Realization Utility. with Reference-Dependent Preferences

Transcription

1 Realzaton Utlty wth Reference-Dependent Preferences Jonathan E. Ingersoll, Jr. Yale School of Management Lawrence J. Jn Yale School of Management PO Box 0800 New Haven CT September 18, 01 Forthcomng, The Revew of Fnancal Studes We thank our colleagues at the Yale School of Management and the semnar partcpants at the LBS Trans-Atlantc Doctoral Conference, the Unversty of Massachusetts, Amherst, and the 01 Western Fnance Assocaton Annual Meetng for helpful dscussons. In partcular we would lke to thank Nck Barbers, Davd Hrshlefer, and an anonymous referee. Jn acknowledges support from a Whtebox Advsors grant.

2 Realzaton Utlty wth Reference-Dependent Preferences Jonathan E. Ingersoll, Jr. and Lawrence J. Jn Yale School of Management We develop a tractable model of realzaton utlty that studes the role of reference-dependent S- shaped preferences n a dynamc nvestment settng wth renvestment. Our model generates both voluntarly realzed gans and losses. It makes specfc predctons about the volume of gans and losses, the holdng perods, and the szes of both realzed and paper gans and losses that can be calbrated to a varety of statstcs, ncludng Odean s measure of the dsposton effect. Our model also predcts several anomales ncludng, among others, the flattenng of the captal market lne and a negatve prce for dosyncratc rsk. JEL classfcaton: G0, G11, G1 Keywords: Realzaton Utlty, Prospect Theory, Dsposton Effect

3 Realzaton Utlty wth Reference-Dependent Preferences Jonathan E. Ingersoll, Jr. and Lawrence J. Jn How do nvestors decde about and evaluate ther own nvestment performance? Standard economc theory posts that nvestors maxmze the expected utlty of ther lfetme consumpton stream by dynamcally adjustng ther portfolo allocatons based on ther current wealth and ther expectatons of the future. Although ths way of modelng nvestors behavor may be close to realty for sophstcated nvestors, t s questonable whether less sophstcated nvestors behave ths way. A growng amount of research shows that ndvdual nvestors do not always behave n the ways that expected utlty theory predcts. In partcular, the ndependence axom seems troublesome as does the assumpton of rsk averson, at least for losses. The latter assumpton s not a requrement of expected utlty theory, but t or somethng lke t s mportant for most equlbrum models whch follow from maxmzng behavor. In contrast to what theores lke the APT or CAPM predct, ndvdual nvestors seem to be partcularly concerned about an asset s change n prce from a reference pont. Behavoral lterature suggests alternatve vews of modelng nvestors behavor. Kahneman and Tversky (1979) have argued that rsk averson does not characterze many choces and proposed an S-shaped utlty functon. Shefrn and Statman (1985) use mental accountng to justfy nvestors concentratng on specfc separate ncdents. Thaler (1999) says A realzed loss s more panful than a paper loss. Barbers and Xong (01) study a model whch assumes that nvestors thnk of ther nvestng experence as a seres of separate epsodes durng each of whch they ether made or lost money and that the prmary source of utlty comes n a burst when a gan or loss s realzed. 1 Frydman et al. (01) fnd evdence usng the neural data that supports ths realzaton utlty hypothess. In ths paper, we use these notons to develop an ntertemporal model of nvestors who have prospect theory s S-shaped utlty and who evaluate ther performance ncdent by ncdent based on realzed profts and losses. Our model s a partal equlbrum framework wth an nfnte horzon. An nvestor purchases stocks whose prces evolve as geometrc Brownan motons. At each subsequent pont n tme, the nvestor decdes whether to hold onto hs current nvestment or realze hs gan or loss thereby obtanng an mmedate utlty burst. If he sells, he renvests the proceeds after transacton costs nto another stock. We show that the nvestor s optmal strategy s to wat untl the stock prce rses or falls to certan percentages above or below the purchase prce before sellng. Our model ncludes that of Barbers and Xong (01) as a specal case. Voluntary loss takng can be optmal n a dynamc settng because the subsequent 1 Further dscusson on the psychologcal foundaton of vewng nvestments as epsodes s n ther paper. Kyle, Ou-Yang, and Xong (006) and Henderson (01) study one-tme lqudaton problems wth prospect theory preferences. But renvestment, whch s a key component of our model, s gnored n ther models. 1

4 renvestment resets the reference level and ncreases the lkelhood of realzng future gans. But n the Barbers Xong (01) model, utlty s pecewse lnear. As a drect consequence, they predct that nvestors voluntarly realze gans but never voluntarly sell at a loss, whch s clearly unrealstc and nconsstent wth the data. In our model, wth an S-shaped functon, margnal utlty decreases wth the magntude of both gans and losses. Ths means that lfetme utlty can be ncreased by takng frequent small gans along wth occasonal larger losses because the latter have less total dsutlty than the utlty of the former, and realzng losses resets the reference level for future gans. The dsposton effect, an emprcally robust pattern that ndvdual nvestors have hgher propenstes to realze gans than to realze losses, 3 follows naturally from ths result but t s a dynamc result. What has been commonly argued n both theoretcal and emprcal lteratures s that an S- shaped utlty functon leads to the dsposton effect because rsk seekng over losses nduces nvestors to retan ther postons and gamble on the future whle rsk averson over gans nduces the opposte. However, ths s a statc argument. Extrapolatng ths reasonng perod by perod would mply losses are never realzed; the dsposton effect should be nfnte. In our dynamc realzaton utlty model exactly the opposte s true. Investors naturally want gans, but an S- shaped utlty helps to generate voluntary losses and thereby reduces the magntude of the dsposton effect to the observed level. We calbrate our model n two parts. Frst we show that the magntudes and frequences of realzed gans and losses and the frequences of paper gans and losses as observed n the tradng data of Odean and others are consstent wth the type of smple two-pont strategy our model predcts. In partcular, Odean reports that 54% of round-trp trades are realzed gans wth an average sze of 8%; the remander are losses averagng 3%. Also condtonal on a trade, nvestors realze 15% of possble gans and only 10% of possble losses. Usng those average realzed gan and loss szes, our model makes the very accurate predcton that 58% of sales should be gans, and nvestors should have propenstes of 14% and 11% to realze gans and losses, respectvely. In addton, we propose a modfed form of Tversky-Kahneman utlty that generates the two optmal sales ponts, 8% and 3%, ether alone or n a mxture of heterogeneous nvestors. Our model also has a varety of other emprcal mplcatons and predctons. For nstance, nvestors may be rsk-seekng n some crcumstances due to the opton value nherent n realzng losses; ths helps explan a flatter securty market lne and the negatve prcng of dosyncratc rsk as shown n Ang, Hodrck, Xng, and Zhang (006). It may also help explan why nvestors appear to hold portfolos that appear under-dversfed. The plan of our paper s as follows. In Secton 1, we lay out a specfc ntertemporal reference-dependent realzaton utlty model and present ts soluton and basc nsghts. Secton 3 The dsposton effect for ndvdual nvestors has been found n the U.S., Israel, Fnland, Chna, and Sweden, by Odean (1998), Shapra and Veneza (001), Grnblatt and Keloharju (001), Feng and Seasholes (005), and Calvet, Campbell, and Sodn (009), respectvely. It s also documented for U.S. mutual fund managers, the real estate market, and the exercse of executve stock optons, by Frazzn (006), Genesove and Mayer (001), and Heath, Huddart, and Lang (1999).

