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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

• f 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

• f 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 pt and dividend yt

supply per investor S ≥ 0

• 2. mean-variance demand

Ah,t(p) = (Eh,t[pt+1 + yt+1] − Rp)

• (a Vh,t[pt+1 + yt+1 − Rp])
• 3. temporary equilibrium

S =

• h nh,tAh,t(p)
• pt
• Rt = pt + yt − Rpt−1
• 4. evolution of fractions

nh,t+1 ∝ exp(βπh,t), where profit of type h is πh,t = RtAh,t−1−Ch 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 Et[pt+1 + yt+1] − Rpt = a S Vt[pt+1 + yt+1 − Rpt]

◮ For IID dividends, yt ∼ N(y, σ2 y)

pt+1 + y − Rpt = a S V[pt+1 + yt+1 − Rpt] and the unique non-bubble solution is pf = y r − aS r σ2

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

◮ adjust price given the forecast for the return variance ˆ

σ2

Rt|It−1 ◮ for IID dividends, yt ∼ N(y, σ2 y)

Ef

t−1[pt] = y

r − a S r ˆ σ2

Rt|It−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 At(pt) = E[pt+1 + yt+1] − Rpt aˆ σ2

Rt|It−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

◮ adjust price given the forecast for the return variance ˆ

σ2

Rt|It−1 ◮ for IID dividends, yt ∼ N(y, σ2 y)

Ef

t−1[pt] = y

r − a S r ˆ σ2

Rt|It−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 At(pt) = E[pt+1 + yt+1] − Rpt aˆ σ2

Rt|It−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

◮ adjust price given the forecast for the return variance ˆ

σ2

Rt|It−1 ◮ for IID dividends, yt ∼ N(y, σ2 y)

Ef

t−1[pt] = y

r − a S r ˆ σ2

Rt|It−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 At(pt) = E[pt+1 + yt+1] − Rpt aˆ σ2

Rt|It−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:

pt = 1 R H

h=1 nh,t Eh,t[pt+1 + yt+1] − S a ˆ

σ2

Rt|It−1

• ◮ Evolution of fractions:

nh,t+1 = exp(βπh,t) H

h′=1 exp(βπh′,t)

πh,t = (pt + yt − Rpt−1)Eh,t−1[pt + yt] − Rpt−1 a ˆ σ2Rt−1|It−2 − Ch

◮ EWMA estimation of variance of return

µt = (1 − wµ)Rt + wµµt−1 ˆ σ2

Rt+1|It = (1 − wσ)(Rt − µt)2 + wσˆ

σ2

Rt|It−1

where Rt = pt + yt − Rpt−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: Vh,t = σ2

Dynamics:

◮ Price:

pt = 1

R

H

h=1 nh,t Eh,t[pt+1 + yt+1] − S a ˆ

σ2

Rt|It−1 ◮ Fractions: nh,t+1 = exp(βπh,t)

H

h′=1 exp(βπh′,t) ◮ Profit:

πh,t = (pt + yt − Rpt−1) Eh,t−1[pt+yt]−Rpt−1

aˆ σ2Rt−1|It−2

− Ch 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: Vh,t = σ2

Dynamics:

◮ Price:

pt = 1

R

H

h=1 nh,t Eh,t[pt+1 + yt+1] − S a ˆ

σ2

Rt|It−1 ◮ Fractions: nh,t+1 = exp(βπh,t)

H

h′=1 exp(βπh′,t) ◮ Profit:

πh,t = (pt + yt − Rpt−1) Eh,t−1[pt+yt]−Rpt−1

aˆ σ2Rt−1|It−2

− Ch 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:

µt = (1 − wµ)Rt + wµµt−1 ˆ σ2

Rt+1|It = (1 − wσ)(Rt − µt)2 + wσˆ

σ2

Rt|It−1

Dynamics:

◮ Price:

pt = 1

R

H

h=1 nh,t Eh,t[pt+1 + yt+1] − S a ˆ

σ2

Rt|It−1 ◮ Fractions: nh,t+1 = exp(βπh,t)

H

h′=1 exp(βπh′,t) ◮ Profit:

πh,t = (pt + yt − Rpt−1) Eh,t−1[pt+yt]−Rpt−1

aˆ σ2Rt−1|It−2

− Ch Feedbacks:

◮ the same as in Brock-Hommes

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

This Model

Assumptions:

◮ positive outside supply: S > 0 ◮ time-varying expectations of variance: ◮ together they imply time-varying fundamental price

Dynamics:

◮ Price:

pt = 1

R

H

h=1 nh,t Eh,t[pt+1 + yt+1] − S a ˆ

σ2

Rt|It−1 ◮ Fractions: nh,t+1 = exp(βπh,t)

H

h′=1 exp(βπh′,t) ◮ Profit:

πh,t = (pt + yt − Rpt−1) Eh,t−1[pt+yt]−Rpt−1

aˆ σ2

Rt−1|It−2

− Ch Feedbacks:

◮ the same as in Brock and Hommes ◮ high past volatility decreases current price ◮ high past volatility decreases fundamental price, i.e.,

expectations of fundamentalists

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

Two-types model: Fundamentalists vs. Chartists

Variance estimate σ2

t−1 := ˆ

σ2

Rt|It−1 using EWMA:

σ2

t = (1 − wσ)(Rt − µt)2 + wσσ2 t−1 ,

µt = (1 − wµ)Rt + wµµt−1 Perceived fundamental price: pf

t :=

• y − a S σ2

t−1

• /r

Expectations:

◮ fundamentalists:

Ef

t [pt+1] = pf t ◮ chartists:

Ec

t [pt+1] = pf t + g(pt−1 − pf t−1),

g ≥ 1 Model in deviations xt := pt − pf

t ◮ price:

