C.V. Starr Center for Applied Economics

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1 ECONOMIC RESEARCH REPORTS Imperfec Knowledge and Asse Prce Dynamcs: Modelng he Forecasng of Raonal Agens, Dynamc Prospec Theory and Uncerany Prema on Foregn Exchange by Roman Frydman & Mchael D. Goldberg RR#: June 2003 C.V. Sarr Cener for Appled Economcs Deparmen of Economcs Faculy of Ars and Scence New York Unversy 269 Mercer Sree, 3 rd Floor New York, New York

2 Imperfec Knowledge and Asse Prce Dynamcs: Modelng he Forecasng of Raonal Agens, Dynamc Prospec Theory and Uncerany Prema on Foregn Exchange* Roman Frydman** and Mchael D. Goldberg*** Frs Draf: 25 Aprl 2003 **** Ths Draf: 10 June 2003 JEL: F31 Keywords: Exchange Raes, Rsk Premum, Imperfec Knowledge, Indvdual Raonaly, Expecaons, Prospec Theory * The auhors are graeful o Ned Phelps, Sephan Schulmeser and George Soros for smulang dscussons on a range of opcs relaed o hs paper. The auhors are ndebed o Charles Engel, Halna Frydman and Mke Woodford for nsghful commens and suggesons ha subsanally mproved he paper, and o Marcella Frydman and Ken Murphy for edoral advce. The auhors would also lke o hank Ben Bernanke, Jan Gross, Kaarna Juselus, Danel Kahneman, Andreas Park, Andrzej Rapaczynsk, Paul Romer and Larry Summers for helpful suggesons on and reacons o an earler draf of hs paper, and o he C.V. Sarr Cener for Appled Economcs a New York Unversy and he Regnald F. Akns Char a he Unversy of New Hampshre for her suppor. **Deparmen of Economcs, New York Unversy, e-mal: roman.frydman@nyu.edu. ***Deparmen of Economcs, Unversy of New Hampshre, and Insue of Economcs, Unversy of Copenhagen, e-mal:mchael.goldberg@unh.edu ****The frs draf of hs paper appeared under he le Imperfec Knowledge Expecaons, Dynamc Prospec Theory and Uncerany Prema on Foregn Exchange: Toward a General Theory of Asse-Prce Dynamcs Do no crculae or ce whou permsson from he auhors.

3 Absrac Models usng he Raonal Expecaons Hypohess (REH) are wdely recognzed o be nconssen wh he observed behavor of prema n fnancal markes, as well as oher feaures of asse prce dynamcs. Moreover, many reasons have been advanced as o why he REH canno generally represen, even approxmaely, he expecaons behavor of ndvdually raonal agens. In hs paper, we develop a new model of he equlbrum premum n he foregn exchange marke ha replaces he REH wh he Imperfec Knowledge Forecasng (IKF) framework. Because we manan ha agens mus cope wh mperfec knowledge and ha hey are no grossly rraonal, our IKF approach mposes only qualave condons on he formaon of ndvdual forecasng models and her updang. We also develop a dynamc exenson of he orgnal formulaon of Kahneman and Tversky s prospec heory. We fnd ha under IKF and dynamc prospec heory, he equlbrum premum on foregn exchange s posvely relaed o he gap beween he aggregae forecas of he exchange rae and s hsorcal benchmark level. We es hs mplcaon, usng survey daa on he German mark-u.s. dollar exchange rae, and fnd ha he behavor of he ex ane premum on foregn exchange s conssen wh our model of he premum.

4 Of course, compared wh he precse predcons we have learn o expec n he physcal scences, hs sor of mere paern predcons s a second bes...i am anxous o repea, we wll sll acheve predcons whch can be falsfed and whch herefore are of emprcal sgnfcance... Ye he danger of whch I wan o warn s precsely he belef ha n order o be acceped as scenfc s necessary o acheve more. Ths way les charlaansm and more. To ac on he belef ha we possess he knowledge,...whch n fac we do no possess, s lkely o make us do much harm...i confess ha I prefer rue bu mperfec knowledge, even f leaves much ndeermned and unpredcable, o a preence of exac knowledge ha s lkely o be false. (Excerps from he Nobel lecure of Fredrch Hayek, 1978, p.33). 1 Inroducon The almos unversal use of he Raonal Expecaons Hypohess (REH) esfes o he wdespread belef among economss ha he REH provdes he soluon o he endurng problem of modelng he expecaons of raonal agens. Ye s precsely n sengs n whch agens expecaons maer mos, such as asse markes, ha REH-based models encouner her greaes dffcules. In he foregn exchange marke, Dornbusch and Frankel surmsed ha The chef problem wh he overshoong heory, and ndeed wh he more general raonal expecaons approach, s ha does no explan well he shorer-erm [long-swngs] dynamcs (Dornbusch and Frankel, 1988, p. 16). 1 Bu unl recenly, such observaons have been largely gnored despe research of he pas wo decades ha ndcaes ha he mplcaons of he REH approach are nconssen wh much of he emprcal evdence on exchange raes and oher asse prces. 2,3 1 For early sudes documenng he dffcules of he REH approach n explanng he observed movemens of he erm srucure of neres raes and sock prces, see Shller (1979, 1981). 2 For examples of recen surveys of anomalous behavor (by REH sandards) of exchange raes see Frankel and Rose (1995) and Engel (1996) and references heren. 3 I should also be noed ha he REH does no, n general, represen, even approxmaely, he expecaons of ndvdually raonal agens. For an early dscusson of he 1

