A test of the Modigliani-Miller invariance theorem and arbitrage in experimental asset markets Gary Charness and Tibor Neugebauer

A test of the Modigliani-Miller invariance theorem and arbitrage in experimental asset markets

Gary Charnessa & Tibor Neugebauerb

a. UCSB

b. University of Luxembourg

July 4, 2015

Abstract: Modigliani and Miller (1958) show that a repackaging of asset return streams to equity and debt has no impact on the total market value of the firm if pricing is arbitrage-free. We test the empirical validity of this invariance theorem in experimental asset markets with simultaneous trading in two shares of perfectly correlated returns. Our data support value invariance for assets of identical risks when returns are perfectly correlated. However, exploiting price discrepancies has risk when returns have the same expected value but are uncorrelated, and we find that the law of one price is violated in this case. Discrepancies shrink in consecutive markets, but seem to persist even with experienced traders. In markets where overall trader acuity is high, assets trade closer to parity.

Keywords: Modigliani-Miller theorem, asset market, twin shares, experiment, limits of arbitrage, perfect correlation, experience, cognitive reflection test, risk aversion

JEL Codes: C92, G12

1. Introduction

In their seminal paper, Modigliani and Miller (1958, 1963) showed mathematically that the

market value of the firm is invariant to the firm’s leverage; different packaging of contractual

claims on the firm’s asset returns does not impact the total market value of the firm’s debt and

equity. The Modigliani and Miller – henceforth MM – value-invariance theorem suggests that

the law of one price prevails for assets of the same “risk class”. The core of the theorem is an

arbitrage proof; if two assets, one leveraged and one unleveraged, represent claims on the same

cash flow, any arising market discrepancies are arbitraged away by homemade (un-)leveraging.

But due to its assumption of perfect capital markets and the no-limits-to-arbitrage condition

(requiring perfect positive correlation of asset returns), the MM theorem has not been tested in a

satisfactory manner on real-world market data. Thus, its empirical significance has been

unclear.1

Nevertheless, such a test is feasible in the laboratory, and providing an empirical test of

the MM theorem is a particular purpose of this laboratory study. Since perfect return correlation

is rare in naturally-occurring equities, we also check how limits of arbitrage affect the empirical

validity of the MM theorem with regards to cross-asset pricing. In particular, we address the

question of whether a perfect positive correlation between asset returns is necessary for the

empirical validity of value invariance or if the same expected (rather than identical) future return

1 The perfect-capital-markets assumption requires among other things that no taxes and transaction fees are levied, and that the same interest rate applies for everyone. Lamont and Thaler (2003a) present several real-world examples where the law of one price is violated. They argue that these violations result from limits of arbitrage. There was an early objection concerning the applicability of value invariance in relation to the variation of payout policy. Modigliani and Miller (1959) replied to this objection by stating that the dividend policy is irrelevant for the value of the company. However, it is now well-accepted that dividends impact empirical valuations (for a recent discussion of the dividend puzzle, see DeAngelo and DeAngelo 2006). The dividend-irrelevance theorem was thus empirically rejected and is considered as being of theoretical interest only. Still, the value-invariance theorem and its proof remain well-accepted in the profession even without an empirical test. It is noteworthy that the put-call-parity theorem (which is conceptually similar to value invariance, as it makes use of the idea of a self-financing replicating portfolio) finds some support with real-world data (Stoll 1969).

1

is sufficient. The data indeed suggest that perfect correlation is necessary for the law of one

price to prevail.

Our design follows the standard line of experimental asset markets research of Smith,

Suchanek and Williams (1988), featuring multi-period cash flows, zero interest rates, and a

repetition of markets with experienced subjects.2 However, in contrast to the standard, single-

asset market approach of Smith et al. and in line with MM, we have simultaneous trading taking

place in two shares of the same “risk class”. These twin shares, which we call the A-share and

the B-share, are claims on the same underlying uncertain future cash flows. So, returns of the A-

share and the B-share are perfectly correlated and any price discrepancies that might arise can be

arbitraged away at no risk. (In a second treatment where returns of the A-share and the B-share

are uncorrelated, we study the impact of limits to arbitrage). However, the stream to

shareholders of A-shares and B-shares differs by a constant amount, i.e., the synthetic value of

debt as we discuss below. Owed to our implementation with zero interest rates, the A-share and

B-share can represent accounting “leveraged” and “unleveraged “equity streams. By comparing

the market prices of shares, we thus present a very simple test of the MM theorem.3 At any point

in time when the price deviates from parity, in other words if the difference between the A-share

and the B-share is not the same as the synthetic debt value, each market participant can exploit

the price discrepancy. Since short-selling and borrowing is costless a trader can make a riskless

arbitrage gain by homemade leveraging or unleveraging. Exploited pricing discrepancies

thereby undo the divergence of market values.

2 See Palan (2013) for a recent literature survey. The literature is mainly concerned with the measuring of mispricing in the single asset market. The conclusion is that confusion of subjects is a main source of the price deviations from fundamentals in the laboratory (e.g., Kirchler et al. 2012). When assets are simultaneously traded in two markets, mispricing seems to be lower than in the single asset market (Ackert et al 2009, Chan, Lei and Veseley 2013).3 We show in Section 3 how our design matches the MM theorem.

2

Our data provide support for the MM theorem as average prices are close to parity, even

though some price discrepancies and deviations from the risk-neutral value seem to persist

throughout the experiment. We notice that perfect correlation is essential for value indifference

as we control for variations in correlation. Besides the perfect-correlation treatment with twin

shares we consider independent draws of dividends of the two simultaneously traded shares in

our control no-correlation treatment. Here, A-share and B-share have the same expected

dividend and idiosyncratic risk as in the perfect-correlation treatment, but there are limits to

arbitrage since an asset swap has risk. We find a clear treatment effect: comparing pricing

across treatments, the (leveraged) A-share is less highly priced than the (all-equity) B-share in

most of the markets of our no-correlation treatment, suggesting a risk premium.

We also observe a higher level of price discrepancies in the no-correlation treatment.

With perfect correlation our measures of cross-asset price discrepancy and deviation from

fundamental value indicate smaller deviations from the theoretical benchmarks than in the no-

correlation treatment. Hence, although an arbitrage-free equilibrium cannot be supported in

absolute quantitative terms for the perfect-correlation treatment, our data provide rather strong

qualitative support for the equilibrium through the comparison of our treatments. That said, as

with evidence observed with experienced subjects in single asset market studies (e.g., Haruvy,

Lahav, and Noussair 2007; Dufwenberg, Lindqvist, and Moore 2005), the price deviation from

fundamental values declines in consecutive markets in both treatments. The movement towards

the theoretical benchmarks, nonetheless, seems to be more rapid in the perfect-correlation

treatment than in the no-correlation treatment, both in decline of price discrepancies and in

deviation from fundamental value. Nevertheless, some potential price discrepancies persist in

3

both treatments, even with experienced subjects, perhaps due to noise (Shleifer and Summers

1990).

We investigate the impact of traders’ acuity, as measured by the cognitive reflection task

(CRT; Frederick, 2005), on the level of price discrepancy; the literature suggests that smart

traders search and eliminate price discrepancies.4 Our measure correlates with the reduction in

price discrepancy on the overall sample. While there are some price discrepancies in markets

with overall high trader aptitude, these are substantially less common and smaller.

More generally our study contributes to the modest experimental literature that tries to

evaluate the market’s ability to reduce or eliminate arbitrage opportunities.5 The observation of

persistence in price discrepancies confirms earlier empirical results. O’Brien and Srivastava

(1993) replicate portfolios of options, stocks and cash in a multiple-asset experimental market

with two stages and information asymmetries. The authors report that if the information

asymmetry cannot be resolved, price discrepancies frequently persist. Froot and Daborah (1999)

study cases with frictions of twin shares that are traded in different real world exchanges; they

report persistent deviations from parity in pricing, creating limited potential for arbitrage since

significant risks remain.6 Lamont and Thaler (2003b) report a related result with limits to

arbitrage, finding that the market value of the sum of the parts in corporate carve-outs deviates

significantly from the market value of the entire company (see also Gromb and Vayanos 2010).

Oliven and Rietz (2004) investigate the data of the 1992 IOWA presidential election

market (IEM), a large-scale experiment conducted for several months on the Internet. Arbitrage

opportunities in this market were quite easy to spot; if the value of the market portfolio deviated

4 See the discussions in Shleifer (2000) and Lamont and Thaler (2003a).5 See the surveys in Cadsby and Maynes (1998), and Sunder (1995).6 For instance, mineral oil companies Royal Dutch and Shell share their joint future cash flows at a ratio 3:2. Price discrepancies arise as their shares usually do not trade at this 3:2 parity. Limits to arbitrage arise from exchange rate risks as shares trade on different exchanges and from margin risks due to the exposure to market volatility.

