A behavioral Model of financial Crisis Taisei Kaizoji International Christian University, Tokyo...
-
Upload
shonda-flowers -
Category
Documents
-
view
225 -
download
0
Transcript of 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