Contracting in Delegated Portfolio Management: The Case of Alternative Assets. June 14, 2012

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1 Conrcing in Dlgd Porolio Mngmn: h Cs o lrniv sss WEI LI nd SHISH IWI Jun 4, Drmn o Finnc, Hnry B. ii Collg o Businss, h Univrsiy o Iow, Iow Ciy, I 54-. Phon: ; Fx: ; Emil: wi-li@uiow.du. Drmn o Finnc, Hnry B. ii Collg o Businss, h Univrsiy o Iow, Iow Ciy, I 54-. Phon: ; Fx: ; Emil: shish-iwri@uiow.du.

2 src h yicl orolio mngmn conrc in rdiionl invsmns lik muul unds rsmls linr shring rul wih mngmn xrssd s roorion o sss undr mngmn. On h ohr hnd, in h cs o lrniv invsmns lik hdg unds nd riv quiy unds, such conrcs yiclly includ n oion-y incniv in ddiion o h roorionl ss-sd. h susnil yous on rsuling rom such conrcs hv ld criics o sk i h wo lus wny conrcs r in h clins s inrss. Or r hy simly vidnc o h mrk owr wildd y h lrniv ss mngrs? his r xlors condiions undr which h irs s oucom my chivd vi linr conrcs in h dlgd orolio mngmn sing. h nlysis suggss h nchmrk-linkd linr conrc cn chiv h irs s oucom whn h mngr s orolio llocion is no sd on riv inormion, nd h invsor is l o noislssly osrv h ss rurns. Nx, h r xmins h limiions o linr conrcs whn h orolio is invsd in oqu sss h r rlivly illiquid or rivly hld. W show h h irs s oucom is no longr sil in his cs nd h scond-s oiml conrc urs n oion-y comonn. h rliv imornc o h oiony comonn is n incrsing uncion o h orolio s ociy. Furhr, h rincil s uiliy loss rom rsricing h wigh o h oion-y comonn o zro is incrsing in h ss s ociy. hs rsuls hl rovid rionl or h orm o conrcs osrvd in h cs o lrniv invsmns lik hdg unds.

3 Conrcing in Dlgd Porolio Mngmn: h Cs o lrniv sss hr is wid disriy in h kind o orolio mngmn conrcs w osrv in sings whr h invsor rincil dlgs h mngmn o hr sss o n invsmn mngr gn. For xml, in h cs o rdiionl invsmns including muul unds, h yicl conrc rsmls linr shring rul wih n dvisory xrssd s roorion o sss undr mngmn. On h ohr hnd, in h cs o lrniv invsmns lik hdg unds nd riv quiy unds, such conrcs yiclly includ rormnc-sd incniv in ddiion o h roorionl ss-sd. ncdol vidnc suggss h h roorionl is wn nd rcn o h sss undr mngmn whil h incniv yiclly quls rcn o h rois. hs rrngmns hv on rsuld in susnil rwrds or h o rorming invsmn mngrs lding som o wondr whhr h s r in c xcssiv. Mor gnrlly, criics hv wondrd whhr h wo lus wny conrcs yicl in h cs o h lrniv ss mngmn indusry r in c in h clins s inrss? Or r hy simly vidnc o h mrk owr wildd y h lrniv ss mngrs h llows hm o imos rrngmns h r in hir own s inrss? Whil i is crinly lusil h xising rrngmns my du o lss hn rcly comiiv indusry srucur, n imorn unrsolvd qusion is: Wh should h orm o h comnsion conrc look lik in h snc o such mrk imrcions? Givn h growing imornc o lrniv sss in insiuionl orolios, hs qusions hv com incrsingly rlvn in rcn yrs. Morovr, h issu hs rcd considrl scruiny in h rmh o h rcn inncil crisis in 8 long wih clls or incrsd rgulion o hdg unds. h gol o his r is woold. On, i xlors condiions undr which linr conrcs cn oiml in h dlgd orolio mngmn sing. wo, i ims o xlin h osrvd dirncs in h orm o h orolio mngmn conrcs sd on h nur o h sss hld in h undrlying In h cs o muul unds, currn rgulions rsric rormnc-linkd incniv s o symmric, i.., incnivs or ovr-rormnc should mirror h nlis or undrrormnc.

4 orolio. s w discuss low, xn horicl modls hv diiculy in xlining h dirncs in h kind o conrcs osrvd cross h vrious sgmns o h ss mngmn indusry. ky sc o our nlysis is h i dos no rly on mrk imrcions such s imrc comiion in h hdg und or riv quiy scor o xlin h osrvd conrc urs. Our nlysis rlis on h dirnc in h comosiion o orolios o hdg unds nd riv quiy unds comrd o rdiionl muul unds. Hdg unds nd riv quiy orolios yiclly invs in rlivly illiquid or rivly hld scuriis. By conrs, rdiionl muul und orolios yiclly invs in ulicly rdd, rlivly liquid scuriis. h liquid nd rnsrn nur o such orolios mks i sil o imlmn suil nchmrks o vlu mngr rormnc. s w susqunly show, h rliv lck o liquidiy nd ociy o h lrniv ss orolios mks i considrly hrdr or n invsor o nchmrk hir rormnc, mking h rincil-gn rolm hrdr o rsolv. Inuiivly, wih n incrs in h diiculy o nchmrking rormnc, mor lxil orm o conrc coms dsirl. W considr rooyicl dlgd orolio mngmn rolm in which h gn h orolio mngr mngs orolio h is invsd in singl risky ss nd risk r ss. W gin wih vry gnrl rmwork h llows us o idniy condiions undr which h irs s oucom my chivd vi linr orolio mngmn conrcs. W show h osrviliy o ss rurns is criicl ur h llows or linr conrcs o oiml in his sing. W urhr show h h gnrl rmwork cn ccommod numr o rcn modls in h lirur nd hus hls xlin numr o rcn rsuls in his r. Eqully imorn, h nlysis hls idniy h condiions h limi h usulnss o linr conrcs in h mor rlisic dlgd orolio mngmn nvironmn. s discussd ov, riculr sing o inrs is h cs whn h orolio mngr invss in illiquid or non-ulic sss, mking h orolio rlivly oqu. In h scond r o h nlysis w xmin how h ss ociy imcs h orm o h conrc. W modl h dgr o rnsrncy or ociy o orolio o sss vi h corrlion o h orolio s rurns wih h rori 3

5 ulicly osrvd nchmrk. W irs show h h iliy o imlmn h irs s soluion is criiclly dndn uon h dqucy o h vill inormion rgrding h rurns o h sss in h orolio in qusion. h imlmnion involvs h us o rormnc nchmrk gins which h orolio mngr is vlud. h rori nchmrk rcks h irs s orolio srgy. h iliy o consruc n ccur nchmrk or his uros is criiclly dndn on h invsor s iliy o osrv h rurns o h sss in h orolio. Inuiivly, wih rc osrviliy o ss rurns, h invsor cn drmin h rminl vlu o h irs s nchmrk orolio nd nliz ny dviion rom his nchmrk vlu. his mks i sil o norc h irs s oucom in his sing. h ov rsul nurlly lds o h qusion: Wh i h invsor cn only consruc noisy nchmrk, i.., nchmrk h rcks h irs s srgy only imrcly? o nswr his qusion w considr h cs whn h invsor s inormion ou h rurns o h undrlying sss in h orolio is lss hn rc. his cs is modld y ring h corrlion wn h risky ss nd h nchmrk s ing sricly lss hn on. W show h in his cs h gn s llocion o h civly mngd risky ss is lss hn in h irs s cs. In ohr words, h rsnc o noisy nchmrk, which is isl yroduc o h chrcrisics o h undrlying sss, lds o or shirking y h orolio mngr, rliv o h irs s oucom. his rsul highlighs ky limiion o linr conrcs in h dlgd orolio mngmn sing whn rc osrviliy o ss rurns is no sil, s is on h cs. r hr conrcing soluions h cn hl rilly ddrss h ov shorcomings o h sndrd linr conrcs? nd i so, how similr r hs lrniv conrcs o h kind o conrcs w osrv in rcic in h cs o sy, hdg unds nd riv quiy unds? W urn nx o n xminion o his qusion. Our nlysis suggss wo mjor rsuls. On, h scond-s oiml conrc in his sing urs n oion-y comonn in ddiion o comonn h is linr in rormnc. wo, For xml, in h xrm cs, n quiy indx und is likly o hv corrlion clos o on wih rsc o sy, h S&P 5 indx. 4

