Working Paper Series Brasília n. 184 Apr p. 160


 Godwin Alexander
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2 ISSN CGC.38.66/5 Working Papr Sris rasília n. 84 Apr. 9 p. 6
3 Working Papr Sris Editd by sarch Dpartmnt Dpp Editor: njamin Miranda Tabak Editorial Assistnt: Jan Soia Moita Had o sarch Dpartmnt: Carlos Hamilton Vasconclos Araújo Th anco Cntral do rasil Working Paprs ar all valuatd in doubl blind rr procss. production is prmittd only i sourc is statd as ollows: Working Papr n. 84. Authorizd by Mário Msquita, Dputy Govrnor or Economic Policy. Gnral Control o Publications anco Cntral do rasil Scr/Surl/Dimp SS Quadra 3 loco EdiícioSd º andar Caixa Postal rasília DF razil Phons: and Fax: Th viws xprssd in this work ar thos o th authors and do not ncssarily rlct thos o th anco Cntral or its mmbrs. Although ths Working Paprs otn rprsnt prliminary work, citation o sourc is rquird whn usd or rproducd. As opiniõs xprssas nst trabalho são xclusivamnt dos autors não rltm, ncssariamnt, a visão do anco Cntral do rasil. Ainda qu st artigo rprsnt trabalho prliminar, citação da ont é rqurida msmo quando rproduzido parcialmnt. Consumr Complaints and Public Enquiris Cntr anco Cntral do rasil Scr/Surl/Diat SS Quadra 3 loco EdiícioSd º subsolo rasília DF razil Fax: Intrnt: http//www.bcb.gov.br/?nglish
4 havior Financ and Estimation isk in Stochastic Portolio Optimization José Luiz arros Frnands * Juan Ignacio Pña ** njamin Miranda Tabak *** Th Working Paprs should not b rportd as rprsnting th viws o th anco Cntral do rasil. Th viws xprssd in th paprs ar thos o th authors and do not ncssarily rlct thos o th anco Cntral do rasil. Abstract Th objctiv o this papr is twoold. Th irst is to incorporat mntal accounting, lossavrsion, asymmtric risktaking bhavior, and probability wighting in a multipriod portolio optimization or individual invstors. Whil ths bhavioral biass hav prviously bn idntiid in th litratur, thir ovrall impact during th dtrmination o optimal asst allocation in a multipriod analysis is still missing. Th scond objctiv is to account or th stimation risk in th analysis. Considring 6 daily indx stock data ovr th priod rom 995 to 7, w mpirically valuat our modl ATE havior sampl Adjustd Tchniqu against th traditional Markowitz modl. Kywords: havior, Portolio Optimization, sampling JEL Classiication: G, G. * Univrsidad Católica d rasília and anco Cntral do rasil GrênciaExcutiva d isco da Ára d Política Montária ** Dpartamnto d Economía d la Emprsa, Univrsidad Carlos III d Madrid, España. *** Univrsidad Católica d rasília and anco Cntral do rasil Dpartamnto d Estudos Psquisas 3
5 In a standard asst allocation procdur, onc th risk tolranc, constraints, and inancial goals ar st, th output is givn by a manvarianc optimization Markowitz, 95; Fldman and isman,. Unortunatly this procdur is likly to ail or individuals, who ar suscptibl to bhavioral biass. For instanc, in rspons to shorttrm markt movmnts and to th dtrimnt o th longtrm invstmnt plan, th individual invstor may rquir his asst allocation to b changd. Frnands t al. [7] suggst that arly liquidation o a long trm invstmnt may b th caus o momntum. In trms o motional biass, svral mpirical studis Tvrsky and Kahnman, 99 hav shown that, whn daling with gains, agnts ar riskavrs, but whn choics involv losss, agnts ar risksking asymmtric risktaking bhavior. Morovr, in a wid varity o domains, popl ar signiicantly mor avrs to losss than thy ar attractd to samsizd gains. Lossavrsion Schmidt and Zank, 5 is a rlvant psychological concpt that has bn importd to inancial and conomic analysis, and it rprsnts th oundation o prospct thory. Th currnt paradigm o individual bhavior in inanc thory is basd on xpctd utility maximization and riskavrsion, which has bn undr attack in rcnt yars du to its dscriptiv inaccuracy. Exprimntal psychologists hav dmonstratd that popl systmatically dviat rom