GAMING PERFORMANCE FEES BY PORTFOLIO MANAGERS* Dean P. Foser and H. Peyon Young We show ha i is very difficul o devise performance based compensaion conracs ha reward porfolio managers who generae excess reurns while screening ou managers who canno generae such reurns. Theoreical bounds are derived on he amoun of fee manipulaion ha is possible under various performance conracs. We show ha recen proposals o reform compensaion pracices, such as posponing bonuses and insiuing clawback provisions, will no eliminae opporuniies o game he sysem unless accompanied by ransparency in managers posiions and sraegies. Indeed here exiss no compensaion mechanism ha separaes skilled from unskilled managers solely on he basis of heir reurns hisories. * This paper is forhcoming in he Quarerly Journal of Economics. We are indebed o Pee Kyle, Andrew Lo, Andrew Paon, Tarun Ramadorai, Krishna Ramaswamy, Neil Shephard, Rober Sine and wo anonymous referees for helpful suggesions. An earlier version was eniled The Hedge Fund Game: Incenives, Excess Reurns, and Performance Mimics, Working Paper 07 041, Wharon Financial Insiuions Cener, Universiy of Pennsylvania, 2007.
I. Background Incenives for financial managers are coming under increased scruiny because of heir endency o encourage excessive risk aking. In paricular, he asymmeric reamen of gains and losses gives managers an incenive o increase leverage and ake on oher forms of risk wihou necessarily increasing expeced reurns for invesors. Various changes o he incenive srucure have been proposed o deal wih his problem, including posponing bonus paymens, clawing back bonus paymens if laer performance is poor, requiring managers o hold an equiy sake in he funds ha hey manage, and so forh. These apply boh o managers of financial insiuions, such as banks, and also o managers of privae invesmen pools, such as hedge funds. 1 The purpose of his paper is o show ha, while hese and relaed reforms may moderae he incenives o game he sysem, gaming canno be eliminaed. The problem is especially acue when here is no ransparency, so invesors canno see he rading sraegies ha are producing he reurns for which managers are being rewarded. In his seing, where managerial compensaion is based solely 1 For a general discussion of managerial incenives in he financial secor see Bebchuk and Fried (2004) and Bebchuk and Spamann (2009). The lieraure on incenives and risk aking by porfolio managers will be discussed in greaer deail in secion 2. 2
on hisorical performance, we esablish wo main resuls. Firs, if a performancebased compensaion conrac does no levy ou of pocke penalies for underperformance, hen managers wih no superior invesmen skill can capure a sizable amoun of he fees ha are inended o reward superior managers by mimicking he laer s performance. The poenial amoun of fee capure has a concise analyical expression. Second, if a compensaion conrac imposes penalies ha are sufficienly harsh o deer risk neural mimics, hen i will also deer managers of arbirarily high skill levels. In oher words, here exis no performance based compensaion schemes ha screen ou risk neural mimics while rewarding managers who generae excess reurns. This conrass wih saisical measures of performance, some of which can discriminae in he long run beween exper and non exper managers. 2 2 There is a subsanial lieraure on saisical ess ha discriminae beween rue expers and hose who merely preend o be expers. In finance, Goezmann, Ingersoll, Spiegel, and Welch (2007) propose a class of measures of invesmen performance ha we discuss in greaer deail in secion 6. Somewha more disanly relaed is he lieraure on how o disinguish beween expers who can predic he probabiliy of fuure evens and imposers who manipulae heir predicions in order o look good (Lehrer, 2001; Sandroni, Smorodinsky, and Vohra, 2003; Sandroni, 2003; Olszewski and Sandroni, 2008, forhcoming). Anoher paper ha is hemaically relaed is Spiegler (2006), who shows how quacks can survive in a marke due o he difficuly ha cusomers have in disinguishing hem from he real hing. 3
Our resuls are proved using a combinaion of game heory, probabiliy heory, and elemenary principles of mechanism design. One of he novel heoreical elemens is he concep of performance mimicry. This is analogous o a common biological sraegy known as mimicry in which one species sends a signal, such as a simulaed maing call, in order o lure poenial maes, who are hen devoured. An example is he firefly Phouris versicolor, whose predaceous females imiae he maing signals of females from oher species in order o arac passing males, some of whom respond and are promply eaen (Lloyd, 1974). Of course, he imiaion may be imperfec and he arges are no fooled all of he ime, bu hey are fooled ofen enough for he sraegy o confer a benefi on he mimic. 3 In his paper we shall apply a varian of his idea o modeling he compeiion for cusomers in financial markes. We show ha porfolio managers wih no privae informaion or special invesmen skills can generae reurns over an exended period of ime ha look jus like he reurns ha would be generaed by highly skilled managers; moreover, hey can do so wihou any knowledge of 3 Biologiss have documened a wide range of mimicking reperoires, including males mimicking females, and harmless species mimicking harmful ones in order o deer predaors (Alcock, 2005). 4
