How To Test The Theory Of Money And Money

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1 Testng alternatve theores of fnancal decson makng: a survey study wth lottery bonds Patrck ROGER 1 Strasbourg Unversty LARGE Research Center EM Strasbourg Busness School 61 avenue de la forêt nore STRASBOURG CEDEX, FRANCE proger@unstra.fr November 2009 Abstract In ths paper, we present the results of a smple, easly replcable, survey study based on lottery bonds. It s amed at testng whether agents make nvestment decsons accordng to expected utlty, cumulatve prospect theory (Tversky-Kahneman, 1992) or optmal expectatons theory (Brunnermeer and Parker, 2005, Brunnermeer et al., 2007) when they face skewed dstrbutons of returns. We show that more than 56% of the 245 partcpants obey optmal expectatons theory. They choose a dstrbuton of payoffs whch s domnated for second-order stochastc domnance and whch would not be chosen accordng to cumulatve prospect theory, for a large range of parameter values. Our results frst cast doubt on the relevance of varance as a measure of rsk; they show the mportance of skewness n decson makng and, more precsely, they emphasze the attractveness of the best outcome, an essental feature of optmal expectatons theory. The rankng of outcomes, used n cumulatve prospect theory, seems nsuffcent to characterze the way people dstort belefs. As by-products of ths study, we llustrate that agents use heurstcs when they choose numbers at random and have, n general, a poor opnon about the ratonalty of others. JEL classfcaton: D03, D81 Keywords: Lottery bonds, optmal belefs, probablty dstorton, rsk averson 1 I thank partcpants to the Internatonal AFFI Meetng, Brest, May 2009, and AFSE Congress, Pars, September I want to thank more specfcally Sylvan Marsat and Benjamn Wllams at Unversty of Clermont- Ferrand for ther effcent help n the emprcal part of the paper. 1

2 I Introducton Standard economc theory assumes relatvely smple rules to descrbe human behavor. Agents are supposed to manage any quantty of nformaton they receve, accordng to Bayes rule, and to take decsons wthout emotons or dstorted belefs. Ther objectve s to maxmze the expectaton of ther utlty functon (henceforth EU model). There s now some evdence that agents make systematc errors, especally n assessng probabltes. The probablty of very good outcomes tends to be overvalued (Alpert and Raffa, 1982; Buehler et al., 1994; Wensten, 1980), ntroducng an optmstc bas. The consequence s a suboptmal allocaton of wealth. For example, households portfolos are not well dversfed. A part of ther wealth s nvested n mutual funds (and then well dversfed) but another part, n general non neglgble, s concentrated on a few stocks (Calvet et al., 2007, Goetzmann and Kumar, 2008, Mtton and Vorknk, 2007, Polkovnchenko, 2005). Moreover, portfolos are often based toward lottery-type stocks wth postve skewness. Barbers and Huang (2008), Bal et al.(2009), Kumar (2009) and Mtton and Vorknk (2007) recently publshed papers focused on that problem. Barbers and Huang (2008) show that stocks wth postvely skewed returns can be overprced on markets populated by nvestors obeyng cumulatve prospect theory (CPT n the followng). It s especally the case f the return on skewed securtes s ndependent of the returns on other securtes and f the supply of skewed stocks s small relatve to the global market supply. Kumar (2009) shows the exstence of sgnfcant lnks between nvestment behavor and lottery play behavor. He shows that nvestors used to play (unfar) state-lotteres also prefer lottery-lke stocks. Ths observaton s renforced durng economc downturns. Bal et al. (2009) show that stocks exhbtng at least one very hgh return n the past month are overprced. Ths effect s robust when controllng for dosyncratc volatlty. Mtton and Vorknk (2007) analyze the behavor of more than 60,000 retal nvestors and show that those who are not dversfed select a few hghly skewed stocks. It then seems that gamblng and nvestment behavors cannot be dsentangled because of the preference for skewness and/or the attractveness of the best outcome. Brunnermeer and Parker (2005) and Brunnermeer et al. (2007) developed a theory of optmal expectatons (OET n the followng) to take nto account ths optmstc bas. In the second paper, Brunnermeer et al. (2007) consder a smple one-perod, two-dates model; they assume that agents behave optmally gven ther belefs, and choose portfolos maxmzng the expected present value of future utlty flows. Roughly speakng, the felcty of agents s composed of ex ante and ex post utlty. Ex ante, t s optmal to dstort belefs n an optmstc 2

3 way. However, ths dstorton comes at a cost, lyng n a sub-optmal portfolo choce and a lower ex post expected utlty. The authors call optmal belefs the subjectve assessment of probabltes whch maxmzes an average of ex ante and ex post utltes. In a complete market framework wth a fnte number of states of nature, they show that the optmal portfolo contans the rsk-free asset, and the most skewed asset. Concernng optmal belefs, they prove that the probablty of only one state s overvalued, the probabltes of the other states beng undervalued. In cumulatve prospect theory, dstorton of belefs s lnked to payoffs, only through the rankng of gans and losses. Consequently, n a fnte state space, the outcomes of two comonotonc prospects are weghted dentcally and ndependently of the values of the outcomes; only rankng matters. Concernng the dstortons of belefs, our results show that a large proporton of agents not only take nto account the rankng, but also the values of the outcomes, especally the largest one. In ths paper, we present a survey study to test whether the attractveness of the best outcome s really an mportant component of the decson makng process or f agents behave accordng to the EU or CPT models. The test s based on a questonnare askng partcpants to choose among dfferent random outcomes of lottery bonds. These securtes are well suted to address the queston we are dealng wth. Frst, they exst n many countres for more than two centures, and are, even today, very popular (Green and Rydqwst, 1997, Gullen and Tschoegl, 2002, Lévy-Ullmann, 1896, Mllar and Gentry, 1980, Pfffelmann and Roger, 2005, Rdge and Young, 1998, Tufano, 2008). Second, contrary to the dstrbuton of stock returns whch s unknown, the dstrbuton of lottery bond returns s, n general, perfectly known. The possble outcomes are gven objectve probabltes. Thrd, most people who never nvested n lottery bonds easly understand how payoffs are defned because they bet, at least occasonally, on state lotteres lke the lotto game 2. We consder two desgns for the lottery bonds. They dffer by the way the amount dstrbuted through the lottery s defned. The frst bond s desgned to study the decson makng process and to answer our man queston concernng the choce of EU, CPT or OET. The ndvdual amount receved by wnners through the lottery s known n advance and the remanng amount to be shared among all subscrbers (ncludng the wnners) s random. The second bond s amed at controllng for the mnmum requred level of ratonalty, that s, the complance to the frst-order stochastc domnance prncple. In ths case, the global amount 2 Skewness-seekng nvestors are labeled Lotto nvestors by Mtton and Vorknk (2007). 3

