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23/05/02 Stock Markets as a 'Beauty Contest' 1

Why Do Stock Markets Form Price Bubbles?

Shyam SunderYale School of Management

Indian Institute of Management, KolkataJanuary 10, 2003


Possible Reasons

• Some people are not smart• Some people manipulate the market• Some people think that others are not smart

—the greater fool hypothesis• There is no such thing as a bubble ex ante• What do mean by “price bubble?”• How would you recognize a price bubble?


Valuation of Securities

• Value = net present value of future cash flows• Future is uncertain, value depends on investor

beliefs about the future cash flows• How far into the future?

– From now to liquidation of the the security– From now to investment horizon (sale)– Sale price depends on beliefs of other investors about

the cash flows after the sale– Return to formalization later (if you want)


Simple Example• Investor A believes the net present value of cash flows

from now on to be $100 ( fundamental value)• Investor should buy below $100 and sell above• What if A also believes that tomorrow, others will believe

the net present value to be $150• Considering second order beliefs, it may not necessarily be

best for A to sell at $110. A may be better off buying at that price, with the expectation of being able to sell at a higher price

• What is the best thing to do for an investor whose first and second order beliefs are not identical?


Stories of Common Knowledge Assumption

• In a great deal of business modeling, common knowledge is routinely assumed

• What happens when it breaks down?


Emperor Has No Clothes


Emperor’s Clothes

• The scoundrels made people believe that the clothes will be invisible only to the incompetent and the stupid

• People thought that others believed it• Nobody wants to be seen as stupid or

incompetent by others, lose his/her job• Visibility of clothes was private, it was easy

to fake seeing the clothes


Emperor’s Clothes (Contd.)

• Scenario 1: Everyone was privately convinced of their incompetence, and cheered to deny it publicly

• Scenario 2: People did not believe they were incompetent just because they could see the naked emperor, but believed that others so believed, and cheered to avoid being seen as stupid


What about the Child?

• The child did not know the link between visibility and competence

• Child was innocent, and said what he saw

• People know children to be innocent• People knew that people knew this


Stock Market

• Stock Market is like a newspaper beauty contest

• John Maynard Keynes, (1936)


Newspaper Beauty Contest

Which Face is the prettiest?


Which Face is the prettiest? Which Face is the prettiest? Which Face is the prettiest?

Which face will they judge to be the prettiest?


Beliefs About Others’ Beliefs

• Common elements to the two stories about the emperor‘s clothes and the stock market

• Central role of what we believe about others, and about their beliefs


Closing the Gap in Beliefs• Is it possible for our first and higher order beliefs to

differ?• When they do differ, what does it take to get them to

converge? Aumann (1976).• Do they actually converge? Why or why not?• When the beliefs of all around you are wrong, does it

pay to hold on to the right beliefs? Fight them or join them?

• Prior experimental results• What are the consequences for theories of bubbles?


Bubbles in Economic History

• “Self-evident” bubbles: Galbraith, Kindleberger• Econometric studies of long term recorded data:

stock prices move sufficiently closely with dividend over the long run to reject bubbles

• Contemporary investors knew more than what the econometricians have in their data

• What changes in fundamentals can justify the 90 percent price drop during the Great Crash?


Beliefs and Observation

• Theories of valuation are a function of beliefs• In the field it is difficult enough to know the first

order investor beliefs, and their diversity• Second and higher order beliefs are practically out

of reach• In lab we might have a better, though still limited,

chance to know beliefs, perhaps even open a gap between the first and higher order beliefs and observe their consequences under controlled conditions


Stock Markets as a ‘Beauty Contest’: Investor Beliefs and Price Bubbles sans Dividend Anchors

Shinichi Hirota and Shyam SunderYale University

Indian Institute of Management, KolkataJanuary 10, 2003


Bubble Stories

• Shiller (2000)– Irrationality of agents

• Blanchard and Watson (1982), Tirole (1985)– Discount rate bubbles with infinite horizon.


The Purpose of Our Study

• Explore an alternative story

• What happens if investors cannot arrive at stock prices through backward induction from future dividends?


Short-Term InvestorStandard textbook treatment:

1 1( ) ( )(1 ) (1 )t t t t

t tE D E P

P Vr r

If the security matures at t+m, fundamental value is defined as:

1 22

( ) ( ) ( )...