5 examnes the propertes of the derved value functon and analyzes the optmal sales polces. Secton 3 provdes a detaled calbraton of our model to several emprcal regulartes. Secton 4 analyzes voluntary loss realzaton n a more general context. Secton 5 presents further model applcatons and predctons. Secton 6 gves some concludng remarks and drecton for future research. A summary of the mportant notaton, all the proofs, and some more detaled techncal consderatons are ncluded n the Appendx. 1. A Realzaton Utlty Model wth Tversky-Kahneman Utlty In ths secton we present a smple, specfc model of ntertemporal realzaton utlty. Our nvestor takes postons n a seres of purchases, buyng a number of shares and later sellng hs entre poston and renvestng t. 4 Each realzed gan or loss contrbutes a burst of utlty, and our nvestor acts to maxmze the expectaton of the sum of the dscounted values of these bursts. We assume the nvestor apples narrow framng when he evaluates hs gans. Ths assumpton sde-steps any complcatons that mght arse from dversfcaton or rebalancng motves. 5 Narrow framng can be justfed f the nvestor derves realzaton utlty only when both the purchase and sale prces of the asset are salent, and, therefore, evaluatng ndvdual assets s the applcable settng for studyng realzaton utlty. As a result, even when the nvestor holds multple stock postons smultaneously, narrow framng allows us to study each sequence of purchases and sales separately. Secondly, we assume that a utlty burst s receved only at the tme when a gan or a loss s realzed. As wth prospect theory, we normalze utlty so that gans and losses contrbute postve and negatve utlty, respectvely. 6 Whle t s assumed that utlty depends prmarly on the sze, G of the gan or loss, t seems reasonable that the reference level, R, mght also have a separate effect. In partcular, a gan or loss of a gven sze probably has a greater utlty mpact, ether good or bad, the smaller s the reference level; e.g., the gan or loss of $10 s felt more strongly when the reference level s $100 than when t s $500. Therefore, we denote the utlty burst functon as a functon of both varables, U(G, R). In ths paper we assume that U(G, R) s homogeneous of degree n G and R UGR (, ) RuGR ( / ). (1) 4 Our model restrcton of full lqudaton s an emprcally plausble one for ndvdual nvestors. Feng and Seasholes (005) document that ndvdual nvestors tradng through a large Chnese brokerage house durng lqudated ther full poston 80.35% of the tme when sellng. Shapra and Veneza (001) report that approxmately 80% of round trps on the Tel Avv Stock Exchange n 1994 conssted of a sngle purchase followed by a sngle sale of the entre holdng. Kausta (010) reports a smlar result for hs Fnnsh data though he does not provde specfc numbers. 5 For nstance, the nvestor mght have an ncentve to sell a losng stock to purchase a wnnng stock. Another example could be the ncentve of purchasng a dversfed fund. These consderatons are outsde the scope of ths paper though some related dscusson s provded n the last two sectons of the paper. 6 Typcally the centerng of utlty s arbtrary and has no effect on expected utlty maxmzaton. In some models lke ths, the nvestor mght be able to choose to take no acton at all so f no acton s presumed to gve a utlty of zero, then the centerng chosen here can affect partcpaton n the market. 3

6 Ths assumpton s mportant for keepng the model tractable, but t also focuses utlty on rates of return rather than dollar changes whch s n keepng wth the general emphass n Fnance. Expressed n ths way, the scalng parameter gauges the mpact of the reference level on utlty bursts measured as rates of return. We study a Merton-type partal equlbrum economy n contnuous tme wth an nfnte horzon. At t 0, the nvestor chooses ether to stay out of the market whch earns hm a utlty of zero or to nvest n one of a number of dentcally dstrbuted stocks. The stock prce evolves accordng to a geometrc Brownan moton, ds / S dt d, where µ and are the growth rate and logarthmc standard devaton, respectvely, and s a standard Brownan moton. At each subsequent pont n tme, t, the nvestor chooses ether to hold hs nvestment for a longer tme or to sell hs entre poston and realze a utlty burst. When he sells, he pays a proportonal transacton cost, k s, and renvests the net proceeds after payng a second proportonal transacton cost, k p reducng hs nvestment to X (1 ks) X /(1 kp) KX. Between realzaton dates, the t t t nvestment value follows the same geometrc Brownan moton as the underlyng asset dx / X dt d. () In a statc prospect theory settng, the reference level s essentally a parameter of the utlty functon defnng the status quo. However, n our dynamc model, we must address how t s updated and exactly how the gan or loss s measured relatve to t. The smplest rule s R s set at the net purchase prce, as defned above, and remans constant between sales. 7 That s, when the nvestor sells hs stock for X t, he resets hs reference level to KX t untl the next sale. However, ths s a subjectve matter and could dffer from nvestor to nvestor; there are other ways that the new reference level mght be set. For nstance, an nvestor mght vew t as the gross amount nvested ncludng the purchasng cost,.e., R = (1k s )X t. It mght also be some ntermedate level partcularly f the transacton costs have dfferent components such as a bdask spread and a commsson. Most brokerage accounts show the purchase prce whch would tend to emphasze the net nvestment as the reference level. On the other hand, the tax cost bass ncludes the purchasng cost whch would tend to emphasze the gross nvestment as the reference level. In our analyss, we assume the smplest case that nvestor fully accounts for costs and sets the reference level to the net amount nvested, KX. 8 A related ssue s how the nvestor evaluates hs utlty burst upon a sale. Agan, there are several ways he mght do so dependng on hs subjectve vew of the transacton costs. For example, f he gnores costs completely, then the gan s the gross sales value less the reference 7 Throughout ths paper we assume the reference level s constant between sales. More generally, t mght grow determnstcally at a constant rate (lke the nterest rate), or evolve stochastcally over tme. It could also be updated based on recent hstory of the stock prce. 8 The analyss here s largely unchanged for dfferent ways of settng the reference level. If an nvestor adopts the gross cost vew, then equaton (5) below has V(KX, (1k s )X) as the second term on the rght-hand sde. Presumably, an nvestor would not adopt a gross cost vew for settng the new reference level as well as recognzng both costs n assessng the gan as ths would double count the purchasng costs. However, the only requrement for our model s that the nvestor sets hs reference level consstently over tme. Barbers and Xong (01) also adopt the net cost nterpretaton whch n ther notaton s (1k)X wth k beng the round-trp transacton cost and only consder the full recognton of transacton costs n determnng gan sze. 4