4 pad through the lottery s known n advance (but the ndvdual gan s random due to a parmutuel feature). The remanng amount shared by all subscrbers s then not random. In fact, as we cannot defne an ncentve compatble payment scheme for the partcpants (wthout assumng that one theory s better than the others), the questons related to ths second bond allow to select respondents that provde answers compatble wth frst-order stochastc domnance. Our results show that more than 55% of partcpants behave lke OET nvestors, exhbtng a preference for the random payoff wth the hghest possble outcome. It s mportant to notce that the possble random payoffs of our frst bond have the same expected value and that the varance of returns s the hghest for the payoff wth the largest outcome. Moreover, our desgn s such that the payoff ncludng the hghest possble outcome s domnated by all other choces n the sense of second-order stochastc domnance. It then renforces our result n favor of OET. Our analyss also provdes two by-products. The frst one concerns the random choce of numbers. We llustrate that, at the aggregate level, people do not choose numbers at random, even when they are expected to do so 3. Ths result s n lne wth most studes on state lotteres. These papers show that the dstrbuton of numbers actually chosen by players s not unform (see, for example, Farrell et al., 2000, Roger and Brohanne, 2007, among others) because they use common heurstcs to select numbers. The second sde-result s lnked to the assumpton that ratonalty s common knowledge (Aumann, 1976). It s a strong assumpton and many examples show that t does not represent the way people are thnkng. The most famous example s the beauty contest (frst ntroduced by J.M. Keynes, 1936, chapter 12, p ), translated by H. Mouln (1986) n numercal terms. Players have to choose a number between 0 and 100 and the wnner s the one who chooses the number closest to a gven percentage (say a) of the mean choce of players. The 3 For example, Boland and Pawtan (1999) show that people have dffcultes to choose numbers randomly, even n very smple tasks. 4 As stated by John Maynard Keynes (1936): Or, to change the metaphor slghtly, professonal nvestment may be lkened to those newspaper compettons n whch the compettors have to pck out the sx prettest faces from a hundred photographs, the prze beng awarded to the compettor whose choce most nearly corresponds to the average preferences of the compettors as a whole; so that each compettor has to pck, not those faces whch he hmself fnds prettest, but those whch he thnks lkelest to catch the fancy of the other compettors, all of whom are lookng at the problem from the same pont of vew. It s not a case of choosng those whch, to the best of one's judgment, are really the prettest, nor even those whch average opnon genunely thnks the prettest. We have reached the thrd degree where we devote our ntellgences to antcpatng what average opnon expects the average opnon to be. And there are some, I beleve, who practce the fourth, ffth and hgher degrees. 4

5 Nash equlbrum of the game s that everybody chooses 0 when a < 1 5. However, all experments show that most people are far from choosng 0 (Thaler, 1998, Nagel, 1995). Our results also confrm that a non neglgble percentage of partcpants have a poor opnon about the ratonalty of others. To sum up, we test three assumptons n ths paper: 1) When facng postvely skewed dstrbutons, nvestors do not behave lke rsk averse expected utlty maxmzers. In partcular, they do not use a mean-varance crteron. Postve skewness s a hghly weghted decson crteron and, more precsely, the probablty of the hghest possble outcome s overvalued. 2) People are not choosng numbers at random even when they are expected to do so. They have common preferred numbers, a sub-optmal characterstc n a par-mutuel game. 3) When the dstrbuton of payoffs depends on the decson of others, agents have a tendency to consder that other people are not fully ratonal (and they seem rght!). The paper s organzed as follows. Secton II descrbes the two lottery bonds used n the survey study. Ths secton s wrtten n such a way that the reader can thnk about hs/her possble answers and can partcpate to the survey. Secton III presents the theoretcal analyss of the bonds and explans what theoretcal choces should be, accordng to the three theores under examnaton. Secton IV presents the emprcal results and secton V concludes. II Desgn of the lottery bonds Lottery bonds are n general fxed-rate bonds (wth coupon rate r) ssued by a state or a frm. However, r apples to the global ssue, not to the ndvdual subscrbers. If N one-year bonds are ssued, each wth a $1 face value, the ssuer repays B = (1+r)N at the maturty date. A part of ths amount s redstrbuted by means of a lottery. For example, B s dvded n two parts such that B = B 1 +B 2. n < N bonds are drawn at random and ther holders share B 1 (equally or not, dependng on the desgn of the lottery). The remanng amount, B 2, s then shared equally among the N subscrbers, or among the N n losers. 5 Some guessng games nvolve negatve feedback. Players have to fnd the closest number to 100 p x mean. For p = 2/3 t s easy to see that the equlbrum choce s 60 (see Sutan and Wllnger, 2009) 5

6 In most cases, the ssuer bears no rsk snce t repays B whatever happens n the random draw. The rsk s entrely borne by subscrbers. Lottery bonds are then unusual fnancal assets snce ssuers voluntarly ntroduce randomness n payoffs. In the two followng subsectons, we descrbe and characterze the two lottery bonds used n the survey study. The two bonds dffer only by the way the amount pad through the lottery s defned. In the second subsecton, we formalze the payoffs and ntroduce the notatons used n the theoretcal analyss of secton 3. II-1 Descrpton of the lottery bonds The two bonds are desgned as follows. A bank ssues N (equal to 1,000,000 n the questonnare) unts of a lottery bond, each bond beng sold $1. The subscrber of one bond has to choose a number between 1 and 10. At the maturty date, the bank pays an nterest rate r (5% n the questonnare) on the global amount ssued, then repayng (1+r)N ($1,050,000). However, the bank frst draws one number at random between 1 and 10, say j (we say that seres j has been drawn). The two bonds dffer n the way gans of wnnng subscrbers are defned. - For the frst bond, the ssuer pays $1 to each of the subscrbers of seres j and shares equally the remanng amount among all subscrbers, ncludng the wnnng ones. For example, f r = 5%, N = 1,000 and f 150 subscrbers have chosen the wnnng seres, they wll receve $1.90. After havng pad the wnners $1 each, the bank shares the remanng $900 among the 1,000 subscrbers, each one recevng $ The second bond follows dfferent rules. 10% of the ntal amount ssued s devoted to wnners, the remanng beng shared among all subscrbers, ncludng the wnnng ones. Wth the same data as before, a wnner would receve $100/150 + $0.95 because the 150 wnners have to share $100. The remanng amount s constant because the global amount won through the lottery s ndependent of the number of wnners. Consequently, all losng seres receve $0.95, whatever the number of wnners s. The man queston addressed n ths paper s to know how people choose a seres when they get nformaton about choces of former subscrbers. Table 1 shows the ndvdual payoffs receved by subscrbers of the frst lottery bond, dependng on the seres they nvested n and on the seres whch has been drawn. In ths example, the number of bonds s 1,000,000 and the nterest rate pad by the bank s 5 %, so the bank repays $1,050,000 at the maturty date. The frst lne ndcates the number of subscrbers n each seres and the frst column dentfes 6

7 the possble states of nature (the seres number drawn at random by the ssuer). There are then 10 states of nature. The followng columns gve the payoffs receved by a subscrber of a gven seres n each state. For example, 1.95 s the fnal payoff obtaned by a seres-1 subscrber f number 1 s drawn. As there are 100,000 subscrbers n ths seres, each of them frst receves $1 and the remanng $950,000 are shared equally among all the partcpants, so each subscrber receves $0.95. It explans the amounts appearng n the correspondng lne. When number 2 s drawn, the seres-1 subscrber receves $0.9 because there were 150,000 subscrbers n seres 2. The remanng amount s $900,000 shared by the 1,000,000 subscrbers. The same calculatons justfy the other amounts n the table. Table 1: Payoffs of bond 1 The seresk column contans the payoffs receved at the maturty date by a subscrber of seres k when the number drawn at random s the one appearng n the frst column and the same lne. Seres1 Seres2 Seres3 Seres4 Seres5 Seres6 Seres7 Seres8 Seres9 Seres Table 2 shows the ndvdual payoffs receved by the subscrbers of the second lottery bond, dependng on the seres they nvested n and on the seres whch has been drawn. Table 2 s bult as table 1, the number of subscrbers n each seres and the coupon rate beng the same. For example, 1.95 s the fnal payoff obtaned by a seres-1 subscrber f seres 1 s drawn. As there are 100,000 subscrbers n ths seres, each of them frst receves $1 ($100,000 shared by 100,000 wnners) and the remanng $ are shared equally among all the partcpants. Each losng subscrber then receves $0.95. It explans the amounts appearng n the correspondng lne. When a dfferent number s drawn, the seres-1 subscrber receves $0.95, because the remanng amount s stll $ , shared among the 1,000,000 subscrbers. The essental dfference between the two bonds s that the payoff receved by a losng-seres doesn t depend on the losng number, all losers recevng $0.95. In other words, each bond 2 7

8 s characterzed by only two possble payoffs, a wnnng one or a losng one. Only the wnnng amount s lnked to the number of subscrbers n the correspondng seres. Table 2: Payoffs of bond 2 The seresk column contans the payoffs receved at the maturty date by a subscrber of seres k when the number drawn at random s the one appearng n the frst column and the same lne. Seres1 Seres2 Seres3 Seres4 Seres5 Seres6 Seres7 Seres8 Seres9 Seres II-2 Formal presentaton of payoffs The payoffs of bond 1 can be formalzed as follows. Let X ( j ) the payoff receved by a seres- subscrber when the number j s drawn. Denote N the global number of subscrbers. 10 N = N where Nk s the number of seres-k subscrbers k = 1 k (frst lne of tables 1 and 2). The ssuer repays (1 + rn ) but, accordng to the precedng rule, the ndvdual payoff X s defned by: N (1 + j r) f j X ( j) = N N 2 + r f j = N (1) Denote Z the random varable defned on the ten states of nature by: Zk ( ) = (1 + r) 1 θ ( k), k= 1,...,10 (2) Ω where θ ( k) = Nk / N and 1 Ω s the ndcator functon of the set of states of nature. X can be wrtten as X = Z +1 {} wth 1 {} beng the ndcator functon of state. Ths decomposton wll be useful to study the moments of payoffs n the next secton. Suppose that you have to choose a seres to nvest n. Obvously, f the numbers N are unknown, one can reasonably expect ndfference between the 10 seres, whch all generate an j 8