6 h rliv imornc o h oion-y comonn is n incrsing uncion o h ss s ociy. In riculr, h wigh o h oion y comonn in h oiml scond-s conrc incrss wih ss ociy. Furhr, h rincil s uiliy loss rom rsricing h wigh o h oion-y comonn o zro, is incrsing in h ss s ociy. hs rsuls rovid n inuiiv jusiicion or h dirncs wn h kinds o orolio mngmn conrcs yiclly osrvd in h dirn sgmns o h ss mngmn indusry. In riculr, h rsuls rovid rionl or h oionlik incniv conrcs osrvd in h lrniv ss mngmn indusry including hdg unds nd riv quiy unds. Our r is rld o n xnsiv lirur h xlors h rincil-gn rolm in h dlgd orolio mngmn conx. Following h ionring work o Holmsröm nd Milgrom 987, numr o sudis hv dmonsrd h oimliy o linr conrcs in h gnric rincilgn sing. By conrs, Soughon 993 highlighs n imorn shorcoming o linr conrcs in h dlgd orolio mngmn sing. yiclly, in his sing, h orolio mngr irs xrs cosly or o oin riv inormion signl nd susqunly drmins h orolio llocion. By chnging h orolio llocion h mngr cn lr h scl o h rsons o h signl hry limiing h civnss o linr conrcs in ddrssing h mngril or undrinvsmn rolm. Bsd on similr inuiion, dmi nd Plidrr 997 highligh h limiions o nchmrk-linkd linr conrcs. Similrly, Srks 987 inds h symmric conrc dos no limin h or undrinvsmn rolm in sing whr h rori risk shring nd or incnivs r oh o concrn. Soughon 993 nd Bhchry nd Plidrr 985 xlor qudric conrcs wihin scuriy-nlys conx in which h orolio mngr simly rvls his inormion o h invsor. Howvr, s nod y Soughon 993, such conrcs r no sil in h mor rlisic dlgd orolio mngmn sing. In rcn r Ou-Yng 3 dmonsrs h oimliy o linr conrcs in riculr orm o h dlgd mngmn rolm. On h ohr hnd, Li nd iwri 9 do Soughon s 993 rmwork o show h n rorily dsignd oion-y onus conrc cn in c 5

7 usd o imrov icincy nd such conrc domins ll symmric conrcs. h rsn r xnds h lirur long wo dimnsions. On, i hls clriy h condiions undr which h irs s oucom my chivd vi linr conrcs in h dlgd orolio mngmn sing, nd y xnsion h condiions undr which such conrcs r su-oiml. wo, i hls xlin h osrvd dirncs in h nur o h orolio mngmn conrcs cross dirn kinds o invsmn vhicls. Imornly, h r hls xlin h xisnc o symmric oion-y conrcs in h lrniv ss univrs including hdg unds nd riv quiy unds wihou ling o mrk ricions such s high nry coss nd imrc comiion in h indusry. 3, 4 h rs o h r is orgnizd s ollows. Scion I xlors h condiions undr which h irs s oucom my chivd vi linr conrcs in h dlgd orolio mngmn sing. I shows h ky condiion is h ssumion h h mngr s orolio llocion is no sd on riv inormion, nd h h invsor cn noislssly osrv h ss rurns. Scion II xmins h conrcing rolm in h cs whr h mngr s orolio is invsd in rlivly oqu sss. h scion lso rsns rsuls rom numricl nlysis h xmins how h orm o h oiml conrc chngs nd how h rincil s uiliy vris wih chng in h undrlying orolio s ociy. Concluding rmrks r rsnd in Scion III. I. Conrcing in Dlgd Porolio Mngmn wih Coml ss Inormion W gin y idniying som ky urs o h inormion nvironmn in h dlgd orolio mngmn rolm h r ssnil or linr conrcs o licl in such sing. Our sic rgumn is uil on siml osrvion, which w illusr in his scion using sciic rmwork. h inuiion is no rsricd o h sciic su. Mor rodly, h undrlying inuiion in is gnrl 3 In rld work Ds nd Sundrm show h in rmwork wih dirnil mngril iliy nd imrc comiion in h mrk or mngrs/dvisors, invsor wlr is gnrlly highr undr rgim whr only h oion y onus rormnc incniv is llowd in h conrc rliv o rgim whr only h ulcrum is llowd. Sudis h ocus xclusivly on h risk king incnivs includ Crnr nd Grinl nd imn 989 whil Plomino nd Pr 3 xlor sing which srcs rom h risk shring concrn. 4 Whil no h ocus o h rsn r, numr o sudis hv xlord h conomics o h commonly osrvd high-wr mrk conrcs in hdg unds. hs includ Gozmnn, Ingrsoll, nd oss 3, Pngs nd Wsrild 9, nd Ln, Wng, nd Yng. 6

8 orm llows us o comr svrl modls, including Holmsröm nd Milgrom 987, Soughon 993, nd Li nd iwri 9. I lso llows us o uncovr h ssnil sourc o h dirnc in h ky rsuls o Ou-Yng 3 nd Li nd iwri 9. I.. h Bsic Modl In his scion, w show h whn h ss ric rocss is rcly osrvl, linr conrc cn chiv h irs s soluion. his rsul srvs s h nchmrk or our min nlysis in h nx scion whr ril osrviliy o h ss ric is llowd. In h currn scion w lso discuss how such rsul is rld o som rcn rsuls in h lirur. In h modl, rincil invsor conrcs wih n gn mngr o mng hr wlh. h gn hs ccss o n invsmn ooruniy s chrcrizd y risk-r ss wih gross rurn rlizd h nd o h riod, nd s o risky sss wih gross rurn dnod y h rndom vcor,. h gn dcids h llocions dnod y h vcor,, o ch o h risky sss, wih h rs o h wlh ing invsd in h risk-r ss. hroughou, w will ssum h h orolio s rminl vlu, which w dno y W, is osrvl y oh ris nd hror cn conrcd on. In gnrl, w hv h ollowing rsul. Proosiion. L h gn s comnsion schm, dnod y S, k h ollowing orm: S = W W P, whr h rm W P is h yo o h rincil undr h irs s soluion. ssum h h orolio rormnc W is ully drmind y h gn s orolio llocion vcor,, h sss rlizd rurn, nd h risk r r,. 5 ssum h ll h sss in h orolio r ulicly rdd nd hror h invsor cn osrv h ss rurn,. Furhr, ssum h h irs s ss llocion vcor,, is sd on ulic inormion. h irs s oucom is chivd y h conrc in, nd h gn will volunrily choos h irs s orolio llocion. 5 W surss h im suscri or rviy. ll o h ims,, nd cn in c im vrying. 7

9 Proo. Undr h ssumions o h roosiion, h rincil s yo in h irs s scnrio, W P, dnds on,, nd. In ohr words, WP is sd only on ulic inormion givn h ssumion ou h irs s llocion,. hror, h conrc in is sil. I is clr h undr h comnsion schm, S, dscrid in Equion, h rincil s yo, W P, is givn y, W = W S = W, which is in c h irs s oucom. hror, h irs s oucom is chivd. P P h c h h gn chooss h irs s orolio llocion cn shown ccordingly. h ky ssumions undrlying h ov rsul r h h mngr s orolio llocion is no sd on riv inormion, nd h rincil is l o osrv h ss rurns. hs ssumions nsur h h rincil cn drmin h rori irs s yo, W P, o dmndd rom h gn s r o h orolio mngmn conrc, which in urn nsurs h h irs s orolio llocion is rlizd. I is imorn o no h h condiions in h roosiion do no rul ou h ossiiliy h h mngr my hv surior iliis comrd o h invsor. For insnc, h mngr my l o imlmn rds h s ossil rnscion ric whil minimizing ric imc. Furhrmor, h roosiion s ssumions sill llow or h ossiiliy h h mngr s cions r unosrvd y h invsor. In gnrl, h invsor cnno inr h mngr s orolio llocion x os. Furhr, h mngr lon rs h riv cos o his cions. Nvrhlss, h morl hzrd rolm is comlly rsolvd undr h ov ssumions. lso, no h h conrc sciid in is linr in h inl oucom, W. In riculr, undr h conrc, h mngr s comnsion is drmind y comring h orolio s rormnc, W, o nchmrk, W P. dmidly, h ov roo is hurisic in nur. In h chnicl ndix w includ dild discussion h irs rorly ss u h rolm nd hn drivs ullr vrsion o Proosiion which lis mor rodly o rincil-gn rolms yond h dlgd orolio mngmn sing. No h h roosiion clrly lis o h rivil cs whr h gn s cion cn vriid x os, cus in his cs, h comnsion schm in is sd on osrvls nd hror is 8

