How To Understand Propect Theory And Mean Variance Analysis

Transcription

1 Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 Enrco De Gorg (Swtzerland ), Thorten Hen (Swtzerland) Propect theory and mean-varance analy: doe t make a dfference n wealth management? Abtract We how that propect theory a valuable paradgm for wealth management. It decrbe well how nvetor perceve rk and wth approprate modelng t can be made content wth ratonal decon makng. Moreover, t can be repreented n a mple reward-rk dagram o that the man dea are ealy communcated to clent. Fnally, we how on data from a large et of prvate clent that there are conderable monetary gan from ntroducng propect theory ntead of mean-varance analy nto the clent advory proce. Keyword: behavoral fnance, propect theory, rk profle, mean-varance analy. JEL Clafcaton: D03, D14, D81, G11. Introducton Behavoral Fnance reearcher have amaed evdence that the propect theory of Kahneman and Tverky (1979) provde a better decrpton of nvetor choce than the mean-varance model of Markowtz (1952). For recent urvey of th evdence ee, for example, Camerer (1995), De Bondt (1998) and Barber and Thaler (2003). Propect theory ha been appled to explan low partcpaton n equty market (Benartz and Thaler, 1995; Barber, Huang and Thaler, 2006), the dpoton effect (Shefrn and Statman, 1985) 1, nuffcent dverfcaton (Barber, Huang and Thaler, 2006), hgh tradng actvte (Gome, 2005) and nvetor preference for potvely kewed payoff dtrbuton (Barber and Huang, 2008). Depte the growng conenu among reearcher that propect theory uperor to mean-varance for decrbng ndvdual preference, the mean-varance model of Markowtz (1952) reman the ndutry tandard n wealth management. The adherent of mean-varance analy do not adapt propect theory for (at leat) the followng reaon: (I) propect theory aocated wth rratonal decon whle mean-varance analy beleved to lead to ratonal decon; (II) propect theory more complcated than mean-varance analy whch mpoe burden both on the computatonal kll and on the communcaton of the analy to clent; (III) under tandard mplfyng aumpton lke normally dtrbuted return, propect theory and mean-varance analy almot concde hence belever n normalty ee no pont to adopt propect theory. Enrco De Gorg, Thorten Hen, We are grateful to Mla Wnter and Deter Nggeler for ther reearch atance. 1 The dpoton effect the obervaton that nvetor tend to hold loer poton too long and ell wnner poton too earler. Recently, Hen and Vlceck (2005) and Barber and Xong (2008 a) how that propect theory doe not explan the dpoton effect. Barber and Xong (2008 b) ntroduce a model where nvetor get utlty from realzed gan and loe and ue th model to explan the dpoton effect. 122 The goal of th paper to addre all the above argument n favor of mean-varance analy and to how that none of them are well hold. We alo go further and how that the uperorty of propect theory compared to mean-varance for decrbng ndvdual preference tranlate nto a gnfcant addtonal monetary value to real-world nvetor. Hence we argue that propect theory can be ntroduced a a worth-whle nnovaton n wealth management. We ntroduce the man reult of th paper n three part. Frtly, t may well be that propect theory lead to ratonal decon whle mean-varance analy doe not. Indeed, mean-varance preference mght lead to volaton of monotoncty,.e., mean-varance nvetor mght dplay a preference for maller payoff when return are not normally dtrbuted. Th the cae wth a large number of aet clae, from tock to alternatve nvetment and tructured product. Secondly, propect theory can be formulated n a mple reward-rk way mlar to mean-varance analy,.e., the propect theory analy can be dplayed n a mple reward-rk dagram whch can be nterpreted n the ame way a the meanvarance dagram,.e., hgher reward mple hgher rk and optmal portfolo are thoe whch maxmze reward gven a rk contrant. Moreover, nce propect theory decrbe how nvetor perceve rk, the nteracton between clent and fnancal advor facltated by propect theory, a the rkreward dagram more meanngful to the clent. Fnally, we do an emprcal exerce to how that mean-varance portfolo are neffcent for real nvetor, who are n fact bet decrbed by propect theory. Th due to the fact that the oberved dtrbuton of return trongly devate from normalty even for tandard aet clae. Thu mean-varance analy and propect theory do not concde n realworld applcaton. We how that the added value delvered to clent when ung propect theory n-

