A currency crisis and its perception with fuzzy C-means

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Information Sciences 178 (2008) 1923–1934 www.elsevier.com/locate/ins

A currency crisis and its perception with fuzzy C-means Ibrahim Ozkan a,b, I.B. Tu¨rksßen a,c,*, Naci Canpolat b a

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada M5S 3G8 b Department of Economics, Hacettepe University, Beytepe, Ankara, Turkey c Department of Industrial Engineering, TOBB-Economics and Technology University, Ankara, Turkey Received 19 October 2005; received in revised form 13 December 2007; accepted 15 December 2007

Abstract In this paper, we attempt to analyze currency crises within the decision theory framework. In this regard, we employ fuzzy system modeling with fuzzy C-means (FCM) clustering to develop perception based decision matrix. We try to build a prescriptive model in order to determine the best approximate reasoning schemas. We use the underlying behavior of the market participants during the crisis. With this analysis, we form the dictionary catalogs to construct a perception based payoff matrix. As an illustrative example, we used data from Turkish economy that covers two currency crises. The results show that market participants’ dictionary catalogs based on perception knowledge extracted from the first crisis help participants to perceive the rise in market uncertainty. When the expectations are revised accordingly a speculative attack becomes inevitable. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Currency crises; Fuzzy system modeling; Perception based decision making

1. Introduction Since the early 1970s, the world economy witnessed an increase in frequency and severity of currency crises originated from industrial as well as from developing countries. Each passing crisis increased the variety and the frequency of currency crises and hence generated a plethora of theoretical models. In spite of their many insightful contributions, these theoretical models of currency crises cannot be considered fully successful in determining the causes and the timing of currency crises.1 In historical perspective, the dynamics of currency crises appears to be elusive due to the evolutionary nature of the market participants. While it is already difficult to understand and predict daily fluctuations in foreign exchange markets, currency crises understandably present additional difficulties. The difficulties lie in the very nature of the crises. By * Corresponding author. Address: Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada M5S 3G8. Tel.: +1 416 978 1278/+90 312 292 4068; fax: +1 416 946 7581. E-mail addresses: [email protected] (I. Ozkan), [email protected], [email protected] (I.B. Tu¨rksßen), [email protected] (N. Canpolat). 1 For seminal contributions, see: [25,28], for the evaluation and classification of the currency crises see [12,22].

