Asset Pricing under Heterogeneous Expectations and Risk Aversion

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and Hommes (1998) and relax hypotheses of no supply of risky asset and a normalized risk aversion, using two types of expectations, fundamentalist.
Asset Pricing under Heterogeneous Expectations and Risk Aversion Cristiana Benedetti Fasil ‡ and Marji Lines† ‡

European University - Fiesole, † University of Udine

Abstract In this paper we reconsider the asset pricing model introduced in Brock and Hommes (1998) and relax hypotheses of no supply of risky asset and a normalized risk aversion, using two types of expectations, fundamentalist and trend-follower. The consequent changes in price behavior and composition of agents in the market are studied through local bifurcation analysis with reference to the standard scenarios of the original model. It is shown that the conditional variance beliefs activated by positive supply may give rise to a “volatility sunspot”. The model with differentiated risk aversion suggests that in a market composed of strategies reflecting different risk attitudes, the majority strategy is that held by the most risk averse. For intermediate parameter values the market is heterogenous even in the steady state, as in BH. However the majority gets stronger the greater the difference in risk aversion, the larger the supply of risky asset, the higher the conditional variance beliefs and, in extreme cases, the less risk averse are driven entirely from the market.

1 Introduction Brock and Hommes (1998, hereafter BH) were among the first to apply local bifurcation theory to an analytical framework in which heterogeneous agents adapt their beliefs over time. Some extensions of their approach of investigating dynamics of asset-pricing models through bifurcation theory are by Gaunersdorfer (2000 and 2001), Hommes (2001 and 2005), Chiarella and He (2002) and Diks and Van der Weide (2003). Gaunersdorfer (2000) remarks the crucial role played by β, the strategy switching parameter called the intensity of choice, in the bifurcation route to chaotic price fluctuations. In particular, for large values of β the limit set is close to a homoclinic orbit which is a common feature of a system with chaotic dynamics. In Gaunersdorfer (2001) the agents’ choice is not only based upon an evolutionary measure of performance but also on market conditions. She finds volatility clustering in the model due to intermittency as trajectories have chaotic price fluctuations mixed with intervals of almost periodic 1

cycles. She also finds coexistence of strange attractors and the locally stable fundamental steady state. Empirical support for the model’s validity is given in the comparison of the statistical properties of the simulated model with time series of German, British and Austrian stock markets. Hommes (2001) captures the risk taken by the agents, before the investment decision, by adjusting the performance measure with the conditional variance. The resulting dynamics of the model and the generated time series are close to those shown by BH. Chiarella and He (2002) depart from BH’s approach as they study the impact of learning schemes with different lag lengths in the formation of expectations. Diks and Van der Weide (2003) generalize the discrete choice model introduced by BH with a continuous choice model in which beliefs are continuous variables. The beliefs distribution is a deterministic function of the observed past variables and agents’ strategies are modelled as random variables. This generalization models the joint dynamics of the beliefs distribution and the observable prices, introducing an endogenous uncertainty. Hommes, Huang and Wang (2005) add an element of real trading to BH by including a market maker into the model with only two types of traders, fundamentalists and trend followers. Once again the bifurcation analysis centers on β, as the parameter increases the fundamental steady state loses stability, non-fundamental steady states become stable then lose stability, a homoclinic bifurcation of the stable and unstable manifolds of the fundamental steady state is followed by chaotic dynamics. Another recent and independent model with a market maker is due to Farmer and Shareen (2002). They study the dynamics of individual trading strategies modelled as signal processing problems for which noise and price autocorrelation are elements. They point out the essential role of the market maker: a risk neutral market maker leads to problems of equilibrium stability which can be overcome by assuming a risk averse market maker. In our extension of BH we relax the assumptions of a supply of risky asset equal to zero and a normalized risk aversion parameter for all trader types. We explore the consequent changes in the model’s dynamical structure through local bifurcation analysis. The positive supply of outside shares per trader and differentiated risk aversion parameters for different types of traders potentially affect the fundamental price, realized profits (in the performance measure) and fraction of agents using a given strategy which, in turn, affect the type and stability of price fluctuation dynamics. The paper is organized as follows. The next section presents the asset pricing model under the assumption of a positive supply of risky asset. In Section 3 we study how the conditional variance beliefs and positive supply affect the asymptotical dynamical behavior of the model through numerical simulations and we introduce the notion of a “volatility sunspot”. In section 4 analytical results are given for the model with differentiated risk aversion. Numerical simulations of 2

this extended model are presented in section 5 and confirm that the majority strategy in heterogenous markets is that held by the most risk averse. In section 6 we conclude.