10 norcl. s nohr milir xml whr h roosiion lis, ssum h h gn s cion cnno vriid x os, u h gn is risk nurl. Considr, or insnc, h rmwork xlicily nlyzd y Shvll 979. I is shown y Shvll h in on o h irs s soluions or his cs, W P = k, whr k is consn. hror, h conrc in Proosiion ks h orm, S = W k, which is sil undr h ssumion h h rormnc W is x os vriil nd hror cn conrcd on. hror, h irs s oucom or h rincil is chivd y h conrc, o summriz, in his scion, w hv shown h linr conrcs cn rlid on o chiv h irs s oucom in dlgd orolio mngmn. In gnrl, h ky ssumions h r h cor o h rgumn r: h irs s orolio srgy is sd on ulic inormion. h rincil cn osrv h rurns on h vill sss. Undr h ov ssumions, linr conrcs cn oiml in h dlgd orolio mngmn conx. I.B. licion o cn suls in Dlgd Porolio Mngmn W nx ly Proosiion o hl shd ligh on som o h rcn rsuls in h lirur on dlgd orolio mngmn. Firs, w show h Ou-Yng s 3 min rsuls cn quickly drivd s corollris o Proosiion. W hn xmin h rlion o h rsuls in h rvious scion o ohr rlvn rmworks in h lirur, in riculr, Holmsröm nd Milgrom 987, Soughon 993, Li nd iwri 9, nd Edmns nd Gix 9.. riculr s o sciicions or uncionl orms nd disriuions In h sing xmind y Ou-Yng 3, ll ssumions undrlying Proosiion r sisid. In riculr, h irs s orolio llocion is sd on ulic inormion nd h rincil cn osrv h ss rurns. h ully sciid modl in Ou-Yng s r includs h ollowing ssumions or h ric rocsss, h wlh rocss, h cos uncion, nd h uiliy uncions. h S. 9

11 risk-r r is consn nd dnod y r. h ric-rocss o h risky sss is dscrid y h ollowing gomric Brownin moion: dp = dig[ P] μd σdb, whr μ is consn vcor in N, σ is consn mrix in N d wih linrly indndn rows, nd B is d N dimnsionl sndrd Brownin moion. h orolio mngr chooss h dynmic llocions o h risky sss nd h risk-r ss ovr im. h llocions r howvr, no osrvd y h invsor. L h dollr moun invsd in h risky sss im rrsnd y, h wlh rocss W } or h orolio srgy } is hn givn y. { { dw = [ rw h] d σ db, 3 whr h μ r, nd dnos h uni vcor. h insnnous cos uncion is sciid s c,, W = k γw, 4 whr k is n N N mrix, nd γ is n consn. h gn s rrnc ovr wlh is dscrid y U W W =, nd h rincil s uiliy is U W W =. Whil Ou-Yng dos no rsn h irs s soluion, w rovid soluion in ndix I. Brily, w no h h mngr s im criny quivln uiliy long h oiml h is rndom rocss. s mr o ormliy, w cn us his rocss o susiu ou h mngr s riciion consrin. his susiuion lds o rsmn o h originl dynmic rogrmming rolm, which hn llows or xook sndrd soluion. coml rsnion o soluion nd h ss h ld o i r rovidd in ndix I. W summriz h rlvn comonn o h rsul in h ollowing roosiion. Proosiion. h irs s soluion o h oiml orolio olicy is givn y = k σσ h, 5

12 whr γ r γ =. h invsor s yo is r r W P = F dig P dp 6 whr F is consn. h oiml conrc ollows s corollry o Proosiion. 6 Comining Equion nd Equion 6, n oiml conrc in his sing cn xrssd s S W W F W W dig P dp, 7 = P = which is xcly h sm s Equion in horm in Ou-Yng 3, g 85. s Ou-Yng nos, h conrc in Equion 7 my viwd s consising o hr rs: ixd, roorionl, nd c onus or nly h dnds on h orolio rormnc rliv o n civ nchmrk. h onus is symmric in orm rsul h rs o rovid horicl suor or h currn rgulion h rsrics h rormnc-sd s or invsmn comny dvisrs o o h symmric orm. Ou-Yng urhr dvlos som ohr orms o h oiml conrc in h cs whr h cos uncion c is consn. hs rsuls r conind in his horm. 88. For hs css, considr h ollowing comnsion schdul or h gn: λ, S = S 8 whr λ is non-zro consn. I cn rdily shown h ll conrcs in h orm o 8 r oiml. h drivion o h ov rsul is rovidd in ndix I S Proosiion. in ndix I. By rsricing h vlu o h rmr, λ in Equion 8 o h inrvl,, w rriv horm in Ou-Yng. I is inrsing o no, howvr, h hr is no nd o rsric h rmr λ in n inrvl s Ou-Yng did, s long s λ is no zro. 6 Whil Proosiion in Scion I is drivd undr som sciic ssumions h r no dircly licl o h currn siuion, h sm rsul cn lso drivd in comlly gnrl sing h indd mds Ou-Yng s su s sciic cs. h gnrl drivion is rovidd in ndix I.

13 I.C. Ohr Frmworks From h ov discussion, i is vidn h h oiml conrc drivd y Ou-Yng 3 is in rincil idnicl o h conrc w driv in Proosiion nd is siiliy hings on h ssumions lisd h nd o Scion I. Imornly, h morl hzrd rolm is comlly rsolvd undr hs ssumions. s ky oin o drur, in h rmwork dod y Soughon 993 nd Li nd iwri 9, h mngr xnds or o collc riv inormion h is corrld wih h ss rurns. Consqunly, linr conrcs r no longr oiml in h rmwork o hs wo sudis. W us Soughon s rmwork o illusr h oin. h mngr hs ccss o risk-r ss wih gross rurn nd risky ss wih gross rurn, μ, whr μ is h mn nd hus E =. On o h ky ssumions md y Soughon is h h mngr cn osrv signl h is corrld wih h ru rurn: 7 I = ε, 9 whr ll vrils r ssumd o joinly norml. Furhrmor, E ε =, nd E ε =. h inormion rcision is curd y, / σ ε ρ = σ, which is lso idniid s h mngr s or. h uiliy uncions o h mngr nd h invsor r ssumd o xonnil: whr W nd B { } U W = x W V ρ, U W = x W, B B W r h nd-o-riod wlh o h gn nd h rincil, rscivly, nd V ρ is h disuiliy o or. h irs s oucom, h yo o h rincil is h ollowing uncion o nd I : 7 W do noion h dirs rom Soughon 993 in ordr o consisn wih h ohr r o his r.