(1 ) (1 ) (1 )t t t t t t m

t m

E D E D E DF

r r r

Equating price to fundamental value needs assumptions


Representative Investor ModelAll investors within and across generations homogenous

Assumption R1: Investors form rational expectations, i.e., they use all available information to form their expectations, and know that the price equation (1) holds every period.

By RE, investor knows at t that equation (1) applies at t+1 also, and forms his expectation of the price at t+1

1 2 1 21

( ) ( )(1 ) (1 )t t t t

tE D E P

Pr r

1 2 1 21

( ( )) ( ( ))( )

(1 ) (1 )t t t t t t

t tE E D E E P

E Pr r


Using homogeneity of agents, and the law of iterated expectations

1 2 2( ( )) ( ),t t t t tE E D E D

1 2 2( ( )) ( ),t t t t tE E P E P

2 21

( ) ( )( )

(1 ) (1 )t t t t

t tE D E P

E Pr r

2 21

( ) ( )( )

(1 ) (1 )t t t t

t tE D E P

E Pr r

1 22

( ) ( ) ( ) ( )...

(1 ) (1 ) (1 ) (1 )t t t t t t k t t k

t k k

E D E D E D E PP

r r r r

Through repeated substitution, terminal price remains:


Assuming finite maturity eliminates the terminal price and equate current price to the fundamental value:

Assumption R2: The security matures in finite time (t+m); after paying dividend Dt+m, the security is worthless.

1( )

( )(1 )t t m

t t mE D

E Pr

1 22

( ) ( ) ( )...

(1 ) (1 ) (1 )t t t t t t m

t tm

E D E D E DP F

r r r

With assumptions R1 and R2, current price equals Ft , derived from a finite sequence of rational expectations and corresponding backward inductions anchored in future dividends


If R1 does not hold beyond some period t+k, we cannot substitute expectations of dividends beyond that time for Pt+k and current price would not be equal to the fundamental value

If R2 does not hold, the price equation includes an undetermined terminal price which cannot be eliminated from the price equation

Price indeterminacy gives rise to the “rational bubbles” (Blanchard and Watson 1982, Tirole 1985): price bubble must grow at the discount rate, even if it is indeterminate

1 1( ) (1 )( )t t t t tE P F r P F


Heterogeneous Investor ModelWithout representative investors, expectations of the future generations of investors cannot be substituted by the expectations of their predecessors in (4), leaving higher order expectations in the equation.

Assumption H1: Investors can form higher order expectations.

1 1 2 1 22 2

( ) ( ( )) ( ( ))(1 ) (1 ) (1 )t t t t t t t t

tE D E E D E E P

Pr r r


To substitute for these higher order expectations, we use a common knowledge assumption about rational expectations, and formation of higher order expectations.

Assumption H2: Assumption R1 (that investors can form rational expectations) is common knowledge.

Assumption H3: Assumption H1(that investors can form higher order expectations) is common knowledge

Through successive substitutions,.

1 1 2 1 1 1 12

( ) ( ( )) ( (...( ( ))...)) ( (...( ( ))...))...

(1 ) (1 ) (1 ) (1 )t t t t t t t t k t k t t t k t k

t k k

E D E E D E E E D E E E PP

r r r r


Note that R2 is not sufficient in heterogeneous investor environment. We use a common knowledge assumption about maturity of the security

Assumption H4: Assumption R2 (that the security matures at time (t+m)) is common knowledge among all investors.

This assumption allows us to replace the higher order expectation of the terminal price by the higher order expectation of the terminal dividend:

1 1 2 1 12

( ) ( ( )) ( (...( ( ))...))...

(1 ) (1 ) (1 )t t t t t t t t m t m

t m

E D E E D E E E DP

r r r

This price is not necessarily equal to the fundamental value (based only on the first order expectations); the two backward inductions can be different


Is Price Equal to the Fundamental Value?

• Even with these strong assumptions, the price need not be equal to the fundamental value

• Besides, there is the problem of whether the aggregate behavior of the market corresponds to the predictions made on the basis of such strong assumptions about rationality, higher order beliefs, common knowledge, and backward induction through multiple layers of such expectations

• Securities with longer maturity and duration, high growth, greater uncertainty and asymmetric information about dividends should be more susceptible to bubbles

• Investors and market behavior under such conditions: an empirical issue, we address experimentally


Keynes: Stock Market as a Newspaper Beauty Contest

• Stock price is determined not by investor’s own beliefs, but by their beliefs about others’ beliefs, …

• He should pick the face that others think that others think that others think that others think that …

• What is the anchor for the reader’s choice?