how he skilled managers acually produce such reurns. 4 Of course, a mimic canno reproduce a skilled manager s record forever; insead he reproduces i wih a cerain probabiliy and pays for i by aking on a small probabiliy of a large loss. In pracice, however, his probabiliy is sufficienly small ha he mimic can ge away wih he imiaion for many years (in expecaion) wihou being discovered. Our framework allows us o derive precise analyical expressions for: i) he probabiliy wih which an unskilled manager can mimic a skilled one over any specified lengh of ime; and ii) he minimum amoun he mimic can expec o earn in fees as a funcion of he compensaion srucure. The paper is srucured as follows. In he nex secion we review he prior heoreical and empirical lieraure on performance manipulaion. In secion 3 we inroduce he model, which allows us o evaluae a very wide range of 4 I should be emphasized ha mimicry is no he same as cloning or replicaion (Ka and Palaro, 2005; Hasanhodzic and Lo, 2007). These sraegies seek o reproduce he saisical properies of a given fund or class of funds, whereas mimicry seeks o fool invesors ino hinking ha reurns are being generaed by one ype of disribuion when in fac hey are being generaed by a differen (and less desirable) disribuion. Mimicry is also disinc from sraegy sealing, which is a game heoreic concep ha involves one player copying he enire sraegy of anoher (Gale, 1974). In our seing he performance mimic canno seal he skilled manager s invesmen sraegy because if he knew he sraegy hen he oo would be skilled. 5
compensaion conracs and differen ways of manipulaing hem. Secion 4 shows how much fee capure is possible under any compensaion arrangemen ha does no assess personal financial penalies on he manager. In secion 5 we explore he implicaions of his resul hrough a series of concree examples. Secion 6 discusses manipulaion proof performance measures and why hey do no solve he problem of designing manipulaion proof compensaion schemes. In secion 7 we derive an impossibiliy heorem, which shows ha here is essenially no compensaion scheme ha is able o reward skilled managers and screen ou unskilled managers based solely on heir rack records. Secion 8 shows how o exend hese resuls o allow for he inflow and ouflow of money based on prior performance. Secion 9 concludes. II. Relaed lieraure The fac ha sandard compensaion conracs give managers an incenive o manipulae reurns is no a new observaion; indeed here is a subsanial prior lieraure on his issue. In paricular, he wo par fee srucure ha is common in he hedge fund indusry has wo perverse feaures: he fees are convex in he level of performance, and gains and losses are reaed asymmerically. These feaures creae incenives o ake on increased risk, a poin ha has been 6
discussed in boh he empirical and heoreical finance lieraure (Sarks, 1987; Carpener, 2000; Lo, 2001; Hodder and Jackwerh, 2007). The approach aken here builds on his work by considering a much more general class of compensaion conracs and by deriving heoreical bounds on how much manipulaion is possible. Of he prior work on his opic, Lo (2001) is he closes o ours because he focuses explicily on he quesion of how much money a sraegic acor can make by deliberaely manipulaing he reurns disribuion using opions rading sraegies. Lo examines a hypoheical siuaion in which a manager akes shor posiions in S&P 500 pu opions ha maure in 1 3 monhs, and shows ha such an approach would have generaed very sizable excess reurns relaive o he marke in he 1990s. (Of course his sraegy could have los a large amoun of money if he marke had gone down sufficienly.) The presen paper builds on Lo s approach by examining how far his ype of manipulaion can be aken and how much fee capure is heoreically possible. We do his by explicily defining he sraegy space ha is available o poenial enrans, and how hey can use i o mimic high performance managers. 7
A relaed srand of he lieraure is concerned wih he poenial manipulaion of sandard performance measures, such as he Sharpe raio, he appraisal raio, and Jensen s alpha. I is well known ha hese and oher measures can be gamed by manipulaing he reurns disribuion wihou generaing excess reurns in expecaion (Ferson and Siegel, 2001; Lhabian, 2000). I is also known, however, ha one can design performance measures ha are immune o many forms of manipulaion. These ake he form of a consan relaive risk aversion uiliy funcion averaged over he reurns hisory (Goezmann, Ingersoll, Spiegel, and Welch, 2007). We shall discuss hese connecions furher in secion 6. Our main conclusion, however, is ha a similar possibiliy heorem does no hold for compensaion mechanisms. A firs his may seem surprising: for example, why would i no suffice o pay fund managers according o a linear increasing funcion of one of he manipulaion proof measures menioned above? The difficuly is ha a compensaion mechanism mus no only reward managers according o heir acual abiliy, i mus also screen ou managers who have no abiliy. In oher words, he mechanism mus creae incenives for skilled managers o paricipae and for unskilled managers no o paricipae. This urns ou o be considerably more demanding because managers of differen skill levels have differen opporuniy coss and herefore differen incenivecompaibiliy consrains. 8
III. The model Performance based compensaion conracs rely on wo ypes of inpus: he reurns generaed by he fund manager and he reurns generaed by a benchmark porfolio ha serves as a comparaor. Consider firs a benchmark porfolio ha generaes a sequence of reurns in each of T periods. Throughou we shall assume ha reurns are repored a discree inervals, say a he end of each monh or each quarer (hough he value of he asse may evolve in coninuous ime). Le r f be he risk free rae in period and le reurn of he benchmark porfolio in period, where X be he oal X is a nonnegaive random variable whose disribuion may depend on he prior realizaions x1, x2,..., x 1. A fund ha has iniial value s0 0 and is passively invesed in he benchmark will herefore have value s0 X by he end of he 1 T h T period. If he benchmark asse is risk free hen X 1 r. Alernaively, X may represen he reurn on a f broad marke index such as he S&P 500, in which case i is sochasic, hough we do no assume saionariy. Le he random variables Y 0 denoe he period by period reurns generaed by a paricular managed porfolio, 1 T. A compensaion conrac is ypically 9
