Measuring the degree to which probability weighting affects risk-taking. Behavior in financial decisions

Transcription

1 Joural of Face ad Ivestmet Aalyss, vol., o.2, 202, -39 ISSN: (prt verso), (ole) Iteratoal Scetfc Press, 202 Measurg the degree to whch probablty weghtg affects rsk-takg Behavor facal decsos Fabo Mattos ad Phlp Garca 2 Abstract The paper vestgates the mportace of probablty weghtg facal decsos ad exames the degree to whch rsk-takg behavor devates from expected utlty theory the presece of probablty weghtg. A group of professoal traders partcpates a expermet, whose data are used to calculate rsk ad ucertaty premums. Ths framework allows measurg ad dsetaglg the mpact of probablty weghtg o rsk perceptos ad behavor. Several fdgs emerge. Professoal traders exhbt probablty weghtg whch has a substatal ad heterogeeous effect o behavor. Probablty weghtg affects traders perceptos of ther ow rsk atttude more tesely tha t affects ther actual behavor. Fally, rsk-averse or rsk-seekg behavor s more tese uder codtos of ucertaty tha t s uder codtos of rsk. These fdgs Uversty of Matoba, e-mal: fabo_mattos@umatoba.ca 2 Uversty of Illos at Urbaa-Champag, e-mal: p-garca@llos.edu Artcle Ifo: Receved : February, 202. Revsed : March, 202 Publshed ole : May 3, 202

2 2 Measurg the Degree to whch probablty weghtg affects rsk-takg are cosstet wth prevous studes, but provde ew sghts o several dmesos of tradg decsos, ad offer sghts to market movemets. JEL classfcato umbers: D03, D8, G02 Keywords: probablty weghtg, behavor, rsk premum, rsk percepto, traders Itroducto The tradtoal framework to vestgate behavor s the expected utlty theory. However, expermetal evdece has demostrated ts assumptos are ofte volated whe decsos are made uder codtos of rsk (Schoemaker []; De Bodt ad Thaler [2]; Starmer [3]; Hrschlefer [4]; Barbers ad Thaler [5]). As a cosequece, researchers have developed alteratve theores to expla choce. I facal applcatos prospect theory developed by Kahema ad Tversky [6] ad Tversky ad Kahema [7] appears to offer the most promsg o-expected utlty theory for explag decso makg (Barbers ad Thaler [5]). Prospect theory dffers from the expected utlty paradgm that choce s flueced by probablty weghtg ad loss averso. Probablty weghtg reflects the oto that decso makers use trasformed probabltes rather tha objectve probabltes to make choces. Loss averso posts that decsos are made terms of gas ad losses rather tha fal wealth, ad dvduals react dfferetly to gas ad losses. The choce model uder prospect theory has two fudametal compoets: a weghtg fucto that reflects a o-lear trasformato of probablty, ad a value fucto that corporates loss averso. Several studes suggest that probablty weghtg plays a mportat role behavor. Tversky ad Kahema [7] dscuss a fourfold patter of decso makg frequetly foud emprcal work,.e. rsk averso for gas ad rsk seekg for losses at hgh probabltes, ad rsk seekg for gas ad rsk

3 Fabo Mattos ad Phlp Garca 3 averso for losses at low probabltes. Ths patter caot be explaed solely by a value fucto; probablty weghtg must be corporated (Tversky ad Kahema [7]). I facal settgs Fox et al. [8] coduct a laboratory expermet wth professoal traders stock optos markets ad fd that ther decsos exhbt probablty weghtg. Lager ad Weber [9] demostrate that probablty weghtg ca make dvduals behave dfferetly tha ther rsk prefereces suggest. They show behavor characterzed by rsk averso for gas ad rsk seekg for losses, as mpled by a typcal value fucto, ca chage dramatcally whe probablty weghtg s take to accout. Further, they provde evdece that models corporatg probablty weghtg are more cosstet wth observed behavor, whch s also le wth Blavatskyy ad Pogreba [0], Daves ad Satchell [], ad Mattos et al. [2]. Despte the mportace of probablty weghtg decso makg, o attempt has bee made to measure the degree to whch behavor of real decso makers ca chage ts presece. A umber of studes use laboratory expermets to elct weghtg fuctos ad estmate ther parameters. Whle estmated parameters provde formato o the magtude of devato from objectve probabltes, they do ot offer a measure of how rsk-takg behavor chages the presece of probablty weghtg. I addto, prevous studes fal to address mportat ssues. Apart from Fox et al. [8], o studes have performed expermets wth professoal traders to obta ther value ad weghtg fuctos. Ths s a mportat pot sce expermets wth studets ad real decso makers ca yeld dstct results (Hagh ad Lst [3]; Alevy et al., [4]). The paper vestgates the mportace of probablty weghtg amog professoal traders ad exames the degree to whch rsk averso s modfed the presece of probablty weghtg. We use a group of fftee propretary traders the CME Group ad a ovel method focusg o rsk ad ucertaty premums. Ths approach s based o Daves ad Satchell [5], who theoretcally vestgate rsk premums to descrbe the degree of rsk averso. I the preset

4 4 Measurg the Degree to whch probablty weghtg affects rsk-takg study the approach s used to emprcally explore the mportace of probablty weghtg usg real decso makers. Traders partcpate a expermet, whose outcomes provde formato about ther rsk atttude ad degree of probablty weghtg. The tradeoff method adopted by Abdellaou [6] ad Abdellaou et al. [7] s used to elct value ad weghtg fuctos uder rsk (whe probabltes of ucerta evets are kow) ad ucertaty (whe probabltes of ucerta evets are ukow). Based o ther elcted value ad weghtg fuctos, rsk ad ucertaty premums are calculated to detfy the mpact of probablty weghtg o behavor. Three premums are calculated: expected utlty () premum, stadard premum ad behavoral premum. The premum s the tradtoal rsk premum whch assumes that probabltes are treated learly. The stadard premum cosders the effect of probablty weghtg o rsk atttude ad reflects whether dvduals perceve themselves to be rsk averse or rsk seekg. The behavoral premum llustrates actual behavor whe probablty weghtg ad the value fucto fluece decsos. Ths paper cotrbutes to the lterature two ways. Frst, t gathers data from expermets wth real decso makers (professoal traders) to dscuss tradg behavor ad uses them to emprcally vestgate the degree to whch probablty weghtg affects tradg decsos. The framework of rsk ad ucertaty premums offers a ovel approach that allows measurg ad dsetaglg the mpact of probablty weghtg o rsk perceptos ad behavor. Secod, results provde sghts o several dmesos of how behavor devates from expected utlty theory due to probablty weghtg. These fdgs ca help uderstad dvdual tradg behavor ad provde sghts to market movemets. 2 Theoretcal Framework Prospect theory s used to vestgate tradg behavor. The choce model s

