Incentive Contracts in Delegated Portfolio Management

Transcription

1 Incntiv Contrcts in Dlgtd Portfolio Mngnt WEI LI * nd ASHISH TIWARI * * My 8 W r grtful to to nonyous rfrs nd Rn Uppl (th ditor) for thir hlpful conts. * Dprtnt of Finnc, Ourso Collg of Businss Adinistrtion, Louisin Stt Univrsity, Bton Roug, LA Phon: (5) ; Fx: (5) ; Eil: [email protected]. ** Dprtnt of Finnc, Hnry B. Tippi Collg of Businss, Th Univrsity of Io, Io City, IA 54-. Phon: (39) ; Fx: (39) ; Eil: [email protected].

2 Abstrct This ppr nlyzs optil non-linr portfolio ngnt contrcts. W considr stting hr th invstor fcs orl hzrd ith rspct to th ffort nd risk choics of th portfolio ngr. Th ngr s ploynt contrct proiss hr: () fixd pynt, (b) proportionl sst-bsd f, (c) bnchrk-linkd fulcru f, nd (d) bnchrk-linkd option-typ bonus incntiv f. W sho tht th option-typ incntiv hlps ovrco th ffort-undrinvstnt probl tht undrins linr contrcts. Mor gnrlly, find tht for th st of contrcts considr, ith th pproprit choic of bnchrk it is lys optil to includ bonus incntiv f in th contrct. W driv th conditions tht such bnchrk ust stisfy. Our rsults suggst tht currnt rgultory rstrictions on sytric prfornc-bsd fs in utul fund dvisory contrcts y b costly.

3 Whn invstors dlgt th portfolio ngnt dcision to profssionl ngr, thy r fcd ith th clssic probl of dsigning ngnt-f contrcts. Nithr th ngr s infortion nor th ffort xpndd by hi is dirctly obsrvbl by invstors. Consquntly, th pproprit contrct should otivt ngr to xrt costly ffort to gthr infortion rlvnt to scurity pyoffs. Th contrct should lso induc th ngr to subsquntly us this infortion in choosing portfolio ith th dsirbl risk chrctristics. Iportntly, s rcognizd by Stoughton (993), th ffort incntivs nd th risk-tking incntivs intrct in prticulr y in th dlgtd portfolio ngnt stting, nd thrfor, th solutions offrd by th gnric principl-gnt litrtur r not rdily pplicbl. For xpl, Stoughton shos tht, du to th fdbck ffct of risk incntivs on th ffort incntiv, contrcts in hich th ngnt f is linrly rltd to portfolio rturns ld to n undrinvstnt in ffort xpndd by th ngr nd such n undrinvstnt probl cnnot b ddrssd by incrsing th ngr s stk. Furthr, s shon by Adti nd Pflidrr (997), this undrinvstnt probl cnnot b rsolvd by th us of bnchrk-djustd linr copnstion contrcts. A nturl qustion thn is hthr non-linr portfolio ngnt contrcts cn hlp ovrco th ffort-undrinvstnt probl fcd by invstors hil t th s ti providing th pproprit risk incntivs. Furthror, if non-linr contrct is to b ployd, ht for should it tk? Stoughton (993) shos tht ithin scurity-nlyst contxt in hich th portfolio ngr siply rvls his infortion to th invstor, qudrtic contrcts proposd by Bhttchry nd Pflidrr (985) hlp ovrco th ffort undrinvstnt probl. Hovr, s Stoughton points out, such contrcts cnnot b iplntd in th or rlistic portfolio ngnt nvironnt hr invstors dlgt th portfolio choic dcision to th ngr. Hnc, th qustion of th optil ngnt f contrct in this nvironnt rins lrgly unnsrd. Ths rsults contrst ith thos fro th stndrd gncy litrtur tht suggst tht linr contrcts r optil in scond bst sns undr firly gnrl conditions (s, for xpl, Holstro nd Milgro (987)). Th diffrnc sts fro th structur nd th tiing of th dlgtd portfolio ngnt probl undr hich th ngr first xpnds costly ffort to cquir n infortion signl, nd thn chooss th portfolio lloction bsd on th signl. Iportntly, th ngr cn ltr th scl of his rspons to th signl, thrby controlling both th lvl nd th voltility of th portfolio rturn. This structur liits th ffctivnss of linr contrcts.

4 In this ppr dopt Stoughton s frork in ordr to nlyz th optil contrcting probl. W xin odl in hich risk-vrs portfolio ngr proposs n ploynt contrct hil soliciting funds fro rprsnttiv potntil invstor. Th contrct spcifis th ngr s copnstion, hich consists of (i) fixd pynt, (ii) proportionl ngnt f qul to portion of th ssts undr ngnt, (iii) sytric or fulcru prfornc incntiv f tht yilds bonus or pnlty s function of th portfolio prfornc rltiv to bnchrk, nd (iv) n option-typ prfornc f tht yilds bonus dpnding on th portfolio prfornc rltiv to bnchrk, ithout corrsponding pnlty in th vnt of undrprfornc. Onc th contrct trs r grd upon by th invstor, th ngr invsts th funds on th invstor s bhlf. Th ngr s invstnt opportunity st consists of () th rkt indx portfolio rprsnting th pssiv coponnt of th ngr s portfolio, (b) risky sst tht rprsnts th ctivly ngd coponnt of th portfolio, nd (c) th riskfr sst. Th ngr first undrtks th lvl of costly ffort in ordr to collct infortion bout th ctivly ngd risky sst s pyoff. His ffort lvl dtrins th prcision of th infortion. Conditionl on this infortion, h thn chooss th portfolio lloction. Our nlysis procds in to stps. In th first stp, tk th contrct s bing xognous, nd study ho th ngril ffort is influncd by th contrct coposition. In th scond stp of th nlysis, ddrss th optil contrct choic probl. Our nlysis lds to thr in conclusions. On, th ngril ffort incntiv is criticlly dpndnt on th choic of n pproprit bnchrk in th dsign of th option-typ prfornc f. To, hn th bnchrk is ppropritly chosn th us of n option-typ bonus incntiv f cn otivt th portfolio ngr to xpnd incrsd ffort. Thr, for th st of contrcts considrd hr, it is lys optil to includ bonus incntiv f in th ngril contrct. W ssss th conoic gnitud of th potntil fficincy gins by clibrting th odl to piriclly obsrvd vlus for th ky prtrs bsd on spl of U.S. quity utul funds. W find tht th inclusion of th option-typ incntiv f in th ngril contrct lds to n iprovnt in infortion production fro th ngr of ovr 9 prcnt rltiv to th optil linr contrct.

5 To highlight th rlvnc of th bnchrk in th bonus incntiv f for ngril ffort, donstrt th xistnc of to in scnrios chrctrizd by th choic of th bnchrk. In th first cs, th bnchrk is spcifid such tht it closly trcks th invstnt styl of th ngr, nd thrby, succssfully purgs th fctors tht r byond th ngr s control, but tht hv n ipct on th portfolio outco. In this cs, th option-typ prfornc f cn nhnc th ngr s ffort incntiv, nd thrfor, hlp itigt th ffort undrinvstnt probl. In th scond cs, th bnchrk fils to purg th fctors tht r byond th ngr s control. In this scnrio th us of th bonus f is countrproductiv nd in ffct ncourgs th ngr to shirk ffort nd rly solly on luck in ordr to bt th bnchrk. Thrfor, if th bnchrk is not chosn ith cr, th option-typ bonus incntiv f is ctully dtrintl to th ngr s ffort incntiv. Whil th us of ll dsignd option-typ bonus prfornc f cn iprov th ffort incntivs of th portfolio ngr, it distorts th risk incntivs nd lds to sub-optil risk shring btn th ngr nd th invstor. Such concrns ld to nturl qustion: Ar th rsultnt gins fro th iprovd ffort choics of portfolio ngrs sufficintly lrg to offst th cost of such n incntiv sch? W sho nlyticlly tht for th st of contrcts considrd by us, it is lys optil to includ bonus prfornc incntiv f in th ngril contrct. In contrst to th xtnsiv thorticl litrtur on th principl-gnt contrcting probl ithin th convntionl corport stting, thr r rltivly f studis tht r spcific to th dlgtd portfolio ngnt nvironnt. Our nlysis ks nubr of contributions to this prticulr strnd of th thorticl litrtur. First, obtin nlyticl rsults tht provid n xplicit link btn th ngril ffort choics nd option-typ prfornc incntiv fs. In contrst to prvious studis such s Strks (987), sho tht in stting hr th portfolio ngr optilly dtrins both th ffort lloction nd th portfolio lloction, th us of th option-typ prfornc incntiv fs cn in fct iprov outcos rltiv to th cs ith pur fulcru prfornc fs. Furthror, our study 3

6 diffrs in iportnt ys fro th gnrl litrtur in th principl-gnt stting, hr th focus hs bn inly on th risk incntivs crtd by th option-typ incntiv contrct. Scond, in contrst to xtnt rsults tht suggst tht bnchrk-bsd copnstion schs r t bst irrlvnt (.g., Adti nd Pflidrr (997)), our nlysis highlights th criticl rol plyd by th choic of bnchrk in otivting ngril ffort. Th ky diffrnc is tht Adti nd Pflidrr nlyz ngril copnstion function tht is linr in th portfolio rturns. In contrst, xin or gnrl st of copnstion contrcts tht llos for coponnt tht is convx function of th portfolio rturns, in ddition to coponnt tht is linr in th portfolio rturns. Our rsults ith rspct to th significnc of th bnchrk choic r consistnt ith th idsprd us of bnchrk-bsd prfornc vlution, ithr xplicitly or iplicitly, by invstors. Finlly, our nlysis provids n nsr to th long-stnding qustion of hthr rgultory rstrictions on th for of prfornc-bsd fs in, sy, utul funds r thorticlly justifid. 3 Our nlyticl rsults suggst tht univrsl prohibition of sytric prfornc-bsd fs is costly fro socil lfr prspctiv. 4 In this rspct our rsults provid justifiction for th xistnc of such bonus prfornc-bsd fs in utul funds prior to th 97 ndnt tht bnnd th. Intrstingly, our nlysis lso provids thorticl justifiction for th piricl findings of Golc nd Strks (4). Thy docunt tht utul funds tht r forcd to ltr thir ngril contrcts in Expls of th xtndd litrtur includ Ross (4), Chn nd Pnncchi (5), nd Hr, Ki, nd Vrrcchi (). Studis hich pririly focus on th ngr s risk incntivs in th portfolio ngnt stting includ Crpntr () nd Grinbltt nd Titn (989). 3 In 97 Congrss brought invstnt copny dvisrs undr th purvi of th Invstnt Advisrs Act of 94. At th s ti it ndd Sction 5 of th Act to xpt ths dvisrs fro th gnrl prohibition ginst prfornc-bsd fs. Hovr, it rquird tht prfornc-bsd fs should b sytricl in trs of rrds or pnltis round th chosn bnchrk, nd thrfor ruld out th us of bonus or option-typ prfornc incntiv fs. 4 Whil th nlysis in this ppr is bsd on th cs hr th ngr hs bsolut rkt por, our rsults on th fficincy gins lso hold in th lss xtr css hr th invstor is bl to shr portion of th gin. Of cours, gin in th fficincy of infortion production is socilly dsirbl, rgrdlss of ho it is dividd up. 4