based on a comparison beween he reurns Y and he reurns X generaed by a suiably chosen benchmark. I will be mahemaically convenien o express he reurns of he managed porfolio as a muliple of he reurns generaed by he benchmark asse. Specifically, le us assume ha X 0 in each period, and consider he random variable M 0 such ha. (1) Y MX A compensaion conrac over T periods is a vecor valued funcion : R 2T T1 R ha specifies he paymen o he manager in each period 0,1,2,...,, T as a funcion of he amoun of money invesed and he realized sequences x ( x1, x2,..., x T ) and m ( m1, m2,..., m T ). We shall assume ha he paymen in period depends only on he realizaions x,..., 1 x and m,..., 1 m. We shall also assume ha he paymen is made a he end of he period, and canno exceed he funds available a ha poin in ime. (Paymens due a he sar of a period can always be aken ou a he end of he preceding period, so his involves no real loss of generaliy. The paymen in period zero, if any, corresponds o an upfron managemen fee.) 10
This formulaion is very general, and includes sandard incenive schemes as well as commonly proposed reforms, such as posponemen and clawback arrangemens, in which bonuses earned in prior periods can be offse by maluses in laer periods. These and a hos of oher variaions are embedded in he assumpion ha he paymen in period, ( mx, ), can depend on he enire sequence of reurns hrough period. Le us consider a concree example. Suppose ha he conrac calls for a 2% managemen fee ha is aken ou a he end of each year plus a 20% performance bonus on he reurn generaed during he year in excess of he risk free rae. Le he iniial size of he fund be s 0. Given a pair of realizaions ( mx, ), le s s ( m, x) be he size of he fund a he sar of year afer any upfron fees have been deduced. Then he managemen fee a he end of he firs year will be 0.02m1xs 1 1and he bonus will be 0.2( mx 1 rf) s1. Hence (2) 1 [.02 mx 1 1.2( mx 1 1 1 rf 1) ] s1. Leing s2 s1 1 and coninuing recursively we find ha in each year, 11
(3) [.02 mx.2( mx 1 r ) ] s. f Alernaively, suppose ha he conrac specifies a 2% managemen fee a he end of each year plus a one ime 20% performance bonus ha is paid only a he end of T years. In his case he size of he fund a he sar of year is s ( m, x) s (.98) m x 1 0 s s 1s1. The managemen fee in he h year equals (4) 1 ( mx, ).02 s( mxmx, ) [.02(.98) mx s s] s0 1s. The final performance bonus equals 20% of he cumulaive excess reurn relaive. o he risk free rae, which comes o.2[ mx (1 r )] s0 f 1 T 1 T IV. Performance mimicry We shall say ha a manager has superior skill if, in expecaion, he delivers excess reurns relaive o a benchmark porfolio (such as a broad based marke index), eiher hrough privae informaion, superior predicive powers, or access o payoffs ouside he benchmark payoff space. A manager has no skill if he canno deliver excess reurns relaive o he benchmark porfolio. Invesors should no 12
be willing o pay managers wih no skill, because he invesors can obain he same expeced reurns by invesing passively in he benchmark. We claim, however, ha under any performance based compensaion conrac, eiher he unskilled managers can capure some of he fees inended for he skilled managers, or else he conrac is sufficienly unaracive ha boh he skilled and unskilled managers will no wish o paricipae. We begin by examining he case where he conrac calls only for nonnegaive paymens, ha is, ( mx, ) 0 siuaion where ( mx, ) 0 for all mx,,. (In secion 7 we shall consider he for some realizaions m and x.) Noe ha nonnegaive paymens are perfecly consisen wih clawback provisions, which reduce prior bonuses bu do no normally lead o ne assessmens agains he manager s personal asses. Given realized sequences m and x, define he manager s cu in period o be he fracion of he available funds a he end of he period ha he manager akes in fees, namely, (5) c ( m, x) ( m, x)/ m xs ( m, x). 13
By assumpion he fees are nonnegaive and canno exceed he funds available, hence 0 c ( m, x) 1 for all mx,. (If mxs ( m, x ) 0 we le c ( m, x) 1 and assume ha he fund closes down.) The cu funcion is he vecor valued funcion 2T T1 c: R [0,1] such ha cmx (, ) ( c0( mx, ), c1( mx, ),..., c( mx, )) T for each pair ( mx, ). In our earlier example wih a 2% end of period managemen fee and a 20% annual bonus, he cu funcion is (6) c ( m, x ) 0 0 and 1 rf c ( m, x).02.2[1 ] mx for 1 T. Proposiion 1. Le be a nonnegaive compensaion conrac over T periods ha is benchmarked agains a porfolio generaing reurns X ( X1, X2,..., X T ) 0, and le c be he associaed cu funcion. Given any arge sequence of excess reurns m 1 here exiss a mimicking sraegy M 0 ( m) ha delivers zero expeced excess reurns in every 0 period ( EM [ ] 1), such ha for every realizaion X x of he benchmark asse, he mimic s expeced fees in period (condiional on x ) are a leas c ( m, x )[(1 c ( m, x )) (1 c ( m, x ))][ x x ] s. (7) 0 1 1 0 14
Noe ha, in his expression, he facor [(1 c0( m, x)) (1 c 1( m, x))] is he fracion lef over afer he manager has aken ou his cu in previous periods. Hence he proposiion says ha in expecaion he mimic s cu in period, c ( m, x), is he same as he cu of a skilled manager who generaes he excess reurns sequence m wih cerainy. The difference is ha he mimic s cu is assessed on a fund ha is compounding a he rae of he benchmark asse ( 1s x s ), whereas he skilled manager s cu is based on a porfolio compounding a he higher rae mx s s. I follows ha he skilled manager will earn more 1s han he mimic in expecaion. The key poin, however, is no ha skilled managers earn more han mimics, bu ha mimics can earn a grea deal compared o he alernaive, which is no o ener he marke a all. To undersand he implicaions of his resul, le us work hrough a simple example. Suppose ha he benchmark asse consiss of risk free governmen bonds growing a a fixed rae of 4% per year. Consider a skilled manager who can deliver 10% over and above his every year, and is paid according o he sandard wo and weny conrac: a bonus equal o 20% of he excess reurn plus 15
