International Christian University, Tokyo
Advance in Computational Social Science
National Chengchi University, Taipei,
November 3, 2010

• 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

160
140
120
100
80
60
40
20
Period I Period II
0
02.
01.
199
8
29.
05.
199
8
21.
10.
199
8
18.
03.
9
199
11.
08.
199
9
04.
01.
200
0
30.
05.
0
200
20.
10.
200
0
19.
03.
200
1
10.
08.
200
1
10.
01.
200
2
06.
06.
200
2
29.
10.
200
2
Internet Stock Index Non-Internet Stock Index
Internet Stocks:
10 (1998/1/2)
140 (2000/3/6): Peak
3 (2002/10/9): Bottom
14
1/50
Non-Internet Stocks:
10 (1998/1/2)
14 (2000/3/3): Peak
9.8 (2002/10/9)
1.4
0.7
Period I: 1998/1/2-2000/3/9
Period II: 2000/3/10-2002/12/31

15
10
5
Period I Period II
0
-5
-10
-15
0
2.
01
.19
98
2
8.
05
.19
98
1
9.
10
.19
98
1
5.
03
.19
99
0
5.
08
.19
2
99
8.
12
.19
99
1
9.
05
.20
00
1
1.
10
.20
0
00
7.
03
.20
01
3
0.
07
.20
01
2
6.
12
.20
01
2
1.
05
.20
02
1
1.
10
.20
02
Internet Stock Non-Internet Stock
Period I: 1998/1/2-2000/3/9
Period II: 2000/3/10-2002/12/31
Internet 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 0.13

Probability Distributions of Price-Changes
0.5
0.45
0.4
0.35
0.3
0.25
Non-Internet Stock
Internet Stock
0.2
0.15
0.1
0.05
0
-7.9 -6.6 -5.2 -3.9 -2.5 -1.1 0.2
1.6
3.0
4.3
5.7
Abnormality of Internet stocks
• leptokurtic
• fat-tailed

• 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.

“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 young fund manager
Inexperienced Investors and Bubbles
Robin Greenwood and Stefan Nagel (2008), forthcoming Journal of Financial Economics.

•
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)

(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)

Participants’ beliefs about prices are adaptive.
Haruvy, E., Lahav, Y., and C. Noussair (2009)

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)

Answer I: Noise Trader Approach
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.
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.

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

•
Bubble asset : x
1 ex. Internet stocks
• Non-bubble asset : x
2 ex. Large stocks like utility stocks
•
Risk-free asset : x f ex. Government bonds, fixed time deposits

•
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))

Rational traders
E W t

1
)


2
V W t

1
)
E ( W ) : The expected value of wealth
V ( W ) :
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.

Rational traders max

E ( W )


2 max

[ x
1
E ( p
1
)
V

( W ) x
2
E ( p
2
)
 u ( W ) x f
]
 s .
t .

2
[

11
( x
1
)
2 p
1
( x
1

 
12 x
1
)
 x
1 x
2 p
2
( x
2
 
21 x
1 x
2
 x
2
)

  q ( x f
22
( x
2
)
2
] }
 x f
)

0
  exp(
 
W ) x
1
: x
2
: p
1
: p
2
: q :
Demand for bubble asset
Demand for non-bubble asset
Price of bubble asset
Price of non-bubble asset
Price of risk-free asset
 ii
 ij
:
: Variance of the asset price change
E ( p i
) :
Covariance of the asset price change
Expected Price of bubble asset

The rational investor’s demands


 x
1




 x
2 x f



1
A
{

 E ( p
2
)

 x f
1
A
{

 E ( p
1
)

 p
1 q p
2 q p
1 ( x
1
 q x
1
)




22


 E ( p
2
)




11


 E ( p
1
)
 p
2 ( x
2
 x
2
) q p
2 q



12
} p
1 q



21
} where
A


11

21

12

22

The rational investors’ aggregate excess demands

M x
2
1 t 
M
A
[

2
2
( (
 p

)
  p
1 t
/ )
 
12
( (
 p
2 t

1
)
  p
2 t
/ )]

M
A
[

1
2
( (
 p
2 t

1
)
  p
2 t
/ )
 
21
( (
 p

)
  p
1 t
/ )] where
  it x it
 x it

1
 p jt

1
 p jt

1
 p jt
E (
 p jt

1
)

E p jt

1
)

E p jt
)

Their preference: s j




1,
1, for holding bubble aeest for holding risk free asset
( j

1, 2,..., N )
Random Utility Function:

U
1

U
1
 
1
 U
2

U
2
 
2

: Random variable i
U : Deterministic part i

Noise-Trader’s Random Utility Function:

U

 U

  s

H

    s

H



 

 i
: Random variable
• Average preference: s

1 N 
N j

1 s j s 1)
• Strength of Herding: 
• Momentum:
H
H t
 
H t

1
  
)( r t

1
 r f
), H t

0

H
0
( Adaptive expectation)

The probability that a utility-maximizing noise trader will choose each alternative:

 P


P

 exp[

U

] exp[

U

 
U

] exp[

U

] exp[

U

 
U

] under the Weibull distribution:
( )

Pr[
 i
 x ]
    x ]]
McFadden, Daniel (1974)

Transition Probabilities:



 ( ) t
( ) t exp[
 exp[ (
 s t

H t

)] s t

H t
] exp[ ( s t

H t exp[
 exp[
 s t

H t
] s t

H t
] exp[ (
 s t

H t
)]
)]

The Master Equation: p s t

1
)

( )

[

( t
 
) ( t
  

( ) ( )]