5 Fabo Mattos ad Phlp Garca 5 based o a fucto V( x ) wth two compoets (equato ): a value fucto v ad a probablty weghtg fucto wp ( ), where x s the x argumet of the value fucto, ad p s the objectve probablty dstrbuto of x. V( x ) v( x ) w( p ) () The value fucto measures value terms of gas ad losses (chages wealth) wth respect to a referece pot. The shape that typcally arses from prospect theory s s-shaped, allowg for rsk-averse behavor (cocavty) the doma of gas (x>0), ad rsk-seekg behavor (covexty) the doma of losses (x<0) (Fgure ). 3 Rsk seekg the loss doma has emprcal support ad arses from the dea that dvduals dslke losses to such a degree (loss averso) that they are wllg to take greater rsks to make up ther losses. Value Fucto w(p) Weghtg fucto p Fgure : Prospect Theory s value ad weghtg fuctos 3 Fgure assumes that the referece pot s zero.

6 6 Measurg the Degree to whch probablty weghtg affects rsk-takg A secod compoet of prospect theory s a probablty weghtg fucto, whch was developed from observatos that dvduals do ot treat probabltes learly. Emprcal evdece shows probabltes ca be overweghed or uderweghted, meag dvduals make decsos based o perceved probabltes that are ether larger or smaller tha really exst. For example, Fgure shows the weghtg fucto of a perso who cosstetly uderweghs probabltes, meag that w p p for the whole probablty scale. 4 If the dvdual s able to clearly dstgush probabltes ad use them objectvely, there s o curvature the weghtg fucto, represeted by the lear dotted le Fgure. I ths stuato, w p p equato () ad rsk-takg behavor s determed solely by the rsk prefereces the value fucto. However, whe objectve probabltes are ot used, the w p p ad decsos are based o trasformed probabltes ad the value fucto. The effect of the weghtg fucto decso makg depeds o ts structure ad stregth. For stace the weghtg fucto Fgure depcts a dvdual who uderestmates the lkelhood of ucerta evets ad thus beleves that probabltes are smaller tha actual. I ths stuato a perso s less wllg to take rsks. Now, cosder the value fucto Fgure, whch shows rsk averso for gas ad rsk seekg for losses. I ths stuato the weghtg fucto ehaces the rsk averso for gas ad reduces (or elmates) the rsk seekg for losses. Cosequetly, the presece of probablty weghtg actual behavor ca dffer from what mght be expected based o the rsk atttude observed the value fucto. 4 I emprcal studes, a varety of shapes have bee detfed.

7 Fabo Mattos ad Phlp Garca 7 3 Prevous Studes Emprcal evdece suggests probablty weghtg s a mportat determat of dvdual behavor. I facal vestmet settgs several studes show how t ca lead to patters of behavor whch dffer from those based solely o rsk ad loss averso. Evdece also exsts that models corporatg probablty weghtg yeld results cosstet wth observed behavor. Levy ad Levy [8] vestgate whether rsk averso characterzes vestors ad the effect of probablty weghts o rsk premum. They coclude that rsk averso s ot preset over the etre wealth doma, ad behavor may be explaed ether by rsk atttude or the presece of a probablty fucto. They argue that eve f dvduals are rsk averse, they ca stll act as rsk-seekg vestors due to probablty weghtg. I some stuatos, ther results dcate that probablty weghts ca ehace rsk averso. Blavatskyy ad Pogreba [0] ad Lager ad Weber [9] troduce probablty weghtg to exted the aalyss of the effect of myopc loss averso o vestmet decsos. Blavatskyy ad Pogreba [0] demostrate probablty weghtg ca make vestors crease the proporto of rsky assets ther portfolos, whch s the opposte cocluso reached by Berkelaar et al. [9] ad Hwag ad Satchell [20] who cosdered just the effect of myopc loss averso. Hece ths stuato probablty weghtg leads vestors to buy more rsky assets as opposed to buy less rsky assets whe oly myopc loss averso s cosdered. Smlarly, Lager ad Weber [9] fd that myopc loss-averse vestors who also trasform probabltes may decde to crease rather tha decrease the proporto of rsky assets ther portfolos. Weghtg fuctos for professoal optos traders have bee vestgated by Fox et al. [8]. They coduct two expermets ad ther fdgs dcate vestors exhbt probablty weghtg. The frst expermet focuses o prcg ad matchg prospects over gas wth kow objectve probabltes. Ther results yeld a lear weghtg fucto whch dcates vestors prce

8 8 Measurg the Degree to whch probablty weghtg affects rsk-takg rsky prospects by ther expected actuaral value accordg to expected utlty theory. A secod expermet volves prcg prospects over gas wth ukow probabltes ad assessg the probabltes of ucerta evets. The results for both decso weghts ad judged probabltes reveal subaddtvty, meag vestors weghtg fuctos are ot lear ad expected utlty theory s volated the presece of ucerta prospects. Hece, whe vestors evaluate prospects weghtg fuctos are affected by whether probabltes are kow or ukow. The revewed studes llustrate a extesve lterature that shows probablty weghtg s a mportat compoet of decso makg. A atural exteso s to explore how much probablty weghtg affects rsk-takg behavor, ad Hlto [2] ad Daves ad Satchell [5] propose a theoretcal framework to perform ths task usg rsk premums. The ext secto presets vestor data used to vestgate the mportace of probablty weghtg ad the extet to whch behavor s modfed ts presece. Ths s followed by a secto whch dscusses procedures based o Hlto [2] ad Daves ad Satchell [5] that are used to measure the effect of probablty weghtg. 4 Research Method ad Data 4. Subjects ad expermetal procedure Decso makg s vestgated a sample of fftee propretary traders. They are all male, have a college degree ad trade agrcultural cotracts at the CME Group. Ther age rages from 23 to 54 years old, wth a average (meda) age of 3.8 (3.0). The most expereced subject has bee tradg for 30 years, whle the least has 5 moths of market experece. The average (meda) tradg experece s 7.2 (5) years. Amog the traders, twelve trade futures ad optos, two trade oly futures, ad oe trades oly optos. I terms of tradg platform, eght trade oly the

