Asset Pricing in Financial Markets

Cognitive Biases, Ambiguity Aversion and Asset Pricing in Financial Markets E. Asparouhova, P. Bossaerts, J. Eguia, and W. Zame April 17, 2009

The Question

The Question Do cognitive biases (directly) affect asset prices?

The Question Do cognitive biases (directly) affect asset prices? Evidence abound that people are subject to cognitive biases. In the psychology literature, biases have been documented in experiments where subjects have no choice but to directly reveal them. In financial markets agents biases, even if revealed in choices, might not be directly revealed in prices, if at all

Motivation

Motivation 1 Consider agents who improperly Bayesian update.

Motivation 1 Consider agents who improperly Bayesian update. 2 Those agents are likely to face market prices that contradict their computations. How do agents react to those prices?

Motivation 1 Consider agents who improperly Bayesian update. 2 Those agents are likely to face market prices that contradict their computations. How do agents react to those prices? 3 Well known that when confronted with conflicting expert opinion, people perceive ambiguity. Fox and Tversky (1995) refer to the phenomenon as comparative ignorance.

Motivation

Motivation 5 Agents who perceive ambiguity and are averse to it (Ellsberg (1961)) actively seek ambiguity neutral positions for a wide range of prices.

Motivation 5 Agents who perceive ambiguity and are averse to it (Ellsberg (1961)) actively seek ambiguity neutral positions for a wide range of prices. Confirmed in experiments (Bossaerts, Ghirardato, Guarnaschelli, Zame (2008)). 6 As a result, the cognitive biases that cause agents to perceive ambiguity in the first place, while reflected in portfolio choices, would not be directly reflected in prices.

Hypotheses

Hypotheses 1 Some agents are insensitive to prices

Hypotheses 1 Some agents are insensitive to prices 2 Those agents hold ambiguity neutral portfolios

Hypotheses 1 Some agents are insensitive to prices 2 Those agents hold ambiguity neutral portfolios 3 Those agents trade less that price-sensitive agents

Hypotheses 1 Some agents are insensitive to prices 2 Those agents hold ambiguity neutral portfolios 3 Those agents trade less that price-sensitive agents 4 Price quality improves the higher the number of price-sensitive agents

Experiment and Results

Experiment and Results In experiments subjects trade two Arrow-Debreu assets whose value is determined by a difficult Bayesian updating problem.

Experiment and Results In experiments subjects trade two Arrow-Debreu assets whose value is determined by a difficult Bayesian updating problem. We use a Monty Hall-like games to determine the payoffs of the assets. We find that many agents cannot solve the updating problem (as proxied by price-insensitivity) Those who do not solve the problem achieve (significantly) more balanced positions and trade (marginally) less than the subjects who solve the problem.

Literature Kluger and Wyatt (2004) conjecture that prices are right if at least two agents know the prices due to Bertrand competition. Coval and Shumway (2005) present evidence of short term effect of behavioral biases on prices on the Chicago Board of Trade. Maciejovsky and Budescu (2005); Bossaerts, Copic, and Meloso (2008) on financial markets facilitating social cognition.

Ambiguity Aversion

Ambiguity Aversion Two-date, two-state economy. States r(ed) and b(lack); corresponding Arrow securities R and B, in equal aggregate supply.

Ambiguity Aversion Two-date, two-state economy. States r(ed) and b(lack); corresponding Arrow securities R and B, in equal aggregate supply. Objective probabilities of the two states not common knowledge but can be computed; π R and π B. If all agents are expected utility maximizers, and subjective probabilities equal the objective probabilities, prices equal expected values (which equal the objective probabilities).

