The CAPM In Thin Experimental Financial Markets

The CAPM In Thin Experimental Financial MarketsPeter Bossaerts‡and Charles Plott§This version:28July2000‡California Institute of Technology and CEPR§California Institute of Technology1The CAPM In Thin Experimental Financial MarketsPeter Bossaerts and Charles PlottAbstract:We report on small-scale experiments of simple,repeated asset markets in two risky securities and one risk-free security.As in large-scale experiments,steady convergence towards the CAPM is discovered,but the process is slower and convergence halts before reaching the actual equilibrium.There is evidence that subjects gradually move up in mean-variance space,in accordance with the CAPM.Yet,adjustment stops,presumably because of subjects’hesitance in the face of market thinness.This hesitance can be optimal because of the multidimensional nature of the desired trades.Because of market thinness,subjects have difficulty implementing bundles of trades in a set of parallel markets based on the MUDA trading mechanism,essentially an electronic version of the Chicago futures markets. JEL Classification:G12,C92,D59.Keywords:Capital Asset Pricing Model(CAPM),Experimental Economics,Financial Markets,Equilibrium,Equili-bration.2The CAPM In Thin Experimental Financial Markets¶Peter Bossaerts and Charles Plott1IntroductionThis paper studies the extent to which the Capital Asset Pricing Model(CAPM)of Sharpe(1964)and Lintner(1965) explains pricing and trading in simple,thin experimentalfinancial markets.The CAPM predicts that equilibrium prices will be set such that expected returns in excess of the riskfree rate will be proportional to the covariance with aggregate risk.Aggregate risk is measured by the return on the market portfolio.The CAPM also predicts that investors will attempt to trade up to a mean-variance efficient portfolio.Since in equilibrium the market portfolio is the only mean-variance efficient portfolio that is entirely invested in risky securities,investors will eventually all hold the market portfolio(plus riskfree securities).The focus on the CAPM is motivated by two considerations.First,the CAPM is the prototype of modern asset pricing theory.Its main pricing prediction,that equilibrium expected returns will be proportional to covariance with aggregate risk,is shared with virtually all other asset pricing models that have been taken to the data(e.g.,Merton (1973),Lucas(1978),Breeden(1979)).The meaning of aggregate risk differs across models.In the CAPM,it is the return on the market portfolio.In Lucas’and Breeden’s models,it is aggregate consumption.CAPM’s main allocational prediction,that investors should hold the same portfolio of risky securities,is recovered or extended in these other models.In the CAPM,this portfolio is the market portfolio;in Merton’s model,hedge portfolios are added to the market portfolio.That is,the models all predict the portfolio separation property where investors’holdings can ¶Thefinancial support of the National Science Foundation and the California Institute of Technology Laboratory for Experimental Research in Economics and Political Science is gratefully acknowledged.Address:m/c228-77,California Institute of Technology,Pasadena, CA91125;Phone(626)395-4067;e-mail:pbs@ and cplott@.3be reduced to the same set of basic portfolios.Ross(1978)provides further analysis on portfolio separation.Second,in an experimental setting risk is fairly limited,so that mean-variance approximations to subjects’actual preferences may be sufficiently reliable.Of course,this is ultimately an empirical issue.Under mean-variance prefer-ences,the CAPM holds in equilibrium.Whence our focus on the CAPM.See also Judd and Guu(2000)for theoretical evidence that the CAPM obtains when risk is small.In large-scale(20to60subjects)experimentalfinancial markets,the CAPM explains many facets of pricing and trading.See Bossaerts and Plott(1999),Bossaerts,Plott and Zame(2000).Prices generally fully equilibrate to the CAPM,and,while individual holdings are quite noisy,transaction-by-transaction price changes are clearly driven by a mean-variance optimal aggregate demand.It is not obvious that thesefindings translate to thin markets.In large-scale markets,competitive pressures must reduce the relevance of strategic behavior.When only10-15subjects are present, strategic considerations may keep markets from moving to their full competitive equilibrium.Our experiments are meant to shed light on this.1Wefind that prices do move towards the CAPM,but very slowly.Also,unlike in the large-scale experiments,the convergence process halts before the actual equilibrium is reached.Still,there is evidence that subjects gradually move up in mean-variance space,in accordance with the CAPM.Yet,adjustment stops as if subjects perceive too little time to fully complete their trades.Subjects’hesitance could be rational given market thinness.This is because subjects generally have to trade combinations of securities in order to improve their positions,yet in thin markets, it is difficult to implement combined trades.Partial execution of portfolio rebalancing can lead to inferior positions.Whence the hesitance.1We should mention Levy[1997],who also reported favorable results from a small-scale CAPM experiment.His argument in support of the CAPM is based on his observation that the average return/beta relationship is positive,and that volatility does not provide incremental explanatory power of the cross-section of average returns.We are more demanding,by testing the full prediction of the CAPM,that the average return/beta relationship is proportional,and not only linear.The proportionality