Equilibrium Return and Agents Survival in a Multiperiod Asset - - PowerPoint PPT Presentation

Introduction LLS Simulations Trick Support of Simulations Conclusions

Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Mikhail Anufriev Pietro Dindo

CeNDEF , Faculty of Economics and Business University of Amsterdam Complex Markets Meeting

Marseille 05 October 2006

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Idea of Microscopic Simulations and analytic challenge

Microscopic Simulations

Levy, Levy and Solomon (2000)

Microscopic Simulation of Financial Markets: from investor behavior to market phenomena

◮ ...The great advantage of the MS methodology is that

systems that do not yield analytical treatment can easily be studied.

◮ ...A disadvantage of the MS approach is that it is more

difficult to reach general conclusions from simulations than from analytical results...

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Idea of Microscopic Simulations and analytic challenge

Microscopic Simulations

Levy, Levy and Solomon (2000)

Microscopic Simulation of Financial Markets: from investor behavior to market phenomena

◮ ...The great advantage of the MS methodology is that

systems that do not yield analytical treatment can easily be studied.

◮ ...A disadvantage of the MS approach is that it is more

difficult to reach general conclusions from simulations than from analytical results...

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Idea of Microscopic Simulations and analytic challenge

Microscopic Simulations

Levy, Levy and Solomon (2000)

Microscopic Simulation of Financial Markets: from investor behavior to market phenomena

◮ ...The great advantage of the MS methodology is that

systems that do not yield analytical treatment can easily be studied.

◮ ...A disadvantage of the MS approach is that it is more

difficult to reach general conclusions from simulations than from analytical results...

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Idea of Microscopic Simulations and analytic challenge

Plan of the Talk

◮ Review of the LLS model and their results

◮ Levy, Levy and Solomon (1994, Economic Letters),

Levy and Levy (1996, The Financial Analyst Journal), Zschischang and Lux (2001, Physica A)

◮ Trick allowing the analytical treatment of the model

◮ Anufriev, Bottazzi and Pancotto (2006, Journal of Economic

Dynamics and Control), Anufriev and Bottazzi (2006, CeNDEF WP)

◮ Application of the trick to the LLS model

◮ Anufriev and Dindo (2006, Advances in Artificial Economics) Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Idea of Microscopic Simulations and analytic challenge

Plan of the Talk

◮ Review of the LLS model and their results

◮ Levy, Levy and Solomon (1994, Economic Letters),

Levy and Levy (1996, The Financial Analyst Journal), Zschischang and Lux (2001, Physica A)

◮ Trick allowing the analytical treatment of the model

◮ Anufriev, Bottazzi and Pancotto (2006, Journal of Economic

Dynamics and Control), Anufriev and Bottazzi (2006, CeNDEF WP)

◮ Application of the trick to the LLS model

◮ Anufriev and Dindo (2006, Advances in Artificial Economics) Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LLS model

◮ N agents trade 2 assets in discrete time

◮ stock: price Pt, dividend Dt ◮ bond: price 1, interest rate rf

◮ max expected power utility with risk aversion γ ◮ ex post returns are used to predict next return

ht+1 = Pt+1 Pt − 1 + Dt+1 Pt

◮ each of past L returns can occur with equal probability 1/L ◮ Walrasian market clearing

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LLS model

◮ N agents trade 2 assets in discrete time

◮ stock: price Pt, dividend Dt ◮ bond: price 1, interest rate rf

◮ max expected power utility with risk aversion γ ◮ ex post returns are used to predict next return

ht+1 = Pt+1 Pt − 1 + Dt+1 Pt

◮ each of past L returns can occur with equal probability 1/L ◮ Walrasian market clearing

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LLS model

◮ N agents trade 2 assets in discrete time

◮ stock: price Pt, dividend Dt ◮ bond: price 1, interest rate rf

◮ max expected power utility with risk aversion γ ◮ ex post returns are used to predict next return

ht+1 = Pt+1 Pt − 1 + Dt+1 Pt = rt+1 + yt+1

◮ each of past L returns can occur with equal probability 1/L ◮ Walrasian market clearing

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

Important features of the LLS model

◮ share of investor’s wealth invested to the risky asset is

independent of his wealth

◮ price and wealth are determined simultaneously and

endogenously ⇓

◮ “natural” dynamics is constant price growth

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

Important features of the LLS model

◮ share of investor’s wealth invested to the risky asset is

independent of his wealth

◮ price and wealth are determined simultaneously and

endogenously ⇓

◮ “natural” dynamics is constant price growth

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LL: Price with homogeneous investors

L = 15 for all investors, γ = 1

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LL: Price with heterogeneous investors

