Impact of Investor's Varying Risk Aversion on the Dynamics of Asset Price Fluctuations

1
Impact of Investor's Varying Risk Aversion on the
Dynamics of Asset Price Fluctuations

• Baosheng Yuan and Kan Chen
• National University of Singapore
• June 13, 2005

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Outline

• Introduction
• Demand Function with Power Utility Function
• Model with Constant Risk-Aversion
• Model with Dynamic Risk-Aversion
• Simulation Results
• SFI Model with CRA
• SFI Model with DRA
• Summary and Future Work

3
Introduction Agent-Based Modeling

• Purpose of the study
• Impact of investors changing risk-aversion on
  price dynamics
• Build a model with DRA to explain and reproduce
  the key stylized facts
• Impact factors to price fluctuations can be
  studied with ABM
• Investors price estimation and their market
  beliefs (Arthur 1997, DeLong 1990)
• Investors ability of acquiring and processing
  market information
• Investors response to price changes (Lux 1999,
  Caldarelli 1997)
• Investors fluctuating risk-aversion attitudes

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Introduction Challenges of ABM

• Inherent
• Agents are intelligent, self-interested, and
  adaptive
• Interdependence of investors price forecasting
• Investors fluctuating sentiments (i.g.,
  risk-aversion)
• Price fluctuates far away from fundamental
  value (Delong 1999)
• Technical
• Price setting can not be determined deductively
  (Arthur 1997)
• No analytical aggregate price setting available
  for power utility function

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Introduction Challenges of ABM

• Empirical
• Few causality analysis of price dynamics and
  investors changing risk aversion attitude or
  sentiment
• Difficult to reproduce all the key stylized facts
• Excess volatility or fat-tails of return
  (Mandelbrot 1963, Fama 1965, Bouchaud 2000,
  Mantegna 1999, and Cont 2001)
• Kurtosis may not be as large as that of real
  market data for example STI model generated
  very low kurtosis (lt4) )
• The dependence of Kurtosis on time lag may not
  be consistent with that of real data
• Volatility clustering ( Engle 1982, Baillie 1996,
  Chou 1988, Schwert 1989, Poterba 1986, and Chen
  2005)
• For example STI model generated no significant
  volatility clustering

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Introduction Our Study

• Introduce DRA and model it with a simple random
  walk process
• Derive an approximated price setting formula
  with power utility function
• Show DRA as a major driving force for excess
  price fluctuation and associated volatility
  clustering
• DRA model reproduces the key stylized facts
• The time scaling properties (of the stylized
  facts) of DRA model is consistent with that of
  real data

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Demand Function with Power Utility

• Model assumptions
• Agents are consumption based, expected utility
  optimizers
• Agents risk preferences are characterized by
  Power utility function with investor-specific and
  time-dependent risk aversion coefficients
  (indices)
• (1)
• Where ?i,t is the risk aversion index of agent i
  at time t and Ci,t the consumption

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Demand Function with Power Utility (Contd)

• Demand function
• Agent i optimizes his total utility over the
  current and next time step
• (2)
• where f (zi,t x,?i,t) (1 x rei,t x zi,t) 1-
  ?i,t rei,t is i-th agents conditional
  expected net return and zt,i is conditionally
  Gaussian
• Approximate the Gaussian integral with sum of the
  roots of Hermite polynomial (see our full paper
  for details)
• (3)

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Model with Constant Risk-Aversion

• Price formula
• (4)
• (5)
• Price Estimation
• (6)
• where MAi,j,t is the jth moving-average price
  predictor of agent i at time t, and ei,j, the
  prediction error N(0,?pd).

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Model with Const. Risk-Aversion (contd)

• Update of the variance of estimation error
• (7)
• where ? (0lt?ltlt1) is a weighting constant.
• Dividend process
• dtdt-1rdet (8)
• where et is an i.i.d. Gaussian with zero and
  variance ?d rd the average dividend growth
  rate.