5 The wdespread belef ha he REH s he model for he expecaons of raonal agens has recenly gven rse o a vew ha o explan he observed asse-prce dynamcs deparures from raonaly" need o be nroduced no models of asse markes. 4 We beleve, however, ha a dfferen facor may be key o undersandng he apparenly anomalous behavor of asse prces. In hs paper, we develop a new model of he equlbrum premum n he foregn exchange marke. The key assumpon underlyng our approach (dubbed he Imperfec Knowledge Forecasng, IKF, framework) s ha economc agens n formulang and revsng her forecasng models are lmed by mperfec knowledge. 5 Whle we manan ha agens mus cope wh mperfec knowledge, we do no dffer n he presumpon ha, on he whole, economcagens are no grossly rraonal, n he sense ha hey do no pass up endlessly prof opporunes. Consequenly, he IKF framework n conras o he exan approaches o he modelng of expecaons, ncludng he REH mposes only qualave condons on he formulaon of ndvdual forecasng models and her revsons. 6 The second dsncve feaure of our approach s ha we develop a dynamc exenson of he orgnal formulaon of Kahneman and Tversky s prospec heory (Kahneman and Tversky, 1979 and Tversky and Kahneman, 1992). Ths dynamc prospec heory assumeshaagensarenoonly loss averse, bu ha hey are more sensve o changes n poenal losses han o changes n poenal gans of he same sze. We fnd ha under dynamc prospec heory, boh bulls and bears hold open posons n foregn assumpons under whch each of he ndvdually raonal agens would choose o use only one common model and oher relaed epsemologcal problems beseng he REH approach, see Frydman (1982), Frydman and Phelps (1983), Frydman (1983) and Phelps (1983). 4 See Akerlof (2002), and references heren, for a recen dscusson on he mporan role of rraonaly for undersandng asse prce dynamcs. 5 Ths IKF framework has s roos n Keynes (1936), whose deas on he behavor of asse markes nfluenced much of he analyss n hs paper. Hayek s (1948) penerang crque of socals plannng and hs nsghs on he use of knowledge n socey have also guded us n he developmen of our approach. 6 Ths qualave approach s pu forh n Frydman and Goldberg (2003a). In ha paper we argue ha f an economs were o characerze fully ndvdual forecasng models and her revsons (.e. n erms of fxed rules or generalzaons of hose rules ha depend on observable varables or facors), hen he resulng formulaons would be, n general, nconssen wh he posulae of ndvdual raonaly. See he concludng remarks below for furher dscusson of hs pon. 2

6 exchange only f hey expec a posve excess reurn on her open posons, o compensae hem for her exra sensvy o he poenal losses. Ths resul leads o a new, momenary, equlbrum condon for he foregn exchange marke, whch we call uncerany adjused uncovered neres rae pary (UAUIP). The UAUIP condon mples ha equlbrum n he foregn exchange marke s assocaed wh an aggregae premum on foregn exchange dubbed an equlbrum uncerany premum ha depends on he relave assessmens of bulls and bears concernng he poenal losses from foregn exchange speculaon. In our model, equlbrum s defned, a each pon n me, for a gven se of ndvdual forecasng models. Revsons of hese models, hen, cause he equlbrum uncerany premum o move over me. To model hs movemen, we follow he IKF framework and mpose only qualave assumpons on he process by whch ndvdual forecasng models are revsed. To hs end, we buld on an dea pu forh by Keynes (1936) and assume ha he gap beween agens condonal forecass of he exchange rae and s perceved hsorcal benchmark plays a key role n how agens revse her forecasng models and her assessmens of he poenal losses. We fnd ha hese ndvdual gap effecs lead o a posve relaonshp beween he equlbrum uncerany premum and an aggregae measure of he gap. We es hs mplcaon, usng one-monh forecass of he German mark- U.S. dollar exchange rae from Money Marke Servces Inernaonal (MMSI), andshowhaheobservedmepahofheex ane premum on foregn exchange s ndeed conssen wh our gap effec. We conjecure ha hs new emprcal fndng ha he gap from he hsorcal benchmark plays an mporan role n undersandng he dynamcs of he premum on foregn exchange may also be mporan n undersandng he behavor of equlbrum prema n oher asse markes. We also show ha our IKF-based model provdes a smple explanaon of he sgn reversals ha have been observed n foregn exchange prema. These posve fndngs sand n sharp conras o he wdely known dffcules of sandard REH models of he rsk premum n explanng he me pah, volaly and sgn reversals of foregn exchange prema over he modern perod of floang. 7 Alhough prospec heory has been ncorporaed recenly no models of asse prces, all of he exan applcaons make use of he represenave- 7 For revew arcles, see Lews (1995) and Engel (1996). 3

7 agen assumpon and he REH. 8 The REH-based models e expecaons rgdly o he model formulaed by an economs, hereby resrcng he updang of expecaons o he arrval of new nformaon on macroeconomc fundamenals. Because hese fundamenals are much less volale han asse reurns, he REH models ofen rely on modfcaons of preferences o generae a greaer varance of asse reurns. 9 Ths consderaon leads Barbers, Huang and Sanos (2001) o adop a parcular nerpreaon of Kahneman and Tversky s (1979) noon of reference dependence, called a house-money effec: how loss averse he nvesor s depends on hs pror nvesmen performance (Barbers, Huang and Sanos, 2001, p. 2). To compare he mplcaons of a house-money effec wh hose of our gap effec, we formalze a house-money effec n he conex of our model of foregn exchange speculaon. In sharp conras o he mplcaon of our gap effec, we show ha a house money effec mples a negave relaonshp beween he prema on foregn exchange speculaon and he expeced gap. Ths negave relaonshp s rejeced by emprcal evdence. 10 The leraure n fnance and economcs has also produced many models wh expecaons ha depar from he REH. In movng away from he REH whch arbues o agens a forecasng rule based on one specfc model economss mus confron hs key queson: whch non-reh forecasng models should be arbued o agens, gven ha he unverse of such models s, n prncple, unbounded? To lm he unverse of poenal forecasng models, exan deparures from he REH have reled on mporan nsghs from behavoral economcs concernng he behavor of agens n real-world markes. 11 Alhough hese approaches are superor o he REH models from an emprcal pon of vew, hey share one crucal feaure wh he REH approach: hey arbue o agens specfc quanave forecasng models and fxed mechansms for how 8 See Benarz and Thaler (1995), Barbers, Huang and Sanos (2001), Barbers and Huang (2001) and references heren. 9 Barbers, Huang and Sanos (2001) recognze ha, even wh he assumpon of loss averson, he sandard, REH-based seup s unable o explan he man perplexng feaures of he aggregae daa. 10 Ths fndng, however, does no necessarly mply ha a house-money effec s rrelevan; may smply mean ha he gap effec s sronger han a house-money effec. 11 Early approaches o modelng deparures from he REH n fnancal markes nclude he semnal sudes by Frankel and Froo (1987) and Delong, Shlefer, Summers and Waldman (1990a,b). For more recen sudes, see Gournchas and Tornell (2001) and Hong and Sen (2002, 2003). 4