Rxt = nf

t Ef t [xt+1] + nc t Ec t [xt+1] = gnc t xt−1 ◮ fractions:

nc

t+1 = 1

• (1 + exp(β(πf

t − πc t ))) ◮ profit differential: πf t − πc t = Rt −gxt−2 aσ2

t−2 − C

◮ excess return:

Rt = xt − Rxt−1 + (yt − y) + aS

r (Rσ2 t−2 − σ2 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

Two-types model: Fundamentalists vs. Chartists

Variance estimate σ2

t−1 := ˆ

σ2

Rt|It−1 using EWMA:

σ2

t = (1 − wσ)(Rt − µt)2 + wσσ2 t−1 ,

µt = (1 − wµ)Rt + wµµt−1 Perceived fundamental price: pf

t :=

• y − a S σ2

t−1

• /r

Expectations:

◮ fundamentalists:

Ef

t [pt+1] = pf t ◮ chartists:

Ec

t [pt+1] = pf t + g(pt−1 − pf t−1),

g ≥ 1 Model in deviations xt := pt − pf

t ◮ price:

Rxt = nf

t Ef t [xt+1] + nc t Ec t [xt+1] = gnc t xt−1 ◮ fractions:

nc

t+1 = 1

• (1 + exp(β(πf

t − πc t ))) ◮ profit differential: πf t − πc t = Rt −gxt−2 aσ2

t−2 − C

◮ excess return:

Rt = xt − Rxt−1 + (yt − y) + aS

r (Rσ2 t−2 − σ2 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

Two-types model: Fundamentalists vs. Chartists

Rxt = gnc

t xt−1

nc

t+1 =

• 1 + exp
• β
• Rt

−gxt−2 aσ2

t−2 − C

−1 Rt = xt − Rxt−1 + (yt − y) + aS

r (Rσ2 t−2 − σ2 t−1)

σ2

t = (1 − wσ)(Rt − µt)2 + wσσ2 t−1

µt = (1 − wµ)Rt + wµµt−1

◮ Deterministic skeleton, yt ≡ y ◮ Fundamental steady state: x∗ = 0, R∗ = 0, µ∗ = 0, σ2∗ = 0 ◮ Locally stable steady state for all values of β ◮ No non-fundamental steady states

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

Simulations with stochastic dividend, Low Beta: β = 1

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, Low Beta: β = 2

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, High Beta: β = 3.5

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, High Beta: β = 5

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, High Beta: β = 10

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Variance of price vs. variance of fundamental price

Parameters: r = 0.1, S = 0.1, a = 1, g = 1.2, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5 Variance over 1000 periods after 1000 transient periods (one simulation) Beta Dividend Fundamental Price Price β = 1 0.2608 0.0051 0.0033 β = 2 0.2608 0.0052 0.0016 β = 3.5 0.2608 0.0053 0.0202 β = 5 0.2608 0.0058 0.0320 β = 10 0.2608 0.0055 0.0407

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

Simulations with stochastic dividend

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

Simulations with stochastic dividend: Variance

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

Summary of a simple model

◮ dynamics is consistent with EMH for low β

◮ demand for the risky asset falls as price exhibit high fluctuations

◮ dynamics exhibit excess volatility for high β

◮ price deviations

◮ endogenous switches between positive and negative attractors

(without external noise)

◮ dynamics of fundamental price is driven by the dividends and

slightly amplified by the price

◮ large price change (in any direction) leads to a drop of

fundamental 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

Alternative form of the same model

Variance of price estimate σ2

t−1 := ˆ

σ2

pt|It−1 using EWMA:

σ2

t = (1 − wσ)(pt − µt)2 + wσσ2 t−1 ,

µt = (1 − wµ)pt + wµµt−1 Perceived fundamental price: pf

t :=

• y − a S (σ2

y + σ2 t−1)

• /r

Expectations:

◮ fundamentalists:

Ef

t [pt+1] = pf t ◮ chartists:

Ec

t [pt+1] = pf t + g(pt−1 − pf t−1),

g ≥ 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

Bifurcation Diagram

Parameters: r = 0.1, S = 0.1, a = 1, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0

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

Simulations with stochastic dividend, Low Beta: β = 2

Parameters: r = 0.1, S = 0.1, a = 1, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, Middle Beta: β = 2.5

Parameters: r = 0.1, S = 0.1, a = 1, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, Higher Beta: β = 3

Parameters: r = 0.1, S = 0.1, a = 1, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Simulations with stochastic dividend, High Beta: β = 5

Parameters: r = 0.1, S = 0.1, a = 1, wµ = wσ = 0.9 i.i.d. dividends y = 1, σy = 0.5

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

Alternative Dividend Process

AR(1) dividend yt = y + ρ(yt−1 − y) + εt, where εt ∼ N(0, σ2

ε) ◮ Fundamental Price

pf

t =

ρ R − ρyt + R (R − ρ)r

• (1 − ρ)y − a S

R R − ρσ2

ε

• ◮ Perceived Fundamental Price

Ef

t−1[pt] =

ρ R − ρyt + 1 + r (R − ρ)r

• (1 − ρ)y − aσ2

e

• 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

Consequences of the bubble’s burst

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

Summary

◮ simple model with fundamentalists and chartists ◮ agents estimate variance on the past data and update the

perceived fundamental price

◮ creates additional feedback: from past price through volatility to

the fundamental price

◮ after bubble busts, perceived fundamental price may stay lower

than the actual fundamental price for a long period of time

Mikhail Anufriev EDG, University of Technology, Sydney Heterogeneous Beliefs in an Asset Pricing Model with Endogenous Fundamentals