the CIO s optmal strategy wthout gvng up ther prvate nformaton. Both nvestment rules descrbed above nterfere wth the asset manager s desre to maxmze hs ndvdual performance, on whch hs compensaton depends. Furthermore, when the nvestment choces of the managers are not always fully observable, these ad hoc rules are not enforceable. In contrast, we propose to change managers ncentves by ntroducng a return benchmark aganst whch the managers are evaluated for the purpose of ther compensaton. When ths benchmark s mplemented n the rght way, t s n the managers own nterest to follow nvestment strateges that are more) n lne wth the objectves of the CIO. In Secton I, we assume that nvestment opportuntes are constant. Ths allows us to focus on the loss of dversfcaton and on dfferences n preferences n a parsmonous framework. We then add market-tmng skll and horzon effects n Secton II. Both sectons assume that the CIO knows the managers rsk atttudes. Ths assumpton s relaxed n Secton III. Perhaps one of the most nterestng questons s why the CIO should hre multple asset managers to begn wth. Sharpe 1981) argues that the decson

4 1852 The Journal of Fnance to employ multple managers may be motvated by the desre to explot ther specalzaton or to dversfy among asset managers. Alternatvely, Barry and Starks 1984) argue that rsk-sharng consderatons may be a motvaton to employ more than one manager. In Secton II, nvestment opportuntes are tme-varyng, consstent wth the emprcal evdence that equty and bond returns are to some extent predctable. 2 Ths allows sklled managers to mplement actve strateges that generate alphas, when compared to uncondtonal passve) return benchmarks. Ths specfc nterpretaton of alpha may seem unconventonal, but t avods the queston of whether asset managers do or do not have prvate nformaton. Treynor and Black 1973), Admat and Pflederer 1997), and Elton and Gruber 2004) assume that managers can generate alpha, but do not explctly model how managers do so. Cvtanć, Lazrak, Martelln, and Zapatero 2006) assume that the nvestor s uncertan about the alpha of the manager and derve the optmal polcy n that case. We explctly model the tme-varaton n nvestment opportuntes and assume that the resultng predctablty can be exploted by sklled managers to generate value. Apart from the tactcal aspect of return predctablty, tme-varaton n rsk prema can also have mportant strategc consequences. After all, when asset returns are predctable, the optmal portfolo choce of the CIO depends on hs nvestment horzon. 3 Ths then requres dynamc optmzaton to fnd the optmal composton of the CIO s portfolo. The resultng portfolo choce s referred to as strategc as opposed to myopc or tactcal). The dfferences between the strategc and myopc portfolo weghts are called hedgng demands as they hedge aganst future changes n the nvestment opportunty set. These hedgng demands are usually more pronounced for longer nvestment horzons of the CIO. As the remuneraton schemes of nvestment managers are generally based on a relatvely short perod, ther portfolo weghts wll be vrtually myopc. The CIO, n contrast, usually has a long-term nvestment horzon. Ths leads to a thrd msalgnment of ncentves. When uncondtonal benchmarks are used to overcome costs nduced by dfferences n nvestment horzons, a key queston s whether 1) the benchmark and/or 2) the strategc allocaton to the dfferent asset classes exhbt horzon effects. Most strategc asset allocaton papers take a centralzed perspectve and assume that the tactcal and strategc aspects are n perfect harmony. 4 Once nvestment management s decentralzed, tactcal and strategc motves are splt between the managers and the CIO, respectvely. We show that both the strategc allocaton, that s, the allocaton to the varous asset classes, and 2 See, for example, Ang and Bekaert 2005), Lewellen 2004), Campbell and Yogo 2006), Torous, Valkanov, and Yan 2005), van Bnsbergen and Kojen 2007), and Lettau and Van Neuwerburgh 2007) for stock return predctablty, and Da and Sngleton 2002) and Cochrane and Pazzes 2005) for bond return predctablty. 3 See, for nstance, Km and Omberg 1996), Brennan, Schwartz, Lagnado 1997), Campbell and Vcera 1999), Brandt 1999, 2005), Aït-Sahala and Brandt 2001), Campbell, Chan, and Vcera 2003), Jurek and Vcera 2006), and Sangvnatsos and Wachter 2005). 4 Consder, for nstance, Brennan, Schwartz, Lagnado 1997), Campbell, Chan, and Vcera 2003), and Jurek and Vcera 2006).

5 Optmal Decentralzed Investment Management 1853 the optmal benchmarks exhbt strong horzon effects. When nvestment managers are not constraned by a benchmark, the horzon effects n the strategc allocaton are less pronounced, mplyng that the strategc allocaton and optmal benchmarks should be desgned jontly. Our paper also relates to the standard prncpal-agent lterature n whch the agent s effort s unobservable. In the delegated portfolo management context, the agent should exert effort to gather the nformaton needed to make the rght portfolo decsons, as explored by Ou-Yang 2003). 5 We abstract from explctly modelng the effort choces of the asset managers. Instead, the managers add value by tmng the market, whch we assume the CIO cannot do. The agency problem arses because the nvestment managers, whose actons are not always fully observable, wsh to maxmze ther annual compensaton, whereas the CIO acts n the best nterest of the benefcares of the frm. When desgnng the benchmarks, the CIO faces a trade-off between 1) allowng the nvestment managers to realze the gans from market tmng and 2) correctng the msalgnments of ncentves descrbed above. As a result, the nvestment problem we solve s nontrvally more dffcult than the problem wth a CIO and a sngle nvestment manager. The strategc allocaton of the CIO results from a jont optmzaton over the benchmark and the strategc allocaton to the asset managers. In the prncpal-agent lterature above, t s common practce to assume that the preferences of the agents the nvestment managers) are known to the prncpal the CIO). We extend ths lterature by also consderng the realstc case n whch the prncpal has lmted knowledge about the agents preferences. As mentoned before, we assume that the CIO knows the cross-sectonal dstrbuton of nvestment managers rsk appettes, but does not know where n ths dstrbuton a gven manager falls. We derve approxmate) closed-form solutons for the strategc allocaton to the asset classes. In partcular, we show that uncertanty about the managers rsk atttudes propagates as a form of background rsk Goller and Pratt 1996)), whch effectvely ncreases the rsk averson of the CIO. Alternatvely, lmted knowledge of the managers rsk atttudes can be nterpreted as a form of Bayesan parameter uncertanty see, for example, Barbers 2000) and Brennan and Xa 2001)). For ease of exposton, we confne attenton to a tractable constant relatve rsk averson preference structure and a realstc lnear class of performance benchmarks that are assumed to satsfy the partcpaton constrant of the asset managers. Fnally, our work relates to the organzatonal lterature of Dessen, Garcano, and Gertner 2005), who nvestgate a general manager the CIO) who attempts to acheve a common goal whle provdng strong performance-lnked compensaton schemes to specalsts the nvestment managers) to overcome the moral hazard problem. They show that to acheve the common goal, ndvdual ncentves may have to be weakened. A common way to algn ncentves s to gve the managers a share n each other s output. Our results ndcate that n the 5 Stracca 2005) provdes a recent survey of the theoretcal lterature on delegated portfolo management.

6 1854 The Journal of Fnance portfolo management settng, cross-benchmarkng, where the benchmark of an asset manager ncludes assets from other classes, s not requred. 6 The paper proceeds as follows. In Secton I, we present the model n a fnancal market wth constant nvestment opportuntes. Secton II extends the fnancal market by allowng for tme-varaton n expected returns. In Secton III, we generalze our framework by consderng the problem of a CIO who s uncertan about the managers rsk atttudes. Secton IV concludes. I. Constant Investment Opportuntes A. Fnancal Market and Preferences We assume that the fnancal market contans 2k + 1 assets wth prces denoted by S, = 0,...,2k. The frst asset, S 0, s a rskless cash account, that evolves accordng to: ds 0t = r dt, 1) S 0t where r denotes the constant) nstantaneous short rate. The remanng 2k assets are rsky. We assume that the dynamcs of the rsky assets are gven by geometrc Brownan motons. For = 1,...,2k, wehave ds t = r + σ S ) dt + σ dz t, 2) t where denotes a 2k-dmensonal vector of, for now, constant prces of rsk and Z sa2k-dmensonal vector of ndependent standard Brownan shocks. All correlatons between asset returns are captured by the volatlty vectors σ. The volatlty matrx of the frst k assets s gven by 1 = σ 1,..., σ k ) and for the second k assets by 2 = σ k+1,..., σ 2k ). The CIO, who acts n the best nterest of the benefcares of the frm, employs two asset managers. The managers ndependently decde on the optmal composton of ther portfolos usng a subset of the avalable assets. The frst asset manager has the mandate to manage the frst k assets and the second manager has the mandate to nvest n the remanng k assets. We explctly model the preferences of both the CIO and the nvestment managers. Intally, the preference structures are assumed to be common knowledge. We assume that the preferences of the CIO and of the two asset managers can be represented by a CRRA utlty functon, so that each solves the problem ) 1 max E t W 1 γ T x s) s [t,t ] 1 γ, 3) where γ denotes the coeffcent of relatve rsk averson, T denotes the nvestment horzon, and = 1, 2, C refers to the two asset managers and the CIO, 6 For a treatment of decentralzed nformaton processng wthn the frm, see Vayanos 2003).

7 Optmal Decentralzed Investment Management 1855 respectvely. The vector x denotes the optmal portfolo weghts n the dfferent assets avalable to agent. Accordng to equaton 3), the preferences of the CIO and the nvestment managers may be conflctng along two dmensons. Frst, the rsk atttudes are lkely to be msmatched. Second, the nvestment horzon used n determnng the optmal portfolo choces are potentally dfferent. The remuneraton schemes of asset managers usually nduce short, say annual, nvestment horzons. Ths form of manageral myopa tends to be at odds wth the more long-term perspectve of the CIO. The dfference n horzons s partcularly mportant for CIOs wth long-term mandates from penson funds and lfe nsurers. For now, we assume that nvestment opportuntes are constant. Secton I.B solves for the optmal portfolo choce when nvestment management s centralzed, mplyng that the CIO optmzes over the complete asset menu. Obvously, n ths case, all msalgnments of ncentves mentoned before are absent. However, when the nvestment management frm has a rch nvestment opportunty set and a substantal amount of funds under management, centralzed nvestment management becomes nfeasble. In Secton I.C, we ntroduce asset managers for each asset class assumng that the asset managers are not constraned by a benchmark. In Secton I.D, the asset managers are then evaluated relatve to a performance benchmark, and we show how to desgn ths benchmark optmally. The proofs of the man results are provded n Appendces A to C. B. Centralzed Problem As a pont of reference, we consder frst the centralzed problem n whch the CIO decdes on the optmal weghts n all 2k + 1 assets. The nstantaneous volatlty matrx of the rsky assets s gven by = 1, 2 ). The correspondng optmal portfolo s gven by x C = 1 ) 1, 4) γ C wth the remander, 1 x Cι, nvested n the cash account. The utlty derved by the CIO from mplementng ths optmal allocaton s J 1 W, τ C ) = 1 W 1 γ C expa 1 τ C ), 5) 1 γ C where τ C = T C t and a 1 = 1 γ C )r + 1 γ C 2γ C ) 1. When nvestment opportuntes are constant, the CIO s optmal allocaton s ndependent of the nvestment horzon, as shown by Merton 1969, 1971). Suppose that the asset set contans sx rsky assets. The frst three rsky assets are fxed ncome portfolos, namely, a government bond ndex and two Lehman corporate bond ndces wth Aaa and Baa ratngs, respectvely. The remanng three rsky assets are equty portfolos made up of frms sorted nto value, ntermedate, and growth categores based on ther book-to-market rato.