9 Fabo Mattos ad Phlp Garca 9 pt, two trade oly electroc, ad fve trade both pt ad electroc. Fally, sx traders trade oly cor, two trade oly soybeas, two trade oly soybea ol, oe trades oly wheat, three trade cor ad soybeas, ad oe trades cor, soybeas, wheat, soybea ol ad soybea meal. Eve though they work uder the same tradg group, they trade depedetly ad oly for ther ow portfolos. The tradg group provdes techcal support, trag ad offers suggestos. The maagers of the group are former traders ad are avalable to revew trades ad dscuss strateges. Traders are free to trade as they wat, but maagers try to emphasze dscple ad steer them towards resposble rsk maagemet. Profts are used to pay trasacto ad overhead costs. All traders partcpated a framed feld expermet coducted betwee December of 2006 ad May of Framed feld expermets are defed as expermets wth ostadard subject pool ad feld cotext (Harrso ad Lst [22]). 5 I ths study professoal traders are the ostadard subjects ad ther atural work evromet provdes feld cotext. The expermet was coducted the form of computer-based sessos the same tradg room where traders work, ad they kew they were partcpatg a expermet. Traders were seated frot of a persoal computer ad aswered choce questos that appeared o the scree. The expermet was coducted after ther regular tradg hours, so that there would be o dstractos ad they could focus o the questos. Each trader partcpated two sessos, the frst for the expermet uder codtos of rsk ad the secod for the expermet uder codtos of 5 Harrso ad Lst [22] propose a termology wth four types of expermets: covetoal lab expermets, artefactual feld expermet, framed feld expermet, ad atural feld expermet. Covetoal lab expermets are the most tradtoal type, employg a stadard subject pool (studets), abstract framg ad mposed set of rules. Artefactual feld expermets mprove o covetoal lab expermets by usg ostadard subjects. Framed feld expermets mprove o artefactual feld expermets by addg feld cotext. Natural feld expermets are the most ovatve type as they use ostadard subjects who perform the expermetal tasks ther ow evromet, but subjects do ot kow they are a expermet.

10 0 Measurg the Degree to whch probablty weghtg affects rsk-takg ucertaty. The expermet allowed geeratg data sets of value ad probablty pots, whch were used to ft value ad weghtg fuctos as descrbed the ext sectos. 4.2 Elctato of value ad weghtg pots The tradeoff method proposed by Wakker ad Deeffe [23] s used to elct value ad weghtg pots the ga ad loss domas. The frst step s to determe probablty p, referece outcomes R ad R *, ad the startg outcome x 0. Those values are set by the expermeter such that x * 0 R R, ad they are held fxed through the whole expermet. The desg of the expermet s crtcal for a good assessmet of values ad probablty weghts (Hershey et al., [24]). The choces related to the decso cotext ad also the dmeso of outcomes ad probabltes are made based o coversatos wth the maager of the traders partcpatg the expermet, alog wth the expermetal procedures adopted by Abdellaou [6]. The expermet should be as close as possble to the subjects evromet; hece the curret study t reflects tradg decsos commoly expereced futures markets. Traders are asked to choose betwee two tradg * strateges x p; R x, p R yeldg dfferet moetary outcomes,, ad ; where x, R, x, ad R * represet possble gas or losses ad p s the probablty assocated wth the outcomes. Gve x, x s elcted such that the * subject s dfferet betwee prospects x, p; R ad x, p; R. Based o umbers dscussed wth the maager of the tradg group partcpatg ths study, small traders usually make gas (losses) a rage betwee US$800 ad US$,000 per trade, whle large traders ca make (lose) up to US$5,000 per trade. Therefore, the tal step of the elctato procedure x 0 s set to $,000 ( $,000), whch the creases (decreases) from x ( x ) through x ( x )

11 Fabo Mattos ad Phlp Garca accordg to each trader s choces durg the expermet. The values of R ad * R are set to $500 ( $500) ad $0, respectvely. The elctato of each outcome the sequece x,..., x s obtaed through a teratve procedure whch elcted outcomes are derved from observed choce rather tha assessed by subjects. After the sequece of outcomes x,..., x s obtaed t s possble to use the same procedure to elct probabltes p,, p. I the probablty elctato process subjects are asked a ew seres of choce questos, ad probablty subject s dfferet betwee the certa outcome p s determed such that the x ad a prospect x, p ; x 0. The process to assess probabltes s also based o a teratve procedure whch elcted probabltes are derved from observed choce. I the expermet uder rsk two sequeces of te outcomes are elcted: x, x2,..., x0 the ga doma, ad x, x2,..., x 0 the loss doma. Addtoal sequeces of e probabltes are assessed: p, p 2,, p 9 the ga doma, ad p, p 2,, p 9 the loss doma. So for each trader each doma there are te pars of outcomes ad value pots vx probabltes ad weghts w x, to detfy ther value fuctos, ad e pars of p, to detfy ther weghtg fuctos. p To explore decso-makg uder ucertaty, the expermetal procedure follows Abdellaou et al. [7]. It s smlar to the procedure descrbed above, except that probabltes are ot provded. Gas ad losses are affected by the occurrece of ucerta evets E represetg some occurrece wth whch traders are famlar. Further, a extra step s added to elct probablty fuctos sce partcpats eed to make ther ow assessmet of the probabltes of those evets. Thus subjects frst eed to judge the probablty of the ucerta evet, geeratg a choce-based probablty whch wll dffer for each dvdual. The choce-based probabltes are used to elct weghtg fuctos. Based o Abdellaou et al. [7] ad coversatos wth the tradg maager of the group