2 Asset pricing model In this section we briefly describe the main features of the asset pricing model in Brock and Hommes (1998) (henceforth referred to as BH) and our extensions. The key characteristic of their adaptive belief system framework is the heterogeneity of the agents with respect to their expectations of future returns. In the following we consider only two types of agents: fundamentalists who believe that the asset price is always on its fundamental value and trend followers who believe that the asset price will increase or decrease at a constant rate. The financial market consists of two investment choices: a risk-free asset and a risky asset. The former has an elastic supply and is available with gross return1 R > 1, the latter instead pays an uncertain dividend. The price per share (ex dividend) of the risky asset at time t is pt and {yt } is the stochastic dividend process of the risky asset which is assumed to be iid. The dynamics of wealth (denoted by W ) is described by Wt+1 = RWt + (pt+1 + yt+1 − Rpt )zt ,

(1)

where zt is the number of purchased shares of the risky asset at time t and Rt+1 = (pt+1 + yt+1 − Rpt ) is the excess return per share. The conditional expectation and the conditional variance operators at time t, denoted as Et and Vt , are based on a publicly available information set of past prices and dividends, It = {pt , pt−1 , . . . , yt , yt−1 , . . .}. Let Eht and Vht be the beliefs or forecast of investor type h regarding the conditional expectation and variance. The investor is assumed to be a myopic mean variance maximizer so that the demand for the risky asset by type h, zht , solves a max U = Eht [Wt+1 ] − Vht [Wt+1 ], zht 2

(2)

where a ≥ 0 represents the risk aversion parameter. In BH this parameter is assumed to be equal for all agents. In the third section we develop the model under the assumption of differentiated risk aversion parameters for different types of agents. The gross return R is simply given by 1 + r, where r is the fixed rate of return of the risk-free asset. 1

3

Solving problem (2) under the assumption in BH that the beliefs about the conditional variance are constant over time and the same for all investors: Vht = σ 2 , gives zht =

Eht [pt+1 + yt+1 − Rpt ] Eht [pt+1 + yt+1 − Rpt ] = . aVht [pt+1 + yt+1 − Rpt ] aσ 2

(3)

The supply of risky asset and the fraction of investor of type h at time t are denoted, respectively, by zst and nht . Equilibrium of demand and supply implies P h nht zht = zst . As assumed also in Hommes, Huang and Wang (2005), henceforth referred to as HHW, we develop the model with a fixed and constant supply over time, so the equilibrium equation becomes H X h=1

nht

Eht [pt+1 + yt+1 − Rpt ] = zs , aσ 2

(4)

where H is the number of different trader types. The market equilibrium pricing equation is then H X

nht Eht [pt+1 + yt+1 ] − zs aσ 2 = Rpt .

(5)

h=1

The quantity zst aσ 2 can be interpreted as the risk premium investors require for holding a risky asset. This equation differs from the basic framework in BH in which the equilibrium equation is independent of the the risk premium term (since zst = 0). We can then explore how the risk premium and, in particular, the conditional variance beliefs affect the system’s long run dynamics. Define the variable giving deviation from the fundamental price, xt = pt − p∗t , where p∗t is the fundamental price2 . BH assume that the beliefs of the trader type h are described by the following form Eht = (pt+1 + yt+1 ) = Et (p∗t+1 + yt+1 ) + fh [xt−1 , . . . , xt−L ]

(6)

that is, the dividend expectations are the same for all investors but agents believe that the price can deviate from its fundamental value by a deterministic function fh , which may differ among traders types. This function, fht = fh [xt−1 , . . . , xt−L ], depends upon the past deviation from the fundamental and represents the linear forecasting strategy used by trader type h: Eht [xt+1 ] = fht = gh xt−1 + bh . 2

(7)

BH and HHW develop the model under the efficient market hypothesis, with only rational P∞ 2 2 agents, and derive the fundamental price p∗ = k=1 y−zRskaσ = y−zrs aσ .

4

We restrict our analysis to a typical case with fundamentalists, i.e f1t = 0, and trend followers, i.e. f2t = gxt−1 (see BH for other cases). Then the equilibrium market equation can be rewritten in terms of deviations from the fundamental price: Rxt =

H X

nht Eht [xt+1 ] =

h=1

H X

nht fht =

h=1

H X

nht gh xt−1 + bh .