8 1856 The Journal of Fnance Table I Constant Investment Opportuntes Ths table gves the estmaton results of the fnancal market n Secton I over the perod January 1973 through November 2004 usng monthly data. The model s estmated by maxmum lkelhood. The asset set contans government bonds Gov. bonds ), corporate bonds wth credt ratngs Baa Corp. bonds, Baa ) and Aaa Corp. bonds, Aaa ), and three equty portfolo ranked on ther book-to-market rato growth/ntermedate Int. )/value). Panel A provdes the model parameters and Panel B portrays the mpled nstantaneous expected returns r + ) and correlatons. In determnng, we assume that the nstantaneous nomnal short rate equals r = 5%. Panel A: Model Parameters Source of Rsk Z 1 Z 2 Z 3 Z 4 Z 5 Z Gov. bonds 13.5% Corp. bonds, Baa 8.2% 5.6% Corp. bonds, Aaa 9.1% 2.7% 2.4% Growth stocks 3.7% 6.3% 0.3% 16.5% 0 0 Int. stocks 3.6% 6.8% 0.3% 11.7% 7.3% 0 Value stocks 3.6% 7.7% 0.1% 10.4% 6.8% 5.9% Panel B: Impled Parameters Expected return Correlaton Gov. bonds 9.5% 100% 82% 93% 20% 23% 22% Corp. bonds, Baa 10.1% 82% 100% 92% 37% 43% 45% Corp. bonds, Aaa 9.1% 93% 92% 100% 29% 34% 34% Growth stocks 10.9% 20% 37% 29% 100% 88% 80% Int. stocks 14.0% 23% 43% 34% 88% 100% 93% Value stocks 15.7% 22% 45% 34% 80% 93% 100% The model s estmated by maxmum lkelhood usng data from January 1973 through November The nomnal short rate s set to 5% per annum. Fnally, to ensure statstcal dentfcaton of the elements of the volatlty matrx, we assume that s lower trangular. The estmaton results are provded n Table I. Panel A shows estmates of the parameters and. Panel B shows the mpled nstantaneous expected return and correlatons between the assets. In the fxed ncome asset class, we fnd an expected return spread of 1% between corporate bonds wth a Baa versus Aaa ratng. In the equtes asset class, we estmate a hgh value premum of 4.8%. The correlatons wthn asset classes are hgh, between 80% and 90%. Furthermore, there s clear dependence between asset classes, whch, as we show more formally later, mples that the two-stage nvestment process leads to neffcences. C. Decentralzed Problem wthout a Benchmark We now solve the decentralzed problem n whch the frst asset manager has the mandate to decde on the frst k assets and the second asset manager

9 Optmal Decentralzed Investment Management 1857 manages the remanng k assets. Nether of the asset managers has access to a cash account. If they dd, they could hold hghly leveraged postons or large cash balances, whch s undesrable from the CIO s perspectve. 7 The CIO allocates captal to the two asset managers and nvests the remander, f any, n the cash account. The optmal portfolo of asset manager when he s not constraned by a benchmark s x NB = 1 x + 1 x ι ) x MV, 6) γ γ where x = ) 1 and x MV = ) 1ι ι ) 1ι. 7) The optmal portfolo of the asset managers can be decomposed nto two components. The frst component, x, s the standard myopc demand that optmally explots the rsk return trade-off. The second component, x MV, mnmzes the nstantaneous return varance and s therefore labeled the mnmum-varance portfolo. The mnmum-varance portfolo substtutes for the rskless asset n the optmal portfolo of the asset manager. The two components are then weghted by the rsk atttude of the asset manager to arrve at the optmal portfolo. The CIO has to decde how to allocate captal to the two asset managers as well as to the cash account. We call ths decson the strategc asset allocaton. The nvestment problem of the CIO s of the same form as n the centralzed problem, but wth a reduced asset set. In the centralzed settng the CIO has access to 2k + 1 assets. In the decentralzed case, each asset manager combnes the k assets n hs class to form hs preferred portfolo. The CIO can then only choose between these two portfolos and the cash account. The nstantaneous volatlty matrx of the two rsky portfolos avalable to the CIO s gven by = 1 xnb 1, 2 xnb 2 ). Thus, the optmal strategc allocaton of the CIO to the two asset managers s x C = 1 ) 1, 8) γ C wth the remander, 1 x Cι, nvested n the cash account. Note that n ths case x C s a two-dmensonal vector, contanng the strategc allocaton to both managers, as opposed to a 2k-dmensonal vector wth the weghts allocated to each of the assets as n equaton 4). Throughout the paper, utlty costs of decentralzed nvestment management are calculated at the centralzed level. In other words, we use the value functon of the CIO the prncpal) to measure utlty losses. 7 A smlar cash constrant has been mposed n nvestment problems wth a CIO and a sngle nvestment manager e.g., Brennan 1993) and Gómez and Zapatero 2003)).

10 1858 The Journal of Fnance The value functon of the CIO wth decentralzaton s gven by J 2 W, τ C ) = 1 1 γ C W 1 γ C expa 2 τ C ), 9) where τ C = T C t and a 2 = 1 γ C )r + 1 γ C 2γ C ) 1. It s straghtforward to show that the value functon n equaton 5) the centralzed problem) s larger than or equal to the value functon n equaton 9) the decentralzed problem). Ths follows from the fact that the two-stage asset allocaton procedure reduces the asset set of the CIO. The CIO can only allocate funds between the two managers, whch does not provde suffcent flexblty to always acheve the frst-best soluton. The two-stage asset allocaton results n the frst-best outcome only when the asset managers already happen to mplement the proper relatve weghts wthn ther asset classes. In ths case, the CIO can use the strategc allocaton to scale up the asset managers weghts to the optmal frm-level allocaton. A set of suffcent condtons for ths to hold s gven by 1 2 = 0 k k, 10) x ι = γ, 11) wth = 1, 2. Note that even when asset classes are ndependent, that s, Condton 10) holds, the frst-best allocaton s generally not attanable. If asset classes are ndependent and when managers do not have access to a cash account, managers allocate ther funds to the effcent tangency portfolo and the neffcent mnmum-varance portfolo of ther asset classes. Condton 11) ensures that the nvestment n the mnmum-varance portfolo equals zero. If both condtons are satsfed, the CIO s optmal strategc allocaton to the managers s gven by γ /γ C, = 1, 2. Fgure 1 llustrates the soluton of the decentralzed portfolo problem for a CIO who hres two nvestment managers wth equal rsk averson of 10. Panel A shows the MV fronter of the bond manager, the MV fronter of the stock manager, and the CIO s optmal lnear combnaton of these two fronters. The decentralzed MV fronter crosses the MV fronter for stocks at the preferred portfolo of the stock manager, and t crosses the MV for bonds at the portfolo chosen by the bond manager. Panel B compares the decentralzed MV fronter wth the centralzed MV fronter. As argued above, the decentralzed MV fronter les wthn the centralzed MV fronter. The welfare loss due to decentralzed nvestment management can be nferred from the dfference n Sharpe ratos.e., the slope of the lnes n MV space through the pont 0, r) and tangent to the centralzed and decentralzed MV fronter, respectvely). Fnally, panel B also dsplays the portfolo choces of the CIO for both the centralzed and decentralzed scenaros. The results clearly show that the CIO nvests more conservatvely n the decentralzed case. In fact, t can be shown n general that the optmal decentralzed portfolo s more conservatve than the optmal centralzed portfolo.

11 Optmal Decentralzed Investment Management 1859 Fgure 1. Decentralzed nvestment management problem. Ths fgure shows a decentralzed asset allocaton problem n whch a CIO delegates portfolo decsons to a stock and a bond manager. Both asset managers have a rsk averson coeffcent of γ 1 = γ 2 = 10. The bond manager nvests n government bonds and corporate bonds wth Aaa and Baa ratngs. The stock manager nvests n growth, ntermedate, and value stocks. Panel A shows the mean-varance fronter for stocks and for bonds. The decentralzed mean-varance fronter ntersects the stock and bond meanvarance fronters at the preferred portfolos of the bond and the stock manager. The CIO allocates money to the two managers and a rskless asset that pays 5% per year. Panel B compares the meanvarance fronter of the decentralzed nvestment problem wth that of the centralzed nvestment problem and depcts the optmal portfolo choces of the CIO for the CIO s rsk averson level γ C, equal to 2, 5, and 10.

12 1860 The Journal of Fnance Fgure 2. Losses from decentralzed nvestment management. Ths fgure depcts the dversfcaton losses due to decentralzed nvestment management as a functon of the rsk averson of the nvestment managers. The CIO has a rsk averson coeffcent γ C = 5 n Panel A and γ C = 10 n Panel B. The horzontal axes depct the rsk appettes of the asset managers. The losses are computed by takng the rato of the annualzed certanty equvalents acheved under decentralzed and centralzed nvestment management after whch we subtract one and multply by 10,000 to express the losses n bass ponts per year. For example, 160 bass ponts mples a loss n terms of certanty equvalents of 1.6% of wealth per year. In Fgure 2, we show the utlty losses nduced by decentralzed nvestment management for varous combnatons of manageral rsk atttudes. The coeffcent of relatve rsk averson for the CIO equals γ C = 5 n Panel A and γ C = 10 n Panel B. We defne the utlty loss as the decrease n the annualzed

13 Optmal Decentralzed Investment Management 1861 certanty-equvalent return at the frm level. Interestngly, ths loss s not mnmzed when the rsk averson of the asset managers s equal to that of the CIO. In fact, the cost of decentralzed nvestment management s mnmzed for a rsk averson of 3.3 for the stock manager and 5.7 for the bond manager, regardless of the rsk averson of the CIO. Even though the locaton of the mnmum s not dependent on the rsk averson of the CIO, the utlty loss ncurred obvously s. When the rsk averson of the CIO equals fve, the mnmum dversfcaton losses are eght bass ponts per year n terms of certanty equvalents. Ths number drops to four bass ponts when the rsk averson of the CIO equals 10 because he moves out of rsky assets and nto the rskless asset. The utlty loss can ncrease to bass ponts even n ths smple example for dfferent rsk atttudes of the nvestment managers. Fnally, note that when the CIO s forced to hre a bond manager who does not have the optmal rsk averson level, ths may nfluence the CIO s preferred choce of stock manager and vce versa. Fgure 3 dsplays the portfolo compostons of the bond manager n Panel A and of the stock manager n Panel B as functons of ther rsk averson. Recall that the managers do not have access to a rskless asset. Fgure 4 shows the fracton of total rsky assets that s allocated to the stock manager as a functon of hs and the bond manager s) rsk averson. The bond manager receves one mnus ths allocaton. The allocaton of captal between the rskless and the rsky assets depends on the rsk averson of the CIO and s not shown. D. Decentralzed Problem wth a Benchmark We now consder the decentralzed nvestment problem n whch the CIO desgns a performance benchmark for each of the nvestment managers n an attempt to algn ncentves. We restrct attenton to benchmarks n the form of portfolos that can be replcated by the asset managers. Ths restrcton mples that only the assets of the partcular asset class are used and that the benchmark contans no cash poston. There s no possblty and, as we show later, no need for cross-benchmarkng. We denote the value of the benchmark of manager at tme t by B t and the weghts n the benchmark portfolo for asset class by β. The evoluton of benchmark s gven by db t = r + β B ) dt + β dz t, 12) t where β ι = 1, for = 1, 2. We assume that the asset managers derve utlty from the rato of the value of assets under ther control to the value of the benchmark. They face the problem ) ) 1 1 γ WT max E t. 13) x s ) s [t,t ] 1 γ B T Ths preference structure can be motvated n several ways. Frst, the remuneraton schemes of asset managers usually contan a component that depends

14 1862 The Journal of Fnance Fgure 3. Portfolo compostons wthout a benchmark. Ths fgure dsplays the portfolo composton of the bond manager n Panel A and the stock manager n Panel B as functons of ther coeffcents of relatve rsk averson when they are not restrcted by a benchmark. The asset managers do not have access to a rskless asset. on ther performance relatve to a benchmark. Ths s captured n our model by specfyng preferences over the rato of funds under management to the value of the benchmark, n lne wth Browne 1999, 2000). Second, nvestment managers often operate under rsk constrants. An mportant way to measure rsk