9 expected payoff (1 + r) because the ndcator functons 1 {}, = 1,...,10, have the same probablty dstrbuton. But what would be your choce f you are told the numbers N j and you are the last one-unt subscrber? For example, wth the data of table 1, would you choose seres 8 wth 50,000 subscrbers, seres 2 wth 150,000 or a seres wth an ntermedate number of subscrbers? The random payoffs of bond 2 can be formalzed n the same way. Let Y ( j ) denote the payoff receved by a seres- subscrber when the number j s drawn and stll denote N the global number of subscrbers. Y s defned by: r f j Y ( j) = 0.1N r+ f j = N (3) Y can then be wrtten as: Y 0.1 = (0.9 + r) 1Ω + 1 {}. θ Suppose you have to choose a seres to nvest n. As before, f the numbers of subscrbers N are unknown, one can expect ndfference between the 10 seres. But what would be your choce f you are told the dstrbuton of θ and you are the last one-unt subscrber? III Theoretcal analyss III-1 Lottery bond 1 We remarked before that f the numbers N j are unknown, the potental subscrbers should be ndfferent between seres. What s changed when the nformaton about the dstrbuton θ becomes avalable? The followng proposton shows that, f the dstrbuton of frequences s gven, the expected return on bond 1 remans equal across seres. Proposton 1 1) The expected payoff of an nvestment n any seres of bond 1 s equal to 1+r. 2) The varance of seres- return, condtonal on a dstrbuton D ( N,..., N ) bonds already sold, s decreasng wth N and gven by: = of 1 10 j 9

10 V D N j 2N X = j= 1 N N (4) Proof 1) Denote E D X the expected payoff receved by a subscrber of seres, condtonal on a gven dstrbuton D. 1 N j N E D X E D Z = + 1{} = 1+ r r = (1 + r) 10 j N N The frst moment s ndependent of frequences and s then not a crteron for a ratonal nvestor to decde. 2) Usng result of pont (1), we can wrte Z = E ( X ) θ wth θ ( ) = N / N. We then get: D ( 1{} θ ) 2 VD X E = D 2 = E D 1 {} + E D θ 2E D θ 1 {} (5) N N = N = 1 N The last term of the second equalty, 2E D θ 1 {}, s equal to 0.2 θ ( ). Consequently, VD X s a decreasng functon of N. Pont 2 of proposton 1 shows that a mean-varance nvestor would choose to play wth the crowd, a not so ntutve result. But, f we consder the case where all subscrbers choose the same number, the ssue becomes a rsk-free asset, payng 1 + r, whatever the number drawn by the bank. It explans why playng wth the crowd s varance reducng. Expected utlty maxmzaton Consder now the general case of a rsk-averse nvestor and denote U her utlty functon, assumed strctly ncreasng and strctly concave. The followng proposton generalzes the precedng results and shows that ths nvestor always chooses the most popular number, that s the one for whch N s maxmum. Proposton 2 10

11 Let U denote a strctly ncreasng and strctly concave utlty functon. If N < N j then j ED U( X ) < E D U( X ) Proof: X = Z +1 {} Assume wthout loss of generalty that the values of Z are ranked n ncreasng order, correspondng to a rankng of the N n decreasng order. We know that X ( ) s the maxmum possble value of X. It means that selectng number, when buyng a bond, transfers the -th outcome of Z at the rght tal of the probablty dstrbuton of X (snce wnnng always generates a better outcome than losng, whatever the losng number s!). It mples that transferrng the lowest possble outcome to the rght tal by addng $1 s always preferred by a rsk averse agent, due to the decreasng margnal utlty assumpton. But, as the lowest value of Z corresponds to the hghest value of N, a rsk averse nvestor always prefer to bet wth the crowd. Proposton 2 then ndcates that rsk averse expected utlty maxmzers should choose unambguously n 2 f they are facng the dstrbuton of frequences gven n tables 1 and 2. Optmal expectatons Accordng to optmal expectatons theory, descrbed here n a one-perod framework wth a fnte number of states (Brunnemeer et al., 2007), agents maxmze the followng expectaton: where the decson varables are to calculate ex ante utlty) and ( * * * ( ) + Π ( ) Π ) 1 2 E U C E U C (5) * Π and * C. * Π s the vector of subjectve probabltes (used * C s the optmal consumpton vector. Π stands for the vector of objectve probabltes, correspondng to ex post utlty. Maxmzng expresson (5) means * * that agents choose smultaneously (,C ) Π n order to optmze an average of ex ante and ex post expected utlty. There s then a trade-off between an optmstc dstorton of belefs whch maxmzes the ex ante utlty and the penalty comng from a sub-optmal allocaton. In fact, wth dstorted belefs, choosng * C does not lead to a maxmum of E * U( C ). Π 11

12 Brunnemeer et al. (2007) get a two-fund separaton result when the rato of Arrow-Debreu securty prces dvded by state probabltes s constant across states of nature. The optmal portfolo conssts to nvest a part of wealth n the rsk-free asset and the remanng amount n one and only one of the most skewed securtes. In our choce context, the prces of all seres are equal, the probabltes are equal and there s no aggregate rsk. Therefore, the prces of Arrow-Debreu securtes are dentcal. Accordng to ths two-fund separaton result, agents should choose number 8 whch s the most postvely skewed portfolo. It also corresponds to the lowest N. Fgure 1 llustrates the non monotonc lnk between the skewness of payoffs and the number of subscrbers usng the data of table 1. It appears that seres 8 s the most postvely skewed seres. As we saw that varance s also hgher for seres 8, choosng ths seres means a strong preference for skewness and llustrates the attractveness of the hghest outcome. Fgure 1: Skewness of bond-1 payoffs for the ten seres, as a functon of the number of subscrbers (data of table 1) Cumulatve prospect theory Suppose now that agents obey CPT. They maxmze a value functon, dependng on gans and losses, calculated wth respect to a reference pont. Two natural choces are avalable for the reference pont; the ntal prce of the bond, or the ntal prce captalzed at rate r. The latter s often used when prospect theory s appled to fnancal decsons (see, for example, Barbers et al., 2001). The two reference ponts lead to the same results for the problem at hand, smply because all seres payoffs nclude 9 possble losses and 1 possble gan, whatever the reference pont s. In CPT, gans and losses are loaded by decson weghts, obtaned by dstortng the cumulatve (or decumulatve) dstrbuton functon of payoffs. The weghtng functon w s defned by Tversky and Kahneman (1992) as: 12

