bubbles in foreign exchange markets

Macroeconomic Dynamics, 11 (Supplement 1), 2007, 102–123. Printed in the United States of America.DOI: 10.1017/S1365100507060257

BUBBLES IN FOREIGN EXCHANGEMARKETS

It Takes Two to Tango

ALAN KIRMAN AND ROMAIN FABIO RICCIOTTIGREQAM, Universite d’Aix-Marseille IIIEcole des Hautes Etudes en Sciences Sociales

RICHARD LEON TOPOLCREA, CNRS

We consider a model in which foreign and domestic traders buy the assets of both of twocountries. Speculators in both countries use chartist or fundamentalist rules for forecastingthe exchange rate. Demand for the assets of each country is determined by these forecasts.Perceptions of the fundamentals in each country are not necessarily the same. Rules areused with a certain probability depending on an agent’s previous experience with them.The demands of the fundamentalist and chartist agents in the two countries determine thetemporary equilibrium exchange rate at each point in time. This is unique under certainassumptions. With traders of both nationalities there is no need, as in other models, for anexogenous supply of foreign exchange. The model produces realistic features of theequilibrium exchange rate series. Periods in which the exchange rate tracks thefundamentals of one of the countries alternate with others in which bubbles appear.

Keywords: Exchange Rates, Forecasting Rules, Herding, Temporary Equilibrium,Switching, Fundamentals, Bubbles

1. INTRODUCTION

For a long time the prevailing view of exchange rate formation was that exchangerates were essentially determined by “fundamentals,” however the latter might bedefined. Yet many of the stylized facts concerning the observed time series of ex-change rates can be better explained by models that relax some of the assumptionsof fundamentals-based models, which are closely related to the “efficient markets”hypothesis. This is true, in particular, for models that take explicit account of theway in which individuals form their expectations and the way in which that processis influenced by the other individuals in the market, and this will be the subject ofthis paper.

Considerable doubt has been cast on the theoretical consistency and the empir-ical validity of fundamentals-based models and in particular the efficient markets

Address correspondence to: Alan Kirman, Universite d’Aix-Marseille lll, IUF, GREQAM, EHESS, 2 Rue de laCharite, 13002 Marseille, France; e-mail: [email protected].

c© 2007 Cambridge University Press 1365-1005/07 $18.00 102

BUBBLES IN FOREIGN EXCHANGE MARKETS 103

hypothesis. In financial markets in general the price of an asset is assumed toreflect the fundamentals, which are given by the discounted sum of the streamof earnings that that asset will yield over time. In foreign exchange markets,which are what will concern us here, this can be translated into saying that thefundamentals are reflected in the real interest differential between the countriesthat emit the various currencies. Yet it has long been observed that asset prices havehigher volatility than the underlying fundamentals. This “excess volatility” puzzlehas generated a large literature. There are those who argue that there is indeedexcess volatility [see, e.g., Le Roy and Porter (1981), Shiller (1981), Blanchardand Watson (1982), and for a recent comprehensive analysis, Lo and MacKinlay(1999)]. The discussion has not been completely one-sided, however, and it wassuggested, in reply to the first criticisms, that it may be the testing procedure thatis at fault [for a discussion see Campbell and Shiller (1987) or West (1987) forexample] and that prices do, in fact, contain all pertinent information. The overallresult of the debate seems to have been that the underlying model is indeed flawedand the next step is to seek possible explanations and alternatives.

In the specific exchange rate context, the standard model, based on the funda-mentals of the system, is the so-called real interest differential model, which wasgiven its definitive expression by Hooper and Morton (1982). As in most of themodels developed in the eighties and nineties, the fundamentals used are somesubset of the money supply, real income, the expected inflation rate, the nominalrate of interest, and the current account balance. Models of this sort performedvery poorly from a forecasting point of view, as indicated by Meese and Rogoff(1983, 1988) and Edison (1991). Doubts were cast on the existence of a consis-tent relation between exchange rates and fundamentals. With the improvementof econometric techniques, models were developed that took into account thenonstationarity of the exchange rate and fundamental time series. Cointegrationmodels such as those used by Chinn and Meese (1995), Mark and Choi (1997),MacDonald and Marsh (1997), and Mark and Sul (2001) seem to establish a long-run equilibrium relation between exchange rate and fundamentals, as do the errorcorrection models used by Campbell and Schiller (1987). Although these modelshave been challenged because of the fragility of the results [see Diebold et al.(1994) and Sephton and Larsen (1991)], even if accepted, they raise an importantproblem. Expectations play a key role in any exchange rate model, and if therelation between exchange rates and fundamentals is truly a long-term one howcan agents who often have relatively short horizons be expected to use this relationto determine their forecasts?

