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Mikhail Anoufriev∗
Asset Pricing Modelwith
Heterogeneous Time Horizons
(Joint work with Giulio Bottazzi∗)
∗S.Anna School of Advanced Studies, Pisa, Italy
Sant’Anna School of Advanced Studies
Pisa, 2 December 2003
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Plan of the talk
• Capital Asset Pricing Model
• Agent-Based Models
• Investment Horizons
• The Model
• Conclusion
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1. Capital Asset Pricing Model
What is it about?
Goal: to ”explain” the asset prices
Result: in equilibriumfor any risky asseti
E[ρi] − R = βi(E[ρm] − R)
whereρm is the return ofmarket portfolio
Consequence:separation theorem
Consequence:all agents have the samemarket portfolio, i.e. the same share of anyrisky asset among other risky assets
Model is equilibrium, static and ”homogeneous”
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1.1. Assumptions
Market
• asset market is complete and competitive• assets are divisible• unlimited short sales are allowed• no taxes• no transaction costs
Agents’ Expectations
• homogeneous expectations: all agents share a common probability distribu-tion of returns for risky assets
• common investment planning horizon
Agents’ Behavior
• mean-variance utility functions express the preferences of investors• agents hold the portfolio during one period
Remark
Equilibrium structure is changing with
• information
• investors’ preferences
• investors’ endowments
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1.2. Intertemporal CAPM
Contributions
• Merton, 1973
• LeRoy, 1973
• Lucas, 1978
Features
• endogenous expectations
• price function:asset price is a function of past realization of returns
• assumptions are even more restrictive
For exampleLeRoy assumes
• utility function with constant absolute risk aversion
• all investors are identical
• each investor can make an unbiased forecast of the price for the next period, inother wordsprice functions derived under hypothesis of rational expectations
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1.3. Summary of theory evolution
• Mean-Variance Analysis: Markovitz (1956, 1959), Tobin (1958)
• CAP model: Sharpe (1964), Lintner (1965), Mossin (1966)
• Intertemporal CAPM: Merton (1973)
• Rational Expectations as a micro-foundation: LeRoy (1973), Lucas (1978)
• Agent-Based Models
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2. Agent-Based Modeling
Driving forces (LeBaron, 2000)
• unsatisfactory assumptions of classical models
• empirical puzzles
– equity premium puzzle
– excess volatility
– persistence in volatility
– using of technical trading
• availability of financial data
• findings in experimental financial markets
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Common ingredients
• bottom-upstructure: from interaction of heterogeneous agents to properties ofmarket dynamics
• both ”fundamentalists” and ”irrational” agents:
noise traders, chartists, ZI traders
• generation of non-trivial price dynamics: cyclical or chaotic
Examples
• simulations:
– Gode and Sunder, 1993– Arthur et al., 1997– Chiaromonte, Dosi and Jonard, 1999– Levy, Levy and Solomon, 2000– Kirman and Teyssiere, 2000– Bottazzi, Dosi and Rebesco, 2002
• analytical studies:
– Grossman and Stiglitz, 1980– De Long et al., 1991– Brock and Hommes, 1998– Chiarella and He, 2002
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What we are interesting in here?
• what is the ”minimal” sufficient source of heterogeneity
• coexistence of heterogeneous traders in CAPM framework
– agents can differ in predictors
– agents can differ in preferences (for example, they might have different riskaversion coefficients)
– agents can differ in investment horizons
– agents can differ in wealth dynamics
• role of assumption about utility function
• role of market architecture
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Key references
• Brock, W.A. and C.H. Hommes, (1998). “Heterogeneous Beliefs and Routes toChaos in a Simple Asset Pricing Model”,Journal of Economic Dynamics andControl 22, 1235-1274
• Bottazzi, G. (2002). “A Simple Micro-Model of Market Dynamics. Part I: the”Large Market” Deterministic Limit”,L.E.M. Working Paper, Sant’Anna Schoolfor Advanced Studies.