13 (, ) w p β = p β ( ) 1 p + (1 p) β β β wth β < 1. Τhe weghtng functon s dfferent for gans and losses. For losses, decson weghts are obtaned by applyng w to the cumulatve dstrbuton functon. For gans, t s appled to the decumulatve dstrbuton functon. Moreover, the parameter β may be dfferent for gans and losses. The values estmated by Tversky and Kahneman (1992) were β + = 0.61 and β = In our framework, all states have the same probablty 0.1. When outcomes are ranked n ncreasng order, the weght of the k-th outcome s thenπ k = w 0.1 k, β w 0.1 ( k 1), β. The weght of the unque gan s equal to w(0.1, β + ). We observe here that weghts are determned only by the rankng of outcomes, not by ther values. It s also mportant to notce that the unque gan would allow to use a unque weghtng functon, as n rank-dependent utlty models. In ths framework, the weght β of the gan s 1 w(0.9, ) and the decson weghts defne a probablty measure on the set of states of nature. Ths case wll be examned hereafter. The value functon s defned, for a gan/loss x by: α x f x 0 vx ( ) = > α λ( x) elsewhere λ > 1 s the loss averson coeffcent and α < 1characterzes the curvature of the value functon. v s then concave for gans and convex for losses. Investors then choose number to maxmze 10 * π k 1 kvx k x = ( ( ) ) where * x s the reference pont. To understand what would be the choce of a CPT nvestor, we report on table 3 some elements of comparson for the two seres wth the lowest number of former subscrbers, N = 50, 000 (seres 8) and N = 60,000 (seres 5). We observe that the two cumulatve j dstrbuton functons are equal for the 8 lowest payoffs, as shown n the two frst lnes of table 3. 13

14 Table 3: Payoffs of seres 5 and 8 The payoffs of the two seres are ranked n ncreasng order and the correspondng weghts are calculated wth the parameters estmated by Tversky and Kahneman (1992), except β whch s equal for gans and losses. We then haveα = 0.88; λ = 2.25; β = Seres Seres Weght The dfference between the two seres comes from the two hghest payoffs whch are (1;1.99) for seres 5 and (0.99; 2) for seres 8. The thrd lne gves the decson weghts for β = The rato of weghts for the two hghest payoffs s around 1.7. It s then an ncentve to choose seres 8 wth the hghest payoff. However, two other varables are mportant n the choce. The loss averson coeffcent, equal to 2.25 n the table, favors seres 5 and largely compensates the lower weght. Moreover, the gan n the state wth the hghest payoff (0.94 or 0.95 wth a reference pont of 1.05) s much hgher than the absolute value of the loss n the other state (0.05 or 0.06). As α < 1, the ncrease of the value functon between 0.05 and 0.06 s much larger than the absolute value of the decrease between 0.95 and The consequence s that a CPT agent would prefer seres 5 nstead of seres 8 for a large range of parameters. Moreover, f α s decreased, then ncreasng the curvature of the value functon, the optmal CPT choce goes to a seres wth a larger number of subscrbers. If the loss averson coeffcent s decreased, the result s unchanged (the optmal choce s not seres 8) as long as loss averson stays above 1.2, an unusual value n expermental studes. Fgure 2 shows the CPT evaluaton of the 10 seres wth the ntal parameters. Usng a weghtng functon only based on the dstorton of cumulatve dstrbuton functons, as n the rankdependent expected utlty model, nduces a weght of 0.25 for the gan whch favors seres n 8. However, t s not enough to compensate loss averson and the effect of decreasng margnal utlty. Seres 5 wth subscrbers s the optmal choce n ths case, as long as the loss averson coeffcent s greater than

15 Fgure 2: Value functon of CPT for each seres. The horzontal axs gves the number of subscrbers and the vertcal axs the value functon of CPT wth α = 0.88, λ = 2.25, β = The reference pont s r = To summarze the analyss, usng the data of table 1, agents should choose seres 2 f they are rsk averse expected utlty maxmzers, seres 8 f ther behavor s drven by optmal expectatons theory and, fnally, seres 2 or 5 f they obey cumulatve prospect theory wth usual parameter values. For CPT, changng the parameters could lead to other choces but keepng reasonable values leads to a choce dfferent from seres 8. Ths result may seem surprsng snce CPT s often used to justfy the partcpaton to unfar state lotteres. However, n state lotteres, the probablty of wnnng s very low, gvng a more mportant role to the dstorton of belefs. Here, the objectve probablty of the wnnng state s only multpled by 1.7. It s not enough to make the choce of the most skewed alternatve optmal. III-2 Lottery bond 2 The analyss of bond 2 s much more smple. We saw n equaton (3) that: r f j 0.1N Y ( j) = (0.9 + r) 1Ω + 1{} = 0.1N N r+ f j = N Losng seres generate the same payoff 0.9+r but the payoff of the wnnng seres s nversely proportonal to the number of subscrbers of ths seres. Beng gven a dstrbuton of frequences, the optmal choce of the last subscrber of the ssue s always the seres wth the lowest number of former subscrbers. In fact, the correspondng payoff domnates the others n the sense of frst order stochastc domnance. Lookng more closely at two seres and j wth frequences N and N j wth N N j <, we observe that the payoffs are 0.9+r wth 15

