Do investors trade too much? A laboratory experiment
Joao da Gama Batista, Domenico Massaro, Jean-Philippe Bouchaud, Damien Challet, Cars Hommes
DDo investors trade too much?A laboratory experiment
Jo˜ao da Gama Batista a , Domenico Massaro b,c Jean-Philippe Bouchaud d , Damien Challet a,e , Cars Hommes c,f December 14, 2015 a Laboratoire de Math´ematiques et Informatique pour la Complexit´e et les Syst`emes,CentraleSup´elec, Universit´e Paris-Saclay b Department of Economics and Finance, Universit`a Cattolica del Sacro Cuore, Milano c CeNDEF, Amsterdam School of Economics, University of Amsterdam d Capital Fund Management, Paris e Encelade Capital SA, Lausanne f Tinbergen Institute
Abstract
We run experimental asset markets to investigate the emergence of excess trad-ing and the occurrence of synchronised trading activity leading to crashes in theartificial markets. The market environment favours early investment in the riskyasset and no posterior trading, i.e. a buy-and-hold strategy with a most probablereturn of over 600%. We observe that subjects trade too much, and due to themarket impact that we explicitly implement, this is detrimental to their wealth.The asset market experiment was followed by risk aversion measurement. We findthat preference for risk systematically leads to higher activity rates (and lower finalwealth). We also measure subjects’ expectations of future prices and find that theiractions are fully consistent with their expectations. In particular, trading subjectstry to beat the market and make profits by playing a buy low, sell high strategy.Finally, we have not detected any major market crash driven by collective panicmodes, but rather a weaker but significant tendency of traders to synchronise theirentry and exit points in the market.
JEL codes:
C91, C92, D84, G11, G12.
Keywords:
Experimental Asset Markets, Trading Volumes, Crashes, Expectations,Risk Attitude.
Acknowledgments:
This work was partially financed by the EU “CRISIS” project (grantnumber: FP7-ICT-2011-7-288501-CRISIS), Funda¸c˜ao para a Ciˆencia e Tecnologia, and theMinistry of Education, Universities and Research of Italy (MIUR), program
SIR (grant n.RBSI144KWH). a r X i v : . [ q -f i n . GN ] D ec Introduction
Financial bubbles and crises are potent reminders of how far investors’ behaviour maydeviate from perfect rationality. Many behavioural biases of individual investors arenow well documented, such as the propensity for trend following or extrapolative ex-pectations (Greenwood and Shleifer, 2014), herding behavior (Cipriani and Guarino,2014), disposition effect (Grinblatt and Keloharju, 2001), home bias (Solnik and Zuo,2012), over and underreaction to news (Barber and Odean, 2008); see Barberis andThaler (2003) and Barber and Odean (2013) for comprehensive overviews.A well established fact about individual trading behaviour which is in stark contrastwith the predictions of rational models is the tendency of individual investors to tradetoo much (Odean, 1999). Many investors trade actively, speculatively, and to theirdetriment. Odean (1999), Barber and Odean (2000), Odean and Barber (2011) andBarber et al. (2009) among others show that the average return of individual investorsis well below the return of standard benchmarks and that the more active traders usuallyperform worse on average. In other words, these investors would do a lot better if theytraded less. Moreover, as noted in Barber and Odean (2013), individual investors makesystematic, not random, buying and selling decisions.Based on the empirical evidence mentioned above, the goal of this paper is to studythe emergence of excess trading in an experimental financial market where trading isclearly detrimental for investors’ wealth. We aim to gain a deeper understanding notonly of why agents trade, but also how correlated is their activity. In particular, wewant to study whether synchronisation of trading activity leads to unstable marketbehaviour, such as crashes driven by panic, herding and cascade effects.Market laboratory experiments now have a rather long history. The most influen-tial paradigm for multi-period laboratory asset markets was developed in Smith et al.(1988). The asset traded in their experiment has a known finite life span and pays astochastic dividend at the end of each period. The fundamental value of the asset fallsdeterministically over time, and lacking a terminal value the asset expires worthless.A salient result is that asset prices in the experimental markets follow a “bubble andcrash” pattern which is similar to speculative bubbles observed in real world markets. This seminal work has spawned a large number of replications and follow-ups, see Palan In fact, Smith (2010) blames a failure of backward induction for the existence of bubbles in thesesimple experimental markets and he suggests that “price bubbles were a consequence of . . . homegrownexpectations of prices rising” . Oechssler (2010) shares this view and argues that “backward inductionis only useful when there is a finite number of periods which most asset markets don’t have. Subjectsare told that they trade assets on a market so they probably expect to see something similar to what theysee on real markets: stochastic processes with increasing or at least constant trend in most cases.”
Seee.g. Noussair and Powell (2010), Giusti et al. (2012), Breaban and Noussair (2014) and St¨ockl et al.(2015) for previous experimental studies on markets with (partly) increasing fundamental values. buy low and sell high , while the decisionsof being inactive and hold cash (shares) depends on whether they expect price returnslower (higher) than average.The outline of this paper is as follows. Section 2 describes the experimental motiva-tion and set-up. The precise instructions given to our subjects are detailed in AppendixC. Section 3 explains the theoretical benchmark to which we want to compare the ex-perimental results. In particular, we show that (risk-neutral) rational agents shouldfavour, in the present market setting, a buy-and-hold strategy. We then summarize ourmain results in Section 4, which includes a refined statistical analysis of agents’ tradingactivity in Section 4.1. In Section 4.2 we relate individual risk attitude to activity rateand final wealth, while in Section 4.3 we link price forecasts and trading behaviour.Section 5 offers our conclusions, open questions, and future experiments.
The basic idea of our experiment is to propose to subjects a simple investment “game”where they can use the cash they are given at the beginning of the experiment to investin a fictitious asset that will – they are told – increase in value at an average rate of m = 2% per period. The asset “lives” for an indefinite horizon, i.e. the game may stoprandomly at each time step with probability p = 0 .
01. The game is thus expected tolast around 100 time steps. Subjects are also informed that random shocks impact theprice of the asset in each period. In the absence of trade, price dynamics are described4y p t +1 = p t · exp( m + sη t ) , (1)where m = 2%, η t is a noise term drawn from a Student’s t-distribution with 3 degrees offreedom and unit variance, as commonly observed in financial markets (Gopikrishnanet al., 1998; Jondeau and Rockinger, 2003; Bouchaud and Potters, 2003), and s is aconstant that sets the actual amplitude of the noisy contribution to the evolution ofthe price (i.e. the price volatility) and is chosen to be s = 10%. These numberscorrespond roughly to the average return and the volatility of a stock index over aquarter. Therefore, in terms of returns and volatility, one time step in our experimentroughly corresponds to three months in a real market, and 100 steps to 25 years.If an amount w is invested in the asset at time t = 0, the wealth of the inactiveinvestor will accrue to w T = w exp (cid:104) mT + ξs √ T (cid:105) (2)at time T , where the second term of the exponential implies random fluctuations of rootmean square (RMS) s per period with ξ defined as a noise term with zero mean and unitvariance. The numerical value of the term in the exponential is therefore equal to 2 ± ξ for T = 100, leading to a substantial most probable profit of e − ≈ fully rational decision in the presence of risk-neutral agents is to buyand hold the asset until the game ends; for the students participating in the experiment,the most probable gain would represent roughly EUR 160, a very significant reward forspending two hours in the lab. In other words, the financial motivation to “do the rightthing” is voluntarily strong.In order to make the experiment more interesting, and trading even more un-favourable, the asset price trajectory is made to react to the subjects decisions, ina way that mimics market impact in real financial markets. The idea is that while abuying trade pushes the price up, a selling trade pushes the price down (for a shortreview on price impact, see e.g. Bouchaud (2010)). As an agent submits a (large)buying or selling order at time t , the price p t +1 at which the transaction is going tobe fully executed is (a) not known to him at time t and (b) adversely impacted by thevery order that is executed. It is made very clear to the subjects that their transactionorders will be executed at the impacted price, meaning that the impact will amount for Mathematically, the average of the exponential of a Student distribution is infinite, because ofrare, but extreme values of η t . In order not to have to deal with this spurious problem, we impose acut-off beyond | η | = 10, with no material influence on the following discussion. p t +1 = p t · exp ( m + sη t + I t ) . (3)The term I t is the price impact caused by all the orders submitted at time t , which wemodel as: I t = N t N B t − S t B t + S t , (4)where N t is the number of subjects who submitted an order at time t and N is thetotal number of subjects in a given market session (i.e. the “depth” of our market). B t and S t , in currency units, are the total amount of buying orders and selling orders,respectively. Note that for a single buying (or selling) order, the impact is given by1 /N , i.e. around 3% for a market with 30 participants (and less if the market involvedmore participants, as is reasonable). On the other hand, if all the agents decide to buy(or sell) simultaneously, the ensuing impact factor would be 100%.With the introduction of market impact, subjects have to guess if the observed pricefluctuations are due to “natural” fluctuations, i.e. to the noise term they are warnedabout at the beginning of the game, or if they are due to the action of their fellowsubjects. This was meant to provide a potentially destabilizing channel, where mildsell-offs could spiral into panic and crashes.We are therefore interested in studying whether excess trading emerges in a marketenvironment which clearly penalises trading, and whether synchronisation of tradingactivity, such as avalanches of selling orders triggered by panic, results in big marketcrashes. The experiment was programmed in Java using the PET software and it was conductedat the CREED laboratory at the University of Amsterdam in May 2014. In the begin- We interpret I t as the permanent component of the price impact, which we assume to be