7 ordr to coply ith th n rstrictions tht nt into ffct in 97, lost both ssts nd shrholdrs in th ftrth. 5 Th rindr of th ppr is orgnizd s follos. Sction prsnts th odling frork. In Sction, nlyz th ngr s optiiztion probl undr contrct hich includs th optiontyp prfornc f. A ky rsult of this sction shos tht th bonus incntiv f hlps itigt th ffort-undrinvstnt probl tht plgus th linr contrct. Sction 3 nlyzs th optil contrct choic probl by siultnously tking into ccount th ipct of th thr ky spcts of th dcision probl, nly risk shring, th ngr s ffort incntiv, nd th ngr s portfolio choic. This sction lso prsnts nuricl rsults bsd on clibrtion of th odl using dt on spl of U.S. utul funds. Our conclusions r prsntd in Sction 4.. Modl Spcifiction W considr on-priod sttionry odl in hich profssionl portfolio ngr ho cn xrt costly ffort to gnrt privt infortion ts rprsnttiv invstor ho hs invstnt cpitl W. At th strt of th priod, th ngr proposs contrct to th invstor undr hich th ngr invsts W on th invstor s bhlf. Upon th invstor s ccptnc of th trs of th contrct, th ngr voluntrily undrtks lvl of costly ffort to collct infortion bout ssts pyoffs nd procds to k th portfolio lloction dcision. At th nd of th priod, th sst rturns r rlizd nd th totl lth of th ngd portfolio is rturnd to th invstor, hil th ngr is copnstd ccording to th trs of his contrct. W ssu tht th invstor s utility is drivn to hr rsrvtion lvl du to coptition for th ngr s srvics. 6 W furthr ssu tht th invstor cn ltrntivly invst through othr chnnls nd gt crtinty-quivlnt rturn of R. 5 Our rsults coplnt thos of d Mz nd Wbb (7), nd Dittn, Mug, nd Spult (7). Othr rltd studis includ Ds nd Sundr (), Ploino nd Prt (3), nd Ou-Yng (3). 6 This is, for xpl, consistnt ith th rgunt d by Brk nd Grn (4). Nvrthlss, ll our ky rsults ill rin unchngd if th surplus is dividd btn th invstor nd th ngr in so othr prspcifid fshion, including, for xpl, th cs hr th invstor rtins th ntir surplus. 5

8 . Invstnt Opportunity St nd Infortion Th invstnt opportunity st for th ngr consists of thr ssts: riskfr sst ith constnt rturn xprssd in gross trs, R > ; rkt indx portfolio (th pssivly ngd sst) ith f rturn dnotd by R, nd th xcss rturn R R R ; nd rprsnttiv stock (stock, th f ctivly ngd sst) ith rturn dnotd by R R, hich cn b dcoposd s f R = R + β R + R ith E ( ) = nd corr ( R, R ) =. Upon xpnding lvl of ffort, ε, th ngr lrns th vlu of fir-spcific signl, s, tht is corrltd ith th stock s xcss rturn, R. Th ngr s ffort lvl dtrins th infortivnss of th signl, cpturd by ρ = corr( R, s). Th signl s is uninfortiv bout R (i.., corr( R, s) = ). Aftr collcting infortion, th ngr ks his portfolio lloction dcision. For portfolio lloction of, ) to th rkt indx portfolio nd stock, th portfolio rturn rlizd t th nd of th priod is ( R R f + ( + ) R + = β R. () For convninc, ill hncforth dnot vctor = (, ), ith = + β, nd =, nd rfr to this vctor s th ngr s portfolio lloction. W not tht stock in our odl rprsnts ny gnric stock tht th ngr rsrchs. As his signl is fir spcific, nticipt tht th ngr ill b long (or incrs th holdings of) th stocks for hich h rcivs positiv signl, nd short (or rduc th holdings of) th stocks for hich h rcivs ngtiv signl. W dnot th portfolio s xcss rturn R R R, nd th trinl vlu of th portfolio, W WR. 7 f W k th folloing ssuptions bout th rginl distributions for th sst rturns nd th infortion signl. Th rkt xcss rturn is ssud to follo discrt distribution ith th folloing vlus: R = δ ith probbility p, nd R δ = ith probbility p. Th xcss rturn 7 Consistnt ith th typicl prctic in th cs of sy, utul funds, rstrict th portfolio ngr s opportunity st to rul out invstnts in drivtivs contrcts. 6

9 of th ctivly ngd sst, R, cn b ithr δ or δ, ith qul probbility. Th ngr s signl, s, ith qul probbilitis, tks vlu of ithr (good ns bout R ) or (bd ns). Th joint distribution of th thr vribls ( R, R, nd s ) is no uniquly dtrind. Conditionl on th rliztion of s, chrctriz th four stts in th sst-rturn spc by hthr th rkt portfolio nd/or th ctivly ngd sst r up (U) or don (D) in vlu. Thus, thr r ight stts in th odl, four stts for th cs hn s =, nd sytriclly nothr four stts corrsponding to s =. Conditionl on s =, th four stts in th sst-rturn spc hv th folloing distribution: Stt (UU): Stt (UD): Stt (DU): Stt (DD): R = δ nd R δ R = δ nd R δ R = δ nd R δ R = δ nd R δ = ith probbility p ( + ρ) / ; = ith probbility p ( ρ) / ; = ith probbility ( p )( + ρ) / ; = ith probbility ( p )( ρ) /. Clrly, s = rprsnts good ns bout R, indicting conditionl rginl probbility of + ρ for R >. Th probbilitis for th bov four stts conditionl on s = r: p ( ρ) /, p ( + ρ) /, ( p )( ρ) /, nd ( p )( + ρ) /, rspctivly. Not tht th rliztion of bd ns ( s = ) iplis conditionl rginl probbility of + ρ for R <. 8. Utility Spcifiction nd th Cost of Effort Hncforth, ssu CARA prfrncs for both th ngr nd th invstor. Th ngr s utility is givn by U( C, ε ) xp[ ( C V ( ε ))] =, hr C dnots consuption nd V (ε ) rprsnts th cost of 8 It is usul prctic to think of lss ccurt infortion signl s bing quivlnt to th su of or ccurt signl plus rndo nois coponnt. In our discrt-distribution stup, siilr intuition holds. A ngr ith signl of lor ccurcy is prcticlly quivlnt to ngr ith signl of highr ccurcy but ho hs th tndncy to isintrprt hr signl ith crtin probbility. 7

10 ngril ffort, ε. Th invstor s utility is givn by C) = xp[ bc] U p (. Th prtrs nd b r th risk-vrsion cofficints for th ngr nd th invstor rspctivly. In th bov spcifiction of th ngr s utility, th cost function of ffort is ssud to b gnric function. W k th s inil ssuption bout V (ε ) s Stoughton (993): V ( ε ) >,ndv ( ε ). () Not tht fro th ngr s prspctiv, th choic of th ffort lvl boils don to th prcivd trdoff btn th cost nd th bnfits of ffort. Hnc, vlu-bsd dfinition of ngril ffort cn llo for n conoiclly ningful coprison btn th costs nd th bnfits of cquiring infortion. To tht nd, dfin th ffort lvl (ε ), s linr trnsfortion of th gin in th crtinty-quivlnt utility for th ngr fro th cquisition of infortion ith ccurcy ρ, if th ngr r to invst for hislf. 9 Such dfinition of ε tks th folloing sipl for: ( ) ε = ln ρ. (3) dε ρ Clrly, = >. In othr ords, obsrving signl ith highr ccurcy rquirs highr ffort dρ ( ρ ) fro th ngr. It is orth phsizing tht do not los gnrlity fro th spcifiction in (3), bcus th cost function of ffort, V (ε ), is still in gnric for. Givn our spcifiction of th ngril ffort, th scond condition in () hs prcisly th folloing conoic intrprttion: it is incrsingly costly for th ngr to iprov his crtinty-quivlnt utility by cquiring incrsingly ccurt infortion if h r to invst for hislf. 9 If th ngr s xpctd utility is Y, thn th crtinty-quivlnt utility, by dfinition, is ln( Y ). Not surprisingly, th nlyticl rsults do not rly on this ffort spcifiction. By using this spcific stup, r svd fro th troubl of rptitivly citing so for of singl crossing condition so tht th voluntry ffort choic probl hs dfinit solution. W not hr tht, for this intrprttion to b quivlnt to th scond inqulity in (), ε hs to b dfind s in (3), or s its positiv linr trnsfortion. 8

11 .3 Mngril Contrct Th ngril contrct is llod to consist of four coponnts: () th fixd pynt ith vlu F ; (b) th proportionl ngnt f tht pys pr-spcifid portion of th ssts undr ngnt ith contingnt pyoff c W ; (c) th fulcru incntiv f ith pyoff c ( W W ), hr th BL prfornc is surd ginst pr-spcifid bnchrk, W BL ; nd (d) th option-typ bonus incntiv f ith pyoff λ x( W WBO,), hr W BO is nothr pr-spcifid bnchrk. Th subscript BL ( BO ) dnots th bnchrk for th linr (option-typ) f coponnt. W us th functionl nottion W (W ) to dnot th ngr s finl rrd contingnt on his prfornc rprsntd by th trinl portfolio vlu, W. Bsd on th contrct, hv W W ssu tht th bnchrks, W ) = F + c W + c ( W W ) + λ x( W W, ). (4) ( BL BO W BL nd W BO, r thslvs th trinl vlus of portfolios ford by llocting th initil lth ( W ) in diffrnt proportions to th rkt indx portfolio nd th riskfr sst. In prticulr, ssu tht W BL rprsnts th trinl vlu of portfolio tht llocts proportion to th rkt indx nd ( ) BL BL to th riskfr sst. Siilrly, W BO rprsnts th trinl vlu of portfolio ith n lloction of to th rkt indx nd ( ) BO BO to th riskfr sst. Th corrsponding xcss rturns to ths bnchrk portfolios r: BL BL f BL R W W R = R, (5) nd BO BO f BO R W W R = R. (6) In sury, th dinsions of th ngril contrct r cpturd by st of six prtrs: F, c, c, λ, BL, nd BO. Thr dditionl prtrs ( c,, nd θ ) r usd to siplify th Th oftn-obsrvd prctic, hr th s bnchrk is usd for both kinds of fs, cn b cpturd by iposing th constrint: BL = BO. Such constrint hs no ffct on our nlysis of th ngr s incntivs, nd thrfor, cn b ssud s bing prsnt hrvr th rdr finds it dsirbl. 9