a managemen fee of 2%. 5 In his case he excess annual reurn is (1.10)(1.04) 1.04 0.104, so he performance bonus is.20(0.104) 0.0208 per dollar in he fund a he sar of he period. This comes o abou 0.0208/[(1.10)(1.04)] = 0.0182 per dollar a he end of he period. By assumpion he managemen fee is.02 per dollar a he end of he period. Therefore he cu, which is he oal fee per dollar a he end of he period, is 0.0382 or 3.82%. Proposiion 1 says ha a manager wih no skill has a mimicking sraegy ha in expecaion earns a leas 3.82% per year of a fund ha is compounding a 4% per year before fees, and 0.027% afer fees ( 1.04(1 0.0382) 1.00027 ). As becomes large he probabiliy goes o one ha he fund will go bankrup before hen. However, he mimic s expeced earnings in any given year are acually increasing wih, because in expecaion he fund is compounding a a faser rae (4%) han he manager is aking ou he fees (3.82%). The key o proving proposiion 1 is he following resul. 5 Of course i is unlikely ha anyone would generae he same reurn year afer year bu his assumpion keeps he compuaions simple. 16
Lemma. Consider any arge sequence of excess reurns m ( m1,..., m T ) (1,1,...,1). A mimic has a sraegy M 0 ( m) ha, for every realized sequence of reurns x of he benchmark porfolio, generaes he reurns sequence ( mx 1 1,..., mtx T) wih probabiliy a leas 1/ m. 1 T We shall skech he idea of he proof here; in he Appendix we show o execue he sraegy using pus and calls on sandard marke indexes wih Black Scholes pricing. Proof skech. Choose a arge excess reurns sequence m1, m2,, mt 1. A he sar of period 1 he mimic has capial equal o s 0. Assume ha he invess i enirely in he benchmark asse. He hen uses he capial as collaeral o ake a posiion in he opions marke. The opions posiion amouns o placing a fair be ha bankrups he fund wih low probabiliy ( 1 1/m1 ) and inflaes i by he facor m 1 wih high probabiliy ( 1/m 1) by he end of he period. If he highprobabiliy oucome occurs, he mimic has end of period capial equal o mxs 1 1 0, while if he low probabiliy oucome occurs he fund goes bankrup. 17
The mimic repeas his consrucion in each successive period using he corresponding value m as he arge. By he end of he h T period he sraegy will have generaed he reurns sequence ( mx 1 1,..., mtx T) wih probabiliy 1/ m 1 T, and his holds for every realizaion x of he benchmark porfolio. This concludes he ouline of he proof of Lemma 1. Proposiion 1 is now proved as follows. Choose a paricular sequence of excess reurns m 1. Under he mimicking sraegy defined in he Lemma, for every realizaion x and every period, he mimic generaes excess reurns m1, m2,, m 1 wih probabiliy a leas 1/ mm 1 2 m. Wih his same probabiliy he earns c ( m, x )[(1 c ( m, x )) (1 c ( m, x ))][ x x ][ m m ] s. 0 1 1 1 0 Thus, since his earnings are always nonnegaive, his expeced earnings in period mus be a leas c( m, x )[(1 c0( m, x )) (1 c 1( m, x ))][ x1 x] s0. This concludes he proof of proposiion 1. 6 6 This consrucion is somewha reminiscen of he doubling up sraegy in which one keeps doubling oneʹs sake unil a win occurs (Harrison and Kreps, 1979). Our se up differs in several crucial respecs however: he manager only eners ino a finie number of gambles and he canno borrow o finance hem. More generally, he mimicking sraegy is no a mehod for beaing he odds in he opions markes; i is a mehod for manipulaing he disribuion of reurns in order o earn large fees from invesors. 18
V. Discussion Mimicking sraegies are sraighforward o implemen using sandard derivaives, and hey generae reurns ha look good for exended periods while providing no value added o invesors. (Recall ha he invesors can earn he same expeced reurns wih possibly much lower variance by invesing passively in he benchmark asse.) Similar sraegies can be used o mimic disribuions of reurns as well as paricular sequences of reurns. 7 In fac, however, here is no need o mimic a disribuion of reurns. Managers are paid on he basis of realized reurns, no disribuions. Hence all a mimic needs o do is arge some paricular sequence of excess reurns ha migh have arisen from a disribuion (and ha generaes high fees). Proposiion 1 shows ha he will earn as high a cu in expecaion as a skilled manager would earn had he generaed he same sequence. Of course, he fund s invesors would no necessarily approve if hey could see wha he mimic was doing. The poin of he analysis, however, is o show wha 7 Indeed, le M be a nonnegaive random variable wih expecaion EM [ ] m 1. Suppose ha a mimic wishes o produce he disribuion M X in period, where X is he reurn from he benchmark. The random variable M (1 / m ) M represens a fair be. The mimic can herefore implemen M X wih probabiliy a leas 1/ inflaing he fund by he facor m by firs placing he fair be m using he sraegy described in he Lemma. M and hen 19
can happen when invesors canno observe he managers underlying sraegies a siuaion ha is quie common in he hedge fund indusry. Performance conracs ha are based purely on repored reurns, and ha place no resricions on managers sraegies, are highly vulnerable o manipulaion. Expression (7) in proposiion 1 shows how much fee capure is possible, and why i is very difficul o eliminae his problem by resrucuring he compensaion conrac. One common proposal, for example, is o delay paying a performance bonus for a subsanial period of ime. To be concree, le us suppose ha a manager can only be paid a performance bonus afer five years, a which poin he will earn 20% of he oal reurn from he fund in excess of he risk free rae compounded over five years. For example, wih a risk free rae of 4% he will earn a performance bonus equal o 5.20[ s5 (1.04) ], where s 5 is he value of he fund a he end of year 5. Consider a hypoheical manager who earns muliplicaive excess reurns equal o 1.10 each year. Under he above conrac his bonus in year 5 would be.20[(1.10) (1.04) s (1.04) s ].149s, ha is, abou 15% of he amoun iniially 5 5 5 0 0 0 invesed. Le us compare his o he expeced earnings of someone who generaes apparen 10% excess reurns using he mimicking sraegy. The 20