14 W P / = W Φ / / / ρ ρ = W μ I, Φ / / σ ρ whr Φ is consn such h h mngr s riciion consrin is sisid. ll h rmrs nd vrils in h ls row o h ov xrssion r ihr known or osrvl h nd o h gm o h invsor nd hror, cn in rincil conrcd on, wih h xcion o h mngr s signl, I. Du o his c, h conrc in Equion is no sil in Soughon s 993 rmwork. Bsd on similr sing s Soughon, Li nd iwri 9 show h ll orms o symmric conrcs r suoiml. hy dmonsr h n rorily dsignd oion-y onus conrc cn usd o imrov icincy nd such conrc domins ll symmric conrcs in his sing. 8 I is imorn o no h h insigh o Holmsröm nd Milgrom 987 is sd on rmwork chrcrizd y symmric inormion. In h coninuous-im gm nlyzd y hm, h gn incurs cos nd conrols h mn μ o h ouu diusion rocss Z : dz = μ d db, 3 whr B is drilss N-dimnsionl vcor sndrd Brownin moion. n imorn condiion h lds o h oimliy o linr conrc is h ssumion h h rincil gs o osrv only h ol ouu,, whr =,,...,. his is in conrs o h rquirmn o Proosiion, whr Z rm in h conrc sciid y Equion is coningn yo, W P, which dos no dnd on h gn s cions. h only sil coningn yo in h Holmsröm nd Milgrom 987 rmwork h dos no dnd on h gn s cions is consn yo. his is du o h c h h rincil in his rmwork dos no osrv nyhing yond h ol ouu. Clrly, such consn yo cnno 8 I is worhwhil o highligh n imorn disincion wn h rmwork dod in h sudis y Soughon 993 nd Li nd iwri 9 nd h in Holmsröm nd Milgrom 987. Boh o h ormr sudis s hir nlysis on sionry modl in conrs o Holmsröm nd Milgrom 987, who dmonsr h oimliy o linr conrc in dynmic sing. 3

15 qul o W P in h cs in which h gn is risk vrs. his suicinly dmonsrs h h conrc in dos no ly o h Holmsröm nd Milgrom sing. In rcn sudy Edmns nd Gix 9 show h h Holmsröm nd Milgrom rsul on h oimliy o linr conrcs cn chivd in sings h do no rly on ssumions such s xonnil uiliy, coninuous im rmwork, nd Gussin nois. I is nurl o sk whhr his gnrlizion xnds o h dlgd orolio mngmn sing. ky sc o h Edmns nd Gix gnrlizion is h i rlis on rmwork in which inormion is rvld o h gn or h gn s cion is chosn. Furhrmor, whil h modl hs mulil riod srucur, h onil comlxiy rom such srucur is minimizd. his is chivd hrough wo ky ssumions. Firs, h cos uncion o h gn s cion is ssumd o k cuniry orm. hror, h ggrgion o h ovrll cos o h gn o his cions in mulil riods cn chivd y dding u h cuniry cos incurrd in ch individul riod. h scond ssumion is h h nd o ch riod, h rlizion o h oucom, joinly drmind y nur nd h gn s cion, is ulicly osrvd. By hs ssumions, ch riod s gm is suicinly indndn, nd h ggrgion hrough mulil riods cn chivd rhr mchniclly. In srk conrs, s mhsizd y Soughon 993, h inrsing dck c o cions in dirn sg o h gm in dlgd orolio mngmn is wh mks h conrcing rolm riculrly chllnging. Mor sciiclly, h gn in such n nvironmn undrks cosly or rior o h rlizion o noisy signl rld o uur ss yos, nd h is hn rquird o dcid on h ss llocion undr imrc inormion. Hnc, h limiions o h Homsrom-Milgrom rmwork in h conx o dlgd orolio mngmn, irs highlighd y Soughon 993, r sill vlid. II. h Cs o Oqu sss s dmonsrd in h rvious scion, our sic rsul, h h irs s soluion cn imlmnd in h dlgd orolio mngmn sing, dnds criiclly on h suicincy o h vill inormion rgrding h rurns o h rdd sss. h imlmnion involvs 4

16 nchmrk h h mngr is msurd gins. In riculr, h nchmrk rcks h rormnc o h irs s orolio srgy. I lds nurlly o h qusion: wh i h invsor cn only consruc noisy nchmrk, nchmrk h rcks h irs s srgy only imrcly? In his scion w sk nswrs or his qusion. W uild on h sic su in Scion I. For simliciy, w rduc h numr o risky sss rdd y h mngr o. Furhr, w ssum h h risky ss is oqu. Sciiclly, w ssum h insd o osrving h rurn on h risky ss in h orolio, h invsor cn only osrv h rurn on nchmrk rrnc ss h hs h sm mrginl rurn disriuion, u is only imrcly corrld wih h risky ss. h modl nlyzd hr is sionry on. s or, rincil conrcs wih n gn o mng hr wlh. W normliz h iniil invsmn o $. Wih his ssumion, h rurn nd h rminl vlu o h invsmn r idnicl. W ssum h h r o rurn on h risky ss ollows h gomric Brownin moion, so h h ss yo is nvr ngiv. h ginning o h riod, h mngr dcids h orolio wighs,, which r hld consn hroughou h riod. Wih h ov ssumion, h risky ss rurn hs log-norml disriuion. h log r o rurn or h orolio is: w = r r r, 4 whr r is h log rurn o h risky ss nd r is h log risk r r o rurn. W dno h rminl vlu o h orolio y W, nd w hv w W =. W ssum h hr is nchmrk ss, whos rurn is dnod y r, nd his nchmrk ss hls h invsor rck h rormnc o h risky ss. Sciiclly, w ssum h r nd r, r joinly normlly disriud wih h idnicl mrginl disriuion, N μ, σ. W dno h corrlion wn r nd r y ρ, whr < ρ. Givn his disriuionl ssumion, h r o rurn on h risky ss my xrssd s: r ρ μ ρr ρ σε, whr h nois rm, ε, is indndn o r nd hs sndrd = Norml disriuion. 5

17 h rincil is ssumd o risk nurl whil h gn is risk vrs wih h log uiliy uncion: U C = log C, whr C dnos h gn s comnsion. h gn surs disuiliy, V, o mnging h orolio. Ovrll, h gn s uiliy is givn y, U C V, whr h gn s comnsion, C, is sujc o h rsricion C c o ccoun or h limid liiliy ur. h consn c is s o smll osiiv numr. 9 h gn s rsrvion uiliy is ssumd o, U, whr h consn is o inrrd s h gn s ooruniy cos o nring h conrc wih h rincil No h whn ρ =, h rincil cn noislssly osrv h rdd risky ss s rurn, hry sisying h ky rquirmn o Proosiion. I ollows h h irs s soluion cn chivd in h cs whn h corrlion wn h risky ss nd h nchmrk ss quls. h irs s orolio llocion is drmind y h ollowing mximizion rolm: FB = rg mx x σ V μ r, 5 W ssum h h gn s cos uncion is o h orm, V = k. W no h such uncion hs h ollowing roris: V = V =, nd wih suicinly lrg choic o k, h ojciv uncion in 5 is sricly concv. h qusion w ocus on in his scion is: wh hns whn h rincil s inormion is lss hn rc, i.., whn ρ <. W gin y noing h in his cs h nchmrk will no rc nd h irs s soluion will no chivd. Blow w xmin h orm o h scond s oiml conrc in his cs. h conrc is dndn on h osrvls. In riculr, w wri h conrc s C r, w, uncion o h log rurn on h nchmrk ss, r, nd h log orolio vlu, w. ky o h nlysis is h qusion o how h roiliy disriuion o h log orolio rurn, w, dnds on 9 o void h singulriy o h log uncion zro, w ound h comnsion wy rom zro. 6

18 h orolio llocion,, nd h osrvion o h nchmrk log rurn, r. No h, condiionl on osrving r, h disriuion o h risky ss s rurn, r, is N ρ μ ρ r, ρ. W dno r μ = ρ μ ρ. h condiionl disriuion o h log rurn o h orolio, w, is givn y N r μ r, ρ σ. h log.d.. o h rminl orolio vlu, W, is givn y: log W, r ; = log π ρ σ w log w r μ r ρ σ. 6 By h liklihood rio rincil s Holmsröm 979, h soluion o h oiml conrc rolm is chrcrizd y h ollowing s o quions h joinly solv or h comnsion, C, orolio llocion,, nd rmrs, λ, nd ϕ. Firs, h oiml conrc should s h comnsion C o sisy h ollowing quion:, ' U W C = λ ϕ ' log W, r ;. 7 U C hn, h shdow ric o h incniv comiiliy consrin, ϕ is h soluion o h djoin quion, w log W, r ; = E C r, w; log W, r ; log W, r ; φ E log nd h vlu o h rmr λ is h soluion o h riciion consrin: C r, w,, 8 V E log C r, w; V = log. 9 Finlly, h orolio llocion in h scond s soluion is givn y h irs ordr condiion o h gn s rolm: [ h log C r, w; ] V E. = No h h conrc comnsion,c, dnds only on r, nd w, nd no dircly on h mngr s orolio llocion, which is o cours unosrvl o h invsor nd hror cnno 7