2 tead of mean-varance hgh enough to jutfy the effort of ntegratng propect theory nto the wealth management proce 1. Alo, clent mght be wllng to pay an addtonal fee for the ervce offered, gven the uperorty of propect theory aet allocaton from ther perpectve. The remander of the paper tructured a follow. Secton 1 ntroduce propect theory. Secton 2 preent an emprcal analy where we compare propect theory wth the mean-varance analy. The lat ecton conclude. 1. Rk from a Behavoral Fnance perpectve Whle the Modern Portfolo Theory of Markowtz (1952) evolved a a top-down proce whch wa nfluenced by the lmted mathematcal ablte of the 1950 (ee Markowtz, 1991), Behavoral Fnance ha been developed a a bottom-up proce by the fndng of nnumerable controlled laboratory experment. In the Behavoral Fnance baed rk theory, the propect theory of Kahneman and Tverky (1979), averon to loe more mportant than averon to volatlty, whch wa potulated by Markowtz (1952) a the only meaure of rk 2. Moreover, t oberved that nvetor are rk avere when comparng two gan, and rk eekng when they can chooe between a ure lo and a gamble whch gve them the chance to break-even. Fnally, propect theory depart from mean-varance analy nce the former allow nvetor to overweght mall probablte n ther decon. Snce th latter apect however alo a departure from ratonal choce, a formalzed by expected utlty, we wll not conder t here. Hence, the recommendaton baed on propect theory we conder are content wth ratonal choce. Under th aumpton propect theory decrbed by a value functon mlar to the rk utlty of von Neumann and Morgentern The propect theory value functon. The propect theory value functon ha three mportant properte: It defned over gan and loe wth repect to ome natural reference pont. It concave n gan and convex for loe. The functon teeper for loe than for gan. 1 The effort for ntegratng propect theory manly refer to the aet allocaton de. Indeed, propect theory requre more advanced optmzaton technque. However, the current technology allow to olve the propect theory aet allocaton n few econd alo wth large opportunty et. 2 Markowtz (1959) uggeted em-varance a meaure of rk, whch only account for negatve devaton from the mean. Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 Thee properte of the value functon are llutrated graphcally n Fgure 1, where x repreent a gan or a lo wth repect to ome (ubjectve) reference pont and v( x) the propect utlty derved from th gan or lo. Tverky and Kahneman (1992) have propoed the followng pecewe-power value functon: v x x for x 0. x for x 0 Baed on expermental evdence they ugget that the medan rk and lo averon of ndvdual are = 0.88 and = 2.25, repectvely 3. Th value functon ha been under fre both theoretcally (De Gorg, Hen and Levy, 2003; Köbberlng and Wakker, 2005; Reger, 2007) and emprcally (De Gorg, Hen and Pot, 2007). For example, propect theory wth a pecewe-power value functon doe not lead to robut aet allocaton,.e., lght dfference n nvetor lo or rk averon lead to ubtantally dfferent optmal aet allocaton. Conequently, for applcaton of propect theory to portfolo electon, other value functon have been ued; ee, for example, De Gorg and Hen (2006) and Hen and Bachmann (2008, Chapter 2.4.1). Note: The x-ax report gan and loe, whle the y-ax report the correpondng propect theory value. The orgn the reference pont,.e., the reference pont ha zero value. Fg. 1. The propect theory value functon Gven the value functon v(x), the propect theory decon crteron decrbed a follow: For any et of cenaro 1,..., S occurrng wth probabl- 3 Experment alo how a hgh degree of heterogenety between partcpant. In our emprcal analy reported n Secton 3 we don t ue medan value of lo and rk averon, but propect theory calbrated to each ndvdual nvetor. 123

3 Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 te p 0, 1,..., Sa decon leadng to the payoff 124 x, 1,..., S preferred to one leadng to alternatve payoff y, 1,..., S f and only f S 1 1 S pv( x) pv( y ). Hence, nce the value functon ncreang, propect theory content wth expected utlty theory and thu lead to ratonal decon. On the other hand, wth general payoff dtrbuton mean-varance analy doe not lead to ratonal decon nce hgher payoff may come along wth hgher varance, whch the o-called mean-varance paradox. For example the bnary lottery delverng a payoff y > 0 wth a potve probablty p > 0 whle havng a zero payoff otherwe wll not be preferred by mean-varance nvetor to the ure payoff of zero f the probablty p tend to zero whle y tend to nfnty and the expected value py kept contant. In other word, a mean-varance