8 agens updae hese models. Each one of hese formulaon s anamoun o an assumpon of one over-archng model wh a srucure ha s unchangng over me. Such an unchangng srucure may nvolve a fne number of pre-specfed forecasng models along wh a fxed rule governng how agens swch beween models n updang her forecass. 12 For example, a large class of sandard learnng models assumes ha all agens learn and forecas on he bass of a common model and a fxed updang mechansm hroughou he perod of learnng.(e.g., Evans and Honkapohja, 2001). In he foregn exchange marke, Frankel and Froo (1987) assume an unchangng srucure for expecaons and her updang: a represenave agen updaes hs expecaons by swchng accordng o a fxed (Bayesan) rule beween a chars model and a fundamenal model. 13 These behavorally-movaed deparures from he REH are an advance overherehapproach;heydonognorehefachaagensnreal-world markes may change her forecasng models raher han merely respond o he arrval of new nformaon when updang her forecass. However, hese deparures from he REH neglec he fac ha n a world of mperfec knowledge, agens, n general, no only face a choce among exan models whch an economs res o capure by a pre-specfed se of models bu hey also nven new models. Moreover, agens also nven new ways of revsng her models ha dffer from he fxed updang rules arbued o hem by an economs. Because exan deparures from he REH largely gnore he creave aspec of he process of he acquson of knowledge n a marke economy, hey also suffer from anoher dffculy. Though varous forms of rraonales nroduced no hese models are well esablshed, and can be plausbly jusfed on behavoral grounds, he assumpon of an unchangng srucure for expecaons and her updang s acually equvalen o a much sronger assumpon abou he degree of agens rraonaly: assumes agens per- 12 Ths nerpreaon of one model as nvolvng a number (or a class) of pre-specfed models wh a fxed rule governng swches beween models has also been adoped by he REH leraure. For examples, see Engel and Hamlon (1990) and Hansen and Sargen (2001a,b). For a crcal dscusson of hs less resrcve, bu sll REH-based approach o he modelng of expecaons, see Frydman and Goldberg (2002, 2003a) and foonoe 61 below. 13 For a relaed approach ha also assumes an unchangng srucure conssng of speculaors, who form expecaons accordng o he REH and feedback raders, who rade on he bass of smple rules see Delong, Shlefer, Summers and Waldman (1990b). Ths model feaures fully nformed. 5

9 ss, n perpeuy, n he parcular forms of rraonaly arbued o hem. In conras, our IKF approach s predcaed on he presumpon ha under mperfec knowledge economc agens do no adhere o one unchangng srucure n formng and updang her forecasng models. To do so would mply gross rraonaly, n he sense of endlessly passng up apparen prof opporunes. Thus, alhough he exan deparures from he REH shed lgh on parcular epsodes of he emprcal record, hey do no offer a general approach ha can replace he REH as a model of he forecasng process. Ths lack of generaly may explan why many economss are relucan o abandon he REH. The IKF approach ams o provde a general framework for modelng he forecasng process, ha can replace he REH. The mos mporan dsncon beween he non-ikf deparures from he REH and our IKF approach s ha we do no arbue specfc forecasng mechansms o agens and wedonoreaherevsonsofforecasngmodelsasamechancal process. Insead, we recognze ha knowledge s mperfec n general and, hus, economcagensmuschoose from among he myrad of exsng forecasng models and decde on wheher and how o develop new models. Because our poson s ha economc agens are no grossly rraonal, we characerze hs creave process n a qualave as opposed o a quanave manner. Ths qualave approach frees he economs from havng o specfy precsely agens forecasng models and how hese models are revsed. Thus, he IKF approach s conssen wh he creave process of model formulaon and dscovery; s no only compable wh a wde class of exsng models ncludng hose from behavoral economcs bu also allows for forecasng models ye o be nvened. Conrary o he mplc presumpon of he REH approach ha quanave resrcons are needed o mpose dscplne on he analyss we show n hs paper ha qualave resrcons on he revsons of he ndvdualforecasngmodelsareableogeneraemplcaonshacanberejeced by he daa. Moreover, as we demonsrae n hs paper, he qualave IKF approach s able o explan asse marke dynamcs ha he REH models deem anomalous. 6

10 2 An Overvew of he Paper Modelng he dynamcs of he equlbrum premum on foregn exchange nvolves wo seps: a defnon of momenary equlbrum a a pon n me and a specfcaon of he updang of forecasng models, and oher facors, ha move hs momenary equlbrum over me. In secon 3 of hs paper, we use dynamc prospec heory o derve our new momenary equlbrum a a pon n me, UAUIP, whle n secon 4, we use our IKF framework o model he movemen of hs momenary equlbrum over me. Secon 3.1 provdes a formal characerzaon of he ndvdual forecasng models and condonal forecass of he one-perod ahead excess reurn, R +1 = S +1 S FP,whereS and FP denoe he log levels of he exchange rae and he forward premum, respecvely. We represen ndvdual s forecasng model of R +1 n erms of a condonal probably dsrbuon of he one-perod ahead exchange rae, denoed by P(S +1 X,θ ), wherex and θ are he me- nformaon se and se of parameers of hs condonal dsrbuon.. 14 An ndvdual exchange rae forecas a me, s +1, s he mean of hs condonal dsrbuon, evaluaed a x,andθ. The condonal forecas of he reurn on a long (shor) poson of one un of foregn exchange held from me o +1 s hen defned as r +1 = s +1 s fp ( r +1 = fp +s s +1). 15 A a pon n me, our assumpons of ndvdual raonaly and mperfec knowledge mply ha, n general, ndvdual agens use dfferen forecasng models and herefore form heerogeneous forecass. In hs paper we assume ha he heerogeney of forecass enals he presence n he marke of boh bulls and bears speculaors who hold long and shor posons, respecvely. To clarfy he mporan elemens n hs overvew, we replace a dversy of forecasng models whn he groups of bulls and bears wh wo forecasng models, one for a represenave bull and he oher for a represenave bear, denoed by =l,s, respecvely. 16 Because we assume he presence of boh bulls and bears n he marke, 14 As s cusomary, we wll use upper and lower case leers o dsngush beween random varables and her realzaons, respecvely. 15 A long (shor) poson n foregn exchange s one n whch a rse (fall) n he value of foregn exchange leads o a posve reurn on he open poson. 16 In secons 3 and 4 of hs paper, we examne aggregaon under he heerogeney of preferences and forecasng models whn each group of bulls and bears and show ha under mld aggregaon condons, our conclusons reman vald. 7