12 2 Measurg the Degree to whch probablty weghtg affects rsk-takg partcpatg ths study, two types of evets are used. For the elctato of v x evet E s USDA report s bullsh, whle for the elctato of wqe j evet E s the percetage chage of the Dow Joes Idustral Average (DJIA) stock dex over the ext sx moths. Four elemetary evets are defed based o hstorcal performace of the DJIA: DJIA 4%, 4% DJIA 0%, 0 DJIA 4%, ad DJIA 4%. Fve other evets are also defed from all uos of elemetary evets that result cotguous tervals, yeldg a total of e evets. The output of ths expermet s two sequeces of te outcomes: x, x2,..., x0 the ga doma, ad, x2,..., x 0 x the loss doma; ad two sequeces of e choce-based probabltes: E, qe2,, qe9 doma, ad E, qe2,, qe9 each doma there are te pars of outcomes ad values v q the ga q the loss doma. So for each trader ad x, to assess x ther value fuctos, ad te pars of probabltes ad weghts q E, wq j E j to assess ther weghtg fuctos. The elctato procedure descrbed above allows detfyg the presece of probablty weghtg. Parametrc procedures ca be used to ft value ad weghtg fuctos to the data ad measure the mpact of probablty weghtg o decsos usg the framework of rsk premums. 4.3 Rsk premums Rsk premums are used to explore the effect of probablty weghtg o behavor. Rsk premum s defed as the sure amout of moey that a dvdual would requre to be dfferet betwee a ucerta prospect x ad a sure amout EV x r, where x EV s the expected value of prospect x ad r s the rsk premum. Followg the deas of Hlto [2] ad Daves ad Satchell [5],

13 Fabo Mattos ad Phlp Garca 3 we cosder prospect theory s fucto V x ad a value fucto v. Rsk premum the s calculated as the soluto to V x vev x r ad therefore ca be represeted as r EV x v V x EV x CEx, whch s equvalet to the dfferece betwee the expected value of equvaletce x x ad ts certaty. Itutvely, t refers to the amout of moey that vestors are wllg to forego to avod the rsk assocated wth a ucerta prospect. A postve rsk premum s assocated wth rsk averso sce a dvdual requres a sure amout of moey to take rsk. I cotrast, a egatve rsk premum s assocated wth rsk seekg as a dvdual s wllg to pay to take rsk. I the calculato of rsk premums a power value fucto wth a referece pot separatg gas ad losses s adopted (equato 2). Assumg a prospect x that yelds outcomes x wth probablty p ad x wth probablty -p, the two compoets of the rsk premum are gve by EV x px px ad CEx v V x v pvx pv x.itally we calculate expected utlty () rsk premums, assumg o probablty weghtg. premums are expressed equato (3) for gas r (see Appedx for detals of the calculato). r ad equato (4) for losses G L r L r G v x x x G x 0 (2) x 0 x x CE x px px px p EV (3) L G x CE x px px p x p x EV L (4) I a prospect theory framework two addtoal rsk premums stadard ad behavoral ca be developed based o Hlto [2] ad Daves ad Satchell [5]. The stadard rsk premum assumes that probablty weghtg s corporated the CE x compoet but ot the x EV compoet. It shows how dvduals perceve ther ow rsk prefereces relatve to the objectve

14 4 Measurg the Degree to whch probablty weghtg affects rsk-takg expected value of the prospect. The weghtg fucto wp exp l p proposed by Prelec [25] s adopted, where p s the probablty ad ad are respectvely the elevato ad curvature parameters of the fucto. Stadard rsk premums are developed for S gas r usg equatos (5) ad (6), where the superscrpt w r ad losses S G L certaty equvalets dcates that probablty weghts are corporated ther calculato. r s G r s L G w x CE x px px wpx w p EV (5) EV L G x w x CE x px px wp x w p x L The behavoral rsk premum assumes that probablty weghtg s corporated both EV x ad CEx (6) compoets, ad shows rsk behavor, where the evaluato of the prospect x s measured agast a probablty weghted expected value of x. Usg Prelec [25] s weghtg fucto, behavoral rsk B premums are calculated for gas r usg equatos (7) ad r ad losses B G (8), where the superscrpt w expected values ad certaty equvalets dcate the presece of probablty weghtg ther calculato. r B G w G x w x CE x wpx w px wpx w p EV (7) G B w w L L L r EV x CE x w p xw p x w p x w p x L If probablty weghtg s relevat explag behavor, the the three rsk premums wll dffer. I partcular, the dfferece betwee the behavoral rsk premum ad the other rsk premums provdes a reflecto of how much probablty weghtg flueces actual behavor. The dfferece betwee behavoral ad stadard rsk premums dcates the degree to whch a dvdual s actos cotradct ther belefs about ther ow rsk atttude. For (8)

15 Fabo Mattos ad Phlp Garca 5 stace, a perso may beleve hmself to be rsk averse but stll act a rsk-seekg maer. The dfferece betwee behavoral ad rsk premums represets how much actual behavor devates from what s predcted by expected utlty theory because of probablty weghtg. The three rsk premums are calculated for each trader from the data obtaed the framed feld expermet uder codtos of rsk. Pots x ad x are the frst ad last pots of the value fucto elcted each expermet. The coeffcets ad of the value fucto are estmated by fttg a power fucto to the elcted pots. The trasformed probabltes wp ad w p are also geerated the expermet, ad coeffcets ad are estmated by fttg Prelec [25] s fucto to the elcted probablty pots. Sce the magtudes of premums deped o a dvdual s rsk atttude ad the dstrbuto of outcomes, premums calculated uder dfferet stuatos caot be compared absolute terms. Cosequetly the effect of probablty weghtg o behavor s assessed by examg proportoal rsk premums the rsk premums expressed as a proporto of the expected value of the prospect p,$,000; p,. Proportoal rsk premums (PRP) are calculated x j j as PRP r EV, where j, stadard, behavoral ad ga, loss. I the ga (loss) doma proportoal rsk premums are postve (egatve) for rsk-averse dvduals, egatve (postve) for rsk-seekg dvduals, ad zero for rsk- eutral dvduals. 6 The effect of probablty weghtg o proportoal rsk premums s explored through the dfferece betwee proportoal behavoral premum ad proportoal premum, whch ca be decomposed to two factors: the B S dfferece betwee behavoral ad stadard rsk premums PRP PRP ad 6 By costructo, EV 0 ad EV 0. G L