th choic prdictions th classical paradigm implis as individuals ar typically biasd. havioral biass can roughly b groupd in two catgoris: cognitiv and motional, though both typs yild irrational dcisions. caus cognitiv biass huristics lik anchoring, availability, and rprsntativ biass stm rom aulty rasoning, bttr inormation and advic can otn corrct thm. Convrsly, motional biass, such as rgrt and lossavrsion, originat rom impulsiv lings or intuition, rathr than conscious rasoning, and ar hardly possibl to corrct. Lo t al. [5] invstigatd svral possibl links btwn psychological actors and trading prormanc, inding that subjcts whos motional raction to montary gains and losss was mor intns on both th positiv and ngativ sid xhibitd signiicantly wors trading prormanc. Shrin [5] posits that th portolios slctd by invstors whos choics conorm to prospct thory will dir in ky aspcts rom th portolios slctd by invstors whos choics conorm to xpctd utility thory. Th gnral charactr o bhavioral portolios is that thy atur a combination o scuritis that ar vry sa 4
6 with scuritis that ar vry risky, with th ovrall portolio ailing to b wll divrsiid. In this sns, an optimal solution to th asst allocation problm should guid invstors to mak dcisions that srv thir bst intrst. This could b th rcommndation o an asst allocation that suits th invstor s natural psychological prrncs motional biass, vn though it may not maximiz xpctd rturn or a givn lvl o risk. Mor simply, a clint s bst practical allocation may b a slightly undrprorming longtrm invstmnt program to which th invstor can comortably adhr. From a manvarianc optimization prspctiv, bhavioral invstors slct portolios that ar stochastically dominatd. This dos not man that th individual invstors ar irrational in any sns: it is not irrational or popl to anticipat motional ractions and tak thm into account whn making dcisions that try to adjust thir choics to thir prrncs. Howvr, portolio managrs lack th guidlins ncssary or incorporating ths biass during th procss o dtrmining asst allocation. W addrss this issu by valuating whthr managrs should modrat th way clints naturally bhav to countract th cts o bhavioral biass so that thy can it a prdtrmind asst allocation or thy should crat an asst allocation that adapt to clints biass, so that clints can comortabl adhr to th und. In gnral trms, prospct thory and its lattr vrsion cumulativ prospct thory Kahnman and Tvrsky, 979, 99 posits our novl concpts in th ramwork o individuals risk prrncs. First, invstors valuat assts according to gains and losss and not according to inal walth mntal accounting. Scond, individuals ar mor avrs to losss than thy ar attractd to gains lossavrsion. Third, individuals ar risksking in th domain o losss and riskavrs in th domain o gains asymmtric risk prrnc. Finally, individuals valuat xtrm probabilitis in a way that ovrstimats low probabilitis and undrstimats high probabilitis probability wighting unction. This study, as ar as w know, is th irst to considr all thos aspcts in th ramwork o portolio choic. Thr ar conlicting rsults in th inanc litratur on how prior outcoms act th risktaking bhavior o invstors in subsqunt priods. Lossavrsion would prdict that tradrs with proitabl mornings would rduc thir xposur to atrnoon risk, trying to avoid losss and thus guaranting th prvious gains Wbr and Zuchl, 3. Odan [998] and Wbr and Camrr [998] hav shown that invstors ar mor willing to sll stocks that trad abov th purchas pric winnrs than stocks that trad blow purchas pric losrs a phnomnon trmd th disposition ct 5