is equivalent to the prediction that the reward-to-risk ratio(Sharpe ratio)of the market portfolio be the highest possible.It is the latter that we verify in the experimental data.4The remainder of this paper is organized as follows.The next section describes the experimental design.Section3 summarizes the predictions that the CAPM makes about the likely experimental outcomes.Section4presents the experimental evidence.Section5concludes.2Experimental DesignThe main features of the experiments are summarized in Table1.A total of seven experiments were conducted.They are indexed by the date of the experiment.Subjects were recruited from the Caltech student community.Many were familiar with basic investment theory,including mean-variance analysis and the CAPM,through an investment class on campus.About one-third of the subjects were graduate students from the natural sciences.The remainder of the subjects was undergraduates,mostly in their junior and senior years.Subjects in the later experiments(5/19and6/9) were drawn exclusively from those who participated earlier and were thus experienced with the experimental setting, markets,etc.The number of subjects varied from a low of5(4/30experiment)to a high of13(5/13experiment). With onlyfive subjects,there was insufficient trade in the riskfree security to make reliable inference,as documented below.While the details of the experiments are outlined in the paragraphs that follow,a brief summary of the setting might be useful.Each experiment consisted of multiple replications(periods)of the same set of conditions.Three securities were created,denoted A,B and C.They had a life of one period,at the end of which they paid a single dividend/payoffand after which they were removed from the system.The magnitude of the dividends depended upon a random draw of one of three possible states,X,Y and Z.The state was drawn after the period was closed,so during trading there was no insider or asymmetric information in the markets.The magnitude of dividends for each of the securities and each state are in Table1.As can be seen there,the dividend of A varies dramatically with the state, the dividend of B varies much less,and the dividend of C does not vary at all with the state.5All trading took place in an experimental currency called francs,which had a known conversion rate into U.S. dollars(shown in Table1for each experiment).2Each period traders were endowed with units of A and B(shown in Table1)and no units of security C.In addition each trader was endowed with afixed number of francs.It was not possible for holdings of francs,security A or security B to go negative(no short sales)but it was possible for holdings of C to go negative(up to a limit).Thus,an individual could use francs to buy A and B but the ability to do so was limited by the endowment of francs.If the individual wanted to increase holdings of A and B beyond the implicit holdings of francs it could be done by selling units of C and paying the dividend,which was the same for all states. Thus,a sale of C is like borrowing.The amount of the(known)dividend will be paid by the person who sold the unit of C to the person who bought the unit.Of course,to the buyer the difference between the price paid and the certain dividend is a risk free return since payment is guaranteed by the experimenter.The price of C was determined in the market so the risk-free rate was determined simultaneously with other rates of return.Table1contains the relevant parameters for each experiment.At the beginning of each period,all subjects were endowed with400francs and a number of securities A and/or B.Each experiment consisted of three markets and the dividends were identical and public.These are shown in Table1.The three securities had the same expected value. As can be seen,security A had a higher variance than security B.Security C had no variance at all.The dividends were the same in all experiments,apart from experiment6/9,when the parameters were changed.Except in the6/9 experiment,subjects were always endowed with four units of security A and security B and no units of security C(but they could go short in security C).The probability of the states was1/3in all experiments.The parameters for experiment6/9differed from the other experiments in two ways.First,the endowments differed.Three groups were formed.Each subject in Group I was given8units of security A;each subject in Group II 2In the4/30experiment,a new member of the experimental team changed the exchange rate ex post,from0.02to0.005dollar per franc.A high number of occurrences of state X led him to believe that the(five)subjects would have been paid too generously under the original exchange rate.At the beginning of subsequent experiments,it was explicitly announced that such ex-post changes in the terms of payment would not be made.6was endowed with8units of security B;and the(single)subject in Group III was endowed with the market portfolio (4of A and4of B).Groups I and II had an equal number of subjects.This way,the market portfolio consisted of an equal number of securities A and B,as in all other experiments.Second,the dividends were changed.While the skewness of the A security was positive in the other experiments,it was negative in6/9.The distribution of security B was left unchanged,as were the expected values.The changes were made in an attempt to increase the liquidity of the markets and to check the robustness of results from the previous experiments.Each period,earnings were determined as(i)the total payoff(unit dividend times number of units,including the possibility of negative holdings of C)on the securities in inventory