16 6 Measurg the Degree to whch probablty weghtg affects rsk-takg the dfferece betwee stadard ad expected utlty rsk premums PRP PRP (equato 9). S PRP B B S S PRP PRP PRP PRP PRP (9) The dfferece betwee the proportoal behavoral ad rsk premums B ( PRP PRP ) provdes a measure of how much actual behavor would devate from what s predcted by expected utlty theory because of probablty weghtg. B S The factor PRP PRP detfes the degree to whch a dvdual s actos S would cotradct a belef about hs ow rsk atttude. The factor PRP PRP detfes the degree to whch a dvdual s percepto of hs rsk atttude s cosstet wth that predcted by expected utlty theory. For example, cosder a perso wth a PRP whch mples a rsk-averse dvdual s wllg to G gve up 5% of expected value to avod rsk. If ths dvdual exhbts S PRP G 0.09, he perceves hmself to be more rsk averse tha expected utlty would suggest sce he s wllg to gve up 9% of the expected value to avod rsk. B Now cosder PRP 0. 05, dcatg ths dvdual would actually gve up 5% G of the expected value to avod rsk, whch s the same value predcted by expected utlty theory. I ths case the mpact of probablty weghtg emerges oly hs B perceptos about hs rsk atttude, wth PRP PRP 0, B S S PRP PRP 0.4 ad PRP PRP Despte percevg hmself to be more rsk averse tha dcated by expected utlty theory, he would behave cosstet wth expected utlty. 4.4 Ucertaty premums Proportoal ucertaty premums (, stadard ad behavoral) are also calculated for each trader the ga ad loss domas. The method to calculate

17 Fabo Mattos ad Phlp Garca 7 ucertaty premums s the same as explaed the prevous secto, except x, x,,, w p, ad w p for each trader equatos (3) through (8) are based o a data set obtaed from the expermet uder ucertaty. Thus the dffereces betwee ucertaty ad rsk premums arse from dstct value ad probablty pots elcted each expermet, whch reflect dverse behavor uder codtos of rsk ad ucertaty. 5 Results 5. Value ad weghtg fuctos uder rsk Despte cosderable heterogeety across traders, the results dcate the mportace of prospect theory. Estmato of value fuctos uder rsk reveals that traders are essetally rsk averse for gas ad rsk seekg for losses. The meas of the estmated parameters of the value fucto are respectvely 0.87 ad 0.89 the ga ad loss domas, deotg cocavty for gas ad covexty for losses (Table ). More specfcally, the ga doma twelve of fourtee traders show cocave fuctos, whle the loss doma te of fourtee traders dsplay covex fuctos (Appedx II). 7 Elcted value fuctos suggest traders follow the stadard structure prospect theory, but behavor also depeds o the weghtg fucto. Our results show that traders geerally exhbt a verse s-shaped weghtg fucto, cosderg the mea values for the elevato ad curvature parameters both domas (Table ). I partcular, e the ga doma ad eleve the loss doma exhbt a verse s-shaped curve 7 Aswers to elctato questos were vald for trader 9 (gas) ad 2 (losses). Therefore fourtee fuctos were elcted for each doma the expermet, eve though there were fftee traders the sample. Ths problem also affects the calculato of rsk premums for these two traders.

18 8 Measurg the Degree to whch probablty weghtg affects rsk-takg (Appedx II). 8 These fdgs already dcate the presece of probablty weghtg decsos uder rsk ad ucertaty, ad provde a sese of overad uderweghtg. But the magtude of probablty weghtg vary across traders, as ca be see by the elevato ad curvature parameters Appedx II, wth probablty weghtg beg geerally more tese as these parameters are further from oe. Rsk ad ucertaty premums ca be used as a addtoal measure of the mpact of probablty weghtg o decsos the cotext of how much proft a dvdual would be wllg to forgo order to avod rsk or ucertaty. Table : Summary statstcs for estmated parameters of the value ad weghtg fuctos uder rsk Value fucto (a) Weghtg fucto (b) gas losses elevato curvature gas losses gas losses mea st. dev th perc meda th perc (a) A power fucto s estmated as show equato 2. I the ga doma t s cocave (covex) f s less (greater) tha. I the loss doma t s cocave (covex) f s greater (less) tha. Both mea ad meda adjusted R 2 rage betwee 0.98 ad 0.99, ad both mea ad meda MSE rage betwee 0.04 ad (b) The weghtg fucto follows Prelec [25] s fuctoal form: wp exp l p. Both mea ad meda adjusted R 2 rage betwee 0.92 ad 0.98, ad both mea ad meda MSE rage betwee 0.08 ad I the ga doma, the remag traders exhbt s-shaped curves (two traders), cocave curves dcatg complete overweghtg of probabltes (two traders) ad covex curves dcatg complete uderweghtg of probabltes (two traders). I the loss doma, each of the remag four traders exhbts a s-shaped curve, a cocave curve, a covex curve, ad a straght le.

19 Fabo Mattos ad Phlp Garca Proportoal rsk premums Three proportoal rsk premums are calculated: expected utlty () premum, stadard premum ad behavoral premum. As dscussed earler, rsk premums represet the amout of moey that a trader s wllg to forego to avod a prospect yeldg x wth probablty p ad where x ad x wth probablty p, x represet moetary values elcted for each trader the expermet. Proportoal rsk premums are calculated for each probablty p from 0.0 to 0.99 cremets of Fgure2 shows premums for gas ad losses for each trader averaged across all probabltes the terval0.0, Usg a t test ad the ull hypothess that mea premums are equal to zero, behavoral ad premums for all traders are statstcally dstgushable from zero at 5%. Wth regards to stadard premums, the ull hypothess oly fals to be rejected for trader 4 ad the ga doma ad trader ad 3 the loss doma. Focusg frst o ad behavoral premums, both suggest that traders are maly rsk averse for gas ad rsk seekg for losses. Behavoral () premums dcate that te (twelve) traders are rsk averse the ga doma, whle e (te) traders are rsk seekg the loss doma. However, ther magtudes dffer, mplyg the presece of probablty weghtg ca cause the stregth of rsk averso or rsk seekg to vary from what expected utlty theory predcts. For example, premum dcates that trader 6 would forego 2.% of expected value to avod rsk the ga doma, whle the behavoral premum suggests he would actually forego oly 2.6% of expected value to avod rsk (Fgure 2).Cotrastg behavoral ad stadard premums also helps detfy the effect of probablty weghtg o behavor by comparg how a trader actually 9 Probabltes 0 ad are ot used because assumptos of probablty weghtg fuctos mply w(0)=0 ad w()=. 0 Premums calculated for each dvdual probablty are ot preseted for brevty but are avalable upo request.