7 Schrin and Statman, 985. oth works intrprtd this bhavior as vidnc o dcrasd riskavrsion atr a loss, and incrasd riskavrsion atr a gain. Th standard xplanation or th prvious bhavior is basd on prospct thory, and particularly on th act that individuals ar risksking in th domain o losss and riskavrs in th domain o gains asymmtric risk prrnc. Howvr, anothr stram o th litratur ound th opposit bhavior. Thalr and Johnson [99] nam th housmony ct, th bhavior o incrasing risk apptit atr a gain. arbris t al. [] prsnt a modl whr invstors ar lss lossavrs atr a gain whil thy bcom mor lossavrs atr prior losss. Our proposd modl addrsss and clariis th prvious contradiction btwn housmony and disposition ct. Dspit th vast litratur conirming th bhavioral biass associatd with prospct thory, th considration o all thos biass in an asst allocation ramwork is still missing. arbris and Huang [] and arbris t al [] us lossavrsion and mntal accounting Thalr, 999 to xplain aspcts o stock pric bhavior, but do not mploy th ull prospct thory ramwork and don t xamin optimal asst allocation. nartzi and Thalr [995] considr prospct thory to solv th quity prmium puzzl whn invstors ar lossavrs and valuat thir portolios myopically with a horizon o approximatly on yar. Thy also suggst an optimal allocation in quitis rom 3% to 55%. Magi [5] uss bhavioral prrncs to numrically solv a simpl modl o intrnational portolio choic, providing a possibl xplanation or th quity hom bias puzzl, th tndncy o individual invstors to prr its homcountry stocks dspit th gratr prormanc o orign stocks. Davis and Satchll [4] provid a solution or th optimal quity allocation, and xplor mor thoroughly th cumulativ prospct thory paramtr spac that is consistnt with obsrvd quity allocations givn a inancial markt s rturns distributions ovr a onmonth horizon. Shrin [5] considrs htrognous invstors to s th impact o bhavioral concpts in th ramwork o asst pricing. Th irst main goal o this study is to incorporat mntal accounting, lossavrsion, asymmtric risktaking, disposition ct, and probability wighting in portolio optimization in a multipriod stting or individual invstors. W provid a solution or th asst allocation problm, taking into account all bhavioral biass associatd with prospct thory and using a utility unction suggstd in Giorgi t. al., 4 consistnt with both th xprimntal rsults o Tvrsky and Kahnman, and also 6
8 with th xistnc o quilibrium. W also shd mor light on th issu o how prior outcoms act subsqunt risktaking bhavior, invstigating th invstor s risktaking bhavior ollowing a ris, or a all, in th pric o th risky asst. In lin with prospct thory, invstors driv utility rom luctuations in th valu o thir inal walth. In our ramwork, thr is a inancial markt on which two assts ar tradd. A risklss asst, also calld a bond, and a risky asst, also calld a stock undr th assumption o normally distributd rturns or th risky asst. As w ar modling th dcision making procss o an individual invstor, shortslling is not allowd. In ach priod w considr two priods, th invstor chooss th wight o his ndowmnt to b invstd in th risky asst, in ordr to maximiz his utility prospct thory basd. W assum that th invstor acts myopically in a sns that h dosn t discount longtrm wlar whn valuating his utility, and that th rrnc point rlativ to which h masurs his gains and losss or th irst priod is his initial ndowmnt. Although all agnts solv th sam maximization problm in th irst priod, th scond priod dcision dpnds on th rrnc point rlativ to which th agnt masurs th scond priod outcoms gains or losss. W considr two possibl rrnc points: th initial walth or th currnt walth, and analyz both cass. St Amour [6] valuats houshold portolios and his rsults rval that rrncs ar strongly rlvant and statdpndnt. Anothr wllknown issu in asst allocation problms, using Markowitz optimization, is that th output is strongly drivn by th risk/rturn stimation, which usually gnrats vry unstabl portolios. Th most amous problm with this tchniqu is th substitution problm, whr two assts with th sam risk but slightly dirnt xpctd rturns. Th optimizr would giv all th wight to th asst with th highr xpctd rturn, lading to a vry unstabl asst allocation. Th scond goal o this chaptr is to incorporat stimation risk in th portolio allocation bhavioral problm. cnt litratur has trid to ovrcom th prvious problm o lading to unasibl portolios. Th main ocus o thos modls is to ind out how to crat ralistic portolios considring that th valus usd or risk and rturn ar not dtrministic but instad just stimats thy ar stochastic. It should b notd that th misspciication o xpctd rturns is much mor critical than that o variancs Zimmr and Nidrhausr, 3. 7
