Asset Pricing with Heterogeneous Agents, Incomplete Markets and Trading Constraints

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Asset Pricing with Heterogeneous Agents,
Incomplete Markets and Trading Constraints

• Tsvetanka Karagyozova
• 25 October 2007

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Outline

• Motivation/Contribution
• Model
• Calibration
• Results
• Robustness
• Conclusion

Asset Pricing with Heterogeneous Agents
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Motivation

• How do individuals make decisions in the presence
  of risk?
• Parsimonious vs. realistic models
• Can heterogeneous preferences account for some of
  the puzzles recorded in empirical studies?

Asset Pricing with Heterogeneous Agents
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The Problem

• In equilibrium, what determines the differential
  in expected real returns across assets?
• How do we define risk? What are the sources of
  risk?
• CAPM (Sharpe, 1964, Lintner, 1965, Mossin, 1966)
• CCAPM (Lucas, 1978, Breeden, 1979)

Asset Pricing with Heterogeneous Agents
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Problems with the consumption capital asset
pricing model

• Fails to account quantitatively for the
  historical high average rate of return and
  volatility of stocks
• The Equity Premium Puzzle (Mehra
  Prescott,1985).
• The Risk Free Rate Puzzle (Weil, 1989)
• The Excess Volatility Puzzle of stock returns
  (Shiller, 1981 Leroy Porter, 1981 Singleton,
  1980)
• No satisfactory alternative model of stock
  pricing.

Asset Pricing with Heterogeneous Agents
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Issues

• Are models of aggregate savings behavior
  omitting some crucial element?
• What is the cost of the business cycle?
• Kocherlakota, 1996

Asset Pricing with Heterogeneous Agents
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Key Assumptions

• Constant relative risk aversion utility function
• Generalized Expected Utility (Epstein and Zin,
  1989, 1991)
• Habit Formation (Constantinides, 1990 Heaton
  1995, Campbell and Cochrane, 1999)
• Keeping up with the Joneses (Abel, 1990 Gali,
  1994)
• Complete asset markets (Constantinides Duffie,
  1996 Lucas, 1994, Telmer, 1993, Marcet
  Singleton, 1999)
• Costless asset trading (Lucas Heaton, 1995
  Luttmer, 1999 McGrattan Prescott, 2003)

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Contribution

• Allowing for agent heterogeneity is instrumental
  in accounting for the stylized facts of financial
  markets.
• Positive models of human behavior provide
  insights into decades-old economic puzzles.

Asset Pricing with Heterogeneous Agents
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Model Features

• DSGE model
• Endowment economy
• Two assets in the economy risky and risk-free.
• Heterogeneous agents
• Preferences
• Income
• Incomplete markets.
• Trading constraints

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Model Features

• Heterogeneous Preferences
• The consumption of stockholders is more volatile
  and more highly correlated with the excess return
  on the stock market (Mankiew and Zeldes, 1991)
• An attempt to introduce elements of prospect
  theory (Kahneman Tversky, 1979) into a standard
  asset pricing model.
• Barberis, Huang Santos, 2001 extension of the
  prospect theory to a dynamic model

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Asset Pricing with Heterogeneous Agents
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Consumer Preferences Type A agent

• Type A agent (non-investors) maximizes
• is the consumption of Type A agent at time
  t
• ? is the subjective discount factor
• ? is the coefficient of relative risk aversion.

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Consumer Preferences Type B agent

• Type B agent (investors or stockholders)
  maximizes
• consumption of Type B agent at time t
• the risky asset holdings of Type B agent at
  time t
• Xt1 the gain or loss on risky asset holdings
  between t and t1.
• zt measures gains and losses prior to time t a
  state variable.
• b0 exogenous scaling factor.
• The second term measures the utility from
  fluctuations in financial wealth the prospect
  theory term in the utility function.

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The Model Gains and Losses

• Gains and losses refer to changes in financial
  wealth.
• The investor only cares about changes in the
  value of the risky asset.
• Changes in financial wealth are determined once a
  year.
• Pt1 the price of the risky asset price at t1
• Pf,t the price of risk-free asset at t

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The Model Prior Gains and Losses

• Zt historical benchmark level for the price of
  the risky asset.
• StB (Pt -Zt ) gt 0 ? the investor has realized
  gains in his financial wealth in time t
• Define ztZt/Pt.
• The house money effect Thaler and Johnson
  (1990)
• The extent to which an economic agent is loss
  averse depends on his prior stock market
  performance.