Experimental Study• Many empirical studies on the existence of stock

market bubbles– Diverse findings– Fundamental values unknown in the field– Difficult to examine the mechanisms of bubble formation

in the field (observe actions, not strategies)• Experimental research can focus on the mechanism

– Can choose fundamental values to compare with data– Does the stock market bubble occur when the future price

expectations are not anchored by dividends?


One Main Lab Treatment (discuss others, time permitting)

• Terminal value exogenously or endogenously specified

• Exogenously specified terminal value:– Investors are informed of the dividend.– Dividend is paid at the end of Period 15. – With the exogenously specified terminal value, the

price should be equal to this dividend.


Main Treatment: Exogenous Terminal Payoff

15-period security, termination date is common knowledge

Single terminal dividend (may vary across traders)

A separate set of subjects record their price predictions at the beginning of each period, which is announced at the end of each period

Period 1 Period 15


Main Treatment: Endogenous Terminal Payoff

30-Period Security with a terminal dividend to be paid at the end of period 30

The security will “likely” be terminated earlier at experimenter’s discretion (in fact terminated at 15-17 periods)

If terminated earlier, liquidation at average of the prices predicted by the predictors for the period following termination

Period 1 Period 15-17 Period 30


Laboratory Stock Markets

• Double auction market for multiple units of a single security that pays a single liquidating payoff

• 15-17 trading periods (3 minutes each)• Each investor endowed with 10 shares, 10,000

points in “cash”• Predictors write down price predictions before

trading starts each period (know the rules, no endowments, no ability to trade, can watch trading, know the range of terminal dividends)


Trading Screen

Figure 1


Main Treatment: Terminal Payoff

Robustness Variations Endogenous Exogenous

Subsidiary Treatment 1: Potential inequality of the first and higher order beliefs about dividends

Potential for a gap between first and higher order beliefs

Sessions 1, 8 Sessions 3, 5, 6, and 7

Equality between first and higher order beliefs

Sessions 2, 9, 10, and 11 Session 4

Subsidiary Treatment 2: Heterogeneity of pre-written dividends

Non-identical pre-written dividends

Sessions 1, 2, 8, and 9 Sessions 3, 4, 5, and 7

Identical pre-written dividends Sessions 10 and 11 Session 6

Subsidiary Treatment 3:Common Knowledge of pre-written dividends

Dividends common knowledge Sessions 10 and 11

Dividends not common knowledge

Sessions 1, 2, 8, and 9 Sessions 3, 4, 5, 6, and 7

Subsidiary Treatment 4: Verification of proper understanding of the instructions

No questionnaire, answer, verification and correction

Sessions 1, and 2 Sessions 3, 4 and 5

Questionnaire, answer, verification and correction

Sessions 8, 9, 10, 11 Sessions 6 and 7

Subsidiary Treatment 5:Subjects paid by absolute or relative performance

Payoff based on absolute performance

Sessions 1 and 2 Sessions 3, 4, 5, 6 and 7

Payoff based on relative performance

Sessions 8, 9, 10, 11


Conducted Experiments

• 5 sessions for the exogenous terminal values– (Session 3, 4, 5, 6, 7)

• 6 sessions for the endogenous terminal values– (Session 1, 2, 8, 9, 10, 11)

• Yale university, undergraduate students• Money is paid depending on each subject’s earned

points (absolute, RPE in sessions 8-11).• Yale School of Management, B-74 Room• September 21, 2001 – July 12,2002


Exogenously-Specified Terminal Value Sessions

Session 3, 4, 5, 6, 7(Variations: dividend heterogeneity,

common knowledge of dividends, gap in first and higher order beliefs, understanding of instructions)


Figure 3: Stock Prices and Efficiency of Allocations for Session 3 (Exogenous Terminal Payoff Session)

Figure 3: Stock Prices and Efficiency of Allocations for Session 3 (Exogenous Terminal Payoff Session)


Figure 4: Stock Prices and Efficiency of Allocations for Session 4(Exogenous Terminal Payoff Session)


Figure 6: Stock Prices for Session 6 (Exogenous Terminal Payoff Session)


Figure 8: Stock Prices and Efficiency of Allocations for Session 1 Endogenous Terminal Payoff Session)


Endogenously-Specified Terminal Value

Sessions

Sessions 1, 2, 8, 9, 10, 11 (Variations: dividend heterogeneity,

common knowledge of dividends, gap in first and higher order beliefs,

understanding of instructions, RPE)