4

from 100% of the issue price, any trader could make an arbitrage gain by selling or buying at the

issue price. Oliven and Rietz report a substantial number of price discrepancies, but find that

these were quickly driven out. Rietz (2005) reports on a laboratory prediction market

experiment with state-contingent claims. Similarly to the IEM, arbitrage opportunities were

easily spotted, but trading was over 100 minutes rather than 100 days. Rietz concludes that this

market is prone to violate the no-arbitrage requirement. However, if (as in one treatment) the

experimenter automatically eliminates each price discrepancy, this automatic arbitrager was

involved in most trades in the experiment. Rockenbach and Abbink (2006) report that, even

after hours of experience, both students and professional traders left arbitrage opportunities

unexploited in an individual investment-allocation task of cash to options, bonds and stock.

To the best of our knowledge, Levati, Qiu, and Mahagaonkar (2012) propose the only

other available experimental study to test the MM theorem. Their design forecloses any

arbitrage possibility or homemade leveraging and unleveraging.7 Levati et al. examine

evaluations for eight independent lotteries with varying degree of risks in a sequence of

experimental single-asset call auction markets, where the risks represent different levels of

company leverage. In line with our results obtained in the no-correlation treatment, their results

indicate a risk premium on leveraged equity capital. In contrast to our perfect-correlation

treatment, the market data in Levati et al. show no support for value invariance. The authors

acknowledge the foreclosure of any arbitrage possibility as a potential reason for this result.

We provide evidence regarding how limits to arbitrage impact value invariance. Our

main contribution is that we are able to empirically validate value invariance under perfect

correlation. The observation of a treatment effect is also an important contribution of our paper,

7 Stiglitz (1969) proves MM value invariance within a general equilibrium approach, without explicit arbitrage assumptions.

5

as price discrepancies in the market increase substantially when one moves from perfect

correlation to no correlation. Finally, we observe that high trader acuity significantly reduces the

price discrepancy in the market and shares trade closer to fundamentals.

The remainder of the paper is organized as follows. In Section 2 we explain the

experimental design. In section 3 we discuss the MM theorem in light of our design, and in

Section 4 we present our measures of price discrepancies and hypotheses. Our experimental

results are presented in Section 5 and we conclude the paper in Section 6.

2. Design

Following Smith et al (1988), subjects can buy and sell multi-period lived assets in continuous

double auction markets. The assets involve claims to a stochastic dividend stream over a

lifespan of ten periods, = 10, after which these assets have no further value. The instructions

can be found in the Appendix.

Trading occurs in two asset classes named A-shares and B-shares. We follow the Smith

et al. (1988) design, where the dividend paid on an A-share is independently drawn {0, 8, 28 or

60 cents} with equal probability at the end of every period. The possible dividends paid on the

B-share are 24 cents higher, so that the expected dividends per period are 24 cents on A-shares

and 48 cents on B-shares. The interest rate is zero and thus the discounted sum of expected

dividends or fundamental equity value of A-shares and B-shares are initially 240 and 480 cents

and decrease by 24 and 48 cents per period, respectively.

Our treatment variation between subjects is the correlation between the dividends on A-

shares and B-shares. In the perfect-correlation treatment, the B-share pays exactly 24 cents more

in each period than does the A-share. In the no-correlation treatment the dividend on the B-share

is independently drawn {24, 32, 52, 84 cents} with equal probability. Our treatment variation

6

allows us to test the MM invariance theorem under perfect correlation, where shares are perfect

substitutes, and we check if price discrepancies change when we move from perfect positive

correlation to zero correlation.

Each market participant can electronically submit an unlimited number of bids and asks

in the two simultaneous markets. Submitted bids and asks are publicly visible and cannot be

cancelled. The best outstanding bid and ask are available for immediate sale and purchase

through confirmation by the other market participants. Upon a transaction, all bids of the buyer

and all offers of the seller are cancelled in both markets and the price is publicly recorded.

However, the buyer may also have placed offers to sell and the seller may have placed bids to

buy; any such offers and bids remain in the market. So, any arbitrager who wants to exploit a

price discrepancy can immediately buy low and sell high by pressing the buttons.

Market participants were initially endowed with two A-shares and two B-shares, and

1200 cents cash. Traders were able to borrow up to 2400 cents for the purchase of assets and

could short-sell up to four A shares and up to four B shares without facing any margin

requirements. Trading was unaffected by short-sales and borrowings, which were displayed as

negative numbers. Each subject had thus a wide scope for financial decision-making. Since we

were interested in effects of experience, an experimental session involved four consecutive

markets, each designed for nine subjects. The period length in the first market was 180 seconds,

and was 90 seconds in the subsequent markets.8 One of the four markets was chosen (with equal

probability) for payment at the end of the session. A subject determined the payoff-decisive

market by a die role. Subjects received their final cash balance in the payoff-decisive market in

an envelope at their cabin, including the reward for forecasting, plus the payoff from two pre-

8 We allowed more time in the first market for people to get accustomed to the decisions. There is no evidence (including questionnaire reports) that subjects were short of time in the shorter intervals.

7

market tasks (further detailed below) and a show-up fee of $15. In case of a negative cash

balance, the subject’s show-up fee would be correspondingly reduced, but by not more than $5.

Subjects were recruited from a pool of economics and science students at UCSB via

ORSEE (Greiner 2004). Each person drew a number from a tray upon arriving at the laboratory.

This number indicated one’s cabin number. Before the instructions were read for the trading

markets, subjects engaged in two pre-market tasks to reveal to us their relevant personal traits.

Information on payoffs from these tasks was only communicated at the end of the session. The

first task was an investment game to assess the subjects’ degree of risk aversion (Charness and

Gneezy, 2010). They chose an amount 0 $10 to allocate to a risky asset that paid with

equal probability 0 or 2.5 and to a safe asset $10 0, to be paid out with certainty. One

of the nine participants in the asset market would receive the payoff from the first experiment.

The second task was the CRT, where the three questions were asked in a random order.9

Subjects had 90 seconds to answer the questions and were rewarded with $1 per correct answer.

As part of the instructions, subjects had to successfully complete four practice exercises:

a dividend questionnaire, a trading round, a forecasting reward questionnaire,10 and a second

trading round. After the four markets, subjects were debriefed in a questionnaire on personal

9 The questions were: (1) A hat and a suit cost $110. The suit costs $100 more than the hat. How much does the hat cost? (2) If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? (3) In a lake, there is a patch of lily pads. Every day the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of it? 10 Prior to each period, subjects make predictions about the average prices at which the assets will be transacted during the period. By asking to reflect and predict market outcomes prior to the period we aimed at deeply engaging subjects in the experiment. Similar to Haruvy et al. (2007), subjects initially submit a forecast of the average price of each asset for each future period. In our design, however, subjects update only their price forecasts of the current period prior to subsequent market openings. Subjects receive salient rewards for accurate forecasts; the mean percentage deviation of the forecast from the realized average price is subtracted from one and the remainder is multiplied by $6. The deviation in any period is capped at 100%. Periods without transactions do not count in the determination of the payoff; neither in the numerator nor in the denominator. Subjects are rewarded either for the accuracy of initial forecasts or for the accuracy of updated forecasts, with equal probability. The decision is made by computerized random draw after the last market period.

8

details. The experiment was computerized with the software ztree (Fischbacher 2007). Written

instructions were distributed, and verbal instructions were played from a recording.

3. Modigliani-Miller invariance theorem

In this section we discuss the invariance theorem in light of our experimental design, in

particular, the perfect-correlation treatment. Modigliani and Miller (1958, p. 266) assume

identical risk classes “such that the return on the shares … in any given class is proportional to

(and hence perfectly correlated with) the return on the shares ... in the same class...” These

conditions are satisfied in our perfect-correlation treatment.11

To show that the company’s value is invariant to its capital structure, Modigliani and

Miller show that streams on a leveraged position and on an equivalent unleveraged position can

be equalized by home-made leveraging and home-made unleveraging. Correspondingly, we

show with a simple accounting exercise that the streams on our A-share and B-share are

equivalent in the perfect-correlation treatment. In particular, we create a synthetic debt asset that

we linearly write off over the remaining periods. The synthetic debt streams are hidden, pure

accounting streams without real impact.12 Let us think of the A-share as a “leveraged” stream

and the B-share represents a pure equity stream. To be clear on the accounting exercise we

introduce some notation: Let denote the expected return on total assets (on a per share basis).