at the end of the period,plus(ii)the change in cash position of the period(i.e.,end-of-period cash holdings minus the beginning-of-period cash holdings of F400),minus (iii)a pre-determined payment for the endowments given at the beginning of the period,namely1500francs.At the end of the experiment,subjects calculated their earnings and were paid in cash.Any subject that lost money was required to work it off.3This requirement was especially necessary because of the ability of subjects to borrow(sell C).It was thus possible for a subject to borrow money,buy securities and then suffer losses due to unfortunate draws of the state.In order to prevent the possibility that such bankruptcies would destroy the integrity of the incentive system,the subjects signed an agreement to work offlosses.Subjects were informed that the experiments could last approximately three hours.Once assembled at the Cal-tech Laboratory for Experimental Economics and Political Science,instructions,were read to them.Markets were organized as a computerized,continuous double auction,employing the MUDA technology(Plott and Gray(1990)). The experiment started with a trial period in which subjects could familiarize themselves with the technology and procedures by buying and selling with no monetary payoffattached.When the actual experiment started the length of each period was pre-announced andfixed(8minutes in all experiments,except4/30,where it was6minutes). The number of periods in the entire experiment was not pre-announced but it was determined by the duration of the 3Subjects signed an agreement to cover any losses by working in the laboratory at the rate of seven dollars per hour.7experiment,from two to three hours.3CAPM’s Predictions About The Experimental ResultsModern asset pricing theory was developed with the aim of predicting the cross-sectional properties of returns in financial markets such as the ones described in the previous section.It derives necessary properties for returns using a particular logical argument in which equilibrium plays a prominent role.According to the theory,the nature of financial instruments,financial markets and investor preferences impose special conditions on market equilibrium. The essence can be summarized as follows.Investors demand portfolios that are optimal in a decision-theoretic sense. Prices,and,hence the distributional properties of returns,adjust until a given supply of securities and investors’demands are in equilibrium.Therefore,equilibrium prices and returns will reflect the origin of investors’demands, namely,the portfolio-theoretic optimality.This generally implies that average returns on individual securities will increase with covariation with aggregate risk.The CAPM illustrates this in a simple way.Investors demand portfolios that are optimal in the mean-variance sense, i.e.,they minimize variance for a given mean return.At the core of the derivation of the CAPM is the mathematical property that the set of mean-variance efficient portfolios is convex:any weighted average of optimal portfolios is optimal as well.This result facilitates the analysis of equilibrium,as follows.Investors demand mean-variance optimal portfolios.Since the combined(aggregate)demand is a convex linear combination of the individual demands,the aggregate demand is an optimal portfolio as well.In equilibrium,demand must meet supply.Consequently,the portfolio that is supplied to the market must be mean-variance optimal for equilibrium to hold.Thefixed total supply (sum of initial endowments)has become known as the market portfolio.Therefore,the CAPM essentially states that the market portfolio will be mean-variance efficient in equilibrium. Mathematically,mean-variance efficient portfolios are identified with the property that the expected returns(in excess8of the riskfree rate)will be proportional to the covariance with the return on the portfolio.Hence,in the CAPMequilibrium,mean excess returns are proportional to covariance with the return on the market portfolio.Empiricists have either focused directly on the mean-variance optimality of the market portfolio,or on the pro-portional relationship between average excess returns and covariances that mean-variance optimality implies.The former have essentially tested whether the reward-to-risk ratio of the market portfolio is the highest one can get in the marketplace.The reward-to-risk ratio must be the highest possible,otherwise the market portfolio would not be mean-variance optimal.The reward-to-risk ratio is commonly referred to as the Sharpe ratio.Let R F t denote the return on a risk-free security in period t(we will assume throughout that a risk-free asset exists);let R mt be the return on the market portfolio;letσmt denote its volatility(standard deviation).The Sharpe ratio of the market portfolio isdefined to beE[R mt−R F t].