A behavioral Model of financial Crisis Taisei Kaizoji International Christian University, Tokyo...

A behavioral Model of financial Crisis

Taisei KaizojiInternational Christian University, Tokyo

Advance in Computational Social Science

National Chengchi University, Taipei,

November 3, 2010

The Aim

• To propose a behavioral model of bubble and crash.

• To give a theoretical interpretation why bubbles is

born and is unavoidably collapsed

• To give a possible solution on Risk Premium Puzzle

Internet Bubble and Crash in 1998-2002

Internet Stocks:

10 (1998/1/2)

140 (2000/3/6): Peak

3 (2002/10/9): Bottom

Non-Internet Stocks:

10 (1998/1/2)

14 (2000/3/3): Peak

9.8 (2002/10/9)

14

1/50

1.4

0.7

Period I: 1998/1/2-2000/3/9

Period II: 2000/3/10-2002/12/31

0

20

40

60

80

100

120

140

160

Internet Stock Index Non- Internet Stock Index

Period I Period II

- 15

- 10

- 5

0

5

10

15

Internet Stock Non- Internet Stock

Price ChangesInternet Stocks:

Mean Variance

Period I 0.2 4.1

Period II -0.2 2.5

Non-Internet Stocks:

Mean Variance

Period I 0.007 0.01

Period II -0.004 0.02

Covariance:

Period I 0.09

Period II 0.13Period I: 1998/1/2-2000/3/9

Period II: 2000/3/10-2002/12/31

Period I Period II

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

- 7.9 - 6.6 - 5.2 - 3.9 - 2.5 - 1.1 0.2 1.6 3.0 4.3 5.7

Non- Internet Stock

Internet Stock

Probability Distributions of Price-Changes

• leptokurtic• fat-tailed

Abnormality of Internet stocks

Who Invested in Internet Stocks? Empirical Evidence (1)

• Hedge funds:

Ex. Soros Fund Management, Tiger Management,

Omega Advisors, Husic Capital Management, and

Zweig Di-Menna Associates

Brunnermeier and Nagel (2004)

“Hedge Funds and the Technology Bubble,”

Journal of Finance LIX, 5.

Riding Bubble?

“Hedge Funds captured the upturn, but, by reducing their positions in stocks that were about to decline, avoided much of the downturn.”

Brunnermeier and Nagel (2004)

Inexperienced Investors and Bubbles

Robin Greenwood and Stefan Nagel (2008), forthcoming Journal of Financial Economics.

Who Invested in Internet Stocks? Empirical Evidence (2)

Inexperienced young fund manager

Who invested in internet stocks?

• Younger managers are more heavily invested in technology stocks than older managers.

• Younger Managers increase their technology holdings during the run-up, and decrease them during the downturn.

• Young managers, but not old managers, exhibit trend-chasing behavior in their technology stock investments.

Robin Greenwood and Stefan Nagel (2008)

Traders’ Expectations in Asset Markets: Experimental Evidence

Haruvy, E., Lahav, Y., and C. Noussair,

Forthcoming in the American Economic Review (2009)

Who invested in internet stocks? Experimental Evidence

Bubbles and Crashes in Experimental Markets

Haruvy, E., Lahav, Y., and C. Noussair (2009)

(a) The bubble/crash pattern is observed when traders are inexperienced.

(b) The magnitude of bubbles decreases with repetition of the market, converging to close to fundamental values in market 4.

Haruvy, E., Lahav, Y., and C. Noussair (2009)

Experimental Evidence

Participants’ beliefs about prices are adaptive.

Experimental Evidence

The existence of adaptive dynamics suggests the mechanism whereby convergence toward fundamental values occurs.

Haruvy, E., Lahav, Y., and C. Noussair (2009)

Participants’ beliefs about prices are adaptive.

Haruvy, E., Lahav, Y., and C. Noussair (2009)

J. De Long, A. Shliefer, L. Summers and R. Waldmann: Positive Feedback Investment Strategies and Stabilizing Rational Speculation, Journal of Finance 45-2 (1990) pp. 379-395.

Answer I: Noise Trader Approach

In DSSW (1990), rational investors anticipate demand from positive feedback traders. If there is good news today, rational traders buy and push the price beyond its fundamental value because feedback traders are willing to take up the position at a higher price in the next period.

Synchronization Failure:D. Abreu, and M. K. Brunnermeier, Bubbles and crashes, Econometrica 71, 2003, 173–204.