Ambiguity Aversion

Ambiguity Aversion If some agents perceive ambiguity, we assume they have α max min preferences, so that they maximize: U(R, B) = α min{u(r), u(b)}+(1 α) max{u(r), u(b)}

Ambiguity Aversion If some agents perceive ambiguity, we assume they have α max min preferences, so that they maximize: U(R, B) = α min{u(r), u(b)}+(1 α) max{u(r), u(b)} An agent with α max min preferences acts as if with probability α, the worst possible state will occurs, and with probability 1 α, the best possible state occurs. The FOC with the α max min preferences imply that ambiguity averse agents (α > 0.5) balance their portfolio for any price vector in the interval [ 1 α, α ]. α 1 α

Assumptions Agents who compute the correct probabilities do not feel comparative ignorance and do not adopt α max min preferences at any price level they face.

Assumptions Agents who compute the correct probabilities do not feel comparative ignorance and do not adopt α max min preferences at any price level they face. Halevy (2007): 95% of agents who fail at the standard calculation task of reducing compound lotteries avoid ambiguity in the Ellsberg experiment, whereas only 4% of agents who correctly reduce compound lotteries exhibit this behavior. Proportion ρ of agents who compute wrong probabilities feel comparative ignorance when confronted with market prices that do not correspond to their subjective probabilities. Proportion 1 ρ act according to their biased subjective probabilities.

Implications If (some of) the individuals who cannot compute the correct probabilities become price insensitive and those who know how to compute the probabilities change the composition of their portfolio according to the prevailing prices then: The deviation of the market price from the expected value of the asset (mispricing) is negatively related to the number of price sensitive subjects. Given some mispricing, price insensitive subjects hold more balanced portfolios than price insensitive subjects.

The Securities Two Arrow securities, Red and Black; Bond

The Securities Two Arrow securities, Red and Black; Bond Two states of the world: red and black.

The Securities Two Arrow securities, Red and Black; Bond Two states of the world: red and black. The color of the last card in a game of cards determines the state.

The Securities Two Arrow securities, Red and Black; Bond Two states of the world: red and black. The color of the last card in a game of cards determines the state. Last Card Red Stock Black Stock Bond red 0.50 0 0.50 black 0 0.50 0.50 Trading only in Red Stock and Bond; endowments in all 3 securities.

The Card Game

The Card Game Four cards face down

The Card Game Four cards face down One card randomly discarded

The Card Game Four cards face down One card randomly discarded One card face up but never

The Card Game Four cards face down One card randomly discarded One card face up but never Last card randomly chosen

The Card Game Four cards face down One card randomly discarded TRADE PERIOD 1 One card face up but never Last card randomly chosen

The Card Game Four cards face down One card randomly discarded TRADE PERIOD 1 One card face up but never TRADE PERIOD 2 Last card randomly chosen

Timeline Trade Trade

Timeline P(Red)=Prob(Last Card is Red) Trade Trade

Timeline P(Red)=Prob(Last Card is Red) Trade P(Red)= Trade P(Red)=

Timeline P(Red)=Prob(Last Card is Red) Trade P(Red)= 7 12 Trade P(Red)=

Timeline P(Red)=Prob(Last Card is Red) Trade P(Red)= 7 12 Trade P(Red)= 11 16

Experimental Sessions 20 subjects per session 3 minute trading periods Market for Red Stock and Bond open; Black market always closed. Initial endowment Holdings carry over from Period 1 to Period 2 of trading

Trading: jmarkets

Prices: UCLA 50 45 40 35 30 25 20 15 10 5 0 0 500 1000 1500 2000 2500 3000

Prices: Utah 50 45 40 35 30 25 20 15 10 5 0 0 500 1000 1500 2000 2500 3000

Prices: Utah-Caltech 50 45 40 35 30 25 20 15 10 5 0 0 500 1000 1500 2000 2500

Identifying Price-Sensitivity Subjects Correct Price = Expected Payoff of Red Stock. MISPRICE=Market Price-Correct Price. For each subject and each treatment perform the regression

Identifying Price-Sensitivity Subjects Correct Price = Expected Payoff of Red Stock. MISPRICE=Market Price-Correct Price. For each subject and each treatment perform the regression RedStock,m = a + b MISPRICE m + e m Misprice m is avg. mispricing in minute m of trading. RedStock,m is change in holdings in minute m. Those who know the correct probabilities have b < 0