This stream coincides with the fundamental value of a B-share, denoted by .

Following MM, we reproduce the payoff stream of the A-share by homemade leveraging.

Purchasing the B-share with debt and equity has the equivalent value as purchasing the A-share

11 Conditions are similar in our no-correlation treatment, since expected returns of assets are the same as in the perfect correlation treatment. However, returns are independent in the no-correlation treatment so that, in the strict sense of the definition, A-shares and B-shares are not in the identical risk class. Our experimental test checks if the strict implementation of the definition is critical for achieving empirical support for value invariance. 12 The accounting exercise is easy because the interest rate is zero. Since trading costs are zero and the shares need not be repurchased but are simply cash at the end of the market, cash or debt financing has equivalent consequences.

9

with equity. Let us denote the value of the synthetic debt asset by , which represents both the

sum of 24 cash units for each remaining period including the present one and at the same time

the accounting debt payment stream. Upon receipt of the dividend on the B-share at the end of

each period, we account for 24 cash units as debt service. The synthetic debt stream reduces the

expected stream of the B-share. The remaining stream after accounting for debt is identical to

the equity stream on the A-share, which we denote by its fundamental value, ,

(1) ( ) = ( ) = (48 24) × ( + 1) = .

Likewise we can reproduce the dividend stream on the B-share by accounting for

homemade unleveraging the A-share. Going long the A-share and going long the sum of 24 cash

units for each remaining period (representing a synthetic bond) has the equivalent value as going

long the B-share. To equalize cash-flows at the end of the period we account for 24 cash units as

a (principal) payment from the synthetic bond and add this amount to the dividend paid on the A-

share. Thus, we make the two considered streams identical.

(2) + 24 × ( + 1) = = .

Given our suggested treating of the A-share as representing a leveraged stream and the

synthetic bond-stream as the proportional value of debt-service per A-share, the following

equation confirms that the leveraged flow to debt and equity equals the return on assets before

deduction of debt service. Correspondingly, as the debt of the (unleveraged) company B is zero,

(2’) + = = .

So we have shown how to reproduce the payoff streams of both shares with repackaging return

streams to equity and (synthetic) debt. Next we restate the invariance theorem (see MM 1958, p.

268) per share value for our finite-horizon setting:

“[Invariance theorem]. Consider any company [ = , ] and let [ ] stand as before for the expected return on the assets owned by the company (that is, its expected profit before deduction of interest).

10

Denote by [ ] the … [proportional] value of the debts of the company; by [ ] the [per share] market value of its common shares; and by [ + ] the [proportional] market value of … the firm.Then, … [the invariance theorem] asserts that we must have in equilibrium: (3) [ + = ( )],13 for any firm in [the risk] class”.

Note, rather than implying equality of intrinsic values as in (2’), the invariance theorem implies

equality of market values. By letting (. ) represent any bijective function of the expected return

on assets, we allow a deviation of the market value from fundamentals due to behavioral biases.

Modigliani and Miller (1958) proceed with the proof of the invariance theorem by contradiction,

showing that as long as equation (3) does (MM, p. 258) “not hold between any pair of firms in a

class, arbitrage will take place and restore the stated equalities.” Modigliani and Miller employ

the word arbitrage in the sense that investors could buy the cheap asset and sell the expensive

asset, thereby swap the income streams without involving any risk as these (p. 258) “are

identical in all relevant respects.” Thus, a deviation of market prices of the A-share and B-share

by more or less than the value of debt would violate the no-arbitrage condition. This violation is

easy to see;

(4) >< >< + >< + >< ,

where the latter equivalence relationship shows the contradiction, and concludes the proof.

Modigliani and Miller (1958) discuss expected future streams, but the rigorous proof of

the theorem requires identical future streams. One key research question that we raise in our

experimental study is whether empirical validation of the invariance theorem actually requires

identical asset returns. In the laboratory it is not so clear if notable differences can be detected.

On one hand, earlier research has suggested that human behavior may disregard correlations

13 In Modigliani and Miller (1958, p. 268) ( ) = / where is the average cost of capital for an asset in the same risk class. In this sense our formulation is more general than the MM invariance theorem, since market prices can be different from fundamental value. However, our interest rates are zero and our considered horizon is finite.

11

between asset returns (e.g, Kroll et al. 1988). On the other hand, our setting favors a pricing at

fundamentals, since subjects are informed about fundamentals in the instructions; rational and

risk-neutral behavior also favors pricing at fundamental value. Therefore we are tempted to

conjecture (a weak law of one price) that the market price will be the same for assets of similar

risks even if there are limits to arbitrage. Hence, our research question asks if value invariance

can be supported if A-shares and B-shares are not in the identical risk class.

Certainly, there are limits to arbitrage if streams are non-identical since swapping of one

asset position for another one is non-riskless.14 The literature on the limits to arbitrage (for a

recent survey see Gromb and Vayanos 2010) has suggested that prices may be pushed away from

fundamentals by demand shocks, e.g., by noise (Shleifer and Summers 1990). Depending on

demand shocks and risk aversion in the market, Gromb and Vayanos (whose equation 7 we adapt

to our notation) derive the following A-share equilibrium asset price, while exogenously fixing

the price of the B-share at fundamental value;

(5) = + (1 ) ,

where denotes the correlation coefficient of returns on A-shares and B-shares, denotes a

demand shock, indicates the risk aversion of the market, and is the return variance of the A-

share. In the case of perfect correlation (with a unit coefficient), A-share and B-share are priced

at fundamentals suggesting relative pricing at parity. In the case of zero correlation, according to

equation (5), the impact of the demand shock on price deviation can be significant, augmented

by risk aversion and demand shocks. So, the theory on the limits to arbitrage suggests that non-

14 Opposite to perfect correlation, the no-correlation treatment offers diversification possibilities. Assuming risk aversion, utility maximization requires holding both assets in equilibrium for the purpose of diversification of risk. If shares are not priced at parity, however, exploiting of price discrepancies by selling high and buying low can still yield higher utility than diversification of risk. In the course of such trade, prices are pushed back to parity.

12

perfect correlation may induce price discrepancies.15 Laboratory research on this question seems

to be missing, so our paper fills a gap.

4. Measures and hypotheses

To be clear on our testable hypotheses and observations we continue with the above notation.

and denote the fundamental share values, and denote the prices of A-share and B-

share in period , respectively. We measure the relative difference from parity pricing of A-share

and B-share as follows.

(6) = + 1 = ( ) ( )+ ( )The first ratio in (6) relates the value of the “unleveraged” company B to the value of the

“leveraged” company A, where the difference between fundamental values in the second

denominator stands for the synthetic value of debt (2’). The invariance theorem requires pricing

at parity, = 0 (see Hypothesis 1). Thus, market values shall differ by as much but not more

than fundamentals. In our data analyses, market value is the average market price of the period.

We define the measure cross-asset price discrepancy (PD) as follows, where = 10 is

the number of market periods.

(7) = | | /The PD-value is the average absolute percentage deviation from parity pricing during the course

of a market. Price discrepancy indicates potential gains by selling high and buying low based on

15 We note that markets are incomplete in our no-correlation treatment, since a self-financing replicating portfolio cannot be formed. Therefore, in incomplete markets (riskless) arbitrage is impossible and according to theory the law of one price does not hold. However, if market prices confirmed fundamentals or if the deviations from fundamentals were the same in both of our markets, no cross-asset price discrepancy would arise and the law of one price would still obtain.

13

average prices. The value is usually positive even if the average difference from parity pricing is

zero. The PD-value is zero if A-share and B-share prices are equal to fundamentals, or if prices

deviate by the same amount from fundamental value.

Even with zero PD-value, prices can deviate from fundamental value. To measure price

deviations from fundamentals we define the deviation from fundamental value (DF) as the sum

of absolute deviations of the price from the sum of discounted expected dividends in a period.

The DF-value here measures the expected excess return of buying and selling at prices off the

known fundamental value, and so is similar to measures for single asset markets (Palan 2013).16

(8)

The DF-value indicates potential gains by purchasing one share if its average price is below and

selling one share if the price is above the fundamental value.

Potential discrepancies can also arise in submitted orders. If the adjusted bid exceeds the

best outstanding offer of the twin share, an arbitrager can sell high and buy low, thereby realize

an (expected) gain. Under perfect correlation the gain from eliminating a price discrepancy is

riskless (arbitrage); under no-correlation the expected gain is risky as differences in future

dividends can induce losses or gains. We call noisy each limit order that (upon submission)

leads to a potential expected gain of another trader, and measure a trader’s aptitude inversely by

the relative frequency of submitting a noisy limit order. The trader’s aptitude is therefore high if

this number is small and vice versa.