(8)

h=1

The investor beliefs are updated over time and this updating determines the dynamics of the fractions nht of different trader types. The fractions’ evolutionary dynamics are based on a fitness or performance measure, but traders are only boundedly rational, in the sense that most traders choose the strategy that has performed well in the recent past. The fitness functions are assumed to be publicly eht = Uht + εht where available and are a component of the random utility model, U Uht is a deterministic part and εht is an iid noise. Fitness in BH is given by a weighted average of the realized profits of investor type h, that is, Uht = (pt + yt − Rpt−1 )

Eh,t−1 [pt + yt − Rpt−1 ] − Ch . aσ 2

(9)

The first term represents the last period’s realized profit of type h. The parameter Ch ≥ 0 is an average cost per period necessary to use forecasting strategy h. The fundamentalists are assumed to pay a positive cost because they must spend time and resources to collect enough information to determine the fundamental price. The fitness Uht is the observed realized profit. The performance measure can also be expressed in deviation from the fundamental price (see Benedetti Fasil (2005), henceforth referred to as BF, for these and other results): Uht = (xt − Rxt−1 + aσ 2 zs )

fh,t−1 − Rxt−1 + aσ 2 zs − Ch . aσ 2

(10)

The updated agents’ fractions are determined by a logit discrete choice probability function that allows a gradual transition between the agents’ fractions and therefore also between the different strategies, that is, exp(βUh,t−1 ) nht = Zt−1

Zt−1 =

H X

exp(βUh,t−1 ),

(11)

h=1

where Zt−1 is a normalization factor. The parameter β ≥ 0 is called the intensity of choice. It measures how fast agents switch between different predictors, and can be interpreted as a measure of traders’ rationality. The higher β, the more rational are the agents in choosing their strategies. 5

The complete evolutionary ABS in deviation from the fundamental price, with fundamentalists (agents type 1) and trend followers (agents type 2), can be summarized as follows: Rxt =

2 X

nht fht = n2,t−1 gxt−1 ,

(12)

h=1

1 (Rxt−2 − aσ 2 zs )(Rxt−2 − xt−1 − aσ 2 zs ) − C aσ 2 1 U2,t−1 = 2 (xt−1 − Rxt−2 + aσ 2 zs )(gxt−3 − Rxt−2 + aσ 2 zs ) aσ ¤ £ 1 exp β( aσ2 (Rxt−2 − aσ 2 zs )(Rxt−2 − xt−1 − aσ 2 zs ) − C) n1t = Zt−1 n2t = 1 − n1t . U1,t−1 =

(13) (14) (15) (16)

The dynamical properties of the model are more easily investigated by introducing the difference in fractions variable mt = n1t − n2t : à ! i βh 1 mt = tanh − 2 gxt−3 (xt−1 − Rxt−2 + aσ 2 zs ) − C . (17) 2 aσ Notice that the positive supply of risky asset and in general the risk premium affect the performance measure and thus affect the updating of the agents’ fractions which determine the equilibrium price and the dynamical behavior of the system. Finally, although the model with positive supply is more general than the model developed in BH it can be shown (see BF, Appendix A) that the fundamental steady state is the same as that of the original model à ! ³ Cβ ´ eq E1 = (x, m) = (0, m ) = 0, tanh − . (18) 2 However, the non-fundamental steady states are altered: E2 = (x∗2 , m∗ ) and E3 = (x∗3 , m∗ ), where m∗ = 1 − 2 Rg and x∗1 and x∗2 are, respectively, the positive and negative solutions of µ ¶ β ∗ ∗ 2 2 −1 ∗ m = tanh [g(R − 1)(x ) (aσ ) − zs x − C] . 2 Note that the solutions, x∗2 and x∗3 , do not have the same modulus. This possibility (excluded in BH where x∗3 = −x∗2 ) implies that the presence of a positive supply of outside shares determines asymmetry in the model with respect to deviations from the fundamental price (see also the market maker model in HHW). 6

3 Conditional variance and positive supply In their numerical simulations BH normalize the risk premium so that aσ 2 = 1. In what follows we use numerical simulations to investigate how the conditional variance beliefs affect the long run dynamical behavior of the model. (Since both risk aversion and variance are positive and always appear as a product they have the same affect.) In order to compare the model with a positive supply of outside shares directly to the model studied in BH we use the same standard set of parameter values R = 1.1, g = 1.20, w = 0, C = 1 and we set the long run behavior as iterations 5000-6000. Note that the parameter trend g satisfies the inequality R < g < 2R. The lower bound g < R means that the trend followers believe that the asset prices grow faster than the interest rate of the risk-free asset. The upper bound g < 2R ensures bounded dynamics. HHW argue that without this upper bound the system may fall into a bubble solution with trend chasers earning higher profits and the asset price diverging to infinity. We begin by defining a basic bifurcation scenario of the model in BH which will be helpful for discussing the similarities and differences between it and the extended versions. In Figure 13 we provide bifurcation diagrams in price deviations for the same parametric constellation as that used in Figure 3 of BH, over an extended interval of β and for both positive initial conditions (left) and negative initial conditions (right). The basic scenario increasing β from zero is the following: fundamental steady state E1 attracts for low β and loses stability through a pitchfork bifurcation; nonfundamental state E2 or E3 attracts over the next interval, with ever larger deviations, only to lose stability through a supercritical Neimark-Sacker bifurcation (referred to as NS for expediency in what follows); quasi-periodic attractors lying on invariant curves appear and increase in radius and change in form with distance from the critical value until they eventually break down into strange attractors; the bounds of the chaotic dynamics increase until extreme fluctuations are exhausted and the fundamental steady state E1 regains stability and holds it for all higher values of β.