7 level, Gt X t R t. If he fully recognzes transacton costs and compares the net renvested amount to the reference level, then Gt KX t R t. If he vews the gan as the dfference between the net proceeds of the sale and the reference level, then Gt (1 ks ) X t R t. These three cases are covered by defnng the gan as Gt X t R t, and settng the parameter to 1, K, or 1k s, respectvely. Intermedate vews are also possble. We leave the parameter free allowng many nterpretatons. The tme-consstency of these rules together wth the assumptons that () the utlty bursts n (1) are homogeneous of degree n X and R, () the asset value process has stochastc constant returns to scale, and () the nvestment horzon s nfnte jontly guarantee that the future looks the same dependng only on the current nvestment and reference level. Ths smplfes our problem n two ways. Frst, t removes tme as an explct varable. Second, our nvestor always has the ncentve to renvest mmedately upon sellng a poston snce he chose to enter the market n the frst place. Denote the value functon dscounted to tme t by V(X t, R t ). As dscussed above, V does not depend on tme explctly but only on the current nvestment and reference level. By defnton, the value functon s the maxmzed expectaton of the sum of future dscounted utlty bursts (, ) max V X (, ) t Rt t e U Gt R { } t (3) where s the rate of tme preference, G t and R t are the dollar sze and the reference level for the th future gan, respectvely, and t s the random tme t s realzed. In our model, these are stoppng tmes that are endogenously chosen by the nvestor to maxmze hs lfetme expected utlty. 9 To solve the problem posed by (3), we use the tme-homogenety property to rewrte t as a recursve expresson V( Xt, Rt) max t e U( X t Rt, Rt) e V( KX t, KX t) (4) where s the tme untl the next sale. Hereafter, we suppress tme subscrpts for notatonal convenence unless necessary for clarty. At a sale, the value functon before the sale must equal the sum of the utlty burst of the sale and the post-renvestment contnuaton value functon. So upon a sale realzng X before costs, V( X, R) U( X R, R) VKX, KX). (5) Between sales tmes, equaton (4) can be re-expressed usng the law of terated expectatons and Itô's lemma t t 1 0 { de [ V( X, R)]} e XV XV V dt. (6) XX X 9 To complete the specfcaton of ths maxmzaton, we need to assgn a utlty value to the polcy of never executng any sales. The obvous choce n ths case s to assgn ths polcy a utlty value of zero, the same value that would be realzed wth a polcy that allowed sales but never happened to execute any. 5

8 Because U(G, R) s homogeneous of degree n G and R and the asset value process has stochastc constant returns to scale, V must be also homogeneous of degree n X and R and therefore can be wrtten as V( X, R) R v( x), where v s the reduced-form value functon and x X /R s the gross return per dollar of the reference value. 10 The equaton for v s The general soluton to (7) s (7) 1 0 x v xv v ( ) vx ( ) Cx 1 Cx where 1,. (8) Ths s true regardless of the form for u. The utlty of the sales bursts affects only the constants, C 1 and C. Agan due to the homogenety, the optmal sales strategy must be to realze a gan or loss when the stock prce reaches a constant multple,, or fracton,, of the reference level. 11 The upper sales pont,, must exceed 1/ > 1 as otherwse the sale s not a gan after costs. The lower sales pont,, must be less than 1. 1 Applyng the homogenety relaton (1) to the boundary condton (5) yelds the reducedform boundary condtons v( ) u( 1 ) ( K) v(1), v( ) u( 1) ( K ) v(1). (9) Equatng these to the general soluton from (8), we can determne the constants C 1 and C n terms of the polcy varables c( ) u( 1) c( ) u( 1) C1 c ( ) c ( ) c ( ) c ( ) 1 1 c ( ) u( 1) c ( ) u( 1) where ( ) ( ). 1 1 C c K c1( ) c( ) c1( ) c( ) (10) The optmal sales ponts, and, can now be determned ether by maxmzng C 1 + C, 10 V must be postvely homogeneous;.e., 0. Snce V( X0, X0) X 0v(1), utlty s decreasng n the amount of the orgnal nvestment when < 0, and the nvestor would always prefer to reduce hs ntal nvestment and n the lmt not partcpate at all. A postve also assures that U(gR, R) s ncreasng n R for a fxed g; that s, the hgher the reference level, the bgger s the utlty of a gven rate of return. Ths property s smlar to ncreasng relatve rsk averson. 11 See the Appendx for more detals on the constancy of the optmal polcy. 1 A sale at any pont n the range (1, 1/) produces a subjectve loss after accountng for transacton costs. Under a constant polcy wth n ths range, there would never be any sales at a hgher prce as the stochastc process for x s contnuous and begns at 1 after each repurchase when the reference level s set to the net nvestment. But ths means that only losses wth ther negatve utlty bursts would be realzed leadng to a negatve v. Ths could not be the optmal polcy as never sellng gves a utlty of zero as does not partcpatng at all. 6