16 The relative absolute deviation of Stöckl, Huber and Kirchler et al. (2010) is a similar definition of mispricing for the single asset market. However, in their definition the sum of fundamental pricing deviations is divided by the average fundamental value. It is not completely clear to us how that definition can be applied to the case of multiple assets.

tB

tA

t

Bt

Bt

At

At

tPtPtFtPAA

tFtPT

DF 1

14

Our empirical measure of subjects’ acuity is the individual’s CRT score, i.e., the number

of correct answers in the pre-market cognitive reflection task (Frederick 2005). Recent results in

single-asset market experiments have shown systematic effects of CRT scores (Corgnet et al.

forthcoming, Akiyama, Hanaki, and Ishikawa 2013, Breaban and Noussair 2014). Below, we

also measure the effect of gender and risk aversion (as elicited in the investment game) on our

mispricing measures (6) – (8). Recent research from experimental single-asset markets suggests

that price levels may be lower if the share of female traders increases (Eckel and Füllbrunn 2015)

and also if the level of risk aversion increases (Breaban and Noussair 2014).

With these measures at hand we next formulate our testable hypotheses. Our most

efficient benchmark requires prices equal to risk-neutral fundamental value.

Hypothesis 0 (Risk-neutral pricing): There will be no deviations of prices from fundamentals in both share classes, i.e., = 0.

Risk-neutral pricing would require that the price equals fundamental value in each period.

In fact, pricing at fundamentals is sufficient to obtain MM invariance (Modigliani and Miller

1958). However, previous experimental evidence (Palan 2013) shows that pricing deviations

from fundamentals can occur in empirical markets in both directions from above and below.

This evidence does not invalidate the MM law of arbitrage-free pricing, as arbitrage-free pricing

is a necessary, but not sufficient, condition for risk-neutral equilibrium pricing. For arbitrage

free pricing and the MM theorem we require that our A-share and B-share are priced at parity.

Hypothesis 1A (MM invariance theorem): The adjusted market values of A-shares and B-shares will be the same, i.e., = 0. 1B (No cross-asset price discrepancy): = 0.

Hypothesis 1A requires that an A-share and a B-share are valued the same on average.

Hypothesis 1B requires that eventual deviations from fundamentals simultaneously and equally

occur in both shares, so that there are no free lunches even under perfect correlation. Hence, we

15

investigate three levels of market efficiency; the first is based on the absence of differences from

parity pricing on average (H1A), the second is based on the absence of price discrepancies (H1B),

and the third requires that prices coincide with fundamental values (H0).

These levels of market efficiency are tested in two treatment dimensions; first under

perfect correlation where elimination of pricing discrepancies is riskless as dividend streams

differ by a constant and can be perfectly hedged; and second under no correlation where

elimination of pricing discrepancies is risky as dividends are independent of each other. Since

the A-share and the B-share have identical idiosyncratic risks in both treatments, and based on

earlier evidence which suggested no sensitivity of behavior to changes in return correlations

(e.g.., Kroll et al. 1988), one might conjecture the absence of a treatment effect in the laboratory.

However, such a conjecture would violate equation (5) unless market prices confirm

fundamentals. In line with equation (5), thus, we state the following testable hypothesis.

Hypothesis 2. (Hedging effect): The cross-asset price discrepancy (PD) will be larger in the no-correlation treatment than in the perfect-correlation treatment.

The hypothesis suggests that markets can eliminate some price discrepancies in the

absence of risk even without a dedicated (automatic) arbitrager, and so decrease the magnitude

of the PD-value in the presence of perfect correlation.

Anticipating from experimental evidence of mispricing in early markets (e.g., Smith et al.

1988, Duwfenberg et al. 2005), we expect a decrease of mispricing in our consecutive markets.

Hypothesis 3. (Experience effect): The cross-asset price discrepancy (PD) and the deviation from fundamental value (DF) will decrease over time (in consecutive markets).

It is an oft-cited conjecture that smart investors eliminate mispricing in the market (see

e.g., Shleifer 2000). Although our design has no dedicated arbitrager, we can nevertheless

address the predictive power of this argument in our data. We compare the magnitude of

16

arbitrage opportunities vis-à-vis characteristics of market participants, suggesting a relationship

between acuity and aptitude of traders.

Hypothesis 4. (Smart trader effect) The cross-asset price discrepancy (PD) and deviation from fundamental value (DF) will decrease with the market participants’ degree of measured acuity.

5. ResultsIn total, 174 students participated, earning an average of $50 in 3 hours. Each subject

participated in only one session. We have 12 sessions in the perfect-correlation treatment and

eight sessions in the no-correlation treatment. Each session is comprised of four consecutive

markets.17

[Figure 1 here]

4.1. MM invariance theorem

Observation 1. A-share and B-share prices are close to parity and thus only in line with the MM invariance theorem in the perfect-correlation treatment. B-share prices are above parity relative to the A-share price in the no-correlation treatment, suggesting a risk premium.

Support. In Figure 1, we see the difference of A-share and B-share prices relative to

parity for each market, period, and treatment, aggregated separately over the 12 sessions of the

perfect-correlation treatment and the eight sessions of the no-correlation treatment. In the case

of a missing price of A-share or B-share, the period in the session is treated as a missing

observation.18 The prices in the perfect-correlation treatment are close to parity, whereas the

prices in the no-correlation treatment are above parity, indicating a risk premium on the A-share:

17 In the fourth market of one session of the perfect-correlation treatment, one subject submitted an obviously erroneous a bid (instead of an asking price) for $20 on the A-share worth $0.24 that led to a transaction. We have eliminated the period from the data as the transaction impacts our average estimates. In the second market of the two other perfect-correlation sessions, the data for period 10 are missing (due to a server crash). One session of the perfect-correlation treatment and two sessions of the no-correlation treatments had only seven participants.18 On average we have 11% and 5% missing observations in the perfect-correlation and no-correlation treatments, respectively.

17

the lower priced A-share involves a more risky dividend stream as its spread around the mean is

relatively larger than of the B-share.19

Table 1 reports the average differences from parity for each market in both treatments

jointly with the Z-statistics of the two-tailed Wilcoxon one-sample test that indicates significant

deviations from zero. The recorded average results for the perfect-correlation treatment are not

significantly different from parity in any market, = 0; the overall average price difference

from parity per period is 3.67%. By period, we find that only three of 40 (= 4 markets x 10

periods) differences from parity in the perfect correlation treatment are significant at the five

percent level; the pricier share is once the A-share and twice the B-share.20 In the no-correlation

treatment, parity is rejected for each market since the B-share is always pricier than the A-share

(see Table 1); the overall average price difference from parity per period is 23.32 percent in the

no-correlation treatment.

Table 1. Average difference from parity in percent

Treatment Perfect correlation(n = 12)

No correlation (n = 8)

Mann-Whitney test(between treatments),p-value, [Z-statistic]

Market 1 6.98[0.39]

28.37**

[2.24]0.031**

[2.16]

Market 2 2.71[0.98]

32.52**

[2.52]0.003***

[2.93]

Market 3 4.10[1.49]

18.51**

[2.10]0.165[1.39]

Market 40.91

[1.02]13.86**

[2.52]0.064*

[1.85]

Average 3.67[1.53]

23.32**

[2.52]0.004***

[2.78]***p < 0.01, **p < 0.05, *p < 0.10; significant differences from zero within treatment according to two-tailed Wilcoxon signed-ranks tests. Numbers in brackets are z-statistics.

19 The absolute standard deviation of dividends is the same for A-share and B-share, but the standard deviation of the dividend-yield on the A-share exceeds the one of the B-share owing to the differences in value. So in our no-correlation treatment with limits to arbitrage, the dividend on the A-share is more risky justifying a lower price than of the B-share. Levati et al. (2012) reported in a single asset market with foreclosed arbitrage possibilities, a higher market value of “unlevered equity” than of “levered equity”, in line with our results in the no-correlation treatment. 20 The A-share is significantly pricier than the B-share in the first period of the first market (p = 0.041), and the B-share is pricier in period 10 of the first market (p = 0.041) and in period 7 of the third market (p = 0.039).

18

We see that the differences from price parity are much smaller in the perfect-correlation

treatment than in the no-correlation treatment; these differences are four to fifteen times larger in

the no-correlation treatment in the four markets. Overall, the average difference from parity per

period across the treatments is highly-significant, with the p-value of the two-tailed Mann-

Whitney test being 0.004. We conclude that our data are in line with the MM theorem (and thus

with Hypothesis 1A) in the perfect-correlation treatment, and reject Hypothesis 1A for the no-

correlation treatment.