9 Jorion [986] ors a simpl mpirical ays stimator that should outprorm th sampl man in th contxt o a portolio. His main ida is to slct an stimator with avrag minimizing proprtis rlativ to th loss unction th loss du to stimation risk. Instad o th sampl man, an stimator obtaind by shrinking th mans toward a common valu is proposd th avrag rturn or th minimum varianc portolio, which should lad to dcrasd stimation rror. Similar to Jorion, Kmp t al [] assums that th prior man is idntical across all risky assts. Howvr, Kmp s modl considrs stimation risk as a scond sourc o risk, dtrmind by th htrognity o th markt and givn by th standard dviation o th xpctd rturns across risky assts. lack and Littrman [99] postulat that th considration o th global CAPM Capital Asst Pricing Modl quilibrium can signiicantly improv th usulnss o asst allocation modls, as it can provid a nutral starting point or stimating th st o xpctd xcss rturns rquird to driv th portolio optimization procss. Horst t al. [] propos a nw adjustmnt in manvarianc portolio wights to incorporat th stimation risk. Th adjustmnt amounts to using a psudo riskavrsion, rathr than th actual riskavrsion, which dpnds on th sampl siz, th numbr o assts in th portolio, and th curvatur o th manvarianc rontir. Th psudo riskavrsion is always highr than th actual on and this dirnc incrass with th uncrtainty in th xpctd rturn stimations. Manhout [4] also considrs an adjustmnt in th coicint o riskavrsion to insur th invstor against som ndognous worst cas. Finally, Michaud [998] suggsts portolio sampling as a way to allow an analyst to visualiz th stimation rror in traditional portolio optimization mthods, and Shrr [] posits that sampling rom a multivariat normal distribution a paramtric mthod trmd Mont Carlo simulation is a way to captur th stimation rror. Markowitz and Usmn [3] compard th traditional approach to rsampling and thir rsults support th lattr. Frnands t al. [8] valuat svral asst allocation modls and suggst that rsampling mthods typically or th bst rsults. This study prsnts a novl approach ATE havioral sampl Adjustd Tchniqu to incorporat bhavioral biass and stimation risk into manvarianc portolio slction. In a papr clos to ours, Vlck [6] proposs a modl to valuat portolio choic with lossavrsion, asymmtric risktaking bhavior, and sgrgation o risklss opportunitis. His indings suggst that th changs in portolio wights crucially dpnd on th rrnc point and th ratio btwn th rrnc point and th 8
10 currnt walth, and thus indirctly on th prormanc o th risky asst. Our work dirs rom his study as w xplicitly considr all novl aspcts o prospct thory: mntal accounting, lossavrsion, asymmtric risktaking bhavior, and probability wighting unction. W also valuat th inicincy cost o th bhavioral biass and considr a mor gnral orm or th risky asst rturn procss, including stimation risk in th analysis. Considring daily quity data rom th priod rom 995 to 7, w mpirically valuat our modl in comparison to th traditional Markowitz modl. Our rsults support th us o ATE as an altrnativ or dining optimal asst allocation and posit that a portolio optimization modl may b adaptd to th individual biass implid in prospct thory. Th rmaindr o this papr contains th ollowing sctions. Sction A discusss th bhavioral biass considrd and dscribs our modl proposing th bhavioral rsampling adjustd tchniqu ATE. Sction prsnts th mpirical study, dscribing th data and implmntation, and providing th