(1)σmtA test of the mean-variance optimality of the market portfolio,and,hence,the CAPM,can simply be based on verifying whether the difference between the actual maximum Sharpe ratio and the Sharpe ratio of the market is zero.So,provided markets converge to equilibrium,the main pricing prediction that the CAPM makes about pricing is the following.Prediction1The difference between the maximum possible Sharpe ratio and the Sharpe ratio of the market ratio should converge to zero.Security C was riskfree and in zero net supply.Its price was not set,but was to be determined by equilibrating demand and supply.Along with the riskfree security there was cash(used in trading),which earned nothing.Hence, the theoretical equilibrium riskfree rate is zero.Most often,however,we observed a positive interest rate.That in itself would lead one to reject equilibrium outright.However,it appeared that subjects felt cash constrained,and, hence,started to borrow money in order to execute their buy orders,without waiting for any sell orders to befilled.9(As a matter of fact,an understanding of this coordination effort is important to explain the experimental results.) After that,interest rates gradually declined.Far cleaner evidence for the cyclical effect of this“cash in advance”constraint on the interest rate could be found in the large-scale experiments,though.See Bossaerts and Plott(1999).As far as allocations is concerned,the CAPM predicts that investors should all trade up to mean-variance optimal portfolios.In equilibrium,the market portfolio will be the component of any mean-variance optimal portfolio that is only invested in risky securities.Away from equilibrium,investors may demand a different portfolio of risky securities. Since markets may spend quite a bit of time offequilibrium(asset pricing theory makes no prediction about how fast a market equilibrates),we do not take CAPM’s equilibrium prediction(that investors’holdings should be mean-variance optimal).Instead,we limit our prediction to“tendency toward mean-variance optimality:”Prediction2The trade-offbetween mean and variance of subjects’holdings(combinations of securities)should grad-ually improve.Further discussion of the CAPM and its relationship with experimentalfinancial markets can be found in Bossaerts, Kleiman and Plott(2000).Let us turn to the experimental results and determine whether these predictions match the data.3.1The Experimental Results3.2PricingIt is standard practice in the experimentalfinance literature to provide plots of the evolution of the prices of the different securities,in order to gauge evidence of convergence.Figure1provides an example.It depicts the evolution of the prices in the5/13experiment.As is typical in experimentalfinancial markets,one observes within-period convergence patterns.Across periods,however,there are clear trends(e.g.,the price of security B drifts upwards), despite the fact that the environment remained identical across periods.10In contrast with classical experiments with induced demand,however,the dynamics of transaction prices cannot be plotted against clearly defined price levels that equilibrium theory predicts.Indeed,the CAPM equilibrium does not translate into specific equilibrium price levels,absent knowledge of subjects’preferences.That is,we cannot easily interpret the evidence from plots such as Figure1.Therefore,we should focus on a metric that unequivocally characterize the CAPM equilibrium,namely,the difference between the maximum Sharpe ratio and the Sharpe ratio of the market portfolio.However,experiments have made clear in the past that price discovery is a very apparent and distinct phenomenon.One must not expect initial prices to be such that the Sharpe ratio of the market is economically an insignificant distance below the maximum Sharpe ratio.Rather,evidence of convergence must be looked for.The emphasis is on dynamics,i.e.,on tendencies.See Prediction1.To obtain the evolution of the Sharpe ratio difference,we proceeded as follows.At the beginning of each period within an experiment,we waited until all three securities traded at least ing the most recent transaction prices,we computed the maximum Sharpe ratio,as well as the market’s Sharpe ratio.Subsequently,Sharpe ratio differences were re-computed whenever a new transaction took place.When repeated across periods and experiments, such computations produced a plot of the evolution of the difference between the maximum Sharpe ratio and the Sharpe ratio of the market.This metric could not be obtained in the few experimental periods that one of the securities did not trade.For instance,during several periods in the4/30experiment,the riskfree security(asset C) did not trade.So,no transaction prices were recorded,and,hence,the Sharpe ratios could not be calculated.Figure2displays the dynamics of the difference between the two Sharpe ratios.Experiments are delineated with vertical lines and labeled with the date when they took place.The horizontal axis measures time in experimental periods.In other words,each period within an experiment covers one unit of time.Zero is the minimum difference between the two Sharpe ratios.If the difference reaches zero,prices correspond to the prediction of the CAPM.Most often,however,the market’s Sharpe ratio is below the maximum,as suggested by the negative Sharpe ratio differences, suggesting that markets had not equilibrated yet.Figure2exposes a pronounced tendency for the difference between11the Sharpe ratios to diminish over time.This is apparent within experiments,but the trend is most manifest across the seven experiments(which were arranged in chronological order).Because of the scale,within-period convergence is less visible.In fact,Sharpe ratio differences often moved erratically within periods,with no clear sign of convergence. Hence,the trend towards the CAPM is most evident over longer time horizons.As the experience of the subjects grows,by the5/19experiment,the market appears to have reached the CAPM equilibrium.To evaluate the robustness of thisfinding,we altered the payoffstructure(reversing the skewness of the payoffof one of the risky securities),as well as individual endowments(allocating only one of the two risky securities to each subject,instead of the market portfolio),as explained in Section3.Figure2clearly suggests that this has no impact on the results:the market’s Sharpe ratio is as close to the maximum Sharpe ratio in the6/9experiment as it