9 Fgure 1: Determnaton of the Optmal Sales Polces. Ths fgure llustrates the value functon and the optmal polcy for realzng gans and losses. The value n the contnuaton or nosales regon s tangent to the sum of the payoff functon and the contnuaton value at each sales pont, or. whch s the value of v(1), or by applyng the smooth-pastng condton at both sales ponts. 13 The soluton and the optmal - strategy are llustrated n Fgure 1.The no-sales regon runs from to. The value functon exceeds the sum of the utlty burst plus the contnuaton value n ths regon as llustrated; t s tangent to the payoff ncludng contnuaton value at and. For some parameter values, t may be optmal to forgo all losses. In these cases, the contnuaton value s not large enough to offset that dsutlty of realzng a loss. The value functon s stll gven by (8) though C = 0 and, therefore, v(0) = 0. Typcal stock and transacton costs values are = 9%, = 30%, and k s = k p = 1%. How would a realzaton-utlty nvestor trade ths stock? To answer ths queston, we must specfy the burst utlty functon. A reference-scaled verson of the Cumulatve Prospect Theory (CPT) 13 The optmal sales strategy must maxmze v for every value of ts argument n the contnuaton regon, and x = 1 s guaranteed to be n the contnuaton regon snce < 1 < 1/ <. Note that the smooth pastng condton does not smply match the dervatve of v to the margnal utlty of the burst. It must be appled to (9) whch has the contnuaton value as well as the utlty burst on the rght-hand sde. As dscussed n Proposton 1 below, n some cases there s a constraned optmum, = 0, at whch the smooth pastng condton does not apply. Unless > 0 and 1, there s no unque optmum as many strateges lead to nfnte utlty. These transversalty ssues are dscussed n the Appendx. 7

10 utlty proposed by Tversky and Kahneman (199), hereafter called scaled-tk utlty, s g 0 G stk stk L g g U ( G, R) R u ( G/ R) for u ( g) stk g ( ) 0 (11) wth 0 < G, L 1, As for CPT, the parameters G and L determne the nvestor s rsk averson over gans and rsk seekng over losses whle loss averson s measured by. The scalng parameter satsfes 0 mn( L, G ). The upper restrcton on ensures the desred property dscussed earler that U(G, R) s weakly decreasng n R for a fxed G. A nonnegatve s a partcpaton constrant. Tversky and Kahneman (199) estmated the utlty parameters as G = L = 0.88 and =.5. Snce they were not concerned about ntertemporal aspects, they dd not estmate a dscount rate nor dd they consder scalng. However, for such a low level of rsk averson, the transversalty condton s volated and the nvestor wats forever to realze any gan unless s nearly equal to the expected rate of return. 15 For G = L = 0.88 and = 8%, the nvestor never voluntarly realzes losses unless there s lttle loss averson wth close to one. However, voluntary loss takng can be part of the optmal polcy for other utlty parameters. Wu and Gonzalez (1996), for example, estmate = 0.5. Usng utlty parameters, G = L = 0.5, =, and = 5% the optmal strategy does nclude voluntary losses for any less than about Fgure shows the optmal sales strateges, and, plotted aganst for dfferent values of G and L. Both and decrease wth though falls at a much faster rate, and for a large enough loss averson, the nvestor refrans from realzng any losses. Provded losses are realzed, they are always larger than gans n magntude. Ths mght seem counterntutve, but the smaller gans are realzed more frequently, and snce margnal utlty s decreasng wth the magntude of the gan or loss, several small gans more than offset the dsutlty of a sngle loss of the same total sze. One common observaton about the realzaton of gans and losses s the dsposton effect, whch has often been clamed to be a consequence of an S-shaped utlty functon. 16 The 14 L G Settng G and R reduces (11) to the standard case ntroduced n Tversky and Kahneman (199), 0 wth 0 beng ther loss averson parameter. Loss averson then would vary wth R, but the reference level s constant n the orgnal statc nterpretaton. The Barbers and Xong (01) model s the specal case = G = L = The restrcton on comes from the transversalty condton, G 1. Whle the requred dscount rate s large relatve to those usually assumed, many behavoral fnance models do assume that nvestors are qute mpatent. It seems reasonable that utlty derved from tradng gans mght well dsplay more mpatence than utlty for lfetme consumpton. In addton, ths hgh dscount rate could ncorporate the hazard rate descrbng the nvestor s ceasng ths type of tradng. Death s sometmes nserted nto nfnte-horzon models n ths fashon, though here the termnaton of tradng mght be a smple lack of further nterest. 16 Shefrn and Statman (1985) were the frst to argue that an S-shaped utlty functon leads to the dsposton effect. Smlar arguments were made n Weber and Camerer (1998), Odean (1998), Grnblatt and Han (005), and other theoretcal and emprcal papers. 8

11 Fgure : Optmal Sales Polces. Ths fgure llustrates the optmal polces for sellng at gans or losses as a multple or fracton of the reference level. The stock prce parameters are = 9%, = 30%. Transacton costs are k s = k p = 1%. Utlty parameters are = 5%, = 0.5, and G = L = 0.3, 0.5, 0.6, as ndcated. The nvestor fully recognzes transacton costs n assessng hs utlty,.e., = K (1k s )/(1+k p ). effect, whch has often been clamed to be a consequence of an S-shaped utlty functon. 17 The reasonng s that the nvestor realzes hs gans as he s rsk averse and therefore unwllng to gamble about future uncertan gans; however, beng rsk-seekng over losses he wll gamble and postpone realzng them. Ths analyss mplctly assumes somethng lke realzaton utlty because any effects of the unrealzed gans or losses are gnored. But the argument s statc consderng only a sngle sale and gnorng any effects of renvestment, nor does t address the queston of why any losses are ever realzed rather than ther beng contnually postponed. Of course, even gnorng renvestment, the realzaton of gans mght be postponed f the expected change n the stock prce s suffcently hgh so that a larger expected gan n the future offsets ts extra rsk. Conversely, f the mean prce change s negatve, losses mght be realzed early to avod larger expected losses n the future whle gans would be realzed both to avod rsk and to avod smaller expected gans Shefrn and Statman (1985) were the frst to argue that an S-shaped utlty functon leads to the dsposton effect. Smlar arguments were made n Weber and Camerer (1998), Odean (1998), Grnblatt and Han (005), and other theoretcal and emprcal papers. 18 Henderson (01) formalzes ths argument by examnng a dffuson model lke ours that allows only a sngle 9