rsults. Sction C concluds th rsarch by rviwing th main achivmnts. A Th havioral Modl W prsnt a two priod s modl or portolio choic in a stylizd inancial markt with only two assts, whr th invstor s prrncs ar dscribd by cumulativ prospct thory as suggstd by Kahnman and Tvrsky [979] and Tvrsky and Kahnman [99]. In our ramwork, thr is a inancial markt in which two assts ar tradd. A risklss asst, also calld th bond, and a risky asst, th stock. Lt us considr th rturn o th stock in ach priod givn by th ollowing procss: n, with n ~ N,. Th riskr bond yilds a sur rturn o. W assum that th tim valu o th mony is positiv, i.. that intrst rats ar nonngativ. Th prrncs o th invstor ar basd on changs in walth and ar dscribd by prospct thory. W assum that h owns an initial ndowmnt, W normalizd to montary unit, and that h arns no othr incom. Th agnt invsts a proportion o his walth in th stock and  in th bond. Sinc w want to modl th individual invstor s bhavior, w assum that short slling is not allowd. W also assum that th invstor acts myopically, and th rrnc point rlativ to which h 9
11 masurs his gains and losss in th irst priod is his initial walth. Thn, th prcivd gain or loss in th nd o th irst priod is givn by: x ΔW x x [ W W ] n Eq. W As pointd out in Vlck [6] th choic procss undr prospct thory starts with th diting phas, ollowd by th valuation o ditd prospcts, and inally th altrnativ with th highst valu is chosn. During th diting phas, agnts discriminat gains and losss. Thy also prorm additional mntal adjustmnts in th original probability unction p x, dining th probability wighting unction p. asd on xprimntal vidnc, individuals adjust th liklihood o outcoms such that small probabilitis ar ovrwightd and larg probabilitis ar undrwightd. W will considr th probability wighting unction, as in Giorgi t al. [4] givn by: p, Eq. γ p γ γ p p γ whr γ is th adjustmnt actor. Th ollowing graph compars th valus o p and p, considring γ Figur In th valuing phas, th agnts attach a subjctiv valu to th gambl. Lt us assum th valu unction proposd by Giorgi t al. [4], as ollows: v x x, x, i i x x < Eq. 3 whr is th coicint o absolut risk prrnc, > > maks th valu unction stpr in th ngativ sid lossavrsion, and x is th chang in walth or wlar, rathr than inal stats mntal accounting, as proposd by Kahnman and Tvrsky [979]. Also, th valu unction is concav abov th rrnc point and convx blow it asymmtric risk prrnc. It is usul to considr th prvious orm or th valu unction bcaus o th xistnc o a CAPM quilibrium 3 and th ability to rach constant coicints o risk prrnc. Th prvious ormulation is also
12 supportd by th laboratory rsults rom oshdomènch and Silvstr [3]. Th ollowing graph indicats vx whn.88,.5 and Kahnman and Tvrsky suggstd valus Figur In our twopriod modl or portolio choic, th invstor chooss a wight in th risky asst to maximiz his xpctd utility V. His prrncs ar basd on changs in his walth x and ar dscribd by prospct thory. Th total xpctd valu h addrsss to a givn choic o is givn by: d V v x x dx Eq. 4 dx whr v x is th prospct valu o th outcom x, and x is th wightd cumulativ probability associatd with that outcom. Prospct thory is a dscriptiv thory, postulating that, in comparing altrnativs, th invstor will choos th altrnativ that maks V as high as possibl. Lt us thn valuat th invstor s problm in ach priod. A. First Priod In th irst priod, th agnt s problm consists o dining th allocation o his initial walth btwn th two assts tradd in th inancial markt. H maximizs his utility in t by allocating a raction,, o his initial walth 4, W, in th risky asst and  in th riskr asst. W considr that th invstor is a myopic optimizr in th sns that h taks into account only th irst priod rsult. For multipriod horizons, th choics at arlir dats impact th rrnc points at latr dats. This atur allows or complx modling. Howvr, as pointd out in Shrin [5], prospct thory is a thory about invstors who ovrsimpliy, and so, assuming that individuals ar sophisticatd nough to prciv th link btwn thir currnt choics and utur rrnc points is somthing unrasonabl. W also constrain short slling, as it is common or individual invstors modls. Thus, his problm can b givn by d maxv v x x dx dx Eq. 5