11 a well-defned momenary equlbrum requres ha he amoun of capal each speculaor chooses o place a rsk (.e., hs poson sze) s lmed. We model hs decson problem on poson sze n secon 3.2, by augmenng he orgnal formulaon of prospec heory wh an assumpon we call dynamc loss averson. Dynamc loss averson assumes ha as speculaors conemplae larger open posons and, hus, conemplae larger poenal gans and losses hey are more sensve o he ncrease n he poenal losses han o he concoman ncrease n he poenal gans. We nex use he sandard decomposon of nex perod s rae of reurn, R +1 = R +1+R , o represen poenal losses of bulls and bears. Because bulls (bears) vew negave (posve) realzaons of R +1 as poenal losses, we represen hese losses by he varable R+1( R +1). + Furhermore, we represen an ndvdual assessmen of he poenal losses, whch we refer o an expeced loss and denoe by l, as he mean of he loss-par R+1 for bulls and R +1 + for bears of he ndvdual probably dsrbuons. 18 In secon 3.2, we show ha dynamc loss averson mples ha all agens lm he sze of her gambles when an expeced prof opporuny arses. 19 We fnd ha he opmal poson sze for bulls and bears f l and f s, respecvely mples ha speculaors hold open posons (eher long or shor) n foregn exchange, only f hey expec a posve reurn n excess of some mnmum value, whch we call an ndvdual uncerany premum and denoe by up. Thus, he opmal poson sze for bulls and bears, f l = f l ( r +1 l up l ) and f s = f s ( r +1 s up s ), s ncreasng n he condonal forecas of he reurn and decreasng n he uncerany premum. Our soluon for he opmal poson sze also shows ha he ndvdual uncerany prema can be nerpreed as compensaon for poenal losses: ndvdual uncerany prema depend posvely on he degree of loss averson and negavely on agens expeced losses. Ths resul mples ha he opmal posons of bulls and bears, f l = f l ( s l +1,s,fp,l l ) and f s = 17 R +1 + = R +1I(R +1 > 0) and R+1 = R +1I(R +1 < 0), wherei(.) s an ndcaor funcon. 18 Noe ha he expeced losses of bulls and bears l l and l s areboh negave magnudes. We wll refer o decreases n l l and l s as ncreases n he magnudes of he poenal losses. 19 In conemplang models wh heerogeneous expecaons, he behavoral economcs leraure has reled solely on he assumpon of rsk averson o oban wha s called lms o arbrage (e.g., see Barbers and Thaler, 2002). Our analyss wh dynamc prospec heory shows ha lms o speculaon can also be acheved solely on he bass of dynamc loss averson. 8

12 f s ( s s +1,s,fp,l), s are decreasng n he magnudes of he poenal losses. In secon 3.3, we se he oal of long posons equal o he oal of shor posons o derve UAUIP. We fnd ha hs equlbrum relaonshp s characerzed by an equaly beween he marke s expeced reurn (or average opnon) on foregn exchange, r +1 = r +1 l + r +1, s and he aggregae uncerany premum, up = up l up s. The equlbrum uncerany premum, hen, s he uncerany premum of bulls n excess of he uncerany premum of bears. The upward-slopng 45 o lne n fgure1salocusofsuch equlbrum pons n he model. One of he mplcaons of UAUIP s ha he sgn of he equlbrum premum on foregn exchange s deermned by he relave magnudes of r +1 l and r +1, s.e., n equlbrum, f bulls (bears) domnae n deermnng hesgnofheaverageopnon r +1 l > r +1( r s +1 l < r +1) s hen a balance beween long and shor posons n he marke obans when bulls (bears) requre a hgher mnmum reurn for holdng open posons up l >up s (up l < up s ). Fgure 1 assumes bears domnae n erms of he average opnon a he nal equlbrum pon (pon A),.e., r +1 = up < 0. The f l and f s curves n fgure2ploame he oal of long posons held by bulls and he oal of shor posons held by bears as negave and posve funcons of he exchange rae, s, respecvely. The nersecon of hese wo curves a pon A, hen, deermnes he equlbrum exchange rae. The equlbrum a pon A n fgures 1 and 2 s defned for agvenseof forecasng models,.e., for gven P(S +1 X,θ ) s. The logc behnd he slopes and shfs of he f l and f s curves s cenral o our analyss and s mplcaons for he behavor of he uncerany premum over me. Boh shfs n and movemens along hese curves are assocaed wh revsons of forecasng models. In our IKF framework, he process of revsng forecass does no jus enal a mechancal updang of he means of agens s condonal dsrbuons, bu s also compable wh a smulaneous revson of he hgher momens of hese dsrbuons;.under IKF revsons of forecass, n general, nvolve revsons of he forecasng models The sandard REH approach o he updang of forecass s o model hs process as a passve recompuaon of condonal forecass rggered by new realzaons of he varables n agens nformaon ses. The IKF framework no only allows for such convenonal, nformaon-drven, updang, bu s also compable wh revsons of forecass ha nvolve changes n he composon of he se of varables ncluded n he model and/or revsons of he values of he model parameers. 9

13 Fgure 1 % r + 1 B up A

14 Fgure 2 s S f B A B L f ' L f f L, f S

15 In secon 4, we use he IKF framework o model revsons of he ndvdual condonal probably dsrbuons and examne he mplcaons of hs process for he movemens of he equlbrum uncerany premum over me. To llusrae our analyss, consder an auonomous (for a gven s )upward revson of he mean of he condonal dsrbuon for a represenave bull, s l +1, assumng ha fp s gven. 21 Thempacofhsrevsonsshownasa rghward shf of he f l curve n fgure 2. The movemen of he equlbrum from pon A o pon B can be decomposed no wo seps; he shf of f l from A o B, for a gven s, and he resulng movemen of s,alonghef l and f s curves o he new equlbrum a pon B. In secon 4.1, we buld on a profound ye negleced dea pu forh by Keynes (1936) o model he frs-sep shf n f l ( s l +1,s,l l ). Weassumeha an auonomous ncrease n he bulls condonal forecas s l +1 no only causes r +1 l o rse, bu also causes l l o fall (.e. s magnude rses) because of a gap effec: a bull s (bear s) expeced loss depends negavely (posvely) on he dvergence, or gap, beween hs condonal forecas and he perceved hsorcal benchmark level, whch we denoe by s hb. To undersand he logc behnd he gap effec, assume for concreeness ha s > s hb for all agens,.e., all agens beleve ha he foregn currency s overvalued relave o he perceved hsorcal benchmark. As bulls ncrease s l +1, hey herefore expec a greaer overvaluaon of he foregn currency. Ths upward revson can occur despe he assumed belef, on he par of all agens, ha he asse prce wll evenually adjus back o s perceved benchmark level. The forecasng problem all agens face, herefore, s ha he asse prce may move perssenly away from s perceved benchmark over an exended me perod, pror o reverng back. The gap effec mples ha as bulls rase her expecaon of an overvaluaon, hey become more fearful of a movemen back o he hsorcal benchmark. Ths, hen, leads bulls o ncrease her assessmen of he magnude of he poenal losses. As bulls rase her assessmens of he poenal losses, hey smulaneously rase her uncerany prema, up l. Thus, n general, r +1 l up l may rse or fall. However, we assume ha alhough agens loss averson lms her wllngness o rsk capal, he degree of loss averson s no so hgh 21 We explore he movemen of he equlbrum premum followng a change n fp n Frydman and Goldberg (2002), where we fnd ha our IKF model of he uncerany premum sheds new lgh on he forward-premum anomaly. 10