non-zero,meaning that even random trades do affect the long term trajectory of the price. This is clearly atodds with the efficient market theory, within which the price impact of uninformed trades should bezero. We tend to believe that the anchoring to the “fundamental price” in real markets is very weakand is only relevant on very long time scales (Bondt and Thaler, 1985), so that the assumption madehere is of relevance for understanding financial markets. See the discussion in (Bouchaud et al., 2008;Donier and Bouchaud, 2015a) PET software was developed by AITIA, Budapest, and is available at http://pet.aitia.ai. Theexperiment lasted about two hours and before starting the experiment subjects had toanswer a final quiz to make sure they understood the rules of the game (see AppendixC). At the beginning of the experiment each subject is endowed with 100 francs. Eachperiod lasts 20 seconds, during which subjects have to decide whether they want tohold cash or shares in the next period. If subjects did not make any decision withinthe limit of 20 seconds, their current market position would simply carry over to thenext period. If they have cash at period t , they can decide either to use it all to buyshares or to stay out of the market at period t + 1. Conversely, if they have shares atperiod t , they have to choose between selling them all for cash or staying in the marketat period t + 1. Fractional orders are not allowed, therefore each player is either in themarket or out of the market at all times.In order to avoid possible “demand effects”, i.e. trading volumes in the marketsrelated to the fact that participation in the asset market is the only activity available forsubjects (Lei et al., 2001), we introduce a forecasting game in which subjects are askedto forecast the asset price and rewarded according to the precision of their forecast. The information visualised on subjects’ screens include a chart displaying the assetprice evolution together with a table reporting time series of asset prices, returns,individual holdings of cash and shares and price forecasts. Fig. 18 in Appendix Ashows a screenshot of the experiment.In order to mitigate behaviour bias towards the end of the experiment, we implementan indefinite horizon (see Crockett and Duffy (2013) for an example in artificial assetmarkets). In each period of the experiment there is a known constant continuation The smallest market includes 15 subjects while the largest includes 29 subjects. For practical reasons we selected in advance exponential-distributed end-times to ensure thatsessions would not stop too early. In a pilot session we also implemented some markets in which subjects were endowed with 1 shareworth 100 francs. This resulted in many subjects selling shares at the very beginning of the experiment,realising immediately net losses and subsequently engaging in trading activity trying to make at leastsome profits. While the difference arising from the different initial conditions is interesting in itself,we leave it for future work and focus on the case of initial endowment of cash. In a pilot session we implemented a duration of 40 seconds per period. Decision times were,however, well below the threshold of 20 seconds, so we reduced the duration of each time step. Moreover, we remark that practically speaking, the choice of being active or inactive in the marketinvolved the same type of action, i.e. a click on the correspondent radio button (see Appendix C fordetails). There is also a large experimental literature which implements a similar procedure to study in- − p = 0 .
99. Moreover, each subject participates in two consecutive marketsessions, after a short initial practice session to familiarise with the software, so thatlearning mechanisms can be studied. When a market session is terminated, subjects’ final wealth is defined as follows: ifthey are holding cash, their amounts of francs will determine their wealth; if they areholding shares, their wealth is defined as the market value of their shares, i.e. theiramounts of shares times the market price at the end of the sequence. In fact, at theend of the market sequence shares are liquidated at the market price without any priceimpact. At the end of the experiment subjects’ net market earnings are computedas their end-of-sequence balance minus their initial endowment. At the end of theexperiment, each subject rolls a dice to determine which of the two market sessions willbe used to calculate his take-home profits (which also include the earnings from theforecasting game). The exchange rate is 100 francs to EUR 25.Finally, we asked subjects to participate in a second experiment involving the Holtand Laury (2002) paired lottery choice instrument. This second experiment occurredafter the asset market experiment had concluded and was not announced in advance,to minimize any influence on decisions in the asset market experiment.The complete experimental instructions can be found in Appendix C.
In this section we devote attention to the theoretical rational benchmark for this exper-iment and derive the equilibrium of the game under the assumption of full rationalityand common knowledge of rationality. It is worth remarking at this point that thisexperiment is a de facto risk-free opportunity, in the sense that the subjects are paidat the end of the experiment if their net profit is positive but do not owe any amountif their net profit ends up negative.
Let the wealth of the agent be w t at time t . If we assume that the session ends at t = t F , fully rational agents have two possible strategies. The first one is to stay out of finitely repeated games in the laboratory, beginning with Roth and Murnighan (1978). In order to minimise changes in the market environments and facilitate subjects’ learning we usethe same noise realisations in the first and the second market sessions. In this way, for a large number of periods, the average final wealth of a risk-neutral populationwould be approximately the same if agents were given shares instead of cash in the beginning of thesession. t > t = t F which yields an expected final wealth E ( w t F ) = w .The second strategy consists in being fully invested in the market at t = 1 and holdthe shares until t = t F . In fact, in every period holding shares is an equilibrium of thestage game since the strategy “Hold” is the best response to any number of players n that an agent might expect to sell. This strategy yields E ( w t F ) ≈ w · (1 + m + s / t F regardless of the initial condition. The average outcome of the first strategy is at bestzero profit (remember that subjects only keep net profits), while the second strategyprovides large expected profits when the experiment session lasts for a long time.Therefore the rational benchmark, with a risk neutral population, is to get in themarket at t = 1 and hold the shares until the end of the experiment. The longer theduration of the experiment, the larger the expected profits.
Empirical evidence suggests that risk attitude affects trading behaviour. In the followingwe consider the case of risk averse traders using the utility function proposed by Holtand Laury (2002) U ( w ) = 1 − exp ( − αw − r ) α , (5)where α and r are positive parameters. Based on the fact that the utility function inEq. (5) is concave, one could be tempted to consider that for high enough volatility s ,myopic rational traders, i.e. traders who only look one time step ahead when makingtheir decisions, would choose to step out of the market as soon as their wealth reacheda certain level. However, this is not the case because if they decided to sell, the priceof the asset would not only be negatively affected by their action through the impactfactor I t , but also be impacted by the noise sη t . This is due to the fact that transactionsordered at time t are executed at the price realized in t +1, meaning that agents decidingto step out of the market at time t are still subject to price volatility. Consequently, it In fact, given the set of actions A t = { Hold,Sell } available to the agent holding shares, theaction “Hold” vs. “Sell” would result in a price p Ht +1 = p t · exp( m − ( n − /N + sη t ) vs. p St +1 = p t · exp( m − n/N + sη t ), with p Ht +1 > p St +1 . We remark however that, due to the indefinite horizon nature of the game, we can not rule outthe presence of other equilibria on the basis of the Folk theorem. Nevertheless, coordination onsuch equilibria would require an extremely demanding coordination device for subjects. Given thelaboratory environment in which the game is played (not to mention the presence of the noise in theprice process), coordination on these equilibria can be ruled out as a concrete possibility. See also Harrison et al. (2007) and Harrison and Rutstr¨om (2008) for an overview on risk aversionin the laboratory.
9s clear that, for a given time step, it is always a better option to stay in the marketand be affected by the noise factor alone, instead of selling and being affected by boththe noise factor and the negative impact factor. Using Eq. (5) we can also study a few cases of bounded rationality within myopicoptimisation. In particular we compare, for different levels of wealth, the expectedutility given by Eq. (5) in the following anticipated scenarios: all players hold to theirshares, taking into account the volatility of the noise term ( I = 0); all players sell theirshares, taking into account the volatility of the noise term ( I = − I = − I = − .
01 no noise).These scenarios are compared in Fig. 1 below. The magnitude of the noise s defines most of the differences. Fig. 1 gives insight on the possible behaviour of risk- W(t) E [ U ( t + ) ] I = 0I = −1.00I = −1.00 no noiseI = −0.01 no noise (a) RMS s = 0 . W(t) E [ U ( t + ) ] I = 0I = −1.00I = −1.00 no noiseI = −0.01 no noise (b) RMS s = 1 . Figure 1: Expected utility in Eq. (5) for different scenarios with bounded rationality,volatility s = 0 . s = 1 . W ( t ). Thecoloured regions correspond to 1- σ around the average utility.averse players. Comparing the solid lines we confirm the intuition presented above:these solid lines never intersect and it is always better for all agents to stay in themarket (green solid line) than to step out collectively (blue solid line). However, whenwe consider bounded rationality and myopic risk-averse traders, the conclusion may In fact for any number of players n that an agent might expect to step out of the market, theactions “Hold” vs. “Sell” available at time t to an agent holding shares would result in a price p Ht +1 = p t · exp( m − ( n − /N + sη t ) vs. p St +1 = p t · exp( m − n/N + sη t ), with p Ht +1 > p St +1 , meaningthat E [ U (Hold)] = E [ U ( p Ht +1 )] > E [ U (Sell)] = E [ U ( p St +1 )]. The only case in which an agent canreceive a certain amount of wealth with certainty is at the beginning of the experiment by decidingto keep the initial endowment and staying out of the market until the end of the experiment. Thisstrategy would however result in zero profits since subjects can only keep net profits. Fig. 1 is obtained using the parameter values estimated in Holt and Laury (2002) for the power-expo utility function, namely α = 0 .
029 and r = 0 . α = 0 .
106 and r = 0 .