was in the5/19experiment.Still,closer inspection of Figure2reveals that even when the Sharpe ratio of the market is close to the maxi-mal value,the difference may remain nonzero.This is clearest in the5/19experiment:during virtually the entire experiment,markets were close to what the CAPM predicts but remained afinite distance away.In summary:Result1The evidence from Figure2partly supports Prediction1:there is evidence of convergence to the CAPM, but full convergence is often not achieved.There are two major differences in terms of pricing between the large-scale experiments discussed in Bossaerts and Plott(1999)and the small-scale experiments reported on here.First,convergence in large-scale experiments is much faster.Whereas it takes almost an entire experiment,or even a couple of experiments(experience)for markets to come close to the CAPM in the thin markets studied here,convergence in large-scale experiments is a matter of a few periods,or even one period.Second,the convergence process in large-scale experiments never halts.The CAPM ultimately obtains:the difference between the maximum Sharpe ratio and that of the market is eventually (economically)indistinguishable from zero.12We now turn to the dynamics of individual allocations,where we mayfind some clues for the causes of the slow and often halted convergence process.3.3Individual Allocations And Interpreting Halted ConvergenceThe evolution of the Sharpe ratios in Figure2provides evidence that the allocational forces of the CAPM are at work:prices are driven by a mean-variance optimal aggregate demand,for otherwise the market portfolio(aggregate supply)would not move up in mean-variance space.Still,we would like to see more direct evidence that subjects’positions improve in mean-variance terms.Prediction2states that the trade-offbetween mean and variance of subjects’portfolios should gradually improve over time.Wefirst focus on the one experiment where prices where close to the CAPM yet never fully reached the CAPM,the5/19experiment.Figure4displays the evolution of the potential gain for each subject in this experiment.The potential gain is expressed as the difference in expected return between the subject’s actual position and the mean-variance efficient portfolio with the same volatility.In other words,it is the vertical distance between a subject’s position and the frontier in plots like Figure3.Because individual holdings are quite noisy,Figure4also displays the evolution of the average potential gain.Figure4was generated like Figure2: after each transaction,the mean-variance frontier is re-computed,and subjects’positions are re-evaluated.Thefirst observation in each period does not reflect thefirst transaction;at least one transaction in each security is needed. Unlike in Figure2,however,time is measured in number of transactions.Figure4depicts how subjects gradually move in mean-variance space.The picture is very noisy,confirming the findings in large-scale experiments(see Bossaerts,Plott and Zame(2000)).There is weak tendency for individuals to decrease their potential gain as time progresses.Perhaps the clearest evidence is the fact that in the second half of each period,some subjects manage to move onto the frontier of optimal portfolios(where their gain is zero).4The 4The simplest way to do this is to move out of risky securities and cash,investing solely in riskfree securities,which generally traded at a discount to its payoff.13convergence is not always monotone.For instance,in periods1,3and6,subjects’positions become worse before improving.Apparently,subjects did not all attempt to reverse the situation.Some of the noise is eliminated by looking at the average(solid line).The tendency for improvement is clearer.Overall,therefore,Figure4provides some support for Prediction2.An investigation of the evolution of potential gains in other experiments confirmed thesefindings.Instead of presenting further visual evidence,Table2provides statistical evidence for the5experiments where there were more than200transactions.It displays estimates of the intercept,slope and R2in OLS regressions of the change in the average potential gain(defined as in Figure4)onto its level.If the CAPM does not have any predictive power,prices and holdings move randomly,which means that the average potential gain should not display any tendency to revert when high.If the forces of the CAPM are at work,one expects reversion in the average potential gain.That is,the slope should be significantly negative.In all experiments,the slope is highly significant(and the R2s are high as well), confirming the influence of the CAPM on allocations.The significantly positive intercepts in Table2confirm that the CAPM process does not fully converge.For CAPM allocations to really obtain,the average potential gain should revert back to zero,which means that the intercepts reported in Table2should be insignificant.At the same time,Table2does indicate that the distance from CAPM decreases as experience increases:the intercepts are smaller in the later experiments.We conclude:Result2While noisy,subjects’portfolios gradually improve after adverse movements,confirming Prediction2,but they do not tend to full optimum.As an aside,it should be emphasized that the evidence that the average potential gain has the tendency to diminish but does not fully converge to zero is not necessarily a mathematical restatement of thefinding that the Sharpe ratio of the market portfolio tends,but does not fully converge,to the maximum Sharpe ratio.There is no simple, monotonic mathematical relation between the two.Bossaerts,Kleiman and Plott(2000;Appendix)illustrate how the14。