0 < L < 30 for half of investors, γ = 1

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

LPS: Survival of investors with high memory span

three groups: (L, γ) = (256, 1), (141, 1), (10, 1)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

ZL: Sensitivity to initial conditions

three groups: (L, γ) = (256, 1), (141, 1), (10, 1)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Model of Levy, Levy and Solomon and its simulations

ZL: Interplay of memory span L and risk aversion γ

(256, 0.4), (141, 0.6), (10, 0.4) and (256, 0.6), (141, 0.4), (10, 0.4)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Property of power utility of wealth

share of invested wealth xt,n does not depend on the current wealth of agent

◮ investment function

xt,n = fn(It) = fn(ht−1, ht−2, . . . , ht−L)

◮

xt ht-1 ht-2 xt

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Property of power utility of wealth

share of invested wealth xt,n does not depend on the current wealth of agent

◮ investment function

xt,n = fn(It) = fn(ht−1, ht−2, . . . , ht−L)

◮

deterministic system in terms of

◮ investment shares, xt,n ◮ price return, rt+1 = Pt+1/Pt − 1 ◮ dividend yield, yt+1 = Dt+1/Pt ◮ agents’ relative wealth shares in the aggregate wealth,

ϕt,n = Wt,n

m

Wt,m

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Property of power utility of wealth

share of invested wealth xt,n does not depend on the current wealth of agent

◮ investment function

xt,n = fn(It) = fn(ht−1, ht−2, . . . , ht−L)

◮

deterministic system in terms of

◮ investment shares, xt,n ◮ price return, rt+1 = Pt+1/Pt − 1 ◮ dividend yield, yt+1 = Dt+1/Pt ◮ agents’ relative wealth shares in the aggregate wealth,

ϕt,n = Wt,n

m

Wt,m

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Existence of equilibria

Specification of dividend process: Dt+1 = Dt (1 + g + ǫt+1)

◮ If g ≤ rf, then price return is rf and dividend yield is 0.

See Levy, Levy, Solomon (1994) with g = 0

◮ If g > rf, then price return is g, all survivors invest the same

share x∗

⋄, which together with dividend yield y∗ satisfies

g − rf y∗ = x∗

⋄

1 − x∗

⋄

. geometry: x∗

⋄ = (g − rf)

• (y∗ + g − rf) (EML)

...and: x∗

⋄ = fn(g + y∗, g + y∗, . . . , g + y∗)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Existence of equilibria

Specification of dividend process: Dt+1 = Dt (1 + g + ǫt+1)

◮ If g ≤ rf, then price return is rf and dividend yield is 0.

See Levy, Levy, Solomon (1994) with g = 0

◮ If g > rf, then price return is g, all survivors invest the same

share x∗

⋄, which together with dividend yield y∗ satisfies

g − rf y∗ = x∗

⋄

1 − x∗

⋄

. geometry: x∗

⋄ = (g − rf)

• (y∗ + g − rf) (EML)

...and: x∗

⋄ = fn(g + y∗, g + y∗, . . . , g + y∗)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Existence of equilibria

Specification of dividend process: Dt+1 = Dt (1 + g + ǫt+1)

◮ If g ≤ rf, then price return is rf and dividend yield is 0.

See Levy, Levy, Solomon (1994) with g = 0

◮ If g > rf, then price return is g, all survivors invest the same

share x∗

⋄, which together with dividend yield y∗ satisfies

g − rf y∗ = x∗

⋄

1 − x∗

⋄

. geometry: x∗

⋄ = (g − rf)

• (y∗ + g − rf) (EML)

...and: x∗

⋄ = fn(g + y∗, g + y∗, . . . , g + y∗)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Existence of equilibria

Specification of dividend process: Dt+1 = Dt (1 + g + ǫt+1)

◮ If g ≤ rf, then price return is rf and dividend yield is 0.

See Levy, Levy, Solomon (1994) with g = 0

◮ If g > rf, then price return is g, all survivors invest the same

share x∗

⋄, which together with dividend yield y∗ satisfies

g − rf y∗ = x∗

⋄

1 − x∗

⋄

. geometry: x∗

⋄ = (g − rf)

• (y∗ + g − rf) (EML)

...and: x∗

⋄ = fn(g + y∗, g + y∗, . . . , g + y∗)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Existence of equilibria

Specification of dividend process: Dt+1 = Dt (1 + g + ǫt+1)

◮ If g ≤ rf, then price return is rf and dividend yield is 0.