16 1 1 probablty 0.9 but seres pays 0.1 N more than seres j wth probablty 0.1. It N N j s as f you were gven for free a lottery tcket payng ths amount wth probablty 0.1. Whatever your preferences are (obeyng frst order stochastc domnance), you accept the lottery tcket. Bond 2 s used to ntroduce a screenng process n the survey study. Due to the problem addressed n the paper, there s no ncentve compatble payment scheme because we have no a pror about whch choce s the rght one for bond 1. However, we can suspect that the answers of partcpants whch do not obey frst-order stochastc domnance are hghly questonable. Smply, some students may not be motvated by the exercse. They then answer at random. The emprcal secton s then focused on partcpants havng provded answers compatble wth frst order stochastc domnance. Nevertheless, we provde n the appendx the table of answers for the complete sample. III-3 Ratonalty as common knowledge Assume now that you have to choose a seres to nvest n bond 2, after one mllon other subscrbers. You also know that one more mllon subscrbers wll choose after you, wth updated nformaton about sales. If ratonalty s common knowledge, t s not dffcult to see that the equlbrum sharng of bonds at the end of the process should be an equal sharng across seres. In fact, for bond 2, t s always optmal to choose the seres wth the lowest frequency when you are the last subscrber. It mples that, as long as the frequency of a gven seres s lower than 200,000, you can choose ths seres. The followng ratonal subscrbers wll stop choosng a gven seres when the frequency wll reach 200,000. Beyond ths threshold, ths choce becomes sub-optmal because there s at least another seres wth a lower number of subscrbers. Consequently, f you beleve that ratonalty s common knowledge, you can choose at random f the current sharng of bonds s the one gven n table 1 or 2. For bond 1, the story s a lttle bt dfferent. If you thnk that others are lke you, you should choose the same answer to questons 3 and 5 f you are a rsk-averse expected utlty maxmze, assumng that the followng subscrbers wll also bet wth the crowd. The optmal choce f you obey OET s, as for bond 2, to nvest at random, assumng the others also obey OET. Obvously, f you consder that a proporton of agents s rsk-averse, you wll never choose the hghest frequency seres, antcpatng that t wll be chosen by these expected utlty maxmzers. 16

17 III-4 The random choce of numbers Fnally assume that you have no nformaton about the sharng of bonds across former subscrbers. You are only told the way the bank wll remburse the ssue. In ths case, the probablty dstrbuton of returns s equal n each seres, ether for bond 1 or for bond 2. Your choce then should be random n the set of ten seres. Therefore, the dstrbuton of choces at the aggregate level should be unform. We show n the next secton that t s not the case. IV The survey study IV-1 The questonnare The study was realzed durng dfferent fnance courses n two French unverstes (Unversty of Strasbourg and Unversty of Clermont-Ferrand) and two busness schools, EM Strasbourg Busness School (France) and HEC Lausanne (Swtzerland). 337 students partcpated, enrolled n economcs, fnance or accountng programs at the MSc level 6. The complete questonnare s provded n the appendx. The Englsh verson was used n Lausanne (all courses beng taught n Englsh at the MSc level) and a French verson n the other programs. The numbers of students n the dfferent locatons are provded n table 4. Table 4: Orgn of partcpants Unversty or Busness School Number of students EM Strasbourg Busness School 41 Unversty of Strasbourg 126 HEC Lausanne 74 Unversty of Clermont-Ferrand 96 TOTAL 337 The partcpants had to answer 6 questons, dvded nto three groups of 2 questons, related to the lottery bonds presented n the precedng secton. In each par of questons, the frst one s related to bond 1 and the second to bond 2. The requred answers were smply numbers between 1 and 10 correspondng to the choce of a seres number. For the frst two questons, partcpants were only told the characterstcs of the bonds, wthout any other nformaton, ether on the table of payoffs or on the choce of former subscrbers. For example, bond 1 was presented as follows. 6 For the Unversty of Strasbourg, 86 (40) students come from the MSc n Fnance (Actuaral Studes), n Lausanne, 69 (5) come from the MSc n Fnance (Actuaral Studes), n Clermont Ferrand, there were 43 students from the MSc n Fnance, 39 from the MSc n Accountng and Control and 14 from the MSc n Economc Analyss. 17

18 A bank (called bank1) ssues 1,000,000 one-year bonds at a prce of 1 each. The bank repays 1,050,000 at the end of the year (a 5% nterest rate on the ssue). When buyng one bond, subscrbers choose an nteger number between 1 and 10. On the repayment date, the bank draws at random a (lucky) number between 1 and 10 and frst repays 1 to each subscrber havng chosen the lucky number. The remanng amount s equally shared among all subscrbers, ncludng the wnnng ones. As mentoned before, wth no nformaton other than the way payoffs are defned, nvestors should be ndfferent between the seres; we then expect a random choce for the two frst questons. After havng answered the two frst questons, partcpants receved the nformaton summarzed n table 5 (t corresponds to the frst lne of tables 1 and 2). It was provded on a slde, so all partcpants knew that everybody was recevng the same nformaton. For questons 3 and 4, t was specfed that the respondent was about to buy the last bond of the ssue. In other words, everybody was able to nfer the fnal dstrbuton of payoffs. Table 5: Informaton about choces of former subscrbers (shown publcly on a Powerpont slde) Quanttes already bought n each seres Seres1 Seres2 Seres3 Seres4 Seres Seres6 Seres7 Seres8 Seres9 Seres A slght modfcaton was ntroduced for the students at unversty of Clermont-Ferrand. They were shown table 5 for questons related to bond 1 (questons 3 and 5) and table A2 (see the appendx) for questons 4 and 6. In table A2, the quanttes of bonds are the same but the numbers assocated wth these quanttes are dfferent. We then deal wth the same set of cumulatve dstrbutons of returns but the number dentfyng a gven dstrbuton s not the same. The dea was to control for a possblty of nerta n the answers. We saw, n the precedng secton, that the ratonal answer for queston 4 s seres n 8 but, t s also the answer to queston 3 for agents obeyng optmal expectatons theory. Consequently, we had to check f some students were choosng the same number to answer the two questons, smply by applyng a law of least effort. In fact, nobody n the subsample (96 students) chose (8, 8) to answer questons 3 and 4. There s then no reason to thnk that t s dfferent for the other subsamples. In the subsample of 96 students, the ratonal answer to queston 4 was n 4. We then also checked whether some students could have been consdered ratonal by nerta, 18