12 Wth concave realzaton utlty, the dsutlty of a loss can never be offset by the beneft of recoverng that loss n subsequent gans of the same total sze, but as seen n Fgure losses wll be realzed as well as gans wth an S-shaped utlty functon. Losses are substantally less common than gans snce s much farther from 1 than s. 19 However, n sharp contrast wth the statc argument that an S-shaped utlty leads to the dsposton effect, the S-shape actually serves to reduce the dsposton effect n a dynamc context. As G and L decrease and the S- shape becomes more pronounced, the optmal gan pont,, s affected only a lttle whle the loss pont,, ncreases dramatcally, reducng the dsposton effect. The reason s that realzng a loss resets the reference level for future possble gans, and ths can more than offset the drect dsutlty of the loss. That s, the realzaton of a loss s, n some sense, the purchase of a valuable opton. When G s small, ths opton effect can be substantal snce the margnal utlty of small gans s very large makng the dsutlty of losses affordable. Wth ntertemporal realzaton utlty, an S-shaped utlty functon does not create the dsposton effect; t actually reduces t, explanng why any voluntary losses are realzed rather than none. 0 As the loss averson parameter ncreases, losses become more panful, and drops dscontnuously to zero as shown n Fgure. The dscontnuous change occurs because ths maxmzaton problem s not a standard convex optmzaton. As llustrated n Fgure 3, the reduced value functon, v(1), s not a concave functon of. Both an nteror local maxmum and a corner local maxmum at zero are possble and ether can be the global maxmum. The hgh margnal dsutlty of repeated small losses together wth transacton costs makes loss takng suboptmal for hgh values of near 1. On the other hand for low, the contnuaton value, whch s proportonal to (K), s very small and cannot offset the dsutlty of a loss. Ths makes avodng losses altogether ( = 0) better than takng a large loss. Only for ntermedate values of s the contnuaton value possbly suffcent to offset the dsutlty of a loss. So v(1) attans ts local maxmum value at ether = 0 or an ntermedate value. Fgure 3 llustrates both optmum types for G = L = 0.5 and = 0.3. The ntal reduced value functon after any sale and repurchase, v(1), s plotted aganst for three values of. The upper sales pont s fxed at ts dstnct optmal value n each case. For =.5, t s optmal to sell for a loss at = For =.56, there s an optmum at = 0.147, but ths s only a local maxmum as never sellng at a loss provdes hgher utlty as shown. For the crtcal value of.531, both sellng for a loss at = and never sellng for a loss provde the same expected utlty. Therefore, the lower sales pont,, does not decrease smoothly to zero as ncreases; t lqudatng sale wth no renvestment. She fnds that losses are voluntarly realzed only f < 0. In contrast to her model, our paper shows that renvestment s mportant, and as a drect consequence, there s voluntary realzaton of losses even wth a postve of emprcally relevant magntude Snce returns are lognormal, the proper dstance comparson s n n ; also 0, so even f the log dstances were equal, gans would be realzed more often. 0 Some research suggests that prospect theory may not lead to the dsposton effect, e.g., Barbers and Xong (009), Kausta (010), and Hens and Vlcek (011). In contrast wth our model, none of these papers consder renvestment. 10

13 Fgure 3: The Value Functon for Scaled-TK Utlty. The reduced value functon for scaled- TK utlty measured mmedately after a sale and reference level reset, v(1), s plotted aganst dfferent loss sales ponts,. The gan sales pont,, s fxed at ts optmal value. The sold lne shows the value functon for =.531, the dashed lne shows the value functon for =.5, and the dotted lne shows the value functon for =.56. The other parameters are = 9%, = 30%, k s = k p = 1%, G = L = 0.5, = 0.3, = 5% = 1k s. For =.5, the two-pont polcy ( = 0.183, = 1.037) s optmal. For =.56, the one-pont polcy ( = 1.036) s optmal. For the crtcal value =.531, the two pont polcy ( = 0.166, = 1.036) and the one-pont polcy ( = 1.036) have the same expected utlty. drops dscontnuously from to 0 as passes the crtcal value of.531. A smlar change n regme for s true for the other parameters. The two regmes are characterzed n Proposton 1; a proof s suppled n the Appendx. Proposton 1: Scaled-TK utlty has both an upper and a (non-zero) lower optmal sales pont f and only f λ s less than the crtcal value where ( 1 ) ( ) * (1 ) G 1 * * G 1 * 1 L 1 * * ( L 1) * 1 0 ( ) ( ) K K G G * * * 0 ( ) ( ) K K L L * * * (1) determne and. If s greater than ths crtcal value, only gans are realzed. The soluton s stll characterzed by (8), (9), and (10) wth C set to 0. As approaches ts transversalty upper lmt, 1, 0, and voluntary losses are never realzed. 11

14 The Barbers Xong (01) model s a specal case of scaled-tk utlty wth L = G = = 1. For ths model, or ndeed any realzaton utlty model wth pece-wse lnear utlty for gans and losses and 0 1, the crtcal value s less than 1. Therefore, losses are never realzed voluntarly.. The Value Functon and Optmal Sales Polces The value functon or ts reduced-form equvalent, v, measures the present value of the nvestor s utlty bursts and gves a pont estmate of the beneft of hs strategy. It serves the role of the derved utlty functon n a standard Merton-type portfolo problem. Fgure 4 presents the reduced value functon, v, measured at the tme of any renvestment,.e., v(1), plotted aganst the asset s expected rate of return,, and standard devaton,. The default utlty parameters are =, = 5%, = 0.3, G = L = 0.5. For the and graphs, the other parameter s set to = 30% or = 9%, respectvely. For comparson purposes, each value Fgure 4: The Value Functon. The ntal optmzed value functon for scaled-tk utlty plotted aganst and. The default parameters are = 9%, = 30%, k s = k p = 1%, = 0.3, =, = 5%, G = L = 0.5. The nvestor fully recognzes transacton costs n assessng hs renvestment gans,.e., = K (1k s )/(1+k p ). 1