20 20 Measurg the Degree to whch probablty weghtg affects rsk-takg behave to how they perceve they would behave. The presece of probablty weghtg has a stroger mpact o stadard premums, whch show a larger magtude tha the other premums. Returg to the prevous example of trader 6 the ga doma, the stadard premum suggests he perceves he would forego 52.2% of hs expected value to avod rsk, whch s larger tha both ad behavoral premums. proportoal rsk premum behavoral stadard traders proportoal rsk premum behavoral stadard traders (a) Gas: postve (egatve) premums dcate rsk averso (seekg). Losses: postve (egatve) premums dcate rsk seekg (averso). (b) All premums are statstcally dstgushable from zero at 5% (t test), except for stadard premum for traders 4 ad. (c) All premums are statstcally dstgushable from zero at 5% (t test), except for stadard premum for traders ad 3. Fgure 2: Proportoal rsk premums averages across probabltes (a), Gas (b), Losses (c) Table 2 presets the dfferece betwee behavoral ad premums (B ) ad ts decomposto to the dffereces betwee behavoral ad

21 Fabo Mattos ad Phlp Garca 2 stadard premums (B S) ad stadard ad premums (S ) based o equato (9). All dffereces are calculated wth the terval 0.0,0.99 ad the averaged across all probabltes. A t test s adopted to test the ull hypothess that mea dffereces betwee premums are equal to zero. The ull hypotheses ca be rejected at 5% almost all cases, wth the excepto of B- for trader 3 (losses) ad trader 4 (gas), ad B-S ad S- for traders (losses) ad 3 (gas ad losses) (Table 2). Falure to reject the ull hypothess mples that probablty weghtg has o mpact o average rsk takg. I the ga doma postve (egatve) dffereces betwee premums dcate less (more) rsk takg. I the loss doma postve (egatve) dffereces betwee premums dcate more (less) rsk takg. Three ma fdgs emerge. Frst, the effect of probablty weghtg terms of more or less rsk takg seems to be almost evely splt the ga doma ad leas towards more rsk takg the loss doma. The dfferece B dcates less (more) rsk takg for sx (seve) traders the ga doma, ad for fve (eght) the loss doma. Secod, the dfferece betwee behavoral ad premums (B ) s relatvely small compared to ts two compoets B S ad S. Thrd, the dffereces betwee behavoral ad stadard premums (B S) ad stadard ad premums (S ) ted to have opposte sgs. These last two pots suggest the strogest mpact of probablty weghtg happes to traders percepto of ther ow rsk atttude. Although they geerally perceve themselves to be much more rsk averse or rsk seekg tha expected utlty dcates, ther actual behavor s relatvely closer to expected utlty predctos. For example, the ga doma trader 3 beleves hmself to be more rsk takg tha expected utlty theory suggests (S =-0.565), but behaves wth more rsk averso tha he beleves to have (B S=0.538). The magtude of these effects s smlar, meag the devato of actual behavor from expected utlty due to probablty weghtg s relatvely small (B =-0.027).

22 22 Measurg the Degree to whch probablty weghtg affects rsk-takg Table 2: Dffereces proportoal rsk premums averages across probabltes (a) Trader Gas (b) Losses (b) B B S S B B S S * * 0.024* * 0.029* * * 2.434* * 0.056* 0.32* * * 0.538* * * 0.92* * * -0.06* 0.630* * * 0.49* -0.53* -0.08* 0.239* * * -0.3* 0.396* 0.03* * 0.32* * 0.239* -0.25* 0.02* * 0.050* * -0.28* 0.75* 0.036* -0.29* 0.255* 9 /a /a /a -0.08* 0.080* * * -0.2* 0.95* 0.043* 0.553* -0.50* 0.004* 0.086* * 0.007* * -0.87* 0.349* /a /a /a * * * * 0.404* * 0.250* * * * 0.22* 0.005* 0.355* * Number of traders who exhbt (c): less rsk takg more rsk takg same rsk takg 2 2 (a) The ull hypothess that dffereces betwee premums are equal to zero s tested wth a t test. The * dcates the ull hypothess ca be rejected at 5%. (b) B : dfferece betwee behavoral ad premums, B S: dfferece betwee behavoral ad stadard premums, S : dfferece betwee stadard ad premums. (c) Gas: postve (egatve) dfferece dcates less (more) rsk takg. Losses: postve (egatve) dfferece dcates more (less) rsk takg. If the dfferece s foud to be statstcally equal to zero, t s cosdered that there s o chage rsk takg. The effect of probablty weghtg s further vestgated for dfferet rages of probablty, sce prevous research dcates that behavor ca chage at

23 Fabo Mattos ad Phlp Garca 23 dfferet levels of probablty. A example s the fourfold patter, showg that both rsk averso ad rsk seekg ca happe the ga ad loss domas depedg o the probabltes volved (Tversky ad Kahema [7]. Other examples are Etchart-Vcet [26] ad Etchart-Vcet [27], who fd evdece that behavor at small probabltes dffers from that at hgh probabltes. 2 Table 3 shows the dffereces betwee proportoal rsk premums, B ad ts decomposto to B S ad S. All dffereces are calculated wth four tervals 0.0,0.25, 0.26,0.50, 0.5,0.75, 0.76,0.99 ad the averaged across all probabltes wth each terval. As dscussed prevously, these probabltes refer to the prospect p,$,000; p, where x vares across traders but s always postve ad greater tha $,000 the ga doma (for losses x s more egatve tha $,000). Hece hgher probablty p meas hgher chaces of gettg the smallest ga (loss) x x. A t test s adopted to test the ull hypothess that average dffereces wth each terval are equal to zero, ad results dcate the ull hypothess ca be rejected at 5% 320 of the 336 tervals. The dffereces betwee premums Table 3 dcate that the magtude of the mpact of probablty weghtg ca vary ad eve swtch sgs for dfferet probabltes. For example, the ga doma trader 2 shows B becomg more egatve as the tervals move towards hgher probabltes. Ths result suggests that probablty weghtg makes trader 2 take more rsk tha expected utlty predcts, ad ths effect becomes more proouced as probabltes crease. I the loss doma trader shows S swtchg sgs as the tervals I the ga doma rsk averso (seekg)s foud at hgh (low) probabltes. I the loss doma rsk seekg (averso) s foud at hgh (low) probabltes. 2 Etchart-Vcet [26] fds evdece that the magtude of losses affects probablty weghtg at small probabltes, but ot at hgh probabltes. Etchart-Vcet [27] also fds that the payoff structure of gambles affects probablty weghtg at moderate ad hgh probabltes, but ot at small probabltes.