19 that s by selectng the same answer for questons 3 and 4. More precsely, we counted the number of students havng chosen the same number for questons 3 and 4, whatever ths number was. Only 3 students made such a choce over the subsample of 96 students. Consequently, the results presented n the followng cannot be nvaldated wth ths argument. In the thrd sequence of two questons (questons 5 and 6), the rule was that partcpants had to choose a seres number wth the same nformaton as n questons 3 and 4, but they were told that one mllon bonds were stll to be sold to other subscrbers after ther own choce. Moreover, partcpants were also nformed that the future subscrbers would get updated nformaton about sales at the tme of ther own purchase. Therefore, partcpants had to buld expectatons about the decson rules of future subscrbers. These two last questons are devoted to analyze the opnon of partcpants about the ratonalty of others, as n the usual beauty contest. To present the results concernng questons 3 to 6 n a smple way, we use the numberng of seres n table 5 for all the subsamples. Obvously, for questons 1 and 2, partcpants were not shown table 5 or table A2, so we keep the numbers they used to answer. IV-2 Results IV-2-1 Attractveness of the best outcome As mentoned before, the data provded n table 5 mply that agents obeyng frst-order stochastc domnance must choose seres n 8 at queston students over 337 made ths choce. They are called ratonal n the followng even f wrong answers to ths queston can smply be due to a lack of motvaton to partcpate. Table 6 shows the percentage of ratonal answers n each tranng program. Table 6: Numbers of «ratonal» partcpants across tranng programs 19 Ratonal answers Total Percentage "ratonal" HEC Lausanne ,08% EM Strasbourg Busness School ,93% Unversty of Strasbourg ,43% Unversty of Clermont Ferrand ,54% Total ,70% The lower percentage n Clermont-Ferrand s possbly due to the more complcated task students had to manage wth dfferent data for bonds 1 and 2. Apart from ths, the results are

20 not really surprsng. We fnd the hghest percentages n the two Busness School programs n Strasbourg and Lausanne. The other programs are part of Departments of Economcs and Management Scence. Unversty students are less used (and possbly less motvated) to partcpate n such surveys than students n Busness Schools 7. It may explan the sgnfcant dfference between the proportons. The detaled answers to the other 5 questons are gven n table 7 8, panel A. The lnes of the table are ranked accordng to the number of hypothetcal subscrbers n the seres, not accordng to the seres number. The man pont concerns queston 3. There s a clear preference for seres n 8 snce 56.61% of respondents chose ths seres. The second mportant observaton s the 22.31% of partcpants havng chosen seres 2. They are ether rsk averse expected utlty maxmzers or CPT nvestors. The eght other possble answers have almost neglgble frequences snce they gather around 20%. However, as mentoned n secton 3, these answers could possbly be attrbuted to CPT nvestors wth dfferent parameters or weghtng functons. Our results show a strong preference, not only for skewed returns (all returns are skewed n our study) but for the most skewed return, and for the random payoff wth the hghest outcome. Ths result s then clearly n lne wth the predctons of optmal expectatons theory. It s also nterestng to come back to the comparson of cumulatve dstrbuton functons of seres 8 and 5 (provded n table 3), correspondng to the two frst lnes of table 7. It s remarkable that 56.61% of partcpants chose seres 8 and only 2.89% seres 5 when the dfference between the two s only a swap of one cent between the two hghest outcomes. It confrms the attractveness of the hghest outcome, as predcted n OET. Panel B of table 7 provdes the proportons of choces 2 and 8 n the dfferent locatons. The proportons n the two unverstes are very close to each other. No sgnfcant dfference appears n the dstrbutons. On the contrary, we observe a hgher (lower) proporton of expected utlty maxmzers n the sample of HEC Lausanne (EM Strasbourg) but t s dffcult to nterpret these dfferences, takng nto account the number of students n each subsample. 7 For example, students n Busness Schools are used to evaluate courses and teachers, a not so common practce n French unverstes. 8 Table A3 n the appendx gves (for completeness of nformaton) the answers of the global sample, ncludng the partcpants qualfed as rratonals. 20