mimic s sraegy runs for five years wih probabiliy 5 (1.10).621, hence his expeced bonus is abou (.621)(.149) s0.0925s0. Thus, wih a five year posponemen, he mimic earns an expeced bonus equal o more han 9% of he amoun iniially invesed. Now consider a longer posponemen, say en years. The probabiliy ha he mimic s sraegy will run his long is 10 (1.10).386. However, he bonus will be calculaed on a larger base. Namely, if he mimic s fund does keep running for en years, he bonus will be 10 10 10.20[(1.10) (1.04) (1.04) ] s0.472s0. Therefore he expeced bonus will be approximaely (.386)(.472 s0).182s0 or abou 18% of he amoun iniially invesed. Indeed, i is sraighforward o show ha under his paricular bonus scheme, he expeced paymen o he mimic increases he longer he posponemen is. 8 I is, of course, rue ha he longer he posponemen, he greaer he risk ha he fund will go bankrup before he mimic can collec his bonus. Thus posponemen may ac as a deerren for mimics who are sufficienly risk averse. 8 T T T The bonus in he final period T is.20[(1.10) (1.04) (1.04) ] and he probabiliy of earning i is (1.10) T. Hence he expeced bonus is.20[(1.10) (1.04) (1.04) ] /(1.10).20[1 (1.1) ][1.04], which is increasing in T. T T T T T T 21
However, his does no offer much comfor for several reasons. Firs, as we have jus seen, he posponemen mus be quie long o have much of an impac. Second, no all mimics need o be risk neural; i suffices ha some of hem are. Third, here is a simple way for a risk averse mimic o diversify away his risk: run several funds in parallel (under differen names) using independen mimicking sraegies. Suppose, for example, ha a mimic runs n independen funds of he ype described above, each yielding 10% annual excess reurns wih probabiliy 1/1.1 = 0.091. The probabiliy ha a leas one of he funds survives T for T years or more is 1 (11/1.1 ) n. This can be made as close o one as we like by choosing n o be sufficienly large. 9 VI. Performance measures versus performance paymens The preceding analysis leaves open he possibiliy ha performance conracs wih negaive paymens migh solve he problem. Before urning o his case, however, i will be useful o consider he relaionship beween saisical 9 A relaed poin is ha, in any large populaion of funds run by mimics, he probabiliy is high ha a leas one of hem will look exremely good, perhaps beer han many funds run by skilled managers (hough no necessarily beer han he bes of he funds run by skilled managers). Correcing for mulipliciy poses quie a challenge when esing for excess reurns in financial markes; for a furher discussion of his issue see Foser, Sine, and Young (2008). 22
measures of performance and performance based compensaion conracs. Some sandard measures of performance, such as Jensen s alpha or he Sharpe raio, are easily gamed by manipulaing he reurns disribuion. Oher measures avoid some forms of manipulaion, bu (as we shall see) hey do no solve he problem of how o pay for performance. Consider, for example, he following class of measures proposed by Goezmann e al. (2007). Le 1 1 x be ux ( ) (1 ) a consan relaive risk aversion (CRR) uiliy funcion wih 1. If a fund delivers he sequence of reurns M (1 r ),1 T, one can define he f performance measure (8) Gm T m ( ) (1 1 1 ) ln[(1/ ) ] 1 T, 1. A varian of his approach ha is used by he raing firm Morningsar (2006) is (9) G m T m 2 1/2 *( ) [(1/ ) 1/ ] 1 1 T. These and relaed measures rank managers according o heir abiliy o generae excess reurns in expecaion. Bu o ranslae hese (and oher) saisical measures ino moneary paymens for performance leads o rouble. Firs, 23
paymens mus be made on realized reurns; one canno wai forever o see wheher he reurns are posiive in expecaion. Second, if he paymens are always nonnegaive, hen he mimic can capure some of hem, as proposiion 1 shows. Moreover, if he paymens are allowed o be negaive, hen hey are consrained by he managers abiliy o pay hem. 10 In he nex secion we shall show ha his leads o an impossibiliy heorem: if he penalies are sufficien o screen ou he mimics, hen hey also screen ou skilled managers of arbirarily high abiliy. VII. Penalies Consider a general compensaion mechanism ha someimes imposes penalies, ha is, ( mx, ) 0 for some values of mx, and. To simplify he exposiion we shall assume hroughou his secion ha he benchmark asse is risk free, ha is, x 1 r for all. Suppose ha a fund sars wih an iniial f amoun s 0, which we can assume wihou loss of generaliy is s0 1. To illusrae he issues ha arise when penalies are imposed, le us begin by considering he one period case. Le (1 rf 1) m 0 be he fund s oal reurn in 10 Noe ha if paymens are linear and increasing in he performance measure (8), hen arbirarily large penalies will be imposed when m is close o zero. 24
period 1, and le ( m) be he manager s fee as a funcion of m. The wors case scenario (for he invesors) is ha m 0. Assume ha in his case he manager suffers a penaly (0) 0. There are wo cases o consider: i) he penaly arises because he manager holds an equiy sake of size (0) in he fund, which he loses when he fund goes bankrup; or ii) he penaly is held in escrow in a safe asse earning he risk free rae, and is paid ou o he invesors if he fund goes bankrup. The firs case he equiy sake would be an effecive deerren provided he mimic were sufficienly risk averse and were prevened from diversifying his risk across differen funds. Bu an equiy sake will no deer a risk neural mimic, because he expeced reurn from he mimic s sraegy is precisely he riskfree rae, so his sake acually earns a posiive amoun in expecaion, namely (1 r ) (0), and in addiion he earns posiive fees from managing he porion f 1 of he fund ha he does no own. Now consider he second case, in which fuure penalies are held in an escrow accoun earning he risk free rae of reurn. For our purposes i suffices o consider he penaly when he fund goes bankrup. To cover his even he amoun placed in escrow mus be a leas b (0) /(1 r f 1) 0. Fix some m* 1 25