Bubbles and Crashes: Lux, Economic Journal, 105 (1995)Kaizoji, Physica A (2000)

Master Equation Approach: Noise traders’ herd behavior

Collective Behavior of a large number of agents Weidlich, W. and G. Hagg (1983) Concept and Models of a Quantitative Sociology, Springer.

The Setting of the Model

Assets traded

• Bubble asset :

　　　　 ex. Internet stocks

• Non-bubble asset:

　　　　 ex. Large stocks like utility stocks

• Risk-free asset:

ex. Government bonds, fixed time deposits

1x

2x

fx

Agents

•Rational traders (Experienced managers)

(i) They hold a portfolio of three assets.

(ii) Capital Asset Pricing Model (CAPM).

(Mossin(1966), Lintner (1969))

• Noise traders (Inexperienced managers)

(i) They hold either risk-free asset or bubble asset.

(ii) Maximization of random utility function of

discrete choice (MacFadden (1974))

The Setting of the Model

Rational traders

1 1( ) ( )2t tE W V W

:)(WV

:)(WE The expected value of wealth

The variance of wealth

References:

John Lintner, The JFQA, Vol. 4, No. 4. (1969), 347-400.

Jan Mossin, Econometrica, Vol. 34, No. 4. (Oct., 1966), pp. 768-783.

0)()()(..

}])()([2

])()([max

)(2

)(max

222111

222221212112

2111

2211

ff

f

xxqxxpxxpts

xxxxxx

xpExpEx

WVWE

:

:

:

:

:

2

1

2

1

q

p

p

x

x

:)(

:

:

i

ij

ii

pE

)exp()( WWu

Demand for bubble asset

Demand for non-bubble asset

Price of bubble asset

Price of non-bubble asset

Price of risk-free asset

Variance of the asset price change

Covariance of the asset price change

Expected Price of bubble asset

Rational traders

The rational investor’s demands

)()(

})()({1

})()({1

222

111

211

1112

22

122

2221

11

xxq

pxx

q

pxx

q

ppE

q

ppE

Ax

q

ppE

q

ppE

Ax

ff

2221

1211

Awhere

The rational investors’ aggregate excess demands

2 121 2 1 1 1 2 1 2

2 212 1 2 1 2 1 1 1

[ ( ( ) / ) ( ( ) / )]

[ ( ( ) / ) ( ( ) / )]

t t t t t

t t t t t

MM x E p p q E p p q

A

MM x E p p q E p p q

A

where

1it it itx x x

1 1jt jt jtp p p

1 1( ) ( ) ( )jt jt jtE p E p E p

Noise Traders

1,( 1,2,..., )

1,j

for holding bubble aeests j N

for holding risk free asset

Their preference:

1 1 1

2 2 2

U U

U U

i : Random variable

iU : Deterministic part

Random Utility Function:

U s H

U s H

Noise-Trader’s Random Utility Function:

1

1( 1 1)

N

jj

s s sN

• Average preference:

• Strength of Herding:

: Random variablei

1 1 0 0(1 )( ),t t t f tH H r r H H

• Momentum: H

( Adaptive expectation)

Probabilities

McFadden, Daniel (1974)

( ) Pr[ ] exp[ exp[ ]]iF x x x

exp[ ]

exp[ ] exp[ ]

exp[ ]

exp[ ] exp[ ]

UP

U U

UP

U U

The probability that a utility-maximizing noise trader will choose each alternative:

under the Weibull distribution:

Transition Probabilities:

exp[ ( )]( )

exp[ ] exp[ ( )]

exp[ ]( )

exp[ ] exp[ ( )]

t tt

t t t t

t tt

t t t t

s Hp s

s H s H

s Hp s

s H s H

Ns

2

The Master Equation:

1( ) ( ) [ ( ) ( ) ( ) ( )]

[ ( ) ( ) ( ) ( )]t t t t t t

t t t t

p s p s w s s p s s w s p s

w s s p s s w s p s

[( / 2) ] ( ) ( ) (1 ) ( )2

(( / 2) ) ( ) ( ) (1 ) ( )2

( ) 02

t t t t t t

t t t t t t

t t t t

Nw s N s w s N p s s p s

Nw s N s w s N p s s p s

Nw s s for s s

Representative Noise-Trader’s Behavior:

1 1[ tanh( ) ]tt t t t t

s s s s H s

1 12t t t t

QNQ n n s s

The noise trader’s Aggregate Excess demands

1tX

tX

０

a

bA

A

B

B

)0,0.2(),(;

)0,9.0(),(;

HlineBB

HlineAA

)tanh( Hss

)0,1( HBimodal: Unimodal: )0,1( H

Market-clearing Conditions:

2 121 2 1 1 1 2 1 2

12 22 1 1 1 1 2 1 2

[ ( ( ) / ) ( ( ) / )]2

02

[ ( ( ) / ) ( ( ) / )] 0

t t t t tt

t

t t t t t

QN MM x s E p p q E p p q

A

QNs

MM x E p p q E p p q

A

21 1 1 1

122 2 1

1 1

[ ( )]

[ ( )]

[tanh( ) ]

(1 )( )

t tt

t tt

tt t t

t t t f

p q s E p

p q s E p

s s H s

H H r r

Market-clearing prices ：

0)tanh()( sHssK

0)(

esss

sK

0)(

esss

sK

for maximum

for minimum

0)(

ess

st

s

sP

0)(

2

2

ess

st

s

sP0

)(2

2

ess

st

s

sP

: peaks

for maximum for minimum

Maxima of stationary probability density distribution

: peaks

)(sPst

)3.0,8.0(),(;

)3.0,8.0(),(;

HlineBB

HlineAA

)tanh( Hss

Unimodal: )0,1( H

])1([

)4.0,8.1(),(;

)4.0,8.1(),(;

2 sCosh

HlineBB

HlineAA

)1(

cs

Phase Transition

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-1 -0.8 -0.6 -0.4 -0.20.01 0.210.410.610.81-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.8 -0.6 -0.4 -0.20.01 0.210.410.610.81

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.8 -0.6 -0.4 -0.20.01 0.210.410.610.81

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.410.61 0.81

Mechanism of bubble and crash

0.0;9.0 H 0.0;2.1 H

05.0;3.1 H1.0;3.1 H

Bubble birth

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.410.61 0.81

Mechanism of bubble and crash (continued)

-0.15

-0.1

-0.05

0

0.05

0.1

-1 -0.8 -0.6 -0.4 -0.20.01 0.210.410.610.81

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.8 -0.6 -0.4 -0.20.01 0.210.410.610.81

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.410.61 0.81

1.0;3.1 H 01.0;3.1 H

05.0;3.1 H1.0;3.1 H

Crash!!

A Example of the Model Simulation:

s Price on bubble asset

Return Moment H

11

12

4.1;

0.09

- 1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1 101 201 301 401 501 601 701

0

2

4

6

8

10

12

14

16

1 101 201 301 401 501 601 701

- 0.004

- 0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

1 101 201 301 401 501 601 701

Price on non-bubble asset

1.1; 0.9

2

2.2

2.4

2.6

2.8

3

3.2

3.4

1 101 201 301 401 501 601 701

5)(;10)( 21 pEpE

02.0H (Crash)

- 1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1 201 401 601 801 1001 1201

- 0.003

- 0.002

- 0.001

0

0.001

0.002

0.003

0.004

0.005

1 201 401 601 801 1001 1201

1 11

1

( ) /[ ] f

E p p qE r r

p

Risk Premium Puzzle:

ff rtrErtr )]([)( 11

Expected risk premium:

E[r] = Risk Premium + Risk Free Rate

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

1 101 201 301 401 501 601

Expected Discount Rate:

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

1 101 201 301 401 501 601 701

Realized Return:

The period of bubble:

After burst of bubble:

ff rtrErtr )]([)( 11

ff rtrErtr )]([)( 11

)]([ 1 trE )(1 tr

A Model Simulation: Risk Premium

In Summary

• Conditions for a bubble’s birth:

(i) to appear any new market and

(ii) a large number of inexperienced investors start to trade

the bubble assets that are listed in the new market.

• Under the above conditions, as noise traders’ herd behavior destabilize the rational expectation equilibrium, and give cause to a bubble.

• As long as the noise traders adopt a return momentum strategy, bubbles burst as a logical consequence.

• After all, the rational investors can make a profit from a long-term investment, while the noise traders lose money.

Example I: Internet Bubble

NASDAQ:

1503 (1998/1/5)

5048 (2000/3/6): Peak

1140 (2002/9/23): Bottom

0

1000

2000

3000

4000

5000

600019

71/2

/519

74/2

/519

77/2

/519

80/2

/519

83/2

/519

86/2

/519

89/2

/519

92/2

/519

95/2

/519

98/2

/520

01/2

/520

04/2

/520

07/2

/5

NASDAQ

S&P500 S&P500:

927 (1998/1/5)

1527 (2000/3/20): Peak

800 (2002/9/30): Bottom

3.3

1/5

1.6

1/2