4.2. Pricing discrepancies and fundamental pricing deviation

Observation 2. Absolute difference from parity and the fundamental pricing deviation are reduced in the perfect correlation treatment relative to the no-correlation treatment.

Support. The data in Table 2A and Table 2B record the averages of PD-value and DF-

value by treatment, respectively.21 From Table 2A we see that by buying the lower-priced share

and selling the higher-priced twin share (both at the average price in a randomly-determined

period), the average riskless return would be 23 percent of the trading value in the first market of

the perfect-correlation treatment. This number is relatively large if compared to the second

market where the average PD-value drops to one half of the first market.22 Relative to the no-

correlation treatment, however, this amount is rather small. As indicated by the one-tailed

Mann-Whitney test recorded in the table, the PD-value is larger overall (at the 10% significance

level) than in the no-correlation treatment.

21 These values have been computed according to equations (7) and (8). A missing average price of A-share or B-share in a certain period is treated as a missing observation in the corresponding session.22 The decrease from the first to the second market is significant at the one percent level (Z = 2.353, p = 0.009, one-tailed Mann-Whitney test, n = 12).

19

Table 2A: Average absolute deviation from parity PD in percentPerfect

correlation(n1 = 12)

Nocorrelation

(n2 = 8)

Mann-Whitney test(between treatments),p-value, [Z-statistic]^

Market 1 22.92*** 39.76** 0.094*

[1.32]

Market 2 12.16*** 33.09** 0.015**

[2.16]

Market 3 12.14*** 23.31** 0.167[0.97]

Market 4 14.52*** 20.37** 0.300[0.39]

Total 16.28*** 29.13** 0.076*

[1.43]p-value (one-tailed Page trend test)^

0.242[-.70]

0.011**

[-2.30]***p < 0.01, **p < 0.05, *p < 0.10; significant differences from zero according to

Wilcoxon signed-ranks tests. ^One-tailed tests. Numbers in brackets are z-statistics.

Table 2B: Average deviation from fundamental pricing DF in percentPerfect

correlation(n1 = 12)

No correlation

(n2 = 8)

Mann-Whitney test(between treatments),p-value [Z-statistic]

Market 1 24.18*** 42.94** 0.158[1.00]

Market 2 13.18*** 31.94** 0.020**

[2.05]

Market 3 13.90*** 22.02** 0.158[1.01]

Market 4 19.61*** 17.44** 0.379[0.386]

Total 17.84*** 28.58** 0.116[1.20]

p-value (one-tailed Page trend test)^

.055*

[-1.60].001***

[-3.30]***p < 0.01, **p < 0.05, *p < 0.10; significant differences from zero according to two-tailed Wilcoxon signed-ranks tests. ^One-tailed tests. Numbers in brackets are z-statistics.

We see in Table 2B that the results for the DF-value point in the same direction. The

overall average deviation from the fundamental value is 18 percent in the perfect-correlation

treatment and 29 percent in the no-correlation treatment. Nevertheless, the differences between

the two treatments are only significant (at the five percent level) for the second market. The

results suggest that pricing relative to fundamentals is (relatively) independent of whether there

20

are perfect arbitrage conditions. In line with the literature, both treatments trend towards the

fundamental value with repetition. We conclude that the data rather support than reject

Hypothesis 2 and Observation 2, particularly for Market 2, although the evidence is not

statistically significant for all markets. In terms of absolute numbers of arbitrage and mispricing,

Hypothesis 0 and Hypothesis 1B must be rejected given that no session and market involves a

zero PD-value or zero DF-value.

Observation 3. The consecutive-markets analysis suggests reduced price discrepancieswith experienced subjects and faster convergence in the perfect-correlation treatment than in the no-correlation treatment.

Support. Eyeballing the data in Tables 2A and 2B, our measures indicate a decrease of price

discrepancies in the first three consecutive markets, but not always from the third to the fourth

market.23 The non-parametric Page trend test (n = 12, k = 4) as reported in the bottom line of

Table 2A suggests no significant decrease of the PD-value over four markets in the perfect-

correlation treatment. However, the test results reported in Table 2B do suggest a significant

decrease for the deviation from fundamentals at the 10 percent level. The results of the one-

tailed Page test are significant at the five percent level for the no-correlation treatment indicating

a reduction over time of the PD-value and DF-value. We also find that the perfect-correlation

treatment exhibits a significant decline at the five percent level in PD-value and DF-value

between the first and the second markets, but not between any other consecutive markets.24

23 In the fourth market in some sessions, subjects push prices to irrationally high levels in late periods. Even if these data points were ignored, however, the potential arbitrage gains would not be reduced from the level of Market 3. 24 The p-values of the one-tailed Wilcoxon signed ranks test that suggest a decline between first and second market are as follows; PD-value (p = 0.010, Z = 2.316), and DF-value (p = 0.009, Z = 2.353).

21

On the other hand, the main decline in the no-correlation treatment occurs from Market 2

to Market 3.25 The data suggest a more rapid convergence to the theoretical benchmarks under

conditions of riskless hedging, i.e., when an exploitation of price discrepancies is riskless.

4.3. Smart traders

Observation 4. Individual performance (i.e. payoff rank) correlates with measured acuity.

Support. Table 3 reports regression results. As market payoffs change according to the drawn

dividends, we use the subject’s average payoff-rank within the group as individual performance

measure rather than the realized payoff. The subject with the highest payoff in the first market is

assigned the first rank, the second-highest payoff the second rank, etc. For each market the

individual rank is computed, and the average of the four ranks is used as the subject’s individual

performance measure.26 Here we apply the individual measures that we obtained in the

investment game and the cognitive reflection task. As stated above, we measure individual

acuity by the CRT score, i.e., the individual’s number of correct answers. The average CRT

score in our sample was 0.97. As a risk-aversion measure we use the individual percentage

allocation to the safe asset in the investment game; the average allocation was 60 percent.

Finally, we consider gender, assigning value one for a female investor and zero otherwise. The

share of females is 49 percent in our subject pool.27

25 The PD-value (p = 0.034, Z = 1.820) and the DF-value (p = 0.034, Z = 1.820) decrease significantly at the five percent level between Market 2 and Market 3 only, but not between Market 1 and Market 2 (p = 0.116, Z = 1.193), (p = 0.104, Z = 1.260). 26 Note the alternative measures of individual gains from trading or price forecasting lead to very similar ranks, therefore, to similar statistical results and to the same conclusions.27 As in many previous studies (see Charness and Gneezy, 2012), our measure finds that females are significantly more financially risk averse than males (Z = 2.366, p = 0.018, two-tailed test). We also find, perhaps surprisingly, that males have significantly higher CRT scores (Z = 3.838, p < 0.001, two-tailed test).

22

Table 3. Regression results

Variable RankedPayoff

Rel. freq of noisy

limit orderPD DF

Intercept 5.47***

[12.9]0.063***

[3.22]0.200[0.48]

0.351[0.76]

0.346[1.12]

CRT-score -0.525***

[-3.51]-0.011***

[-2.84]-0.333*

[-1.86]-0.376*

[-1.92]-0.396***

[-2.99]

Female 0.670**

[2.71]-0.005[-0.77]

-0.104[-0.47]

-0.050[-0.19]

0.168[1.03]

Risk aversion 0.002[0.66]

-0.000[-0.64]

0.005[0.81]

0.004[0.65]

0.003[0.67]

#obs. 174 174 20 20 20

#clusters 20 20

R-squared .169 .045 .227 .228 .461

The first column records the outcomes of the cluster regression with robust standard errors of the within-group ranked individual payoff and the relative frequency of a limit order that opens up an arbitrage opportunity on the individual CRT-score, Risk Aversion, and FEM gender dummy. The other columns record the OLS regression results of measures (6)-(8) on the group average of the individual measures and the consensus treatment variable.The results of two-tailed t-tests are indicated [Z-statistics in brackets]; ***p < 0.01, **p < 0.05, *p < 0.10.

The outcomes of the ordinary least square regression of individual performance measure

on CRT score, Female, and Risk aversion are recorded in the first column of Table 3.

Accordingly, the former two explanatory variables are significant determinants of the individual

performance in this regression, but the risk aversion measure is not significant. We find no

significant differences across treatments.28 The evidence clearly supports Observation 4.

Instead of measuring performance by payoff rank as in Observation 4, we can measure

performance by the number of bad decisions in trading. We classify an individual’s limit order

as noise if it induces an opportunity for others to realize an expected gain by swapping assets,

given the outstanding orders. Across all subjects we find that the number of noisy limit orders

decreases in the course of the consecutive markets. For each individual we compute the ratio of

28 The two-tailed Mann-Whitney between treatment test returns the following p-values on the individual data 0.559 (risk aversion), 0.809 (Female) and 0.987 (CRT score), and similar results on the session data 0.165 (risk aversion), 0.754 (Female), and 0.529 (CRT score).