16 as o preven agens from changng he sze of her posons n he same drecon as he change n her condonal forecass,.e., r +1 l up l rses above r +1 s up s. Ths ncrease n he expeced excess reurn leads bulls o ncrease her long posons, as depced by he shf of f l from pon A o B n fgure 2. A he nal exchange rae, hen, he oal of long posons s greaer han he oal of shor posons, causng s o rse. In secon 4.2, we model he second sep movemen of s from s nal equlbrum level o s new equlbrum level a pon B. An ncrease n s leads o revsons of he dsrbuon of nex perod s reurn, R +1,forboh bulls and bears. In general, hese revsons nvolve no only changes n he means, r +1 l and r +1, s bu also changes n l l and l, s and, hus, changes n up l and up s. The slopes of he f l and f s curves n fgure 2 follow from wo qualave assumpons on he updang of s +1 as a consequence of changes n s. 22 Under hese assumpons, he rse n s causes r +1 l up l o fall and r +1 s up s o rse, hereby workng o reesablsh equlbrum and a balance beween he oal of long and shor posons. The equlbrum movemens of r +1 and up are shown n fgure 1 as a movemen from pon A o pon B. The fgure assumes ha he nal frs-sep ncrease n s l +1 (and herefore n r +1) l was large enough so ha, afer he equlbrum movemen n s, he average opnon no only ncreases, bu s algebrac sgn changes from negave o posve. Equlbrum, hen, requres ha he aggregae uncerany premum no only ncrease, bu ha s algebrac sgn also change from negave o posve, as he domnan wegh n he average opnon shfs from bears o bulls. Thus, our IKF model of he uncerany premum provdes a smple explanaon of he sgn reversals n foregn exchange prema ha have been observed n he leraure. 23 The shfs n fgures 1 and 2 also llusrae anoher mplcaon of our IKF model of he equlbrum uncerany premum, whch we refer o as an aggregae gap effec: up ncreases wh he expeced gap from benchmark levels, gap = s +1 s hb. 22 The frs assumpon s ha d s +1 ds < 1, whch we need for sably of he momenary equlbrum. The second qualave assumpon s d s +1 ds > 0.We refer o hs assumpon as an average opnon effec. We movae hs assumpon by drawng on he exensve evdence n he behavoral economcs leraure ha hs assumpon capures he behavor of agens n he real world markes. 23 See Lews (1995) and Mark and Woo (1998) for he nably of sandard REH models o explan he observed sgn reversals n foregn exchange prema. 11

17 In secon 4.3, we formulae a model of foregn exchange prema wh a house-money effec and dynamc loss averson, and show ha hs model mples a negave relaonshp beween he equlbrum uncerany premum and he expeced gap from benchmark levels. Secon 5 provdes emprcal evdence n suppor of our IKF model wh an aggregae gap effec. Fgure 3 llusraes hs new emprcal fndng well. The fgure plos sx-monh averages of up, usng he MMSI daa, and he expeced gap, n a sample ha spans he perod from January 1983 hrough December Alhough no a sascal es, he me plos are raher suggesve ha he relaonshp beween up and gap s ndeed posve. Formal ess reveal ha such a posve relaonshp s suppored a very hgh sgnfcance levels. Thus, we are able o rejec he IKF model wh ahousemoneyeffecnfavorofheikfmodel whagapeffec;.bu see foonoe 10 above. Secon 6, whch concludes he paper, dscusses he relave mers of he qualave IKF and he quanave REH approaches o he modelng of he forecasng process n a world of mperfec knowledge. 3 Modelng he Decsons of Speculaors a a Pon n Tme In hs secon we model he decsons of speculaors n he foregn exchange marke under he assumpons of a dversy of ndvdual forecasng models and a heerogeney of expecaons. The nroducon of heerogeneous expecaons rases a problem ha has been noed n behavoral-fnance models such as Delong, Shlefer, Summers and Waldman (1990a,b) and Shlefer and Summers (1990): heerogeney of forecass and unlmed shor sellng requre lms o speculaon n order o oban a well-defned equlbrum To oban a measure of gap, we used he Bg Mac PPP exchange rae repored n he Aprl 1993 edon of he Economs (whch was 2.02), and hen used nflaon dfferenals (based on CPI seres from he IFS daa bank) o esmae he PPP exchange rae boh forwards and backwards. 25 A dealed examnaon of hs problem s ousde he scope of hs paper. In Frydman and Goldberg (2003b) we examne hs problem whn he conex of he moneary models of Dornbusch (1976) and Frankel (1979). We fnd ha under heerogeneous expecaons, equlbrum n hs class of models s no longer characerzed by uncovered neres rae pary. 12

18 Fgure 3 Sx-Monh Averages up gap

19 To solve hs problem, he leraure assumes ha all speculaors are rsk averse. 26 We show n hs secon ha lms o speculaon can also be obaned solely on he bass of prospec heory, whch assumes ha uly funcons are specfed over he change n wealh, provded ha he orgnal formulaon of prospec heory s exended o he dynamc seng of fnancal marke speculaon. We begn wh a descrpon of he speculave decson n he foregn exchange marke under he sandard assumpon of perfec capal mobly. A every pon n me, boh domesc and foregn speculaors operang n he foregn exchange marke face a decson on wheher o ake a long or a shor poson n foregn exchange. The assumpon of perfec capal mobly mples ha only pure speculaon maers,.e., he acvy of borrowng capal so as o ake smulaneously a shor poson n he borrowed currency and a long poson of equal sze n he oher currency. 27 The ex pos value of he reurn on a long poson of one un of foregn exchange from me o +1, whch we denoe by r +1, can be approxmaed as follows: r +1 =(s +1 s ) fp (1) Snce r +1 s he ex pos reurn from a shor poson of one un of foregn exchange, he decson of speculaor on wheher o ake a long or shor poson n foregn exchange depends on hs ndvdual forecas a me of r +1 n equaon (1), r +1. If agen were rsk neural, hen he would ake a long (shor) poson n foregn exchange f r +1 > 0 ( r +1 < 0). In developng our model, we manan ha a every pon n me bulls and bears coexs n he marke. Snce he forecass of bulls and bears are necessarly of he oppose sgn, he coexsence of bulls and bears n he marke necessarly mples a heerogeney of expecaons. 26 Ths resul, ha rsk averson leads o lms o arbrage, svewedasoneofheman pllars of behavoral fnance (see Shlefer and Summers, 1990 and Barbers and Thaler, 2002). Snce n a world of mperfec knowledge no speculaor knows he rue fundamenal value, we wll use he erm lms o speculaon. 27 Perfec capal mobly mples ha pure speculave flows are nfnely elasc wh respec o he expeced reurn on foregn exchange, elmnang he need o consder capal flows orgnang from all oher sources. I s based on he assumpon of no barrers o he nernaonal flow of capal, ncludng no consrans on shor sellng. See Mundell (1963) and Dornbusch (1976). 13