19 dircly conrcd on. Howvr, hrough Equion 7, h orolio llocion,, in h scond s soluion, dos inlunc C s rmr. hror, w us h noion C w, r ; o rck his dndnc in h ov quion sysm. r h orolio llocion is drmind, is vlu is luggd in o solv or C. From Equion 7, w hv h h oiml conrc ks h orm: C w, r = c λ φ = c λ φ log W, r ; c w r μ r μ r w r μ r ρ σ 3 ρ σ c nol ur o h ov conrc is h h mngr s comnsion is sd on comrison o h orolio s rormnc, w, gins h nchmrk: r μ r. Hr, is h orolio llocion in h scond s soluion, nd i dos no vry wih h mngr s o-quilirium orolio llocion. No lso h h ov scond s conrc is convx. Proosiion 3. Whn h rincil s inormion is no rc i.., ρ <, h shdow ric o h incniv comiiliy consrin is osiiv. h is, ϕ > irs s llocion, i.., Proo. S ndix II. < FB.. h orolio llocion is lowr hn h h ov roosiion shows h du o imrc inormion i.., whn ρ is sricly lss hn, h mngr s llocion o h civly mngd orolio will lss hn h in h irs s soluion, which is orm o or shirking in our sing. Hnc, h irs s oucom is chivd only in h scil cs wih rc osrviliy o h risky ss s rurn, i.., whn ρ =. II.. Convrgnc o Firs Bs W nx xmin how h scond s oucoms convrg o h irs s s h corrlion wn h risky ss nd h rrnc nchmrk ss rochs. From h rsuls in Scion, w know h in h scil cs whn ρ =, h irs s oucom is chivd. h conrc in Equion ks 8

20 h ollowing orm in his cs: CFB w, r = W x FB μ r λ, whr μ = ρ μ ρ nd λ is chosn so h h gn s riciion consrin is inding h r quilirium. Givn h ρ =, w hv, μ = r r =. s shown in Scion, his conrc will ld o h irs s llocion, nd h quilirium, h conrc will y h gn consn moun which is consisn wih oiml risk shring in his sing. Furhr, w cn rsric h gn s ymn o nonngiv in h ov conrc s such consrin will no inding h quilirium. W now urn o h cs whr ρ. o comrl wih h scond s conrcs discussd rlir, w consrin h conrc C w, r wy rom zro. h is, w considr h conrc FB [ W x μ r λ ] C w, r = c c. No h, givn his conrc, h ni o h FB FB mngr rom dviing rom h irs s soluion, is oundd y uncion o h dviion o his orolio llocion rom h irs s llocion, FB. h cos o h mngr o dviing rom h irs s is h ossiiliy o ging h minimum ymn, c. For ny ε > nd > ε, w FB hv h cos incrsing o ininiy s c rochs zro uniormly. hror, s c, h gn s orolio llocion choic whn cing h ov conrc lso rochs h irs s. In ohr words, whn w choos c o suicinly smll, h gn s orolio choic is rciclly h sm s h irs s soluion. Givn h h scond s soluions or h css wih ρ < should wkly domin h oucom rom h conrc, C FB, h scond s soluions will convrg o h irs s soluion s ρ rochs. W summriz h conclusion in h ollowing roosiion. Proosiion 4: s ρ, nd c II.B. Numricl nlysis, h scond s soluion rochs h irs s. How do h rmrs in h oiml conrc, h mngr s orolio llocion, nd h rincil s uiliy chng wih chng in h corrlion wn h risky ss nd h nchmrk ss, ρ? In riculr, how do h conrc rmrs, h orolio llocion nd h rincil s uiliy chng s 9

21 ρ dclins susnilly low, i.., s h mngr s orolio rogrssivly coms mor oqu? W rly on numricl nlyss o ddrss hs qusions. o clir h modl, w ssum h h ss rurn in h modl hs h sm sisicl chrcrisics s h rod mrk indx. Using h monhly U.S. -ill nd vlu-wighd mrk indx rurns or h riod 963:-:9, w g h ollowing nnulizd sisics: vrg risk r r o 5.9%, vrg mrk xcss rurn qul o 5.86%, nd mrk voliliy o 5.6%. W urhr ssum h h gn s rsrvion uiliy is qul o % o h iniil sss undr mngmn. h cos uncion rmr, k, is s qul o. For h uros o comrison, w sudy h oucoms undr h ollowing hr conrcs s h corrlion coicin, ρ, vris wn nd : h scond s oiml conrc, h oiml linr conrc, nd c h rcicl incniv conrc wih n oion-lik onus. h ls conrc is conrc h is similr o h kind o incniv conrc osrvd in rcic. h scond s oiml conrc is s dscrid in. o cili comrison, w now rwri his conrc in h ollowing orm: Cwr, = F β w r μ r μ r γ w r μ r W cn inrr h irs rm F in h ov conrc s h ixd slry comonn. h scond comonn cn inrrd s linr comonn, nd h hird comonn s convx comonn. W no h h rmrs β nd γ in h ov xrssion r rld y: β / γ =. Furhrmor, h linr comonn is linr in h orolio s rormnc. Howvr, h comonn is in h orm o h inrcion wn h orolio rormnc nd h rormnc o h nchmrk. hror, h y-rormnc snsiiviy in his cs vris wih h nchmrk rormnc. W imos h consrin, s in our nlyicl sudy, h h mngr s comnsion is no lss h rdrmind smll numr, c, which is s qul o.. Such consrin is imosd in ll hr conrcs. s rousnss chck w lso clird h modl using h mrk sisics or h ollowing su-riods: 96:7 :3; 995: :3; c : :3; d 963: 987:; nd 988: :3. In ch cs h rsuls r quliivly similr o h rsuls or our s cs rsnd hr.

22 h linr conrc w considr is conrc h rls h mngr s comnsion o h orolio rormnc in linr shion. h is, Cwr, = F ϕ w w, 3 whr F is h ixd slry, φ is h y-rormnc snsiiviy, nd w = r r r. W s h quilirium llocion,. Finlly, or h rcicl incniv conrc, w k h ov linr conrc s h sring oin nd dd comonn h rsmls h usul oion-y onus : Cwr, = F ϕ w w γ w w. 4 h rsuls o h numricl nlysis r rord in l. In ch nl o h l, h irs column liss h vlu o h corrlion wn h nchmrk nd h risky ss rurn, whil h scond column rors h rincil s uiliy in xcss o h iniil invsmn nd s rcng o h iniil invsmn. h hird column shows h corrsonding ss llocion o h civly mngd ss inducd y h conrc. In h ollowing wo Pnl B or hr Pnls nd C columns, h conrc rmrs F, β, γ r lisd undr h il slry, linr, nd qudric or oion. h ls wo columns o Pnl ror h rliv wighs o h wo conrc comonns: linr vrsus qudric. h rliv wighs r sd on h rliv vriion in h linr nd qudric comonn o h mngr s comnsion s h rormnc o h civly mngd ss nd h nchmrk ss vris cross h join disriuion o h risky ss rurn nd h nchmrk rurn. For sy comrison, h ls row o ch nl liss h irs s oucoms in rms o h orolio llocion nd h rincil s uiliy. From Pnl, i is clrly sn h undr h scond s conrc, whn h corrlion coicin, ρ, is lss hn on, hr is n undrinvsmn in h civly mngd risky ss. In h irs s oucom, h llocion o h risky ss is 46.77% wih h rsuling rincil s uiliy 8.75%. h inducd llocion in h scond s cs whn ρ = is 89.8%. h rincil s uiliy in his cs dros o 6.5%, i.., h rincil surs uiliy loss o mor hn wny iv rcn s h nchmrk