21 Table 7 Panel A: Answers of «ratonal» partcpants The fgures are provded n percentage but the number of answers n each column vares from 241 and 244 (a few students left some questons unanswered). The answers for Q4 are not provded because, by constructon, all ratonal respondents chose seres n 8 for ths queston (except those n Clermont-Ferrand for whch the ratonal answer was n 4). BONDS BOUGHT Q1 Q2 Q3 Q5 Q ,30 8,26 56,61 20,25 27, ,03 13,22 2,89 4,13 4, ,60 15,29 3,72 8,68 9, ,37 9,09 4,13 7,85 5, ,77 15,29 2,89 6,61 6, ,47 9,09 1,24 2,48 5, ,05 5,37 1,24 2,48 2, ,88 7,44 2,07 2,48 1, ,73 4,96 2,89 6,61 7, ,79 11,98 22,31 38,43 30,74 TOTAL 100,00 100,00 100,00 100,00 100,00 Panel B: Percentages of choces 2 and 8 n the dfferent locatons EMS= EM Strasbourg Busness School, LAU = HEC Lausanne, UDS = Unversty of Strasbourg, UCF = Unversty of Clermont-Ferrand. N s the number of ratonal partcpants n each subsample. The last lne gathers all answers that do not correspond to choces 2 and 8. EMS LAU UDS UCF N Seres 8 82,35% 46,67% 54,44% 52,46% Seres 2 5,88% 31,67% 23,33% 19,67% Other 11,76% 21,67% 22,22% 27,87% IV-2-2 Opnons about the ratonalty of others Questons 5 and 6 were desgned as questons 3 and 4, except that partcpants were told that one mllon bonds were stll to be sold after ther own choce, the next subscrbers choosng wth updated nformaton about sales. As mentoned n the precedng secton, concernng Q6, f partcpants were thnkng that other subscrbers are ratonal they should be ndfferent between all solutons. We then expect a unform dstrbuton of choce. However, assumng that other subscrbers are not completely ratonal, and have a one-step reasonng, leads you to play wth the crowd, expectng that the others wll stay n the low frequency seres. In the same way, wth a two-step reasonng, you 21

22 should stay n the low frequency seres. Fgure 3 shows that answers to queston 6 corresponds to agents manly usng a one or two-step reasonng. 141 partcpants (58%) chose numbers 2 or 8 correspondng to the two hghest frequences. Obvously, the unform dstrbuton hypothess s rejected at conventonal levels. Fgure 3: Bar chart of answers to queston 6 Concernng questons 3 and 5, the panel A of table 8 summarzes the frequences of choce of the 242 partcpants havng answered the two questons. The two seres wth 100,000 former subscrbers have been aggregated snce they pay exactly the same payoffs. The frst column (row) gves the answer to queston 3 (5). We observe that among the 54 partcpants havng chosen the hghest frequency at queston 3, 39 (72.2%) stay on that choce for queston 5. It s ratonal snce they want to bet wth the crowd and hope that the others wll do the same. We also observe the behavor already mentoned for queston 6. Around one thrd of partcpants (47) obeyng OET use a one step reasonng and swtch to the hghest frequency seres at queston 5. They expect that the others wll contnue to bet on the lowest frequency seres (whch then becomes a hgh frequency seres!). 35 partcpants (25%) stay on ther choce, usng a two-step reasonng. The remanng 40% have a more sophstcated behavor by choosng other seres. One more tme, t s clear that the condtonal dstrbuton of answers to queston 5 (condtoned on the choce of the lowest frequency seres to queston 3) s not 2 unform. The devaton from the unform dstrbuton ( χ > 140 ) 9 s essentally due to the partcpants usng a one-step or a two-step reasonng. They account for more than 75% of the 2 χ value. 9 2 To calculate the χ statstc, we grouped the two seres wth 120,000 and 130,000 subscrbers due to the low frequences of these seres. 22

23 If we compare now the answers at questons 3 and 6 (table 8, panel B), almost the same comments can be done. We also reject the unform dstrbuton assumpton for queston 6, 2 condtonal on the choce of the lowest frequency at queston 3 ( χ 116 ). Table 8: Answers to the pars of questons (3, 5) and (3, 6) Panel A: Frequences of pars of choces for questons 3 and 5 Q3\ Panel B: Frequences of pars of choces for questons 3 and 6 Q3\ To sum up, t appears that the choces of partcpants do not confrm the assumpton that ratonalty s common knowledge. The most frequent answers correspond to a one-step or two-step reasonng. Ths result s ndeed not surprsng snce t s the most common values found n the lterature for the depth of reasonng 10. IV-2-3 Heurstcs n random choce of numbers We expect random choces to questons 1 and 2 because partcpants have no nformaton about choces of former subscrbers. Fgure 4 shows the bar chart cumulatng choces for these two questons. In case of a unform dstrbuton, we should have frequences around 10 See Nagel (1999) for a survey on beauty contest games. 23

24 48.3 whch s the mean frequency of the unform dstrbuton. It appears that the actual answers are far from a unform dstrbuton. Ths hypothess s clearly rejected at the 1% 2 level. The χ value s for a 1% crtcal value of Fgure 4: Bar chart of answers to questons 1 and 2 The popularty of numbers 7 and 5, representng respectvely 15.94% and 12.62% of choces for questons 1 and 2, can be explaned by comparable results for state lotteres. For example, n the French lotto game, players have to choose 5 numbers between 1 and 49 and (ndependently) a lucky number between 1 and 10. The sponsor of the game draws at random the wnnng combnaton and the lucky number. Snce the start of ths verson of the game n October 2008, number 7 has been drawn 14 tmes 11 (see the detaled results n table 9). For these partcular draws, the mean proporton of wnners of the lucky number s 16.53% when 10% s expected f players choose ther numbers at random 12. For number 5, the mean proporton s 13% n a set of 10 draws, the mnmum and maxmum proportons beng 12.16% and 13.79%. The man dfference between the results of the lotto game and ours concern number 1. However, ths dfference can possbly be justfed by the fact that partcpants, consderng all choces as equvalent, select number 1 whch s obvously the frst n the lst of possble choces. In some sense, selectng number 1 could be nterpreted as a random choce. But even f we share these answers between the ten numbers, the dfference wth a unform dstrbuton remans sgnfcant. 11 For the 155 frst draws up to 09/26/ The data on French lotto draws are provded on and the percentage of lucky number wnners s reported on 24