10e different, in particular when the random fluctuations of the market, which impactthe price at each time step, are not taken into account. Consider for example theutilities, computed without taking into account the random fluctuations, for the casesin which only one agent sells (red dashed line) and everyone sells (blue dashed line),and compare them with the case in which everyone stays in the market (green solidline). In these scenarios, there will be a value of wealth W ( t ) above which, in the mindof these boundedly rational agents, it pays off to sell whatever they are holding. Thispoint depends on the agent thinking either that he will be the only one selling or thateveryone will, as well as on the magnitude of the random fluctuations or noise s . Thesevalues are critical thresholds which we represent as a function of s in Fig. 2 throughnumerical simulations. As we would expect from the concavity of the utility function l l l l l l l l l l l s l o g ( W * ) l l l l l l l l l l ll I=−0.01 no noiseI=−1.00 no noise
Figure 2: Critical threshold values W ∗ , as a function of s , beyond which the single-player-out (in red) and all-out strategies (in black) are triggered (corresponding, re-spectively, to the intersection point of the green/red dotted lines and green/blue dottedlines in Fig. 1).in Eq. (5), the value W ∗ beyond which boundedly rational agents of this sort wouldsell decreases as a function of s when they consider the possibility of everyone sellingat the same time.In the presence of myopic boundedly rational agents of the sort considered in thesection, heterogeneity in risk preferences would lead to threshold strategies in whichtraders, when wealth increases beyond a critical value W ∗ , get nervous and get outof the market. This scenario does not, however, account for the variability amongstagents, which most certainly impacts observed events in real markets and, as we shallsee below, the trading activity observed in our experiment.11 Experimental Results
In this section we report on the trading activity observed in the experiment. Subsection4.1.1 describes results on the volume of trading activity in relation to subjects’ wealth.Subsection 4.1.2 studies synchronization in subjects’ trading activity, while Subsection4.1.3 investigates the presence of common patterns in subjects’ trading behaviour.
It is clear that if all subjects adopted the strategy of buying and holding the sharesuntil the end of the game, the price impact would be I ( t ) ≡ .
75% on average. Remarkably, all groups learnto some extent and trade much less in the second market sessions, leading to a muchhigher average earning of 92%. These results are somewhat in line with findings inthe literature on experimental asset markets showing that sufficient experience with anasset in certain environments eliminates mispricing and the emergence of bubbles (Smithet al., 1988; King, 1991; Van Boening et al., 1993; Dufwenberg et al., 2005; Haruvy et al.,2007; Lei and Vesely, 2009). In fact we observe, as a result of learning, consistentlydifferent behaviour of subjects in the second market sequences when compared to theirbehaviour in the first sequences. The Welch two-sample t-test applied to the final wealthand average activity rate of each subject in first and second runs statistically confirmsthis difference (p-value ≈ − ). Therefore in the following analysis we merge all firstmarket sessions into one dataset and all second sessions into another dataset. Theseaggregated data sets lead to the price time series illustrated in Fig. 3. We remark thatthe durations of the sessions were all different because of the indefinite time horizon.Thus, the number of data points used in the averaging process is not the same for eachtime t but a decreasing step function of t (this is clearly visualised in Fig. 4 below). In practice the average payouts were higher since participants could not incur losses, thereforenegative contributions did not play a role in actual average payouts. On the contrary, Hussam et al. (2008) and Xie and Zhang (2012) find that bubbles can be rekindledor sustained when the market experiences a shock from increased liquidity, dividend uncertainty andreshuffling, or from the admission of new subjects. Moreover, during the first experimental market sessions we experienced network problems withthree computers. Hence the number of subjects included in the pool for the first runs is 198 insteadof 201.
20 40 60 80 . . . . . . c u rr e n c y u n i t s i m p a c t tprice without tradingactual priceimpact (a) First sessions: quantities averaged over 198subjects (18 for the last point). The correlationbetween realized price log returns and price logreturns in the absence of trading is 0 . . . . . . . c u rr e n c y u n i t s i m p a c t tprice without tradingactual priceimpact (b) Second sessions; quantities averaged over 201subjects (23 for the last point). The correlationbetween realized price log returns and price logreturns in the absence of trading is 0 . Figure 3: Average price time series for first and second market sessions.The lines in green represent the “bare” price time series, i.e. the price evolution thatwould have occurred in the market if all agents had played the buy-and-hold strategy.A comparison with the red lines, depicting the realized prices, immediately reveals thattrading significantly weakens the upwards price trend via the impact factor (blue lines):the average slope (i.e. price trend) is divided by a factor of approximately 2 in the firstsessions and 1 . .
85 in first market sessions and0 .
89 in second market sessions. The higher values of the slope (i.e. price trend) and ofthe correlation coefficient for the second sessions are due to a lower trading activity.Fig. 4 displays the positions of the traders – in the market (green) or out of themarket (red) – throughout the experimental market sequences. Notice that not all thesessions lasted the same number of periods, hence the white space in both Figs. 4a andFig. 4b for large t . In fact, for each case, only one session – the longest – lasted untilthe maximum time t displayed, t = 86 for first sessions (Fig. 4a) and t = 82 for secondsessions (Fig. 4b).Subjects’ positions – in (holding shares) or out of the market (holding cash) – are13 s u b j e c t inout (a) First sessions: 198 subjects. t s u b j e c t inout (b) Second sessions: 201 subjects. Figure 4: Traders’ positions – in the market (green) or out of the market (red) – in firstsessions (above) and in second sessions (below).14 . . . . . c u rr e n c y u n i t s a c t i v i t y r a t e subjectfinal wealthaverage activity rate (a) First sessions: 198 subjects and 69-85 peri-ods. The average final wealth was 100 .
75 units ofcurrency, the average activity rate 29% and thecorrelation between the two − . . . . . . c u rr e n c y u n i t s a c t i v i t y r a t e subjectfinal wealthaverage activity rate (b) Second sessions: 201 subjects and 63-81 pe-riods. The average final wealth was 191 .
97 unitsof currency, the average activity rate 12% and thecorrelation between the two − . Figure 5: Activity rate and final wealth in first sessions (left) and in second sessions(right).mostly intermittent, which implies excessive trading. However, comparing Figs. 4a and4b, we observe that when the same subjects play for a second time, some of themactually learn the optimal strategy, which translates into “green corridors” in Fig. 4b.Fig. 5 shows that the distribution of average trading activity changes when the sameset of people play the game for the second time, and relates it to agents’ final wealth.The number of people who keep trading to a minimum increases significantly in secondsessions, where only a few outliers keep trading activity above 40%, i.e. they changedtheir market positions in more than 40% of the periods. In both cases, the final wealthof the agents is strongly anti-correlated with average trading activity, which is expectedsince trading is costly. In fact, if a trader decides to buy shares at period t and to sellthem at period t + 1, he will, on average, end up with less cash than he started becauseof his own contribution to price impact.The average wealth of the subjects in first and second sessions is shown in Fig. 6 asa function of time, while Fig. 7 shows the average components of wealth over time.Fig. 6 shows once more that subjects fared much better in second sessions, in whichtheir average final wealth was approximately twice their initial endowment, than infirst sessions, in which the vast majority did not even break even. In what concerns theaverage components of wealth over time, there are also differences between first sessions15
20 40 60 80 t c u rr e n c y u n i t s W t + s W t W t W t − s W t (a) First sessions, 198-18 subjects. t c u rr e n c y u n i t s W t + s W t W t W t − s W t (b) Second sessions, 201-23 subjects. Figure 6: Average wealth for first sessions (left) and second sessions (right). . . . . . c u rr e n c y u n i t s n u m b e r o f s h a r e s t W t C t S t (a) First sessions, 198-18 subjects. . . . . . . c u rr e n c y u n i t s n u m b e r o f s h a r e s tW t C t S t (b) Second sessions, 201-23 subjects. Figure 7: Average wealth W t (black), average cash C t (green) and average number ofshares S t (blue) in first sessions (left) and in second sessions (right).16Fig. 7a) and second sessions (Fig. 7b). In first sessions, where overall trading activityis high (Fig. 5a), the average wealth does not follow the upward trend one would expectin a set-up with “guaranteed” average growth of 2% per period. In fact, players trade somuch that they keep eroding their wealth when they sell and affording fewer and fewernumber of shares when they buy. This accounts for a negative impact bias on averageand results in very low earnings at the end of the session, as shown in Fig. 6a. On theother hand, in second sessions, where overall trading activity is much lower (Fig. 5b),the average wealth does increase with time. Nevertheless the number of shares ownedeventually decreases, which is due not only to excessive trading but also to the fact thatsome subjects cash in their earnings before the end of the experiment and stay out ofthe market from that point onwards.This is particularly visible in Fig. 7b, when a surge in price triggers selling ordersover several periods which result in a higher average amount of cash and, naturally, ina lower average number of shares. Although this is also visible in the middle of the timeseries in first sessions (Fig. 7a), the difference between the two cases is that most of theresulting cash is eventually reinvested in the first sessions, while in the second sessionsthis does not happen: cash holdings consistently increase after the initial investmentphase (Fig. 7b).As we showed in Sec. 3 the optimal strategy in our experimental set-up would be tocollectively buy-and-hold and reap the benefits from the baseline average return of 2%per period in the absence of trading. We see that the behaviour of the agents is veryfar from this benchmark, even in the second sessions, in