3.1 Zero outside shares Consider first, the long run behavior of the price deviation, xt−1 , as a function of the conditional variance beliefs, σ 2 . The first simulations assume, as in BH, a 3

All figures produced www.dss.uniud.it/nonlinear.

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Figure 1: Bifurcation diagrams no supply: left, positive initial conditions; right, negative initial conditions. supply of outside shares zs at zero. Figure 2 presents two bifurcation diagrams obtained for positive initial conditions4 for deviations from fundamental, but for different intensity of choice: β = 3.4 and β = 3.7 in the left and in the right diagram, respectively. We set a = 1, so that the limit set observed in Figure 1 left, at β = 3.4 [at β = 3.7] is the same observed in Figure 2, left [right] at σ = 1. In general, both show that the higher the conditional variance the higher the price deviations. This result occurs for all the intermediate values of the intensity of choice. In other words, if the agents switch fast between the two trading strategies and they expect increasing conditional variance then the price will persistently deviate from its fundamental values. We call this a second moment or “volatility sunspot”, if agents expect high variability in the risky asset price, it will actually exhibit this high variability and, not surprisingly, the degree of variability increases with increasing beliefs σ 2 and increasing intensity of choice β. For these values of β, the original model has a quasi-periodic attractor and strange attractor respectively. Overall, the wide range of conditional variance beliefs does not change the attractor type with zero supply of the risky asset: for β = 3.4 only quasi-periodic attractors appear while for β = 3.7 the attractor begins with positive radius and appears to have chaotic dynamics from the start. Differences in types of fluctuating limit sets are more evident in the the fraction of agents variable. For β = 3.4 the fraction of fundamentalist attractors form 4

If the initial conditions are negative the bifurcation diagrams are analogous at those presented in Figure 2 but with negative price deviations.

8

a solid band, and trajectories also suggest quasi-periodic motion. For β = 3.7 the fraction has an uneven distribution typical of chaotic attractors. Moreover these two bifurcation diagrams suggest that the long run behavior of n1 is independent of the conditional variance. However, for higher β (that is, the faster agents switch between the two forecasting strategies) the higher is the variability in the fundamentalists fraction. In Figure 2, bottom left, n1 varies over a small range, while in bottom right, it varies over a much wider range. In the first case the market is composed prevalently of trend followers and the bounded switching between the two fractions drives bounded price deviations. In the second case the market composition is very changeable. There are phases characterized by mostly trend followers that follow phases characterized by mostly fundamentalists. The complicated dynamic in the agents’ fractions drives the wider price fluctuations observable in Figure 2.

3.2 Positive supply The positive supply of risky asset introduces asymmetry in the dynamics with respect to initial conditions. The parameter values for the bifurcation diagrams in Figure 3 are the same as before except zs = 0.1. We have price deviations (top) and fraction of fundamentalists (bottom), for positive [negative] initial conditions on the left [right]. Because a positive supply introduces asymmetry diagrams were obtained for different values of the intensity of choice. For β = 2.9 in the original model the attractor is the steady state x∗2 , while in Figure 3, top left with σ = 1 there appears to be quasi-periodic motion on an invariant closed curve that encloses the value of x2 of the model with zs = 0. In the more realistic case with positive supply the beliefs about conditional variance have a profound effect on the long-run dynamics of the model. In fact, with fixed β but varying σ, the diagram is qualitatively similar to the basic bifurcation scenario of the original model over β after the pitchfork bifurcation. We refer to diagrams similar to Figure 3 left as foreshortened scenarios since, with respect to the basic scenario in Figure 1 left, the first attractor - the fundamental value - has already lost stability in this version, but the rest of the scenario follows. The attractor for low values of the conditional variance beliefs is E2 , the stable non-fundamental steady state. The value of E2 increases with increasing variance beliefs until stability is lost through a supercritical NS bifurcation and quasi-periodic orbits arise. For 0.71 < σ 2 < 2.97 the price deviation moves between near the fundamental value and optimistic states. Consistent with the previous results, the higher is the conditional variance the more the price may deviate from its fundamental value. Near σ 2 = 1.25 chaotic dynamics replace the quasi-periodic dynamics. Finally, for σ 2 ≈ 2.97 the price deviation suddenly disappears and for very high values of the conditional variance beliefs, the price does not diverge from its fundamental 9

Figure 2: Conditional variance diagrams: left, β = 3.4; right, β = 3.7. Top, xt−1 ; bottom nt−1 .