23

the number of noisy limit orders to total limit orders. This ratio measuring trader’s aptitude is

next compared to the individual CRT score.

Observation 5. The trader’s aptitude is correlated to the measured individual acuity.

Support. The trader’s aptitude is inversely measured by the relative frequency of submissions of

noisy limit orders to the trader’s total number of submitted limit orders. As characterized above,

a noisy limit order offers an arbitrager’s opportunity to realize an (expected) gain by swapping

assets. The measured trader’s aptitude is regressed on individual characteristics. The second

column of Table 3 records the OLS regression in which we cluster standard errors by

independent group. We find that the individual acuity measured by the CRT score thus explains

much of the bidding data, supporting Observation 5.

Observation 6. When there is high overall trader acuity, price discrepancies are smaller, and the price of assets is closer to parity.

Support. In Table 3 we also record three OLS regressions of our measures (6)-(8) on the group

aggregate personal measures. In these regressions only the average CRT-score shows up

significantly. The differences of price from parity, fundamentals, and price discrepancy decrease

in markets with overall high trader acuity. We have also conducted regressions on the treatment

level. The results indicate similar directions but the significance drops. So, as for Hypothesis 4

and Observation 6, we conclude that acuity could actually affect reduction of price discrepancy;

however, potential discrepancies do not disappear completely.

4.5. Summary of results

We do not observe pricing of assets at fundamental value, but we have support for Hypothesis

1A (MM invariance theorem) in our perfect-correlation treatment where exploitation of a price

24

discrepancy is riskless. This is significant in relative terms when we compare the data to the no-

correlation treatment where exploitation of price discrepancies across assets has risk. Hypothesis

1B (no price discrepancy) is not supported in either of our treatments as price discrepancies seem

to persist. Under perfect correlation, however, convergence to a level where discrepancies

stabilize is fast (achieved by the second market) whereas under no correlation convergence takes

at least more time. Hypothesis 2 (hedging effect) is confirmed; the results on price discrepancies

of the perfect-correlation treatment are closer to the theoretical prediction than of the no-

correlation treatment.

Support for Hypothesis 3 (experience effect) is no surprise; experienced subjects trade

closer to the theoretical predictions than inexperienced subjects. Nonetheless, unlike the results

of the single-asset market (Smith et al., 1988) our data do not suggest a complete convergence on

fundamental values or vanishing price discrepancies in repeated markets. Finally, we find some

support for Hypothesis 4 (smart trader effect) as our measures of noise and mispricing decrease

with an increase in our measure of acuity. To the extent that experience may substitute for initial

acuity, ours confirm other results in the experimental asset markets literature (Palan 2013).

6. Conclusions

Our data support the MM invariance theorem in the following way. As proposed in the arbitrage

proof of the original paper (Modigliani and Miller 1958), we observe the same average market

values for shares in the same risk class. The data suggest that perfect correlation of future

returns (where price discrepancies can be arbitraged away without risk) is required to obtain the

law of one price in our setting. This necessary condition is recommended through our control

treatment where perfect correlation is removed. Despite the fact that earlier research reported an

25

insensitivity of behavior to changes in return correlations (e.g. Kroll et al. 1988), we do find a

significant effect. Under no correlation, i.e. with independent asset returns, we do not obtain the

law of one price for the same idiosyncratic risks of our single assets. With limits to arbitrage, the

market manifests a risk premium. The (“leveraged”) A-share is less pricy than the (“all equity”)

B-share. For existing limits to arbitrage, our data (on the no-correlation treatment) suggest that

leverage may indeed vary the market value of the company. The finding that value invariance

holds under perfect correlation but not under no-correlation is a key contribution of the paper.

Our study provides no general proof of MM, since we consider a zero interest rate and

non-traded debt. Nonetheless, we note that we match the standard setting of Smith et al (1988)

that is known for its quality of producing mispricing among inexperienced subjects. So, we

believe that value invariance will be obtained also in simpler settings, including those with

constant fundamental value.

Our study sheds light on different levels of market efficiency. We find support of market

efficiency for the case of riskless arbitrage. We also find that market efficiency increases with

repetition. Price discrepancies decrease and prices move towards fundamental values with

repetition. We observe that high overall acuity, as measured by the CRT score, reduces price

discrepancy in the market. Deviations from theoretical predictions become smaller when we

look at markets with overall high trader acuity. Nonetheless, the deviations are never entirely

removed. So for the data from our design, we have the sense that Rietz (2005) is correct when

he states that without a dedicated arbitrager taking advantage of any mispricing, the laboratory

market does not completely eliminate price discrepancies.

As is frequently observed with market studies, we conclude, the data suggest a movement

of behavior in the direction of the predictions of financial economic theory, but do not provide

26

full quantitative support of the theoretical benchmark. Hence, as price discrepancies do not

cease during the experiment our data hint to significant potential gains for arbitragers. In the real

world, exploitation of price discrepancies requires the arbitrager to take risk (as has been shown

in the twin share LTCM cases, Shiller 2000). So in the real world, arbitragers may require deep

pockets and patience to lock in their arbitrage gains.

References

Abbink, Klaus and Bettina Rockenbach, 2006. Option pricing by students and professional traders: a behavioural investigation, Managerial and Decision Economics 27(6), 497-510.

Ackert, L.F., Charupat, N., Deaves, R. and Kluger, B.D., 2009. Probability judgment error and speculation in laboratory asset market bubbles. Journal of Financial and Quantitative Analysis 44(3), 719–744.

Akiyama, Eizo, Nobuyuki Hanaki, and Ryuichiro Ishikawa, 2013. It is Not Just Confusion! Strategic Uncertainty in an Experimental Asset Market. Aix Marseilles School of Economics WP 2013 - Nr 40.

Breaban, Adriana and Charles N. Noussair, 2014. Fundamental Value Trajectories and Trader Characteristics in an Asset Market Experiment. (CentER Discussion Paper; Vol. 2014-010).Tilburg: Economics.

Cadsby, Bram C., and Elisabeth Maynes, 1998. Laboratory Experiments in Corporate and Investment Finance: A Survey. Managerial and Decision Economics 19(4/5), Laboratory Methods in Economics, 277-298.

Chan, Kenneth S., Vivian Lei, and Filip Vesely, 2013. Differentiated Assets: An Experimental Study on Bubbles. Economic Inquiry 51(3), 1731–1749.

Charness, Gary and Uri Gneezy, 2010. Portfolio choice and risk attitudes: An experiment.Economic Inquiry 48(1), 133-146.

Corgnet, Brice, Roberto Hernán-González, Praveen Kujal, and David Porter, Forthcoming. The Effect of Earned vs. House Money on Price Bubble Formation in Experimental Asset Markets. Review of Finance.

DeAngelo, Harry, and Linda DeAngelo, 2006. The irrelevance of the MM dividend irrelevance theorem. Journal of Financial Economics 79(2), 293-315.

Dufwenberg, Martin, Tobias Lindqvist, and Evan Moore, 2005. Bubbles and Experience: An Experiment. American Economic Review, 95(5): 1731-1737.

Eckel, Catherine C., and Sascha Füllbrunn, 2015. Thar 'She' blows? Gender, Competition, and Bubbles in Experimental Asset Markets. American Economic Review 105(2), 906-920.

Fischbacher, Urs, 2007. z-Tree: Zurich toolbox for ready-made economic experiments.Experimental Economics 10(2), 171-178.

Frederick, Shane, 2005. Cognitive Reflection and Decision Making. Journal of Economic Perspectives 19(4), 25-42.

Froot, Kenneth A., and Emil M. Dabora, 1999. How are stock prices affected by the location of trade?. Journal of financial economics 53(2), 189-216.

27

Greiner, Ben, 2004. An Online Recruitment System for Economic Experiments. In: Forschung und wissenschaftliches Rechnen 2003. GWDG Bericht 63, 79-93.

Gromb, Denis, and Vayanos, 2010. Limits of Arbitrage: The State of the Theory. Annual Review of Financial Economics 2, 251-275.

Haruvy, Ernan, Yaron Lahav, Charles Noussair, 2007. Traders’ Expectations in Asset Markets: Experimental Evidence. American Economic Review 97(5), 1901-1920.

Kirchler, Michael, Jürgen Huber, Thomas Stöckl, 2012. Thar she bursts – Reducing confusion reduces bubbles. American Economic Review 102(2), 865-883.