15 functon s normalzed to 1 at the values = 9% and = 30%. 1 Naturally the value functon s ncreasng n. On average for hgher, the next trade s more lkely to be a gan and to occur sooner. The relaton s steeper for a larger because the contnuaton value from the renvestment s larger due to scalng. The relaton s also steeper for smaller snce the benefts from future gans are dscounted less heavly. Surprsngly, the value functon s not always strctly decreasng n volatlty as t s for a standard expected utlty maxmzaton; t can be ncreasng or U-shaped. Of course, CPT nvestors are rsk-seekng wth respect to losses, but that s not the reason for ths effect. For example, the value functon s ncreasng n volatlty n the Barbers and Xong (01) model where burst utlty s pece-wse lnear and weakly concave. In our model there are conflctng effects. Changng the three parameters,,, and proportonally s dentcal to a change n the unt of tme and leaves our model unaffected. So an ncrease n can be nterpreted as a proportonal decrease n both and. Decreasng lowers the value functon as just explaned, but decreasng rases the value functon snce the future net postve bursts are dscounted less heavly. As explaned above, the smaller s, the less mportant s the effect. So for small, the value functon s less steeply decreasng or even ncreasng n volatlty. Also the larger s the more mportant s ts effect. So for large, the value functon s agan less steeply decreasng or even ncreasng n volatlty. Fgure 5 presents graphs of the optmal sellng ponts for gans and losses for scaled-tk nvestors. The parameters left unchanged n each graph are set to the default values = 9%, = 30%, k s = k p = 1%, G = L = 0.5, = 0.3, =, = 5%. The dotted lnes show the optmal polces for an nvestor who gnores the renvestment cost n assessng hs gans,.e., = 1k s. The sold lnes show the optmal polces for an nvestor who does recognze ths cost,.e., = (1k s )/(1+k p ). Several features are mmedately evdent. For both types of nvestors, realzed losses are typcally much larger than realzed gans so the basc strategy s to realze a few large losses and many small gans as we have already suggested ntutvely. However, the no-sales regon s wder for an nvestor who recognzes the renvestment cost as reducng hs gan. An nvestor who nternalzes the costs more when assessng hs well-beng s obvously more reluctant to trade. The upper sales pont,, s much less affected by parameter changes than s the lower sales pont,, n most cases. In fact, s largely unaffected by any of the varables except transacton costs and the scalng parameter,. And for, any effect occurs mostly near the transversalty lmt. As approaches ts lmt of 1, drops dscontnuously to zero, and approaches /[( G )]. 1 Standard utlty functons are defned only up to a postve affne transformaton. Realzaton utlty has ts level set so that a gan of zero gves a utlty of zero, but scalng s stll arbtrary. /( 1 ) For > 1, there s no well-defned optmal upper sales pont. Any K provdes nfnte expected utlty. See the Appendx for detals on ths and other transversalty-type volatons. When G s very close to 1, the upper sales pont s senstve to and can be decreasng for low. 13

16 Fgure 5: The Optmal Sales Polces for Scaled-TK Utlty. The optmal sales ponts, and, for scaled-tk utlty plotted aganst varous parameters. The default parameters n each graph are = 9%, = 30%, k s = k p = 1%, G = L = 0.5, = 0.3, =, = 5%. The dotted lnes show the optmal polces for an nvestor who gnores the renvestment cost n assessng hs gans, = 1k s. The sold lnes show the optmal polces for an nvestor who recognzes the renvestment cost n assessng hs gans, = K (1k s )/(1+k p ). 14

17 On the other hand, the optmal loss-takng strategy does vary substantally as the utlty parameters change. Increasng makes losses hurt more so s lowered n response to avod some of them. Smaller L leads to hgher margnal dsutlty for small losses. Ths nduces the nvestor to wat longer to realze a loss as hs rsk-seekng behavor ncreases ( L decreases from 1 to 0). Conversely for low G, the margnal utlty of small gans s qute hgh makng small losses affordable and desrable to set up future gans. So decreases wth G. Smlarly when s small, the pan of realzng a loss s offset more by the lowerng of the reference level for subsequent gans; ths ntensfes loss takng, ncreasng. Wth a hgher, the nvestor s more reluctant to experence a loss just to ncrease future gans. The effect of the subjectve dscount rate,, on the optmal sales polcy s unusual. A more mpatent nvestor wants to realze gans sooner and defer losses longer. We see that s decreasng n as expected; however, s not. All losses are taken voluntarly, and the desre to take gans early nduces a derved wllngness to realze losses n order to set up these future gans. Ths causes to also be ncreasng n at low dscount rates; however, at hgher dscount rates the ntal ntuton domnates because future gans are dscounted more so the beneft of resettng s less. Increasng the transacton costs, k s and k p, naturally wdens the no-sales regon because the costs take part of each gan and ncrease every loss. As the costs go to zero, both and approach 1, and the tradng frequency ncreases wthout lmt. Because margnal utlty becomes nfnte as the gan sze goes to zero, the nvestor takes every opportunty to realze even the smallest of gans. Of course, there s unbounded margnal dsutlty for near-zero losses, but under the optmal strategy approaches 1 slower than does, so there s a net ncrease n the value functon wth frequent trades. There s a smlar result for any utlty functon that s strctly concave for gans even f ther margnal utlty s not nfnte at zero. There s always an ncentve to realze any gan as a seres of smaller ncrements because the margnal utlty s hghest near 0, and n the absence of transactons coasts, there s no sellng penalty to offset ths. However, when the margnal utlty for gans s not nfnte at zero, the loss sales pont,, need not be near 1. In the presence of loss averson ( >1), the margnal utlty of small losses exceeds the margnal utlty of small gans whch precludes an mmedate loss realzaton even n the absence of transacton costs. Changes n have very lttle effect on the sze of optmal realzed gans,. The value functon s strongly ncreasng n, but ths s due to the reducton n the average tme before a sale occurs rather than any sgnfcant change n polcy. The lower sales pont,, s affected more. For large or small, the opton to reset the reference pont by sellng at a loss s less valuable than for ntermedate values of. It s less valuable for large because the asset prce grows quckly enough for gans to be realzed wthout a panful reset. Conversely, wth a very low, there s less value n resettng the reference level snce future gans wll be realzed nfrequently. So s hghest for ntermedate values of. 3 3 Note that ths s very dfferent from the result n Henderson (01). In her model wth nether renvestment nor dscountng, losses are only realzed to avod even larger losses on average when s negatve. 15