15 Optmal Decentralzed Investment Management 1863 Fgure 4. Fracton of rsky funds allocated to equtes wthout a benchmark. Ths fgure dsplays the percentage of total nvestment n rsky assets that s under control of the stock manager as a functon of the rsk averson of the bond and the stock manager. attrbutable to manager s to employ trackng error volatlty. The trackng error s usually defned as the return dfferental of the funds under management and the benchmark. Takng logs of the rato of wealth to the benchmark provdes the trackng error n log returns. Thrd, for nvestment management frms that need to account for labltes, such as penson funds and lfe nsurers, supervsory bodes often summarze the fnancal poston by the rato of assets to labltes, the so-called fundng rato as further descrbed n Sharpe 2002) and van Bnsbergen and Brandt 2007). Hence, the rato of wealth to the benchmark labltes) can be nterpreted as a reasonable summary statstc of relatve performance. 8 When the performance of asset manager s measured relatve to the benchmark, hs optmal portfolo s gven by x B = 1 γ x + 1 1γ ) β + 1 γ 1 x ι ) x MV, 14) where x and x MV are gven n equaton 7). Ths portfolo dffers from the optmal portfolo n the absence of a benchmark n two mportant respects. 8 In addton, Stutzer 2003a) and Foster and Stutzer 2003) show that when the optmal portfolo s chosen so that the probablty of underperformance tends to zero as the nvestment horzon goes to nfnty, the portfolo that maxmzes the probablty decay rate solves an objectve smlar to power utlty wth two man modfcatons. Frst, the nvestor s preferences nvolve the rato of wealth over the benchmark. Second, the nvestor s coeffcent of relatve rsk averson depends on the nvestment opportunty set. Ths provdes an alternatve nterpretaton of preferences over the rato of wealth to the benchmark as well as dfferent coeffcents of relatve rsk averson for the varous asset classes.

16 1864 The Journal of Fnance Frst, the optmal portfolo contans a component that replcates the composton of the benchmark portfolo. It s exactly ths response of the nvestment manager that allows the CIO to optmally desgn a benchmark to algn ncentves. Note that the benchmark weghts enter the optmal portfolo lnearly. Second, when the coeffcent of relatve rsk averson, γ, tends to nfnty, the asset manager tracks the benchmark exactly. Hence, the benchmark s consdered to be the rskless asset from the perspectve of the asset manager. The CIO has to optmally desgn the two benchmark portfolos and has to determne the allocaton to the two asset managers as well as to the cash account. It s mportant to note that x B = x NB when β = x MV. That s, the optmal portfolos wth and wthout a performance benchmark concde when the benchmark portfolo equals the mnmum-varance portfolo. Ths mples that when desgnng a benchmark, the no-benchmark case s n the choce set of the CIO. As a consequence, the optmal benchmark wll reduce the utlty costs of decentralzed nvestment management. More mportantly, when nvestment opportuntes are constant, the benchmark can be desgned so that all neffcences are elmnated. The composton of the optmal benchmark that leads to the optmal allocaton of the centralzed nvestment problem s gven by β = x MV + γ x C ) γ 1 x C ι xnb, 15) where x C are the optmal weghts for the assets under management by manager when the CIO controls all assets as gven n equaton 4) and x NB s gven n equaton 6). The benchmark weghts sum to one because of the restrcton that the benchmark cannot contan a cash poston. The two components of the optmal benchmark portfolo have a natural nterpretaton. The frst component s the mnmum-varance portfolo. As we pont out above, once the benchmark portfolo concdes wth the mnmum-varance portfolo, the benchmark does not affect the manager s optmal portfolo. The second component, however, corrects the manager s portfolo choce to algn ncentves. If the relatve weghts of the CIO and the portfolo of the manager wthout a benchmark.e., x NB ) concde, there s no need to nfluence the manager s portfolo and the second term s zero. However, when the CIO optmally allocates a larger share of captal to a partcular asset n class, the optmal benchmark wll contan a postve poston n ths asset when γ > 1. The rato before the second component accounts for the manager s preferences. If the manager s more aggressve.e., γ 1), the benchmark weghts are more extreme as the manager s less senstve to benchmark devatons. If the nvestor becomes more conservatve.e., γ ), we get x NB = x MV and the benchmark concdes wth the relatve weghts of the CIO. Fnally, the CIO uses the strategc allocaton to the two asset managers to mplement the optmal frm-level allocaton. The optmal weght allocated to each manager s gven by x C ι, wth = 1, 2, and the remander, 1 x C 1 ι xc 2 ι, s nvested n the cash account.

17 Optmal Decentralzed Investment Management 1865 Fgure 5. Composton of the optmal performance benchmarks. Composton of the optmal bond benchmark n Panel A and stock benchmark n Panel B as a functon of the rsk averson of the asset managers. Fgure 5 shows the composton of the optmal benchmarks for the bond manager n Panel A and for the stock manager n Panel B as functons of ther rsk averson. The mechansm through whch the benchmark algns ncentves s partcularly clear for the fxed ncome asset class. Wthout a benchmark, the

18 1866 The Journal of Fnance bond manager nvests too aggressvely n corporate bonds wth a Baa ratng. The optmal benchmark therefore contans a large short poston n the same asset that reduces the manager s allocaton to Baa-rated bonds. For Aaa-rated bonds, the benchmark provdes exactly the opposte ncentve. II. Tme-Varyng Investment Opportuntes A. Fnancal Market In Secton I, nvestment opportuntes are constant through tme and there are only two neffcences caused by decentralzed nvestment management, namely, loss of dversfcaton between asset classes and msalgnments n rsk atttudes. However, the role of asset managers s rather lmted n that they add no value n the form of stock selecton or market tmng. In ths secton, we allow nvestment opportuntes, and n partcular expected returns, to be tme-varyng and drven by a set of common forecastng varables. Ths settng allows asset managers to mplement actve strateges that optmally explot changes n nvestment opportuntes n ther respectve asset classes. These actve strateges can generate alphas when compared to an uncondtonal passve) performance benchmark. Thus, actve asset management can be valueenhancng. Ths extenson of the problem adds several new nterestng dmensons to the decentralzed nvestment management problem. Frst, dfferences n nvestment horzons create another msalgnment of ncentves. The CIO generally acts n the long-term nterest of the nvestment management frm, whle asset managers tend to be more shortsghted, possbly nduced by ther remuneraton schemes. When the predctor varables are correlated wth returns, t s optmal to hedge future tme-varaton n nvestment opportuntes. 9 As a consequence, the myopc portfolos held by the asset managers wll generally not concde wth the CIO s optmal portfolo that ncorporates long-term hedgng demands. Second, when a common set of predctor varables affects the nvestment opportuntes n both asset classes, actve strateges are potentally correlated. Ths mples that even f nstantaneous returns are uncorrelated, long-term returns can be correlated, whch aggravates the loss of dversfcaton due to decentralzaton. Thrd, the role of benchmarks s markedly dfferent compared to the case of constant nvestment opportuntes. For the sake of realsm, we restrct attenton to passve uncondtonal) strateges as return benchmarks. As we dscussed earler, Admat and Pflederer 1997) show that when the asset manager has prvate nformaton, an uncondtonal benchmark can be very costly. After all, the asset managers base ther decson on the condtonal return dstrbuton, whereas the CIO desgns the benchmark usng the uncondtonal return dstrbuton. 10 In ther framework, t follows therefore, that unless the benchmark s set equal to the mnmum-varance portfolo, t 9 See, for nstance, Km and Omberg 1996), Campbell and Vcera 1999), Brandt 1999), and Lu 2007). 10 Although the predctors are publcly observed, we assume that the CIO s tme-constraned or not suffcently specalzed to explot ths nformaton. As such, the condtonal return dstrbuton

19 Optmal Decentralzed Investment Management 1867 nduces a potentally large effcency loss. In our model, n contrast, the benchmark s used to algn ncentves n a decentralzed nvestment management frm. We now consder a more general fnancal market n whch the prces of rsk,, can vary over tme. More explctly, we model X ) = X, 16) where X denotes an m-dmensonal vector of de-meaned state varables that capture tme-varaton n expected returns. Although the state varables are tme varyng, we drop the subscrpt t for notatonal convenence. All portfolos n ths secton are ndexed wth ether the state realzaton, X, or the nvestment horzon, τ, n order to emphasze the condtonng nformaton used to construct the portfolo polces. Most predctor varables used n the lterature, such as term structure varables and fnancal ratos, are hghly persstent. In order to accommodate frst-order autocorrelaton n predctors, we model ther dynamcs as Ornsten Uhlenbeck processes: dx t = κ X t dt + σ X dz t, 17) where Z now denotes a 2k + m)-dmensonal Brownan moton. The volatlty matrx of the m predctors s gven by X = σ X1,..., σ Xm ). We assume agan that only the CIO has access to a cash account. Fnally, we postulate the same preference structures for the CIO and the asset managers as n Secton I.A. We estmate the return dynamcs usng three predctor varables: the short rate, the yeld on a 10-year nomnal government bond, and the log dvdend yeld of the equty ndex. These predctors have been used n strategc asset allocaton problems to capture the tme-varaton n expected returns see the references n footnote 3). The model s estmated by maxmum lkelhood usng data from January 1973 through November The estmaton results are presented n Table II. The estmates of the uncondtonal nstantaneous prces of rsk, 0, are smlar to the results n Table I. The second part of Table II descrbes the responses of the expected returns of the ndvdual assets to changes n the state varables, 1. We fnd that the short rate has a negatve mpact on the expected returns of all assets except for government bonds. Furthermore, the expected returns of assets n the fxed ncome class are postvely related to the long-term yeld, whle the expected returns of assets n the equty class are negatvely related to ths predctor. The dvdend yeld s postvely related to the expected returns of all assets. The estmates of the autoregressve parameters, κ, reflect the hgh persstence of the predctor varables. Fnally, the last part of Table II provdes the jont volatlty matrx of the assets and the predctor varables. remans unknown to the CIO and the condtonng nformaton exploted by the asset managers s equvalent to prvate nformaton.