spite of a significant decreasein activity. The average performance in second sessions is indeed still far from what itwould have been if all subjects used the optimal buy-and-hold strategy, i.e. despite thelearning there is still excess trading activity which translates into detriment of collectivewelfare – since even the “virtuous” agents are adversely impacted by the trading activityof excessively “active” agents. Our subjects trade too much, but can we describe in more detail how correlated theiractivity is? In fact, our initial intuition – that turned out to be quite far from whatactually happened — was that the agents would not trade at the beginning of thegame, letting the price rise from its initial value of 100 to quite high values, say 400(EUR 100), before starting to worry that others might start selling, pushing the priceback down and potentially inducing a panic chain reaction. This would have translatedinto either a major crash, or perhaps smaller downward corrections, but in any case asignificant skewness in the distribution of returns – absent in principle from the bare17rice series which is constructed to be perfectly symmetrical, since the noise term inEq. (3) is symmetric. In fact, as we will see below, the empirical skewness of theparticular realization of the noise turns out to be negative, so the reference point thatwe shall be comparing to must be shifted.We have therefore measured the relative skewness of the distribution of price changes,upon aggregation over time intervals of increasing length, from τ = 1 round to τ = 5rounds. The idea is that a panic spiral would lead to a negative skewness that becomeslarger and larger when measured on larger time intervals, before going back down tozero after the typical correlation time of the domino effect. This is called the “leverageeffect” in financial markets, and is observed in particular on stock indices where thenegative skewness indeed grows as the time scale increases, before decreasing again,albeit very slowly (Bouchaud et al., 2001).In order to reduce the measurement noise, it is convenient to measure the skewnessusing two low-moment quantities. One is 1 / − P ( r τ > m τ ). If this quantity is negative,it means that large negative returns are more probable than large positive returns, asto compensate the excess number of positive returns larger than the mean. Anotheroften used quantity is the mean m t of the returns minus the median, normalised bythe RMS of the returns on the same time scale. Again, if the median exceeds themean, the distribution is negatively skewed (see e.g. Reigneron et al. (2011) for furtherdetails about these estimators of skewness). Both quantities were found to give thesame qualitative results, thus we chose to average these two definitions of skewness andplot them as a function of τ , averaged over all first and second sessions.The result is shown in Fig. 8. The blue dots correspond exactly to the time seriesof bare prices because there is only one (collective) trade in the buy-and-hold strategy,right at the first period, which we discard from the computation. Although the barereturns were constructed using a Student’s t-distribution with 3 degrees of freedom,which by definition is not skewed, we see in Fig. 8 that the bare prices do not have zeroskewness. This illustrates the role of the noise, which gives way to different values ofskewness for bare prices depending on the number of periods of the session. We observein Fig. 8 that the realized skewness of trade impacted returns is typically larger (i.e.less negative) than bare returns, but without any significant time dependence. Thissuggests that buying orders tend to be more synchronized than selling orders, speciallyin the first sessions, but that neither buying nor selling orders induce further buy/sellorders. In short, there is no destabilising feedback loop in the present setting, whichexplains why we never observed any large crash in our experiments.In order to detect more precisely the synchronisation of our agents, we define an18 ll lll lll lll lll −0.2−0.10.0 1 2 3 4 5 t [ − P ( r t > m t ) + p m t − m e d i a n ( r t ) s r t ] lll buy−and−holdfirst runssecond runs Figure 8: Average skewness of price log returns as a function of aggregation length τ in first sessions (red) and second sessions (green), together with the skewness of logreturns in the buy-and-hold strategy, i.e. in the absence of trading (blue).activity correlation matrix A as follows: A ij = 1 T (cid:88) t θ i ( t ) θ j ( t ) − T (cid:88) t θ i ( t ) × T (cid:88) t θ j ( t ) , (6)where θ i ( t ) is the activity of agent i at time t , θ i ( t ) = 0 if s/he is inactive, θ i ( t ) = ± A and study the three largest eigenvalues, corre-sponding to the more important principal components of the subjects’ activity. In orderto detect synchronisation, where a substantial fraction of agents tend to act in exactlythe same way across the experiment, we compute the absolute value of the dot productsof these three eigenvectors (cid:126)v , (cid:126)v , (cid:126)v and the uniform vector (cid:126)e = (1 , , . . . , / √ N . Then,we average the maximum of these three numbers over all runs. It may indeed happenthat the “synchronized” mode does not correspond to the largest eigenvalue of A , whilestill being amongst the three most important ones, and the above procedure allows oneto capture these cases. The resulting values are represented in Fig. 9 for the first andsecond sessions, and compared with a null-hypothesis benchmark, obtained using 1000random bootstrap replicates of the experiments.The dashed lines depict the cases where agents would act completely at random,which would lead to a value of this average maximum overlap at approximately 0 . st sessions 2nd sessions m e a n ( m a x [ | e v * ( ,.., ) / s q r t ( s u b j e c t s ) | ] ) . . . . . . rank ( l ) = ( l ) = Figure 9: Average maximum absolute value of the three dot products between theeigenvectors corresponding to the three largest eigenvalues, and the unit vector, withthe corresponding statistical error bars. All orders are considered except for the firsttime step, in which we expect a natural bias towards synchronization. The dashedhorizontal line around 0 . .
57 for the first sessions and approximately 0 . θ i ( t ) accordingly. This way, when we study thesynchronization concerning only buying orders, we define θ i ( t ) = 0 if agent i is inactiveor sells and θ i ( t ) = 1 if he buys. Likewise, in the case where we look into synchronizationover selling orders only, we set θ i ( t ) = 0 if agent i is inactive or buys and θ i ( t ) = − Fig. 5 shows us, once again, that the average final wealth in first sessions is muchsmaller than in second sessions, which is tied to the higher average trading activity ofthe subjects when they play the game for the first time. In second sessions, we observea number of subjects who kept trading activity very low, increasing their chances ofa positive payout at the end of the experiment. As we discussed in Sec. 4.1.1, this isan indication that the subjects learn. In any case, there are always traders who keeptrading at very high rates and lose money in the process.However, Fig. 5 does not provide insight about common patterns in the behaviourof the subjects. We know from Fig. 4 that at least in second market sessions a numberof subjects use the buy-and-hold strategy or similar, which corresponds to the greenhorizontal “corridors” in the figure. Therefore, we apply clustering techniques to searchfor groups of subjects with similar trading profiles in the data sets. Afterwards, we lookinto the trading activity and trading performance in each cluster.As in Tumminello et al. (2011) and Tumminello et al. (2012), we applied falsediscovery rate (FDR) methods to validate links between subjects to the data set withcomposite data from first sessions and to the data set with composite data from secondsessions. The variable used to establish links (i.e. similarity) between subjects wastheir position – in or out of the market – over time for each subject. The FDR rejectionthreshold was 1%.The clusters are visualised in Fig. 10, which also displays the number of subjects ineach cluster for first sessions and second sessions.The clusters identified in the second sessions are much larger than those identifiedin first sessions, which is expected because the number of “intermittent” players in thegame was lower in second sessions.We see in Fig. 11 that clusters with a lower average trading activity tend to have21 l llll l lll l ll l l lll lllllll (a) First sessions llllllllllllllllllllllllllllllll ll ll lllll l l llll llllllllll ll l ll l ll lll ll ll l l l ll l llll lllllll (b) Second sessions
Figure 10: Clusters for first and second sessions using the FDR algorithm with a thresh-old of 1% applied to positions – in or out of the market – over time. The number ofsubjects in each cluster is displayed in the legends. l llll l l activity rate f i n a l w e a l t h (a) First sessions l l l ll ll activity rate f i n a l w e a l t h (b) Second sessions Figure 11: Average trading activity and final wealth for each cluster represented in Fig.10.a higher average final wealth. A notable example is the cluster number 2 in Fig. 11b,which includes 14 subjects (see Fig. 10b) who kept trading to a minimum in second runsand maximized their returns. Conversely, the cluster number 6 in Fig. 11a consists of2 traders (see Fig. 10a) with very high average trading activity and, as a consequence,low final wealth. 22 .2 Risk attitude and activity rate