18 The standard devaton,, has only a tny mpact on the optmal gan pont. decreases as rses but by amount mperceptble n the graphs. Increasng also lowers the loss sales pont,. The reason s, that for typcal parameter values, hgher volatlty would ncrease the probablty of loss realzaton, and the nvestors responds by lowerng to postpone voluntary loss takng. 3. Model Calbraton In ths secton we calbrate our model usng representatve nvestors to determne f t can explan observed tradng patterns. The model descrbed n the prevous sectons makes specfc predctons about the magntudes, frequences, and relatve proporton of both realzed and unrealzed gans and losses. In ths context we explore an alternate utlty specfcaton whch mproves the model predctons. We also compare our model results to predctons assumng random tradng. The best calbraton s acheved when we consder heterogeneous tradng strateges. If we consder a sngle set of utlty and stock prce parameters, our model predcts that the magntudes of all realzed gans and losses are 1 and 1, respectvely. The frequency of tradng s determned by the tme requred for the nvestment value to rse to R or fall to R from the orgnal reference level R. Paper gans expressed as a percentage of the reference level are dstrbuted over the range 1 to 1. The propertes of these dstrbutons are gven n Proposton. Its proof and that of all later propostons are provded n the Appendx. Proposton : Propertes of Realzed and Unrealzed Gans and Losses. If the asset value has a lognormal evoluton, dx/x = dt + d, and the nvestor trades repeatedly accordng to a constant two-pont polcy, -, then the probabltes that a gven epsode eventually ends wth a gan or a loss are QG QL 1QG ( / ) 1 1 ( / ) (13) where 1 /. The fractons of the tme the asset has an unrealzed gan or loss are 1 (1 )[ n ( 1)] G L G 1. (1 )n ( 1)n (14) The expected duraton of each nvestment epsode s ( 1)n ( 1)n [], 1 ( )( ) (15) 4 When = (so = 0), L Hôsptal s rule gves QG 1QL L 1G n n( / ) and [] = n( ) n( ). 16

19 and n a sequence of consecutve nvestments, the expected number of nvestment epsodes per unt tme s 1/[]. The propertes derved n ths proposton and Proposton 4 below do not depend on the specfc realzaton utlty functon assumed nor even on the maxmzaton of utlty at all. They obtan whenever a specfc two sales pont strategy s employed for assets wth lognormal dffusons. For comparson we want the same type of predctons for nvestors who may trade for lqudty purposes, based on nformaton, or for other reasons. Descrbng the actons of all such nvestors s outsde the scope of ths paper so we smply assume that these nvestors trade stocks randomly n separate epsodes wth each epsode termnated ndependently of the stock prce evoluton. The predctons of ths model are gven n Proposton 3. Proposton 3: Characterstcs of Investment Epsodes for Random Trades. Assume that each asset s prce evolves accordng to a lognormal dffuson and that each tradng epsode termnates wth a sale that occurs accordng to a Posson process whch s ndependent of the evoluton of the stock prce and has ntensty. The tradng epsodes have the propertes gven below. The duraton of each epsode has an exponental dstrbuton wth mean duraton [] = 1/. The average realzed gan and loss are 5 where (1 ) ( 1) ( ) ( ) ( ) ( ) 1 1. (16) The probablty that a gven epsode wll end wth a sale at a gan and the probablty that an unrealzed nvestment s a paper gan are both. (17) Q G G Of course, the probablty that a gven epsode wll end n a loss and the probablty of an unrealzed loss are Q 1Q 1. L L G G Note that the tradng ponts, and, gven n (16) are averages. Whle thresholdrealzaton-utlty nvestors always trade at fxed ratos, the sales of random traders can occur at any prce. 5 1 For =, L Hôsptal s rule gves ( ). The expected value of the upper sales,, s fnte only f > ; s always fnte snce ts realzatons are bounded between 0 and 1. 17

20 One set of tradng statstcs that has garnered consderable attenton was proposed by Odean (1998) to measure the dsposton effect. These nclude the proporton of gans realzed, PGR; the proporton of losses realzed, PLR; and the Odean measure,. PGR s defned as a rato. The numerator s the number of realzed gans summed over all days and all accounts. The denomnator s the total over all days of the number of stock postons showng a gan (realzed or not) n all accounts whch realzed ether a gan or a loss on that day. PLR s smlarly defned for losses. The Odean measure s the rato PGR/PLR. In a gven sample, these statstcs wll be affected by many factors. These nclude the varyng sales thresholds for the dstnct assets held by dfferent nvestors, the number of stocks held n each account, the correlaton between the stocks returns, and the samplng nterval. 6 Proposton 4 derves PGR, PLR, and the Odean measure for the specal case of ndependent and dentcally dstrbuted assets. The number of stocks held per account may vary across nvestors, but each ndvdual stock s traded accordng to the same two-pont or random strategy. Proposton 4: Odean s Statstcs wth a Representatve Investor. Assume that asset returns are ndependent and dentcally dstrbuted wth a lognormal evoluton, dx/x = dt + d, and that all stocks are traded usng the same strategy. Then as the number of sales ncreases, the probablty lmts of PGR, PLR, and the Odean measure are plm PGR Q Q plm PLR G L G ( n n / n 1) G QL ( n n / n 1) L G QL ( n n/ n 1) L L QG n n n G plm PGR Q plm plm PLR Q ( / 1) Q (18) where n and are the average number and varance of the number of stocks held across n accounts, 7 and Q G, Q L, G, and L are the probabltes defned n (13) and (14) f the nvestors are realzaton-utlty nvestors or n (17) f the nvestors are random Posson traders. The statstcs of Proposton 4 are the same as those that would be produced by a sngle representatve nvestor holdng n n / n rather than n stocks. The representatve nvestor s holdng s based hgh relatve to the average because those nvestors holdng more stocks are represented more often n the data. Wth the statstcs derved n Propostons through 4, we can assess how our realzaton utlty model and the random tradng model ft the tradng patterns of ndvdual traders. For comparson we ncorporate data from Odean (1998) and Dhar and Zhu (006) wth the statstcs generated by ths model nto Table 1. The frst row of the table presents Odean s data extracted from hs Tables 1 and 3 ncludng the header text. He reports that 53.8% of sales were realzed 6 The effects of these factors are examned n Ingersoll and Jn (01). 7 If each account holds a fxed number of assets over tme, then n and are the average and varance of account n szes. Under mld regularty condtons, the proposton remans vald for accounts whose szes vary over tme wth n and beng the average and varance of the number of shares held per account across both accounts and tme. n 18