12 nottion throughout th rst of this ppr: c = c + c, = c + λ /, nd λ / θ =. Not first tht c rprsnts th py-prfornc snsitivity of th contrct du to th proportionl f nd th fulcru f. Th tr λ / rprsnts th vrg py-prfornc snsitivity fro th option lik bonus f. In this contxt th ord vrg is to b undrstood purly s thticl forlity. Siilrly, rprsnts th vrg py-prfornc snsitivity of th ntir contrct. Also not tht th tr θ cpturs th rltiv ight of th py-prfornc snsitivity contributd by th option-typ bonus f in th contrct. Whn θ =, hich iplis λ =, th contrct consists xclusivly of th linr coponnts ith no bonus f. As θ bco lrgr, hich cn b chivd by dcrsing c, incrsing λ, or both, th bonus f bcos n incrsingly iportnt coponnt of th ovrll contrct. W ssu tht >, nd θ (,). W llo λ to b ngtiv, in ordr to xin th possibility tht th optil contrct is concv instd of convx. Nvrthlss, our priry focus is on th cs hr λ is non-ngtiv.. Mngril Effort Choic undr Altrntiv Contrct Fors In this sction, study th ngr s portfolio lloction nd ffort choic probl undr diffrnt contrct fors. In ordr to xin ho th contrct for influncs th ngr s dcision, for this prt of th nlysis tk th contrct s bing xognously dtrind. Our priry objctiv t this stg is to invstigt th bility of th option-typ bonus incntiv f to rsolv th ffort-undrinvstnt probl tht xists undr th linr contrct. On of th ky rsults of th ppr is contind in Thor in Sction.4, hich shos tht n ppropritly dsignd bonus f cn in fct b usd to iprov th ngr s ffort incntiv. Th iportnc of th bnchrk choic in th dsign of th bonus f is highlightd in th discussion in Sction.5.. First Bst Solution nd Mngril Effort-Undrinvstnt undr th Linr Contrct W bgin our discussion of th ngril ffort choic by prsnting th clssicl ffortundrinvstnt probl ithin our stting by copring th linr contrct solution ith th first bst

13 solution. A linr contrct corrsponds to th cs ith λ = in xprssion (4). Th solution to th ngr s optiiztion probl is strightforrd. W rport th rsult blo. Proposition : Undr th linr contrct, th ngr s optil portfolio lloctions r, l p ln c p = BL + ; (7) Wδ, l s= = + ρ ln ρ W δ ; nd (7b) =, (7c), l s=, l s= hr th subscript l dnots th linr contrct. Th rsulting ngr s crtinty-quivlnt utility, dnotd by Π l, is 3 ( 4 p( p) ) + ε V ( ε ) Π l = F + cw R f ln l l, (8) hr ε l, th optil ngril ffort choic, is dtrind by th first ordr condition: V ( ε l ) =. (9) As bnchrk cs, nxt xin th contrcting probl hn th ngr s ffort choic, th signl obsrvd by hi, nd his portfolio lloctions r vrifibl nd thrfor cn b contrctd on. This is stting dvoid of orl hzrd. Th optil solution of this probl is coonly rfrrd to s th first bst solution in th principl-gncy litrtur. Th ngr s optiiztion probl is th folloing: x ε,,, W ( W ) E [ U ( W ( W ) V ( ε ))] s.t. W W ( R + R + R ), () =, () f 3 Unlik xpctd utility, th crtinty-quivlnt utility rprsnts th conoic gnts ll-bing in scl tht is coprbl ith th initil lth. W find it or ccssibl to intrprttion, nd thrfor us it throughout th ppr in plc of xpctd utility hn prsnting rsults.

14 E [ U ( W W ( W ))] U ( W R ) p. () Inqulity () rprsnts th invstor s prticiption constrint hr th constnt R rprsnts th opportunity cost of cpitl to th invstor. Th ngr s trinl lth rprsntd by W ( W ) cn b ny surbl function of th trinl portfolio vlu, W. This function in ffct spcifis shring rul btn th invstor nd th ngr. Th functionl for of W ( W ) is t th discrtion of th ngr. Th solution to th optiiztion probl in ()-(), notd blo, is th filir rsult tht th linr shring rul is optil. Th drivtion is oittd. Proposition : Th first bst solution of th principl-gnt probl is: p, fb p ( + b) ln p = ; (3) bwδ, fb s= + ρ ( + b) ln ρ = ; (3b) bw δ = ; nd (3c), fb s=, fb s= b W, fb ( W ) = W + V ( ε ) + W WC + ln E xp W. (4) + b bb + In th xprssions bov, th subscript fb dnots first bst hil W, rprsnts th first bst fb shring rul btn th to prtis. Th crtinty-quivlnt utility for th ngr, dnotd by is: Th first-bst ffort lvl, ε fb + b + b ( R R ) ln( 4 p( p) ) + ε V ( ε ) Π fb, Π fb = W f B B. (5) b b, is dtrind by th first ordr condition: V ( ε fb) = +. (6) b

15 Copring Eqution (9) ith Eqution (6), hv tht V ε ) < V ( ε ) nd thrfor th ( l fb ffort-undrinvstnt probl undr th linr contrct: ε < ε. 4 Notic tht th condition in (9) is l fb indpndnt of th slop cofficints of th linr contrct (both c nd c ) nd th spcifiction of th bnchrk ( BL ). W thus conclud tht, ith linr contrct, nithr chnging th py-prfornc snsitivity of th contrct, nor using bnchrk in th fulcru incntiv f, hlps in rsolving th ffort-undrinvstnt probl, conclusion knon s th irrlvnc rsult. In othr ords, Stoughton s rsult holds in ssntilly th s fshion in th currnt odl.. Chrctriztion of th Mngr s Portfolio Choic Probl undr th Incntiv Contrct W no nlyz th ngr s probl hn fcing contrct in th for of Eqution (4). W solv th ngr s ovrll optiiztion probl in to stps. In th first stp, tk th ngr s ffort lvl s givn, nd solv for th ngr s optil portfolio choics. In th scond stp, plug in th optil portfolio choics nd solv for th ngr s ffort choic. W strt by xining th contribution of th option-typ bonus incntiv f to th ngr s pyoff. Th ngr nds up rning th bonus incntiv f x post (i.., λ x( W W,) > ) if nd BO only if h outprfors th bnchrk, i.., R > R, or quivlntly, BO ( ) R < R. (7) It is clr fro this inqulity tht rning th incntiv f is contingnt on th stt of ntur chrctrizd by th signs of th xcss rturns on th to ssts. Intuitivly, in ordr to otivt th ngr to xrt ffort to iprov his signl prcision, ould nt to rrd hi hn th stt of ntur x post confirs his x nt signl. In othr ords, nt Inqulity (7) to b stisfid in th stts hr R BO = δ hn s =, nd in th stts hr R = δ hn s =. W not tht it is only 4 If dopt th prspctiv tht cofficint b, rprsnting th ggrgt risk vrsion of ll shrholdrs of fund, is uch sllr thn th ngr s risk vrsion cofficint,, thn th diffrnc btn V ( ε l ) nd V ( ε fb ) ill b substntil, highlighting th svrity of th undrinvstnt probl ith linr contrct. 3

16 nturl for th ngr to choos hn s = nd hn s =. Substituting ths vlus for R nd in (7) nd using th fct tht BO R = ± δ, rriv t th folloing inqulity: < δ δ. (8) Considr th cs hn s =, for xpl. Inqulity (8) iplis tht th bonus f option is in th ony in stts (UU) nd (DU), i.., hn th ctivly ngd sst is up in vlu hil th rkt indx portfolio cn b ithr up or don. Th bonus f pyoff is zro in th othr to stts (i.., stts (UD) nd (DD)). Intuitivly, th bov inqulity is stisfid hn th bnchrk lloction to th rkt indx, BO, is sufficintly clos to th ngr s lloction to th indx, t th quilibriu. Whthr th rspctiv portfolio ights r clos nough dpnds on th ngr s portfolio choics, nly, nd, hich nd to solv for. Thrfor, nd to xhust ll possibl css including thos css hr Inqulity (8) dos not hold. Also, nd to confir our intuition tht it is optil to rrd th ngr through bonus f in stts (UU) nd (DU) in th cs hr s =. Mthticlly, th folloing thr scnrios xhust ll th possibilitis bddd in Inqulity (8): 5 Scnrio () Scnrio (b) Scnrio (c) δ BO, δ δ BO >, δ δ BO <. δ Undr Scnrio (b), th ngr s lloction to th rkt indx portfolio is uch highr thn th bnchrk s lloction to th indx portfolio. For th cs s = nd thrfor, in Scnrio (b) th bonus is in th ony in stts (UU) nd (UD), i.., hn R >, rgrdlss of th sign of Undr Scnrio (c), th ngr s portfolio lloction to th rkt indx portfolio is uch lor thn th bnchrk s lloction to th indx portfolio. Consquntly, th bonus f is in th ony in stts R. (DU) nd (DD), i.., hn R <, rgrdlss of th sign of R. W solv for th ngr s optil 5 Including th qulity in Scnrio () is ttr of convntion for th considrtion of copltnss. 4