To measure risk aversion of subjects we use the paired lottery choice instrument ofHolt and Laury (2002). The Holt-Laury paired-lottery choice task is a commonly-usedindividual decision-making experiment for measuring individual risk attitudes. Thissecond experimental task was not announced in advance; subjects were instructed that,if they were willing, they could participate in a second experiment that would last anadditional 10-15 minutes for which they could earn an additional monetary payment.All subjects agreed to participate in this second experiment. In this task subjects choosebetween a lottery with high variance of payoffs (Option B) and lottery with less variance(Option A). As in Holt and Laury (2002) we use the relative frequency of B-choices(“risky” choices) as a measure for a preference for risk. Moreover, in order to make theamounts at stake comparable to what subjects could earn over an average session oftrading periods and to assess whether or not the risk attitudes of subjects depended onthe wealth levels involved, we elicited subjects’ choices for two lotteries, correspondingto two times (2 × ) and ten times (10 × ) the amounts offered by Holt and Laury (2002)in their baseline treatment. Subjects were ex-ante informed that they would throw adice to determine which of the two lotteries would determine their payoff from thisadditional experimental task.Appendix C includes the instructions for the Holt-Laury paired-lottery choice ex-periment.In Fig. 12 we show the distribution of subjects according to their risk aversion, bothfor the 2 × and the 10 × lottery. The blue vertical line shows where a risk neutral subject would be, based on theexpected pay-off differences alone. The fact that the majority of the subjects – 76%for the low-pay-off and 89% for the high pay-off lotteries – have a number of “safe”A-choices larger than 4 indicates that, overall, the participants in our experiment wererisk-averse. Moreover, if we compare the two curves, red and green, we see that thesubjects tend to safer choices when the lottery pay-off is higher, which indicates thatthe risk aversion of our population not only depends but increases with the pay-offlevel. This is corroborated by the Welch two-sample t-test, which states that the meanrisk aversion values for low pay-off and high pay-off are statistically different with a p-value of 1 . − . We also estimate the parameters of the power-expo utility function inEq. (5) using the lottery choices of our pool of subjects. As in Holt and Laury (2002), wefind evidence for increasing relative risk aversion and decreasing absolute risk aversion,i.e. positive estimates for both parameters α and r . Details on the estimation procedure We discarded from our data set the cases in which the subjects chose the safe lottery after havingpreviously chosen the risky option for a lower pay-off advantage. l l ll l l ll l lll llll ll l risk aversion e c d f ll low pay−off lotteryhigh pay−off lottery Figure 12: Risk aversion cumulative distribution function. The x-axis shows the numberof A-choices (“safe choices”) in the paired lotteries. The higher this level, the more risk-averse is the agent. The blue vertical line shows where a risk neutral subject wouldbe.and results are reported in Appendix B.We now relate individual risk attitude, as elicited by the binary lottery choices, totrading behaviour. Previous experimental studies on the relation between elicited riskattitudes and aggregate market behaviour (Robin et al., 2012; Fellner and Maciejovsky,2007) show that the higher the degree of risk aversion the lower the observed marketactivity. On the other hand, Michailova (2010) finds no significant effect of the numberof safe choices in paired binary lotteries on the frequency of trading.Fig. 13 displays average activity rates and levels of final wealth for different riskattitudes in both first and second runs.Our results clearly show that the activity rate decreases and the final wealth in-creases with the level of risk aversion, both in first runs (red and orange) and in secondruns (green and blue). In other words, preference for risk is an important determinantof excess trading in our experiment. Our findings are in line with the empirical evidencesuggesting that risk-loving, overconfident individuals are more willing to invest in stocks(Keller and Siegrist, 2006) and engage in speculative activity (Odean and Barber, 2011;Camacho-Cuena et al., 2012). 24 l l ll l l ll l lll llll ll l ll l ll l l ll l lll llll ll l risk aversion a c t i v i t y r a t e ll l ll l l ll l lll llll ll l ll l ll l l ll l lll llll ll l risk aversion f i n a l w e a l t h llll (a) ll l ll l l ll l lll llll ll l ll l ll l l ll l lll llll ll l risk aversion a c t i v i t y r a t e ll l ll l l ll l lll llll ll l ll l ll l l ll l lll llll ll l risk aversion f i n a l w e a l t h llll (b) Figure 13: Risk attitudes, trading activity and final wealth. Plot (a) displays averageactivity rate and final wealth for each level of risk aversion. Plot (b) shows linearregressions with 95% intervals of confidence of the average activity rate and final wealthon risk aversion. Clearly, more risk averse subjects trade less and end up with a higherfinal wealth.
The fact that the subjects input their price predictions throughout the experimentallows us to have a glimpse of their frame of mind. In fact, trades only tell about theconsequence of the state of mind (i.e. the price expectation) of traders when they areactive. But traders (both in real life and in experiments) are in fact inactive most ofthe time. As a consequence, trades alone are unlikely to be able to explain why tradersare inactive. Since we have both trades and subjects’ price expectations, we are ableto give a consistent picture of activity and inactivity as a consequence of price returnexpectations.In both market sessions, the subjects did not input anything in about 7% of the time,as price prediction was not a mandatory activity (although monetarily incentivised); inthe following, we restrict our analysis to the subjects that did report their predictions.The discussion focuses on the predicted log returns, i.e. from subject i ’s price pre-dictions (cid:98) p i ( t + 1), we compute the predicted log returns (cid:98) r i ( t + 1) = log[ (cid:98) p i ( t + 1) /p ( t )],for all subjects. The average prediction in the first market sessions is − .
01, and +0 . m = 2% inabsence of trading. The percentage of positive predicted returns is 54% in the first run,25nd 58% in the second run. Fig. 14 illustrates the full empirical cumulative distributionfunctions of expectations for both runs and for positive and negative return separately. -4 -2 0 2 - - - - - Return predictions log ( r 0.001 ) e c d f (r) run 1run 2 Figure 14: Reciprocal empirical cumulative distribution functions of negative (left) andpositive (right) expected price returns during the first and second sessions (black andred lines, respectively). The baseline return distribution is plotted in dashed lines. Thejump at r = 0 indicates the respective fraction of positive and negative predictions.The starting point for each of the positive and negative distributions represents thefraction of positive and negative expectations, thus the higher jump at r = 0 for thesecond session reflects the increase in the fraction of positive expected returns.We then check how the distributions of positive and negative expected returns arerelated to the baseline return distribution, determined by the Student noise term sη t inthe price updating rule. The baseline return distribution is plotted in dashed lines inFig. 14; comparison between the latter and empirical distribution of expected returnsgives a first clue about the type of extrapolation rules from past returns that theagents use. The asymptotic power-law tails of Student’s t-distributed variables, such asthe noise η t , are unchanged under summation, by virtue of the central limit theorem.This implies that linear expectations yield expectation distributions with power-lawstails that have the same exponent. On the other hand, panic or euphoria may leadto non-linear extrapolations and thus may modify either the tail exponent of thesedistributions, or even the nature of the tails.The most obvious finding is that the actions of the agents increase the volatility ofthe baseline signal (in dashed lines) as the empirical distribution functions are abovethe baseline signal for both sequences. The amplification of the noise for positive26xpectations is almost the same in the two sequences, while for negative expectationsthere are marked differences between the two runs as the scale of negative expectationswas much larger during the first run. We used robust power-law tail fitting (Clausetet al., 2009; Gillespie, 2015) and determined the most likely starting point of a power-law r min and the exponent α (see Table 1). Quite remarkably, the parameters of thepositive and negative tails are simply swapped between the two runs: thus not only thescale of negative expectations changes, but the nature of largest positive and negativeexpectations also changes. The fitted tail exponent is not far from 3, the one of theStudent noise showing once again the absence of destabilizing feedback loops. r min α run 1 r < r > r < r > r min denotes the most likelystarting point of the power-law.The fact that the subjects have heavy-tailed predictions suggests that they formtheir predictions by learning from past returns, which do contain heavy tails becauseof the Student’s t-distributed noise. We thus hypothesize some relationship betweenpredicted returns and past returns. This is in line with the best established fact aboutreal investors, which is the contrarian nature of their trades: their net investment overa given period is anti-correlated with past price returns (Jackson, 2003; Kaniel et al.,2008; Grinblatt and Keloharju, 2000; Challet and de Lachapelle, 2013). In addition,previous experiments (Hommes et al., 2005) have demonstrated that four simple classesof linear predictors using past returns are usually enough to reproduce the observed pricedynamics.Based on the considerations outlined above, we first the return predictions of eachsubject with a linear model. (cid:98) r ( t + 1) = ω + ω r ( t ) , (7)where ω and ω ∈ R . Price return predictions are fitted separately for each trader,each session, and each possible action. We fit ω and ω simultaneously. Sec. 4.3.1discusses results for ω while Sec. 4.3.2 is devoted to ω . The parameters r min and α for the power-law are not to be confused with the parameters of thepower-expo utility function in Eq. (5). .3.1 Average predictions ( ω ) In order to give a picture of traders’ movements on the market we compute returnexpectations conditionally on the actions of the subjects. There are four possible ac-tions: buying, selling, holding shares and holding cash. Fig. 15 reports the conditionaldistributions of price return predictions, for both sessions. w d e n s i t y BuyStay in (keep shares)Stay out (keep cash)Sell 051015 −0.2 −0.1 0.0 0.1 0.2 w d e n s i t y BuyStay in (keep shares)Stay out (keep cash)Sell
Figure 15: Densities of trader average return prediction ω during the first sessions (leftplot) and second sessions (right plot) for the four types of decisions. Dashed verticallines refer to the baseline return of 2%.The results are qualitatively the same for both market sequences: the conditionaldistributions are clearly separated, as shown in Tab. 2 below; the main difference be-tween the two runs is that the variance of expectations among the population is muchreduced in the second run. ω , 1nd run Sell Hold cash Hold sharesBuy 1 . − . − . − Sell 3 . − . − Hold cash 1 . − ω , 2nd run Sell Hold cash Hold sharesBuy 2 . − . − . − Sell 6 . − . − Hold cash 6 . − Table 2: Tests of the difference of distributions of ω among the subjects, conditionalon two given actions. The table reports the p-values of Mann-Whitney tests for eachpossible pair of actions. 28et us break down the results for each possible action:1. When the subjects hold assets, their expectations are in line with the baselinereturn of 2%.2. When the subjects hold cash, their expectations are significantly lower (essentiallyzero).3. When the subjects make a transaction, however, their expectations of the nextreturns are anti-correlated with their actions, i.e., they buy when they expect anegative price return and vice versa.Thus, the actions of the subjects are fully consistent with their expectations: theydo not invest when they do not perceive it as worthwhile and they keep their shareswhen they have a positive expectation of future gains. The actions of trading subjectsare instead consistent with a “buy low, sell high” strategy. In fact, knowing thattransactions will be executed at the price realized in the next period, subjects submitbuy orders when they expect negative returns and sell orders when they expect positivereturns. ω ) We find that the importance of the coefficient ω , which encodes the linear extrapolationof the past return on future returns, is very weak. Figure 16 reports the conditionaldistributions of past returns’ impact coefficient for both runs, while Tab. 3 reports thep-values of the Mann-Whitney tests between coefficients ω between all state pairs forall subjects. ω , 1st run Sell Hold cash Hold sharesBuy 9 . − . − . − Sell 3 . − . − Hold cash 2 . − ω , 2nd run Sell Hold cash Hold sharesBuy 6 . − . − . − Sell 5 . − . − Hold cash 2 . − Table 3: Tests of the difference of distributions of ω among the subjects, conditionalon two given actions. The table reports the p-values of Mann-Whitney tests for eachpossible pair of actions.The plot shows that during the first run, this coefficient was negative when the agentsdid not act and zero when they did trade. The second run is different: the coefficients29 w d e n s i t y BuyStay in (keep shares)Stay out (keep cash)Sell 01234 −0.2 −0.1 0.0 0.1 0.2 w d e n s i t y BuyStay in (keep shares)Stay out (keep cash)Sell