13 Lt us mak th ollowing drivation: n x. arranging th trms in x, w gt n x. W call and C. Thn, Cn x, and so x > implis C n >. Thn, 6 Eq. ˆ C C C C C V n d n d C V n d n d C C V n d n d V x d x d V dx x dx d x v V C C nc C nc C nc C nc C Cn C Cn x x Whr, or th last stp, w usd 5 : z x z x d φ φ ˆ Obsrv that, i w wr considring a standard utility unction riskavrsion ovr all possibl outcoms, th valu would b givn by: C S V Eq. 7 Morovr, th partial drivativs o V Eq. 6 ar:
14 3 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Eq. 9 } ] [ { Eq. 8 ]} ˆ [ { V V As a consqunc, th ollowing proprtis hold 6, i > V ; ii V or or ; iii < V or >. Equations 6 and 7 clarly yild dirnt wights or th risky asst, considring th rmaining paramtrs ixd. Thus, it is possibl to valuat th cost o inicincy associatd with th bhavioral biass as compard to th standard utility solution. [ ] [ ] Eq. Cost PT PT S S whr S is th risky asst wight givn by th standard utility maximization problm, and PT is th stock wight as dind in our modl. Proposition. Th optimal asst allocation in t, or th risky asst * is such that maximizs th valu unction givn by: C C C C C V C whr: [ ] * * and * C. I w wr considring a standard utility unction, th optimal allocation in t, or th risky asst would thn b givn by:
15 * Lt us irst considr standard valus or th modl s paramtrs 7. Th riskr rat quals th historical annual rturn o th US thrmonth Trasury ill.73%. Th quity xpctd rturn and volatility quals th historical avrag o th MSCI global quity indx and its standard dviation 7.6% and.98%. Th adjustmnt actor in th probability wighting unction quals γ.9. Th coicint o riskavrsion quals 3. Also, as suggstd by Kahnman and Tvrsky,.5 and. Th individual s valus prospct thory and standard as a unction o th prcntag o his walth invstd in th risky asst ar givn in Figur 3. Th individual invstor is xpctd to choos th allocation in th risky asst which maximizs his xpctd valu Figur As can b obsrvd rom th graph, using a standard utility unction, th allocation in th risky asst approachs % thta or which th valu unction rachs its maximum, whil using prospct thory utility, th invstor should allocat 8% o his walth in th stock 8. Th shaps o th graphs ar dirnt, notably or larg allocations in th stock. Th valu unction using standard utility is qual to or gratr than th on or prospct utility. Th rason or this dirnc coms rom th act that in prospct thory, ngativ outcoms ar pnalizd mor as ar risky portolios bcaus individuals ar lossavrs >. In th lossavrsion litratur vidnc suggsts that individuals ar around twic mor snsitiv to losss than thy ar attractd to sam siz gains. For small allocations in stocks, th prospct o losss bcoms lss likly and th valu unctions tnd to coincid. latd to th ct o probability wighting, i w st γ, thus cancling out its ct, w rach th ollowing Figur rprsnting th valu unction: Figur
16 Not that th amount optimally invstd by th bhavioral invstor in th risky asst dcrass to 48%, and so probability wighting tnds to incras th risk apptit. Kahnman and Tvrsky [979] suggst that th ovrwighting o low probabilitis has an ambiguous ct on risktaking, as it can induc riskavrsion in th domain o losss, and risksking in th domain o gains. In our cas, th ovrstimation o th xtrm positiv outcoms probabilitis, shown in Figur 3, is inducing invstors to tak mor risk. Howvr, dspit th cts o lossavrsion and probability wighting, vn i w considr and γ, kping constant th rmaining paramtrs, th valu unctions wouldn t coincid, as can b sn in Figur 5: Figur oth modls would prdict that th invstor should allocat % o his ndowmnts in th stock. Howvr, th valu