20 3.1 Dversy of Forecasng Models To develop our model, we need a formal characerzaon of he creave process by whch ndvdual agens form her forecass of r +1. In general, hs process can be concepualzed as nvolvng hree basc seps: A descrpon of he nformaon se on whch ndvdual agens form forecass The nformaon se used by agen a me, denoed by X,canbedvded no wo basc subses: a subse, X,f, conssng of varables capurng nformaon conaned n fundamenal facors (e.g. money supply or GDP growh) and a subse, X,nf, conssng of non-fundamenal facors (e.g., hose based on echncal radng or marke psychology) consdered relevan by agen for forecasng he reurn o foregn currency speculaon. We noe ha he fundamenal and/or nonfundamenal subses of he nformaon se are ndexed by and hus he composon of eher of hese subses s no resrced o reman unchanged over me. Ths reflecs he recognon ha, n general, ndvdual agens may use dfferen ses of varables n formng her forecass a dfferen pons n me. 28 To sreamlne he exposon, we rea all of he varables n X as random varables, even hough some of hese (parcularly hose n X,nf ) may be qualave n naure or consans, and hus, may have degenerae margnal probably dsrbuons. In wha follows, we assume ha he nformaon se for each agen ncludes S and FP. 29 A formal represenaon of he ndvdual forecasng model To form forecass n a world of mperfec knowledge, ndvdual agens need o form assessmens of he lkelhood of poenal realzaons of nex perod s reurn, R +1. They arrve a such judgemens on he 28 For emprcal evdence ha agens use dfferen varables n her forecas funcons a dfferen pons n me see Goldberg and Frydman (1996b,2001). See also Lyons (2001) and he survey sudes of Cheung and Chnn (2001) and Cheung e al (1999). 29 Ths plausble assumpon smplfes our noaon consderably, n ha allows us o express, n a parcularly smple way, he condonal dsrbuon of R +1, P R,,nerms of he condonal dsrbuon of S +1 : P R, (R +1 X,θ )=P(S +1 X,θ ) S FP. Noe ha f eher S and/or FP were excluded from X, we would need separae noaon for he condonal dsrbuons for R +1 and S +1. Because keepng he wo dsrbuons dsnc would no affec any of our conclusons, we gnore hs possbly. 14

21 bass of he me- realzaons of he varables hey choose o nclude n her nformaon se, and by some formal or nformal procedure hey choose o map hese realzaons no her assessmens of he lkelhood of poenal values of R +1. One way o model hs forecasng process, whch we follow n hs paper, s o represen an ndvdual forecasng model of R +1 n erms of a condonal probably dsrbuon over S We denoe hs laer dsrbuon by P(S +1 X,θ ), whereθ s a se of parameers of hs condonal dsrbuon. An ndvdual (condonal) forecas An ndvdual forecas of R +1 s based on he mean of he condonal dsrbuon P(S +1 X,θ ), evaluaed a he me- realzaons of he varables n X,denoedbyx, and he me- values of he parameers θ : r +1 = E P (S +1 x,θ ) s fp = s +1 s fp (2) Beyond provdng a formal represenaon of he ndvdual forecasng models, he foregong formulaon hghlghs he wo poenal sources of he heerogeney of expecaons: 1) a dversy among he ndvdual condonal dsrbuons, P (..); and/or2)dfferences n he measuremen of he varables n some common nformaon se, X, on whch forecass are based, for example, as n he Lucas (1973) sland model. We noe ha our formulaon of ndvdual forecasng models subsumes he REH-based expecaon funcon as a specal case: under he REH, all agens form expecaons based on he common, somemes referred o as he objecve, probably dsrbuon, P. Thus, o raonalze he heerogeney of expecaons, convenonal REH models mus appeal o dfferences n he measuremen of he varables appearng n he common forecasng model. 31 In conras, heerogeney n he IKF framework can be raonalzed by appealng o he mplcaons of he posulae of economc raonaly n a 30 The formal represenaon of he ndvdual forecasng models as condonal expecaons s analycally convenen. The analyss n hs paper can be exended o ncorporae one of he elemens of prospec heory: ha nsead of aachng probables o prospecs, agens aach decson weghs o hem. See Kahneman and Tversky (1979) and Tversky and Kahneman (1991,1992). Also see foonoe 33 below for a remark on hs pon. 31 See Lyons (2001) for such an approach n he foregn exchange marke. Alhough an exensve dscusson of he dffcules nvolved wh such raonalzaons s ousde he scope of hs paper, we reurn o some of hese problems n secon 5. 15

22 world of mperfec knowledge. For example, Frydman (1982) formally shows ha raonal agens, who do no pass up opporunes for gan, wll no, n general, use a common model and/or rely on he same nformaon when formng forecass n a world of mperfec knowledge. Thus, a heerogeney of forecasng models as well as dfferences n he nformaon used by agens canbelnkedoheraonaly of agens copng wh he fac ha hey mus forecas on he bass of mperfec knowledge. 3.2 Poson Sze a a Pon n Tme: Dynamc Prospec TheoryforaGvenSeofIndvdualForecasng Models In hs subsecon we buld on he semnal formulaon of prospec heory due o Kahneman and Tversky (1979) and Tversky and Kahneman (1992). Kahneman and Tversky developed prospec heory o explan he ofen gross nconssences beween he acual choces of ndvduals who are faced wh rsky or unceran oucomes and he predcons mpled by he convenonal expeced uly hypohess. The ypcal choce problems consdered by Kahneman and Tversky were sac, n ha hey nvolved a rch varey of moneary gambles nvolvng choces beween exogenously fxed oucomes wh respec o he poenal gans and losses. The large leraure n psychology and economcs has presened srkng evdence ha n such suaons, prospec heory s able o explan he observed choces of ndvdual agens much more successfully han he convenonal expeced uly hypohess. Alhough gambles wh exogenously fxed oucomes characerze a large class of decson problems under uncerany, such gambles do no characerze speculaon n fnancal markes. Oucomes from fnancal speculaon are endogenous, n he sense ha, gven speculaors forecasng models of nex perod s reurn, hey mus decde on he amoun of capal o place a rsk. Ths decson wll hen deermne he overall poenal gans and losses. In applyng prospec heory o speculave decsons, however, here s one specal case he case of homogeneous expecaons n whch he decson of how much o gamble when a prof opporuny s perceved need no be modeled. Wh homogeneous expecaons, here s a unque prce a whch agens are wllng o hold a gven supply and hs equlbrum level s ndependen of he sze of speculave posons ousde of equlbrum. Snce recen applcaons of prospec heory o he modelng of asse markes all make use 16