and consider a risk neural mimic who generaes he reurn m*(1 r f 1) wih probabiliy 1/ m * and goes bankrup wih probabiliy 1 1/ m*. To deer such a mimic, he fees earned during he period mus be nonposiive in expecaion, ha is, (10) ( m*)/ m* (0)(11/ m*) 0. Since a mimic can arge any such m *, (10) mus hold for all m* 1. Now consider a skilled manager who can generae he reurn m * wih cerainy. This manager mus also pu he amoun b in escrow, because ex ane all managers are reaed alike and he invesors canno disinguish beween hem. However, his involves an opporuniy cos for he skilled manager, because by invesing b in her own privae fund she could have generaed he reurn m*(1 r ) b. The resuling opporuniy cos for he skilled manager is f 1 m*(1 r ) b(1 r ) b( m* 1) (0). Assuming ha uiliy is linear in money f1 f1 (i.e., he manager is risk neural), she will no paricipae if he opporuniy cos exceeds he fee, ha is, if (10 ) ( m*) ( m* 1) (0) 0. 26
Dividing (10 ) by m *, we see ha i follows immediaely from (10), which holds for all m* 1. We have herefore shown ha, if a one period conrac deers all riskneural mimics, i also deers any risk neural manager who generaes excess reurns. The following generalizes his resul o he case of muliple periods and randomly generaed reurn sequences. Proposiion 2. There is no compensaion mechanism ha separaes skilled from unskilled managers solely on he basis of heir reurns hisories. In paricular, any compensaion mechanism ha deers unskilled risk neural mimics also deers all skilled risk neural managers who consisenly generae reurns in excess of he risk free rae. Proof. Le x 1 r be he risk free rae of reurn in period. To simplify he f noaion we shall drop he x ' s and le ( m ) denoe he paymen (possibly negaive) in period when he manager delivers he excess reurn sequence m. The previous argumen shows why holding an equiy sake in he fund iself does no ac as a deerren for a risk neural mimic. We shall herefore resric ourselves o he siuaion where fuure penalies mus be held in escrow. Consider an arbirary excess reurns sequence m 1. Le he mimic s sraegy M 0 ( m) be consruced so ha i goes bankrup in each period wih probabiliy 27
exacly 1/( m1 m ). Consider some period T. The probabiliy ha he fund survives o he sar of period wihou going bankrup is 1/( m1 1). A he end of period, he mimic earns ( m ) wih probabiliy 1/ m and m ( m,..., m,0,...,0) wih probabiliy ( m 1)/ m. Hence he ne presen value of he 1 1 period paymens is ( ) ( 1) ( 1,..., 1,0,...,0) (11) m m m m. ( m m )(1 r ) (1 r ) ( m m )(1 r ) (1 r ) 1 f1 f 1 f1 f To deer a risk neural mimic, he ne presen value be nonposiive: V 0 ( m ) of all paymens mus (12) V ( m) ( m 1) ( m,..., m,0,...,0) ( m) [ ] 0. ( m m )(1 r ) (1 r ) ( m m )(1 r ) (1 r ) 0 1 1 1 T 1 f 1 f 1 f 1 f (Alhough some of hese paymens may have o be held in escrow, his does no affec heir ne presen value o he mimic because hey earn he risk free rae unil hey are paid ou.) 28
Now consider a skilled manager who can deliver he sequence m 1 wih cerainy. (We shall consider disribuions over such sequences in a momen.) Le B( m ) be he se of periods in which a penaly mus be paid if he fund goes bankrup during ha period and no before: (13) Bm ( ) { : ( m1,..., m 1,0,...,0) 0}. For each B( m ) le (14) b( m) ( m1,..., m 1,0,...,0) /(1 rf ) 0. This is amoun ha mus be escrowed during he h period o ensure ha he invesors can be paid if he fund goes bankrup by he end of he period. The skilled manager evaluaes he presen value of all fuure fees, penalies, and escrow paymens using his personal discoun facor, which for period paymens is ( m) 1/ m (1 r ). s f s 1s 29
Consider any period B( m ). To earn he fee ( m ) a he end of period, he manager mus pu b ( m ) in escrow a he sar of he period (if no before). 11 Condiional on delivering he sequence m, he knows he will ge his back wih ineres a he end of period, ha is, he will ge back he amoun (1 r ) b( m). f For he skilled manager, he presen value of his period scenario is (15) ( m) (1 rf ) b( m) 1b( m) ( m) [( m 1) ( m1,..., m 1,0,...,0)] ( m) ( m 1) ( m1,..., m 1,0,...,0). ( m m )(1 r ) (1 r ) ( m m )(1 r ) (1 r ) 1 f1 f 1 f1 f Now consider a period B( m ). This is a period in which he manager earns a nonnegaive fee even hough he fund goes bankrup, hence nohing mus be held in escrow. The ne presen value of he fees in any such period is ( m)/[( m m )(1 r ) (1 r )]. Thus, summed over all periods, he ne presen 1 f1 f value of he fees for he skilled manager comes o 11 If penalies mus be escrowed more han one period in advance, he opporuniy cos o he skilled manager will be even greaer and he conrac even more unaracive, hence our conclusions sill hold. 30
( m) ( m 1) ( m1,..., m 1,0,...,0) (16) V( m) [ ] ( m m )(1 r ) (1 r ) ( m m )(1 r ) (1 r ) B( m) 1 f 1 f 1 f 1 f ( m). ( )(1 ) (1 ) B( m) m1m rf1 rf Since m 1 and ( m) 0 for all B( m ), we know ha (17) ( m 1) ( m) 0. ( )(1 ) (1 ) B( m) m1m rf1 rf From (16) and (17) i follows ha ( m) ( m 1) ( m1,..., m 1,0,...,0) (18) V( m) [ ]. ( m m )(1 r ) (1 r ) ( m m )(1 r ) (1 r ) 1 T 1 f1 f 1 f1 f Bu he righ hand side of his expression mus be nonposiive in order o deer he risk neural mimics (see expression (12)). I follows ha any conrac ha is unaracive for he risk neural mimics is also unaracive for any risk neural skilled manager no maer wha excess reurns sequence m 1 he generaes. Since his saemen holds for every excess reurns sequence, i also holds for any disribuion over excess reurn sequences. This concludes he proof of proposiion 2. 31