17 portfolio choics ithin ch of ths thr scnrios, strting ith Scnrio (). W thn rriv t th coplt solution of th ngr s optil portfolio lloction probl by chcking for consistncy ithin ch scnrio nd his optil choic cross th scnrios..3 Mngr s Portfolio nd Effort Choics undr Option-typ Bonus F Contrct W ploy th folloing nottion to highlight th rcurring trs in th ngr s objctiv function nd thrfor hlp siplify our xposition: u δ + δ, nd (9) v δ δ. () W furthr dfin N δ N BL BL, nd BO BO δ. Th trs u nd v rprsnt th xcss rturn of th ovrll portfolio R, in th to stts (UU) nd (UD) rspctivly. Th trs th xcss rturns of th bnchrk portfolios, R BL nd N BL nd N BO rprsnt R BO rspctivly, in ths to stts. In Scnrio (), th ngr s objctiv function, conditionl on rciving positiv signl (i.., s = ), is th product of th constnt, xp [ ( F + c W )] R f, nd th folloing iniiztion probl: + ρ in p { xp[ W ( cu + c( u N BL ) + λ( u N BO ))]} u ρ + ( p) { xp[ W ( cu c( u N BL ))]} () ρ + in p { xp[ W ( cv + c( v N BL ))]} v + ρ + ( p) { xp[ W ( cv c ( v N BL ) + λ( N BO v) )]}. Aftr solving () for u nd v, th ngr s optil portfolio choic is dtrind jointly by Equtions (9) nd (). It is qully likly tht th ngr rcivs ngtiv signl ( s = ), nd his portfolio choic probl cn b solvd th s y. In fct, du to th xct sytry, ll th ngr dos in this cs is to rvrs th sign of. W thn cobin th ngr s utility in both css, includ th cost of ffort into his objctiv function, nd thn solv for his ffort choic. Th rsults r surizd in th folloing l. 5

18 L : In th cs of Scnrio (), th ngr s optil portfolio ights r, ic p ln c p BL θ = + BO + ; () W δ, ic s= + ρ θ ln + ln ρ θ = + ; nd (b) W δ W δ, ic = s=, ic. (c) s= W us th subscript ic to dnot tht ths portfolio ights r th solutions to th optiiztion probl undr th incntiv contrct. Th ngr s optil ffort, ε ic is dtrind iplicitly by hr V ( ε ic ) = + θ, ρic (3) εic ρ ic is function of ε ic nd is iplicitly dfind by Eqution (3) (i.., ρ = ). An ic xplicit for of th ngr s utility nd ncssry condition for th ngr s optiiztion probl to fit Scnrio () r givn in th Appndix s xprssions (A) nd (A), rspctivly. W cn s fro Eqution (b) tht th ngr s lloction to th ctivly ngd sst is dtrind by to fctors rprsntd by th to trs in th xprssion for, ic. Th first tr in (b) tks th xct s for s tht in th linr contrct (Eqution (7b)). Conditionl on rciving positiv signl bout R (i.., conditionl on s = ), th ngr invsts or in th ctivly ngd sst hn th signl is or infortiv (i.., if ρ is highr). Th scond tr in (b) is indpndnt of infortion prcision nd dpnds solly on th convxity of th contrct, cpturd by θ. W cn lso s fro Eqution () tht th ngr s lloction to th rkt indx, ic, is snsitiv to th rltiv ights for th to f coponnts (i.., c / nd θ ) in th contrct nd to th corrsponding bnchrk ights (i.., BL nd BO ). Folloing siilr solution tchniqu to tht usd for Scnrio (), solv th ngr s optiiztion probl undr Scnrios (b) nd (c). W rcord th rsults in th folloing l. L : In th cs of Scnrios (b) nd (c), th ngr s optil portfolio ights r 6

19 hr th lst tr in th xprssion for, ic p θ ln + c ln ( p) θ BL θ = + BO + ± ; (4) W δ W δ, ic s= + ρ ln ρ = ; nd (4b) W δ, ic s=, ic s= =. (4c), bov tks th positiv sign in Scnrio (b), nd it tks ic th ngtiv sign in Scnrio (c). Th ngril ffort is iplicitly dtrind by th condition: hr + ρic ζ = ρ ic ζ (+ θ ) / θ, ( ζ, θ ) ζ ( + ζ ) ρic g ( ζ, θ ) / g( ζ, θ ) + V ( ε ) = (5) ic ζ ζ ic g, nd s in (5), ρic =. An xplicit for of ζ + ε th ngr s utility nd ncssry condition for th ngr s optiiztion probl to fit Scnrio (b) r givn in th Appndix s (A3) nd (A4), nd for Scnrio (c) s (A5) nd (A6). Th finl solution for th ngr s portfolio dcision probl involvs copring th dsirbility of th outcos cross th thr scnrios. Th bov ls provid th ncssry conditions on th control vribls for th ngr s probl to fit into on of th thr scnrios. W no driv th corrsponding ncssry nd sufficint conditions, nd thrfor hv coplt dscription of th ngr s optil strtgy. In th folloing proposition, prsnt th solution to th ngr s optiiztion probl s st of ruls tht dtrins th ngr s portfolio choics nd ultitly dtrins th ngr s optil ffort lvl. Proposition : Conditionl on th prcision ( ρ ) of th signl obsrvd by th ngr, his optil portfolio dcision is s follos: () In th cs hn th infortion prcision is sufficintly high such tht + ρ p c ln ln cw δ ( BO ρ p c BL th ngr s optil portfolio dcision, ) is spcifid by L. (, ic, ic ) (6) (b) In th cs tht Inqulity (6) dos not hold, thn ithr 7

20 + ρ p c ln < ln cwδ ( BO BL ) ρ p c nd th ngr s optil portfolio dcision is spcifid by L for Scnrio (b); or + ρ p c ln < ln + cwδ ( BO BL ) ρ p c nd th ngr s optil portfolio dcision is spcifid by L for Scnrio (c). (7) (8) Th optil ffort lvl of th ngr is subsquntly dtrind by ithr Eqution (3) in Scnrio (), or Eqution (5) in Scnrios (b) nd (c). Proof. S Appndix. Proposition spcifis xplicitly nd copltly th ngr s dcision procss in king his portfolio choic ftr obsrving signl ith prcision ρ. Th ngr s optil ffort choic is thn th outco of xiizing his utility using ithr Eqution (3) or Eqution (5) dpnding on hich on is consistnt. 6 At quilibriu ith ρ = ρic, th conditions in th proposition shll ultitly b vid s rstrictions on th contrct spc. With this in ind, it y b hlpful to rrrng (6) nd xprss it s constrint on th choic of bnchrks: c p + ρic + ρic cw δ ( BO BL ) ln ln, ln (9) c p ρic ρic Fro Eqution (3) or (5), th only contrct prtr tht xplicitly ffcts ρ ic is θ. Hnc, ρ ic in (9) is function of θ, BO, nd BL. Thrfor, (9) cn b vid s joint rstriction on th choic of contrct prtrs BO, BL, nd θ. With this undrstnding in ind, r in position to nlyz th ipliction of th contrct for for th ngril ffort choic..4 Motivting th Mngr Using th Bonus F In th folloing thor, prsnt th ffort iplictions of th option-typ bonus f. Thor : Corrsponding to th thr css in Proposition, hv th folloing: 6 Dpnding on th spcifiction of th contrct prtrs, Eqution (3) is sid to b consistnt if it yilds solution for ρ ic tht stisfis Inqulity (6). Siilrly Eqution (5) is sid to b consistnt if it yilds solution for ρ ic tht stisfis ithr (7) or (8). By king us of th rsults in Thor, on cn sho tht t lst on of th to qutions ill b consistnt. In th cs tht both r consistnt, dirct coprison btn th to lvls of rsulting ngril utility ill rsolv ny rining biguity. 8

21 () Aong contrcts tht stisfy th rltion in (9), th ngr s optil ffort lvl, ε ic, is n incrsing function of θ for θ. In prticulr, s θ = corrsponds to th cs of linr contrct, hv ε > ε for θ >. On th othr hnd, in th cs tht θ <, th ngr s optil ffort lvl is ic l lor thn tht in th linr contrct. Tht is, ε ic < ε l for θ <. (b) Aong contrcts tht violt th rltion in (9), nd if xp() ρ ic < (i.., if ρ ic <. 766 ), th xp() + ngr s ffort lvl, ε ic, is dcrsing function of θ for θ solution, nd th scond ordr condition of th ffort choic probl is stisfid. Proof. S Appndix.. W ssu hr tht ε ic is n intrior Thor contins th in rsult of this ppr, nly th rlvnc of th option-typ bonus f in otivting th ngr, nd th iportnc of th bnchrk choic. According to th thor, chnging th rltiv ight of th ngr s bonus f in th ngril contrct (θ ) chngs th ngr s optil ffort choic ( ε ic ). Th prcis rltion btn spcifiction of th bnchrks (i.., on th choic of ε ic nd θ dpnds criticlly on th BL nd BO ). In th cs tht BL nd BO r so chosn tht th rltion in (9) is stisfid, incrsing th rltiv ight of th ngr s bonus f in th contrct (i.., incrsing θ ) lds to highr ffort (i.., n incrs in ε ic ). Undr this scnrio, th dditionl option-typ bonus f otivts th ngr to ork hrdr nd potntilly rducs th cost of ffort-rltd orl hzrd. 7 On th othr hnd, incrsing th rltiv ight of th bonus f in th ngr s copnstion hn th rltion in (9) is violtd, ill in fct ld to rduction in th ngr s ffort. In this cs, th us of th bonus f is countrproductiv nd in ffct ncourgs th ngr to shirk ffort nd rly solly on luck in ordr to bt th bnchrk. It is thrfor iportnt to choos th bnchrks crfully so tht th rltion in (9) is stisfid. W ill rturn to discussion of ths conditions in th nxt subsction. With th rltion in (9) bing stisfid, cn ithr incrs λ 7 It is orth phsizing tht th optil contrct coposition (including th coposition of th bnchrk) tht ould induc ffort t inil cost dpnds, ong othr things, on th ffort cost. W ddrss this issu in th nxt sction hn forlly nlyz th optil contrct choic probl. 9