Kroll, Yoram, Haim Levy and Amnon Rapoport, 1988. Experimental test of the Separation Theorem and the Capital Asset Pricing Model. American Economic Review 78(3), 500-519.

Lamont, Owen A., and Richard H. Thaler, 2003a. Anomalies: The Law of One Price in Financial Markets. Journal of Economic Perspectives 17(4), 191-202.

Lamont, Owen A., and Richard H. Thaler, 2003b. Can the Market Add and Subtract? Mispricing in Tech Stock Carve-Outs. Journal of Political Economy 111(2), 227-268.

Levati, M. Vittoria, Jianying Qiu and Prashanth Mahagaonkar, 2012. Testing the Modigliani-Miller theorem directly in the lab. Experimental Economics 15(4), 693–716.

Miller, Merton H., 1988. The Modigliani-Miller Propositions After Thirty Years. The Journal of Economic Perspectives 2(4), 99-120.

Modigliani, Franco and Merton H. Miller, 1958. The Cost of Capital, Corporate Finance and the Theory of Investment. American Economic Review 48(3), 261-97.

Modigliani, Franco and Merton H. Miller, 1959. The Cost of Capital, Corporate Finance and the Theory of Investment: Reply. American Economic Review 49(4), 655-669.

Modigliani, Franco and Merton H. Miller, 1963. Corporate Income Taxes and the Cost of Capital: A Correction. American Economic Review 53(4), 433-443.

Oliven, Kenneth and Thomas A. Rietz, 2004. Suckers Are Born but Markets Are Made.Management Science 50(3), 336–351.

Palan, Stefan, 2013. A Review of Bubbles and Crashes in Experimental Asset Markets. Journal of Economic Surveys 27(3), 570-588.

Rietz, Thomas A., 2005. Behavioral Mis-pricing and Arbitrage in Experimental Asset Markets.Working Paper, University of Iowa.

Shleifer, Andrei, 2000. Inefficient Markets: An Introduction to Behavioral Finance (Clarendon Lectures in Economics), Oxford University Press.

Shleifer, Andrei and Lawrence H. Summers, 1990. The noise trader approach to finance.Journal of Economic Perspectives 4(2), 19-33.

Smith, Vernon L., Gerry L. Suchanek and Arlington W. Williams, 1988. Bubbles, Crashes, and Endogenous Expectations in Experimental Spot Asset Markets. Econometrica 56(5), 1119-1151.

Stiglitz, Joseph, 1969. A Re-Examination of the Modigliani-Miller Theorem. American Economic Review 59(5), 784-93.

Stöckl, Thomas, Jürgen Huber, and Michael Kirchler, 2010. Bubble measures in experimental asset markets. Experimental Economics 13(3), 284-298.

Stoll, Hans R., 1969. The relationship between put and call option prices. Journal of Finance24(5), 801-824.

Sunder, Shyam, 1995. Experimental Asset Markets. In: The Handbook of Experimental Economics (edited by John Kagel and Al Roth), Princeton University Press, 445-500.

28

Appendix: Instructions

This is an experiment in market decision-making. You will be paid in cash for your participation at the end of the experiment. Different participants may earn different amounts. What you earn depends on your decisions and the decisions of others. The experiment will take place through computer terminals at which you are seated. If you have any questions during experiment, raise your hand and a monitor will come by to answer your question.

I. The SituationIn this experiment, you will participate in a market of 9 participants. The identities of the other market participants will not be revealed to you.

Each market participant will be initially given Cash (1200 units) and Shares (2 units of asset “A” and 2 units of asset “B”). Shares generate Dividends (income) over the 10 periods of the experiment, but have no value at the end of the session. There are four possible dividends that can be paid in a period, and each is equally likely to be paid (random drawing). At the end of EACH period, EACH share will pay the owner a dividend.

When the experiment starts, you will participate in a market where the Shares can be bought and sold between participants. You pay out of your Cash when you buy a share, and you get Cash when you sell a share.

The experiment is divided into 10 consecutive trading Periods. Within each period, the market is open for trading Shares. When a period is over your Cash and Shares will carry over to the next period.

After the payment of the last dividend at the end of period 10, all shares will be worth nothing.Your earnings will be based on the amount of cash that you have at the end of the 10 periods.You accumulate cash by buying and selling shares, and/or by holdings shares and collecting dividends.

II. Share classes and DividendsTrading will occur in two classes of shares, one is called “A” and one is called “B”. The dividend of an “A” share per period can be 0, 8, 28 or 60 Cash Units (CU), with equal chances.The dividend of a “B” share per period will be identical to the one paid on “A” shares plus a fixed 24 CU, i.e., 24, 32, 52, 84 CU. Thus, if an “A” share pays 0, a “B” share pays 24 CU; if an “A” share pays 8 CU, a “B” share pays 32 CU, etc. The dividends will be added to your cash amount immediately.

Note that the average dividend per period per “A” share is 24 CU, since this is the average of 0, 8, 28, and 60, the equally likely dividends that can be paid. That is, over many periods, the expected average dividend per period tends to be 24 CU per “A” share. Likewise, the expected dividend per share of “B” is 48 CU, as this is the average of 24, 32, 52, and 84.

The minimum total dividend that could possibly be paid on an “A” share is 0 (if 0 is drawn in each of the 10 periods) and the maximum that could possibly be paid on an “A” share is 600 (if

29

60 is drawn in each of the 10 periods). Similarly, the minimum total dividend that could possibly be paid on a “B” share is 240 (if 0 is drawn on “A” in each of the 10 periods) and the maximum that could possibly be paid on a “B” share is 840 (if 60 is drawn on “A” in each of the 10 periods).

Understanding expected dividends:

The following table summarizes the sum of remaining dividends per share class:

Period

remaining dividends

incl.same

period

X

Range and Expected

dividend per “A” share

=

range and Sum of expected

dividends per “A” share

Expected dividend per “B” share

range and Sum of expected

dividends per “B” share

1 10 0..24..60 0..240..600 24 + “A“ 240..480..8402 9 0..24..60 0..216..540 24 + “A” 216..432..7563 8 0..24..60 0..192..480 24 + “A“ 192..384..6724 7 0..24..60 0..168..420 24 + “A” 168..336..5885 6 0..24..60 0..144..360 24 + “A“ 144..288..5046 5 0..24..60 0..120..300 24 + “A” 120..240..4207 4 0..24..60 0..96..240 24 + “A“ 96..192..3368 3 0..24..60 0..72..180 24 + “A” 72..144..2529 2 0..24..60 0..48..120 24 + “A“ 48..96..168

10 1 0..24..60 0..24..60 24 + “A” 24..48..84End 0 - 0 - 0

Exercise: To check your understanding of the table, please answer now the quiz questions on your screen. Please inform the instructor if you need help with the exercise.

III. How to Trade Shares?In order to buy shares, you need cash. Alternatively you can borrow cash (with no interest) up to 2400 CU. The cash you own is shown on the screen. In order to sell shares, you need shares.The number of shares you own is indicated at the top of your screen for “A” shares and “B” shares. If you do not own (enough) shares and wish to sell (more) shares anyway, you can borrow to sell up to 4 class “A” shares and up to 4 class “B” shares. If you sell more shares than you own your share count will be negative. For given number of negative shares at the end of the period, the dividend on these negative shares will be subtracted from your cash.

During a period, you may conduct the following actions (see Figure 1 at the end of the Instructions).

1. Submit an ASK (a proposed selling price) for one share. You can offer a share from your share holdings for sale by entering the asking price to sell one share in the space underneath the button ASK. You confirm the ask by a click on the button. The ask is then added to the list of outstanding asks. The outstanding asks are publicly recorded in increasing order, i.e. the best outstanding ask (the cheapest proposed selling price) being placed at the top of the list. Allmarket participants can see this list.

30

Note: you can submit as many asks as you like to sell one share. Upon selling one share, all your outstanding asks (for both share classes) are cancelled. To sell another share you then must again submit an ask.

2. Submit a BID (a proposed buying price) for one share. You can bid to purchase a share by entering your bidding price for one share in the space underneath the button BID. You confirm your bid by a click on the button. The bid is then added to the list of outstanding bids. The outstanding bids are publicly recorded in decreasing order, i.e., the best outstanding bid (highest proposed purchase price) being placed at the top of the list. All market participants can see this list.

Note: you can submit as many asks as you like to sell one share. Upon selling one share, all your outstanding asks (for both share classes) are cancelled. To sell another share you then must submit another ask. The same holds for bids. Upon purchasing one share, all your outstanding bids (for both share classes) are cancelled. If two or more orders (bids or asks) are the same, they are listed in the order of arrival, earlier orders being given priority over later ones.