VIII. Aracing new money The preceding analysis shows ha any compensaion mechanism ha rewards highly skilled porfolio managers can be gamed by mimics wihou delivering any value added o invesors. To achieve his, however, he mimic akes a calculaed risk in each period ha his fund will suffer a oal loss. A manager who is concerned abou building a long erm repuaion may no wan o ake such risks; indeed he may make more money in he long run if his reurns are lower and he says in business longer, because his sraegy will arac a seady inflow of new money. However, while here is empirical evidence ha pas performance does affec he inflow of new money o some exen, he precise relaionship beween performance and flow is a maer of debae. 12 Forunaely we can incorporae flow performance relaionships ino our framework wihou commiing ourselves o a specific model of how i works and he previous resuls remain essenially unchanged. To see why, consider a benchmark asse generaing reurns series X and a manager who delivers excess reurns M relaive o X. Le 12 See for example Gruber, 1996; Massa, Goezman, and Rouwenhors, 1999; Chevalier and Ellison, 1997; Sirri and Tufano, 1998; Berk and Green, 2004. 32
Z Z ( m,..., m ; x,..., x ) be a random variable ha describes how much ne 1 1 1 1 new money flows ino he fund a he sar of period as a funcion of he reurns in prior periods. In keeping wih our general se up we shall assume ha Z is a muliplicaive random variable, ha is, is realizaion z represens he proporion by which he fund grows (or shrinks) a he sar of period compared o he amoun ha was in he fund a he end of period 1. Thus, if a fund sars a size 1, is oal value a he sar of period is (19) Z MXZ. s s s 1s1 Given any excess reurns sequence m 1 over T years, a mimic can reproduce i wih probabiliy 1/ m 1 T for all realizaions of he benchmark reurns. Since by hypohesis he flow of new money depends only on m and x, i follows ha he probabiliy is a leas 1/ m 1 T ha he mimic will arac he same amoun of new money ino he fund as he skilled manager. The quesion of wha paerns of reurns arac he larges inflow of new money is an open problem ha we shall no aemp o address here. However, here is some evidence o sugges ha invesors are araced o reurns ha are seady 33
even hough hey are no specacular. Consider, for example, a fund ha grows a 1% per monh year in and year ou. (The recen Ponzi scheme of Bernard Madoff grew o some $50 billion by offering reurns of abou his magniude.) This can be generaed by a mimic who generaes a monhly reurn of 0.66% on op of a risk free rae of 0.33%. The probabiliy ha such a fund will go under in any given year is 12 1 (1.0066).076 or abou 7.6%. In expecaion, such a fund will say in business and coninue o arac new money for abou 13 years. One could of course argue ha porfolio managers migh no wan o ake he risk involved in such schemes if hey care sufficienly abou heir repuaions. Some managers migh wan o say in business much longer han 13 years, ohers migh be averse o he damage ha bankrupcy would do o heir personal repuaion or self eseem. We do no deny ha hese consideraions may serve as a deerren for many people. Bu our argumen only requires he exisence of some people for whom he prospec of high expeced earnings ouweighs such concerns. The preceding resuls show ha i is impossible o keep hese ypes of managers ou of he marke wihou keeping everyone ou. 34
IX. Conclusion In his paper we have shown how mimicry can be used o game performance fees by porfolio managers. The framework allows us o esimae how much a mimic can earn under differen incenive srucures; i also shows ha commonly advocaed reforms of he incenive srucure canno be relied upon o screen ou unskilled risk neural managers who do no deliver excess reurns o invesors. The analysis is somewha unconvenional from a game heoreic sandpoin, because we did no idenify he se of players, heir uiliy funcions, or heir sraegy spaces. The reason is ha we do no know how o specify any of hese componens wih precision. To wrie down he players uiliy funcions, for example, we would need o know heir discoun facors and degrees of risk aversion, and we would also need o know how heir rack records generae inflows of new money. While i migh be possible o characerize he equilibria of a fully specified game among invesors and managers of differen skill levels, his poses quie a challenge ha would ake us well beyond he framework of he presen paper. The advanage of he mimicry argumen is ha we can draw inferences abou he relaionship beween differen players earnings wihou knowing he deails of heir payoff funcions or how heir rack records arac new money. The argumen is ha, if someone is producing reurns ha earn 35
large fees in expecaion, hen someone else (wih no skill) can mimic he firs ype and also earn large fees in expecaion wihou knowing anyhing abou how he firs ype is acually doing i. In his paper we have shown how o apply his idea o financial markes. We conjecure ha i may prove useful in oher siuaions where here are many players, he game is complex, and he equilibria are difficul o pin down precisely. Wharon School, Universiy of Pennsylvania Universiy of Oxford, Johns Hopkins Universiy, and The Brookings Insiuion 36
Appendix Here we shall show explicily how o implemen he mimicking sraegy ha was described informally in he ex, using pus and calls. We shall consider wo siuaions: i) he benchmark asse is risk free such as US Treasury bills; ii) he benchmark asse is a marke index such as he S&P 500. We shall call he firs case he risk free model and he second case he marke index model. As is cusomary in he finance lieraure, we shall assume ha he price of he marke index evolves in coninuous ime according o a sochasic process of form (A1) dp P d P dw, ha is, P is a geomeric Brownian moion wih mean and variance 2. The reporing of resuls is done a discree ime periods, such as he end of a monh or a quarer. Le 1,2,3... denoe hese periods, and le r denoe he risk free f rae during period. Similarly, le r f denoe he coninuous ime risk free rae during period, which we shall assume is consan during he period and 37