23 of he sandard represenave-agen assumpon (e.g., Benarz and Thaler, 1995, Barbers, Huang and Sanos, 2001 and Barbers and Huang, 2001), hey are able o sdesep he problem of modelng he sze of speculave posons. In hs paper, we assume a heerogeney of forecasng models and expecaons. In hs seng, we fnd ha he orgnal, sac formulaon of prospec heory does no provde a bass for modelng he decson of how much o gamble, gven he ndvdual forecasng models. Ths consderaon leads us o our assumpon of dynamc loss averson, whch mposes a condon on he curvaure of he uly funcon over gans relave o s curvaure over losses. In hs secon, we make use of a parcularly smple verson of dynamc loss averson, whch connecs he degree of loss averson o poson sze, hereby makng preferences dynamc. 32 We show ha wh dynamc loss averson, loss-averse agens lm he sze of her gambles when an expeced prof opporuny arses. Ths resul leads o our new equlbrum condon for he foregn exchange marke, Dynamc Prospec Theory Accordng o Tversky and Kahneman (1991, pp ), prospec heory mples ha rsky prospecs are evaluaed by a value funcon ha has he followng hree basc characerscs: 33 Reference Dependence The carrers of value are gans and losses defned relave o a reference pon and he effec of he level of wealh on uly s assumed o be 32 In applyng prospec heory o fnancal markes, Barbers, Huang and Sanos ( 2001) and Barbers and Huang (2001) also make preferences dynamc by combnng he assumpon of loss averson wh he house-money effec. As we show n hs subsecon, such a specfcaon of dynamc preferences would no by self lm he sze of speculave posons. However, Barbers, Huang and Sanos ( 2001) and Barbers and Huang (2001) embed her verson of dynamc preferences under loss averson no a general uly funcon ha s also defned over he level of consumpon. Snce hs par of he uly funcon s assumed o be concave, hese sudes would n prncple be able o oban lms o speculaon based on he sandard assumpon of rsk averson f he assumpon of homogenous expecaons were dropped. 33 Alhough Kahneman and Tversky resrc her orgnal dscusson o prospecs wh so-called objecve or sandard probables hey noe ha he heory can also be... [appled] o he ypcal suaons of choce, where he probables of oucomes are no explcly gven (Kahneman and Tversky, 1979, p. 263). 17

24 relavely neglgble. Loss Averson The dsuly from losses exceeds he uly from gans of he same magnude. As hey pu, losses loom larger han correspondng gans. Dmnshng Sensvy The margnal value of boh gans and losses decreases wh her sze Benarz and Thaler (1995) exend he noon of loss averson n an nsghful and mporan way by observng ha when decson makers are loss-averse hey wll be more wllng o ake rsks f hey evaluae her performance...nfrequenly (Benarz and Thaler, 1995, p. 75). Ths leads Benarz and Thaler o a noon hey call myopc loss averson, whch combnes loss averson wh frequen evaluaon. We follow Benarz and Thaler by adopng he assumpon of myopc loss averson as one of he key assumpons of our approach. In wha follows we assume ha speculaors on foregn exchange evaluae her performance every perod. The dsncon beween gans and losses plays a cenral role n prospec heory and so we need expressons for agens forecass of he poenal gans and losses from speculaon. Usng he sandard decomposon of R +1,we have r +1 = E P (R +1 x,θ )=E P (R +1 x +,θ )+E P (R+1 x,θ )= r +, +1 + r, +1 (3) Recallng, ha for a bull (bear), posve realzaons of R +1 mply gans (losses) and negave (posve) realzaons mply losses (gans) on an open poson equal o one un of foregn exchange, we can express he oal expeced gan and loss of a bull and a bear as follows: Remark 1 Le g l (g s )andl l (l s ) denoe he expeced gan and loss a me of a bull (bear) who has a one-un long (shor) poson n foregn exchange. Also, le f denoe he number of uns of foregn exchange ha speculaor has decded o gamble, o be referred o as he sze of speculaor s poson. Then he oal expeced gan and he oal expeced loss on a poson of sze f 0 are Noe ha alhough we use upper case G and L o denoe he oal expeced gans and losses, s clear from (4)-(7) ha G and L are no random varables bu raher he values of he oal expeced gans and losses evaluaed a me. 18

25 for a bull: G l L l = g l f l = r +1f +, l = E P (R +1 x +,θ )f l 0 (4) = l l f l = r +1f, l = E P (R+1 x,θ )f l 0 (5) and for a bear: G s L s = g s f s = r +1f, s = E P (R+1 x,θ )f s 0 (6) = l s f s = r +1f +, s = E P (R +1 x +,θ )f s 0 (7) We can now specfy he preferences of a loss-averse speculaor as a funcon of hs expeced gan and loss. Tversky and Kahneman (1992) proposed he followng specfc funconal form for he uly funcon of loss-averse agens, ½ V = G β λ ( L ) α (8) where he parameer λ capures he degree of loss averson, andβ and α deermne he curvaure of he uly funcon over oal gans and losses, respecvely. A value of λ greaer han 1 mples loss averson on he par of agens, namely ha losses loom larger han correspondng gans. Based on expermenal evdence nvolvng one-sho, fxed gambles, Kahneman and Tversky (1992) esmaed he value of λ o be n excess of Snce he curvaure of he uly funcon over poenal gans relave o s curvaure over poenal losses plays no role n sac decson problems, Kahneman and Tversky (1992) se β = α. Under hs assumpon, Kahneman and Tversky found β = α =.88, mplyng ha V s concave over gans and convex over losses. In applyng prospec heory o fnancal markes, Barbers, Huang and Sanos (2001) and Barbers and Huang (2001) use a smpler, lnear verson of he uly funcon n (8): V (G,L )=gf. + λ lf = vg f + vl f = Vg + Vl (9) where we express he uly funcon for a loss averse speculaor n erms of he expeced gan and loss from a one-un poson n foregn exchange, g 35 Ths resul has been found n a wde range of expermenal conexs. See Barbers and Thaler (2002) and references heren. 19