24 24 Measurg the Degree to whch probablty weghtg affects rsk-takg move towards hgher probabltes. The dfferece betwee stadard ad premums s postve for the two lower tervals ad egatve for the two hgher tervals, mplyg he beleves hmself to be relatvely more rsk seekg (averse) tha expected utlty predcts whe probabltes are lower (hgher). Here, Table 3, Appedx 5.3 Value ad weghtg fuctos uder ucertaty The mea value of the estmated parameter of the power fucto dcates cocavty for gas ad losses, suggestg traders are rsk averse both domas (Table 4). However, there s also large heterogeety across traders, ad the mea value the loss doma s hghly flueced by trader 9 (Appedx II). 3 I geeral, traders exhbt cocave value fuctos the ga doma (twelve of fftee traders) ad covex value fuctos the loss doma (e of fftee traders) (Appedx II). Aga, value fuctos elcted the expermet suggest that most traders follow the stadard behavor suggested by prospect theory, wth rsk averso for gas ad rsk seekg for losses. Estmato of weghtg fuctos uder ucertaty reveals a smlar patter compared to the fdgs uder rsk wth respect to ts verse s-shape, as suggested by mea values of the elevato ad curvature parameters (Table 4). However, the ga (loss) doma oly seve (fve) of fftee traders exhbt verse s-shaped curves (Appedx II). Further, estmato results show larger degree of probablty weghtg uder ucertaty, sce estmated elevato ad curvature parameters uder ucertaty are farther from compared to values foud uder rsk (Table 4). I partcular, may traders weghtg fuctos (fve for gas ad seve for losses) showed a 3 If the mea s calculated wthout trader 9 the loss doma, the average value fucto s covex for losses.

25 Fabo Mattos ad Phlp Garca 25 low sestvty to chages probablty. Cosequetly they ted to gve smlar weghts to dfferet probabltes, accoutg for early horzotal curves ther weghtg fuctos as dcated by curvature parameters close to zero (Appedx II). Fally, as detfed for stuatos uder rsk, dvdual results uder ucertaty also show large heterogeety across traders. As dscussed prevously, these dffereces ca have dstct mpacts the calculato of ucertaty premums. Table 4: Summary statstcs for estmated parameters of the value ad weghtg fuctos uder ucertaty Value fucto (a) Weghtg fucto (b) gas losses elevato curvature gas losses gas losses mea st. dev th perc meda th perc (a) A power fucto s estmated as equato 2. I the ga doma t s cocave (covex) f s less (greater) tha. I the loss doma t s cocave (covex) f s greater (less) tha. Both mea ad meda adjusted R 2 are 0.99, ad both mea ad meda MSE are (b) The weghtg fucto follows Prelec [25] s fuctoal form: wp exp l p. Both mea ad meda adjusted R 2 rage betwee 0.97 ad 0.99, ad both mea ad meda MSE rage betwee 0.03 ad Proportoal ucertaty premums The proportoal ucertaty measures are calculated for the expected utlty () premum, stadard premum ad behavoral premum. Ther deftos are the same as dscussed for rsk premums. Fgure 3 shows premums for gas ad

26 26 Measurg the Degree to whch probablty weghtg affects rsk-takg losses for each trader averaged across all probabltes the terval 0.0,0.99. Usg a t test ad the ull hypothess that mea premums are equal to zero, behavoral ad premums for all traders are statstcally dstgushable from zero at 5%. Wth regards to stadard premums, the ull hypothess oly fals to be rejected for trader 6, 9 ad 0 the ga doma ad trader, 4 ad 5 the loss doma. I geeral, the qualtatve results are the same as the expermet uder rsk. Both behavoral ad premums suggest that traders are maly rsk averse for gas ad rsk seekg for losses, but the magtudes of premums ca vary. Behavoral () premums dcate that te (twelve) traders are rsk averse the ga doma, whle seve (te) traders are rsk seekg the loss doma (Fgure 3). Further, devatos from expected utlty theory appear to happe more strogly traders ow percepto of ther rsk atttude (stadard premums) tha ther actual behavor (behavoral premums). proportoal ucertaty premum behavoral stadard traders proportoal ucertaty premum behavoral stadard traders (a) Gas: postve (egatve) premums dcate rsk averso (seekg). Losses: postve (egatve) premums dcate rsk seekg (averso). (b) All premums are statstcally dstgushable from zero at 5% (t test), except for stadard premum for traders 6, 9 ad 0. (c) All premums are statstcally dstgushable from zero at 5% (t test), except for stadard premum for traders, 4 ad 5. Fgure 3: Proportoal ucertaty premums averages across probabltes (a), Gas (b), Losses (c)

27 Fabo Mattos ad Phlp Garca 27 The dfferece betwee behavoral ad ucertaty premums (B ) ad ts decomposto usg equato (9) are also calculated wth the terval 0.0,0.99, ad the averaged across all probabltes. Smlar to rsk premums, three ma fdgs ca be draw here. 4 Two of them are qualtatvely the same as the expermet uder rsk: the dffereces B are relatvely small compared to B S ad S, ad the dffereces B S ad S ted to have opposte sgs. These two pots mply the strogest mpact of probablty weghtg happes o traders percepto o ther ow rsk atttude also uder ucertaty. Fally, oe fdg dffers from the expermet uder rsk. The effect of probablty weghtg o actual behavor seems to lea towards more rsk takg the ga doma ad less rsk takg the loss domas. The dfferece B dcates less (more) rsk takg for fve (e) traders the ga doma, ad for eght (fve) the loss doma. The aalyss of the effect of probablty weghtg for dfferet levels of probablty also reveals smlar results to the expermet uder rsk. Dffereces betwee proportoal ucertaty premums calculated wth the same four probablty tervals as before, ad the averaged across all probabltes wth each terval, dcate that the magtude of the mpact of probablty weghtg ca vary ad also swtch sgs for dfferet probabltes The effect of probablty weghtg uder rsk ad ucertaty Our fdgs show the effect of probablty weghtg s stroger uder ucertaty tha uder rsk. Fgure 4 exhbts dffereces betwee behavoral ad premums (B ) the two expermets. 4 Sce most results are qualtatvely smlar to the expermet uder rsk, calculatos of ucertaty premums are ot preseted for brevty but are avalable upo request. 5 These calculatos are also ot preseted for brevty but are avalable upo request.