Figure 16: Densities of trader conditional return prediction ω during the first sessions(left plot) and second sessions (right plot) for the four types of decisions. Dashedvertical lines refer to the baseline return of 2%.do not seem to depend much on the state, the only clear difference is between holdingcash and holding shares.The lack of influence of this coefficient is confirmed when one measures the averagepredicted return conditional on the action of the subjects, which gives results very closeto ω . We presented the results of a trading experiment in which the pricing function favoursearly investment in a risky asset and no posterior trading. In our experimental marketsthe subjects would make an almost certain gain of over 600% if they all bought sharesin the first period and held them until the end of the experiment.However, market impact as defined in Eq. 4 acts de facto as a transaction cost whicherodes the earnings of the traders. Our subjects are made well aware of this mechanism.Still, when they participate in the experiment for the first time, their trading activityis so high that their profits average to almost zero. They are however found to faremuch better when they repeat the experiment as they earn 92% on average – which isstill much below the performance of the simple risk-neutral rational strategy outlinedabove. We therefore find that, echoing Odean (Odean, 1999; Odean and Barber, 2011), investors trade too much , even in an environment where trading is clearly detrimental30nd buy-and-hold is an almost certain winning strategy (at variance with real marketswhere there is nothing like a guaranteed average return of 2% per period). Our result ispotentially quite important when translated into the real world: unwarranted individualdecisions can lead to a substantial loss of collective welfare, mediated by the mechanicsof financial markets.At the end of the experiment we collect data on traders’ risk attitude by meansof paired lottery choices `a la
Holt and Laury (2002). We observe that, overall, oursubjects are risk-averse and their relative risk aversion increases with the pay-off levelin a way that is quantitatively similar to the results reported in Holt and Laury (2002).We then relate individual risk attitude with the results of the trading experiment andwe observe that the activity rate increases and the final wealth decreases with subjects’preference for risk.Moreover, in each period we also ask subjects to predict the next price of the asset.This provides us with additional information about our controlled experimental market,i.e. we have access not only to the decisions of each trader but also to their expectations.It is important to emphasize that this information would not be available in broker data.In fact, knowing the expectation behind each decision of each trader – including thedecisions to do nothing – is one of the advantages of laboratory experiments comparedto empirical analysis of real data. Using this information, we confirm in Sec. 4.3 that thetraders in our experiment have a contrarian nature, which, together with the patternof excessive trading, is one of the known features of individual traders in real financialmarkets, as discussed in de Lachapelle and Challet (2010) and Challet and de Lachapelle(2013). In particular, it seems that our subjects actively engaged in trading in theattempt to “beat the market”, i.e. trying to make profits by buying the asset whenthey expected the price to be low and selling the asset when they expected the price tomove upwards.Contrary to what happens in real financial markets, we have not observed any“leverage effect” (i.e. an increase of volatility after down moves). Although there isa clear detrimental collective behaviour in all sessions of this experiment, we do notwitness any big crash or avalanche of selling orders that would result from a panicmode. As we discuss in Sec. 4.1.2, the coordination amongst traders was actuallyslightly stronger when buying than when selling, resulting in a positive skewness ofthe returns. In order to induce crashes and study their dynamics, the duration ofthe experiment could be expanded while the time available for each decision could bereduced. The former would increase the probability of a sudden event by increasing thenumber of trials and the amounts at stake, while the latter could contribute to higherstress levels amongst subjects and increased sensitivity to price movements. However,31e believe that a more efficient way to generate panic and herding behaviour would beto reduce the “normal” volatility level while and increasing the amplitude of “jumps” inthe bare return time series. Within the current setting, large fluctuations do not seemsurprising enough to trigger panic among our participants. Another idea, perhaps closerto what happens in financial markets, would be to increase the impact of sell ordersand reduce the impact of buy orders when the price is high, mimicking the fact thatbuyers are rarer when the price is high (on this point, see the recent results of Donierand Bouchaud (2015b) on the Bitcoin). Other natural extensions of this experimentwould include the possibility of fraction orders and hedging, as well as short selling. Weleave these extensions and additional experimental designs to future research.32 eferences
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Economics Letters , 41(2):179–185.Xie, H. and Zhang, J. (2012). Bubbles and experience: an experiment with a steadyinflow of new traders. Cirano scientific series, CIRANO, Montreal.38 ppendixA Additional figures m e a n ( m a x [ | e v * ( ,.., ) / s q r t ( s u b j e c t s ) | ] ) . . . . . . rank ( l ) = ( l ) = (a) Buying orders. m e a n ( m a x [ | e v * ( ,.., ) / s q r t ( s u b j e c t s ) | ] ) . . . . . . rank ( l ) = ( l ) = (b) Selling orders. Figure 17: Average maximum absolute value of the sum of the components of theeigenvectors corresponding to the three largest eigenvalues. On the left we restrictourselves to buying orders, while the result for selling orders is shown on the right.The first time step is excluded from the data set because we expect a natural biastowards synchronization. The dashed horizontal lines around 0 . Estimation of risk aversion parameters
As in Holt and Laury (2002), we use the lottery choices of the subjects to calibrate theutility function described in Eq. (5). In short, we apply a maximum-likelihood methodto find the parameters which maximize the probability that the observed lottery choicesare dictated by Eq. (5).In this spirit, the first step we take is to model the probability P Ri that a subjectchooses the risky lottery out of the pair i of lotteries. As in Holt and Laury (2002), wedefine P Ri := E [ U Ri ] µ E [ U Ri ] µ + E [ U Si ] µ , (8)where E [ U Ri ] = (cid:80) k =1 p k U Rk is the expected utility of the risky lottery and E [ U Si ] is theexpected utility of the safe lottery in the pair i . Each lottery has two possible outcomes k = 1 ,
2, each with probability p k and utility U k given by Eq. (5), parameterized by twonumbers α and r . The parameter µ is a real number which allows one to consider a rangeof scenarios between equiprobable choices ( µ = + ∞ ) and utility maximization ( µ → L ( β, y ) = (cid:89) i ( P Ri ) yi · (1 − P Ri ) − y i , (9)where y i are the observed choices for each lottery pair, i.e. y i = 0 if the subject chose thesafe lottery and y i = 1 if he chose the risky lottery from pair i . In addition, β = [ r, α, µ ]includes all the model parameters in Eq. (8).This way, we have the log-likelihood functionlog[ L ( β, y )] = (cid:88) i y i log( P Ri ) + (1 − y i )log(1 − P Ri ) . (10)Finally, the last step is to find the model parameters that maximize Eq. (10). Weapply the Nelder-Mead algorithm (Nelder and Mead, 1965) and use the bias-correctedand accelerated (BCa) bootstrap method (Efron, 1987) for 95% confidence intervals.The results are summarized in Tab. 4. Quite remarkably, the values of the a-dimensionalparameters r and µ are found to be very close to those reported in Holt and Laury (2002)for their lottery experiments ( µ = 0 . r = 0 . α .
106 [0 . , . r .
345 [0 . , . µ .
114 [0 . , . Experimental instructions and paired lottery task(for online publication only) xperimental Instructions 1. Overview This is an experiment on economic decision making. If you follow the instructions carefully and make good decisions you may earn a considerable amount of money that will be paid to you in cash at the end of the experiment. The whole experiment is computerized, therefore you do not have to submit the paper on your desk. Instead, you can use it to make notes. There is a calculator on your desk. If necessary, you can use it during the experiment. Please do not talk with others for the duration of the experiment. If you have a question please raise your hand and one of the experimenters will answer your question in private. Today you will participate in one or more market “sequences”, each consisting of a number of trading periods. There are two objects of interest in this experiment, shares and cash , the latter denominated in francs . In each period you can trade shares in a market with a computer program, called market maker , and the currency used in the market is francs. You pay francs when you buy shares and receive francs when you sell shares. In each period you will have the opportunity to participate in trading or take no action in the market. Details about how this is done are discussed below in section 3. All trading will be in terms of francs. The cash payment to you at the end of the experiment will be in euros. The conversion rate is 4 francs to 1 euro.