26 and l, mulpled by he sze of he poson, f. Thus, vg + vl denoes he expeced uly from an open poson of one un and Vg + Vl s he expeced uly on a poson of sze f. The uly funcon n (9) capures loss averson, bu gnores he assumpon of dmnshng sensvy. Barbers and Thaler argue ha such a smplfcaon s jusfed because s dffcul o ncorporae all hese feaures no a fully dynamc framework; bu also, s based on Benarz and Thaler s observaon ha s manly loss averson ha drves her resuls (Barbers and Thaler, 2002, p. 26). A uly funcon ha s lnear n oal gans and losses, however, does nogverseolmsospeculaon. Noehaonceaspeculaorformulaes hs forecasng model a me and observes he realzaons of he varables n hs nformaon se, he values of he condonal forecas, r +1, andhe expeced gan and loss, g and l, are pre-deermned from he pon of vew of deermnng he opmal sze of hs poson. Le f o denoe he poson sze of speculaor ha maxmzes hs uly. Thus, f speculaor beleves ha g + λ l < 0, he wll be unwllng o ake an open poson n foregn exchange of any sze (.e., f o =0), whereas f g + λ l > 0, he wll wan o ake an open poson of unlmed sze (.e., f o = ). Thus, o jusfy lms o speculaon, solely on he bass of prospec heory, he uly funcon of loss-averse agens mus be concave n f,.e., he dsuly of losses mus grow faser han he uly of gans as he poson sze ncreases. Gven he uly funcon n (8), hs requres ha β<α,.e., he degree of concavy of V overgansmusbelargerhansdegree of convexy over losses. Ths leads o he followng exenson of prospec heory o he dynamc seng of fnancal-marke speculaon: Dynamc Loss Averson The dsuly of losses grows faser han he uly of gans as he magnudes of losses and gans ncrease proporonaely. Remark 2 We noe ha he assumpon of dynamc loss averson s n he spr of he orgnal formulaon of prospec heory: as he magnudes of losses and gans ncrease proporonaely, he oal change n losses s assumed o loom larger n uly han he oal change n gans. To ncorporae he assumpon of dynamc loss averson n a convenen way, we rean he addve form of he uly funcon n (9) wh respec 20

27 o he oal expeced gan and loss. We also gnore he hrd posulae of prospec heory, bu unlke Barbers, Huang and Sanos (2001) and Barbers and Huang (2001), we develop a specfcaon ha s lnear n he expeced gan and concave n he expeced loss. 36 Such a specfcaon follows from he assumpon ha here exss a posve relaonshp beween he dsuly of losses on a un poson, v l, whch Kahneman and Tversky (1979) assumed o be a consan, and poson sze. The dea behnd hs assumpon s a naural one; agens fear of poenal losses grows as poson sze grows. To keep he analyss smple, whou affecng any of our conclusons, we assume ha v l = λ l = λ 1l λ 2f, 1 <λ <λ max (10) where now he degree of loss averson s a funcon of poson sze,.e., λ = λ 1 + λ f 2 s some consan. 37 Pluggng (10) no (9) yelds ( l he followng expresson ),andλmax for he expeced oal ulyonanopenposonof sze f held by a loss-averse agen: V = gf + λ 1lf λ 2(f ) 2 (11) The assumpon of dynamc loss averson, whch mples concavy over f, ensures ha here s a fne poson sze ha maxmzes speculaor s uly. Dfferenang V wh respec o poson sze, and seng he resul equal o zero, yelds he followng expresson for f o : f o = 1 2λ (g + λ 1l) (12) 2 Noe ha a negave value for f o occurs for speculaor only when g + λ 1l < 0. In hs case, speculaor expecs ha an open poson n foregn exchange wll no ncrease hs uly and herefore decdes o say ou of he marke,.e., f o =0.Gvenhaλ 1 > 1, he speculaor wll say ou of he marke even when g + l > 0,.e., alhough speculaor may 36 All we need s a concave funcon n f. Thus, our man resuls, alhough more complcaed, would be unchanged f we were o manan dmnshng sensvy over boh gans and losses, as n Kahneman and Tversky (1979), and assumed ha he curvaure of V was greaer over gans. 37 Havng made he degree of loss averson a funcon of poson sze, we need o resrc he parameers λ 1 and λ 2 n (10) so ha all values of λ le whn a range beween 1 and λ max. Ths can be done easly under mld assumpons. See Frydman and Goldberg (2003b) for a dervaon of such condons. 21

28 expec o earn a posve reurn from an open poson n foregn exchange, he sze of hs reurn may no be large enough o compensae hm for hs greaer sensvy o losses. The soluon n (12) mples, herefore, ha a posve f o s necessarly assocaed wh g + l > 0. Ths reasonng leads o one of he major conclusons of our analyss: Concluson 1 All dynamcally loss-averse speculaors requre an expeced reurn n excess of some ndvdually deermned posve value n order o ake open posons n foregn exchange. In he nex secon, we connec hs mnmum expeced reurn o he uncerany faced by agens concernng he mng of counermovemens n he exchange rae back o s hsorcal benchmark level. We refer o hs mnmum expeced reurn, herefore, as an ndvdual uncerany premum, up. To oban a smple expresson for up, we rewre (12) as follows: f o = (g + l)+(λ 1 1) l = 1 I(ype ) r +1 up 2λ 2 2λ 2 (13) where I(ype ) s an ndcaor funcon such ha f ype = bull, I =1,and f ype = bear, I = 1. From (13), he mnmum expeced reurn agen needsoakeanopenposons: up = 1 λ 1 l > 0 (14) The expresson n (14) shows ha he ndvdual uncerany premum can be wren as a smple funcon of agen s expeced loss and degree of loss averson as capured by he parameer λ 1 > 1. Agen wll be wllng o ake an open poson n foregn exchange only when hs condonal forecas of nex perod s reurn s large enough o compensae hm for hs greaer sensvy o losses (.e., r +1 >up ). I s useful o compare he speculave decson under dynamc loss averson wh he decson under rsk neuraly. Wh rsk neuraly, a r +1 > 0 ( r +1 < 0) leadsagen o ake a long (shor) poson n foregn exchange of unlmed sze. Bu wh dynamc loss averson, agen lms hs wllngness o comm capal o he speculave game because hs fear of losses grows as poson sze grows. The assumpon of dynamc loss averson, herefore, leads o lms o speculaon. Moreover, as we noed earler, he fear of losses may be large enough so ha even hough a speculaor may expec a posve 22