unctions ar dirnt bcaus, in prospct thory, individuals ar risksking in th loss domain asymmtric risk prrnc. Thus, thy would b mor comortabl in allocating a gratr part o thir walth in th risky asst. Th prospct valu unction is gratr than th standard utility unction. Obsrv that th ct o th asymmtric risk prrnc gos in th opposit dirction o lossavrsion and probability wighting. Whn w diminish th coicint o risk prrnc.5 in both utility unctions, w rduc th ct o asymmtry, and so th valu unctions ar much closr, as can b sn in th ollowing igur Figur Th cts o th bhavioral biass can thus b summarizd as ollows: lossavrsion rducs risktaking, and asymmtric risktaking bhavior inducs risky attituds. Probability wighting has an ambiguous ct on risk. Our intuition is that, in th long run, as th valu unction paramtrs ar changing, ths biass tnd to cancl out, liminating th icincy loss originatd by ach bias. That is why w argu that human biass do not nd to b modratd to rach an icint invstmnt stratgy. Th 5
17 xprimntal rsults o lavatskyy and Pogrbna [6] rval that th ct o lossavrsion is largly nutralizd by th ovrwighting o small probabilitis and undrwighting o modrat and high probabilitis. In ordr to vriy proprty i, Lt us valuat V whil changing and kping constant th othr paramtrs considring 5%. Figur 7 prsnts th graph which indicats that ovr all positiv valus o, th slop o V is positiv. Th valu unction is incrasing in. Thus, whn th risky asst has a highr xpctd rturn, ctris paribus implis a highr valu or th invstor: Figur Considring proprtis ii and iii, Lt us valuat V whil changing and kping constant th othr paramtrs considring 5%. Figur 8 prsnts th graph indicating that ovr all positiv valus o, th slop o V is ngativ, whil or, th slop is null. Whn tnds to ininity, th slop tnds to null. Th valu unction is dcrasing in. Th intuition is that, i th volatility o th risky asst is highr, or th sam allocation, this implis a highr probability o losss rducing th valu o th prospct. In lin with traditional rational invstor, bhavioral individuals also prr highr rturn and lowr risk; mainly bcaus thy ar riskavrs in th gain domain and also lossavrs Figur Now lt us valuat th valus o whn w chang th riskr rat and th xpctd rturn o th risky asst. Sinc many paramtrs ar involvd, it is not possibl to ind closd orm solutions or. Thror, w prsnt numrical rsults or th optimal allocation o walth in t. Figur 9 prsnts th rsults or % < < 5% and < 6%. Th rmaining paramtrs ar ixd.98%, 3,.5, < and Figur 9 6
18 As xpctd, whn th risky asst ors mor attractiv rturns, th agnt gradually invsts mor in th stock. Whn th stock is vry attractiv, th invstor chooss to allocat his ntir walth in th risky asst. Thus, w obsrv that is incrasing in and dcrasing in. Also, whn is highr, th changs in du to a variation in ar smoothr, bcaus in ths cass losss ar lss likly and w approach th standard utility solution. Whn is lowr, th changs in du to a variation in ar mor abrupt, giving ris to xtrm portolio allocations. I w considr that is not known with crtainty, th rsulting portolio would b vry unstabl. Goms [3], in a modl with lossavrs invstors, has ound that individuals will not hold stocks unlss th quity prmium is quit high. W can valuat th xpctd cost o inicincy rlatd to th bhavioral biass associatd to th prospct thory unction, or th sam paramtrs considrd in th prvious analysis, using quation. Th rsult is prsntd in Figur, and its orm is du to th act that, in standard utility unction, th invstor is willing to tak mor risk than with th lossavrs prospct utility. Th cost is du to th act that th xpctd rturn o th stock is gratr than th bond, and th standard utility invstor is allocating a gratr part o his walth in th risky asst than th prospct utility individual. Thus, th cost is incrasing in. Howvr, it is worth noting that th prvious cost is basd on xpctd rturns, which ar stochastic in practic. Th ral cost can just b obsrvd at th nd o th irst priod with th ralization o th stock s rturn. An important insight can b mad rom Figur in trms o th bst practic or asst allocation. As long as th riskr rat is lowr and th xpctd rturn o th stock is highr, th optimal allocation should modrat th invstor s biass in ordr to rach a bttr prormanc. On th othr hand, i th risk prmium is lowr, th modration is lss rlvant, and th optimal allocation may adapt to th individual s biass Figur