22 or dcrs c in ordr to incrs th rltiv ight of th bonus f, θ, nd thrfor induc highr ngril ffort. In fct, hv th folloing corollry of th thor. Corollry. Givn contrct of th for in (4) nd undr ny of th thr scnrios, th ngr s optil ffort lvl, ε ic, is copltly dtrind by th ngr s risk vrsion ( ), th rltiv ight of th option-typ bonus f (θ ) in th ngr s copnstion, nd th functionl for of th ffort cost, V ( ). Th proof of this corollry is idit. It follos by noting tht th first ordr conditions of ffort choic (Equtions (3) nd (5)) cn ultitly b xprssd in trs of only th bov prtrs (i.., nd θ ) nd, of cours, th vribl ε ic itslf. An ipliction of th bov corollry is tht incrsing th ovrll py-prfornc snsitivity of th contrct, i.., incrsing (th su of c, c, nd λ / ), hil kping th rltiv ight of th bonus f constnt, i.., hil kping th rtioθ constnt, ill not ffct th ngr s optil ffort choic. This strngthns Stoughton s rsult undr our stting, nly, tht th vrg slop of th contrct is irrlvnt for th ngr s ffort..5 Th Rlvnc of th Bnchrk W no intrprt condition (9) in ordr to provid or ningful guidlins ith rspct to th bnchrk choic. In this contxt not tht, (9) cn b trnsford to th condition:, ic BO <, l δ δ. Intuitivly, th pproprit bnchrk ight on th pssiv sst ( BO ) should trck th ngr s lloction to th pssiv sst (, ic ) closly nough such tht th bonus f ill b in th ony only if th ngr s x nt signl is confird x post by th rlizd stt of ntur. This, in turn, otivts th ngr to ork hrd to nhnc th prcision of his signl, nd to ppropritly invst in th ctivly ngd sst, in ordr to iprov th chncs of rning bonus f s ll its gnitud. Thrfor, hn th bnchrk is chosn to ccurtly rflct th invstnt styl of th ngr, th us of th bnchrk-linkd incntiv iprovs ngril ffort. W thrfor hv confird th intuition tht otivts our dfinition of Scnrio (). In ssnc, this rsult rprsnts

23 n ppliction of th Holströ (979) infortivnss principl to th dlgtd portfolio ngnt probl. In this sns th us of styl bnchrks, s dvoctd by Shrp (99), cn b quit bnficil, vn fro ngril incntiv stndpoint. On th othr hnd, s discussd prviously undr Scnrios (b) nd (c), bnchrk y involv ight on th pssiv indx portfolio tht is too littl (too uch), rltiv to th ngr s ctul pssiv sst lloction. Th option coponnt of th ngr s bonus f ill no b in th ony if th pssiv indx portfolio hppns to hv fvorbl (unfvorbl) outco, rgrdlss of th prfornc of th ctivly ngd sst. In this cs, thr is littl incntiv for th ngr to iprov his infortion prcision. As rsult, th ngr ill ctully b or rluctnt to xrt ffort, nd consquntly th incntiv f in this for ncourgs th ngr to shirk..6 Undrstnding th Incntiv Effcts of th Bonus Incntiv F Fro Thor, conclud tht th inclusion of n ppropritly spcifid bonus incntiv f in th contrct provids th ngr incntivs to collct costly infortion. To gt sns of th potntil gnitud of this incntiv ffct, lt us tk concrt xpl. Suppos th ngr fcs n nvironnt hr th typicl infortion signl hs corrltion cofficint of 6.8% ith th tru stt of ntur, i.. ρ = 6.8%. 8 Coprd to linr contrct, ho uch hrdr ill th ngr ork if h is givn th To nd Tnty incntiv contrct oftn obsrvd in hdg funds? Tht is, contrct hr c =., λ =., nd thus, θ =. 83. According to (3), th rginl cost of ffort incrss ultifold fro V = in th cs of linr contrct to 5.96 V = in th cs of th bonus incntiv contrct. Thus, th option-typ bonus f not only onotoniclly incrss ngril ffort in qulittiv sns, but lso dos so in vry conoiclly ningful y. 8 This vlu is clibrtd fro U.S. stock utul fund dt nd is usd in th nuricl nlysis in Sction 3.5.

24 3. Optil Contrct Choic 3. Th Mngr s Optiiztion Probl ith Endognous Contrct Choic Aftr studying th ngr s incntiv probl hil tking th contrct s givn, conclud tht, by including th option-typ bonus f ith n ppropritly spcifid bnchrk, th ngr ill b otivtd to ork hrdr to collct vlubl infortion. Hovr, s hv not considrd th cost of including th bonus f, it still lvs unnsrd th qustion: Ar incntiv contrcts dsirbl? To nsr this qustion nd to tk into ccount th invstor s prticiption constrint, hich dictts trdoff ong th coponnts of th contrct. W focus on contrct fturing option-typ bonus incntiv fs ith th ppropritly chosn bnchrk tht is consistnt ith th ngr s lloction to th pssiv sst. To dirctly cptur th trdoff btn th bonus f nd th fixd slry, cn solv th binding prticiption constrint of th invstor (s in Inqulity ()) for F, nd plug it, togthr ith th ngr s dcisions solvd in L, into th ngr s crtinty-quivlnt utility function. As rsult, r bl to forult th optil contrct probl for th ngr s th folloing unconstrind xiiztion probl, ith n xplicit objctiv function, nd fiv control vribls, nly, c, c, BL, λ, nd BO. Th ngr s objctiv function rprsnting his crtintyquivlnt utility is: x c,c,bl,λ,bo W ( R R ) f ( p V ( ε ic ) ln + p θ ) / (+ θ ) / ( p) ( p) (+ θ ) / ( Wδ θ ( cbo cbl )) ( W δ θ ( c c )) + ρic ρic ( θ )ln + ( + θ )ln ln[ D + D + D3 + D4 ], θ + θ b hr ε ic is dtrind by Eqution (3) nd ic xp ( θ ) / εic xp BO BL (3) ρ =. Th drivtion of th bov xprssion is providd in th Appndix. Th xprssions for D, D, D 3, nd D 4 r givn by (A)-(A4) in th ppndix. Blo, chrctriz th solution of this optiiztion probl.

25 3. Th Optil Linr Contrct W bgin by srching for th optil contrct ithin th st of ll linr contrcts, cs of intrst by itslf. Tht is, no ipos th constrint, λ = (i.., θ = th ngr s objctiv is givn by: ). Th rsulting siplifid xprssion for hr ( R R ) V ( ε ) ( ln p + ln( p) ) x W f l c,c,bl (3) ( ln( + ρl ) + ln( ρl )) ln[ D + D + D3 + D4 ], b ε is dtrind by Eqution (9). Only th lst tr in th bov xprssion dpnds on th l contrct cofficints. Thrfor, th srch for optil contrct prtrs c, c, nd iniizing th rithtic vrg of four positiv trs ( D, D, D, D ) 3 nd BL boils don to 4 9 hos gotric vrg, through dirct coputtion, is p p)( + ρ )( ρ ). As th rithtic vrg cnnot b lss ( l l thn th gotric vrg nd th lttr is indpndnt of ll control vribls of th optiiztion probl in (3), th gotric vrg provids nturl lor bound for th iniiztion, hich cn b ttind hn D = D = D3 = D4. Th ncssry nd sufficint condition is: b c =, nd c BL =. W hv rrivd t th folloing proposition. + b Proposition 3: In th optil linr portfolio ngnt contrct, b c = nd c =. + b According to Proposition 3, th bnchrk-linkd fulcru f should not b usd in th optil linr contrct. It thrfor confirs th bnchrk irrlvnc rsult of Adti nd Pflidrr (997). Th rit hr is tht th rsult is rrivd t dirctly fro th optiiztion of th ngr s objctiv function nd thrfor it tks into ccount ll of th forntiond contrcting issus siultnously. 9 Th xprssions for ths trs in th cs of linr contrct r givn by (A)-(A4) in Appndix C. Th othr possibility is to hv c b nonzro but BL =. Hovr, in our odl, stting BL = is quivlnt to slcting n pproprit cobintion of th fixd f nd th proportionl sst-bsd f. Thrfor, by skipping th cs ith BL =, do not iss out on ny contrcts in our optil st. Th rsult in Proposition 3 is nticiptd by cobining nd copring rsults in Propositions nd. 3

26 3.3 Th Optil Incntiv Contrct W hv th for of th optil linr contrct ccording to Proposition 3. Ar thr circustncs undr hich this contrct is th optil contrct ithin th brodr contrct spc xind in this ppr? W nsr this qustion in th folloing thor. Thor : In th optil ngril contrct, λ >. Mor spcificlly, th optil linr contrct in Proposition 3 doints ll contrcts ith λ <, nd thr r contrcts ith λ > tht doint th optil linr contrct. Proof. S Appndix. According to Thor, it is lys optil for th invstor to spcify contrct tht includs th option-typ bonus incntiv f coponnt. W prov th thor by studying th rginl ffct on th ngr s objctiv function of introducing sll ount of bonus incntiv f in th optil linr contrct. W sho tht t th rgin, th bnfits of including th bonus incntiv f, in trs of th iprovd ffort incntivs, outigh th potntil costs in trs of suboptil risk shring nd distortions in th ngril risk incntivs. Clrly, du to th ntur of this rgunt, Thor is qulittiv rsult. Although in thory it is lys dsirbl to includ th option-typ bonus f in th contrct, th conoic significnc of such f is not dirctly ddrssd in th thor. Hovr, bsd on th discussion in Subsction.6., kno tht th ngril ffort incntivs inducd by th optiontyp bonus f contrct r indd conoiclly ningful. W provid furthr vidnc on this issu bsd on clibrtion of th odl in Subsction Altrntiv Fors of th Portfolio Mngnt Contrct Evn though th diffrnt lnts of th gnric contrct tht xin r quit siilr to ht is obsrvd in prctic, it is rsonbl to sk hthr our rsults continu to hold hn othr fors of fulcru f or incntiv fs r llod. For xpl, is th option-typ incntiv f still optil hn th fulcru f is sy, convx (concv) in gins nd concv (convx) in losss? Siilrly, cn th nonlinr coponnt of th contrct tk th siplr for of lup-su bonus pynt for xcding 4