20 1868 The Journal of Fnance Table II Tme-Varyng Investment Opportuntes Ths table shows the estmaton results of the fnancal market n Secton II over the perod January 1973 through November 2004 usng monthly data. The model s estmated by maxmum lkelhood. The asset set contans government bonds Gov. bonds ), corporate bonds wth credt ratngs Baa Corp. bonds, Baa ) and Aaa Corp. bonds, Aaa ), and three equty portfolo ranked on ther book-to-market rato growth/ntermedate Int. )/value). In determnng 0, we assume that the nstantaneous nomnal short rate equals r = 5%. We report 1 rather than 1 as the former expresson s easer to nterpret. The short rate, the yeld on a 10Y nomnal government bond, and the dvdend yeld are used to predct returns. Source of Rsk Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z Gov. Baa Aaa Growth Int. Value κ Short rate Y yeld DP Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Gov. bonds 13.2% Corp. bonds, Baa 7.7% 5.4% Corp. bonds, Aaa 8.7% 2.6% 2.4% Growth stocks 3.1% 5.8% 0.2% 16.5% Int. stocks 2.9% 6.2% 0.1% 11.7% 7.2% Value stocks 2.8% 7.1% 0.2% 10.4% 6.7% 5.8% Short rate 1.1% 0.1% 0.0% 0.3% 0.1% 0.1% 2.3% Y yeld 0.0% 0.0% 0.0% 0.1% 0.1% 0.0% 0.0% 1.3% 0 DP 3.0% 6.7% 0.1% 14.0% 2.5% 0.9% 0.0% 0.6% 4.7% B. Centralzed Problem We frst solve agan the centralzed nvestment problem n whch the CIO manages all assets. Ths soluton serves as a pont of reference for the case n whch nvestment management s decentralzed. The centralzed nvestment problem wth affne prces of rsk has been solved by, among others, Lu 2007) and Sangvnatsos and Wachter 2005). We denote the CIO s nvestment horzon by τ C. The optmal allocaton to the dfferent assets s gven by x C X, τ C ) = 1 ) 1 X ) + γ C 1 γ C ) 1 X Bτ C ) CτC ) + Cτ C ) ) X ), 18) where expressons for Bτ C ) and Cτ C ), as well as the dervatons of the results n ths secton are provded n Appendx B. The optmal portfolo contans two components. The frst component s the condtonal myopc demand that optmally explots the rsk return trade-off provded by the assets. The second

21 Optmal Decentralzed Investment Management 1869 component represents the hedgng demands that emerge from the CIO s desre to hedge future changes n the nvestment opportunty set. Ths second term reflects the long-term perspectve of the CIO. The correspondng value functon s gven by J 1 W, X, τ C) = 1 1 γ C W 1 γ C exp { A τ C) + B τ C) X + 1 } 2 X C τ C) X, 19) wth the coeffcents A,B, and C provded n Appendx B. In Fgure 6, we llustrate the composton of the optmal portfolo for dfferent nvestment horzons when the coeffcent of relatve rsk averson of the CIO equals ether γ C = 5 n Panel A or γ C = 10 n Panel B. Focusng frst on the fxed ncome asset class, we fnd substantal horzon effects for corporate bonds. At short horzons, the CIO optmally tlts the portfolo towards Baarated corporate bonds and shorts Aaa-rated corporate bonds to take advantage of the credt spread. At longer horzons, the fracton nvested n Baa-rated bonds ncreases even further, whle the allocaton to Aaa-rated corporate bonds decreases. Swtchng to the results for the equtes asset class, we detect a strong value tlt at short horzons due to the hgh-value premum. The optmal portfolo contans a large long poston n value stocks and large short poston n growth stocks. However, as the nvestment horzon ncreases, the value tlt drops, consstent wth the results of Jurek and Vcera 2006). 11 C. Decentralzed Problem wthout a Benchmark We now solve the decentralzed problem when the CIO cannot use the benchmark to algn ncentves. In general, the optmal portfolos of the asset managers depend on both the nvestment horzon and the state of the economy. However, to make the problem more tractable and realstc, we assume that the nvestment managers are able to tme the market and explot the tmevaraton n rsk prema, but gnore long-term consderatons. That s, asset managers mplement the condtonal myopc strategy x NB X ) = 1 x X ) + 1 x X ) ) ι x MV, 20) γ γ where x X ) = ) 1 X ) and x MV = ) 1ι ι ) 1ι. 21) 11 Ths result s also n lne wth the fndngs of Campbell and Vuolteenaho 2004), who explan the value premum by decomposng the CAPM beta nto a cash flow beta and a dscount rate beta. The cash flow component s hghly prced, but largely unpredctable. The dscount rate component demands a lower prce of rsk but s to some extent predctable. Campbell and Vuolteenaho 2004) show that growth stocks have a large dscount rate beta, whereas value stocks have a large cash flow beta. Ths mples that from a myopc perspectve, value stocks are more attractve than growth stocks. However, the predctablty of growth stock returns mples that long-term returns on these assets are less rsky, makng them relatvely more attractve.

22 1870 The Journal of Fnance Fgure 6. Optmal portfolo choce n the centralzed problem. Ths fgure depcts the optmal allocaton to government bonds, corporate bonds wth ratngs Baa and Aaa, and three stock portfolos ranked based upon ther book-to-market ratos growth, ntermedate, and value). The horzontal axs depcts the nvestment horzon of the CIO n months. The coeffcent of relatve rsk averson of the CIO equals γ C =5nPanelAandγ C = 10 n Panel B. Ths partcular form of myopa can be motvated by the relatvely short-sghted compensaton schemes of asset managers. Snce the average hedgng demands for 1-year horzons are neglgble, we abstract from the managers hedgng motves n ths part of the problem.

23 Optmal Decentralzed Investment Management 1871 The CIO does account for the long-term perspectve of the frm through the strategc allocaton. However, we assume that the CIO mplements a strategc allocaton that s uncondtonal, that s, ndependent of the current state. At each pont n tme, the allocaton to the dfferent asset classes s reset towards a constant-proportons strategc allocaton, as opposed to constantly changng the strategc allocaton dependng on the state. In order to decde on the strategc allocaton, the CIO maxmzes the uncondtonal value functon max x C τ C ) EJ 2W, X, τ C ) W ), 22) where J 2 denotes the condtonal value functon n the decentralzed problem above. Obvously, the CIO s horzon, τ C, nfluences the choce of the strategc allocaton. To revew the setup of ths decentralzed problem, the asset managers mplement actve strateges n ther asset classes usng condtonng nformaton but gnore any long-term consderatons. The CIO, n contrast, allocates captal uncondtonally to the asset classes, but accounts for the frm s long-term perspectve. In order to determne the uncondtonal value functon, we evaluate frst the condtonal value functon of the CIO, J 2, for any choce of the strategc allocaton. In Appendx B, we show that the condtonal value functon s exponentally quadratc n the state varables: J 2 W, X, τ C ) = W 1 γ C 1 γ C exp { Aτ C, x C ) + Bτ C, x C ) X X Cτ C, x C )X }. 23) One aspect of the CIO s problem s partcularly nterestng. The actve strategy mplemented by the asset managers, x NB, s affne n the predctor varables: x NB X) = ζ0 NB + ζ1 NB X. As a consequence, the mpled wealth dynamcs faced by the CIO are gven by dw t W t = r + σ W X ) X ) ) dt + σ W X ) dz t, 24) where σ W X) = x 1C ζ01 NB + ζ 11 NBX) 1 + x 2C ζ02 NB + ζ 12 NBX) 2. Snce the asset managers condton ther portfolos on the state varables, the CIO has to allocate captal to two assets that exhbt a very partcular form of heteroskedastcty. Hence, despte the homoskedastc nature of the fnancal market, the CIO s confronted wth heteroskedastc asset returns n the decentralzed nvestment management problem. We solve for the optmal strategc asset allocaton numercally see Appendx B for detals). In Fgure 7, we present the strategc allocaton to the fxed ncome and equtes classes for dfferent nvestment horzons. The preference parameters are set to γ C = 10 and γ 1 = γ 2 = 5. The strategc allocaton to the asset classes exhbts substantal horzon effects and margnally overweghts equtes. Recall that the strategc allocaton to the asset classes s ndependent of the state varables, by constructon, because t s uncondtonal.

24 1872 The Journal of Fnance Fgure 7. Optmal strategc allocaton n the decentralzed problem wthout a benchmark. Ths fgure dsplays the optmal allocaton to the fxed ncome and equty asset classes n the absence of a benchmark. The horzontal axs depcts the nvestment horzon of the CIO n months. The preference parameters have been set to γ C = 10 and γ = 5, wth = 1, 2. Fgure 8 provdes the annualzed utlty costs from decentralzed asset management for dfferent rsk atttudes of the nvestment managers. The nvestment horzon equals ether T = 1 year n Panel A or T = 10 years n Panel B. The utlty costs are large and ncreasng n the horzon of the CIO. For relatvely short nvestment horzons, the costs closely resemble the case wth constant nvestment opportuntes, wth an order of about 40 to 80 bass ponts per annum. In contrast, for longer nvestment horzons, the utlty costs are substantally hgher, around 200 to 300 bass ponts per annum. Note that the rsk atttudes of the managers, for whch the costs of decentralzed nvestment management are mnmzed, depend on the CIO s nvestment horzon. D. Decentralzed Problem wth a Benchmark We show n Secton I.D that when nvestment opportuntes are constant, a performance benchmark can be desgned to elmnate all neffcences nduced by decentralzed asset management. Ths secton reexamnes ths ssue for the case of tme-varyng nvestment opportuntes. We restrct attenton to uncondtonal benchmarks, meanng the benchmark portfolo weghts are not allowed to depend on the state varables. 12 Uncondtonal benchmarks have the advantage that they are easy to mplement. Moreover, nvestment managers followng an uncondtonal benchmark do not have to trade excessvely, whch 12 See also Cornell and Roll 2005).

25 Optmal Decentralzed Investment Management 1873 Fgure 8. Utlty costs of decentralzed nvestment management wthout a benchmark. Ths fgure gves a comparson of certanty equvalents followng from the centralzed and decentralzed nvestment management problem when there s no benchmark and the nvestment horzon s 1 year n Panel A and 10 years n Panel B. The horzontal axes depct the rsk appettes of the asset managers. The coeffcent of relatve rsk averson of the CIO equals 10. The losses are computed by takng the rato of the annualzed certanty equvalents acheved under decentralzed and centralzed nvestment management after whch we subtract one and multply by 10,000 to express the losses n bass ponts per year. could be the case wth a condtonal benchmark. Condtonal benchmarks are more flexble and may therefore reduce further or even elmnate the costs of decentralzaton. The performance benchmark of asset manager s gven by a k-dmensonal vector of uncondtonal portfolo weghts, β, wth β ι = 1. Snce the benchmark

26 1874 The Journal of Fnance s chosen uncondtonally, asset managers can outperform ther benchmark.e., generate alpha) by properly ncorporatng the condtonng nformaton. The benchmark dynamcs are db t = r + β B X ) ) dt + β dz t. 25) t To solve for the optmal benchmark, we frst determne the optmal response of the asset managers to ther benchmarks. The optmal condtonal myopc strategy of the nvestment managers wth a benchmark s gven by x B X ) = 1 γ x X ) + 1 1γ ) β + 1 γ 1 x X ) ι ) x MV, 26) where x X) and x MV are gven n equaton 21). The CIO chooses the uncondtonal) benchmarks and determnes the uncondtonal) strategc allocaton to the asset classes by maxmzng the uncondtonal expectaton of the condtonal value functon, max x C τ C ),β 1 τ C ),β 2 τ C ) EJ 3W, X, τ C ) W ). 27) The condtonal value functon, J 3, s agan exponentally quadratc n the state varables and the coeffcents are provded n Appendx B. Note that both the strategc allocaton and the benchmarks are allowed to depend on the CIO s horzon. We use numercal methods to solve for the optmal benchmarks and allocatons to the two asset classes see Appendx B for detals). Panel A of Fgure 9 shows the optmal performance benchmarks for dfferent nvestment horzons of the CIO. The CIO s rsk averson equals 10 and the managers rsk averson s set to 5. At short horzons, or f the CIO behaves myopcally, the optmal benchmarks are smlar to when nvestment opportuntes are constant. However, the benchmark portfolos exhbt strong horzon effects. For nstance, n the equtes asset class, the myopc benchmark renforces the value tlt already present n the equty manager s myopc) portfolo. The long-run benchmark, n contrast, antcpates the lower rsk of growth stocks and provdes an ncentve to reduce the value tlt. Ths llustrates how performance benchmarks can be used to ncorporate the CIO s long-term perspectve n the short-term portfolo choces of the asset managers. Panel B of Fgure 9 provdes the correspondng strategc allocaton to both asset classes for dfferent nvestment horzons. Recall that when nvestment opportuntes are constant, the centralzed allocaton s always more rsky than the decentralzed allocaton wthout a benchmark. When nvestment opportuntes are tme varyng, we fnd the ntal allocaton wth a benchmark to be smlar to and even somewhat more conservatve than) the allocaton wthout a benchmark. However, for longer nvestment horzons of the CIO, the optmal strategc allocaton of the CIO s tlted substantally towards equtes.