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Model with Dynamic Risk-Aversion

• Key Ingredient Agents are heterogeneous with
  dynamic risk aversion (DRA)
• Model DRA each agents risk-aversion index
  follows a bounded random walk with a constant
  variance ? 2
• ?i,t ?i,t-1? zi,t ?i,t??0, ?u (9)
• where zi,t is an i.i.d. Gaussian noise N(0,1)
  ? is a constant, ?0(gt0) is the lower boundary,
  and ?u the upper boundary. After t time steps
• (10)
• where Si,t ?tt1zi,t

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Model with Dynamic Risk-Aversion (Contd)

• Price equation
• (11)
• Range of DRA indices
• Empirical studies Mehra Prescott(1985), Friend
  and Blume(1975), Levy, et al (2000) reported
  for a typical investor ? ?0,2
• Mehra Prescott (1985) used ? with upper
  boundary of 10
• To explain Equity Premium Puzzle of NYSE over
  50 years post war (with consumption based model
  in Cochrane 2005) ? is required to be 250!
• In our model ? ?1.0e-5, 50

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Simulation Results System setup

• Daily setting parameters
• N100, M2
• ?i,0 ?0.2, 2, ?i,t ?1.0e-5,20
• ?pd 3, ? 1/250 Li,j ?2,250
• r4, rd2, ?d2
• Experiment with different variance ?2 of DRA

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Simulation Results Price and Trading Volume

• CRA both price and trading volume show little
  fluctuation
• DRA leads to increased price/trading volume
  fluctuation
• The degree of fluctuation is directed related to ?

Simulated prices for ? (from bottom) 0, 0.005,
0.01, 0.015, and 0.02
Trading volume for ? 0 (bottom), and 0.01
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Simulation Results Excess Volatility

• CRA model close to Gaussian
• DRA model close to real data (DJIA)
• DRA model produces significant fat-tails as those
  of real data
• DRA gives correct time scaling the larger the
  time lag is, the less leptokurtic it gives

PDF of returns of different time periods
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Simulation Results Excess Volatility (Contd )
Table 1. Statistics for DRA model and real data
(DJIA)
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Simulation Results Excess Volatility DRA
The kurtosis of simulated series for different ?
of DRA indices
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Simulation Results Autocorrelation Function

• ACF (absolute-valued return) for real data
• Starts low
• Decays slowly
• ACF for CRA close to Gaussian
• ACF for DRA close to real data

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Simulation Results Volatility Clustering

• Measure of VC
• Use conditional return distribution (Chen et al
  2005), i.e. S.D. of current return v.s. absolute
  return of the previous period
• Simulated results
• CRA model produces little VC close to that of
  Gaussian
• DRA produces significant VC in good agreement
  with real data (DJIA)
• Time scaling of VC in DRA model is consistent
  with real data

S.D. of return v.s. the absolute return in the
previous period
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SFI Market Model with CRA

• Assumptions (Arthur et al. 1997, LeBaron 1999,
  LeBaron 2005)
• Agents are (CARA negative power) utility
  optimizers with homogeneous Constant Risk
  Aversion (CRA)
• Agents are adaptive exploratory learners
• Use best predictors among the multiple
  predictors
• Drop out the worst predictors and develop new
  predictors with G.A.
• Price predictors
• Each predictor contains a market recognizer and a
  price forecaster
• Market condition recognizer incorporates any
  market variables.
• Price forecaster is a linear combination of the
  current price and dividend

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SFI Market Model with CRA (Contd)

• Key results of SFI model with CRA
• Key observations of SFI model with CRA
• Slow learning leads to rational expectation
  regime
• Fast learning leads to limited fluctuation in
  excess volatility
• The excess volatility and the associated
  clustering are far less than that of real data.

Table 2. Return and volume statistics collected
for 25 experiments after 250,000 periods. (Arthur
1997)
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SFI Model with DRA

• SFI model with DRA
• Basic model assumptions/structure unchanged
• Agents are heterogeneous in risk aversions
• Key new ingredient of DRA is incorporated in the
  model

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SFI Model with DRA (contd)

• Results of SFI-CRA/DRA
• CRA is close to Gaussian process
• DRA produces significant fluctuation that
  resembles real time series
• DRA E.V. and V.C. are in good agreement to real
  data
• Key implication
• DRA is a main driving force for the price
  dynamics
• VC is controlled by parameter ? - variance of DRA
• DRA is independent of baseline model

Price series for ? (from bottom) 0, 0.01, 0.02,
0.03, 0.04, and 0.05
The kurtosis for different variance ?
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Summary and Future Work