27 prfornc thrshold (nd zro othris) rthr thn th option-typ f considrd hr? Our rsults suggst tht our conclusion rgrding th optility of th option-typ f is robust to ths xtnsions. W bgin by stting th folloing proposition tht shos tht ll fors of th fulcru f contrcts, not just th linr fulcru f considrd hr, ld to th idnticl lvl of ngril ffort. Proposition 4: Considr th gnrl fulcru f contrct for: W W ) = F + c W + g( W W ) (3) ( BL hr g (x) is n odd function (i.., x R, g( x) = g( x) ). Th ngr s optil ffort choic is dtrind by Eqution (9), nd is thrfor indpndnt of th for of th fulcru f. Proof. S Appndix. Proposition 4 iplis tht ll fors of fulcru f contrcts ill rsult in th s ngril ffort lvl undr our stting. It is clr thn tht ny non-linr fulcru f contrct is dointd by th linr fulcru f du to th forr s () suboptil risk shring fturs, nd (b) suboptil ngril risk incntiv (i.., portfolio lloction incntiv). Hovr, s hv shon rlir, th option-typ bonus f hn usd in ddition to th linr fulcru f, cn in fct iprov th ngril ffort nd utility. Thrfor, th optility of th option-typ incntiv f is robust to or gnrl fors of fulcru f rrngnts. W nxt xin hthr non-linr coponnt in th for of sipl lup-su bonus ould suffic in liu of th option-typ bonus incntiv considrd by us. W stt th nsr to this qustion in th for of th folloing proposition. Proposition 5: Considr contrct tht dds to our gnrl contrct n dditionl it tht corrsponds to lup-su bonus f if th ngr outprfors bnchrk, nd is zro othris. Tht is, considr contrct ith th folloing copnstion structur: W W ) = F + c W + c ( W W ) + λ x( W W,) + ξ { W > W } (33) ( BL BO BP W dopt d Fintti s nottion nd us th s sybol for st nd its indictor function. 5

28 Hr WBP is th bnchrk for th lup-su bonus f. Th ngr s optil ffort choic undr this contrct is th s s tht undr contrct (4). Thrfor, dding lup-su bonus coponnt to th contrct dos not ltr th ngr s ffort incntiv. Proof. S Appndix. Fro th bov proposition, hn rstricting λ to b zro, it follos tht dding lup-su bonus f to linr contrct dos not iprov th ngr s ffort lvl. For tht givn lvl of ffort, th linr contrct in Proposition 3 ill chiv th optil risk shring nd ld to th pproprit ngril risk incntiv. Consquntly, unlik th option-typ incntiv f, th inclusion of sipl lup su bonus f to linr contrct ill in fct b dtrintl to th ngr s utility. Thrfor, th conclusion in Thor tht λ > in th optil contrct, still holds. 3.5 Nuricl Rsults In this sction provid n ssssnt of th potntil fficincy gins fro th doption of th optiontyp bonus incntiv f. For th sk of siplicity, ill ssu in this subsction tht th fund ngnt contrct dos not includ bnchrk-linkd fulcru f. To clibrt th odl, ssu tht th typiclly obsrvd linr contrct in th utul fund industry is th optil linr contrct, nd th typiclly obsrvd portfolio lloction to stocks by quity funds is th ngr s optil rspons to th linr contrct. Spcificlly, ssu tht th optil linr contrct hs slop of prcnt. Th ngr s lloction to th rkt indx portfolio is ssud to b 9 prcnt. In ordr to gt rsonbl stit of th quilibriu vlu of th infortion prcision prtr, ρ, us dt on spl of stock utul funds fro th CRSP Survivor-Bis Fr US Mutul Fund Dtbs for th priod Th spl includs,839 ctivly ngd funds blonging to th Aggrssiv Groth, Groth nd Inco, nd Long Tr Groth ctgoris. In th contxt of th odl, ith probbility ( + ρ) /, th ngr chivs positiv Jnsn s lph, x post. For our spl of funds find tht th Jnsn s lph bsd on gross rturns is positiv for 58.44% of th funds. Stting this figur qul to ( + ρ) / yilds vlu of 6.88 prcnt for ρ. W ssu tht 6

29 this is th optil choic of infortion prcision for th ngr undr linr contrct. Th rkt xcss rturn s positiv in 39 out of 56 onths during th bov priod, i.., prcnt of th ti. Accordingly, st p = Th onthly rkt risk priu s 6 bsis points nd th onthly rkt voltility s 4. prcnt during this priod. This corrsponds to vlu of 6.3 prcnt for th nnulizd squr root of th scond ont for th rkt xcss rturn, hich us s th vlu for prtr δ. Whn dding bonus f to linr contrct, cn utiliz xprssions (-c) nd rrit Condition (8) s BO, l + ρ < ln. Givn th bov clibrtion, this condition is stisfid cwδ ρ if.3 <. 67. In othr ords, s long s th bnchrk ight on th rkt portfolio flls in th < BO intrvl btn.3 nd.67, th corrsponding bonus f iprovs th ngril ffort. Whil not vry rstrictiv, this condition dos iply tht, for typicl utul fund, th us of bonus f ith th risk-fr rt s th bnchrk ill hurt th ngr s ffort incntiv. To furthr chrctriz th optil contrct, spcify th folloing infortion cost function: ˆ ρ [ ] [ ] ρ V ( ρ) = γ ln + ρ + ln ρ ln (34) + ˆ ρ ˆ ρ ˆ ρ ˆ ρ hr γ is positiv prtr, nd ρˆ is prtr tht flls in th intrvl [, ]. W us th prtr γ to cptur diffrnt lvls of th infortion rltd costs. Th prtr ρˆ rprsnts th highst possibl infortion prcision nd st it qul to.5. W not tht th bov cost function stisfis ll thorticl rquirnts in th contxt of our odl nd llos for n xplicit solution for th optil ngril ffort. W initilly choos vlu of γ to fit th piriclly obsrvd vlu for ρ, i.., 6.88 prcnt for linr contrct. W find tht th optil contrct undr this stting consists of proportionl f of.74 prcnt of th ssts undr ngnt, nd bonus f qul to 7.35 prcnt of th outprfornc ginst th bnchrk. Th optil bnchrk involvs n 84 prcnt lloction to th rkt (portfolio) nd 6 prcnt to T-bills. This contrsts ith th optil linr contrct ith proportionl sst-bsd f of prcnt. 7

30 Bsd upon th bov prtrs find tht th infortion prcision undr th optil bonus f contrct incrss to 3.9 prcnt fro th linr contrct lvl of 6.88 prcnt, n conoiclly significnt iprovnt. W nxt clcult th crtinty-quivlnt utility for th ngr undr thr scnrios: th optil linr contrct, th optil contrct ith th option-typ bonus coponnt, nd th first bst solution. W find tht undr th optil bonus contrct, th ngr s crtinty-quivlnt utility is 4.6 prcnt of th initil ssts undr ngnt ( W ). This rprsnts n iprovnt of ovr prcnt of th ngr s crtinty-quivlnt utility undr th linr contrct (. prcnt of W ). 3 W nxt xprint ith diffrnt choics for th infortion cost cofficint γ hil kping ll othr odl prtrs unchngd. Pnl A of Figur dpicts th ngr s crtinty-quivlnt utility, xprssd s proportion of th initil ssts ( W ), undr th optil incntiv contrct, th optil linr contrct, nd in th first bst solution rspctivly, for vrying lvls of γ. As cn b sn, th ngr s utility is consistntly highr undr th incntiv contrct coprd to th linr contrct, thrby rducing th gp btn th contrct-inducd ngr utility nd th first-bst lvl of utility. Pnl B of Figur shos th coposition of th optil bonus contrct s function of th cost of infortion. Th coposition of contrct is chrctrizd by th vrg py-prfornc snsitivity () nd th rltiv ight on th bonus f (θ ). Initilly, s infortion bco costly, it bcos incrsingly iportnt to otivt th ngr to xrt ffort lding to lrgr rltiv ight of th option-typ bonus f in th contrct. This is ccopnid by corrspondingly highr vrg py-prfornc snsitivity to offst th shiftd risk incntiv of th ngr. Hovr, s th infortion cost incrss furthr, th quilibriu lvl of infortion prcision bcos sll. At this point, ligning th ngr s risk incntiv bcos th doinnt concrn. Thrfor, s dcrs in th rltiv ight ssignd to th bonus f long ith n incrs in th vrg py-prfornc snsitivity of th contrct. 3 Rcll tht in our nlysis hv ssud tht th ngr hs rkt por nd thus cpturs ll of th fficincy gins. Undr th ltrntiv ssuption tht th gins r cpturd by th invstor, th option typ bonus f contrct lds to n iprovnt in th invstor s crtinty-quivlnt rturn of ovr 5 prcnt. 8

31 W nxt xplor th ngr s crtinty-quivlnt utility nd th optil contrct coposition undr vrying lvls of invstor risk vrsion, hil kping th ngr s risk vrsion nd ll othr prtrs fixd. In prticulr, llo th rtio of th invstor s risk vrsion cofficint ( b ) to th ngr s risk vrsion cofficint ( ) to vry fro lo of. to high of.. Th rsults r prsntd in Pnls C nd D of Figur. Pnl C of th figur dpicts th ngr s crtinty-quivlnt utility, xprssd s proportion of th initil ssts ( W ). As th invstor bcos lss risk vrs, th ovrll invstnt opportunity st looks or ppling to hr. Thrfor, s highr surplus lft ovr to th ngr hn th invstor is lss risk vrs. In Pnl D of th figur s tht th rltiv ight on th bonus f bcos highr s th invstor bcos lss risk vrs. This is du to to rsons. First, s cn b sn fro Eqution (6), s th invstor bcos lss risk vrs, i.., s b bcos sllr, th optil ngril ffort in th first bst solution bcos highr. At th s ti, th optil ngril ffort fro linr contrct stys unchngd, s shon in Eqution (9). Thrfor, th undrinvstnt probl in ngril ffort undr linr contrct bcos or svr nd hnc, th nd to us th bonus f to otivt th ngr bcos or urgnt. Furthror, ith dclining invstor risk vrsion, concrns ovr xcssiv ngril risk-tking s consqunc of th inclusion of bonus f in th contrct, bco lss iportnt. In Pnl D lso s tht th ovrll py-prfornc snsitivity of th contrct dcrss s th invstor bcos lss risk vrs. This is bcus th lss risk-vrs invstor optilly brs or risk. Stoughton (993) considrs truth rvltion chnis to ovrco th orl hzrd probl in th contxt of dlgtd portfolio ngnt. Whil th chnis cnnot b iplntd in th rlistic portfolio ngnt nvironnt, nvrthlss dpt it to our stting to srv s bnchrk for vluting th fficincy of th incntiv contrct. Undr this chnis th ngr siply rports his forcst bout th xcss rturn on th ctivly ngd sst to th invstor ho thn iplnts th portfolio lloction bsd on th rportd forcst. Th ngr rcivs pyoff qul to φ + d if h is x post corrct, nd φ d othris. Th copnstion cn thus b spcifid by to 9

32 prtrs: th rithtic vrg of th to lvls, dnotd by φ, nd th diffrnc of th to lvls, dnotd by d. Th ppndix provids brif nlysis of this chnis. Th rdr is rfrrd to Stoughton (993) for dditionl dtils. W find tht, ithin th rng of prtr vrition considrd in Figur, hil th ngr s ffort undr th infortion rvltion chnis (dnotd by th subscript ir ) is oftn highr coprd to tht undr th optil incntiv contrct, th diffrnc is vry sll. In fct, th rtios ρ / ll fll into th intrvl [.99,.]. This diffrnc bcos noticbl only hn th ir ρ ic ggrgt risk vrsion of invstors bcos coprbl to tht of th ngr. If for instnc tk th rtio b / s.5, vlu highr thn ht considrd in Figur (Pnls C nd D), th rtio ρ ir / ρ ic is.8, hich is noticbly highr thn thos rtios considrd rlir but still vry clos to. Intrstingly, in this cs, th rtio of th ngr s crtinty-quivlnt utility undr th infortion rvltion chnis to tht undr th incntiv contrct ( Π ir / Π ic ) is only.66. Tht is, th ngr is uch bttr off undr th optil incntiv contrct thn undr th infortion rvltion chnis dspit th fct tht th lttr gnrts highr ngril ffort. On th othr hnd, hn th invstor pprochs risk nutrlity, th rtio Π / Π incrss. For xpl, it rchs.6 for th cs hr ir ic b / =.. This highlights th intuition tht by svring th fdbck rltion btn portfolio lloction nd infortion collction, th infortion rvltion chnis is quit ffctiv in otivting th ngr. Hovr, hn th invstor bcos or risk vrs, such chnis provs infrior to th incntiv contrct s it dos not llo th invstor to ffctivly shr th portfolio risk ith th ngr. 4. Conclusion Th qustion of ho to bst dsign contrcts in th dlgtd portfolio ngnt stting hs long nggd cdics, rgultors, nd th invstnt counity. Thr considrtions ply iportnt rols in dtrining th dsirbility of contrct in this stting. W nt to find contrcts tht () provid th portfolio ngr ith th right incntivs to xrt costly ffort to collct infortion, (b) ld th 3

33 ngr to choos portfolio ith th dsirbl risk chrctristics, nd (c) provid ffctiv risk-shring btn th ngr nd th invstors. In this study, ddrss th optil contrct dsign probl by ffctivly considring ll thr of th bov issus. W strt ith siplifying ssuption on th joint distribution of sst rturns nd infortion tht introducs substntil sytry into th structur of th odl hil prsrving th richnss of th contrcting probl s intrinsic dcision structur. Our ky rsults cn b surizd s follos. On, hn th bnchrk is ppropritly chosn to purg th ipct of fctors byond th ngr s control, th us of th option-typ bonus prfornc f in th contrct lds to incrsd ffort fro th portfolio ngr. To, ith poorly chosn bnchrk, th option-typ incntiv f ctully hurts th ngr s ffort incntiv. Thr, for th gnrl st of contrcts considrd hr, ith th pproprit choic of bnchrk, it is lys optil to includ n option-typ incntiv f in th contrct. W driv th conditions tht th bnchrk ust stisfy. Our rsults highlight th iportnc of bnchrk choic in th dsign of prfornc-bsd fs in portfolio ngnt contrcts. Furthr, thy suggst tht univrsl prohibition on th us of sytric prfornc-bsd fs in utul fund dvisory contrcts y in fct b hrful to shrholdr intrsts. 3

34 Appndix Proof of L. Th first ordr condition, hich is lso th sufficint condition of optiiztion for u in Exprssion () is p + ρ c + λ = { xp[ W ( cu + c ( u N BL ) + ( u N BO ))]} p λ. ρ c Solving for u gt p + ρ c + λ ln + ln + ln p ρ c c u = + N BL + θn BO. W Siilrly, cn solv th optiiztion probl for v. W gt p + ρ c + λ ln ln ln p ρ c c v = + N BL + θn BO. W u + v u v c + λ + θ Fro Equtions (9) nd (), hv = nd =. Not tht =. Dirct δ δ c θ coputtion thn lds to Equtions () nd (b) in th l. Eqution (c) is du to sytry. It is thn strightforrd to coput th ngr s crtinty-quivlnt utility, hich rport blo: Πic, = F + cw R f + (( θ )ln( θ ) + ( + θ )ln( + θ )) ( θ ) / (+ θ ) / p ( p) xp( Wδ ( )) θ cbo c BL ln (A) (+ θ ) / ( θ ) / + p ( p) xp( Wδ θ ( cbo cbl )) + x (( θ )ln( + ρ) + ( + θ )ln( ρ) ) V ( ε ). ε Th first ordr condition of th bov xiiztion probl of ffort choic ε is: θ + θ dρ V ( ε ) + =. + ρ ρ dε d ρ Totl diffrntition of Eqution (3) yilds. Aftr substituting th rsult in th bov xprssion nd dε upon furthr siplifiction, rriv t Eqution (3). To b consistnt, nd, ic nd, ic to fit in Scnrio (). Inqulity (8), ftr plugging in th solutions for, ic nd, ic, turns into th folloing condition: + ln ρ p c c λ ln cwδ ( BO BL) ln +. (A) ρ p c c Appndix for L. Th ngr s optil crtinty-quivlnt utility for Scnrio (b) is Π ic, b = F + cw R f + [( θ )( ln( θ ) ln p) + ( + θ )( ln( + θ ) ln( p) )] + W δ θ c c (A3) ( ) BL BO ( θ ( ) ( ) ( ) ( ) ) ) / (+ θ ) / (+ θ ) / ( θ ) / + ρ ρ + + ρ ρ V ( ε ). + x ln ( ε A ncssry condition for th ngr s optiiztion probl to fit Scnrio (b) is: 3

35 + ρ p c c λ ln < ln cw δ BO + ( BL) + ln. (A4) ρ p c c Th ngr s optil crtinty-quivlnt utility for Scnrio (c) is: Π ic, c = F + cw R f + [( θ )( ln( θ ) ln( p) ) + ( + θ )( ln( + θ ) ln p) ] + W δ θ c c (A5) ( ) BL BO ( θ ) / (+ θ ) / (+ θ ) / ( θ ) / (( + ρ) ( ρ) + ( + ρ) ( ρ) ) V ( ε ). + x ln ε A ncssry condition for th ngr s optiiztion probl to fit Scnrio (c) is + ρ c p c λ ln < cw δ BO + ( BL) ln + ln. (A6) ρ c p c Notic tht th optiiztion probl of ε is idnticl in (A3) nd (A5). Th first ordr condition of th optiiztion probl, ftr siplifiction, is givn by Eqution (5). Proof of Proposition : Aftr Ls nd, th ngr s choic of portfolio ights is still not yt copltly solvd for, bcus conditions (A), (A4) nd (A6) r not utully xclusiv. First, hn + ρ ln is in th intrvl ρ p c c + λ p c c + λ ln cw δ( BO BL) ln, ln cwδ ( BO BL) + ln (A7) p c c p c c th ngr nds to copr th portfolio solution ccording to Scnrio () dscribd by L ith tht ccording to Scnrio (b) dscribd by L. For ny givn lvl of ρ tht stisfis (A9), copr th ngr s utility undr th optil portfolio choic for Scnrio () (Eqution (A) ithout tking th xiiztion on ε ) ith tht for Scnrio (b) (Eqution (A3)). W find tht th ngr ill choos to go ith Scnrio () rthr thn Scnrio (b) if nd only if + ρ p c ln ln cwδ ( BO BL), (A8) ρ p c hich is xctly t th iddl point of th intrvl in (A7). Siilrly, copr th ngr s intrior optil utility for Scnrio () ith tht for th Scnrio (c) (Eqution (A5)). W rriv t th condition tht th ngr ill choos to go ith Scnrio () rthr thn Scnrio (c) if nd only if, + ρ p c ln ln + cwδ ( BO BL). (A9) ρ p c Aftr this, thr is no or biguity in th ngr s portfolio dcision. Cobining (A8) nd (A9), rriv t (6) in th proposition. Th rst follos. Proof of Thor : Cs (): Through diffrntition of Eqution (3) in Sction.3, hv ε ic ρ = >, 3 θ θ ( ρ) + ρ V ( εic ) for θ >, s V is positiv. As for th cs hr λ < (i.. θ < ), notic tht fro Eqution (3) hv V ( ε ic ) < /. By copring this inqulity ith Eqution (9), conclud tht ε ic < ε l. Cs (b): Th proof of th corollry in this cs ks us of th folloing cli. ( θ Cli: for < ζ < xp() nd <θ <, ( )( ) / ( θ ) / (+ θ ( + + ) ( )( ) / + θ θ ζ ζ θ ζ + ζ ) / ) >. ( θ )/ ( θ ) (+ θ ( ) Proof of th cli: To prov th cli is to prov ( ) ( ) ( ) ) / + θ θ ζ + ζ > θ ζ ( + ζ ) ( θ ) (+ θ ) Notic tht ( + ζ ) > ( + ζ ) bcus ζ >, nd θ > +.. Thrfor, strongr sttnt thn our 33