27 Optmal Decentralzed Investment Management 1875 Fgure 9. Optmal performance benchmarks and strategc allocaton. Panel A portrays the composton of the optmal performance benchmarks for dfferent nvestment horzons of the CIO. Panel B presents the correspondng optmal strategc asset allocaton to the asset classes. We plot the benchmark for the stock and bond manager n the same graph, but there s stll no crossbenchmarkng. That s, the benchmark weghts n both asset classes each sum up to 100%. The horzontal axs depcts the nvestment horzon of the CIO n months. The preference parameters are γ C = 10 and γ = 5, wth = 1, 2. Fgure 10 presents the utlty gans generated by an optmally chosen benchmark. The CIO s coeffcent of rsk averson equals 10 and the horzon s set to T = 1 year n Panel A and T = 10 years n Panel B. For the 1-year horzon, the value added by the benchmark s lmted to approxmately 20 bass ponts.

28 1876 The Journal of Fnance Fgure 10. Value generated by an optmally chosen benchmark. Ths fgure gves a comparson of certanty equvalents followng from the decentralzed problem wth and wthout an optmally chosen benchmark. We present the annualzed gans n bass ponts from usng the benchmark optmally. The nvestment horzon of the CIO equals 1 year n Panel A and 10 years n Panel B. The horzontal axes depct dfferent rsk appettes of the asset managers. The coeffcent of relatve rsk averson of the CIO equals 10. However, when the nvestment horzon ncreases to 10 years, the beneft of an optmally chosen benchmark ncreases as the asset managers become less conservatve. We conclude that uncondtonal performance benchmarks are sgnfcantly value enhancng. Ths extends the results of Admat and Pflederer 1997) concernng the role of performance benchmarks n delegated portfolo management problems. In case of multple asset managers, performance benchmarks can be

29 Optmal Decentralzed Investment Management 1877 useful n algnng ncentves along at least three dmensons, namely, dversfcaton, preferences, and nvestment horzons. III. Unknown Rsk Appettes of the Managers In the prevous sectons, we assume that the CIO s able to observe the managers rsk averson levels n decdng on the strategc allocaton and n constructng the performance benchmarks. In realty, the CIO usually has relatvely lmted nformaton about the managers preferences. Even though past performance or current portfolo holdngs can be nformatve about the managers rsk atttude, exact nference s often nfeasble. In ths secton therefore, we generalze our framework by explctly modelng the CIO s uncertanty about the managers preferences. Specfcally, we focus on the mpact of the unknown rsk averson levels of the asset managers on 1) the strategc allocaton to each of the asset classes, 2) the utlty costs of decentralzaton, and 3) the value of optmally desgned performance benchmarks. We model the CIO s uncertanty wth respect to the managers rsk atttudes by assumng that the CIO has a pror dstrbuton over the rsk atttudes of the managers. It s mportant to note that even when the CIO does not wsh to mplement optmally desgned benchmarks, the CIO needs ths pror dstrbuton to decde the strategc allocaton to each of the asset classes. We then examne the extent to whch the mplementaton of optmal benchmarks s effectve n algnng ncentves when the CIO can use no more nformaton than hs pror belefs to desgn the benchmarks. We assume that the CIO s pror over the managers coeffcents of relatve rsk averson s gven by a normal dstrbuton truncated between 1 and More formally, the pror s gven by [ exp 1 ] 2 γ µ γ ) γ 1 γ µ γ ) γ f γ ) = [ exp 1 ], γ 1, 10) 1, 10), γ µ γ ) γ 1 γ µ γ ) dγ 1 dγ 2 28) wth γ = γ 1, γ 2 ). The parameters µ γ and γ allow us to vary the average rsk appettes of the asset managers as well as the precson. 14 The off-dagonal elements of γ allow for correlatons between the rsk atttudes of the managers. Note that when γ 1,1) and γ 2,2) tend to nfnty, the pror converges to an unnformatve unform pror on the nterval 1, 10). Note further that wthn our 13 Increasng the upper bound of ths truncated normal dstrbuton to, for example, 15 or 20 does not affect our qualtatve results. 14 Note that the truncated normal dstrbuton s skewed f µ γ does not equal the average of the upper and lower truncaton ponts. In ths case, changng µ γ affects the precson and, lkewse, changng γ has an mpact on the average rsk atttude. To analyze the mpact of uncertanty about the managers preferences by varyng γ, we focus our dscusson predomnantly on a symmetrc pror wth µ γ = 5.5. The results for alternatve, skewed pror dstrbutons are reported for completeness and are qualtatvely smlar.

30 1878 The Journal of Fnance model, the CIO could potentally learn about manageral preferences through the volatlty matrx of the managers portfolo returns Merton 1980)). We consder learnng about the managers preferences to be beyond the scope of ths paper, however, and we therefore assume that the uncertanty about the managers preferences s not allevated or resolved durng the course of the nvestment perod. In order to determne the optmal strategy of the CIO, we ntegrate out the uncertanty about the managers rsk averson levels. Ths results n a strategc asset allocaton and performance benchmarks that are robust to a range of preferences of the asset managers. In Secton III.A, we determne the optmal strategc allocaton and the costs of decentralzaton for dfferent prors over the managers preferences. Next, we examne n Secton III.B the extent to whch optmal performance benchmarks are useful n reducng the utlty costs nduced by decentralzaton. Fnally, Secton III.C ntroduces trackng error volatlty constrants, whch are often observed n the nvestment management ndustry to constran asset managers. A. Decentralzed Problem wthout a Benchmark We frst consder the case n whch the asset managers are not remunerated relatve to a benchmark. These managers adopt the strateges gven n equaton 6). The CIO determnes the strategc allocaton by maxmzng ) 1 max E t W 1 γ C x T C 1 γ C, 29) C where the expectaton s taken wth respect to both the uncertanty n the fnancal market and the rsk appettes of the asset managers. We can smplfy the problem by frst condtonng on the managers rsk averson levels γ ) and then applyng the law of terated expectatons: max E x C E t )) 1 W 1 γ C T 1 γ C γ. 30) C The nsde expectaton, condtonal on the managers preferences and possbly the state varables at tme t, can be determned n closed-form for any strategc allocaton x C usng the arguments n Sectons I and II. To develop the man ntuton, we focus ntally on the case of constant nvestment opportuntes. The condtonal expectaton s then gven by ) 1 E t W 1 γ C T 1 γ C γ C = 1 1 γ C W 1 γ c t expax C, γ )τ C ), 31) where ax C, γ ) = 1 γ C )x C γ ) + r) γ C1 γ C ) 2 x C γ ) γ ) x C and τ C = T C t. Gven the pror over the managers rsk appettes, t s straghtforward to optmze numercally) over the strategc allocaton. Along these lnes we can determne 1) the optmal strategc allocaton to both asset classes and 2) the

31 Optmal Decentralzed Investment Management 1879 utlty costs nduced by decentralzaton for varous pror dstrbutons over the managers rsk averson levels. Even though the results n the remander of ths secton are determned numercally, we can llustrate the mpact of not knowng the managers preference parameters usng an accurate approxmaton. The CIO s frst-order condton wth respect to the strategc allocaton, x C, s gven by 1 E W 1 γ c t expax C, γ )τ C ) ax ) C, γ ) = ) 1 γ C x C If the term expax C, γ )) n equaton 32) were constant, 15 the optmal strategc allocaton would be gven by 16 x approx C = 1 E )) 1 E ). 33) γ C It s straghtforward to show that when the rsk appettes of the managers are ndependent, t follows that ) 1 Var 0 γ [ 1 b E ) = E )E ) + ) b ] Var 0 b b, 2 34) where b = x x ι)x MV, = 1, 2. In other words, b s a long short portfolo that s long the speculatve portfolo and short the mnmum-varance portfolo. We now dscuss the last two matrces on the rght-hand sde of equaton 34) n turn. The frst matrx shows that the covarance matrx of managed portfolo returns ncreases as a result of the uncertanty about the managers preferences. Ths nduces the CIO to reduce the strategc allocaton to each of the asset classes. If the uncertanty about the managers rsk atttudes s equal across managers, ths effect s symmetrc across asset classes. However, the second matrx depends on the propertes of the asset class, whch mples that even f the CIO has the same nformaton about the managers rsk atttudes, the relatve allocatons to the asset classes change as the uncertanty about the managers rsk atttudes ncreases. Usng the approxmaton n equaton 33), we can approxmate the value functon as expax C, γ )) exp a x approx C, γ )) expãγ )). 35) Ths approxmaton allows us to solve the frst-order condton 32) n closed form: xc 1 Eexpãγ )) )) 1 Eexpãγ )) ), 36) γ C 15 Ths s the case, for nstance, f we consder a 0 th order expanson n γ = Eγ ). 16 We normalze τ C = 1. γ 2

32 1880 The Journal of Fnance Table III Strategc Allocaton wthout Benchmarks when Rsk Atttudes Are Unknown Ths table gves the strategc allocaton of the CIO to the asset classes when the rsk atttudes of the managers are unknown and there are no benchmarks. The pror of the CIO over the rsk averson level of each of the managers s a truncated normal dstrbuton wth parameters µ γ and σ γ, truncated below at 1 and truncated above at 10. µ γ = 3.1 µ γ = 5.5 µ γ = 7.3 Bonds Stocks Bonds Stocks Bonds Stocks Panel A: Constant Investment Opportuntes σ γ = 0 19% 30% 22% 22%) 36% 36%) 24% 37% σ γ = 1 18% 27% 22% 22%) 36% 36%) 23% 37% σ γ = 2 18% 26% 21% 21%) 33% 33%) 23% 36% σ γ = 3 18% 27% 21% 21%) 31% 31%) 22% 33% σ γ = 25 unform) 20% 28% 20% 20%) 28% 29%) 20% 28% Panel B: Tme-Varyng Investment Opportuntes T = 1) σ γ = 0 27% 31% 36% 37% 39% 38% σ γ = 1 23% 27% 35% 37% 38% 38% σ γ = 2 22% 26% 29% 34% 35% 37% σ γ = 3 22% 27% 27% 31% 30% 35% σ γ = 25 unform) 24% 29% 24% 29% 24% 29% Panel C: Tme-Varyng Investment Opportuntes T = 10) σ γ = 0 33% 35% 47% 51% 51% 57% σ γ = 1 23% 26% 26% 42% 51% 56% σ γ = 2 23% 25% 25% 30% 26% 36% σ γ = 3 23% 25% 25% 28% 25% 31% σ γ = 25 unform) 24% 26% 24% 26% 24% 26% whch s smlar to before except that the covarance matrx and expected returns are weghted by the scaled) value functon of the CIO, expãγ )). In the emprcal applcaton, we treat the uncertanty about the rsk averson levels of both managers symmetrcally and assume ndependence: µ γ 1) = µ γ 2) and γ = σγ 2 I, wth I denotng a 2 2 dentty matrx. We consder pror dstrbutons wth mean parameters µ γ = 3.1, 5.5, and 7.3 and uncertanty parameters σ γ = 0, 1, 2, 3, and 25. Note that when µ γ = 5.5 the dstrbuton s symmetrc as 5.5 s the average of the truncaton ponts 1 and 10. When σ γ = 25, the CIO effectvely has a unform pror over γ, and the parameter µ γ has no further mpact. The results are summarzed n Tables III and IV. In Table III we compute the optmal strategc allocatons wthout benchmarks. In Table IV we report the correspondng costs of decentralzed nvestment management. Each table has three panels, one for constant nvestment opportuntes Panel A) and two panels for tme-varyng nvestment opportuntes, wth the CIO s nvestment horzon equal to ether T = 1 Panel B) or T = 10 Panel C).