Figure 9: Stock Prices and Efficiency of Allocations for Session 2 (Endogenous Terminal Payoff Session)


Figure 10: Stock Prices and Efficiency of Allocations for Session 8(Endogenous Terminal Payoff Session)


Figure 11: Stock Prices and Efficiency of Allocations for Session 9(Endogenous Terminal Payoff Session)


Figure 12: Stock Prices for Session 10(Endogenous Terminal Payoff Session)


Figure 13: Stock Prices for Session 11 (Endogenous Terminal Payoff Session)


Figure 14: Dispersion of Investor Profits


Inferences

1. When it is easy for the investors to backward induct the value of the security, the security prices converge to the equilibrium level derived from the fundamental values of individual investors.


Inferences

2. When it is difficult for investors to backward induct the value of the security,

a) Prices may deviate from the fundamental values to form bubbles, and

b) Prices are indeterminate and free-floating.


Formation of Price Expectations

• If the expectation of future price cannot be formed on the basis of the investors’ own knowledge of dividends, how do they form these expectations?

• In the absence of anchors that might be provided by common knowledge of dividends and of expectation formation process, backward induction becomes difficult, perhaps impossible.

• How do investors value securities when backward induction is not possible

• Soccer: goalkeeper versus striker


When You Can’t Do Backward Induction

• Try forward induction• Keynes (1936, 148)

• “It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain. … For this reason, the facts of the existing situation enter, in a sense disproportionally, into formation of our long-term expectations”

• Is he hinting at technical analysis?• We examined the price prediction data


Analysis of Predictions

1( ) ( )t t t tE P P D P

Backward induction: Fundamental Model, 0 < 1:

Forward induction: First order adaptive model, 0 < 1:

1 1 1( ) ( ) ( ( ))t t t t t t tE P E P P E P

Forward induction: Trend model, 0 :

1 1( ) ( )t t t t tE P P P P


Backward vs.Forward Induction with Exogenous Payoff Data

Fundamental versus first order adaptive model:

Adjusted. R2 =0.76, N=102, a1 = 1 and 0 < a2 <1, the data support the fundamental over the adaptive model

1 1 1(0.166) (0.041)( ) ( ) 1.009( ( )) 0.176( )t t t t t t t tE P E P P E P D P

Fundamental versus Trend Model:

Adjusted R2 =0.36, N=102; b1 = 0 and 0 < b2 <1, the data support the fundamental model over the trend model

1 1(0.144) (0.045)( ) 0.091( ) 0.179( )t t t t t tE P P P P D P


Backward vs.Forward Induction with Endogenous Payoff Data Fundamental versus first order adaptive model:

Adjusted R2 =0.66, N=166; a1=1 and a2<0, neither the fundamental nor the adaptive model fits the data well

Fundamental versus Trend Model:

Adjusted R2 =0.43, N=166; b1 0 and b2 = 0, the data support the trend model over the fundamental model

1 1 1(0.154) (0.038)( ) ( ) 1.212( ( )) 0.087( )t t t t t t t tE P E P P E P D P

1 1(0.172) (0.028)( ) 0.508( ) 0.024( )t t t t t tE P P P P D P


Inferences

3. When it is difficult for investors to backward induct the value of the security, price expectations are consistent with forward induction; when backward induction is easy to do, price expectations are consistent with backward induction.


Inferences

4. Allocative efficiency is high when backward induction is easy to do, and unpredictable when backward induction is difficult.

5. The cross-sectional dispersion of investor wealth increases with the size of bubbles.


Concluding Remarks

• Difficulty or impossibility of backward induction from an anchor tends to give rise to price bubbles

• Stock prices become indeterminate.• When securities mature within the investment

horizon of investors, we do not see bubbles• Without bubbles, allocative efficiency is high and

wealth dispersion is low; with bubbles, it becomes unpredictable


Concluding Remarks

• Bubbles more likely to occur in markets for securities with long maturity and duration

• It has long been known that securities with high-growth and new technologies, and short-horizon investors are more susceptible to price bubbles (their dividend anchors are difficult to pin down)

• Free floating prices supported purely by higher order expectations

• Distributional effects of bubbles and public policy


Concluding Remarks

• Conjecture: Investors resort to forward induction from whatever data they can use when the door to backward induction is blocked

• History and accounting numbers as a source for forward induction


Thank You

• Paper and the presentation will be available at www.som.yale.edu/faculty/sunder/research

• Please send comments to • [email protected]

http://www.som.yale.edu/faculty/sunder/research







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