10

value. This scenario is similar for all the intermediate values of the intensity of choice.

Figure 3: Bifurcation diagrams: left, positive initial conditions; right, negative initial conditions. Top, xt−1 ; bottom, nt−1 lower. For β = 4.5 in the original model all non fundamental solutions have lost stability and the fundamental price is the limit set (for positive and negative initial conditions). With positive supply the analogous attractor, at σ = 1 appears to be chaotic for negative initial conditions. Although it is difficult to see from the diagram, for very low values of the conditional variance the attractor is, in fact, E1 . Stability of the fundamental steady state is lost immediately to what appear to be chaotic dynamics on strange attractors, followed by invariant curves on 11

which lie quasi-periodic attractors. The bifurcation diagram qualitatively resembles the foreshortened scenario run backwards and we call it the reverse scenario. The price fluctuates between near its fundamental value and pessimistic phases. In particular when σ 2 ≈ 1.5 the price starts to deviate persistently from its fundamental value. Another important feature is that for increasing values of the variance the price deviation consistently increases but the range of deviation becomes smaller. Thus the more agents expect the price to fluctuate in optimistic phases (with positive initial values) the wider is the fluctuation, but in pessimistic phases (with negative initial values) the fluctuations widen to a maximum value and then decrease. Then, for σ 2 ≈ 4.56 the system regains a non-fundamental steady state due to a backwards supercritical NS bifurcation. After this point the higher the conditional variance beliefs the higher the price deviations even if the price ceases to fluctuate, directly opposite of the case for positive initial values. The shift from quasi-periodic to chaotic dynamics is again more obvious in the fraction of fundamentalists variable, Figure 3, bottom. For positive initial conditions the stable steady state at low values of the conditional variance is characterized by n1 ≈ 0.1, so that the market is composed of only 10% fundamentalists. After the critical value for the NS bifurcation the fraction of fundamentalists varies between one and phases where it is close to zero even if it never reaches exactly zero. The change from quasi-periodic attractors on invariant curves to chaotic dynamics on strange attractors occurs around σ 2 ≈ 1.2 after which the attractor increases to cover almost the entire interval although low values are more frequently visited. Even when the system recovers its fundamental steady state E1 , n1 ≈ 0.05 and heterogeneity persists. If initial conditions are negative, the scenario differs. For very low values of the conditional variance the attractor is a stable steady state with n1 ≈ 0. A positive variance leads to fluctuations, first chaotic and then quasi-periodic, until the fundamental state becomes stable. In particular, if σ 2 < 1.5 the fraction of fundamentalists may vary from one to close to zero. After these values n1 exhibits decreasing ranges of variation and the price deviation also exhibits decreasing fluctuations. Therefore the more amplitude in the switching in the fractions, the higher are the price fluctuations. After the backwards supercritical NS bifurcation the stable steady state is characterized by a heterogeneous market and price deviations always negative. If the supply of risky asset is set at higher values the bifurcation scenario shifts left and fluctuations characterize a smaller part of the diagrams. We conclude that the main effects of positive supply are two-fold. First, it introduces dependency of the model dynamics on the conditional variance beliefs. Whatever assumption one makes for these beliefs is relevant for the resulting price dynamics and must be motivated. Second, it introduces asymmetry in the dynamical behavior with respect to positive or negative initial conditions, which further complicates the 12

effects of the conditional variance assumption. The scenario in Figure 4 (setting σ 2 =a = 1) takes zs as the bifurcation parameter. On the left at zs = 0 we have the same attractor as in Figure 1, left, at β = 2.9. At zs = 0.1 we have the same attractor found in Figure 2, left, at σ = 1. Not only does the existence of positive supply mean that conditional variance assumptions affect the qualitative dynamics, but the total supply also affects the long-run dynamics. It would appear that only small ranges of supply lead to fluctuating asymptotic sets for price deviations. Most values zs ∈ (0, 1) lead to the fundamental price if initial conditions are positive, to deviations at x∗3 if they are negative. As for the scenario, in Figure 4 left the diagram is qualitatively similar to the foreshortened scenario for conditional variance and positive initial conditions, Figure 3 left. In Figure 4 right, with negative initial values a compacted and distorted version of the reverse scenario appears.

Figure 4: Bifurcation diagrams: left, positive initial conditions; right, negative initial conditions.