A possible resolution of this difficulty may lie in the fact that the standardmodel says little about the way in which the market processes the information thatis available into prices. One way of thinking of markets is to consider them asalgorithms that transform information into prices. In the case that will concern usin this paper, the foreign exchange market, Lyons (2001) has taken this problemseriously and shows how information may be transformed into prices as a result ofthe order flows observed by traders or market makers. Thus, we can see how the

104 ALAN KIRMAN ET AL.

information from those who give orders is transformed through the market intoprice signals. Such transmission may be far from perfect and, in particular, thedifferent horizons of the individuals in the chain will be important.

What is it that generates the orders that, in turn, generate current prices? Indi-viduals have preferences and forecasts as to future prices. This, coupled with theirwealth, will generate their demand. What Lyons is suggesting is that the demandsalone will not explain the evolution of prices, because these are intermediated byindividuals who themselves take positions. This complicated process may lead tosome of the features that one observes in actual markets. Lyons focuses on theprocess of equilibration, how the arrival of orders that then enter a dealer’s orderbook will convey information to the dealer, and how that, in turn, will affect theprices quoted. However, Lyons’s emphasis is on the idea that this complicatedprocess of transmission will lead to efficient outcomes in the standard sense. Thusthe presence of varying expectations will not perturb the market from its truecourse. Yet more attention has been paid to the idea that some individuals not onlymay not forecast on the basis of fundamentals, but also may have an importantinfluence on the evolution of prices.

Such considerations gave rise to a series of models in which agents forecastpurely on the basis of previous exchange rates, which are frequently referred to as“chartist” models. These models were contested by theorists on the grounds that, iffundamentals really do determine exchange rates in the long run, then arbitrageurswould take positions that would remove any profits that chartists might have made.Nevertheless a series of papers starting with DeLong et al. (1990) showed thatchartists could, under quite reasonable assumptions, consistently make money,and this was confirmed by LeBaron (1999).

The empirical evidence is mixed, Curcio et al. (1997) arguing that chartist ortechnical rules are not profitable, whereas Dooley and Schafer (1983), Sweeney(1986), Levich and Thomas (1993), and Neely et al. (1997) suggest that technicalrules can be shown to be profitable. More recently, Dewachter (2001) showed thatMarkov switching regime models could be profitable.

This raises a natural question. Do traders actually use technical rules? Allen andTaylor (1990) and Menkhoff (1997) conducted surveys in London and Frankfurt.The former found that 90% of traders used at least a chartist component in theirshort-term forecasts and that a majority of them found that the chartist componentwas at least as important as the fundamentals. In the long run, agents using methodsbased solely on fundamentals composed 30% of the total, whereas, on the otherhand, 86% found that fundamentals were more important than extrapolations.Menkhoff’s results are strikingly similar.

We are then faced with a paradox. There seems to be some evidence of a long-term relation between exchange rates and fundamentals, and agents acknowledgethis, but in the short run they overwhelmingly use chartist methods and there issome evidence that these are indeed profitable. The only reasonable answer seemsto be to use models in which fundamentalists and chartists coexist, as suggestedby Frankel and Froot (1988) and Goodhart (1988).


In this paper we not only allow for the coexistence of chartists and funda-mentalists but also allow for individuals switching between the two types offorecasts, as in Kirman (1993) and Kirman and Teyssiere (2002). Whereas in thelatter model the supply of foreign currency from the foreign investors was ex-ogenous, in the model presented here this supply is endogenous and results fromthe choices of fundamentalist or chartist foreign investors. As shown by Kirmanand Teyssiere (2000), such models capture certain features of the real data, suchas long memory. They also eliminate some of the contradictions inherent in theevidence by allowing traders to vary the importance they attach to fundamentalsover time. The choice, made by the investors, either domestic or foreign, betweenthe forecasting rules representing their beliefs about the functioning of the market,that is, fundamentalist or chartist, is probabilistic. To be more precise, it is madeby a drawing from a Logit probability law that is exponential, and where theexponent is positively proportional to the discounted cumulative gain obtainedfrom using the rule in question. Thus, the probability is an increasing functionof the cumulative gain. Then, every investor revises his or her beliefs accordingto an exploration-exploitation process. Such a process emphasizes the forecastingstrategy that has been the most profitable in the past—exploitation—but the choiceof the other strategy is still possible—exploration. Finally, the investor demandfor foreign currency is determined according to a standard model of the theoryof finance [Grossman (1976)]. Thus we concentrate on the way in which individ-uals change their expectations and how this then changes their demands. We donot study the precise way in which markets equilibrate, but assume that pricesadjust to clear the market in each period. Thus we analyze a series of temporaryequilibria.

