A behavioral Model of financial Crisis Taisei Kaizoji International 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 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/ Period I: 1998/1/2-2000/3/9 Period II: 2000/3/ /12/31 Period IPeriod II

Price Changes Internet Stocks: Mean Variance Period I Period II Non-Internet Stocks: Mean Variance Period I Period II Covariance: Period I 0.09 Period II 0.13 Period I: 1998/1/2-2000/3/9 Period II: 2000/3/ /12/31 Period IPeriod II

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 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

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 The expected value of wealth The variance of wealth References: John Lintner, The JFQA, Vol. 4, No. 4. (1969), Jan Mossin, Econometrica, Vol. 34, No. 4. (Oct., 1966), pp

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 where

The rational investors’ aggregate excess demands where

Noise Traders Their preference: : Random variable : Deterministic part Random Utility Function:

Noise-Trader’s Random Utility Function: Average preference: Strength of Herding: : Random variable Momentum: ( Adaptive expectation)

Probabilities McFadden, Daniel (1974) The probability that a utility-maximizing noise trader will choose each alternative: under the Weibull distribution:

Transition Probabilities:

The Master Equation:

Representative Noise-Trader’s Behavior: The noise trader’s Aggregate Excess demands

Bimodal:Unimodal:

Market-clearing Conditions:

Market-clearing prices ：

for maximum for minimum : peaks for maximum for minimum Maxima of stationary probability density distribution : peaks

Unimodal:

Phase Transition

Mechanism of bubble and crash Bubble birth

Mechanism of bubble and crash (continued) Crash!!

A Example of the Model Simulation: s Price on bubble asset Return Moment HPrice on non-bubble asset (Crash)

Risk Premium Puzzle: Expected risk premium: E[r] = Risk Premium + Risk Free Rate

Expected Discount Rate: Realized Return: The period of bubble: After burst of bubble: 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 NASDAQ S&P500 S&P500: 927 (1998/1/5) 1527 (2000/3/20): Peak 800 (2002/9/30): Bottom 3.3 1/ /2