nvetor may not take a potve payoff even t wthout payment (.e., free), a he perceve rk trctly a varance. Note that th of hgh practcal relevance nce applyng the meanvarance crteron to tructure product may mply a tuaton a n the mean-varance paradox. A mple tructured product wth captal protecton ha no potental for loe but tll t ha a potve varance, and may thu be undervalued by meanvarance analy A reward-rk perpectve on propect theory. A fundamental prncple n fnancal economc that very ueful n the communcaton wth clent that there no reward wthout rk. In the mean-varance framework, the reward-rk tradeoff mplemented ung the dea that nvetor who dere to ncreae the expected return of ther nvetment mut accept return whch devate more trongly from the mean. Actually, the real groundbreakng dea of Markowtz (1952) wa the uggeton of a mple reward-rk dagram. That he had choen the mean return for the reward and the tandard-devaton for the rk ax wa more for convenence, becaue at the tme t wa not poble to effcently deal wth hgher moment of the return dtrbuton. Here we ugget a dfferent perpectve on mplementng the reward-rkprncple (ee De Gorg, Hen and Mayer (2006) for a detaled decrpton). From the nvetor pont of vew, the reward of an nvetment not t expected return a n the mean-varance analy but the expected return over h reference pont, t average gan. It defned a the um of all portfolo return over the nvetor reference pont, weghted wth the correpondng probablte a perceved by the nvetor. More precely, the average gan defned a: pt S 1 0 p v R RP, where RP the nvetor reference pont, R the return of the portfolo n tate and RP. Repectvely, the rk of the nvetment not the devaton from the expected return a n the meanvarance analy but the expected portfolo return below the nvetor reference pont. Th the portfolo average lo,.e. pt 1 S 1 0 p v RP R, where, a above, the nvetor lo averon. Note that average gan and average loe are expreed n utlty term to account for nvetor rk atttude over gan and loe. Moreover, the average lo multpled by mnu one to obtan a potve meaure of rk. Fnally, the average lo normalzed by the nvetor lo averon, nce doe not decrbe nvetor atttude on loe, but the nvetor tradeoff between gan and loe. Indeed, the utlty over the average gan and loe PT pt pt and play the ame role of varance averon n the mean-varance model. Graphcally, the return-rk perpectve can be repreented a n Fgure 2. Hence changng the degree of lo averon,.e., the lope of the traght lne n Fgure 2, dfferent portfolo on the propect theory effcent fronter can be elected. Note: Rk meaured a the abolute value of the normalzed value of lo, whle reward the value of gan. The trade-off gven by lo averon, whch determne the optmal aet allocaton on the propect theory effcent fronter. Fg. 2. Reward-rk dagram of propect theory R

4 2. Propect theory and mean-varance analy Even when return are normally dtrbuted, t not clear whether propect theory and mean-varance analy delver the ame et of effcent portfolo. Indeed, whle t clear that propect theory decon only depend on mean and varance when return are normally dtrbuted (nce any rk utlty ntegrated over a normal dtrbuton only depend on mean and varance), n general, propect theory doe not mply varance averon nce propect theory nvetor are rk eekng over loe. Levy and Levy (2004) how that the propect theory effcent et a trct ubet of the mean-varance effcent et under the condton that return are normally dtrbuted and portfolo are formed wthout retrcton, e.g., no hort-ale contrant. Moreover, n th cae, the ubet of mean-varance effcent portfolo whch are propect theory neffcent mall. However, t well known that for mot aet the aumpton of normally dtrbuted return ha weak emprcal upport. Moreover, ndvdual nvetor often face hort-ale contrant. Therefore, we could expect relevant dfference between the propect theory effcent et and the meanvarance effcent et when more realtc aumpton are made concernng return dtrbuton and portfolo retrcton. Whether th dfference large depend, for example, on how return dtrbuton depart from the normal dtrbuton and on how hgher moment of the dtrbuton mpact the propect theory value functon. The latter pont obvouly alo related to the nvetor degree of lo averon and rk averon or rk eekng behavor on gan and loe repectvely. Our emprcal analy addree the followng queton: Aumng that propect theory the correct model to decrbe nvetor preference, what the added value n monetary term when an nvetor chooe an optmal portfolo from the propect theory effcent