2. Sequences of trading periods
As mentioned, today's experiment consists of one or more sequences, with each sequence consisting of an uncertain number of periods. Each period lasts 20 seconds. In each period you have to decide if you want to buy shares, sell shares, or hold either your francs or shares, i.e., take no action in the market. The amounts of your shares and francs will be shown on your computer screen. At the beginning of each period a computer program will spin a virtual roulette wheel visualized on your screen colored in blue for a proportion of 99 percent and colored in pink for the remaining 1 percent. If the black pointer ends up in the blue region, the sequence will continue with a new, 20-seconds period. If, instead, the black pointer ends up in the pink region, the sequence will end and your franc balance and the amount of shares for the sequence will be final (see Figure 1). Thus, at the start of each period, there is a 1 percent chance that the period will be the last one played in the sequence, and a 99 percent chance that the sequence will continue with at least one more period.
Figure 1. Left panel: experiment continues to next period. Right panel: experiment ends. f less than 50 minutes have passed since the start of the first sequence, a new sequence will begin. You will start the new sequence just as you started the first sequence. If more than 50 minutes have passed since the beginning of the first sequence then the current sequence will be the last sequence played, meaning that the next time the roulette spin ends up with the black pointer in the pink region the sequence will end and the experiment will be over. If, by chance, the final sequence has not ended by the three-hour period for which you have been recruited, we will gradually increase the pink region in the roulette wheel until the chance that the sequence will continue equals 0.
3. Market and trading rules
The market works as follows. At the beginning of the experiment, each participant will be given an initial endowment of 100 francs. In each period shares can be traded with the market maker. In particular, in each period each participant is allowed to hold either only shares or only francs. Therefore, during each period you may choose to Sell all your shares to the market maker in exchange for francs, if you are holding shares; Buy shares from the market maker by investing all your francs, if you are holding francs; Hold either your shares or francs, and take no action in the market. Therefore, you cannot buy additional shares if you are already holding shares. Vice versa, you cannot sell shares if you are holding francs. Trading within each period t occurs according to the following mechanism. First, at the beginning of each period t , the computer program spins the roulette wheel. Depending on the outcome of the roulette spin, we can distinguish two cases. CASE 1. Roulette spin ends up in the blue region: market continues to next period The price per unit of share in period t , denoted by P t , is announced by the market maker and visualized on your computer screen. You can then decide whether to sell shares (if you are holding shares), buy shares (if you are holding francs), or hold either your shares or francs, and take no action in the market. In each period t , if you decide to sell or buy, your transaction will take place in the next period at price P t+1 . Therefore, if you decide to sell shares in period t , you will receive an amount of francs in period t+1 given by your amount of shares in period t times the price of shares in period t+1 , which is: Francs in period t+1 = amount of shares in period t × price of shares in period t+1 If you decide to buy shares in period t , you will receive an amount of shares in period t+1 given by your amount of francs in period t divided by the price of shares in period t+1 , which is: Shares in period t+1 = amount of francs in period t / price of shares in period t+1 If you decide to hold either your shares or your francs in period t , the amount of shares or francs that you own will simply carry over to the next period t+1 . otice that shares need not to be bought or sold in integer units. For example, suppose that in period t you own 127.65 francs and you decide to buy shares. Suppose then that the unit price of shares announced in period t+1 turns out to be 172.50. This means that you will receive 127.65/172.50 = 0.74 shares in period t+1 . CASE 2. Roulette spin ends up in the pink region: experiment ends The market maker announces the price per unit of share in period t , denoted by P t , and all the buying and selling orders placed in the previous period t-1 are executed at price P t but it will not be possible to make any other decision to buy, sell or hold either shares or francs. Your final market earnings will then be computed as explained in section 4. The timing of trading is summarized in Figure 2 Figure 2. Timing of trading
How the price is determined
At the end of each period t , the market maker will collect the buy and sell orders and use them to determine the price P t+1 for the next period t+1 . The percentage change between the price in period t+1 and the price in period t , also called return , is approximately given by the sum of the following components a) A constant positive term equal to 2% b)
A “trade impact factor” which depends upon the difference between the total amounts of buy and sell orders from the participants in the market. In particular: - The higher the amount of buy orders in one period, the higher the price in the next period. Therefore, each buy order has a positive impact on price. In particular, if you and all other participants in the market decide to buy shares at the same time, the trade impact factor will be +100%. -
The higher the amount of sell orders in one period, the lower the price in the next period. Therefore, each sell order has a negative impact on price. In particular, if you and all other participants in the market decide to sell shares at the same time, the trade impact factor will be -100%. Therefore the trade impact factor can take a maximum value of +100% and a minimum value of –100%. If all participants in the market decide to hold their shares or francs, the trade impact factor is zero. c)
Price shocks that can take positive and negative values with the same probability. Given that the percentage change of prices from one period to the other is approximately given by the sum of the terms listed above (a + b + c), in case of no trading activity by any participant, price grows on average by approximately 2% every period. Figure 3 reports examples of typical price patterns in markets without any trading activity at any period, i.e. if all participants held their shares until the end of the experiment. All numbers in Figure 3 are provided only to give EXAMPLES, they SUGGEST NOTHING about the duration and price realizations of the experiment you are about to start.
Figure 3. Examples of price patterns without any trading at any period . Your earnings Net market earnings When a sequence is terminated, that is whenever the roulette spin ends up in pink region, your end-of-sequence balance will be computed. If you are holding francs when the sequence is terminated, your amount of francs will determine your end-of-sequence balance. If you are holding shares when the sequence is terminated, the market value (in francs) of your shares, given by the amount of your shares times their price in the end of the sequence, will determine your end-of-sequence balance. Your net market earnings will then be given by your end-of-sequence balance minus 100 francs, corresponding to the initial endowment that you received at the beginning of the experiment: Net market earnings = end-of-sequence balance – initial endowment. Therefore your earnings from participating in the market will be given by your end-of-sequence balance in francs minus your initial 100 francs endowment.
Forecast earnings
In addition to the money that you can earn from participating in the market, you can earn money by accurately forecasting, in each period t , the future price of shares in period t+1 . You will earn a forecast prize of 0.10 Euro per period if your forecast of the shares’ price is within the interval [0.95 × realized price, 1.05 × realized price]. For example, if the realized price in period t is p t = 100, you will earn the forecast prize if your forecast for p t is within the interval [95, 105]. If, for example, the realized price in period t is p t = 200 you will earn the forecast prize if your forecast for p t is within the interval [190, 210]. Total earnings
Your total earnings for participating in today's experiment will equal the net market earnings that you have at the end of the sequence plus any money that you receive for the forecast task. If your net market earnings are negative, or smaller than the show up fee of 7 Euro, then your earnings from participating in the market will be zero and you will only receive the show up fee of 7 Euro Earnings from trade = max(net market earnings, show up fee) plus the money you earned for the forecasting task. If you participate in more sequences, one of them will be randomly selected and your earnings will equal the total earnings in the selected sequence. As mentioned, the cash payment to you at the end of the experiments will be in Euro. The conversion rate is 4 francs to 1 Euro. . The computer screen Below is a sample screen for a fictitious player 1 at the start of period 1.
Figure 1. Example of screenshot
In every period, after the roulette spin, each player must perform two tasks: Decide whether to buy shares, hold either shares or francs, or sell shares by clicking on the corresponding radio button, i.e., BUY, HOLD, SELL. As explained in section 3, you can decide to buy shares only if in that period you are holding francs, and to sell shares only if in that period you are holding shares. Therefore, in every period, the only active buttons on your screen will be the buttons corresponding to your available actions; Enter a forecast of the shares’ price in the next period. Please use the dot symbol to separate decimals (example: 10.32). The box for “Player Actions” is located in the bottom-left corner of the screen. After making your choices, you have to submit your decisions by clicking on the “Submit” button. The box in the bottom-right corner of the screen named “Player Information” reports the following information: The amount of shares you own in the current period The amount of francs you own in the current period Your wealth (in francs) in the current period, given by your francs (if you are holding francs) or your shares times the current price (if you are holding shares) The rest of the screen allows you to track results from previous periods. The graph in the “Market price evolution” box on the upper-left corner of the screen reports a graphical representation of the shares’ price over time. The table contained in the “Information table” box on the upper-right corner of the screen displays additional information about the results in the experiment and it is supplemental to the graph in the left part of the screen. The first column of the table shows the time period. The most recent period is always at the top. The second and the third columns show espectively the price of shares and the returns, which represent the percentage change in shares price between the current period and the previous. A positive return, say in period 10, means that the price increased from period 9 to period 10, while a negative return in period 10 means that the price decreased from period 9 to period 10. The fourth column reports your forecasts (made in the previous period) of the price in the current period. For example, in period 6 the number at the top of the fourth column will report the forecast you entered in period 5 for the price of the shares at period 6. The fifth and sixth columns report respectively your amounts of shares and francs. Finally, the seventh column of the table shows whether in each period you earned the forecast prize or not. The status bar at the bottom of the screen contains information about the status of the experiment and monitors how much time, out of the 20 seconds constituting the duration of each period, you have to take your decisions. If time is up before you make your choices, the computer program will select HOLD as default action, i.e., you will hold either your shares or francs, and use your previous period forecast. . Final Quiz It is important that you understand these instructions. Before continuing with the experiment, we ask that you consider the following scenarios and provide answers to the questions asked in the spaces provided. The numbers used in the quiz are merely illustrative; the actual numbers in the experiment may be quite different. You may find it useful to consult the instructions to answer some of these questions.