See Levy, Levy, Solomon (1994) with g = 0

◮ If g > rf, then price return is g, all survivors invest the same

share x∗

⋄, which together with dividend yield y∗ satisfies

g − rf y∗ = x∗

⋄

1 − x∗

⋄

. geometry: x∗

⋄ = (g − rf)

• (y∗ + g − rf) (EML)

...and: x∗

⋄ = fn(g + y∗, g + y∗, . . . , g + y∗)

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Equilibrium Market Line

Investment Share Dividend Yield EML 1 S UII UI I II

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Stability of equilibria (g > rf)

The equilibrium, where M agents survive, is stable if and only if

• 1. the equilibrium investment shares of the non-surviving

agents are such that x∗

⋄

• 1 − 2(1 + g)/(g − rf)
• < x∗

m < x∗ ⋄

• 2. after eliminating all non-surviving agents, the behavior of

survivors generates stable dynamics ...that is with:

◮ lower (in absolute value) relative slope of the investment

function and the EML, f ′/l′, at the equilibrium

◮ higher memory span L Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Analytical results: Stability of equilibria (g > rf)

The equilibrium, where M agents survive, is stable if and only if

• 1. the equilibrium investment shares of the non-surviving

agents are such that x∗

⋄

• 1 − 2(1 + g)/(g − rf)
• < x∗

m < x∗ ⋄

• 2. after eliminating all non-surviving agents, the behavior of

survivors generates stable dynamics ...that is with:

◮ lower (in absolute value) relative slope of the investment

function and the EML, f ′/l′, at the equilibrium

◮ higher memory span L Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

1st Condition on the Equilibrium Market Line

Investment Share Dividend Yield EML 1 S UII UI I II

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Summary of Analytical Results

• 1. equilibria are located on the 1−D curve, the EML
• 2. agents with low risk aversion γ can “invade” the market
• 3. agents with high memory span L can “survive” in the stable

equilibrium

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Summary of Analytical Results

• 1. equilibria are located on the 1−D curve, the EML
• 2. agents with low risk aversion γ can “invade” the market
• 3. agents with high memory span L can “survive” in the stable

equilibrium

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions How to overcome an obstacle

Summary of Analytical Results

• 1. equilibria are located on the 1−D curve, the EML
• 2. agents with low risk aversion γ can “invade” the market
• 3. agents with high memory span L can “survive” in the stable

equilibrium

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Fresh view on the LLS simulations

Application 1: one agent

Investment Share Dividend Yield Aα EML 1 10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 50 100 150 200 250 Price Time L=10 L=20

Levy and Levy, 1996

◮ for small L: equilibrium is unstable fluctuations ◮ for high L: “erratic” fluctuations due to random dividend

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Fresh view on the LLS simulations

Application 1: one agent

Investment Share Dividend Yield Aα EML 1 10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 50 100 150 200 250 Price Time L=10 L=20

Levy and Levy, 1996

◮ for small L: equilibrium is unstable fluctuations ◮ for high L: “erratic” fluctuations due to random dividend

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Fresh view on the LLS simulations

Application 2: two agents (α′ < α)

Investment Share Dividend Yield Aα Aα’ EML 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Wealth share Time L’=20 L’=30

Zschischang and Lux, 2001

◮ “Looking [...] at the interplay of risk aversion and memory span,

it seems to us that the former is the more relevant factor [...]: groups which were fading away before became dominant when we reduced their degree of risk aversion.”

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Fresh view on the LLS simulations

Application 2: two agents (α′ < α)

Investment Share Dividend Yield Aα Aα’ EML 1 0.1 1 10 100 1000 10000 100000 1e+06 1e+07 50 100 150 200 250 Price Time L’=20 L’=30

Zschischang and Lux, 2001

◮ “It also appears that when adding different degrees of risk

aversion, the differences of time horizons are not decisive any more, provided the time horizon is not too short”

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Final remarks

Conclusions

◮ simulations of the LLS model are explained ◮ agents with constant shares do not necessary dominate

(cf. Zschischang and Lux, 2001)

◮ existence of the equity premium depends on the relation

between the dividend growth rate g and the risk-free interest rf

◮ equilibria are located in the one-dimensional curve, the

Equilibrium Market Line

◮ the stability of the equilibria is related to the memory span

and risk aversion of the agents

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model

Introduction LLS Simulations Trick Support of Simulations Conclusions Final remarks

Conclusions

◮ simulations of the LLS model are explained ◮ agents with constant shares do not necessary dominate

(cf. Zschischang and Lux, 2001)

◮ existence of the equity premium depends on the relation

between the dividend growth rate g and the risk-free interest rf

◮ equilibria are located in the one-dimensional curve, the

Equilibrium Market Line

◮ the stability of the equilibria is related to the memory span

and risk aversion of the agents

Mikhail Anufriev, Pietro Dindo CeNDEF , University of Amsterdam Equilibrium Return and Agents’ Survival in a Multiperiod Asset Market: Analytic Support of a Simulation Model