et ntead of choong from the mean-varance effcent et? A dcued before, Levy and Levy (2004) how that n the cae of normally dtrbuted return and no retrcton on portfolo, th added value zero,.e., the optmal aet allocaton for a propect theory nvetor belong to the mean-varance effcent et. Therefore, Levy and Levy (2004) conclude that a propect theory nvetor hould not determne the propect theory effcent et (a th more complex to do), but mply optmze the propect theory value functon over the mean-varance effcent et. Doe th reult hold n general? We ue data from 792 prvate nvetor. For each nvetor n our dataet we calbrated an extended veron of the propect theory value functon ung Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 the BhFS rk profler 1,2. We denote by V the calbrated propect theory value functon for nvetor 1,...,792. Therefore, we do not ue the medan parameter, a uual n the behavoral fnance lterature, a the nvetor dplay a hgh degree of heterogenety n our dataet. Ung the calbrated value functon we calculated for each nvetor two dfferent aet allocaton: 1) the optmal propect theory aet allocaton from the propect theory effcent et (portfolo 1); 2) the aet allocaton on the mean-varance effcent et wth the hghet value gven the nvetor propect theory value functon (portfolo 2). The optmzaton algorthm to fnd the optmal aet allocaton n the propect theory effcent et decrbed n De Gorg, Hen and Mayer (2007). For nvetor 1,..., 792, we denote by R 1 the (random) return of portfolo 1 above and by R 2 the (random) return of portfolo 2. Snce any potve lnear tranformaton of a value functon delver the ame optmal aet allocaton a the orgnal value functon, the dfference n utlty level can be made a mall a poble for any two portfolo, and thu dfference n utlty are not very nformatve. Moreover, utlty level for two dfferent nvetor cannot be compared n general. Therefore, for each nvetor we compare the certanty equvalent of the two optmal aet allocaton ntead of the propect theory value. The certanty equvalent correpond to the rk-free payoff that delver the ame propect theory value a the rky portfolo,.e., the rk-free return r uch that V (r) = V (R), where R the (random) return of the rky portfolo. Note that certanty equvalent are not affected by any potve lnear tranformaton of the value functon. For k 1, 2 and 1,..., 792 let r k be the certanty equvalent of portfolo k. Obvouly, r1 r2 for all, nce portfolo 1 propect theory effcent and V an ncreang functon. We call the dfference r r1 r2 the added value n monetary term for ung portfolo 1 ntead of portfolo 2. Fgure 3 how the dtrbuton of annualzed monetary added value n bae pont (bp) for our dataet wth 792 nvetor. 1 BhFS tand for Behavoral Fnance Soluton, a pn off frm of the Unverty of Zurch that tranfer reearch n Behavoral Fnance nto the bankng ndutry, for detal ee: 2 The extenon of propect theory ued n th work relate to a dfferent pecfcaton of the value functon n order to olve the robutne problem of the pecewe-power functon uggeted by Tverky and Kahneman (1992). 125

5 Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, r Fg. 3. Dtrbuton of monetary added value baed on a comparon between the propect theory effcent portfolo and mean-varance effcent portfolo wth the hghet propect theory value The average added value from nvetng n the propect theory optmal portfolo (portfolo 1) ntead of the optmal aet allocaton on the mean-varance effcent fronter (portfolo 2) 0.88 bp. About 10% of the nvetor experence an added value whch larger than 2 bp, whle about 1% of the nvetor have an added value between 5 and 24 bp per annum. We run a mple lnear regreon model to analyze the relatonhp between nvetor charactertc (reference pont, lo averon, rk averon) and the monetary added value for holdng the propect theory effcent portfolo ntead of the meanvarance portfolo. We found that lo averon, the reference pont and the apraton level have a tattcally gnfcant mpact on r 1. The dfference between the apraton level and the reference pont potvely related to nvetor rk averon. When th dfference hgher, nvetor dplay, on average, a hgher rk tolerance nce they are wllng to take more rk on order to acheve a hgher average return above the reference pont. In our dataet, the dfference between the apraton level and the reference pont, a well a the apraton level telf, are potvely related to the added 1 The apraton level dffer from the reference pont and ued to calbrate nvetor rk averon. The apraton level hgher than the reference pont and determne the average