Differences among them
In Bottazzi (2002)
1. Agents make prediction aboutreturns, not about prices
2. Agents are heterogeneous in the estimation of risk
3. Agents do not change own type
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3. Investment Horizons
Hypothesis
The investment decision of a trader strongly depends on the time horizon on which hisjudgment about the profit and risk of a given investment is based
Agent-based literature
• Agents are ”myopic”, i.e. they participate to the market in order to maximize thewealth at the next period
• Do the results of models change if one introduces additional heterogeneity in termsof investment time horizons?
• Is it enough to introduce heterogeneity at the level of investment time horizons tobreak the predictions of standard representative-agent models?
Classical literature
• Merton, but there is no dynamics indeed...
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4. The Model
4.1. Goals of the paper
• Search for the ”minimal” heterogeneity to generate non-trivial price dynamics
• Qualitative consequences of the introduction of heterogeneity in time horizons
4.2. Conclusions
• Heterogeneity only in investment horizons is not enough
• In general, price dynamics is strongly affected by investment horizons
• Dynamics does depend on the way how agents estimate risk
• Overestimation of risk by fundamentalists leads to unstability of the system
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 5 10 15 20 25 30 35 40 45 50
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4.3. Model setup
Bond B
• price1
• constant interest rate0 < R < 1
Stock A
• price pt return ρt,t+η = (pt+η − pt)/pt
• constant dividendD
Timing
• beginning of periodt
• agenti has Ai,t−1, Bi,t−1
• agent makes predictions Eit [ρt,t+ηi
] and V it [ρt,t+ηi
]
• agent maximizes expected utility Eit [Wi,t+ηi
] − βi
2V i
t [Wi,t+ηi]
• the demand function of the agenti ∆Ai,t(p) = −Ai,t−1 + Ai,t(p)
• the market clearing condition is∑N
i=1 ∆Ai,t(p) = 0
• solution is price pt
• now agenti has Ai,t, Bi,t
• dividendD and interest rateR are paid
• end of periodt
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4.4. Modeling different time horizons
Agent with a time horizonη > 0 periods
1. – does not correct the portfolio between timet andt + η
2. – participates in the market at each period
– continuously corrects the portfolio composition
– takes into account the fact that the portfolio will be revised each period
3. – participates in the market at each period
– continuously corrects the portfolio composition
– does not take into account the fact that the portfolio will be revised
– so he maximizes at periodt his expected wealth at periodt + η
Wt+η = (1 − xt)Wt(1 + R)η + xtWt
(1 + ρt,t+η +
p((1 + R)η − 1)
pt
)xt share of wealth invested in the risky asset
p = D/R fundamental price
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Thus
• demand for the stock
Ai,t(p) =Ei
t [ρt,t+ηi] + ((1 + R)ηi − 1)( p
p− 1)
βip V it [ρt,t+ηi
]
• the market clearing condition∑N
i=1 ∆Ai,t(p) = 0
• the market clearing condition A = 〈Ai,t(p)〉i A = ATOT /N
• new price of the risky assetpt is a solution of
p2 A − p
⟨Ei
t [ρt,t+ηi] − Ri
βiV it [ρt,t+ηi
]
⟩i
−
⟨pRi
βiV it [ρt,t+ηi
]
⟩i
= 0
whereRi = (1 + R)ηi − 1 > 0
– non-linear pricing equation (cf. Brock-Hommes model)
– unique positive root
– in the situation of complete certainty, the only solution isp
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4.5. Two classes of agents
Chartists behave as econometricians
• λ ∈ [0, 1) – length of the agent’s memory
• one period forecast
Ect [ρt,t+1] = RMA
t−1 = (1 − λ)∑∞
τ=2 λτ−2ρt−τ
V ct [ρt,t+1] = V MA
t−1 = (1 − λ)∑∞
τ=2 λτ−2[ρt−τ − RMAt−1]
2
• recursive relation