23 ss s corrlion dros rom rc o zro. h undrinvsmn rolm is lss svr s h corrlion, ρ incrss. Indd, s ρ incrss, oh h llocion o h risky ss nd h rsuling rincil s uiliy incrs grdully. Furhrmor, s ρ rochs, oh h risky ss llocion nd h rincil s uiliy convrg o hir corrsonding vlus in h irs s oucom. h scond s conrc nds o chiv h win ojcivs o roviding h rori or incniv o h mngr nd chiving h rori risk shring wn h wo ris. s ρ dcrss, h rormnc o h nchmrk rrnc ss is lss inormiv ou h mngr s cions, i.., his risky ss llocion. s rsul, s ρ dcrss, i.., s h orolio coms incrsingly oqu, h linr comonn o h conrc coms lss civ, nd h conrc rlis mor on h convx comonn o moiv h mngr o invs in h risky ss. Indd, s sn rom Pnl o h l, h rliv wigh o h convx comonn incrss susnilly s ρ dcrss. Whn comring h linr conrc oucoms Pnl B wih h scond s oucoms Pnl, w cn s h hr is susnil uiliy loss or h rincil whn ρ is low. For insnc, whn ρ =, h uiliy dcrss rom 6.5% in h scond s cs o 3.86% in h linr conrc cs uiliy loss o ovr 4%. h loss o uiliy in h cs o linr conrcs is lrgly du o h undrinvsmn in h risky ss. For insnc, whn ρ =, h llocion o h risky ss is 89.8% in h cs o h scond s soluion, whil i is only.65% in h cs o h linr conrc. s ρ incrss, h llocion o h risky ss in h linr conrc cs incrss susnilly wih h rsuling dclin in h rincil s uiliy loss. For xml, whn ρ =. 98, h uiliy loss undr h linr conrc is lss hn % rliv o h scond s cs. his rsul highlighs h c h s h undrlying orolio coms lss oqu h linr conrc coms mor rciv. By Proosiion 4, h rincil s uiliy convrgs o irs s in oh css whn c rochs zro. Som rsidul dirnc rmins du o h consrin h ll conrc ymns hv o lrgr hn osiiv consn, c.

24 W nx considr h oucoms undr h rcicl incniv conrc Pnl C which urs h oion-lik comonn in ddiion o h linr conrc comonn. W no h h undrinvsmn rolm in h linr conrc is llvid o crin dgr y h inclusion o h oion-y comonn in h rcicl incniv conrc. In h cs whn ρ =, h orolio llocion o h risky ss incrss rom.65% in h linr conrc cs o 7.4% in h cs o h conrc wih h oion-lik comonn. h corrsonding invsor s uiliy incrss rom 3.86% in h linr conrc cs o 4.33% in h cs o h rcicl incniv conrc. h gin in h invsor s uiliy is much lss or highr vlus o ρ. Indd, whn ρ gs clos o, h oion-lik comonn is no longr usul. In c, s sn rom Pnl C o l I, whn ρ =.9, h oion-lik comonn coicin is clos o zro, nd h coicin coms zro whn ρ =. 95 or whn ρ =.98. his is n inrsing rsul s i conirms h s h ociy o h mngr s orolio dclins, h oion-lik comonn is no longr ndd o moiv h mngr, nd h linr conrc comonn suics. Convrsly, i is rcisly in h cs o non-rdiionl, rlivly oqu sss h h oion-y comonn is dsirl conrc ur. Concluding mrks his r ddrsss wo rld issus in h conx o dlgd orolio mngmn. Firs, i xlors h condiions undr which irs s oucoms my chivd vi linr conrcs in his sing. h nlysis shows h nchmrk-linkd linr conrc cn chiv h irs s oucom whn h mngr s orolio llocion is no sd on riv inormion, nd h rincil is l o noislssly osrv h ss rurns. Inuiivly, wih rc osrviliy o ss rurns, h invsor cn drmin h irs s orolio oucom nd nliz ny dviion rom his nchmrk vlu. Scond, h r xlors h limiions o h linr conrc whn h orolio is invsd in oqu sss h r illiquid or h r rivly rdd. his is cs o riculr inrs s i chrcrizs h orolios o numr o lrniv or non-rdiionl invsmn vhicls such s 3

25 hdg unds nd riv quiy unds. h rliv ociy o h orolios in hs css conrius o h nchmrks or such sss ing rlivly noisy. W xmin how h oiml conrc chngs s h corrlion wn h mngr s orolio nd h rrnc nchmrk orolio dclins. h nlysis suggss h in h snc o rc osrviliy, i.., whn h corrlion wn h mngr s orolio nd h nchmrk orolio is lss hn, h irs s oucom is no longr sil. h scond-s oiml conrc in his sing urs n oion-y comonn in ddiion o comonn h is linr in rormnc. Morovr, h rliv imornc o h oion-y comonn is n incrsing uncion o h orolio s ociy. Furhr, h rincil s uiliy loss rom rsricing h wigh o h oion-y comonn o zro is incrsing in h ss s ociy. hs rsuls hl xlin h dirncs in h conrcs osrvd in h rdiionl ss mngmn scor nd h lrniv ss mngmn rn. In riculr, hy rovid rionl or h oion-lik incniv conrcs osrvd in h lrniv ss mngmn indusry. 4

26 rncs dmi,.., nd P. Plidrr Dos I ll dd U? Bnchmrks nd h Comnsion o civ Porolio Mngrs. Journl o Businss 7, Bhchry, S., nd P. Plidrr Dlgd Porolio Mngmn. Journl o Economic hory 36, -5. Crnr, J. N.. Dos Oion Comnsion Incrs Mngril isk i? Journl o Finnc 5, Gozmnn, W.N., J. Ingrsoll, nd S.. oss, 3, High-wr mrks nd hdg und mngmn conrcs. Journl o Finnc 58, Ds, S.., nd. K. Sundrm.. F Sch: Signling, isk-shring, nd h Imc o F Srucurs on Invsor Wlr. viw o Finncil Sudis 5, Edmns, lx, nd Xvir Gix. 9. rciliy in Incniv Conrcing, NBE Working Pr No. 5545, Nionl Buru o Economic srch, Inc. Grinl, M., nd S. imn dvrs isk Incnivs nd h Dsign o Prormnc-Bsd Conrcs. Mngmn Scinc 35, Holmsröm, Bng, 979, Morl hzrd nd Osrviliy, h Bll Journl o Economics,, Holmsröm, B., nd P. Milgrom ggrgion nd Linriy in h Provision o Inrmorl Incnivs. Economric 55, Ln, Yingcong, N. Wng, nd J. Yng.. h conomics o hdg unds. Working r, Columi Univrsiy. Li, C. W., nd. iwri. 9. Incniv Conrcs in Dlgd Porolio Mngmn. viw o Finncil Sudis, Mss, M., nd. Pgiri. 9. Incnivs nd Muul Fund Prormnc: Highr Prormnc or Jus Highr isk king? viw o Finncil Sudis, Ou-Yng, H. 3. Oiml Conrcs in Coninuous-im Dlgd Porolio Mngmn Prolm. viw o Finncil Sudis 6, Plomino, F., nd. Pr. 3. isk king nd Oiml Conrcs or Mony Mngrs. ND Journl o Economics 34, Pngs, S., nd M.M. Wsrild. 9. High-wr mrks: High risk is? Concv comnsion, long horizons, nd orolio choic. Journl o Finnc 64, -36. Shvll, S isk Shring nd Incnivs in h Princil nd gn lionshi. Bll Journl o Economics, Srks, L Prormnc Incniv Fs: n gncy horic roch. Journl o Finncil nd Quniiv nlysis, 7-3. Srcc, L. 6. Dlgd Porolio Mngmn: Survy o h horicl Lirur. Journl o Economic Survys 5, Soughon, N Morl Hzrd nd h Porolio Mngmn Prolm. Journl o Finnc 48,

27 l h l documns h numricl rsuls on h oucoms o using hr ys o conrcs undr dirn scnrios whr h corrlion, ρ, wn h rurn on nchmrk osrvd y h rincil, nd h rurn on h orolio mngd y h gn, vris rom o.98. h l rors, or ch conrc, h rincil s xcd uiliy n o h iniil invsmn scond column xrssd s rcng o h iniil invsmn, h gn s llocion wigh on h civly mngd risky orolio hird column, nd h conrc comonns. Pnls, B nd C corrsond o h scond s conrc, h linr conrc, nd h rcicl incniv conrc, rscivly. h conrcs r dscrid in mor dil in Scion II.B. in h x. Columns 4, 5 nd/or 6, undr h suil Conrc, ror h coicins in h conrc: coicin F undr Slry, coicin β undr Linr, nd/or coicin γ undr Qudric in Pnl nd undr Oion in Pnl C. o cili inrrion, in Pnl, w lso ror h rliv wighs o h wo vril conrc comonns. h rliv wighs r sd on h rliv solu vriion in h linr nd qudric comonn o h mngr s comnsion s h rormnc o h civly mngd risky ss nd h nchmrk ss vris cross h join disriuion o h risky ss rurn nd h nchmrk rurn. h rliv wighs r rord in Columns 7 nd 8 undr h suil liv Wighs. h ky inus or h clirion r sd on h ollowing nnulizd U.S. mrk sisics or h riod 963:-:9: vrg risk r r o 5.9%, vrg mrk xcss rurn qul o 5.86%, nd mrk voliliy qul o 5.6%. Pnl. Scond Bs Conrc Conrc liv Wighs ρ Uiliy llocion Slry Linr Qudric Linr Qudric 6.5% 89.8% % 83% % 9.4% % 8% % 9.3% % 69%.75 7.% 4.% % 56%.9 7.6% 9.5% % 4% % 7.6% % 33% % 7.63% % 3% Firs s 8.75% 46.77% 6