[

( t
 
) ( t
  

( ) ( )] t t






(
[( N t

/ 2)
 s t
]


( ) t

( ) t

N
(1
 s p s t t
2
(( N t

/ 2)
 s t
)

( ) t

( ) t

N
(1
 s p s t t
2 t
  s t
)

0 for s
 t
  t
N
2
 s

2
N

Representative Noise-Trader’s Behavior:
 s t

1
 s t

1
 s t
 
[ tanh(
 s t

H t
)
 s t
]
The noise trader’s Aggregate Excess demands
Q n
 t
 n
 t

1

QN
2
 s t
 s t

1


Bimodal:
(
 
1 , H

0 )
Unimodal:
(
 
1 , H

0 )
X t

1 a B
A s
 tanh(
 s

H )
X t ０
A
B b
AA line ; (

, H )
BB line ; (

, H )

( 0 .
9 , 0 )

( 2 .
0 , 0 )

Market-clearing Conditions:






1 t

M x
2 t
QN
2
 s t


M
A
[

2
2
E
 p

)
  p / )
1 t
 
12
( (
 p
2 t

1
)
  p q
2 t
/ )]

QN
2
 s t

0

M
A
[
 
12
E
 p

)
  p
1 t
/ )
 
1
2
E
 p
2 t

1
)
  p
2 t
/ )]

0

Market-clearing prices ：
 






 p
1 t
2 t

 s t

H t
  q q
H
[
[
 t
 s E ( p t
 
2 t

1
)]
[tanh(


1
1
 s t

E (
 p
(1

)( r t

1
 r f
)

)] s t

H t
)
 s t
]

Maxima of stationary probability density distribution P st
( s )

P st
 s
( s ) s
 s e

0 : peaks
 2
P st
 s
2
( s ) s
 s e

0 for minimum
 2
P st
 s
2
( s ) s
 s e

0 for maximum
K ( s )
 tanh(
 s

H )
 s

0 : peaks

K ( s )
 s s
 s e

0

K ( s )
 s s
 s e

0 for maximum for minimum

Unimodal: (
 
1 , H

0 )
AA line ; (

, H )

BB line ; (

, H )

( 0 .
8 , 0 .
3 )
( 0 .
8 ,

0 .
3 ) s
 tanh(
 s

H )

Phase Transition
AA line ; (

, H )
BB line ; (

, H )

( 1 .
8 , 0 .
4 )

( 1 .
8 ,

0 .
4 )
Cosh
2
[
 s
 
(
 
1 ) ]
  s c

(
 
1 )


Mechanism of bubble and crash
 
0 .
9 ; H

0 .
0
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
 
1 .
2 ; H

0 .
0
0.2
0.15
Bubble birth
0.1
0.05
0
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.05
-0.1
0.3
0.2
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.1
-0.2
-0.3
 
1 .
3 ; H

0 .
1
0.2
0.15
0.1
0.05
0
-0.05
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.1
-0.15
-0.2
 
1 .
3 ; H

0 .
05

Mechanism of bubble and crash (continued)
 
1 .
3 ; H

0 .
1
0.1
 
1 .
3 ; H

0 .
01
0.3
0.2
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.1
-0.2
-0.3
0.05
0
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.05
-0.1
-0.15
0.3
0.2
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.1
-0.2
-0.3

Crash!!

1 .
3 ; H
 
0 .
1
0.2
0.15
0.1
0.05
0
-0.05
-1 -0.8 -0.6 -0.4 -0.2 0.01 0.21 0.41 0.61 0.81
-0.1
-0.15
-0.2
 
1 .
3 ; H
 
0 .
05

A Example of the Model Simulation: s
0.8
0.6
0.4
0.2
0
-0.2
1 101 201 301 401 501 601 701
-0.4
-0.6
-0.8
-1
Return Moment H
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002
1 101 201 301 401 501 601 701
-0.004
Price on bubble asset
16
14
12
10
8
6
4
2
0
1 101 201 301 401 501 601 701
 
1.1;
 
0.9

11

12

4.1;

0.09
E ( p
1
)

10 ; E ( p
2
)

5
Price on non-bubble asset
H
 
0 .
02 (Crash)
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1 101 201 301 401 501 601 701

0.005
0.004
0.003
0.002
0.001
0
-0.001
1
-0.002
-0.003
-0.4
-0.6
-0.8
-1
0.8
0.6
0.4
0.2
0
-0.2
1 201 401 601 801 1001 1201
201 401 601 801 1001 1201

Risk Premium Puzzle: r
1
( t )
 r f

E [ r
1
( t )]
 r f
Expected risk premium:
[ ]
1 r f

(
1
)
 p
1
/ q p
1


E[r] = Risk Premium + Risk Free Rate

A Model Simulation: Risk Premium
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
1
-0.04
101 201 301 401 501 601
Expected Discount Rate: E [ r
1
( t )]
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002
1 101 201 301 401 501 601 701
-0.004
Realized Return: r
1
( t )
The period of bubble:
After burst of bubble: r
1
( t )
 r f

E [ r
1
( t )]
 r f r
1
( t )
 r f

E [ r
1
( t )]
 r f

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 longterm investment, while the noise traders lose money.

Example I: Internet Bubble
6000
5000
4000
3000
2000
1000
0
NASDAQ
S&P500
NASDAQ:
1503 (1998/1/5)
5048 (2000/3/6): Peak
1140 (2002/9/23): Bottom
3.3
1/5
S&P500:
927 (1998/1/5)
1527 (2000/3/20): Peak
800 (2002/9/30): Bottom
1.6
1/2