21 Table 1: Summary Statstcs for Reference-Dependent Realzaton Utlty Model wth Scaled Tversky-Kahneman Utlty The table reports: 1, 1: percentages above and below the reference level for realzed gans and losses, Q G : fracton of epsodes that end n realzed gans, G : fracton of stocks wth unrealzed paper gans, []: average holdng perod n tradng days (50 per year), PGR, PLR: proportons of gans and losses realzed, and PGR/PLR: Odean s measure. Asset parameters are = 9% and = 30%. The accounts szes are fxed wth n n / n 8.0. Utlty parameters are = and = 5% (except = 8% for G = 0.88 and = 10% for G = 1 to avod a transversalty volaton). Transacton costs are k s = k p = 1%, and the nvestor accounts for both costs n hs subjectve vew of hs realzed gans, = K. Odean s data s taken from Tables 1 and 3 of hs 1998 paper. Dhar and Zhu s data s from the notes to ther Table 3. The Ft to Odean s, row uses Odean s estmates of and to compute the other values usng Propostons and 4. Each Posson Model row choose---s to match one of Odean s estmates of,, Q G, or [] and computes the other values usng Propostons 3 and 4; the observed G cannot be matched as the Posson model cannot gve values less than 50% when > /. 1 1 Q G G [] PGR PLR Odean data 7.7%.8% 53.8% 41.9% % 9.8% 1.51 Dhar & Zhu data % 46.5% 1 13.% 6.4%.06 Ft to Odean s, 7.7%.8% 57.7% 50.7% % 10.9% 1.8 Random Tradng (Posson) Model 1 1 Q G G [] PGR PLR = %.8% 58.7% 58.7% % 1.5% 1 = % 17.4% 55.9% 55.9% % 1.5% 1 = % 15.% 54.9% 54.9% % 1.5% 1 = % 1.4% 53.8% 53.8% % 1.5% 1 G 1 L 1 G 0.88 L 0.88 G 0.5 L 0.88 G 0.5 L 1.0 G 0.5 L 0.5 Realzaton Model wth Scaled-TK Utlty 1 1 Q G G [] PGR PLR = 0 or % never 100% 7.1% % 0 = % never 100% 16.6% % 0 = % never 100% 7.7% % 0 = % never 100% 7.3% % 0 = 0 3.9% 13.5% 80.6% 1.5% % 3.4% 10. = % 45.3% 93.8% 9.5% % 1.0% = 0 3.8% 6.3% 64.9% 36.7% 7 0.% 7.3%.74 = % 8.% 87.6% 15.6% %.1% 1.64 = 0 4.0% 47.3% 95.9% 6.5% % 0.6% = % 75.8% 98.3% 4.9% % 0.3%

22 gans wth an average sze of 7.7%; the remanng trades were losses averagng.8%. The average holdng perod was 15 months whch we have expressed as 31 tradng days. 8 Paper gans and losses composed 41.9% and 58.1% of the unrealzed postons. PGR and PLR were 14.8% and 9.8%. Dhar and Zhu s (006) data are taken from ther Table 1 and the note to ther Table 3. Gans were realzed on 65.8% of trades, but paper gans composed only 46.5% of unrealzed postons. PGR and PLR were 13.% and 6.4%. 9 The average holdng perod was 1 days. They do not report the average szes of realzed gans and losses. The dfferences n ths data can probably be attrbuted to the perods studed. Durng the Dhar-Zhu perod, , the market rose 113% wth only mnor correctons whle durng Odean s perod , the market rose only 89% and suffered two major downturns. So Dhar-Zhu traders would have reached ther -ponts more frequently whle Odean s traders would have had more opportuntes to sell at losses. To determne f any calbraton s feasble, the data n the thrd row uses just Odean s average sales prce ratos as estmates for and. The remanng values are determned from them and the stock evoluton parameters usng Propostons and 4. Ths ft s not optmzed; we have smply chosen an asset comparable to a typcal share of stock wth = 9% and = 30%. The ft for Q G and G, and therefore the correspondng loss statstcs, do seem reasonable allowng for samplng error and heterogenety of assets and nvestors n the actual sample. 30 To compute PGR, PLR, and, we need account sze nformaton. Goetzmann and Kumar (008), usng the same data set as n Barber and Odean (000), provde more detals about portfolo szes. They gve the percentages of accounts of varous szes n ther Table 1 from whch we compute approxmate values of n = 4.1 and n = 4.0 gvng n n / n 8.0. For a smlar data set, Barber and Odean (000) report that the average number of stocks per account s 4.3; Dhar and Zhu (006) gve average account szes of 4.4 and 4. for nvestors whose occupatons they dentfy as professonal and nonprofessonal. The next four rows of the table llustrate the ft of a tradng model based on random Posson trades to Odean s data. As ncreases, both average sales ponts, and, approach 1. Under random tradng, Q G and G must be equal, and both fall from 100% to 50% as ncreases from 0 to. 31 Whle the ndvdual statstcs can be matched, they cannot be ft smultaneously. 8 Ths calculaton assumes 15 months s an exact fgure of 31.5 days. The actual value could range from 30 to 33 days due to roundng. 9 These are reported n ther note to Table 3 usng Odean s method of aggregaton. In ther Table, Dhar and Zhu (006) report smple averages across nvestors for PGR and PLR of 38% and 17%, respectvely. Computng PGR and PLR frst at the nvestor level and then averagng across nvestors puts relatvely more weght on nvestors who have fewer stocks n ther accounts, and these nvestors typcally have hgher PGR and PLR. For nstance, suppose Q G = G = 0.5. Then for an equal mx of nvestors who hold and 6 stocks, PGR s 0.5 and The average PGR s 0.33, but usng equaton (18) wth n = 4 and n =, the aggregated PGR s Our ftted value of 174 days for [] dffers from both Odean s and Dhar-Zhu s though t s between them. All of the statstcs n the last sx columns except for [] depend only on the rato / so ncreasng and proportonally wll reduce [] and leave the others unchanged. In our analyss below t s only the relatve holdng tmes for dfferent accounts that matters not the level. 31 If <, then both Q G and G rse from 0 to 50% as ncreases. In ths case Odean s value of Q G = 53.8% cannot be matched. Conversely, for the parameters used here, G cannot be matched to Odean s value of 41.9%. 0