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Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals Mikhail Anufriev Economics Discipline Group University of Technology, Sydney Nonlinear Economic Dynamics conference Siena, 4 – 6 July 2013 Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Introduction Heterogeneous Agent Models of financial markets ◮ co-existence of different trading strategies (fundamentalists and chartists, trend-followers) in financial markets ◮ reproduce several stylized facts of financial markets ◮ generate endogenous booms and busts, i.e., persistent deviations of price from the fundamental price ◮ feedback from the relative success of trend-following strategy to price dynamics Idea of this paper: ◮ introduce an additional feedback: from past market volatility to the fundamental price ◮ market crushes can be very severe, they can be followed by long undervaluation of the assets Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Introduction Heterogeneous Agent Models of financial markets ◮ co-existence of different trading strategies (fundamentalists and chartists, trend-followers) in financial markets ◮ reproduce several stylized facts of financial markets ◮ generate endogenous booms and busts, i.e., persistent deviations of price from the fundamental price ◮ feedback from the relative success of trend-following strategy to price dynamics Idea of this paper: ◮ introduce an additional feedback: from past market volatility to the fundamental price ◮ market crushes can be very severe, they can be followed by long undervaluation of the assets Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion CARA framework with heterogeneous expectations 1. two assets ◮ riskless: risk-free interest rate r + 1 = R ◮ risky: price p t and dividend y t supply per investor S ≥ 0 2. mean-variance demand � A h , t ( p ) = ( E h , t [ p t + 1 + y t + 1 ] − Rp ) ( a V h , t [ p t + 1 + y t + 1 − Rp ]) 3. temporary equilibrium � S = h n h , t A h , t ( p ) R t = p t + y t − Rp t − 1 p t � � 4. evolution of fractions n h , t + 1 ∝ exp ( βπ h , t ) , π h , t = R t A h , t − 1 − C h where profit of type h is β ∈ [ 0 , ∞ ) is the intensity of choice Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Fundamental Price ◮ The fundamental price is the solution under rational (homogeneous) expectations E t [ p t + 1 + y t + 1 ] − Rp t = a S V t [ p t + 1 + y t + 1 − Rp t ] ◮ For IID dividends, y t ∼ N ( y , σ 2 y ) p t + 1 + y − Rp t = a S V [ p t + 1 + y t + 1 − Rp t ] and the unique non-bubble solution is p f = y r − aS r σ 2 y ◮ present value of the expected dividends adjusted for risk Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Conditional Fundamental Price Conditional fundamental price is the price which fundamentalists expect to be at the “rational” market given past price history σ 2 ◮ adjust price given the forecast for the return variance ˆ R t | I t − 1 ◮ for IID dividends, y t ∼ N ( y , σ 2 y ) t − 1 [ p t ] = y r − a S E f σ 2 r ˆ R t | I t − 1 ◮ present value of the expected dividends adjusted for perceived risk ◮ forward solution of the equilibrium equation under assumption that the demand of all traders is given by A t ( p t ) = E [ p t + 1 + y t + 1 ] − Rp t σ 2 a ˆ R t | I t − 1 Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Conditional Fundamental Price Conditional fundamental price is the price which fundamentalists expect to be at the “rational” market given past price history σ 2 ◮ adjust price given the forecast for the return variance ˆ R t | I t − 1 ◮ for IID dividends, y t ∼ N ( y , σ 2 y ) t − 1 [ p t ] = y r − a S E f