As in the original model, the initial state of the market, whether the price of the asset is above or below its fundamental value, determines the phase for all future price evolution (there is no crossing of the fundamental price boundary). Moreover, given the asymmetry introduced by positive supply of the risky asset, the initial state also determines whether the foreshortened scenario is forward or reverse and the most common steady state. 13

4

Differentiated risk aversion parameters

In this section we relax the assumption of equal risk aversion parameters for the two types of agents. In the real world it is reasonable to believe that different traders have different risk attitudes, that the market may be composed of both risk loving and risk averse agents. Let a1 and a2 denote the risk aversion parameters for fundamentalists and trend followers, respectively. The system in deviation from the fundamental price is: Rxt =

2 X

nht fht = n2,t−1 (gxt−1 )

(19)

h=1

U1,t−1 =

1 (Rxt−2 − a1 σ 2 zs )(Rxt−2 − xt−1 − a1 σ 2 zs ) − C a1 σ 2

1 (xt−1 − Rxt−2 + a2 σ 2 zs )(gxt−3 − Rxt−2 + a2 σ 2 zs ) 2 a2 σ ¤ £ exp β( a11σ2 (Rxt−2 − a1 σ 2 zs )(Rxt−2 − xt−1 − a1 σ 2 zs ) − C)

U2,t−1 = n1t =

Zt−1 n2t = 1 − n1t ,

(20) (21) (22) (23)

with the difference in fraction given by5 Ã βh 1 ¡ mt = tanh (Rxt−2 − a1 σ 2 zs )(Rxt−2 − xt−1 − a1 σ 2 zs )a2 2 a1 a2 σ 2 − (xt−1 − Rxt−2 − a2 σ 2 zs )(gxt−3 − Rxt−2 + a2 σ 2 zs )a1

¢

! i −C . (24)

The presence of two risk averse parameters complicates the analytical tractability of the model. However, it can be shown that the fundamental steady state is ¶ µ ¢ ¡β 2 2 eq (25) E1 = (0, m ) = 0, tanh [(a1 − a2 )σ zs − C] . 2 Compare (25) to the fundamental steady state for the original model in (18), which is the same for the extended model with positive supply of risky asset but equal risk aversion. An essential difference is immediately obvious. If a1 6= a2 , not only the risk aversion parameters, but also conditional variance and supply of risky asset all influence the equilibrium difference in fractions meq . In particular, the 5

See appendices in BF for derivations of the main equations and steady states.

14

effects of the cost of deriving a fundamentalist forecasting strategy will be reduced or amplified according to whether the fundamentalists are more or less risk averse than the trend followers. In the likely case that those who willingly spend to acquire information are more anxious about risk, a1 > a2 and in the fundamental steady state the difference in fractions will vary with these 4 parameters. For simplicity let information costs C = 0. The difference in agents’ fractions is positive if a1 > a2 . That is, fundamentalist are more numerous in E1 if they are more risk averse. Trend followers are more numerous if they are more risk averse (a1 < a2 , meq < 0). The market is dominated by the type of agent which is more risk averse. For C 6= 0, fundamentalist will dominate if they are sufficiently more risk averse with respect to information costs, conditional variance beliefs and supply of risky asset ( that is, (a1 − a2 )σ 2 , zs2 > C). If risk aversion parameters are equal, the steady state reduces to that of BH (18). The non-fundamental steady states are (x∗2 , m∗ ) and (x∗3 , m∗ ), where m∗ = 1 − 2 Rg and x∗1 and x∗2 are the solutions of ³ ©£ ¤ m∗ = tanh β2 (a1 σ 2 )−1 (−R + R2 ) + (a2 σ 2 )−1 (−g + R + Rg − R2 ) (x∗ )2 ª´ ∗ 2 −gzs x + (a1 − a2 )σ zs − C . (26)

5 Simulations of differentiated risk Numerical simulations suggest that the bifurcation scenarios of long run price behavior for many combinations of a1 and a2 are only shifted with respect to the case of equal risk aversion parameters. For β - the intensity of choice, the bifurcation parameter studied in BH, assuming differing risk aversions leads to shifting of the scenario in the case of a supply of risky asset equal to zero (as in BH) but also for some parameters values in the case of positive supply. We present below only a few interesting cases for other bifurcation parameters (for many others see BF).

5.1 Variance beliefs as bifurcation parameter In this section we again take the conditional variance beliefs as bifurcation parameter, for the case in which agents using a fundamentalist strategy are the more risk averse. The compounding of asymmetric shifting effects of negative initial conditions and those due to differing risk aversion coefficients give a distorted bifurcation scenario. Consider Figure 5, with negative initial conditions and β = 4.5, 15

Figure 5: Scenario with risk-averse fundamentalists: left, xt−1 ; right, n1 . zs = 0.1, a1 = 10 and a2 = 0.2, that is, fundamentalists are much more risk averse than the trend followers. With respect to Figure 3 right (in which a1 = a2 = 1 so that conditional variance and risk aversion have the same effect) the reverse scenario is shifted right and the area with the chaotic and quasi-periodic attractors contracts. It is interesting to note that after the backwards supercritical NS bifurcation the long run behavior of the two examples are completely different. In Figure 3 right, after the critical value the price deviation is characterized by a negative non-fundamental steady state which decreases in value as σ 2 increases and the fraction of fundamentalists remains at n1 ≈ 0.1. In Figure 5 the nonfundamental steady state E3 increases and eventually converges to the fundamental steady state. The fraction of fundamentalists at the non-trivial steady state remains n1 ≈ 0.1. However, after the price reaches its fundamental value, the fraction using the fundamentalist strategy starts to increase until, for high values of the conditional variance beliefs (not shown) it converges to n1 = 1. Therefore if agents expect very high conditional variance, trend followers will be forced out of the market and only risk averse fundamentalists remain.