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The Model Utility from Gains and Losses

• ztlt1 (prior gains or neither gains nor losses)
• StBPt Rt1-StBPt Rf,t
  Rt1 ztRf,t
• v(Yt1, StB,zt)
  for
• StBPt Rf,t(zt -
  1)?StBPt (Rt1-ztPt Rf,t) Rt1ltztRf,t
• where ? gt1
• zt gt1 (prior losses)
• StBPtRt1-StBPtRf,t
  Rt1 Rf,t
• v(Yt1,StB,zt) for
• ?(zt)(StBPtRt1-StPtRf,
  t) Rt1 lt Rf,t
• where ?(zt) ? k(zt -1) and k gt 0,

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Utility from Gains and Losses
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The Model The dynamics of the benchmark level, zt

• Assumptions
• The benchmark level, zt, responds sluggishly to
  changes in the price of financial asset over
  time.
• where
• fixed parameter set in such a way that in
  equilibrium, the median value of zt 1.
• h a measure of investor's memory.

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Budget Constraints

• Budget Constraints
• where
• bond holdings of agent i for i A,B in
  period t
• income of agent i for i A,B in period t

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Short-Sale and Borrowing Constraints

• No Ponzi schemes
• Borrowing Constraint
• where
• Short-sale constraint

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Endowment and Dividend Processes

• Aggregate endowment normalized to 1.
• where dt is the per-share real dividend
• Population normalized to 1.

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Market Equilibrium

• Endogenous variables
• The equilibrium distributions of
• Consumption
• Asset holdings
• Asset prices

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First Order (Kuhn-Tucker) Conditions

• Bond holdings
• Either
• or
• Equity holdings Agent A
• Either
• or

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FOC, Equity Holdings, Agent B

• Either
• or
• where
• for zt 1
• for zt gt1

for
for
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Market-Clearing Conditions

• Consumption
• Where p is the proportion of Type A agents in the
  population and is the aggregate endowment
  (income).
• Bonds zero net supply
• Stocks normalized to 1

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State Variables

• Exogenous
• At every t, given by ln(ytA) ln(ytB) ln(dt )
• The income of Type i agent for i A, B follows
• ln(yti) ?i ?i ln(yt-1i) eti
• eti ? Niid(0,(se t )2)
• The aggregate dividend
• ln(dt) ?i ?i ln(dt-1) et
• eti ? Niid(0,(se t )2)
• Endogenous
• The elements of wealth Bt-1i , St-1i ,
• Prior investment outcomes zt-1

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Model Calibration

• Parameter Values

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Model Calibration Income and Dividend Processes

• Incomes PSID data 1968-1997
• Panel survey data, annual
• Stockholders 35 of population
• Non-investors 65 of population.
• Income processes of Type A and B agents
• Dividends Net dividends, NIPA tables 1929-2006.
• Solution algorithm Parameterized Expectations
  Approach (Marcet, 1989 Marcet Singleton, 1999)

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Income and Dividend Processes (Cont.)
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Results
Asset Pricing with Heterogeneous Agents
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Excess Stock Return Volatility
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Consumption Processes
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Income Processes
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Equilibrium Distribution of z
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Prior Stock Market Performance and Agent Bs
Consumption
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Income, Stock and Bond Holdings
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Income, Stock and Bond Holdings
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Scatter plot
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Robustness Prospect Theory
Asset Pricing with Heterogeneous Agents
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Robustness Prospect Theory
Asset Pricing with Heterogeneous Agents
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Robustness Borrowing Constraints
Asset Pricing with Heterogeneous Agents
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Robustness Investors memory
Asset Pricing with Heterogeneous Agents
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Conclusion

• In equilibrium, individuals hold different
  portfolios.
• Individuals trade on financial markets to smooth
  their consumption.
• Time-varying risk premium of stocks over bonds.
• Low risk free rate and low correlation of
  consumption with stock returns.
• Accounting for loss aversion and prior stock
  market performance are instrumental in matching
  the empirical moments of asset returns.