3. Immediate BUY – accept an ask: The best outstanding ask of the other market participants is marked on your screen. You can accept the asking price (i.e., entering in a purchase agreement of a share with the seller) by clicking on the button Immediate BUY below the list of outstanding asks.

4. Immediate SELL – accept a bid: The best outstanding bid of the other market participants is marked. You can accept the bid (i.e., entering in a sale agreement of a share with the buyer) by clicking on the button Immediate SELL below the list of outstanding asks.

Note: Your own orders are displayed in blue, while the other orders are visible to you in black.You cannot accept your own orders. You cannot purchase shares if the ask exceeds your cash plus credit line. If your holding of “A” shares is -4, you cannot sell any further “A” shares. If your holding of “B” shares is -4, you cannot sell any further “B” shares.

IV Transaction and price announcementUpon acceptance of a bid or ask, via Immediate BUY or Immediate SELL, a transaction is completed. The accepted order is the transaction price. The transaction price is recorded on your screen in between the lists of bids and asks. The more recent prices are listed first. The latest prices are also recorded for each share class in the middle of the screen below the cash amount.

Upon transacting the price is debited from the buyer’s cash balance and credited to the seller’s cash balance. The purchased share is added to the buyer’s share holdings and subtracted from the seller’s share holding.

Note: Next you participate in a Practice Session of trading. You trade for 3 minutes on your screen with the other participants. There are NO payoff consequences linked to trading in the Practice Session. During the round please practice submissions of bids and asks, immediate selling and buying. You may want to practice selling more shares than you own to end up with a

31

negative share count. You may also want to practice buying more shares than you can pay with your own money to end up with a negative cash balance. During the Practice Session all your actions will have no payoff consequences.

V. The forecasting of future pricesAt the beginning of the first period, you are asked to forecast the average prices of all future periods for both share classes “A” and “B” (see Figure 2). At the beginning of each following period you will be able to revise and update your forecasts (see Figure 3). You earn additional CU depending on the accuracy of your forecast relative to the average price. So you will want to forecast the prices as closely as possible. The payment will be based either on first forecasts or on updated forecasts.

The average price in a period is the sum of the prices on all transactions during the period divided by the number of transactions.

We measure your squared forecasting error, i.e., the squared (relative) difference of your forecast from the average price.

Forecasting error = (price forecast- realized average price) / realized average price.Squared forecasting error = forecasting error x forecasting error

If for any reason no transaction occurs in some period, your forecast of that period is not considered.

Question :Assume the price forecast is 110 and the realized price is 100. What is the forecasting error, and what is the squared forecasting error? If the price forecast is 150 and the realized price is 100. What is the forecasting error, and what is the squared forecasting error?

Exercise: Please make the computation yourself on your screen, and enter the right answer. The on screen calculator may help you. Please inform the instructor if you need help with the exercise.

Note that the forecasting error of 50% is 5 times the forecasting error of 10%, but the squared forecasting error (25%) is 25 times the squared forecasting error (1%) of the latter. Larger deviations are punished relatively more than smaller deviations by squaring. However, if the forecasting error exceeds 100%, the squared forecasting error is counted as 100%. Your forecast cannot be worse than 100% wrong.

The accuracy of your forecasts is the average absolute forecasting error of the two share classes. We compute the average absolute forecasting error for each share class at the end of period 10 and take the average. Your reward for forecasting accuracy is then

32

Reward for forecasting accuracy = 600 cash units (1 - average absolute error)

If the forecasting error on each share class is always 10%, so the average squared error is 1%, the reward is 99% of 600. If the average squared error is 25%, the reward is 75% of 600. The minimum reward for forecasting accuracy is zero.

At the end of period 10, the squared forecasting errors are reported for both the first forecasts and the updated forecasts. The computer makes a binary random decision to determine if you are paid for the first forecasts or the updated forecasts.Note: It is required that you SAVE your UPDATED FORECASTS by pressing the button or no changes in your forecasts will take place. (It is not sufficient to change the numbers in the cells, please press the SAVE button)!

VI. Information[Consensus treatment only:]29 You will receive onscreen information on the price forecasts of the other participants in your group. You will be informed about the median forecast among the updated nine forecasts for each share class at the beginning of the period. The median forecast can, in fact, be your own forecast. Four forecasts are smaller or equal and four forecasts are larger or equal to the median forecast of the share class in the market (see Figure 1).

You will receive real-time updates on bids, asks and prices for both share classes. Information regarding the two share classes “A” and “B” are given on the screen on the left-hand and on the right-hand side, respectively. You will receive summary information about the prices at opening of the period, the high, the low and the average price during the period. Recall the average price is the one you forecast before the market opens.

In each period you will be reminded on screen about the expected future dividends, and the sum of expected dividends for the remaining periods. Finally, the realized past dividends are shown.The latest paid out dividend of the prior period is highlighted.

The past prices are shown in a table on the bottom of the screen, including the prices at opening, closing, the high, low and average of each past period. Alternatively to the past prices, you receive past information on your share and cash holdings at the end of the period, buys and sells during a period, and the past period dividends. You can alternate the past information with the past prices by clicking on the button (see Figure 1).

VII. Endowment and earningsThe experiment involves 4 rounds of 10 periods of trading and forecasting. Each trading period in the first round will last 180 seconds, and 90 seconds in the later rounds. Before the first round of 10 periods starts, you will be given another Practice Session during which you can practice making offers and transactions. The Practice Session will allow you again to trade three minutes without any payoff consequences.

29 We also varied whether there was a consensus forecast, which is the median forecasted average prices of A-shares and B-shares is revealed at the market opening. We implemented the consensus treatment since we thought that a consensus would help to decrease price discrepancies. However, since there was virtually no difference in the results across consensus conditions, we skip over it and pool the data.

33

At the beginning of each of the 4 rounds, you will be endowed with shares and cash. You will receive 2 “A” and 2 “B” shares and 1200 cash units. When you run out of cash, you will be able to borrow up to 2400 cash units to purchase shares. When you run out of shares, you will be able to sell 4 borrowed shares of “A” and 4 borrowed shares of “B”. Once again, note that if you have borrowed shares, your share count is negative. For each negative share at the end of the period, the dividend to be paid on one share will be subtracted from your cash.

At the end of the experiment, cash units CU will be converted to USD, at an exchange rate of $1= 100 CU. Your final payment will be equal to the cash you had at the end of the decisive repetition, adding a fee of $15 for participating. The final payment will be made to you in private; you will receive an envelope delivered to your seat in exchange for your signed receipt.

VIII. Summary

1. You will be given an initial amount of Cash and Shares at the very beginning.2. Each “A” share pays the owner a dividend of either 0, 8, 28 or 60 CU at the end of EACH

of the 10 trading periods. Each of these amounts is equally likely to be drawn at the end of the period. The average dividend per period per “A” share is 24 CU. The dividend of the “B” share is 24 CU higher than the dividend of the “A” share, thus the average dividend of the “B” share is 48.

3. You can submit offers to BUY shares and offers to SELL shares. You can make immediate trades by buying at the lowest ask (offer to sell) or selling at the highest bid (offer to buy).

4. There are 10 market trading periods and 10 forecasting periods per round. At the end of period 10, after the last dividend payment the shares expire and are worth nothing. Your final cash balance is the eligible earnings from market participation for your payment in the experiment. Your eligible earnings from forecasting are computed either from your first forecasts or from your updated forecasts. A computer draw determines the decisive forecasts; the first forecasts or the updated forecasts are selected with equal probability.

5. You will participate in 4 rounds of 10 periods. At the end of the experiment one repetition is selected for payment. The participant #1 will roll the die. The first outcome of the die roll (smaller or equal 4) will determine the payment-decisive round for every participant in the experiment.

6. Note that if you borrow cash or shares you may end a round with a negative cash balance.If a round is chosen for payment in which you incur losses, these losses may reduce your show up fee. The first $5 of any losses you may incur will lower your show up fee of $15. In fact, the payoffs you have collected before will be untouched.

7. The instructions are over. If you have any question, raise your hand and consult the monitor. Otherwise, please wait for the following Practice Session of three minutes.

34

Figure 1: Screen shot of trading screen

35

Figure 2: Screen shot of first forecasting task

36

Figure 3: Screen shot of updating forecasts

37

Figure 4: Screen shot of payoff screen

38

Figure 1. Evolution of relative difference from price-parity of A-shares and B-shares

-0.20

0.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

Perfect correlation Market 1 No correlation-0.20

0.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

Perfect correlation Market 2 No correlation

-0.20

0.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

Perfect correlation Market 3 No correlation

-0.20

0.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

Perfect correlation Market 4 No correlation

39