19 W can also analyz th chang in th allocation o th stock whn w vary th lossavrsion in th risktaking bhavior. Th rsult is shown in Figur, or < < 4. Obsrv that, as long as th invstor is much mor avrs to losss than h is attractd to gains, th allocation in th risky asst is lowr. Whn. 5, th allocation in th risky asst corrsponds to 8%, as prviously mntiond Figur Dimmock[5] has alrady shown that a highr lvl o lossavrsion lads to lowr quity xposur, and htrognity in th coicint o lossavrsion has th ability to xplain puzzling aturs o houshold inancial bhavior. A. Scond Priod In ordr to valuat th scond priod allocation choic o th invstor, Lt us kp som paramtrs ixd:.98, 3,.5 and. Atr th invstor has mad his irst priod dcision in t, th stat o natur ralizs in t, whn h is acd with his scond priod problm. Again, h must allocat his walth in th two possibl assts in th inancial markt, bond and stock, to maximiz his utility. Lt us considr th sam normal distribution or th rturn o th risky asst. Th invstor s walth position at t quals his position in t plus th rturn o his portolio in th scond priod. Whil all agnts solv th sam maximization problm in th irst priod, in th scond priod, it will dpnd on th rrnc point to which h masurs his gains and losss in th ramwork o prospct thory. In our modl, thr ar two candidats or th invstor s rrnc point at t : his initial walth at t W or his walth at th nd o th irst priod, t W. I h masurs his gains and losss rlativ to his walth at t his currnt walth, h trats ach gain and loss sparatly. On th othr hand, i h considrs his initial walth as th rrnc point, h adds up th outcoms gains and losss, that is, h nts his positions. Th prvious distinction is rlvant in prospct thory. Th valu unction is concav in th domain o gains and convx in th loss domain asymmtric risk bhavior. First, Lt us considr as th invstor s rrnc point his currnt walth at t. In this cas, th maximization problm h will solv in th scond priod is th sam as th on or th irst priod. Thus, w can stat th ollowing proposition. 8
20 * Proposition. Th optimal asst allocation in t, or th risky asst, i th agnt masurs his gains and losss rlativ to his currnt walth, is such that maximizs th * sam valu unction o th irst priod. * W can obsrv that an individual who masurs his gains and losss rlativ to his currnt walth is actually solving th sam maximization problm in ach priod. That is why th allocation in th risky asst might b th sam. This is not surprising; as h is not using past inormation to updat his blis about th assts, his prrncs ar similarly unactd. Nxt, lt us analyz th invstor s maximization problm i h valuats his gains and losss rlativ to his initial walth. I h has an initial walth position o W and his walth riss in th irst priod to W and alls in th nxt priod to W 5, h valus his position at t as a gain o 5, and not as a gain o ollowd by a loss o 5. In th scond priod, th agnt s problm consists o dining th allocation o his walth W btwn th two assts tradd in th inancial markt. H maximizs his utility in t by allocating a raction,, o his walth W in th risky asst and  in th risklss asst. As w did in th irst priod analysis, w also constrain short slling. d maxv v x x dx dx Lt us mak th ollowing drivation: x W [ n ] W and W W [ ], whr priod. So x W [ ] is th rturn o th stock in th irst [ n ] arranging th trms in x and considring W, w gt x [ n ] Lt us call [ ] [ ] ] [. 9
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