Mispricing by Treatment Experiment Treatment Mean Absolute Number of (T < 1.65) Number of (T > 1.9) Mispricing Subjects Subjects Caltech 1 3.13 7 2 2 5.54 3 2 3 3.40 2 2 4 1.25 4 6 Utah 1 3.50 4 1 2 11.79 1 2 3 11.28 0 1 4 7.24 1 0 Utah-Caltech-1 1 3.35 5 1 2 4.75 2 0 3 1.62 2 0 4 1.89 5 3 UCLA-3 1 4.86 2 1 2 5.07 3 0 3 2.53 3 3 4 2.90 3 0 Utah 1 7.50 2 0 2 7.95 3 3 3 5.88 6 2 4 4.19 2 2 Utah-Caltech-2 1 8.78 1 0 2 2.58 2 0 3 2.77 3 1 4 3.38 3 0

Mispricing and Price-sensitivity Result The correlation between the number of price-sensitive subjects and average absolute mispricing equals -0.53 (st. error=0.146). Thus, the larger the number of subjects who appear to know the probabilities, the closer the prices to their theoretical levels.

End-of-game Imbalances by Treatment I i = a + bx i + e i I i = Holdings Red Holdings Black of subject i. X i = t i M i. t i =t-stat from RedStock,t = a + b MISPRICE t + e t M i = MISPRICE i.

End-of-game Imbalances by Treatment I i = a + bx i + e i I i = Holdings Red Holdings Black of subject i. X i = t i M i. t i =t-stat from RedStock,t = a + b MISPRICE t + e t M i = MISPRICE i. Theory: b < 0.

End-of-game Imbalances by Treatment b b Treatment all t i (t i > 1.9) included excluded 1 0.057-0.032 (0.159) (0.172) 2-0.141-0.274 (0.099) (0.117) 3-0.070-0.035 (0.094) (0.125) 4-0.027 0.088 (0.188) (0.228)

End-of-game Imbalances: All Treatments I it = I i + ɛ it, I i = E(I it i) X it = X i + ξ it, X i = E(X it i) I i = a + bx i + e i corr(ɛ it, ξ it ) = ρ b b All all t i (t i > 1.9) Treatments included excluded -1.198-1.713 (0.646) (0.762) R 2 0.07 0.108

Mid-game Imbalances by Treatment b b Treatment all t i (t i > 1.9) included excluded 1 0.186 0.135 (0.127) (0.132) 2-0.165-0.268 (0.082) (0.100) 3-0.184-0.156 (0.075) (0.099) 4-0.163-0.085 (0.167) (0.208)

Mid-game Imbalances by Treatment b b All all t i (t i > 1.9) Treatments included excluded -1.423-2.230 (0.583) (0.777) R 2 0.156 0.265

Imbalance Price-sensitivity Relation Result Consistent with the the prediction of the theory, price-insensitive subjects hold more balanced portfolios than price-sensitive subjects both mid-game and at conclusion of the trading periods.

Number of Trades by Treatment N i = a + bx i + e i b b Treatment all t i (t i > 1.9) included excluded 1 0.102 0.027 (0.101) (0.280) 2-0.349-0.417 (0.146) (0.190) 3 0.139 0.103 (0.174) (0.205)) 4-0.059 0.146 (0.291) (0.367)

Number of Trades: All Treatments b b All all t i (t i > 1.9) Treatments included excluded -0.481-0.693 (1.135) (1.398) R 2 0.004 0.007

Number of Trades Price-sensitivity Relation Result The hypothesis that numbers of trades of price sensitive and price insensitive subjects are equal cannot be rejected.

In Conclusion... Hypothesis Cognitive biases may not directly affect prices in financial markets because they may translate into perception of ambiguity, and hence, price insensitivity. Evidence Large number of price insensitive subjects exists Price insensitive tend to hold balanced portfolios The price quality improves with the number of price sensitive subjects.