33 Optmal Decentralzed Investment Management 1881 Table IV Costs of Decentralzed Investment Management f Rsk Atttudes Are Unknown Ths table gves the costs of decentralzed nvestment management when the rsk atttudes of the managers are unknown and there are no benchmarks. The pror of the CIO over the rsk averson levels of each of the managers s a truncated normal dstrbuton wth parameters µ γ and σ γ, truncated below at 1 and truncated above at 10. The losses are computed by takng the rato of the annualzed certanty equvalents acheved under decentralzed and centralzed nvestment management after whch we subtract 1 and multply by 10,000 to express the losses n bass ponts per year. µ γ = 3.1 µ γ = 5.5 µ γ = 7.3 γ C = 5 γ C = 10 γ C = 5 γ C = 10 γ C = 5 γ C = 10 Panel A: Constant Investment Opportuntes σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) Panel B: Tme-Varyng Investment Opportuntes T = 1) σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) Panel C: Tme-Varyng Investment Opportuntes T = 10) σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) We focus our dscusson on the pror dstrbuton wth µ γ = 5.5, snce ths dstrbuton s symmetrc. The results n Table III ndcate that an ncrease n the uncertanty about the managers rsk averson leads to a decrease n the optmal allocaton to both asset classes. Ths mples that uncertanty about the managers preferences effectvely ncreases the rsk averson of the CIO. Not knowng the managers preferences consttutes a form of background rsk, whch reduces the nvestor s appette for fnancal rsk. 17 The results can also be nterpreted as a form of Bayesan parameter uncertanty. Ths ntuton can easly be derved from equatons 33) to 36). The effect s quanttatvely strong, especally for the equty class. If the pror changes from known preferences no uncertanty) to a unform pror between 1 and 10, the CIO reduces the allocaton to the equty asset class by 25% 50% of the total allocaton. Fnally, to verfy the 17 See, for nstance, Goller and Pratt 1996).

34 1882 The Journal of Fnance accuracy of our approxmaton, we also present n Panel A n parentheses the approxmate optmal strategc allocaton usng equaton 36). We conclude that our approxmaton has a very hgh level of accuracy, lendng further credblty to the ntutve nsghts t offers. We report n Table IV the utlty costs ncurred by the CIO as a result of decentralzaton for rsk averson parameters of the CIO equal to γ C = 5 and γ C = 10. The utlty costs are annualzed and measured n bass ponts. The costs of decentralzed nvestment management are generally ncreasng n the uncertanty about the managers preferences. The mpact of ths uncertanty on the utlty costs s economcally sgnfcant. In most cases, the costs double when we move from known levels of rsk averson to a unform pror dstrbuton over the levels of rsk averson. For nstance, n Panel B wth µ γ = 5.5 and γ C = 10, the utlty costs ncrease from 59 to 109 bass ponts per annum. These results mply that the common, yet unrealstc, assumpton that the preferences of the manager the agent) are known to the CIO the prncpal) can grossly understate the problem and have serous consequences for optmal polces, partcularly n the case of tme-varyng nvestment opportuntes and a long nvestment horzon for the CIO see Panel C of Table III). Note that there are exceptonal cases n whch the costs of decentralzaton are slghtly decreasng n the uncertanty about the preferences of the managers. If the CIO assgns a hgh pror probablty to hgh-cost managers to begn wth, whch s the case when µ γ = 7.3 and σ γ s low see, for nstance, Fgure 8), ncreasng σ γ wll ncrease the probablty of allocatng captal to lower-cost managers. Ths n turn can lead to a decreasng relatonshp between the costs of decentralzaton and the uncertanty about the managers preferences. However, ths effect s quanttatvely neglgble and up to only one bass pont per year. B. Decentralzed Problem wth a Benchmark We now examne how effectve benchmarks are n algnng ncentves f the CIO does not know the rsk averson levels of the managers. Table V presents the optmal strategc allocaton when the asset managers are remunerated relatve to optmal performance benchmarks. The man effects are n lne wth Table III. The optmal strategc allocaton to both asset classes decreases as the uncertanty about the managers rsk appettes ncreases. We also fnd that the mplementaton of optmal benchmarks can lead to ether an ncrease or decrease n the strategc allocaton relatve to the problem wthout benchmarks, dependng on the CIO s pror belefs. In the prevous subsecton, we argue that the neffcences caused by decentralzaton are generally aggravated when the rsk appettes of the managers are unknown Table IV). The value of an optmally desgned benchmark Table VI) depends on the followng two effects. Frst, compared to the case of known rsk appettes, the amount of nformaton that can be used to desgn the optmal benchmarks s lower because rsk appettes are now unknown. Ths suggests that the value of an optmal benchmark dmnshes. Second, the neffcences

35 Optmal Decentralzed Investment Management 1883 Table V Strategc Allocaton wth Benchmarks when Rsk Atttudes Are Unknown Ths table gves the strategc allocaton of the CIO to the asset classes when the rsk atttudes of the managers are unknown and the optmal benchmarks are mplemented. The pror of the CIO over the rsk averson levels of each of the managers s a truncated normal dstrbuton wth parameters µ γ and σ γ, truncated below at 1 and truncated above at 10. µ γ = 3.1 µ γ = 5.5 µ γ = 7.3 Bonds Stocks Bonds Stocks Bonds Stocks Panel A: Constant Investment Opportuntes σ γ = 0 24% 32% 24% 32% 24% 32% σ γ = 1 21% 26% 24% 31% 24% 32% σ γ = 2 19% 24% 23% 29% 24% 31% σ γ = 3 19% 24% 21% 27% 23% 29% σ γ = 25 unform) 20% 25% 20% 25% 20% 25% Panel B: Tme-Varyng Investment Opportuntes T = 1) σ γ = 0 31% 35% 40% 34% 41% 33% σ γ = 1 25% 28% 38% 33% 41% 33% σ γ = 2 23% 25% 31% 30% 37% 33% σ γ = 3 23% 25% 27% 28% 31% 31% σ γ = 25 unform) 24% 26% 24% 26% 24% 26% Panel C: Tme-Varyng Investment Opportuntes T = 10) σ γ = 0 34% 61% 47% 66% 50% 67% σ γ = 1 23% 28% 26% 43% 50% 66% σ γ = 2 23% 25% 25% 30% 26% 36% σ γ = 3 23% 25% 25% 27% 25% 30% σ γ = 25 unform) 24% 25% 24% 25% 24% 25% that can potentally be mtgated by the benchmarks are also much larger. Therefore, there s more scope for the benchmarks to have value-added. Ths explans why, for low levels of uncertanty, there s a small) negatve relaton between the value of benchmarks and the level of uncertanty about the rsk appettes. In these cases the frst effect domnates. However, as the uncertanty about the rsk averson levels ncreases, the value of the benchmarks also generally ncreases and exceeds the value for known preferences because then the second effect domnates. As explaned before, there are exceptonal cases n whch the costs of decentralzaton are slghtly decreasng n the uncertanty about the managers preferences e.g., when µ γ = 7.3). In such cases, ncreasng the uncertanty about the managers preferences does not suffcently enlarge the scope for mprovement by optmally desgned benchmarks. As a result, the fact that the benchmarks are based on less nformaton domnates and the value of an optmally desgned benchmark decreases n the uncertanty about the managers rsk averson levels.

36 1884 The Journal of Fnance Table VI Value of Optmal Benchmarks when Rsk Atttudes Are Unknown Ths table gves a comparson of certanty equvalents followng from the decentralzed problem wth and wthout an optmally chosen benchmark. We present the annualzed gans n bass ponts from usng the benchmark optmally. The pror of the CIO over the rsk averson levels of each of the managers s a truncated normal dstrbuton wth parameters µ γ and σ γ, truncated below at 1 and truncated above at 10. µ γ = 3.1 µ γ = 5.5 µ γ = 7.3 γ C = 5 γ C = 10 γ C = 5 γ C = 10 γ C = 5 γ C = 10 Panel A: Constant Investment Opportuntes σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) Panel B: Tme-Varyng Investment Opportuntes T = 1) σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) Panel C: Tme-Varyng Investment Opportuntes T = 10) σ γ = σ γ = σ γ = σ γ = σ γ = 25 unform) When the CIO s nvestment horzon s longer, for example T = 10, the results n Panel C may gve the mpresson that benchmarks become less effectve n algnng ncentves when rsk appettes are unknown. It s mportant to emphasze, however, that the results presented for ths case consttute a conservatve lower bound on the value of benchmarks. It s common practce n the nvestment management ndustry to have the opportunty to revse the benchmark annually. We consder a sngle, uncondtonal benchmark that s held constant for 10 years, whch s the absolute mnmum of what optmally desgned benchmarks can actually acheve. In case of annual rebalancng, or an effectve 1-year horzon, Panel B shows that the benchmarks are ndeed more effectve the more uncertan the CIO s about the managers rsk preferences. To summarze, we fnd that uncertanty about the managers rsk preferences has a strong effect on the optmal strategc allocaton to the dfferent asset classes. We show that ths uncertanty ncreases the costs of decentralzed nvestment management even further. We also show that optmally desgned performance benchmarks become more effectve to overcome these costs.

37 Optmal Decentralzed Investment Management 1885 C. Rsk Constrants Apart from desgnng optmal return benchmarks, the CIO can also employ rsk constrants n order to change or to restrct the behavor of asset managers. These rsk constrants can be formulated ether n terms of absolute rsk n absence of a benchmark or n terms of relatve rsk when the asset manager s renumerated relatve to a benchmark. Absolute rsk constrants restrct the total volatlty of the portfolo return. Relatve rsk constrants lmt the volatlty of the portfolo return n excess of the benchmark return, as n Roll 1992) and Joron 2003). We assume that the volatlty constrants have to be satsfed at every pont n tme. In modern nvestment management frms, rsk management systems montor the rsk exposures of portfolo holdngs frequently, whch makes t plausble to presume that rsk constrants have to be satsfed contnuously. For ease of exposton, we focus ntally on the fnancal market of Secton I n whch nvestment opportuntes are constant. The nstantaneous volatlty of the portfolo return s gven by σ A x ) = x x, whch s the portfolo s absolute rsk. The nstantaneous volatlty of the portfolo return n excess of the benchmark relatve rsk) s gven by σ R x ) = x β ) x β ), whch s also called the trackng error volatlty. Usng these defntons for absolute and relatve rsk, we mpose rsk lmts of the form σ j x ) φ j, 37) wth j = A, R. To ensure that the optmzaton problem of the asset managers s well defned, we assume that σ A x MV ) φ A, whch states that the lmt on absolute rsk must exceed the volatlty of the mnmum varance portfolo. In the case of relatve rsk constrants, we requre that φ R 0, snce we restrct attenton to benchmarks that can be replcated by the managers. A relatve rsk lmt of φ R = 0 mples that the asset manager must exactly mplement the benchmark portfolo. We focus on the effect of mposng ether one of these constrants, but not both. 18 Whenever the unconstraned portfolo choce n absence of a benchmark does not volate the absolute rsk constrant, ths portfolo remans optmal for manager. However, once the absolute rsk constrant s volated, Appendx C shows that the optmal portfolo equals x NB 1 ξ ) = γ 1 + ξ ) x x + 1 ι ) x MV, 38) γ 1 + ξ ) where x and x MV are gven by equaton 7) and ξ > 0 satsfes σ A x NB ξ )) = φ A. Ths soluton shows that the absolute rsk constrant nduces an effectve ncrease n rsk averson. The results n Fgure 2 then mply that absolute rsk constrants can mtgate neffcences whenever the nvestment manager s 18 Joron 2003) nfers n addton the effect of mplementng both absolute and relatve rsk constrants.