saisfies r f. Wihou loss of generaliy we may assume ha each period is of lengh one, in which case e 1r f. r f Mimicking sraegies will be implemened using pus and calls on he marke index, whose prices are deermined by he Black Scholes formula (see for example Hull, 2009). Lemma. Consider any arge sequence of excess reurns m m1 m T (,..., ) (1,1,...,1) relaive o a benchmark asse, which can be eiher riskfree or a marke index. A mimic has a sraegy M 0 ( m) ha, for every realized sequence of reurns x of he benchmark asse, generaes he reurns sequence ( mx,..., m x ) wih probabiliy a leas 1 1 T T 1/ m. 1 T Proof. The opions wih which one implemens he sraegy depend on wheher he benchmark asse is risk free or he marke index. We shall rea he risk free case firs. 38
Fix a arge sequence of excess reurns m ( m1,..., m T ) (1,1,...,1). We need o show ha he mimic has a sraegy ha in each period delivers he reurn m(1 rf) wih probabiliy a leas 1/ m. A he sar of period, he mimic invess everyhing in he risk free asse (e.g., US Treasury bills). He hen wries (or shors) a cerain quaniy q of cash ornohing pus ha expire before he end of he period. Each such opion pays one dollar if he marke index is below he srike price a he ime of expiraion. Le be he lengh of ime o expiraion and le s be he srike price divided by he index s curren price; wihou loss of generaliy we may assume ha he curren price is 1. Le denoe he cumulaive normal disribuion funcion. Then (see Hull, 2009, secion 24.7) he opion s presen value is e r f v, where (A2) v sr, 2 [(ln f / 2) / ] and he probabiliy he pu will be exercised is (A3) p s 2 [(ln / 2) / ]. 39
Assume ha he value of he fund a he sar of he period is w dollars. By selling q opions he mimic collecs an addiional e r f vq dollars. By invesing everyhing (including he proceeds from he opions) in he risk free asse, he can cover up o q opions when hey are exercised provided ha r f e wvqq. Thus he maximum number of covered opions he mimic can wrie is q we /(1 v). r f He chooses he ime o expiraion and he srike price s so ha v saisfies v1 1/ m. Wih probabiliy p he opions are exercised and he fund is enirely cleaned ou (i.e., paid o he opion holders). Wih probabiliy 1 p he opions expire wihou being exercised, in which case he fund has grown by he facor r me f over he ime inerval. The mimic eners ino his gamble only once per period, and he funds are invesed in he risk free asse during he remaining ime. Hence he oal reurn during he period is m (1 r ) wih probabiliy 1 p f and zero wih probabiliy p. We claim ha p v ; indeed his follows immediaely from (A2) and (A3) and he assumpion ha r f. Therefore, if he mimic had w 1 0 dollars in he fund a he sar of period, hen by he end of he period he will have m(1 rf) w 1 dollars wih probabiliy a leas 1/ m 1 v and zero dollars wih 40
probabiliy a mos 1 1/ m. Therefore afer T periods, he will have generaed he arge sequence of excess reurns ( m1,..., mt ) wih probabiliy a leas 1/ m 1 T, as assered in he lemma. Nex we consider he case where he benchmark asse is he marke index. The basic idea is he same as before, excep ha in his case he mimic invess everyhing in he marke index (raher han in Treasury bills), and he shors asseor nohing opions raher han cash or nohing opions. (An asse or nohing opion pays he holder one share if he marke index closes above he srike price in he case of a call, or below i in he case of a pu; oherwise he payou is zero.) As before he mimic shors he maximum number of opions ha he can cover, where he srike price and ime o expiraion are chosen so ha he probabiliy hey are exercised is a mos 1 1/ m. Wih probabiliy a leas 1/ m, his sraegy increases he number of shares of he marke index held in he fund by he facor m. Hence, wih probabiliy a leas 1/ m, i delivers a oal reurn equal o m ( P / P ) m x for every realizaion of he marke index. 1 41
I remains o be shown ha he srike price s and ime o expiraion can be chosen so ha he preceding condiions are saisfied. There are wo cases o consider: 2 2 r f and r f. In he firs case he mimic shors an asseor nohing pu, whose presen value is (A4) v sr, 2 [(ln f / 2) / ] and whose probabiliy of being exercised is (A5) p s 2 [(ln / 2) / ]. (See Hull, 2009, secion 24.7.) From our assumpion ha r f 2, i follows 2 f ha p v,which is he desired conclusion. If on he oher hand r, hen he mimic shors asse or nohing calls insead of asse or nohing pus. In his case he analog of formulas (A4) and (A5) assure ha p 24.7). v (Hull, 2009, secion 42
Given any arge sequence ( m1, m2,..., mt ) (1,1,...,1), his sraegy produces reurns ( mx 1 1,..., mtx T) wih probabiliy a leas 1/ m for every realizaion x 1 T of he benchmark asse. This concludes he proof of he lemma. We remark ha he probabiliy bound 1/ m 1 T is conservaive. Indeed he proof shows ha he probabiliy ha he opions are exercised may be sricly less han is required for he conclusion o hold. Furhermore, in pracice, he pricing formulas are no compleely accurae for ou of he money opions, which end o be overvalued (he so called volailiy smile ). This implies ha he seller can realize an even larger premium for a given level of risk han is implied by he Black Scholes formula. Of course, here will be some ransacion coss in execuing hese sraegies, and hese will work in he opposie direcion. While i is beyond he scope of his paper o ry o esimae such coss, he fac ha he mimicking sraegy requires only one rade per period in a sandard marke insrumen suggess ha hese coss will be very low. In any even, i is easy o modify he argumen o ake such coss ino accoun. Suppose ha he cos of aking an opions posiion is some fracion of he opion s payoff. In order o inflae he fund s reurn in a 43
given period by he facor m 1 afer ransacion coss, one would have o inflae he reurn by he facor m m / (1 ) before ransacion coss. To illusrae: he ransacion cos for an ou of he money opion on he S&P 500 is ypically less han 2% of he opion price. Assuming he exercise probabiliy is around 10%, he payou if i is exercised will be abou en imes as large, so will be abou 0.2% of he opion s payoff. Thus, in his case, he mimicking sraegy would achieve a given arge m ne of coss wih probabiliy.998 / m insead of wih probabiliy 1/ m. 44
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