28 Pnl B. Linr Conrc Conrc ρ Uiliy llocion Slry Linr 3.86%.65% % 3.9% % 5.7% % 3.% % 44.94% % 83.99% % 8.36% Firs s 8.75% 46.77% Pnl C. Prcicl Incniv Conrc Conrc ρ Uiliy llocion Slry Linr Oion 4.33% 7.4% % 7.6% % 7.% % 9.34% % 43.8% % 83.99% % 8.36% Firs s 8.75% 46.77% 7

29 ndix I chnicl ndix Drivion o h rsuls o Scion I undr gnrl rmwork Blow, w driv h rsuls corrsonding o hos o Scion I undr gnrl rmwork. s will dmonsrd, our rsuls do no rly on h sciic orm o uiliy uncion or h rincil nd h gn, ssumions rgrding how h gn s cion cs h disriuion o rojc s yo, nd hror h disriuionl ssumions rgrding h s vrils, nd c h choic o dynmic rmwork nmly, whhr i is coninuous-im or discr-im modl. Considr n conomy in which rincil hirs n gn o crry ou usinss ndvor. h oucom o h ndvor is wlh disriuion long s sc nd im. W ssum h h ir o msur sc nd sigm lgr, dnod y Ω, F, curs ll h dimnsions nd ll h rlvn disincions in h s sc nd im wihin h modl. o cur h id h h gn s cion conrius o h oucom, w dno i in uncionl orm s W : D, whr is h sc o ll sil cions y h gn, nd D is h sc o ll ossil wlh disriuions long s sc nd im. h is, D consiss o rl vlud F - msurl uncions wih domin on Ω, nd sil cion y h gn lds o wlh disriuion W D. W ssum h h rincil hs uiliy uncion, U P : D. Such gnrl uiliy uncion rnks ll sil wlh disriuions h rincil onilly gs ou o h join vnur. h gn hs uiliy uncion ovr h comnsion h rcivs nd h cion h chooss. Formlly, U : D. In h rincil-gn sing, h gn s cion hs o volunrily chosn. hror, n rorily dsignd conrc is ndd o lici h ror cion rom h gn. W viw h conrc rom h gn s rsciv. s h oucom o ny cion h gn chooss, hr r h rsuling disriuions on h msur sc Ω, F o h comnsion, which w dno y D S. h is, similr o W, S is uncion rom o D. W dno h sc o ll sil comnsion-sd conrcs y S. h gnric conrcing rolm cn sd s h ollowing oimizion rolm: S S, S su U W S,. s.. S = rg mx U S,,. U S, U..3 For conrc o rovid moivion, i hs o link h comnsion wih h gn s cion. Givn h h gn s cion is no osrvl, his link is in gnrl, no dirc. For insnc, S my drmind y indircly hrough W. hror, dnding on h x os inormion h rincil hs, h sil conrc sc S is in gnrl ror sus in h sc o ll uncionls rom o D. 8

30 In h ov, U is h gn s rsrvion uiliy, nd Inquliy.3 is h gn s riciion consrin. Hr, w ssum h h rincil ks h nir surlus nd h h gn s uiliy is drivn o h rsrvion lvl. h ddiionl consrin. is h gn s incniv comiiliy consrin. h noion o rgmx in. is o undrsood s h s o ll soluions o h mximizion rolm. hror, is s-vlu uncion such h or ny S S, S. s scil cs, i S is lwys rducd o singl oin or ll cndid conrcs, S, hn is uniquly drmind y S hrough consrin.. In his cs, h only conrol vril or h oimizion rolm will S, h choic o h conrc. In h wy w s h rolm in., w do h convnion h mong ll cions or which h gn is indirn, h will choos h on h is mos nicil o h rincil. Such convnion coms unncssry i w do h ssumion h h gn ks h nir surlus nd h rincil s uiliy is drivn o rsrvion lvl. W ssum h irs s soluion xiss. h is, h ollowing oimizion rolm hs ls on soluion., S D mx U W S,.4 For sy rrnc, w dno h s o irs s soluions y s.. U S, U..5 9 Q FB : {,,,, solvs h oimizion rolm in. nd. } QFB S S D S For h ollowing discussion, w will in gnrl us h noions nd S or h rsuling quniis in h irs s soluion. h irs s soluion is yiclly undrsood s h oiml soluion h cn chivd whn h gn s cion cn x os vriid nd hror conrcd on. lhough his cs is wll undrsood in h lirur, w mk som rmrks in ordr o cili our lr comrison. No h wih sligh us o noion, S in.4 nd.5 is oin in h sc in D, nd i drmins only on comnsion disriuion or h gn. In conrs, conrc S in.-.3 is uncion rom o D, nd dirn cions y h gn cn onilly ld o dirn comnsion disriuions or him. Howvr, his inconsisncy cn sily rsolvd ollowing h gnrl roocol o viwing consn numr s consn uncion, y viwing h comnsion disriuion S hror, oin in D s quivln o h consn uncion h ms ny cion o h sm comnsion disriuion S, wih h undrsnding h hr is n imlici dimnsion o h conrc h hrns o unish h mngr whn his cion dvis rom h oiml cion,. his dimnsion o h conrc is no xlicily sd in h rolm, u i is viwd s rivil in inuiion. Comring h oimizion rolms in.4-.5 nd.-.3, w no h h dirnc sms rom h ddiionl consrin, nmly, h incniv comiiliy consrin., or h conrcing rolm in.-.3. his consrin is h ky cor h cn ld o onil morl.6

31 hzrd rolm. h onil morl hzrd lds o loss in icincy i nd only i consrin. is inding. For our uros, riculr s o conrcs r o inrs. hy k h orm: S = W W S.7 or som ir, S S = Q FB. W dno h s o conrcs king h orm in.7 y S. h is, { S S S, wih S = W W S or som ir, S Q } FB..8 I will com clr lr h his s o conrcs cn inuiivly undrsood s ing quivln o h rincil slling h rojc o h gn. h ky qusion ou s S is whhr i is my. In cs i is no my, w hv h ollowing rsul: S S Proosiion.. I hr xiss conrc,, hn h incniv comiiliy consrin in. is no inding nd h irs s soluion is chivd y his conrc. In c, or ny in h s o S, h ir, S is irs s soluion. I w ssum, in ddiion, h oh U nd U, r coninuous nd sricly incrsing in wlh, w hv h S. h is, oiml rsons o conrc S or h gn. is n Proo. Givn h diniion o S, hr xiss, S Q FB such h W S = W S, or ny..9 h is, h disriuion o h rincil s yo, W S, dos no dnd on h gn s cion, nd w hv U W S = U W S, whr h righ-hnd sid is y diniion h soluion o h oimizion in h rincil chivs h irs s oucom rgrdlss o h gn s cion. For h gn, cus is drmind y., w hv U S, U S, U. In summry, h ir, S sisis h consrins in.5 nd chivs h mximum vlu o h ojciv uncion in.4. I hus quliis s soluion or h oimizion rolm in h roo o h ls clim is lso srighorwrd, u is omid or h sk o rinss. Q.E.D. 3 I h rincil cn x os dmnd h yo h lwys mchs h irs s oucom, sh cn sll h rojc o h gn in rurn or ric qul o W. In gnrl, howvr, W is rndom vril. Indd, whn h gn is risk vrs, h quniy W should vry wih h undrlying ss in such wy h i chivs h oiml risk shring wn h rincil nd h gn h quilirium. Nvrhlss, his coningn yo is in no wy dndn on h gn s 3 s his roosiion dos no ssum h uniqunss o h irs s soluion, i mks no clim ou whhr, S =, S. 3