Unlike most models, in which the changing expectations of individuals causemovements in the exchange rate [see Goeree and Hommes (2000), Lux (1998)],we consider both sides of the market; that is, we think of a situation in whichthere are two countries and there are domestic and foreign traders who will takepositions in each other’s assets. This avoids some of the artificial features found inthe single-country trading models and allows us to examine the coordination thatmay occur between the expectations in the two countries. In each country there arewhat are considered as domestic fundamentals by their inhabitants, and these maybe strongly related. To be more explicit about the two types of expectation thatmay prevail in the tradition of Frankel and Froot (1990) and the literature basedon their idea, we define them as follows. There is a “fundamentalist” point of viewthat takes the position that there is an equilibrium value of the exchange rate basedon fundamentals and that the rate will tend towards that value. There is also a“chartist” point of view that simply forecasts on the basis of an extrapolation fromprevious values. We will give an explicit form to these extrapolations, but anyrule that conditions only on past values of exchange rates will serve the purpose.There will be chartists and fundamentalists in each country and the proportionswill depend on the success that individuals have had, in the past, from followingthe two forecasting rules.


Recall that a standard formulation of the exchange rate model can be written

st = αE(�st+1 | It ) + βzt , (1)

where st is the current or spot exchange or its logarithm, E(�st+1|It ) is theexpected change in the exchange rate, �st+1 = st+1 − st , conditioned on currentinformation, It , and zt is a vector of fundamentals.1 The important part of thismodel will be Et(�st+1), which can be interpreted as the market forecast basedon a weighted average of the two forecasting rules. The weights will depend onthe relative number of traders using each rule and the market forecast will notnecessarily be the same in the two countries. Furthermore, the market forecastswill evolve as the proportions of fundamentalists and chartists change over time.

Having developed our basic model, we will then analyze the effect of theevolving beliefs of the traders in the two countries on the exchange rate, andwe will examine the uniqueness of the equilibrium and the dynamic evolutionof the exchange rate. We will finally simulate the model in order to evaluate ourtheoretical conclusions and to get an idea as to the relative influence of the variousparameters on the exchange rate time series.

2. THE MODEL

We assume the following:

There is one domestic exchange market on which are traded two currencies, domestic d

and foreign f , whose values are linked by the exchange rate s. The variable s is theprice of one unit of foreign currency in units of domestic currency, so that, from thepoint of view of a domestic investor, a devaluation of her currency means an increasein the value of s. For convenience, we define the exchange rate in terms of foreigncurrency as e = 1

s.

There are n domestic and m foreign investors; the former measure their wealth in unitsof d and the latter in units of f (investors of one kind are all identical, i.e., they havethe same risk aversion and the same utility functions).

Investors can hold one of two views as to the evolution of the exchange rate.

Consider the case of the domestic investor who wishes to forecast the value ofthe exchange rate of the next period, which is not observed when the expectationis determined, st+1 (the case of the foreign investor is symmetric).

If the investor is a fundamentalist, he or she believes there is a fundamentalvalue st that is observed before the expectations are determined and to which theequilibrium exchange rate will revert. We attribute the following form to thesebeliefs:

E (st+1 | It , F ) = st +Mf∑j=1

νj (st−j − st−j−1) withMf∑j=1

νj = 1. (2)


If the investor is a chartist, his or her forecast as to the future value of the ex-change rate will be an extrapolation of its past values. We give these extrapolationsthe following form:

E (st+1 | It , C) =Mc∑j=1

hj st−j withMc∑j=1

hj = 1 (3)

Now define the following variables at time t :

ρ is the dividend in foreign currency paid on one unit of foreign currency;r is the interest rate on domestic assets;st is the exchange rate of the current period t ;f i

t is the demand by individual i for foreign currency;di

t is the demand by individual i for domestic currency.

The wealth of individual i at time t is determined by his or her investments inforeign and domestic assets and what he or she earned on them. That is,

Wit = (1 + r) di

t−1 + st (1 + ρ)f it−1. (4)

At each point in time the individual’s demands for foreign and domestic assetsmust satisfy the budget constraint

Wit = di

t + stfit . (5)

The discounted financial gain for individual i in period t is given by

git = Wi

t − (1 − ω) Wit−1, (6)

where ω is the discount factor. Explicitly, it reads

git = (1 + r) di

t−1 + (1 + ρ) f it−1st − (1 − ω)

[di

t−1 + f it−1st−1

],

git = (r + ω) di

t−1 + [(1 + ρ) st − (1 − ω) st−1] f it−1.

Note that if ω = 0, then the financial gain in the period is

git = rdi

t−1 + ρf it−1 + f i

t−1 (st − st−1) , (7)

which is the sum of the interests rdit−1 + ρf i

t−1 paid on the investor i’s portfolio(di

t−1, fit−1) and the capital gain f i

t−1 (st − st−1). And the cumulative gain is givenby

Git = Wi

t − (1 − ω)t−1 Wi1,

where Wi1 is the wealth of individual i at the beginning of period one (before she

chooses the rule to be used).However, the gains in each period are determined by the demands for domestic

and foreign currency in the previous period, and these are, in turn, determined bythe forecast that the individual made as to the exchange rate in the next period. Thelatter depends on whether he or she was following the fundamentalist or chartist


rule. As already mentioned, the choice of rule is made by a drawing from a Logitprobability law. It is a stochastic individual behavioral learning process. We wantto keep a total of the gains obtained up to period t by following the fundamentalistrule and similarly for the chartist rule. Define a random variable θ i

t that will takeon two values F and C; that is,

θ it = F with probability pi

t (F ),

θ it = C with probability 1 − pi

t (F ) = pit (C).

Now we can define an indicator function for the random variable, and this is simply

It (F ) = 1 if θ it = F and 0 if θ i

t = C,

(8)It (C) = 1 if θ i

t = C and 0 if θ it = F.

This leads to the gains for an individual using each forecasting rule as

Git (F ) =

t∑r=1

Ir

(θ ir−1 = F

)gi

t

(θ it−1, sr , sr−1

),

(9)

Git (C) =

t∑r=1

Ir

(θ ir−1 = C

)gi

t

(θ it−1, sr , sr−1

),

where git (θ

it−1, sr , sr−1) = rdi

t−1 + ρf it−1 + f i

t−1(st − st−1) is the investor i’sfinancial gain at the current period, which is a function of his or her beliefs θ i

r−1and of the equilibrium exchange rate sr−1, as shown further in the paper, and withGi

0(F ) = Gi0(C) = 0.

An alternative model—not investigated in the paper—would be one in whicheveryone is able, ex post, to calculate what his or her gains would have been hadhe or she used the other rule. In this case all agents would attribute the same gainsto each rule, that is,

Git (F ) =

t−1∑r=1

(Gf

r − Gf

r−1

),

(10)

Git (C) =

t−1∑r=1

(Gc

r − Gcr−1

),

where Gfr and Gc

r denote the gain that would have been obtained by an individualusing the rules F and C, respectively, in the period t − 1. Of course, this dependson the fact that the individual’s gains depend only on the forecasting rule chosenand not on his or her wealth, as will be the case for the class of utility functions thatwe will consider in this paper. Such a rule corresponds to a collective behaviorallearning process.

The first piece of information that investor i is the rule θ it he or she has drawn.

What is the remaining information I it of a domestic investor i at time t? He or she


has observations of the past values of his or her demands for foreign and domesticassets in the previous periods, the vector of observed exchange rates up to periodt − 1 and the cumulated gains that he or she has realized from using each of thetwo forecasting rules, fundamentalist and chartist. Thus we have

I it = {

dit−1, f

it−1, St−1 = (s1, . . . , st−1),G

it−1(F ),Gi

t−1(C)}. (11)

The total information available to the individual i once a forecasting rule has beendetermined is given by (I i

t , θit ). The total information of a foreign investor j has

the same form, with the foreign currency replaced by the domestic and vice versa,and the exchange rate replaced by its inverse.

We now turn our attention to the derivation of the individual i’s demand forforeign assets and, given his or her budget constraint, this will also determine hisor her demand for domestic assets.

2.1. The Demand of a Domestic Investor for Foreign Currency

Consider the set of domestic investors D. Any i in D determines his or heroptimal demand for risky currency according to the standard Grossman (1976)model, which states that the investor maximizes the expected utility of his or herwealth in the next period t + 1. So the investor has the following problem:

Maxf itE

(U

(W i

t+1

) | I it , θ

it

)(12)

where the investor’s wealth in the next period, t + 1, is given by

W it+1 = (1 + ρ) st+1f

it + (

Wit − stf

it

)(1 + r)

= (1 + r) Wit + f i

t [(1 + ρ) st+1 − (1 + r) st ] . (13)

In this paper we choose, on the one hand, the very simple mean-variance utilityfunction,

E(U

(W i

t+1

)∣∣I it , θ

it

) = E(W i

t+1

∣∣I it , θ

it

) − µD Var(W i

t+1

∣∣I it , θ

it

)(14)

where µD is the measure of risk aversion. The expectations of the investor dependon whether he or she is following the fundamentalist or the chartist rule.

Hence, using first-order conditions, we can write the demand for agent i as

f it ≡ f i

t

(st , θ

it

) =(1 + ρ

)E

(st+1

∣∣I it , θ

it

) − (1 + r

)st

2µD

(1 + ρ

)2Var

(st+1

∣∣I it , θ

it

) . (15)

As for the CARA utility function, the optimal demand does not depend on theagent’s wealth in the current period t .

Clearly the expectations in equations (12) and (15) are conditioned on whichrule the agent is following, and recalling the definition of the indicator functionfor the random variable given in equation (7), we can write

f it

(st , θ

it

) = f it (st , F ) It

(θ it−1 = F

) + f it (st , C) It

(θ it−1 = C

). (16)


Recall, finally, that the demand for the individual i when he or she is using therule F or C is given by replacing the expectation in (13) by the expectations foreach of the rules that were defined in (2) and (3):

(i) for the fundamentalist,

E (st+1 | It , F ) = st +Mf∑j=1

νj (st−j − st−j−1) with

Mf∑j=1

νj = 1; (2)

(ii) for the chartist,

E (st+1 | It , C) =Mc∑j=1

hj st−j withMc∑j=1

hj = 1. (3)

2.2. The Demand for Domestic Assets by a Foreign Investor

The Demand for Domestic Assets by a Foreign Investor is obtained in a preciselyanalogous manner for each investor j in the set of foreign investors F . Thewealth of an individual is now measured in foreign currency, for whom the riskyinvestment is in domestic assets, the safe rate of interest is that in the foreigncountry, ρ, and the exchange rate et = 1

st. Thus we can write the foreign investor’s

demand for domestic currency as

djt ≡ d

jt

(et , θ

jt

) =(1 + r

)E

(et+1

∣∣I jt , θ

jt

) − (1 + ρ)et

2µF(1 + ρ)2Var(et+1

∣∣I jt , θ

jt

) (17)

and, as in the case of the domestic investor, he or she will use one of the two rulesgiven by (2) and (3), with st replaced by et . Thus we have

djt

(st , θ

jt

) = djt (st , F )It

(θ

j

t−1 = F) + d

jt (st , C)It

(θ

j

t−1 = C). (18)

2.3. The Market Equilibrium

Consider first the aggregate demand for foreign currency of the domestic investors.This is given by

�t ≡∑i∈D

f it

(st , θ

it

) =∑i∈D

f it (F )I i

t

(θIt−1 = F

) +∑i∈D

f it (C)I i

t

(θ it−1 = C

)(19)

or, if we define the number of domestic investors who use the fundamental ruleat time t as ND,t (F ) and the total number of domestic investors as ND, andxt = ND,t (F )

ND, then we can rewrite aggregate demand as

�t = ND[ft (F )xt + ft (C)(1 − xt )]. (20)

Now define analogously for the foreign investors their total number NF and thenumber of them who are fundamentalists at time t by NF,t (F ) and yt = NF,t (F )

NF.


Then we can write the aggregate demand for domestic assets by foreigners as

t = NF[dt (F )yt + dt (C)(1 − yt )] (21)

The equilibrium exchange rate s∗t is then determined by the equilibrium condi-

tion, given in the domestic currency, of the foreign exchange market,

Dt(s∗t ) − t(s

∗t ) = 0, (22)

where Dt ≡ �ts∗t (t ) is the value in domestic currency of the demand for foreign

(domestic) currency of the domestic (foreign) investors. It also reads

D(s∗t ) − t(s

∗t ) ≡ as∗

t − b(s∗t )

2 −[γ − δ

s∗t

], (23)

where

a = ND[E(st+1 | It , F )xt + E(st+1 | It , C)(1 − xt )]

µDσ 2s (1 + ρ)

> 0,

b = ND(1 + r)

µDσ 2s (1 + ρ)2

> 0,

γ = NF[E(et+1 | It , F )yt + E(et+1 | It , C)(1 − yt )]

µFσ 2e (1 + r)

> 0,

δ = NF(1 + ρ)

µFσ 2e (1 + r)2

> 0.

Having specified our model, we can now turn to the basic analytic results.

3. RESULTS

3.1. Analytical Results

The first result (Proposition 1) establishes the existence and uniqueness of thetemporary equilibrium at each point in time and the second the properties of theprobability that an investor chooses the fundamentalist rule (Proposition 2).

The equilibrium exchange rate

PROPOSITION 1. Let s∗∗ = (δb

)1/3be the exchange rate for which the second

derivative with respect to the exchange rates of the excess demand is equal to zeroand let M

(δb

)be the maximum of the first derivative. Let s1 and s2 be the values

of the exchange rate such that the first derivative of the excess demand equals 0.


Define the following auxiliary variables:

δ

b= NF

ND· µD

µF· σ 2

s

σ 2e

(1 + ρ)3

(1 + r)3,

a

2b= 1

2· (1 + ρ)

(1 + r)[E(st+1|It , F )xt + E(st+1|It , C)(1 − xt )].

The equilibrium exchange rate s∗t exists, is positive, and is unique, unless the

following conditions hold:

(i) M

(δ

b

)> 0,

(ii) 0 <δ

b< s1 < s2 <

a

2b,

(iii)dDt

dst

(s1t

) − dt

dst

(s1t

)< 0,

(iv)dDt

dst

(s2t

) − dt

dst

(s2t

)> 0.

Proof. The equilibrium exchange rate is that for which the excess demand isequal to zero; it is the solution of the polynomial of degree 3 that is the equilibriumcondition (23). The analysis of the variations of the excess demand according tothe parameter values of the demand and the supply of foreign currency gives theconditions to be fulfilled for the equilibrium condition (23) to have either onereal solution, that is, one equilibrium state, or three real solutions, that is, threeequilibrium states (see Appendix).

Interpretation of Proposition 1. Condition (i) can be more easily interpretedin the particular case where the numbers of investors, the interest rates, and theconditional variances of the exchange rates for the two countries are equal. Thens∗∗ = ( δ

b)1/3 is equal to the cube root of the ratio µD

µFof risk aversions and the

quantity M(δb) is the difference between, on the one hand, the expected gain

for the domestic investors with respect to s, and on the other hand, what can beconsidered as an “expected” gain for the foreign investors, defined as the differencebetween the inverse of the square of the expected exchange rate e = 1/s and theinverse of the square of s.

In the particular case of interest, the proposition says that, if the domestic andforeign investors’ expectations and risk aversion are such that M(δ

b) < 0, then the

exchange rate exists, is positive, and is unique.

The probability of choosing a rule. As already mentioned, after the equilibriumexchange rate of the current period has been determined and consequently theinvestor knows his or her gain, each investor chooses the rule he or she is goingto use to determine his or her demand for the next period. The probability ofindividual i being a fundamentalist at time t is given by

P it+1(F ) = eβGi

t (F )

eβGit (F ) + eβGi

t (C), (24)


which can be rewritten as

P it+1(F ) = 1

1 + e−β�Git (F )

, (25)

where

�Git (F ) = Gi

t (F ) − Git (C), (26)

defined as in (9), that is,

Git (F ) =

t∑r=1

Ir

(θ ir−1 = F

)gi

t

(θ it−1, sr , sr−1

),

(9)

Git (C) =

t∑r=1

Ir

(θ ir−1 = C

)gi

t

(θ it−1, sr , sr−1

).

To analyze the properties of the probability of an investor choosing the fun-damentalist rule, we can drop the i superscript in (25), because all individualscalculate the gains from each strategy in the same way. Then the evolution of thetwo groups, the fundamentalists and the chartists, in each country can be deduced.Clearly this will depend on the success of the two strategies. The extreme casesare defined by the extreme magnitudes of the ratio β

µbetween the coefficient of

cumulative gain and the coefficient of risk aversion in the above probability. Theyare illustrated by the following result:

PROPOSITION 2. Investor i has the following extreme probability of choosingthe fundamentalist rule:

Ifβ

µ→ ∞ and

if (i)[(1+ρ)E(st |It−1, F ) − (1+ r)st−1](ρ + st − st−1) > 0 then Pt(F ) ∼= 1

if (ii)[(1+ρ)E(st |It−1, F ) − (1+ r)st−1](ρ + st − st−1) < 0 then Pt(C) ∼= 0

Ifβ

µ→ 0 then Pt(F ) ∼= 1

2.

Proof. This is easily obtained using equations (26), (9), and (7):

e−β�Git (F )

=⎧⎨⎩

e−β�Git−1(F ) · e−βrdi

t · e− β

µ· [(1+ρ)E(st | It−1,F)−(1+r)st−1]

2σ2s (1+ρ)2

(ρ+st−st−1) if θ it−1 = F

.

e−β�Git−1(F ) · e−βrdi

t · e− β

µ

[(1+ρ)E(st | It−1,C)−(1+r)st−1]2σ2

s (1+ρ)2(ρ+st−st−1) if θ i

t−1 = C


What this result says is that, if one rule gets the direction of the change of theexchange rate wrong, and the ratio of the importance the investor attaches to hisor her previous experience, which is given by β, to his or her coefficient of riskaversion, µ, becomes large, then the probability of its being used will tend to zero.On the other hand, if the investor does not attach much importance to the past or isvery risk-averse, the proportions of the two types will fluctuate around one-half.

An alternative. So far, we have been considering the individual’s probabilityof choosing a forecasting rule as dependent on his or her own strategy. We could,on the other hand, as already presented as an alternative, consider that he or sheis capable of evaluating what he or she would have gained from adopting thealternative strategy, and in this case gains would be given by (9)

Git (F ) =

t−1∑r=1

(Gf

r − Gf

r−1

),

(10)

Git (C) =

t−1∑r=1

(Gc

r − Gcr−1

).

Analyzing the consequences of such a modification is the subject for future re-search.

3.2. Simulation Results

We proceeded to run a number of simulations in order to analyze the effects ofvarying the crucial parameters of the model. In the simulated model we introduceda discount factor ω for past gains, which is clearly related to the weight β thatindividuals attach to their previous experience. The higher the discount factor ω,the less likely the individual is to become locked into one of the forecasting rules,as is the case when β is low. Our basic purpose is to study the effect of modifyingthe following parameters of the time series:

The variable β is the weight attached to exploitation as opposed to exploration withrespect to previous gains.

The variable µ is the coefficient of risk aversion.In particular, we will consider the ratio β

µ, which plays a special role in the model.

The variable ν is the rate at which individuals following the fundamentalist rule believethe exchange rate will return to its true value.

The variable ω is the rate at which individuals discount past gains.

The domestic and foreign interest rates, as well as the number of investors ineach country, are kept constant. The fundamental exchange rate in each country istaken as exogenous and different.

According to Proposition 1, when three real values are obtained as equilibria,a “continuity” argument is used to choose the real value that is closest to the


previous unique solution. In the simulations that are shown to clarify the results(Figures 1–4), the parameter ν is the coefficient of the first-order adjustment term,which enters the definition of the fundamentalist expectations, equation (2); andwhen it takes a small value, ν = 0.005, this means that the expectations are almostequal to the fundamental exchange rate.

The main results of the simulations, run using Matlab software over 500 periodsof time, are the following:

(I) The dynamics of the proportions of domestic or foreign investors has the followingcharacteristics:

(A) If β/µ � 1, the proportion changes rapidly from 1 to 0; the agents are almost alleither “fundamentalists” or “chartists.” The value taken by the discount factor ω

used in the calculation of the cumulative gain determines the qualitative aspect ofthe changes:

(1) If all the past gains have the same weight, 1 − ω close to 1, then the number ofchanges of the proportion from 0 to 1, or vice versa, is small in number (1 to 3)and in this case the proportions remain a relatively long time at a given value (0or 1); see simulation S1 in Figure 1.

(2) If only the recent past values of the gains are taken into account, 1 − ω closeto 0, then the number of changes of the proportion from 0 to 1, or vice versa,is very large in number and the proportion oscillates very rapidly between thevalues 0 or 1; see simulation S6 in Figure 3.

(A) If β/µ � 1, the proportions oscillate rapidly about the value one-half, and theagents are almost equally distributed among “fundamentalists” and “chartists.”The value of the discount factor determines the qualitative aspect of the changes,which in this case is not as clear-cut as in the previous case: whatever the valueof the discount factor be, either close to 0 or to 1, the proportions oscillates veryrapidly but the amplitude of the oscillations is small, in the sense that the valuesare taken in the interval between 0.35 and 0.65; see simulations S3 and S8 inFigures 2 and 4.

(II) The dynamics of the equilibrium exchange rate when the expectations are of the“fundamentalist” type and have values close to the fundamental shows the followingcharacteristics:

(A) If β/µ � 1(1) If 1 − ω is close to 1; see simulation S1:

(a) If the proportion of domestic “fundamentalist” investors is close to 1 andthe proportion of foreign “fundamentalists” is small, then the equilibriumexchange rate follows the domestic fundamental exchange rate and its inverseis very different from the foreign fundamental exchange rate.

(b) In the opposite case the exchange rate differs substantially from the nationalfundamentals and the inverse is very close to the foreign fundamentals.

(2) If 1 − ω is close to 0, see simulation S6: the exchange rate exhibits highervolatility than when the discount factor 1 − ω is close to 1.

(B) If β/µ � 1(1) If 1 − ω is close to 1, see simulation S3: the exchange rate never comes close to

the fundamental, as was the case for β/µ � 1.(2) If 1 − ω is close to 0, see simulation S8: the exchange rate shows a behavior

qualitatively equivalent to that obtained when the discount factor is close to 1.


FIGURE 1. Simulation 1.


(III) The comparison between Cases (2.A) and (2.B) shows the important role played,ceteris paribus, by the following:

(A) The ratio β/µ, which has an important influence on the dynamics of the proportionsof fundamentalist investors as well as the dynamics of the exchange rate, whichdiverges from the fundamental when this ratio is relatively weak.

(B) The discount factor, which can induce fast fluctuations with large amplitude ofthe proportion of fundamentalist investors when it is close to 1 (1 − ω close to 0);and furthermore, the volatility of the exchange rate is also increased.

4. CONCLUSIONS

The results that have been obtained here show that, as we suggested at the outset,a rather basic model in which traders of different national origins who trade inthe same market using simple rules to form their expectations can generate com-plicated dynamics at the aggregate level. We should also emphasize the extremesensitivity to certain important parameters of the dynamics of the proportionsof domestic or foreign investors who chose the fundamentalist strategy to de-termine their expectations. This is also true of the equilibrium exchange rate,whose time path varies according to the values of the crucial parameters of theexploration/exploitation process used by the domestic or the foreign investors.The process that is used to revise the beliefs involves the following parameters:risk aversion µ, coefficient β of the cumulative gain which intervenes in theprobability to draw the “fundamentalist” strategy, discount factor ω to calculatethe cumulative gain, etc. Although the particular time path followed will dependon the specific values of the parameters, it is also worth noting that the generalstructural characteristic of the process do not depend on this choice. [The proof ofthis is given in Foellmer et al. (2005)].

Further developments will be to make the choice of the rule, fundamentalist orchartist, endogenous and/or to replace the individual behavioral learning processextensively presented in the paper with a collective behavioral learning processsuggested as an alternative.

NOTE

1. The following notation is used throughout the paper: any stochastic variable is denoted with atilde, e.g., any nonobserved variable such as the exchange rate of the next period st+1, whereas anyrealization of a variable is denoted by the same symbol without any particular sign, e.g., the observedexchange rate value of the current period st or the agent’s information set It .

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APPENDIX: TABLE OF VARIATIONS OFTHE EXCESS DEMAND D −

A.1. CASE 1: M(

δb

)> 0


A.2. CASE 2: M(

δb

)< 0

bubbles in foreign exchange markets

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