5.2

Asset supply as bifurcation parameter

In Figure 6, the supply of outside shares per trader is the bifurcation parameter; on the left for positive initial conditions and β = 2.9, on the right for negative initial conditions and β = 4.5. For brevity we consider only the fraction of fundamentalists variable. In comparison with diagrams of positive supply, but equal risk aversion in Figure 4 (notice the difference in scale of zs axis on left) the main 16

difference is that for values of positive supply beyond those associated with fluctuating behavior, the non-fundamental steady state is the attractor and increases with supply, tending to n1 = 1. That is, as the supply of risky asset increases the market drives out trend followers leaving fundamentalists who are the more risk averse agents. This is not the case with equal risk aversion parameters where a heterogeneous market persists even for high values of supply.

Figure 6: Risk averse fundamentalists: left; positive initial conditions; right, negative initial conditions.

5.3 Risk aversion as bifurcation parameter Once differentiated risk aversion parameters are assumed they no longer affect the dynamical behavior of the system in the same manner as σ. Consider the risk aversion parameter for the fundamentalist strategy as bifurcation parameter. Setting the coefficient for trend followers low (a2 = 0.1), over most of the interval represented in Figure 7 left, fundamentalists are more risk averse than trend followers (positive initial conditions and β = 2.9). First, it is obvious that chaotic fluctuations occur over a much wider range of values for a1 than for σ (see Figure 3, left). Second, the bifurcation scenario does not end with exponentially expanding deviations collapsing to fundamental value. Rather deviations are limited, first increasing and then decreasing until the backward NS bifurcation, at which point E2 regains stability. the constant steady state value declines as E2 converges to the fundamental price for a1 > 19.5 (not shown). In a heterogeneous market with trend followers nearly risk neutral, as fundamentalist risk aversion increases 17

from neutrality, price fluctuations increase in amplitude, reach a peak and decline, collapse onto the non-fundamental steady state and finally, at extreme values, the price remains on its fundamental value. As fundamentalists become very risk averse their market share increases until they eliminate trend followers and drive the price to its fundamental.

Figure 7: Positive supply: left, positive initial conditions; right, negative initial conditions. The scenario in Figure 7 right, obtained for the same negative initial conditions, β and a2 as in Figure 5, again suggests that price fluctuations occur on a much wider range of a1 with respect to σ. Also for negative initial conditions for a1 > 19.4 the price converges to the fundamental value and again the very risk averse squeeze the trend followers out of the market. Finally, we consider the pathological case that trend followers are more risk averse than fundamentalists. The scenario in Figure 8 was obtained for same values as in Figure 7 except a1 = 4.5. The diagrams are characterized by backward and forward NS bifurcations connected by the non-fundamental steady state. In the backward sequence of the NS bifurcation, a2 < a1 , the higher the trend followers risk aversion the smaller the amplitude of price fluctuations. Between the two critical values of the NS bifurcations, for values of a2 ≈ a1 , the non-fundamental steady state is stable. The forward sequence, a2 > a1 , from the critical value is interrupted by the breaking of the invariant curve into strange attractors, the latter then eventually collapse. For extreme values, with the trend followers nearly 15 times as risk averse as the fundamentalists, the attractor is always the fundamental steady state but the fundamentalists are eliminated and the market consists entirely of trend followers. 18

Figure 8: Risk neutral fundamentalists: left, xt−1 ; right, n1 . The strategic tendency to follow the trend is checked by their extreme relative risk aversion. For values of fundamentalist risk aversion closer to neutrality the distance between the backward and forward supercritical NS bifurcations progressively shortens. The closer to neutrality, the narrower the range of stability for the nonfundamental steady state until it disappears altogether and the two invariant curves connect. The opposite occurs for higher risk aversion values for fundamentalists. The non-fundamental steady state is the attractor over a wider range of a2 values and the forward supercritical NS bifurcation shifts right. Increasing a1 amplifies the price deviations.