38 1886 The Journal of Fnance too aggressve. In contrast, when the nvestment manager s too conservatve, absolute rsk constrants can actually aggravate the neffcences. We also show n Appendx C that the optmal portfolo n the presence of a performance benchmark and bndng relatve rsk constrant s gven by x B ξ ) = 1 γ 1 + ξ ) x γ 1 + ξ ) ) β + 1 x ι γ 1 + ξ ) xmv, 39) where x and x MV are gven n equaton 7) and ξ > 0 satsfes σ R x NB ξ )) = φ R. In addton, Appendx C shows that the relatve rsk constrant bnds for an nvestment manager wth rsk averson γ once the benchmark s desgned on the bass of a hgher rsk averson γ, wth γ >γ. Ths mples that the CIO does not requre specfc knowledge of the manager s rsk atttude, more than knowng an upper bound. If the benchmark and relatve rsk constrant are desgned on the bass of ths conservatve upper bound, the relatve rsk constrant bnds for more aggressve managers. The bndng constrant nduces an effectve ncrease n the manager s rsk averson to the level for whch the benchmark s desgned. Combnng these results wth our dscusson of unknown rsk appettes, rsk constrants essentally shft the lower truncaton pont of the CIO s pror over the managers rsk averson levels upwards. All managers, who are more aggressve than the rsk constrant allows, wll behave as an asset manager for whch the constrant bnds on the margn. Hence, rsk constrants effectvely reduce the CIO s uncertanty about the manager s preferences. In case of constant nvestment opportuntes, there s no dsadvantage from selectng tght rsk constrants. However, n the more realstc case of tmevaryng nvestment opportuntes, the same dervaton s vald, albet ξ becomes tme dependent and the constrant wll bnd only at certan ponts n tme. In that case, tght rsk constrants wll reduce the tmng ablty of the asset managers. Therefore, the CIO can optmally determne the strategc allocaton to both asset classes, the benchmarks for each manager, and the rsk constrants for a gven pror over the managers rsk tolerances. Tght rsk constrants ndcate that t s valuable for the CIO to reduce uncertanty about the managers rsk atttude, whle wde rsk constrants ndcate that the CIO prefers to explot the tmng expertse of the managers rather than reducng the uncertanty about ther preferences. IV. Conclusons We address several msalgnments of ncentves generated by decentralzed nvestment management. These msalgnments between a CIO and the asset managers he employs can lead to large utlty costs. One straghtforward soluton s to mplement centralzed nvestment strateges whereby the CIO attempts to manage all assets hmself. However, from an organzatonal pont of vew, decentralzed nvestment management s an nevtable and stylzed fact of the nvestment ndustry. We show n ths paper that the optmal desgn of

39 Optmal Decentralzed Investment Management 1887 an uncondtonal lnear benchmark can be very effectve n mtgatng the costs of decentralzed nvestment management. Ths s even more pronounced when we generalze our model by relaxng the assumpton that the CIO knows the rsk averson levels of the asset managers. The optmal benchmark s derved assumng that the CIO only knows the cross-sectonal dstrbuton of nvestment managers rsk appettes, but does not know where n ths dstrbuton a gven manager falls. For ease of exposton, we confne attenton to CRRA preferences and lnear performance benchmarks. Future work could focus on a more complcated preference structure and/or nonstandard contracts. For example, t seems reasonable that the utlty functon of the CIO s knked as n van Bnsbergen and Brandt 2007). The compensaton scheme for the asset managers may also be nonlnear and/or asymmetrc, as n Browne 1999, 2000), Carpenter 2000), and Basak, Pavlova, and Shapro 2007), for example. Another nterestng extenson would be to assess the asset prcng mplcatons of decentralzed nvestment management. In delegated portfolo choce problems, Brennan 1993), Gómez and Zapatero 2003), Cuoco and Kanel 2006), and Cornell and Roll 2005) llustrate the mpact of delegaton and benchmarkng on equlbrum asset prces. Stutzer 2003b) shows that multple benchmarks mply a factor model wth these benchmarks returns as possbly prced factors. Fnally, we show that not knowng the rsk preferences of the managers to whch the CIO delegates the avalable captal effectvely ncreases the CIO s rsk averson. Snce the amount of captal managed nsttutonally has ncreased dramatcally durng recent decades, t s mportant to further understand the asset prcng mplcatons of unknown rsk preferences. Appendx A: Constant Investment Opportuntes A. Decentralzed Problem wth a Benchmark We solve the decentralzed problem wth the optmally desgned benchmark of Secton I.D. We derve frst the optmal allocatons of the asset managers n the presence of a benchmark. Defne normalzed wealth as w t = W t B 1 t. Recall that the benchmark comprses only postons n the assets avalable to the nvestment managers and no cash. The asset managers are therefore able to replcate the benchmark. The dynamcs of the benchmark are gven n equaton 12). Usng Ito s lemma, the dynamcs of normalzed wealth are dw t = x B β w + β β β x B ) dt + x B β ) dz t. A1) t The correspondng Hamlton Jacob Bellman HJB) equaton s max x B :x B ι=1 J w w x B β + β β β x ) B = w2 J ww x B β ) x B β ) + Jt A2)

40 1888 The Journal of Fnance The frst-order condtons FOC) are 0 = J w w β ) + Jww w 2 x B β ) ξι, and 1 = x B ι, A3) wth ξ denotng the Lagrange multpler. The value functon s of the form J 3 W /B, τ ) = 1 ) W 1 γ expcτ ), A4) 1 γ B wth τ = T t. The soluton of the FOCs s gven by equaton 14). The CIO has to desgn the benchmarks, that s, β, = 1, 2, and decde on the strategc allocaton to the managers and to the cash account. Snce the managers optmal portfolos are affne n the benchmark weghts, see equaton 14)), the benchmark can be desgned to solve for the optmal relatve fractons nvested n the dfferent assets present n the asset classes. The strategc allocaton, x C R 2, can subsequently be used to optmally manage the absolute fractons allocated to the dfferent assets. More formally, the optmal portfolo s gven by [ ] x1c x C = = 1 ) 1, A5) γ C x 2C where x C denotes the allocaton to the assets managed by manager. We use β to solve for the optmal relatve fractons nvested wthn the asset class: The optmal benchmark weghts are gven by [ β = x C γ γ 1 x B = x C x C ι ) 1. A6) x C ι ) 1 1 x x γ γ ι ) x MV )], A7) and the optmal allocaton of the CIO s wealth to the managers s gven by x C ι. Appendx B: Tme-Varyng Investment Opportuntes B. Centralzed Problem The centralzed problem n Secton II.B relates to the portfolo choce problems n Sangvnatsos and Wachter 2005) and Lu 2007). The problem s solved usng standard dynamc programmng technques. The HJB equaton reads J W W r + x C X )) + 1 max 2 J WWW 2 x C x C + J t x C J X KX tr X J ) = 0, B1) XX X + Wx C X J WX

41 Optmal Decentralzed Investment Management 1889 where we omt the ndces of x C X, τ C ) for notatonal convenence and K = dagκ 1,..., κ m ). The affne structure of the fnancal market mples that the value functon s exponentally quadratc n the state varables: JW, X, τ C ) = W 1 γ { C exp Aτ C ) + Bτ C ) X + 1 } 1 γ C 2 X Cτ C )X. B2) Solvng for the FOC of problem B1) and usng equaton B2) to determne the partal dervatves, we obtan x C X, τ C ) = 1 [ ) 1 X ) + X Bτ C ) + 1 CτC ) + Cτ C ) γ C 2 ) )] X, B3) whch we can rewrte as x C X, τ C ) = ζ C 0 τ C) + ζ C 1 τ C)X, wth ζ C 0 τ C) = 1 γ C ) 1 [ 0 + X Bτ C) ], ζ C 1 τ C) = 1 γ C ) 1 B4) [ X CτC ) + Cτ C ) )]. B5) To fnd the coeffcents A, B, and C, we substtute the optmal portfolo nto the HJB equaton B1) and match the constant, the terms lnear n X, and the terms quadratc n X. In what follows, we derve the value functon for any affne polcy, xx, τ) = ζ 0 τ) + ζ 1 τ)x, whch turns out to be useful n subsequent dervatons. The value functon for ths partcular problem s obtaned for ζ 0 τ) = ζ0 Cτ) and ζ 1τ) = ζ1 C τ). The resultng ODEs are Ȧ = 1 γ C ) r + ζ 0 ) γ C1 γ C )ζ 0 ζ tr X C + C ) X ) B X X B + 1 γ )ζ 0 X B, Ḃ = 1 γ C ) [ ζ ζ 1 ] γc 1 γ C )ζ 0 ζ 1 B K B X X C + C ) γ C)ζ 0 X C + C ) + 1 γ C )B X ζ 1, Ċ = 21 γ C )ζ 1 1 γ C 1 γ C )ζ 1 ζ 1 C + C )K C + C ) X X C + C ) + 1 γ C )ζ 1 X C + C ), B6) subject to the boundary condtons A0) = 0, B0) = 0 m 1, and C0) = 0 m m. C. Decentralzed Problem wthout a Benchmark In the decentralzed problem wthout a benchmark n Secton II.C, we frst solve for the myopc, cash-constraned polcy of the managers. The optmzaton problem of the myopc) managers can be smplfed to

42 1890 The Journal of Fnance x NB max :x NB ι=1 E t x NB X ) X ) γ 2 xnb ) X ) xnb X ). B7) As a result, the optmal strategy of the myopc, cash-constraned nvestment managers s x NB X ) = 1 x X ) + 1 x X ) ) ι x MV = ζ0 NB + ζ1 NB X, B8) γ γ where x X) and x MV are gven n equaton 21) and ζ NB 0 ζ NB 1 = 1 ) x MV γ = 1 ) 1 1 x MV γ 1 ι ) 1 ) 0, B9) γ ) ι 1 ) 1. B10) Antcpatng the allocatons of the asset managers, the CIO has to decde on the strategc allocaton. We consder strategc allocatons that are ndependent of the current state of the economy, but that do account for the nvestment horzon of the CIO. We optmze the uncondtonal value functon γ EJ 2 W, X, τ C ) W ), B11) wth J 2 W, X, τ C ) denotng the condtonal value functon, whch s exponentally quadratc n the state varables. After all, f we denote the allocaton to the th asset manager by x C, then the resultng portfolo of the CIO s affne n the state varables: x Impled C = [ x1c ζ NB 01 + ζ NB 11 X ) ] x 2C ζ NB 02 + ζ NB 12 X ) = ζ Impled 0 + ζ Impled 1 X, B12) and the results of Appendx B apply. To determne the uncondtonal value functon, we use Lemma 1. LEMMA 1: Let Y R m 1, Y N0, ), a R m 1, and B R m m. If 1 2B) s strctly postve defnte, then we have Eexpa Y + Y BY )) = exp 1 2 ln deti 2 B) + 1 ) 2 a 1 2B) 1 a. B13) Solvng for the optmal strategc asset allocaton s then reduced to a statc optmzaton of the uncondtonal value functon, whch we perform numercally. D. Decentralzed Problem wth a Benchmark The performance benchmark of manager n Secton II.D s parameterzed by a vector of constant portfolo weghts, β, wth the correspondng dynamcs

43 Optmal Decentralzed Investment Management 1891 specfed n equaton 25). The asset manager s concerned wth wealth relatve to the value of the benchmark. The dynamcs of normalzed wealth, w t = W t B 1 t, are gven by dw t w t = x B X ) X ) + β [ β X ) x BX )]) dt + x B X ) β ) dz t, where x B X) denotes the myopc condtonal portfolo choce of nvestment manager. We frst optmze the managers portfolos when they have no access to a cash account, that s, x B ι = 1. The optmal strategy of the managers s gven by x B X ) = 1 ) x X ) + 1 1γ β x X ) ι ) x MV = ζ0 B γ γ + ζ 1 B X, B14) where x X) and x MV ζ B 0 = 1 γ ) as n equaton 21) and ) 1 1γ β + 1 x MV 1 ι ) ) 1 0, B15) γ ζ1 B = 1 ) x MV ι ) ) 1 1. B16) γ γ The mplcaton of equaton B14) s that the optmal portfolo of the managers s agan affne n the state varables. The CIO selects the optmal constant proportons strategy and the constant benchmarks, β 1 and β 2, to optmze the uncondtonal value functon, that s, equaton B11). Ths yelds [ x1c ζ B 01 + ζ11 B X ) ] x Impled C = x 2C ζ B 02 + ζ B 12 X ) = ζ Impled 0 + ζ Impled 1 X, B17) where ζ0 B and ζ 1 B obvously depend on the choce of the benchmark. The condtonal value functon s exponentally quadratc as n equaton B2), wth ζ 0 τ) = ζ Impled 0 and ζ 1 τ) = ζ Impled 1. The coeffcents satsfy the ODEs gven n equaton B6). To solve for the strategc allocaton and the performance benchmark, we evaluate the uncondtonal expectaton of the condtonal value functon usng Lemma 1. We then optmze numercally. Appendx C: Rsk Constrants We derve n ths secton the optmal allocatons of the asset managers n the presence of ether relatve or absolute rsk constrants as defned n Secton III.C. We assume that nvestment opportuntes are constant. For the case wth absolute rsk constrants, the optmzaton problem of asset manager can be smplfed to