σ 2 r ˆ R t | I t − 1 ◮ present value of the expected dividends adjusted for perceived risk ◮ forward solution of the equilibrium equation under assumption that the demand of all traders is given by A t ( p t ) = E [ p t + 1 + y t + 1 ] − Rp t σ 2 a ˆ R t | I t − 1 Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Conditional Fundamental Price Conditional fundamental price is the price which fundamentalists expect to be at the “rational” market given past price history σ 2 ◮ adjust price given the forecast for the return variance ˆ R t | I t − 1 ◮ for IID dividends, y t ∼ N ( y , σ 2 y ) t − 1 [ p t ] = y r − a S E f σ 2 r ˆ R t | I t − 1 ◮ present value of the expected dividends adjusted for perceived risk ◮ forward solution of the equilibrium equation under assumption that the demand of all traders is given by A t ( p t ) = E [ p t + 1 + y t + 1 ] − Rp t σ 2 a ˆ R t | I t − 1 Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Dynamical System ◮ Temporary equilibrium: p t = 1 �� H � σ 2 h = 1 n h , t E h , t [ p t + 1 + y t + 1 ] − S a ˆ R t | I t − 1 R ◮ Evolution of fractions: � � H n h , t + 1 = exp ( βπ h , t ) h ′ = 1 exp ( βπ h ′ , t ) π h , t = ( p t + y t − Rp t − 1 ) E h , t − 1 [ p t + y t ] − Rp t − 1 − C h σ 2 R t − 1 | I t − 2 a ˆ ◮ EWMA estimation of variance of return µ t = ( 1 − w µ ) R t + w µ µ t − 1 R t + 1 | I t = ( 1 − w σ )( R t − µ t ) 2 + w σ ˆ σ 2 σ 2 ˆ R t | I t − 1 where R t = p t + y t − Rp t − 1 is the realised excess return Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Model of Brock and Hommes, JEDC 1998 Assumptions: ◮ zero outside supply: S = 0 constant fundamental price � ◮ constant expectations of variance: V h , t = σ 2 Dynamics: � H p t = 1 σ 2 ◮ Price: h = 1 n h , t E h , t [ p t + 1 + y t + 1 ] − S a ˆ R R t | I t − 1 � � H ◮ Fractions: n h , t + 1 = exp ( βπ h , t ) h ′ = 1 exp ( βπ h ′ , t ) π h , t = ( p t + y t − Rp t − 1 ) E h , t − 1 [ p t + y t ] − Rp t − 1 ◮ Profit: − C h σ 2 Rt − 1 | It − 2 a ˆ Feedbacks: ◮ past prices affect expectations of chartists ◮ last period price affects profits and fractions of different types ◮ expectations and fractions affect current price Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Model of Brock and Hommes, JEDC 1998 Assumptions: ◮ zero outside supply: S = 0 constant fundamental price � ◮ constant expectations of variance: V h , t = σ 2 Dynamics: � H p t = 1 σ 2 ◮ Price: h = 1 n h , t E h , t [ p t + 1 + y t + 1 ] − S a ˆ R R t | I t − 1 � � H ◮ Fractions: n h , t + 1 = exp ( βπ h , t ) h ′ = 1 exp ( βπ h ′ , t ) π h , t = ( p t + y t − Rp t − 1 ) E h , t − 1 [ p t + y t ] − Rp t − 1 ◮ Profit: − C h σ 2 Rt − 1 | It − 2 a ˆ Feedbacks: ◮ past prices affect expectations of chartists ◮ last period price affects profits and fractions of different types ◮ expectations and fractions affect current price Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals

Introduction Heterogeneous Beliefs Model Dynamics of the Model Extensions Conclusion Model of Gaunersdorfer, JEDC 2000 Assumptions: ◮ zero outside supply: S = 0 constant fundamental price � ◮ time-varying expectations of variance: µ t = ( 1 − w µ ) R t + w µ µ t − 1 R t + 1 | I t = ( 1 − w σ )( R t − µ t ) 2 + w σ ˆ σ 2 σ 2 ˆ R t | I t − 1 Dynamics: � H p t = 1 σ 2 ◮ Price: h = 1 n h , t E h , t [ p t + 1 + y t + 1 ] − S a ˆ R t | I t − 1 R � � H ◮ Fractions: n h , t + 1 = exp ( βπ h , t ) h ′ = 1 exp ( βπ h ′ , t ) π h , t = ( p t + y t − Rp t − 1 ) E h , t − 1 [ p t + y t ] − Rp t − 1 ◮ Profit: − C h σ 2 Rt − 1 | It − 2 a ˆ Feedbacks: ◮ the same as in Brock-Hommes Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals