Amending agent-based asset pricing models : an analysis of the eects of adding risk aversion and ARMA prediction rules to the adaptive belief system introduced by Brock and Hommes (1998)

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An analysis of the effects of adding risk aversion and ARMA prediction rules to the adaptive belief system introduced by Brock and Hommes (1998)

by Yair Naaman 11004908

Supervised by dr. D. Kop´anyi

A thesis submitted in partial fulfillment for the Bachelor of Science of Econometrics

in the

Faculty of Economics and Business Amsterdam School of Economics

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i

Statement of Originality

This document is written by Student Yair Naaman who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This thesis introduces two alterations to the switching mechanism of the existing adaptive belief system, established by Brock and Hommes (1998). The first alteration inserts risk aversion in the preferences of the agents and the second alteration includes the use of ARMA models by the agents to predict the profit of each strategy. The main result is that irrational agents can still survive in the market but the specific dynamics change significantly. Generally, when risk aversion is added the range of the price, is larger but the price is more often closer to the fundamental. Including the ARMA prediction rules lead, in most cases, to aperiodic unstable behaviour of the price.

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About ten years ago, the housing market in the United States of America was enduring a disaster not seen since the Great Depression. The deficiencies of common economic tools to guide policymakers in their decisions suddenly seemed abundantly clear. According to Colander et al. (2014), a major component of these aforementioned deficiencies later seemed to be the assumption of perfect rational expectations of individuals. The idea that a choice of an individual would not be completely rational but bounded in the real world, was already well known in academic work. Simon (1955) already commented negatively on the widespread use of the concept which he calls the perfectly rational ”economic man”. However Friedman (1953) argued that irrational investors would not be able to survive in the market as they would be facing rational investors who accrue higher revenue. A valuable result of models which try to internalise the idea of irrational investors is that they would reject this predominant hypothesis (Farmer et al., 2012). In light of this, models which include a bounded rationality approach to replicate heterogeneous agents are becoming increasingly popular and therefore examining these models has also become increasingly necessary.

A prominent instance of such a model has been established by Brock and Hommes (1998). It calls for four different kinds of agents who use distinctive strategies on the pricing of assets. One type they introduce is the so called fundamentalist who believes that the price of an asset is only determined by the discounted value of the future dividends. Furthermore, they describe trend-followers and contrarians, who believe that the trend will (respectively) continue or reverse. Lastly, there are rational agents who have perfect insight into all information and are thus able to estimate the future price perfectly. Furthermore, it is important to note that all of these agents are able to have a positive or negative bias with regard to the price of an asset. According to the model, the fractional distribution of these agents and more importantly the amount of agents shifting between these different strategies, aptly named the ”intensity of choice”, influence the price fluctuations significantly.

However, in the model of Brock and Hommes (1998), agents choose their strategy by only looking at the profit one period before. This might not be entirely realistic. Therefore, the goal of this paper, is to study the effects on the prices if the agents use more sophisticated approaches to choose their strategies. This is done in two different ways. Instead of agents having this shortsighted view that only past profits are useful in choosing their strategies, investors take the the risk of the strategies into account. Secondly the individuals use econometric methods, in particular ARMA models, to estimate the future performance of the strategies.

The framework of this paper can be summarised in the following way. The specifics of the Brock and Hommes (1998) model is fully elaborated upon in the Chapter 2. Building on this, in Chapter 3, the changes that are made to this model are introduced. Later, in Chapter 4, the simulations in situations of two types of strategies are executed and analysed. Chapter 5 expands on this by

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considering multiple type of strategies at the same time. Lastly, based on these analyses a conclusion is drawn in Chapter 6.

2. Adaptive belief system

This chapter introduces the original model of Brock and Hommes (1998), which is going to be altered and investigated later. The first section introduces the heterogeneity of the model, the second section sets up a fundamental benchmark in terms of which the model is written and the last section explains the different beliefs and how these are incorporated in the system.

2.1 Introducing heterogeneous beliefs to an asset pricing model

A typical asset pricing model discusses a situation that involves an investor who has to decide how much he or she is going to invest in either a risk-free asset or a risky asset.1 The risk-free asset is characterised by being perfectly elastically supplied with a discount rate of R > 1. The risky asset, in contrast, is finitely available and disburses uncertain dividends. Let zt be the amount of risky assets,

ptthe price of the asset and ytthe dividend, which are all defined at time t. The wealth of the investor

in the next period, Wt+1 is defined by:

Wt+1 = RWt+ (pt+1+ yt+1− Rpt)zt. (2.1)

Seeing that the model accounts for heterogeneous agents, it specifies the conditional expectation and conditional variance of the agents by Eht and Vht, where h specifies the type of agent. Agents are

assumed to only care about maximising mean-variance. If a is defined as the risk aversion parameter, the demand zht of type h for the risky asset is determined by:

M axzht{Eht(Wt+1) −

a

2Vht(Wt+1)}.

Solving for the first order condition gives:

zht=

Eht(pt+1+ yt+1− Rpt)

aVht(pt+1+ yt+1− Rpt)

.

1_{Note that it is also possible (and arguably more logical) to regard the budget available to the investor as constant.}

The resulting maximisation problem is then which fraction of the budged should be allocated to the risky asset and which fraction to the risk-free asset. Anufriev, Bottazzi, and Pancotto (2006), for example, define their model along these lines.

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If the conditional variance is presumed to be constant with Vht= σ2, this expression is reduced to:

zht=

Eht(pt+1+ yt+1− Rpt)

aσ2 .

It was already noted that the risky asset is not infinitely supplied. Let this fixed supply be denoted by zs. Combining this with nht, which denotes the fraction of type h at time t, and the market

equilibrium of supply and demand, yields:

H X h=1 nht Eht(pt+1+ yt+1− Rpt) aσ2 = z s_. _(2.2)

H is defined as the total amount of different beliefs. The conditional expectation of Eht(Rpt) is equal to

Rpt, when it is assumed that the value of the future price and dividend (Eht(pt+1+ yt+1)) is estimated

before price pt is public knowledge. Rewriting (2.2) while using this assumption gives:

Rpt= H

X

h=1

nhtEht(pt+1+ yt+1) − aσ2zs.

The scope of this paper is limited to the situation where the outside supply of shares (zs) is equal to 0, so: Rpt= H X h=1 nhtEht(pt+1+ yt+1). (2.3)

The result is thus the standard asset pricing model with a small change namely that the different (irrational) strategies can be taken into account. How these strategies are defined is clarified in Section 2.3.

2.2 Constructing a fundamental benchmark

A useful aspect of the model as proposed by Brock and Hommes (1998), is the adoption of a funda-mental benchmark to relate the actual price to. This is important as it is the rational benchmark, which the introduction of different beliefs influences. The details of the aforementioned benchmark are discussed in this section.

The first step that has to be taken to compute this fundamental solution is to discard the idea of irrationality for a moment. In this case, all agents are perfectly rational and therefore homogeneous.

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It follows that H = 1 and nt= 1∀ t, thus (2.3) can be revised to:

Rp∗_t = Et(p∗t+1+ yt+1), (2.4)

where p∗_t indicates the fundamental price. If it is presumed that the stream of dividends, yt+1, is an

independent and identical distributed (IID) process with expected value ¯y, then: Rp∗_t = Et(p∗t+1) + ¯y.

In the event of R > 1, the only solution that conforms with the transversality condition2 according to Gaunersdorfer, Hommes, and Wagener (2008, p. 7) is:

p∗_t = p∗= y¯ R − 1.

This is the present value of all discounted future dividends. To be able to relate the actual price to this fundamental benchmark, xtis defined as:

xt= pt− p∗t. (2.5)

2.3 Modelling different beliefs

Previously, the types of beliefs already came up. This section discusses how Brock and Hommes (1998) define these strategies and integrate them in the model of section 2.2.

To accomplish this, an important assumption has to be made, namely that traders of type h ex-pect the price to differ from the fundamental price by a deterministic function based on past prices. Representing this mathematically yields:

Eht(pt+1+ yt+1) = Et(p∗t+1+ yt+1) + fh(xt, xt−1, ...., xt−k), (2.6)

where fh is defined as a deterministic function. This function can be specified for the aforementioned

different belief types. For the moment, these beliefs will only be expressed in simple linear form:

fht = ghxt−1+ bh. (2.7)

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This condition is sometimes also called the ’no-bubbles’ condition and implies that no bubbles exists in the market (Giglio, Maggiori, & Stroebel, 2016).

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If (unbiased) fundamentalists are examined, it makes sense that ft= 0, so g = b = 0. In case of

trend-followers g > 0 and adversely for contrarians g < 0. These can be considered to be (respectively) strong trend-followers and strong contrarians when |g| > R. Furthermore, it is possible for all these types of agents to have bias which can be expressed as b 6= 0. Lastly, it is important to note that strategies can have a fixed cost associated with them as it takes more money, time and knowledge to pursue more complicated strategies.

It is possible to rewrite the asset price in term of change from the fundamental price by first utilising (2.3) and (2.6) to get: Rpt= H X h=1 nht[Eht(p∗t+1+ yt+1) + fh(xt, xt−1, ...., xt−k)],

and secondly, combining this with (2.5) to get:

Rxt= H

X

h=1

nhtfh(xt, xt−1, ...., xt−k). (2.8)

The foundations of the strategies have been set but the dynamics of switching strategies still has to be investigated. To be able to do this, some kind of performance measure has to be introduced first, because in the model, agents use this to base their decision to change strategies on. Brock and Hommes (1998) do this by creating a fitness function:

Uht= πht+ ηUh,t−1. (2.9)

Uh,t−1 introduces memory to the fitness function, if η 6= 0. The scope of this paper however is limited

to the case where η = 0. The other function, πht, is the realised profits at time t.3 According to Brock

and Hommes (1998), this implies that the fitness function can be written as: Uht= (pt+ yt− Rpt−1)z(ρht),

but also in terms of change from the fundamental price:

Uht= (xt− Rxt−1)z(ρht) (2.10)

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Hommes, Kiseleva, Kuznetsov, and Verbic (2012) investigate the influence of memory on the stability of the system in more detail.

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The fractions nht can now be expressed as multi-nomial logit probabilities: nht= eβUh,t−1 PH h=1eβUh,t−1 . (2.11)

Brock and Hommes (1997) explain that using a discrete choice model has multiple advantages. Namely, the analytic traceability of the models and that they are easy to generalise to ’models with social interactions’ . The parameter β is called the ’intensity of choice’ parameter. If the extreme β = 0 is looked at, agents do not switch strategies and as a result the fractions are distributed uniformly4_.

Looking at the other extreme, when β −→ ∞, all agents instantly adapt their belief to the best performing strategy. Hommes et al. (2012) state that in this sense, β describes the amount of rationality in the dynamic system.

All in all, the system can be summarised in a h+3 dimensional map (Hommes et al., 2012):

            xt−1 xt−2 xt−3 U1,t−1 .... UH,t−1             7→             1 R PH i=1ni,t(gixt−1+ bi) xt−1 xt−2 πt,1 .... πt,H             (2.12)

3. Alterations to the model

In this chapter, the adjustments to the model of Brock and Hommes (1998) are touched. The focus of this paper is to change the performance measure on which the agents base their choice and study the resulting effects.

The first alteration includes risk aversion into the system. In the paper of Brock and Hommes (1998) agents do not care about the risk of a strategy, while in the real world people are generally risk averse. This phenomenon can be captured by using the standard deviation as this straightforward parameter measures the variability and hence the risk of a time series. Thus, the performance measure can be mathematically defined as:

Uh,t−1= πh,t−1− ασπh, (3.1)

4_{In this case the fractions can be defined as n}

ht=_H1∀t.

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where α can be specified to indicate how much the risk plays a role in the choices of the agents. The h+3 dimensional map of the system becomes:

            xt−1 xt−2 xt−3 U1,t−1 .... UH,t−1             7→             1 R PH i=1ni,t(gixt−1+ bi) xt−1 xt−2 π1,t−1− ασpi1 .... πH,t−1− ασpiH             (3.2)

The second change introduces a more complicated approach. It assumes that agents do not only base their choice on past profits but that they also try to forecast the future profits. There are of course a lot of techniques by which this can be done, but to keep it simple only an ARMA prediction rule is incorporated in this paper. The performance measure U is consequently defined as:

Uh,t−1= ch+ p X i=1 φh,iπh,t−i+ q X i=1 θh,it−1+ i, (3.3)

where the parameters φ and θ are estimated by using the past profits. In practice, it was obvious that the moving average parameters were not significant. Therefore, these were set to zero and that leaves only the autoregressive terms. The resulting map now yields:

            xt−1 xt−2 xt−3 U1,t−1 .... UH,t−1             7→             1 R PH i=1ni,t(gixt−1+ bi) xt−1 xt−2

c1+Pp_i=t−sφ1,iπ1,t−i+ 1,i

....

cH +Pp_i=t−sφH,iπH,t−i+ H,i

            (3.4)

The amount of terms differ through time as the agents use the Yule-Walker approach to construct the models and thereafter pick the model which yield the lowest Akaike Information criterion.

Another question to consider is the amount of memory in the ARMA model, as the sample size used by the agents to make their ARMA-predictions needs to be set. In chapter 4 and 5 the impact of this is investigated in more detail. The parameter defining the amount of periods the agents take into account is s.

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first for a certain amount of periods 1 without the ARMA prediction rules taken into account. Only thereafter the agents use the new prediction rules. This is done so there are actually useful data for the agents to base their predictions on.

4. Two types of strategies

The simplest form of the adaptive belief system uses two types of beliefs. This chapter analyses numerically what the consequences of the alterations to the model are in this simple situation. The two belief types can be summarised mathematically as follows:

   f1,t = g1xt−1+ b1 f2,t = g2xt−1+ b2 (4.1)

Inserting this more specific situation in the general dynamical system derived in chapter 2 yields:                  Rxt = n1,t(g1x2,t−1+ b1) + n2,t(g2x2,t−1+ b2) nh,t = e βUh,t−1 P2 h=1eβUh,t−1 πh,t−1 = _aσ12[(ghxt−2+ bh− Rxt−1)(xt− Rxt−1)] − Ch Uh,t−1 = f (πh,t−1) (4.2)

The changes to the performance measure can be split into the original method and the two cases introduced in chapter 3, which were:

Uh,t−1= πh,t−1− ασpih, (4.3) and: Uh,t−1= ch+ φ X i=t−s πh,t−i+ h,i. (4.4)

To easily spot the fluctuations between the percentage of agents per belief type a new parameter m = n1− n2 is introduced, in the case of two types of agents. To be able to run simulations, the value of

certain parameters need to be assumed; for this chapter, they are defined as: a = 1, σ = 1, R = 1.1, C1= 1 and C2= 0.

1_{For double the sample size to be precise.}

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4.1 Fundamentalists versus trend-followers

The first composition that is analysed contains unbiased fundamentalists and trend-followers. Thus, as was explained in Section 2.3, the belief functions are:

   f1,t = 0 f2,t = 1.2xt−1 (4.5)

The fundamentalists are in this case represented by type 1 and the trend-followers by type 2.

In Figure 4.1a, the price deviations and the distribution of agent beliefs are displayed. This is still the same system as proposed by Brock and Hommes (1998) and thus yields the same results. It can be seen that the agents at first choose the strategy of trend-following because no cost is associated with the strategy. However once the prices start to grow significantly and become overvalued they refer back to the fundamental.

The change of prices with respect to the fundamental is always positive. However this is caused by the starting values of the simulation. If negative starting values are chosen, x will always be negative. The arbitrariness can easily be spotted when an error term is added, which is done in Appendix A.

(a) Without alterations to the original model

(b) With a penalty on variance

Figure 4.1: Price fluctuations and strategy distribution in case of fundamentalists vs trend-followers

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(a) Unaltered system (b) Risk aversion added Figure 4.2: Bifurcation diagrams of fundamentalists versus trend-followers

Now the penalty to variance is introduced to the same situation. Figure 4.1b shows that the amount of bubbles significantly decrease when risk is penalised upon. However the maximum overvaluation of the price does increase and the steepness in case of a bubble does as well. The bifurcation diagrams1, see Figure 4.2, confirm this idea as the maximum value x takes is higher but the points seem to stack closer to the fundamental.

The reason for this becomes quite clear if Figure 4.3 is consulted. The standard deviation of the fundamentalist strategy is consequently higher than that of the trend-following strategy. This causes the agents to prefer the trend-following strategy and subsequently delays the bubbles.

The bifurcation diagrams in Figure 4.2 also show a Hopf-bifurcation occurring at around a value of 3.3, however the bifurcation happens a little earlier when risk aversion is added. The fact that this is a Hopf-bifurcation is easily spotted since it is followed by periodic behaviour of x, as can be seen in Figure 4.1.

Interesting is the amplifying wave behaviour of the upper limit of the price when risk aversion is added. This is not only the case in Figure 4.2b where α is constant and β varies but also the other way around, which is displayed in Figure 4.4.

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The bifurcation diagrams are created by defining the lower and upper value of the parameter we are interested in, in this case β, and subsequently simulating for all the values in between with small increments. For each increment a large time interval is chosen, to be specific in situations without the ARMA prediction rule 1000 periods are simulated but with the ARMA prediction rules 6000 periods, and plotting the last 7

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Figure 4.3: Profit and variance plotted for the two strategies

Figure 4.4: Bifurcation diagram where α varies and β is constant

The second alteration to the model concerns the integration of the ARMA model in the switching rules. These simulations were also done with a β of 3.5. An assumption that needs to be considered for the ARMA simulations is the amount of memory in the system, to be more precise the sample size taken by the agents for their estimations. Two different options are displayed in Figure 4.5.

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(a) The agents look at the previous 100 data points

(b) The agents look at the previous 250 data points

Figure 4.5: Price fluctuations and strategy choice with an integrated ARMA model, β = 3.5

The introduction of this rule as a whole establishes significant bias and very different price behaviour, as the price difference now oscillates around a value of 1.5. Furthermore, for small s, the price oscilla-tions seems to be lower, indicating stability. This seems a bit counter-intuitive as generally long-term predictions are associated with more stable investment decisions. The reason that this is not the case in this situation seems to be that because of the increased horizon, agents choose the fundamentalist strategy quicker. This increase of switching, however, does destabilise the system more instead of less as the status quo in this system is for agents to be a trend-follower.2

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4.2 Fundamentalists vs contrarians

Now, the effect of contrarians instead of trend-followers in the system is studied. Inserting the math-ematical definitions of these beliefs in (4.1), yields:

   f1,t = 0 f2,t = −1.5xt−1 (4.6)

The other parameters are not changed with respect to Section 4.1 with exception of β. The simulation results are captured in Figure 4.6. The price fluctuations are described by a steady period on the fundamental followed by an oscillation around this fundamental. The penalty on variance does result in fewer unstable periods, which is also what happened in the situation of fundamentalists versus trend-followers presented in Section 4.1.

(a) No alterations (b) With penalty on variance, α = 2 Figure 4.6: Differences from the fundamental fundamentalists vs contrarians

The bifurcation diagrams, shown in Figure 4.7 indicate that increasing the risk aversion widens the range of x after the secondary (Hopf) bifurcation. Furthermore without a penalty on variance, some stable phases of x do appear again, after the Hopf bifurcation already appeared for values of β around 6. However, the risk aversion seems to decrease the domain on β that show these stable cycles. Both cases do, however, follow a similar structure because the primary period doubling bifurcation and the Hopf bifurcation occur at around the same time.

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(a) No alterations (b) With penalty on variance, α = 0.5

(c) With penalty on variance, α = 1.0 (d) With penalty on variance, α = 1.5 Figure 4.7: Bifurcation diagrams in the situation of fundamentalists vs contrarians

Figure 4.8 show the results of the integrated ARMA model in the price fluctuations. The stability relies heavily on the sample size chosen. In this case, a larger sample size does imply more stability. This effect is opposite of what happened in the situation of fundamentalists vs trend-followers in Section 4.1.

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Until now, only figures showing unstable situations were displayed. Now, a stable situation is analysed. From the bifurcation diagrams in Figure 4.7 it can be seen that for β = 3 and no penalty on variance the price is in a stable situation as it developed from a pitchfork bifurcation to a stable period-2 solution, taking one of two values, one positive and one negative. The attractors in Figure 4.9 confirm this. However once a significant penalty is added, the system actually loses its stability and becomes more unpredictable but still periodic. The fact that it is still periodic can be seen from the accompanying time series of x in Figure 4.10. The bifurcation diagram in Figure 4.11 shows this gradual loss in stability as the risk aversion increases.

(a) α = 0 (b) α = 2 Figure 4.9: Fundamentalists vs contrarians, β = 3

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Figure 4.10: Time series of x for β = 3 and α = 2.

Figure 4.11: Bifurcation diagram of α.

In this chapter, only situations with two strategies were considered. In the next chapter, situations with more strategies are examined.

5. More types of strategies

Until now, only systems with two types of beliefs have been considered. This chapter looks at systems of 3 and more types of agents. The general system is summarised by (2.12) with parameter H specified to refer to the amount of strategies. For the other parameters nearly the same assumptions as in Chapter 4 are made: a = 1, σ = 1 and R = 1.1. The costs of the strategies, however, do vary in this chapter.

5.1 Three types: Fundamentalists versus biased trend-followers

ver-sus biased contrarians

The three belief types present in this system are mathematically represented by:          f1,t = 0 f2,t = 1.2xt−1− 0.2 f3,t = −1.2xt−1+ 0.2 (5.1)

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(a) No alterations (b) With penalty on variance, α = 1.5 Figure 5.1: Bifurcation diagrams in the situation of fundamentalists vs biased

contrarians and biased trend followers

This is a system containing pure fundamentalists, negatively biased strong trend-followers and pos-itively biased strong contrarians. No cost is associated with any of the strategies. The bifurcation diagrams of this system are displayed in Figure 5.1

Changing the risk aversion does not seem very interesting if we look at the bifurcation diagrams in Figure 5.1. There does not seem to be any significant difference in the range of the price. Further-more, there is a very slight difference in the bifurcation locations but the sequence is the same. A Hopf bifurcation is followed by alternations of stable and unstable periods.

If the time series in Figure 5.2 are considered, the amount of fundamentalist is quite stable and the switching between strategies of mostly contrarians and trend-followers cause the cyclic price fluctua-tions.

The attractors projected in the (xt, xt−1) plane tell a more interesting story. Without risk aversion

it exhibits more stable properties as the cycle of x is defined by a few values. As risk aversion increases the invariant circle closes thus indicating that the price fluctuations of x do not follow up on each other as well, indicating more unpredictability.

Introducing the ARMA prediction rules to the system yields quite different results. This is illus-trated in Figure 5.3a and 5.3b. The strange attractors seem to depend on the time of the simulation, which explains the disorganised nature of the ’invariant circle’. To be more precise, the price starts

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out fairly unstable but as time passes the price oscillates closer to the fundamental. Increasing the sample size results in a smaller price domain but slower convergence.

(a) No alterations (b) With penalty on variance, α = 0.25

(c) With penalty on variance, α = 0.5 (d) With penalty on variance, α = 1.5 Figure 5.2: Time series of x and attractors in the situation of fundamentalists vs

biased contrarians and biased trend followers, β = 40

(a) With integrated ARMA rule, β = 60, s = 500 (b) With integrated ARMA rule, β = 60, s = 500 Figure 5.3: Time series of x and attractors in the situation of fundamentalists vs

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5.2 Four types: Fundamentalists, biased fundamentalists and

con-trarians

In this section, the belief types are as follows:                  f1,t = 0 f2,t = 0.2 f3,t = −0.3 f4,t = −1.1xt−1 (5.2)

Note that there is a cost of C = 0.1 associated with belief type 1, the unbiased fundamentalists.

The most important result regarding implementing risk aversion can be seen in the bifurcation dia-grams in Figure 5.4. The extra risk aversion results in longer period of chaos before returning to a stable section containing four steady states.1 The time series in Figure 5.5 also show that the price cycle shortens if risk aversion is added.

(a) No alterations (b) With penalty on variance, α = 1

1

Note, that after a while (not shown in these diagrams), a chaotic period arises again and from then on it alternates between stable and chaotic periods

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(c) With penalty on variance, α = 5 (d) With penalty on variance, α = 10 Figure 5.4: Bifurcation diagrams in the situation of fundamentalists vs biased

fundamentalists and contrarians

Implementing the ARMA prediction rules heavily stabilises the system, as is shown in figure 5.5c and 5.5d. If the horizon for the agents is increased the price deviations are even smaller. The cycle is time dependant again, just like it was in the situation presented in 5.1 with the ARMA prediction rules. The price converges to the fundamental as time passes.

(a) No alterations (b) With penalty on variance, α = 1

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(c) With integrated ARMA rule, β = 40, s = 200 (d) With integrated ARMA rule, β = 40, s = 400 Figure 5.5: Time series of x and attractors in the situation of fundamentalists vs

biased fundamentalists and trend contrarians

6. Conclusion

In this paper, the existing adaptive belief system as introduced by Brock and Hommes (1998) was investigated. Two separate alterations were introduced and the effect on the price dynamics was ex-amined. All numeric examples are briefly discussed and conclusions with regard to the effects have been drawn.

The first alteration concerned the addition of risk aversion. This change did have a relatively con-sistent effect. First of all in simple situations where only two types of agents were present, the amount of bubbles decreased while the intensity of the bubbles increased. For more strategies, the β for which bifurcation occurred shifted and the cyclic nature of the price changed.

The second change regarded integration of the ARMA prediction rules into the system. In the situation of fundamentalists versus trend-followers, the resulting price fluctuations displayed relatively stable oscillating behaviour, however not around the fundamental. In contrast, in the case of fun-damental versus contrarians, the amount of bubbles increased, resulting in a less stable situation. Furthermore, when more than two types of strategies were considered, the price fluctuations became time dependent. First displaying periodic oscillations around the fundamental but slowly converging to this fundamental.

Thus, the model handled the addition of risk aversion nicely as it produced logical results, most prominently the idea that the price does fluctuate less than in the original model but the occurring bubbles are more severe. The ARMA prediction rule seems to produce very different results, some-times stabilising the system but somesome-times destabilising it, depending on the situation and the time horizon chosen. This alteration produces too sensitive results in the model to be useful and base policy decisions on. This is not surprising as Brock and Hommes (1998) did already emphasise that simple

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strategies perform better and produce clearer results. The reason for this is that the chances that a large fraction of the agents use the same relatively complicated rules are close to none. Furthermore, Anufriev and Hommes (2012) show in laboratory experiments that simple linear forecasting rules do already perform well. The aforementioned research was specifically aimed to investigate the linear rules of the strategies but the same concepts might also apply to the rules of switching. It may be considered to simplify the prediction rules by limiting the lags of the ARMA model to a small amount instead of using the standard AIC criterion to choose the best model.

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Anufriev, M., Bottazzi, G., & Pancotto, F. (2006). Equilibria, stability and asymptotic dominance in a speculative market with heterogeneous traders. Journal of Economic Dynamics and Control , 30 (9-10), 1787–1835.

Anufriev, M., & Hommes, C. (2012). Evolution of market heuristics. The Knowledge Engineering Review , 27 (2), 255–271.

Brock, W. A., & Hommes, C. H. (1997). A rational route to randomness. Econometrica, 65 (5), 1059–1095.

Brock, W. A., & Hommes, C. H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control , 22 (8-9), 1235–1274.

Colander, D., Follmer, H., Haas, A., Goldberg, M., Juselius, K., Kirman, A., . . . Sloth, B. (2014). The financial crisis and the systemic failure of academic economics. The Economics of Economists: Institutional Setting, Individual Incentives, and Future Prospects, 21 (2-3), 344–364.

Farmer, J. D., Gallegati, M., Hommes, C., Kirman, A., Ormerod, P., Cincotti, S., . . . Helbing, D. (2012). A complex systems approach to constructing better models for managing financial markets and the economy. The European Physical Journal Special Topics, 214 (1), 295–324. Friedman, M. (1953). The case for flexible exchange rates. Essays in Positive Economics, 157–203. Gaunersdorfer, A., Hommes, C. H., & Wagener, F. O. (2008). Bifurcation routes to volatility clustering

under evolutionary learning. Journal of Economic Behavior & Organization, 67 (1), 27–47. Giglio, S., Maggiori, M., & Stroebel, J. (2016). No-bubble condition: Model-free tests in housing

markets. Econometrica, 84 (3), 1047–1091.

Hommes, C., Kiseleva, T., Kuznetsov, Y., & Verbic, M. (2012). Is more memory in evolutionary selection (de) stabilizing? Macroeconomic Dynamics, 16 (3), 335–357.

Simon, H. A. (1955). A behavioral model of rational choice. The Quarterly Journal of Economics, 69 (1), 99–118.

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Appendix A: Error terms added in case of

fundamentalists versus trend-followers

In Section 4.1, the price fluctuations are only discussed in deterministic situation. A more realistic model would incorporate a stochastic error terms to simulate unknown shocks. This is not considered in the paper but an example of the results are shown below.

(a) Without alterations to the original model

(b) With a penalty on variance

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fundamentalists versus trend-followers

In Section 4.1, the effects on the price by the AR estimation is discussed. For completeness an example case of the amount of lags chosen by the agents to predict the profit per strategy is shown in Figure 6.2. No significant correlation between the amount of lags and the price itself or the stability of the price is present. There does seem to be a correlation between the sample size taken and the amount of lags.

(a) s = 100

(b) s=250

Figure 6.2: Amount of lags used by agents

Amending agent-based asset pricing models : an analysis of the eects of adding risk aversion and ARMA prediction rules to the adaptive belief system introduced by Brock and Hommes (1998)

by Yair Naaman 11004908

Statement of Originality

Abstract

Contents

2. Adaptive belief system

2.1

Introducing heterogeneous beliefs to an asset pricing model

2.2

Constructing a fundamental benchmark

2.3

Modelling different beliefs

3. Alterations to the model

4. Two types of strategies

4.1

Fundamentalists versus trend-followers

4.2

Fundamentalists vs contrarians

5. More types of strategies

5.1

Three types: Fundamentalists versus biased trend-followers

ver-sus biased contrarians

5.2

Four types: Fundamentalists, biased fundamentalists and

con-trarians

6. Conclusion

Appendix A: Error terms added in case of

fundamentalists versus trend-followers

fundamentalists versus trend-followers