gan above the reference pont the nvetor want to acheve. value, whle lo averon negatvely related to t. To ummarze, nvetor who have a hgher apraton level dplay a hgher rk and lo tolerance, alo obtan a hgher added value from the propect theory effcent portfolo compared to the mean-varance portfolo. For comparon, nvetor n both the lowet 20% quantle for lo averon and the hghet 10% quantle for the apraton level, have an average added value of 1.62 bp (almot twce the average added value over the whole ample). In contrat, nvetor both n the hghet 20% quantle for lo averon and the lowet 10% quantle for the apraton level, have an average added value of 0.70 bp (whch le than the average added value over the whole ample). Whle thee number are mall, conder a bank whch ue the propect theory approach for the 792 clent n our dataet ntead of choong clent optmal portfolo from the mean-varance effcent fronter. Suppoe that a typcal prvate bankng clent hold $1 mllon, reman at the bank for at leat fve year, and aet allocaton are updated annually. Then aumng an nteret rate of 2%, the preent value of the total added value from ung the propect theory effcent portfolo ntead of the meanvarance portfolo $324'950. Settng th n relaton to the current aet under management of $792 mllon, we fnd that the bank could ak an addtonal fee of 4 bp f t ue propect theory effcent portfolo

6 ntead of mean-varance portfolo. Put nto a dfferent perpectve, the added value delvered to a large number of clent worth the addtonal cot of mplementng the propect theory approach. In typcal mplementaton of mean-varance analy n wealth management, clent profle are mapped nto a few mater portfolo on the mean-varance effcent fronter. Rk profle quetonnare baed on mean-varance are degned accordngly. One reaon for th mght be that t dffcult to calbrate clent varance averon, partally becaue varance dffer from nvetor percepton of rk. A (peronalzed) optmal aet allocaton uperor only for thoe nvetor who are able to expre ther volatlty averon, and who advor able to accurately calbrate ther volatlty averon n the frt place. Recently, Da et al. (2008) have propoed a way to determne nvetor averon to varance tartng from a noton of rk whch more famlar to nvetor,.e., the poblty of mng a gven target return or reference pont. Da et al. (2008) tate that nvetor are better calbrated about ther tolerated probablty of mng the target return than about ther varance averon. Smlarly, the BhFS rk profler ued to calbrate the propect theory value functon addree nvetor ung ther own noton of rk, e.g., loe below a target return or reference pont. Therefore, we expect that a move from the current cenaro to a propect theory approach wth Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 more accurate rk proflng offer even more value than what decrbed above. Aumng that propect theory the correct model to decrbe clent preference and that propect theory well calbrated gven that t ncorporate nvetor noton of rk, we now addre the queton of the added value of ung propect theory effcent portfolo ntead of mater portfolo on the mean-varance effcent fronter. We defne fve mater portfolo on the meanvarance effcent fronter and ung the calbrated value functon we obtan for each nvetor n our dataet the mater portfolo wth the hghet propect theory value (portfolo 3). For nvetor 1,...,792, we denote by R 3 the (random) return of portfolo 3 and by r 3 the correpondng certanty equvalent. Recall that r 1 the certanty equvalent of the optmal aet allocaton from the propect theory effcent et. Agan, r1 r3 and for each nvetor 1,..., 792 we defne the added value n monetary term for ung the optmal aet allocaton from the propect theory effcent et ntead of ~ mean-varance mater portfolo a r r. r 1 3 Fgure 4 how the dtrbuton n our dataet of the annualzed value for ~ r n bp. Fg. 4. Dtrbuton of monetary added value (n ba pont) baed on a comparon between propect theory effcent portfolo and mater portfolo on the mean-varance effcent fronter 127