RMAt−1 = λRMA
t−2 + (1 − λ)ρt−2
V MAt−1 = λV MA
t−2 + λ(1 − λ)(ρt−2 − RMAt−2)
2
• long-term forecast
Ect [ρt,t+η] = η Ec
t [ρt,t+1] = η RMAt−1
V ct [ρt,t+η] = η V c
t [ρt,t+1] = η V MAt−1
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Fundamentalists believe in mean-reverting dynamics
• θ ∈ [0, 1] – perception of the reactivity of market
• one period price forecast
Eft [pt+1] = pt + θ(p − pt)
• long-term return forecast
Eft [ρt,t+η] =
(1 − (1 − θ)η
)( p
pt
− 1)
• long-term expected variance
V ft [ρt,t+η] = ηV f
t [ρt,t+1] = ηV MA
t−1
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4.6. Fundamentalists vs. Chartists I
• f1 and f2 are the fractions of fundamentalists and chartists
• all investors have the same time horizon 1
• the market clearing condition
βA p2 V MA
t−1 + p(R + f1θ − f2 RMA
t−1
)− p
(R + f1θ
)= 0
• parameters and notation:
γ = βA/f2 x(t) = γpt
r = (R + f1θ)/f2 y(t) = RMAt
d = rp = (D + f1θp)/f2 z(t) = V MAt
s = dγ
• 3-dimensional systemx(t + 1) =
(y(t) − r +
√(y(t) − r)2 + 4sz(t)
)/2z(t)
y(t + 1) = λy(t) + (1 − λ)(
g(y(t),z(t))x(t)
− 1)
z(t + 1) = λz(t) + λ(1 − λ)(
g(y(t),z(t))x(t)
− 1 − y(t))2
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4.7. The local stability of the system
• system has only one fixed point(γp, 0, 0).
• necessary and sufficient condition for the local stability of this point:
r + λ > 1 r = (R + f1θ)/f2
• system has Hopf bifurcation
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4.8. Fundamentalists vs. Chartists II
• f1 and f2 are the fractions of fundamentalists and chartists
• η1 and η2 are the time horizons of fundamentalists and chartists
• parameter
r =1
η1
f1
f2
(θ + R)Bη1(R, θ) +1
η2
RBη2(R, 0)
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0 5 10 15 20 25 30 35 40
with η increasing the function increases exponentially to the infinity
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5. Discussion
• convergence to the fixed point comes with
– increasing of the length of the agents’ memoryλ
– increasing of the share of fundamentalists on the marketf1
– increasing of the risk-less returnR
– increasing of the perception of fundamentalists about the efficiency of the marketθ
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• price dynamics is strongly affected by investment horizons
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 20 40 60 80 100 120 140 160 180 200
Parameters areR = 0.05, D = 1.0, A = 2, β = 1, λ = 0.9, f1 = 0.2, θ = 0.4, η2 = 1. The initialconditions arep = 1, RMA = 0.01, V MA = 0.0001.
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• effect of increasing of time horizonsη1 andη2 is asymmetric
10
20
30
40
50
60
η1
2 4
6 8
10 12
14
η2
0.08 0.09
0.1 0.11 0.12 0.13 0.14 0.15 0.16
Parameterr as a function ofη1 andη2. The values of other parameters areR = 0.05, f1 = 0.2,θ = 0.3. The curve on the horizontal space confines the closed region in space(η1, η2) where the
fundamental price is unstable fixed point.
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• the moderate time horizons of fundamentalists can bring the system to chaotic behavior.
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 5 10 15 20 25 30 35 40 45 50
Bifurcation diagram. The price support of a1000 steps orbit (after a1000 steps transient) isshown for50 distinct values ofη1 from 1 to 50.
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6. Conclusion
• Heterogeneity only in investment horizons is not enough
• In general, price dynamics is strongly affected by investment horizons
• Different types of population contribute in different way
• Dynamics does depend on the way how agents estimate risk
• Overestimation of risk by fundamentalists leads to unstability of the system