• Summary
• A DRA market model has been developed which
  explains and reproduces the key stylized facts,
  suggesting that
• DRA is the main driving force of excess price
  fluctuation.
• The excess volatility and VC are controlled by
  ?2, the variance of DRA which can be used to
  model fluctuating market sentiment
• DRA model can generate excess price fluctuations
  of various strengths and includes CRA as a
  special case (?20)
• DRA can be incorporated in other market models of
  CRA
• DRA provides new insights into the mechanism of
  price fluctuations governed by investors
  fluctuating sentiments
• Future work
• Real (Non-random walk) dynamic process of DRA
• The impact of investors DRA on asymmetric return
  distribution

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Acknowledgement

• Baosheng Yuan is deeply grateful to Blake LeBaron
  for his helpful suggestions at several points in
  the research!

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References

• Arthur, W. B., J.H. Holland, B. LeBaron, R.
  Palmer, and P. Tayler (1997), Asset pricing
  under endogenous expectations in an artificial
  stock market'', appeared in The Economy as an
  Evolving Complex System II, Reading, Mass
  Addsion-Wesley, Advanced Book Program, 1997.
• Baillie, R. T., T. Bollerslev, and H. O.
  Mikkelsen (1996), Fractionally integrated
  generalized conditional heteroskedasticity'',
  Journal of Econometrics, 74(1), 3-30.
• Bak, P., M. Paczuski, and M. Shubik (1997),
  Price variations in a stock market with many
  agents'', Physica A 246, 430-453.
• Barberis, N., A. Shleifer, and R. Vishny (1998),
  A model of investor sentiment'', Journal of
  Financial Economics, 49, 307-343.
• Bouchaud, J.-P., M. Potters (2000), Theory of
  Financial Risks From Statistical Physics to Risk
  Management, Cambridge University Press,
  Cambridge.
• Brock, A. W. and B. LeBaron (1996), A Dynamic
  structural model for stock return volatility and
  trading volume'', The Review of Economics and
  Statistics, 78, 94-110.
• Caldarelli, G., M. Marslli, and Y.-C. Zhang
  (1997), A prototype model of stock exchange'',
  Europhysics Letters, 40, 479-484.
• Chen, K., C. Jayaprakash, and B. Yuan (2005),
  Conditional probability as a measure of
  volatility clustering in financial time series'',
  arXiv physics/0503157.
• Chou, R. Y. (1988), Volatility persistent and
  stock valuation Some empirical evidence using
  GARCH'', Journal of Applied Econometrics, 3,
  279-294.
• Cochrane, J. H. (2005), Asset Pricing, Rev. ed.,
  Princeton NJ, Princeton University Press.
• Cont, R. (2001), Empirical properties asset
  returns stylized facts and statistical issue,
  Quantitative Finance, 1, 223-236.
• Cont, R. and J.-P. Bouchaud (2001), Herd
  Behavior and aggregate fluctuations in financial
  markets'', Macroeconomic Dynamics, 4, 170-196.
• DeLong, J. B., A. Shleifer, L. H. Summers, and
  R. J. Waldmann (1990), Noise trader risk in
  financial markets'', Journal of Political
  Economy, 98 (4), 703-738.
• Engle, R. F. (1982), Autoregressive conditional
  heteroskedasticity with estimates of the variance
  of the United Kingdom inflation'', Econometrica,
  50(4), 987-1008
• Fama, Eugene F. (1963) Mandelbrot \ the stable
  parentian hypothesis'', in The Random Character
  of Stock Market Prices, P. Costner, Editor, MIT
  Press, Mass., 1969 reprint.
• Fama, Eugene F. (1965), The behavior of
  stock-market prices'', Journal of Business,
  38(1), 34-105.
• Friend, I., and M. E. Blume (1975), The demand
  for risky assets'', The American Economic Review,
  65, 900-922.
• LeBaron, B., W. B. Arthur, R. Palmer (1999),
  Time series properties of an artificial stock
  market'', Journal of Economic Dynamics \
  Control, 23, 1487-1516.
• LeBaron, Blake (2005), Agent-Based
  Computational Finance'', to appear in The
  Handbook of Computational Economics, Vol. II,
  edited by K. L. Judd and L. Tesfatsion.

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Thank you!