44 1892 The Journal of Fnance max x NB A x NB + r γ ) 2 xnb xnb. C1) and the set A s gven by A = x x ι = 1, x x φ A). Consequently, the Kuhn Tucker FOCs are 0 = γ 1 + ξ 1 ) xnb ξ 1 ι C2) 1 = x NB ι, φa 2 xnb xnb, ξ 2 0 C3) ) xnb, C4) 0 = ξ 2 φ 2 A x NB wth ξ 1 and ξ 2 denotng the Kuhn Tucker multplers. In fact, ξ 2 s the multpler for the rsk constrant scaled by a factor γ /2 to smplfy the nterpretaton. If the rsk constrant s not bndng, the managers optmal portfolo s as derved n Secton I.C. Otherwse, the absolute rsk constrant bnds and the optmal portfolo s gven by the soluton to equaton C2) for ξ 2 > 0 so that the rsk constrant holds wth equalty. Ths results mmedately n the optmal portfolo gven n equaton 38). When the asset managers have to satsfy relatve rsk constrants, ther objectve s max x B B x B + β [ β x B ] γ 2 x B β ) x B β ) ), C5) where the set B s gven by B = x x ι = 1, x β ) x β ) φ R ). The FOCs are gven by ) 0 = β γ 1 + ξ1 ) ) x B β ξ1 ι, C6) 1 = x B ι, φ 2 R x B ) β ) x B β, ξ2 0, C7) 0 = ξ 2 φ 2 R x B ) β ) ) x B β, C8) where ξ 1 and ξ 2 ndcate the Kuhn Tucker multplers. Agan, f the relatve rsk constrant s not bndng, the optmal portfolo of Secton I.D. prevals. Otherwse, the optmal strategy of manager s gven by the soluton to equaton C6) wth ξ 2 > 0 so that the relatve rsk constrant s satsfed wth equalty. Ths mples the strategy gven n equaton 39). Fnally, suppose that the benchmark s desgned on the bass of a hgher rsk averson level, say γ, than the manager s rsk averson, denoted by γ.in ths case, the relatve) rsk of the manager s portfolo wll exceed the relatve) rsk that would correspond to a manager wth rsk averson level γ. If the rsk lmt s constructed for a manager wth rsk averson γ, then the relatve rsk constrant wll bnd for the manager wth rsk averson γ. Ths nduces an effectve ncrease n the manager s rsk averson from γ to γ. To show ths, note that the dfference between the optmal portfolo of the manager, who has a

45 Optmal Decentralzed Investment Management 1893 rsk averson γ, and the benchmark weghts, whch are desgned for a manager wth rsk averson γ, s gven by x B γ, β γ )) β γ ) = γ 1 γ 1 γ { x x C x C ι ) ι x ) x MV }. C9) In ths expresson, x B γ, β γ )) denotes the optmal portfolo choce when the nvestor has a coeffcent of relatve rsk averson γ, but s evaluated relatve to a benchmark, β γ ), whch s based on γ. Ths mmedately mples for the relatve rsk of the manager s portfolo: x B γ, β γ )) β γ )) ) x B γ, β γ )) β γ )) ) γ 2 = x B γ, β γ )) β γ )) γ ) x B γ, β γ )) β γ )), C10) that s, the relatve rsk of a more aggressve manager under a benchmark desgned for a more conservatve manager s larger than when the more conservatve manager mplements the strategy, snce γ >γ. Ths mples that when the rsk constrant s satsfed wth equalty for a manager wth rsk averson γ, an unconstraned manager wth rsk averson γ wll mplement a strategy that exceeds the relatve rsk lmt. Consequently, the rsk constrant on the bass of whch the benchmark s desgned wll be bndng and nduces an effectve ncrease n the manager s rsk averson from γ to γ. REFERENCES Admat, Anat R., and Paul Pflederer, 1997, Does t all add up? Benchmarks and the compensaton of actve portfolo managers, Journal of Busness 70, Aït-Sahala, Yacne, and Mchael W. Brandt, 2001, Varable selecton for portfolo choce, Journal of Fnance 56, Ang, Andrew, and Geert Bekaert, 2005, Stock return predctablty, s t there?, Revew of Fnancal Studes 20, Barbers, Ncholas C., 2000, Investng for the long run when returns are predctable, Journal of Fnance 55, Barry, Chrstopher B., and Laura T. Starks, 1984, Investment management and rsk sharng wth multple managers, Journal of Fnance 39, Basak, Suleyman, Anna Pavlova, and Alex Shapro, 2007, Offsettng the ncentves: Benefts of benchmarkng n money management, Workng paper, London Busness School. Basak, Suleyman, Alex Shapro, and Luce Teplá, 2006, Rsk management wth benchmarkng, Management Scence 52, van Bnsbergen, Jules H., and Mchael W. Brandt, 2007, Optmal asset allocaton n asset lablty management, Workng paper, Duke Unversty. van Bnsbergen, Jules H., and Ralph S.J. Kojen, 2007, Predctve regressons: A present-value approach, Workng paper, Duke Unversty. Brandt, Mchael W., 1999, Estmatng portfolo and consumpton choce: A condtonal Euler equatons approach, Journal of Fnance 54, Brandt, Mchael W., 2005, Portfolo choce problems, n Y. Aït-Sahala and L.P. Hansen, eds., Handbook of Fnancal Econometrcs, forthcomng. Brennan, Mchael J., 1993, Agency and asset prcng, Workng paper, UCLA. Brennan, Mchael J., Eduardo S. Schwartz, and Ronald Lagnado, 1997, Strategc asset allocaton, Journal of Economc Dynamcs & Control 21,

46 1894 The Journal of Fnance Brennan, Mchael J., and Xa, Yhong, 2001, Stock prce volatlty and equty premum, Journal of Monetary Economcs 47, Browne, Sd, 1999, Beatng a movng target: Optmal portfolo strateges for outperformng a stochastc benchmark, Fnance & Stochastcs 3, Browne, Sd, 2000, Rsk constraned dynamc actve portfolo management, Management Scence 46, Campbell, John Y., Yeung L. Chan, and Lus M. Vcera, 2003, A multvarate model of strategc asset allocaton, Journal of Fnancal Economcs 67, Campbell, John Y., and Lus M. Vcera, 1999, Consumpton and portfolo decsons when expected returns are tme varyng, Quarterly Journal of Economcs 114, Campbell, John Y., and Tuomo Vuolteenaho, 2004, Bad beta, good beta, Amercan Economc Revew 94, Campbell, John Y., and Motohro Yogo, 2006, Effcent tests of stock return predctablty, Journal of Fnancal Economcs 81, Carpenter, Jennfer N., 2000, Does opton compensaton ncrease manageral rsk appette? Journal of Fnance 55, Cochrane, John H., and Monka Pazzes, 2005, Bond rsk prema, Amercan Economc Revew 95, Cornell, Bradford, and Rchard W. Roll, 2005, A delegated-agent asset-prcng model, Fnancal Analysts Journal 61, Cuoco, Domenco, and Ron Kanel, 2006, Equlbrum prces n the presence of delegated portfolo management, Workng paper, Unversty of Pennsylvana. Cvtanć, Jaka, Al Lazrak, Lonel Martelln, and Fernando Zapatero, 2006, Dynamc portfolo choce wth parameter uncertanty and the economc value of analysts recommendatons, Revew of Fnancal Studes 19, Da, Qang, and Kenneth J. Sngleton, 2002, Expectaton puzzles, tme-varyng rsk prema, and affne models of the term structure, Journal of Fnancal Economcs 63, Dessen, Wouter, Lus Garcano, and Rob Gertner, 2005, Organzng for synerges, Workng paper, Unversty of Chcago. Elton, Edwn J., and Martn J. Gruber, 2004, Optmum centralzed portfolo constructon wth decentralzed portfolo management, Journal of Fnancal and Quanttatve Analyss 39, Foster, F. Douglas, and Mchael Stutzer, 2003, Performance and rsk averson of funds wth benchmarks: A large devatons approach, Workng paper, Australan Graduate School of Management. Goller, Chrstan, and John W. Pratt, 1996, Rsk vulnerablty and the temperng effect of background rsk, Econometrca 64, 5, Gómez, Juan-Pedro, and Fernando Zapatero, 2003, Asset prcng mplcatons of benchmarkng: A two-factor CAPM, European Journal of Fnance 9, Joron, Phlppe, 2003, Portfolo optmzaton wth trackng-error constrants, Fnancal Analysts Journal 59, Jurek, Jakub W., and Lus M. Vcera, 2006, Optmal value and growth tlts n long-horzon portfolos, Workng paper, Harvard Unversty. Km, Tong Suk, and Edward Omberg, 1996, Dynamc nonmyopc portfolo behavor, Revew of Fnancal Studes 9, Lettau, Martn, and Stjn Van Neuwerburgh, 2007, Reconclng the return predctablty evdence, Revew of Fnancal Studes, forthcomng. Lewellen, Jonathan, 2004, Predctng returns wth fnancal ratos, Journal of Fnancal Economcs 74, Lu, Jun, 2007, Portfolo choce n stochastc envronments, Revew of Fnancal Studes 20, Merton, Robert C., 1969, Lfetme portfolo selecton under uncertanty: The contnuous-tme case, Revew of Economcs and Statstcs 51, Merton, Robert C., 1971, Optmum consumpton and portfolo rules n a contnuous-tme model, Journal of Economc Theory 3,

47 Optmal Decentralzed Investment Management 1895 Merton, Robert C., 1980, On estmatng the expected return on the market: An exploratory nvestgaton, Journal of Fnancal Economcs 8, Ou-Yang, Hu, 2003, Optmal contracts n a contnuous-tme delegated portfolo management problem, Revew of Fnancal Studes 16, Roll, Rchard W., 1992, A mean/varance analyss of trackng error, Journal of Portfolo Management 18, Sangvnatsos, Antonos, and Jessca A. Wachter, 2005, Does the falure of the expectatons hypothess matter for long-term nvestors? Journal of Fnance 60, Sharpe, Wllam F., 1981, Decentralzed nvestment management, Journal of Fnance 36, Sharpe, Wllam F., 2002, Budgetng and montorng penson fund rsk, Fnancal Analysts Journal 58, Stracca, Lvo, 2006, Delegated portfolo management: A survey of the theoretcal lterature, Journal of Economc Surveys 20, Stutzer, Mchael, 2003a, Portfolo choce wth endogenous utlty: A large devatons approach, Journal of Econometrcs 116, Stutzer, Mchael, 2003b, Fund managers may cause ther benchmarks to be prced rsk, Workng paper, Unversty of Colorado. Teplá, Luce, 2000, Optmal portfolo polces wth borrowng and shortsale constrants, Journal of Economc Dynamcs & Control 24, Torous, Walter, Rossen Valkanov, and Shu Yan, 2005, On predctng stock returns wth nearly ntegrated explanantory varables, Journal of Busness 77, Treynor, Jack L., and Fscher Black, 1973, How to use securty analyss to mprove portfolo selecton, Journal of Busness 46, Vayanos, Dmtr, 2003, The decentralzaton of nformaton processng n the presence of nteractons, Revew of Economc Studes 70, Wachter, Jessca A., 2002, Portfolo and consumpton decsons under mean-revertng returns: An exact soluton for complete markets, Journal of Fnancal and Quanttatve Analyss 37,