32 cion,, nd is no dircly rld o h rojc s oucom, W. hror, in his sns, h rincil-gn rlion is svrd h oin whn h conrc is signd y oh ris, nd i is lso in his sns h w my inrr i s h rincil slling h rojc o h mngr. By doing so, h gn rs ll h consquncs o his cion, nd hror h incniv rolm is comlly ddrssd. h conrc cn viwd s ing quivln o h rincil slling h rojc o h gn in rurn or coningn ymn o W h rminl d. O cours, in ordr or i o sil comonn o conrc, W cnno coningn on x os unvriil condiions. In h ov nlysis, w ssum h mngr s uiliy ks h orm U P S,. I llows h mngr s cion o c h uiliy in vry gnrl wy. s is yicl in h morl hzrd lirur, h mngr s uiliy cn sciid s uncion o wo vrils, h wg h mngr rcivs nd riv cos du o his cion. h is, h uiliy ks h orm U P S, c, whr c is h riv cos incurrd y cion,. W nx nlyz h cs in which h cos o cion o h mngr is consn h is, c c or som consn, c. Undr his ssumion, h mngr s uiliy is rducd o S, which is dndn on his cion only hrough h consqun yo h U P rcivs. W dno h s o ll sil wlh disriuion o h rojc y W h is, { W } W. W cll h s W mniold i or ny λ nd ny wo inl oucoms, W, W W, h linr cominion W3 = λw λ W is n lmn in W. W hv h ollowing roosiion: Proosiion.. I, in ddiion o h ssumions in Proosiion., w ssum h c c or som consn, c, nd h s W is mniold, hn ll o h ollowing conrcs will chiv h irs s oucom nd hror r oiml conrcs: Proo. Fix h ir S = S λ W W, or ny ixd λ, nd ny, S Q FB. Noic h whn h mngr chooss cion, S Q FB.., w would hv h yo or h mngr nd h or h invsor xcly mch h irs s soluion h is, S = S. hror, w nd only o show h, undr h conrc in., hr is no cion h h mngr would rr ovr. ssum ohrwis, h hr xiss such h U S > U S. Bcus W is mniold, λw λ W cn imlmnd y λ crin cion o h mngr, which w dno s. Noic h λ λ S = S [ λw λ W ] W = S W W = S. nd S = S = S. hror, U S = U S > U S = U S. his conrdics h conclusion in Proosiion. h is h oiml srgy or h mngr whn cing h conrc, S. Q.E.D. λ 3

33 h rsuls in Proosiions. nd. gnrliz h rsuls in Scion I o h x. Drivion o h irs s soluion in Ou-Yng s sing W s mor dild vrsion o h rsul low: Proosiion 3. h irs s soluion o h oiml orolio olicy nd h oiml ymn o h gn r givn y = k σσ h,. nd γ r γ whr =, r r givn y S = c, W, g g d g db,.3 =, nd h rocss { W }, or [, ] is g σ σ..4 W = W h ds db r r s r s s s s W urhr hv h invsor s yo in h irs s soluion s W W S F dig P dp,.5 P = = whr F is consn. Proo: Considr h orolio mngr s xcd uiliy, x,, < E U S c W d,.6 which is mringl ovr h Brownin ilds gnrd y B. horm o Myr s Jcod 977 holds h vry mringl ovr h Brownin ilds cn rrsnd s n Io sochsic ingrl wih rsc o h drilss Brownin moion: dx = θ db, whr { θ ; } is d-dimnsionl dod sochsic rocss nd θ θ d is lmos surly ini. Din h rocss { Ζ } s Ζ = U x = ln x..7 3

34 By Io s lmm, w hv, whr g θ Ζ = Ζ g gd g db,.8. W urhr hv h Ζ = S x c, W, d, nd rom h riciion consrin, Ζ = Ζ. Hnc, w g h ollowing quivln rrsnion o h riciion consrin: S =Ζ c, W, d =Ζ cwd,, ggd gdb..9 Wih h riciion consrin nod in Equion.9, w cn s h irs s rolm s ollows: whr N su { },{ g } s.. dn W S. Givn h Mrkov rocsss, wih E[ U d dw = rw N ] dp = P μ d σdb ;. nd = dw c, W, d h d σdb ; V, W, N, P or h ov oiml conrol rolm s g g d g db, nd g r dd sochsic rocsss, { P ; W ; N} r conrolld g ing h conrols. W din vlu uncion rocss { u },{ g u } [ U N ] V, W, N, P = su E.. h Bllmn quion or his dynmic rogrmming rolm is s ollows:, = su g V,., g whr g, snds or h ckwrd gnring oror, i.., 33

35 , g V = V V V VWW VNN rc PP V dig P σg. NP W rw h V σσ V V dig P σσ dig P N N c, W, NN g g V V WP NN V g σg V NP WN g VPdig P μ dig P σσ σσ σg.3 h irs-ordr condiions o h Bllmn quion wih rsc o h conrol vrils nd g r = V V h V c V V V σσ V V σg, W N N WW WN NN WN NN d σσ P V V WP NP.4 nd d = V N g VNN g VNN VWN σ σ P VNP..5 Conjcur h V, W, N, P ks h ollowing orm: { [ W N ]} V, W, N, P = x 3,.6 wih h oundry condiions =, nd. hn w hv h V = V, V N = V, V 3 = WW = V, V = NN = V, VWN V W =, nd ll h drivivs wih rsc o P r zros. Susiuing ll o hs ino h irs-ordr condiions ov, w g g = σ,.7 nd = k σσ h..8 o drmin h vlus o,, nd, w susiu V ck ino h Bllmn quion: 3 = W r γ N 3 h k g g g g σσ σ g..9 34

36 35 Noic h oh nd g do no dnd on N or W. o limin h N rm rom h righ-hnd sid o h ov quion, w mus hv = wih h oundry condiion, =. hror, =. o limin h W rm, w mus hv, r r γ =.3 wih h oundry condiion =. hror,. = r r γ.3 h uncion in horm is givn y =. h Bllmn quion is sisid y sing. 3 d g g g h g g k s s = σ σσ.3 h ollowing comuion is no ncssry or our drivion o h irs s soluion nd or h drivion o h oiml conrc. I is rsnd hr solly or h uros o ully rlicing Equion in Ou-Yng s horm. Givn h orolio srgy } {, h wlh rocss is drmind y h sochsic dirnil quion: db d h rw dw σ =..33 Using Io s lmm, w cn rwri h quion s: r r r db d h W d σ =..34 hror, = s s s r s s r r db ds h W W σ, or ], [..35 Noic h = db g d g g W c S,,..36 W hus hv

37 r r S W Ζ W k g g h d = γwd σdb σdb γ r r rwd dw rwd σdb = r γ = d r r W σdb γ = r r W μd dig P dp..37 W choos h consn, F, s in Ou-Yng 3, o sor ll h consn rms in h ov quion, nd wri h rsul in shor s: S W = F dig P dp..38 Hnc, h roosiion is rovd. ndix II Proo o Proosiion 3. h roo is dvlod in wo ss. Firs, i φ =, h conrc is h consn conrc. Clrly, h gn will choos =, which is sricly lss hn h in h irs s. his conrdics h c h φ =. hus, φ. Scond, ssum h φ <. Dno h oiml conrc y C. Now, considr h s o conrcs: C β = βc β C α, whr α is s s or o sisy h gn s riciion consrin. h on xrm whn w choos β =, w hv h oiml conrc wih > FB du o h ssumion h φ <. h ohr xrm whn w choos β =, h rsuling conrc is h consn conrc nd h gn will choos =. W cn hror choos vlu o β, such h h rsuling choic o = FB. Noic h his choic o h conrc chivs r risk shring hn conrc C cus i rducs h vriion in h ymn o h gn. I rsuls in r cos-ni rdo h choic o FB. hror, G β > G. his conrdics h ssumion h C is h oiml conrc. Hnc, w conclud h φ >. 36