Question 1:
Suppose that a sequence has reached period 25. What is the chance that this sequence will continue with another period, namely period 26? . Would your answer be any different if we replaced 25 with 7 and 26 with 8? Yes No Question 2:
Suppose that the sequence has reached period 12, the price announced for that period is P = 15 and you own 5.3 shares. If you decide to sell your shares, how many shares are you allowed to sell? . Suppose that you indeed decide to sell in period 12 and that the price announced in period 13 is P = 10. How many francs will you receive? . Question 3:
Suppose that the sequence has reached period 5, the price announced for that period is P = 8 and you own 10 francs. If you decide to buy your shares, how many francs are you allowed to invest? . Suppose that you indeed decide to buy in period 5 and that the price announced in period 6 is P = 20. How many shares will you receive? . Question 4:
Suppose that at the beginning of period t a price P t = 200 is announced and then the roulette spin ends up in the pink region. Suppose that you have 1.5 shares. What will be your net market earnings (in francs) at the end of the sequence? . Question 5:
Suppose that when the experiment ends, i.e., the roulette spin ends up in the pink region, you have 100 francs. What will be your net market earnings (in francs) at the end of the sequence? .
Question 6:
Suppose that in period t you and all other participants decide to hold either your francs or shares and take no action in the market. Do you expect next period’s price P t+1 to Increase Decrease [Turn sheet for final question] Question 7:
Consider the following scenarios a)
In period t you and all other participants have shares. All of you decide to sell. b) In period t you and all other participants have shares. Only you decide to sell. Regarding scenario a), do you expect next period’s price P t+1 to Increase Decrease How big will the “trade impact factor” (see section “How the price is determined”) be in scenario a)? . Regarding scenario b), do you expect the “ trade impact factor” to be: -
Bigger than in scenario a) -
Smaller than in scenario a) Moreover, do you expect: -
Bigger P t+1 in scenario a) than in scenario b) -
Smaller P t+1 in scenario a) than in scenario b) ou will now face an additional task that will give you the chance to earn extra money, which will be added to what you already earned in today’s experiment. You will face two sequences (sequence 1 and sequence 2) of 10 decisions each. Details about the decisions that we ask you to make are described in the following sheets. After you make decisions for both sequence 1 and sequence 2, you will randomly select one of the two sequences by picking a ball from a jar containing balls numbered 1 and 2, and your choice will determine which sequence will be used to determine your payoff. Obviously Sequence 1 and sequence 2 have the same chance of being chosen. When you have completed all your decisions, and you are satisfied with those decisions please raise your hand and you will be called for payment. You may now read the instructions on the following sheets. EQUENCE 1
You will face a sequence of 10 decisions. Each decision is a paired choice between two options labeled “Option A” and “Option B”. For each decision you must choose either Option A or Option B. After making your choice, please record it on the attached record sheet under the appropriate headings.
The sequence of 10 decisions you will face are as follows: Decision Option A Option B 1 Receive €4.00 1 in 10 chances OR Receive €3.20 9 in 10 chances Receive €7.70 1 in 10 chances OR Receive €0.20 9 in 10 chances 2 Receive €4.00 2 in 10 chances OR Receive €3.20 8 in 10 chances Receive €7.70 2 in 10 chances OR Receive €0.20 8 in 10 chances 3 Receive €4.00 3 in 10 chances OR Receive €3.20 7 in 10 chances Receive €7.70 3 in 10 chances OR Receive €0.20 7 in 10 chances 4 Receive €4.00 4 in 10 chances OR Receive €3.20 6 in 10 chances Receive €7.70 4 in 10 chances OR Receive €0.20 6 in 10 chances 5 Receive €4.00 5 in 10 chances OR Receive €3.20 5 in 10 chances Receive €7.70 5 in 10 chances OR Receive €0.20 5 in 10 chances 6 Receive €4.00 6 in 10 chances OR Receive €3.20 4 in 10 chances Receive €7.70 6 in 10 chances OR Receive €0.20 4 in 10 chances 7 Receive €4.00 7 in 10 chances OR Receive €3.20 3 in 10 chances Receive €7.70 7 in 10 chances OR Receive €0.20 3 in 10 chances 8 Receive €4.00 8 in 10 chances OR Receive €3.20 2 in 10 chances Receive €7.70 8 in 10 chances OR Receive €0.20 2 in 10 chances 9 Receive €4.00 9 in 10 chances OR Receive €3.20 1 in 10 chances Receive €7.70 9 in 10 chances OR Receive €0.20 1 in 10 chances 10 Receive €4.00 10 in 10 chances OR Receive €3.20 0 in 10 chances Receive €7.70 10 in 10 chances OR Receive €0.20 0 in 10 chances After you have made all 10 decisions, you will be called in a separate room for payment and we will throw a ten-sided die (the faces are numbered from 1 to 10, and the “0” face of the die will serve as 10) twice, once to select one of the ten decisions to be used, and a second time to determine what your payoff is for the option you chose, A or B, for the particular decision selected. Even though you will make ten decisions, only ONE of these will end up affecting your earnings, but you will not know in advance which decision will be used. Obviously, each decision has an equal chance of being used to determine your earnings. Consider Decision 1. If you choose Option A, then you receive €4.00 if the throw of the ten-sided die is 1, while you receive €3.20 if the throw is 2-10. If you choose Option B, then you receive €7.70 if the throw of the ten-sided die is 1, while you receive €0.20 if the throw is 2-10. The other decisions are similar, except that as you move down the table, the chances of the higher payoff for each option increase. In fact, for Decision 10 in the bottom row, the die will not be needed since each option pays the highest payoff for sure, so your choice here is between €4.00 or €7.70. Please circle your choice for each of the 10 decisions on your record sheet. Notice that you may choose Option A for some decisions and Option B for others. EQUENCE 2
You will face a sequence of 10 decisions. Each decision is a paired choice between two options labeled “Option A” and “Option B”. For each decision you must choose either Option A or Option B. After making your choice, please record it on the attached record sheet under the appropriate headings.
The sequence of 10 decisions you will face are as follows: Decision Option A Option B 1 Receive €20.00 1 in 10 chances OR Receive €16.00 9 in 10 chances Receive €38.50 1 in 10 chances OR Receive €1.00 9 in 10 chances 2 Receive €20.00 2 in 10 chances OR Receive €16.00 8 in 10 chances Receive €38.50 2 in 10 chances OR Receive €1.00 8 in 10 chances 3 Receive €20.00 3 in 10 chances OR Receive €16.00 7 in 10 chances Receive €38.50 3 in 10 chances OR Receive €1.00 7 in 10 chances 4 Receive €20.00 4 in 10 chances OR Receive €16.00 6 in 10 chances Receive €38.50 4 in 10 chances OR Receive €1.00 6 in 10 chances 5 Receive €20.00 5 in 10 chances OR Receive €16.00 5 in 10 chances Receive €38.50 5 in 10 chances OR Receive €1.00 5 in 10 chances 6 Receive €20.00 6 in 10 chances OR Receive €16.00 4 in 10 chances Receive €38.50 6 in 10 chances OR Receive €1.00 4 in 10 chances 7 Receive €20.00 7 in 10 chances OR Receive €16.00 3 in 10 chances Receive €38.50 7 in 10 chances OR Receive €1.00 3 in 10 chances 8 Receive €20.00 8 in 10 chances OR Receive €16.00 2 in 10 chances Receive €38.50 8 in 10 chances OR Receive €1.00 2 in 10 chances 9 Receive €20.00 9 in 10 chances OR Receive €16.00 1 in 10 chances Receive €38.50 9 in 10 chances OR Receive €1.00 1 in 10 chances 10 Receive €20.00 10 in 10 chances OR Receive €16.00 0 in 10 chances Receive €38.50 10 in 10 chances OR Receive €1.00 0 in 10 chances After you have made all 10 decisions, you will be called in a separate room for payment and we will throw a ten-sided die (the faces are numbered from 1 to 10, and the “0” face of the die will serve as 10) twice, once to select one of the ten decisions to be used, and a second time to determine what your payoff is for the option you chose, A or B, for the particular decision selected. Even though you will make ten decisions, only ONE of these will end up affecting your earnings, but you will not know in advance which decision will be used. Obviously, each decision has an equal chance of being used to determine your earnings. Consider Decision 1. If you choose Option A, then you receive €20.00 if the throw of the ten-sided die is 1, while you receive €16.00 if the throw is 2-10. If you choose Option B, then you receive €38.50 if the throw of the ten-sided die is 1, while you receive €1.00 if the throw is 2-10. The other decisions are similar, except that as you move down the table, the chances of the higher payoff for each option increase. In fact, for Decision 10 in the bottom row, the die will not be needed since each option pays the highest payoff for sure, so your choice here is between €20.00 or €38.50. Please circle your choice for each of the 10 decisions on your record sheet. Notice that you may choose Option A for some decisions and Option B for others. ECORD SHEET FOR SEQUENCE 1
Decision 1 Circle Option Choice A B Decision 2 Circle Option Choice A B Decision 3 Circle Option Choice A B Decision 4 Circle Option Choice A B Decision 5 Circle Option Choice A B Decision 6 Circle Option Choice A B Decision 7 Circle Option Choice A B Decision 8 Circle Option Choice A B Decision 9 Circle Option Choice A B Decision 10 Circle Option Choice A B
LAB COMPUTER ID: ECORD SHEET FOR SEQUENCE 2
Decision 1 Circle Option Choice A B Decision 2 Circle Option Choice A B Decision 3 Circle Option Choice A B Decision 4 Circle Option Choice A B Decision 5 Circle Option Choice A B Decision 6 Circle Option Choice A B Decision 7 Circle Option Choice A B Decision 8 Circle Option Choice A B Decision 9 Circle Option Choice A B Decision 10 Circle Option Choice A B