36 θ cli is: + θ ζ <. Notic first, tht + θ θ li = xp() > ζ. W no only nd to prov + θ θ θ d + θ θ d > for θ >. It suffics to prov + θ ln < dθ θ, or quivlntly, + θ θ ln dθ θ θ θ θ <. Notic tht + θ θ d ln =. Thrfor, only nd to prov tht + θ θ ln < for θ θ dθ θ = θ θ 4θ θ >. Tht is, to prov <, hich holds of cours. End of th proof of th cli. θ W no gt bck to th proof of th thor. Through dirct coputtion, hv: ( θ (, ) [( )( ) ( )( )] ) / ( θ ) / (+ θ ) / ( = θ ζ θ θ ζ ζ θ ζ ζ ) / < ζ g ( ζ + ), θ ( θ )/ ζ ζ + θ nd g ( ζ, θ ) = ( ζ ) lnζ > W not tht θ g( ζ, θ ) = θ ζ ( ζ + ). Furthror, + ( θ ) / ( θ ) / (+ θ ) / (+ θ ) / ( ζ ζ ) + ( ζ ζ ) ( θ ) / ( θ ) / (+ θ ) / (+ θ ) / [ ( ) ( )] ( + θ ) ζ + ζ ( θ ) ζ + ζ lnζ ( θ ( ) / ( ) / ( )/ ( ) / { ( ) ( )} [( ) ( )] ) / ( θ ) / (+ θ )/ (+ θ + ) / θ θ + θ θ ζ ζ ζ ζ = ζ + ζ + ζ + ζ lnζ > +. Thrfor, ( θ nd ( ) ( ) ) / ( θ ) / (+ θ ) / (+ θ ζ ζ + ζ ζ ) / = θ = ( θ ( ) / ( θ ) / (+ θ ( ) + ( ) / + θ ζ ζ ζ ) / ) > ζ for θ >. This inqulity togthr ith th inqulity in th cli provd bov, iplis tht g ( ζ, θ ) >. θ ζ Thrfor, hv: g( ζ, θ ) g( ζ, θ ) g( ζ, θ ) g( ζ, θ ) θ ζ ζ θ g ( ζ, θ ) / g( ζ, θ ) = >. θ ζ g ( ζ, θ ) As pr ssuption, th scond ordr condition of th ngr s optil ffort choic probl is stisfid: ρ g( ζ, θ ) / g( ζ, θ ) + V ( ε ) >. ε ζ ζ g( ζ, θ ) / g( ζ, θ ) Thrfor, fro Eqution (7), hv ε ic θ ζ = <. θ ρ g( ζ, θ ) / g( ζ, θ ) + V ( ε ) ε ζ ζ Driving th rducd-for objctiv function for th ngr (Exprssion (3)): Du to sytry, only nd to look t th cs ith s =. By plugging in th ngr s optil portfolio strtgy, spcifid by Equtions (-c) in L into Eqution (), subsquntly pplying Eqution (4), nd finlly cobining th rsults, cn gt th distribution for W W (W ), th invstor s pyoff. Th invstor s crtinty-quivlnt utility cn b rdily coputd, hich lds to n xplicit for of th invstor s prticiption constrint: ε = ε ic ε = ε ic., 34

37 hr ( R R ) ln[ D + D + D + D ] = F c W R, W f F (A) b D = p D = p + ρ G ( λ) (+ c) G ( (+ λ) ( c ) G ρ) G p + ρ θ p ρ θ p ρ θ p + ρ θ F F, (A) F, (A) ( + ρ) λ + c p + ρ θ D3 = ( p) G G p, (A3) ρ θ F ρ + λ c p ρ θ D4 = ( p) G G p. (A4) + ρ θ c b In th bov xprssions, G = xp bwn BL, G = xp[ θbwn BO ], F = θ, nd b F = + θ. Substituting th lft hnd sid of (A) for F + cw RF in th ngr s crtinty-quivlnt utility (i.., A), rriv t (3) in Sction 3.. In th spcil cs hn λ =, nd thrfor θ =, hv b( c) b( c) b( c) b( c) D G p c p c c c = 3 ( ) ( + ) ( ρ) ρ, (A5) b( c) b( c) b( c) b( c) D G p c p c c c = 3 ( ) ( + ) ( ρ) ρ, (A6) b( c) b( c) b( c) b( c) D G p c p c c c 3 = 3 ( ) ( + ) ( ρ) b( c) b( c) b( c) b( c) D G p c p c c c 4 = 3 ( ) ( + ) ( ρ) ρ, nd (A7) ρ, (A8) c hr G3 = xp bw N BL. c Proof of Thor : W first prov tht th optil linr contrct is dointd by contrct ith λ >. Lt us dnot th ngr s objctiv function in xprssion (3) in Sction 3. by Π ( c, c, BL, λ, B ). W chng to th quivlnt st of control vribls: {, c, BL, θ, B }. To prov th thor, nd to first prov tht Π >. (A9) θ b hr nd =, c =, θ = + b Applying th xprssion for Π in Exprssion (3) in Sction 3., gt 4 4 Π ln Π Π D = + i D i, (A) θ θ θ b θ Π + p = p (+ θ ) / ( θ ) / ( p) ( p) ( θ ) / (+ θ ) / xp i= i= xp( Wδ θ ( cbo cbl )) ( W δ θ ( c c )) BO BL, 35

38 + ρ( ε ) ρ( ε ) Π = ( θ )ln + ( + θ )ln V ( ε ). θ + θ ε = ε ic Hrε ic is function of θ iplicitly dfind by Eqution (5). On cn dirctly vrify tht Bcus ln Π θ = Π Π θ b =, c=, θ = b =, c=, θ = + b + b =. (A) Π ε ic solvs th xiiztion probl in (A), hv =. This srvs s th ε nvlop condition. Bcus of th nvlop condition, cn tk th drivtiv of Π ith rspctiv to θ ithout th concrn tht ε ic dpnds on θ. W hv th folloing prtil diffrntition: Π θ b =, c + b =, θ = In th drivtion of th bov qution, usd th fct tht drivtiv for ch ε = ε ic = ln( ρ( ε )) ln( + ρ( ε )). (A) l ic l l ε = ε, hn θ =. Tking prtil D i, suing up th four its, nd thn dividing by D i cn sho tht: D i Di θ b = +, c b =, θ = =. (A3) Plugging (A), (A), nd (A3) into (A), hv Π = ( ln( + ρ( ε l )) ln( ρ( εl ))) >. θ b =, c, = θ = + b W hv thus provd inqulity (A9). Wht is lft is to prov tht th optil linr contrct doints ll contrcts ith λ <. Although intuitiv, th rigorous proof is long nd tdious. W only sktch out th in stps hr. Considr contrct ith λ <. Th ngr ill voluntrily choos th ffort lvl, dnotd by ε ic. For this fixd ffort lvl ( ε ic ), chnging th contrct prtrs in (3) to th st tht is in ccordnc ith th optil linr contrct ill ld to n iprovnt in th objctiv function du to th optil contrct s or pproprit risk incntiv nd bttr risk shring proprtis. Thn on cn prov tht, ith th optil linr contrct prtrs, th objctiv function in (3) is onotonic in ρ for ll css ith ε ic < ε l. Fro Corollry, kno tht ε ic < ε l. Thrfor, incrsing th ffort lvl to ε l lds to furthr incrs in th vlu of th objctiv function in (3). Th bov chin of coprisons lds to th conclusion of th thor. Proof of Proposition 4: Th ngr s objctiv function is: + ρ in p u ρ + in p v ρ { xp[ ( c W u + g( W ( u N )))]} + ( p) { xp[ ( c W u g( W ( u N )))]} + ρ { xp[ ( c W v + g( W ( v N )))]} + ( p) { xp[ ( c W v g( W ( v N )))]} For th ont, dnot tht D xp( ( cu + g( u N BL ))) involvs u. Th probl trnsfors to: in( p( + ρ) D + ( p)( ρ) / D) BL BL =, nd look t th optiiztion probl tht D> BL BL. W r in ffct iniizing th rithtic vrg of to its hos gotric vrg is p ( p)( + ρ)( ρ), 36

39 hich no longr dpnds on th control vribl nd is thrfor th iniu. 4 Th iniiztion probl for v cn b solvd in th s fshion. It is thn strightforrd to chck tht th rsulting ngril utility is th s s in Eqution (8), hich thn lds to th ffort choic condition (9). Proof of Proposition 5: In th folloing proof, ill ssu tht condition () undr Thor is stisfid. Th othr to css follo siilrly. W dnot tht: + ρ ρ xp( ξ ) ρ =. (A4) + ρ ρ xp( ξ ) + Aftr so nipultion, th ngr s portfolio choic probl cn b xprssd in xctly th s for s tht in Exprssion () in Sction.3, xcpt th tr ρ is rplcd by ρ hr. W cn thrfor invok L to solv th portfolio lloction probl. Th proof of th proposition thn follos. Infortion Rvltion Mchnis: Th ngr rports th signl to th invstor nd gts φ + d if h is x post corrct, nd φ d othris. Th ngr s incntiv coptibility condition to rport th signl truthfully is siply: d. Th ngr s incntiv coptibility condition for infortion collction is strightforrd: d d ρir = rg x ln + φ ln [( + ρ) + ( ρ) ] V ( ρ) (A5) ρ Hr th subscript ir dnots th infortion rvltion chnis. It is clr fro (A5) tht only th tr d, nd not φ, otivts th ngr. Th invstor nticipts, corrctly in quilibriu, th signl prcision ρ ir. Strt ith th invstor s pyoff distribution, hich lds to th drivtion of hr xpctd utility. Aftr so siplifiction, th invstor s xpctd utility is th product of xp( b ( WR f φ) ) nd th folloing iniiztion probl: + ρ bd bw u ρ bd bw u ρ bd bw v ρ bd bwv in p ( p) p ( p) u, v. Agin, cn vi th bov s iniizing n rithtic vrg of four trs hos gotric vrg is p ( p)( ρ ), hich dos not dpnd on th control vribls nd hnc, is th iniu. Fro this, th invstor s prticiption constrint iplis: φ = ln p( p)( ρir ) W ( R R f ). Th ngr hs th frdo to choos th b copnstion prtr d. By pplying th invstor s prticiption constrint, th ngr s crtintyquivlnt utility is obtind by th folloing xiiztion: b d d x ln W ( R R f ) ln p( p)( ρir ) ln [( + ρir ) + ( ρir ) ] V ( ρir ). b b b In th bov optiiztion, ρ ir is vid s function of th control vribl d iplicitly dfind by (A5). 4 To b xct, nd to nsur tht ftr solving for D, r bl to obtin solution for u. W cn, for xpl, ssu tht g (x) is continuous. 37

40 Figur Mngr Crtinty Equivlnt Utility As Proportion of Initil Cpitl Optil linr contrct Optil bonus contrct First bst contrct Avrg Slop, Avrg slop (on lft hnd scl) Rltiv ight on bonus f (on right hnd scl) Rltiv Wight on Bonus F, θ Cost of Infortion, γ A. Mngr s Crtinty-Equivlnt Utility Undr Altrntiv Contrcts Mngr Crtinty Equivlnt Utility As Proportion of Initil Cpitl.4 Optil linr contrct Optil bonus contrct First bst contrct Avrg Slop, Cost of Infortion, γ B. Coposition of Optil Bonus Contrct.5.96 Avrg slop (on lft hnd scl) Rltiv ight on bonus f (on right hnd scl) Rltiv Wight on Bonus F, θ Rtio of Risk Avrsion Cofficints, b/ C. Mngr s Crtinty-quivlnt Utility As Function of th Rtio of th Invstor s Risk Avrsion to th Mngr s Risk Avrsion, b / Rtio of Risk Avrsion Cofficints, b/ D. Optil Contrct Coposition As Function of th Rtio of th Invstor s Risk Avrsion to th Mngr s Risk Avrsion, b/

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