28 28 Measurg the Degree to whch probablty weghtg affects rsk-takg Gas Losses rsk ucertaty rsk ucertaty B B trader (a) The ull hypothess that the average (B ) uder rsk s equal to the average (B ) uder ucertaty s tested wth Welch F-test. Results show t ca be rejected at % for all traders, except trader 9 the ga doma ad traders 9 ad 5 the loss doma. trader Fgure 4: Dfferece betwee behavoral ad premums (B ) uder rsk ad ucertaty (a) Gas Losses rsk ucertaty rsk ucertaty.5.5 S -.0 S trader (a) The ull hypothess that the average (S ) uder rsk s equal to the average (S ) uder ucertaty s tested wth Welch F-test. Results show t ca be rejected at % for all traders, except traders 4 ad 5 the ga doma ad trader 0 the loss doma. trader Fgure 5: Dfferece betwee stadard ad premums (S ) uder rsk ad ucertaty (a)

29 Fabo Mattos ad Phlp Garca 29 The ull hypothess that the average (B ) uder rsk s equal to the average (B ) uder ucertaty s tested wth a Welch F-test. The ull hypothess ca be rejected at % for all traders, except trader 9 the ga doma ad traders 9 ad 5 the loss doma. Results dcate that the magtudes of devatos from expected utlty theory are geerally larger uder ucertaty tha they are uder rsk, ad they also seem to be more proouced for gas tha for losses. Fgure 4 also shows the drecto of the effect of probablty weghtg ca chage depedg o the evromet. The sg of B s dfferet uder rsk tha t s uder ucertaty for four (sx) traders the ga (loss) doma. Smlar fdgs ca be see for the effect of probablty weghtg o traders percepto o ther ow rsk atttude. Fgure 5 exhbts dffereces betwee stadard ad premums (S ).The ull hypothess that the average (S ) uder rsk s equal to the average (S ) uder ucertaty s also tested wth a Welch F-test. The ull hypothess ca be rejected at % for all traders, except trader 4 ad 5 the ga doma ad trader 0 the loss doma. Aga, these dffereces ted to be larger uder ucertaty tha uder rsk, ad the drecto of the effect of probablty weghtg also ca chage depedg o the evromet; the sg of S dffers across expermets for fve (sx) traders the ga (loss) doma. 6 Dscusso ad Coclusos The study emprcally vestgates the mpact of probablty weghtg o facal decsos usg a group of professoal traders ad a ovel approach based o rsk ad ucertaty premums. Despte the relatvely small sample sze the expermet, our framework appled to real decso makers offers formatve results. Professoal traders exhbt probablty weghtg ther choces. Ths s cosstet wth the expermet coducted by Fox et al. [8] uder dfferet market

30 30 Measurg the Degree to whch probablty weghtg affects rsk-takg codtos ad usg dfferet types of securtes. The fact that both studes separated by tme ad dffereces the markets examed fd that probablty weghtg s a mportat determat of facal decsos emphaszes ts sgfcace uderstadg the ature of rsk ad the resultat rsk-retur behavor usg prospect theory. Probablty weghtg also has a substatal mpact o behavor. May stuatos exst whch premums chage sg whe probablty weghtg s troduced. For stace, rsk averso (rsk seekg) chages to rsk seekg (rsk averso) the presece of probablty weghtg. I other stuatos probablty weghtg ehaces dramatcally the testy of rsk averso or rsk seekg. Further, behavor s ot homogeous across probabltes. The magtude ad drecto of devatos from expected utlty theory ca vary depedg o the level of probabltes. Ths result s le wth prevous research whch fds evdece of dstct behavor at small ad large probabltes (Harbaugh et al. [28]; Etchart-Vcet [26]; Etchart-Vcet [27]). Harbaugh et al. [28], for example, fd that dvduals are rsk averse over small-probablty gas but rsk seekg over hgh-probablty gas. Coversely, they fd rsk averso over hgh-probablty losses ad rsk seekg over small-probablty losses. Eve though our dscusso s framed terms of devatos from expected utlty theory, t supports the oto that behavor chages over dfferet levels of probablty. Probablty weghtg affects traders perceptos of ther ow rsk atttude more tesely tha t affects ther actual behavor. Stadard premums are geerally of larger magtude tha behavoral premums, meag premums ted to be closer to behavoral premums tha to stadard premums. Ths suggests traders beleve themselves to be more rsk averse or rsk seekg tha ther actual behavor shows, whch mples ther behavor s closer to expected utlty theory tha they perceve t to be. Here t s mportat to ote ths fdg may be related to the evromet whch our traders work. The tradg group ecourages ts traders to trade wth dscple ad resposble rsk maagemet, whch may

31 Fabo Mattos ad Phlp Garca 3 atteuate the mpact of probablty weghtg o behavor eve though t s stll proouced traders perceptos. Ths dea s cosstet wth studes whch fd that market experece (here gaed through ether traders ow experece or obtaed from former traders) reduces devatos from expected utlty theory (Lst [29] ad [30]; Feg ad Seasholes [3]).It s also le wth results by Lu et al. [32] that devatos from expected utlty theory ted to be smaller amog traders who place orders combg calls, puts ad the uderlyg asset, whch are essetally the types of trades used by traders studed here. I addto, Zaloom [33] ad [34] pots out the mportace of dscple ad self-cotrol tradg, hghlghtg that kowg whe to place a trade ad how much to trade are mportat sklls for traders. It s possble that traders lear these sklls through dscussos wth former traders avalable to help them the tradg group. Fally, rsk-averse or rsk-seekg behavor s more tese uder codtos of ucertaty tha t s uder codtos of rsk. Ths fdg s cosstet wth prevous studes cludg Tversky ad Fox [35], who fd that the effect of probablty weghtg s more proouced uder ucertaty tha rsk. They argue that departures from expected utlty theory are amplfed by ambguty. Smlarly, Fox et al. [8] fd that vestors who partcpate ther expermets ted to follow expected utlty theory decsos where objectve probabltes are kow, but depart from expected utlty theory decsos volvg a subjectve assessmet of probabltes. Clearly, larger devatos from expected utlty theory emerge whe dvduals eed to make ther ow assessmets about the lkelhood of evets. Overall, the fdgs support the exstece of bas ad heurstcs whe vestors make decsos uder codtos of rsk ad ucertaty. It also supports the usefuless of the tradeoff method ad behavoral rsk premums to gather expermetal data to emprcally vestgate the mportace of probablty weghtg. By combg procedures developed by Hlto [2] ad Daves ad Satchell [5], we provde a useful measure of the degree that actual behavor