7 Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, 2009 Agan, conder a bank wth our 792 clent and uppoe that the average wealth of a prvate bankng clent $1 mllon. Then, the average added value n dollar term when ung the propect theory optmal allocaton ntead of mean-varance mater portfolo correpond to $ The total added value for all 792 clent $ If clent update ther aet allocaton annually, th fgure refer to annual gan. If clent reman at the bank at leat fve year, we fnd that the bank could ak an addtonal fee of 52 bp f t ue propect theory effcent portfolo ntead of mean-varance mater portfolo. Concluon We ee propect theory a a major breakthrough n decon theory that f modeled carefully can mprove on the mean-varance analy n partcular wth repect to wealth management applcaton uch a clent rk proflng and creatng clent aet allocaton. A careful modelng of propect theory can enure that the decon baed on t are fully ratonal even n the cae of non-normally dtrbuted return Reference 128 lke that of tructured product where meanvarance analy found to fal. Moreover, nothng lot n term of mplcty a propect theory can ealy be formulated n a reward-rk way mlar to mean-varance analy, and thu a mple tool to communcate wth clent can be ued. Fnally, to upport our qualtatve argument, we meaured the clent added value from holdng a propect theory optmal portfolo a compared to a mean-varance aet allocaton. Ung real data, we how that a bank followng the propect theory approach could ncreae t management fee wth 4 bp or 52 bp dependng on the degree of peronalzaton the bank offer to t clent when determnng ther optmal aet allocaton. Note that offerng peronalzed aet allocaton depend on havng the nvetor preference well calbrated. Th poble wth propect theory, beng a model that ue nvetor noton of rk, and more dffcult wth mean-varance. Dependng on a bank aet under management, t can expect to gan conderably from ung propect theory ntead of mean-varance analy. 1. Barber, Nchola and Mng Huang (2008), Stock a Lottere: The Implcaton of Probablty Weghtng for Securty Prce, Amercan Economc Revew, Vol. 98, Barber, Nchola and Rchard H. Thaler (2003), A Survey of Behavoral Fnance, In Handbook of the Economc of Fnance, edted by George M. Contantnde, Mlton Harr, and Rene' Stultz, Elever Scence, North Holland, Amterdam. 3. Barber, Nchola, Mng Huang, and Rchard H. Thaler (2006), Indvdual Preference, Monetary Gamble, and Stock Market Partcpaton: A Cae for Narrow Framng, Amercan Economc Revew, Vol. 96, Barber, Nchola and We Xong (2008 a), What Drve the Dpoton Effect? An Analy of Long-Standng Preference-Baed Explanaton, Journal of Fnance, forthcomng. 5. Barber, Nchola and We Xong (2008 b), Realzaton Utlty, Workng paper, Yale Unverty. 6. Camerer, Coln (1995), Indvdual Decon Makng, In Handbook of Expermental Economc, edted by John H. Kagel and Alvn E. Roth, Prnceton Unverty Pre, Prnceton. 7. Da, Sanjv R., Harry Markowtz, Jonathan Sched and Mer Statman (2008), Portfolo Optmzaton wth Mental Account, Journal of Fnancal and Quanttatve Analy, forthcomng. 8. De Bondt, Werner (1998), A Portrat of the Indvdual Invetor, European Economc Revew, Vol. 42, De Gorg, Enrco G. and Thorten Hen (2006), Makng Propect Theory Ft for Fnance, Fnancal Market and Portfolo Management, Vol. 20, De Gorg, Enrco, Thorten Hen and Jano Mayer (2007), Computatonal Apect of Propect Theory wth Aet Prcng Applcaton, Computatonal Economc, Vol. 29, De Gorg, Enrco, Thorten Hen and Jano Mayer (2006), A Behavoural Foundaton of Reward-Rk Portfolo Selecton and Aet Allocaton Puzzle, Workng paper, Unverty of Lugano and Unverty of Zurch. 12. De Gorg, Enrco, Thorten Hen and Therry Pot (2008), Propect Theory and the Sze and Value Premum Puzzle, mmeo, Unverty of Lugano and Unverty of Zurch. 13. Gome, Francco (2005), Portfolo Choce and Tradng Volume wth Lo Avere Invetor, Journal of Bune, Vol. 78, Hen, Thorten and Kremena Bachmann (2008), Behavoural Fnance for Prvate Bankng, Wley & Son. 15. Hen, Thorten and Martn Vlcek (2005), Doe Propect Theory Explan the Dpoton Effect?, Workng paper, Unverty of Zurch. 16. Kahneman, Danel and Amo Tverky (1979), Propect Theory: An analy of Decon Under Rk, Econometrca, Vol. 47, Köbberlng, Veronka, and Peter Wakker (2005), An Index of Lo Averon, Journal of Economc Theory, Vol. 122, Levy Mohe and Ham Levy (2004), Propect Theory and Mean-Varance Analy, Revew of Fnancal Stude, Vol. 19,

8 Invetment Management and Fnancal Innovaton, Volume 6, Iue 1, Markowtz, Harry (1952), Portfolo Selecton, Journal of Fnance, Vol. 7, Markowtz, Harry (1991), Foundaton of Portfolo Theory, Nobel Lecture. 21. Reger, Marc Olver (2007), Too Rk Avere for Propect Theory, Workng paper, NCCR-FINRISK. 22. Shefrn, Herch and Mer Statman (1985), The Dpoton to Sell Wnner too Early and Rde Loer too Long, Journal of Fnance, Vol. 40, Tverky, Amo and Danel Kahneman (1992), Advance n Propect Theory, Cumulatve Repreentaton of Uncertanty, Journal of Rk and Uncertanty, Vol. 5, Von Neumann, J., and O. Morgentern (1944), Theory of Game and Economc Behavor, Prnceton Unverty Pre. 129