6 Conclusion In this paper we relax some of the hypotheses in the heterogeneous agent asset pricing model formalized by Brock and Hommes (1998) in which agents choose among a finite set of competing forecasting trading strategies on the basis of their mean-variance utility and an evolutionary performance measure. For simplicity we consider two types of trader: fundamentalists and trend followers and use a mixture of analytical and numerical tools to study the price dynamics and the distribution of trader types. The first hypothesis to be relaxed is that of zero supply of the risky asst. It is shown that the supply of risky asset, and in general the risk premium, affect the profits realized in the performance measure and the difference in agents’ fractions 19

and therefore the fundamental price. In general we have shown analytically and in simulations that the assumption of positive supply introduces dependency of the model dynamics on the conditional variance beliefs. Then any assumption about these beliefs will have an effect on the price dynamics. Positive supply also introduces asymmetry in the dynamical behavior with respect to positive or negative initial conditions, which further complicates the effects of the conditional variance assumption. The model with a positive supply presents some interesting features. First of all, it has been shown that with positive information costs and zero memory the market is composed of both trend followers and fundamentalists even if the price has reached its fundamental value. That is, a fraction of fundamentalists remains in the market and they continue to pay positive information costs even if the price does not diverge from its fundamental value (but if the intensity of choice tends to infinity, the fraction of fundamentalists tends to zero). On the other hand, if the price is not on its fundamental value, the market is composed of both types of agents and even if the intensity of choice is high, the fundamentalists cannot drive out the trend followers and therefore the price may persistently diverge from its fundamental value. It is also shown that the higher the conditional variance beliefs or, alternatively, the single risk aversion, the higher the price fluctuations. In particular, without a positive supply and for intermediate values of the intensity of choice, if agents expect high price variability, it will actually exhibit this high and increasing variability a “volatility sunspot”. This is partially true also in the case of positive supply and intermediate values of the conditional variance beliefs. In fact, if the variance beliefs are very high, instead of high price fluctuations the price will stay on its fundamental value for positive initial conditions and on a non-fundamental steady state for negative initial conditions. In general, the above features persist also in the model in which we allow the risk aversion parameter to differ for fundamentalists and trend followers. Relaxing the equal aversion hypothesis leads, primarily, to shifting of asymptotic dynamics for the asset price, observable in the bifurcation scenarios. The interesting phenomena occurs in the agent mix in the market. Allowing for differences in risk attitude raises the opportunity for risk-aversion dominance in the market, that is, contrary to the previous models, agents with risk aversion are able to drive out the type with lower risk aversion. This event is more probable the greater the difference in risk aversion, the higher the conditional variance beliefs, the larger the supply of risky asset.

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7 References Benedetti Fasil C., (2005). Dynamical aspects of a heterogeneous agents model for financial market (unpublished thesis, University of Udine). Bullard J. and Duffy J., (1999). Using genetic algorithms to move the evolution of heterogeneous beliefs,Computational Economics 13, 41-60. Brock W. A. and Hommes C. H., (1998). Heterogeneous beliefs and route to chaos in a simple asset pricing model, Journal of Economic Dynamics & Control 22, 1235-1274. Chiarella C. and He X., (2002). Risk and learning in a simple asset pricing model, Computational economics 19, 95-132. DeLong J. B., Shleifer A., Summers L. H. and Waldmann R. J., (1990). Noise trader risk in financial markets, Journal of Political economy 98, 703-738. Diks C. Van der Weide R., (2003). Heterogeneity as a natural source of randomness, working paper CeNDEF, Univeristy of Amsterdam, and Tinbergen Institute. Farmer J. D. and Shareen J., (2002). The price dynamics of common trading strategies, Journal of Economic Behavior and Organization 49, 149-71. Gaunersdorfer A., (2000). Endogenous fluctuations in a simple asset pricing model with heterogeneous agents, Journal of Economic Dynamic and Control 24, 799-831. Gaunersdorfer A., (2001). Adaptive beliefs and the volatility of asset prices, Central European Journal of Operations Research 9, 5-30. Haltiwagner J., Waldmann M., (1985). Rational expectation and the limits of rationality: an analysis of heterogeneity, American Economy review 75, 326-340. Hommes, C., (2005). Heterogeneous agent models:two simple examples, in: Lines (ed.) Nonlinear dynamical systems in economics, Springer-Wien, New York, 131-164. Hommes C. H., (2006). Interacting agents in finance, New Palgrave Dictionary of Economics, Second edition, edited by Blume L. and Durlauf S., Palgrave Macmillan, forthcoming. Hommes C. H., (2001). Financial market as nonlinear adaptive evolutionary system, Quantitative Finace 1, 149-167. Hommes C. H., Huang H. and Wang D., (2005). A robust rational route to randomness in a simple asset pricing model, Journal of Economic Dynamics & Control 29, 1043-72. Lines, M., ed., 2005, Nonlinear dynamical systems in economics, SpringerWien, New York. Medio, A. and Lines, M., 2001, Nonlinear Dynamics: A Primer, Cambridge University Press, Cambridge

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