0020-0255/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.12.007

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definition, crises’ periods are ‘‘times of turbulence and excessive volatility”. Hence the models developed to explain market behavior under ordinary periods may not serve well during those chaotic times. One may choose to refine the existing models and gather more data to increase the explanatory and predictive power of the current theoretical and empirical models of currency crisis. On the other hand, one may choose to develop newer models with the help of emerging data analysis techniques. This paper in conjunction with our previous work [30] can be considered examples of such attempts. Also, an application of support vector machines to determine the most significant factors in explaining the consequences of currency crises on the economy [32] can be considered as similar example of the usage of new data analysis technique. A particularly important aspect of FSM is its power to capture underlying behavior of historical data with proper analysis and without excessive ad hoc axioms. There are at least two advantages of FSM that attracts researchers: (i) its power of linguistic explanation with resulting ease of understanding, and (ii) its tolerance to imprecise data which provides flexibility and stability for prediction. Because of these features, FSM has been increasingly applied to problems in various areas such as computer science, system analysis, electronic engineering, pharmacology, finance and more recently social sciences (some related examples are [41,39,34,33]). To our knowledge this paper represents the first attempt to analyze currency crisis within a decision theory with an application of FSM framework. To be more precise, in our analysis of currency crises, we adopt Zadeh’s perception based decision approach [45] with an application of the rule based fuzzy system modeling. Accordingly, we attempt to capture the underlying behavior of market participants during the crisis as part of perceptions. Then we analyze how a payoff matrix can be constructed by integrating these perceptions. The rest of the paper is organized in four sections. In Section 2 we explain why we choose to use FSM to investigate currency crises. In Section 3, we obtained dictionary catalogs (fuzzy clusters) by using FSM with FCM using Turkish data from 1990 to 2002 which covers two currency crises. This section also includes the results of the model and payoff matrix construction. In Section 4 we present our conclusions. 2. Perception based decision making In modeling human decision process, one may distinguish the descriptive and prescriptive type approaches. In these approaches, descriptive modeling attempts to identify system structure that capture the behavior characteristics as best as it can, where as the prescriptive modeling attempts to determine the best approximate reasoning schemas that produce the best prediction of system behavior for a given descriptive model.2 In the first phase of this work, we developed a Type 1 fuzzy system model to predict the currency crisis [30,29]. In order to predict the currency crisis, publicly available data are used. A brief review that includes only the fuzzy system modeling part of this modeling is given in the next section. As a next step, we analyze the currency crisis within the decision theory as explained in this paper. Human decision processes depend on the perceived world. At any instance of a decision process, a decision maker faces uncertainties. For instance, since the return on financial assets cannot be known with certainty investment decisions are taken under uncertainty, because it is difficult to assign objective probability values for possible outcomes. Observed values of economic indicators may provide insights to investors and help them to form their expectations. According to mainstream theoretical economics, rational individuals use all available information during the expectation formation process and they optimize the expected value of a well defined objective function under the assumptions of von Neumann and Morgenstern’s expected utility theory [27]. Then a decision becomes a mechanical action without emotions. Basic assumptions of von Neumann and Morgenstern’s theory may not be fulfilled since most real world probabilities are not precise and measurable. Even if it is the measurable case, when there is a tolerance for imprecision which can be exploited through granulation to achieve tractability, interpretability, robustness and economy of communication, there is a rationale which underlie granulation of attributes and use of linguistic variables [17,47]. Furthermore, uncertainty may appear

2

See [2, Chapter 2] for a clear exposition of descriptive and prescriptive modeling in decision making.

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in different forms such as ambiguity, vagueness, discord, imprecision and fuzziness [23]. It is an attribute of information and information is a generalized constraint on the values which a variable is allowed to take [47]. Uncertainty is the key ingredient of the most, perhaps all, real life decision making problems. Under these circumstances it becomes necessary to use uncertainty as a source of information that may be helpful to reasoning. As Zadeh [45] pointed out, information can be analyzed by perception based theory of approximate reasoning which is a generalization of classical reasoning that contains the capability to operate with perception based information. Fuzzy logic and fuzzy sets lay the ground for this kind of information processing and decision making. Following Zadeh’s seminal paper on fuzzy sets [44] and fuzzy decision analysis [6], during 1970s and 1980s the principles of fuzzy theory were applied to classical statistical decision theory. These contributions include ‘‘fuzzy acts” [1,36–38], ‘‘fuzzy events” [1,36–38], ‘‘fuzzy probabilities” [8,42], ‘‘fuzzy utilities” [20,43,31,42], and ‘‘fuzzy information” [1,36–38], ‘‘fuzzy linguistic modeling” [9–11]. With these contributions classical statistical decision theory is transformed into fuzzy decision theory. The importance of formulations of perceptions in fuzzy decision theory and formulations of perception based probabilistic reasoning with imprecise probabilities are articulated by Zadeh [45,46]. Human perception process is a flexible function of experiences. Studies have shown that attention can be directed to objects that are defined on the basis of generic grouping principles based on previous experiences [48]. Previous experiences determine the familiarity of the objects. In most experiments, it is demonstrated that object based attention are stronger for highly familiar objects than for unfamiliar ones [40]. For instance, gestalt perceptual grouping principles which have proximity, similarity, continuity, common movement, and common fate properties are sufficient to define the objects. New objects are clustered based on the similarity to the past learning experience [26]. For instance, similarity can be used as a part of a new classification schema that is the basis for forming new object clusters and approximate reasoning. Models that have properties similar of those grouping principles stated above may provide an approach to describe the behavior of the market participants during such times of crises. In applications of FSM to real life problems, previous approaches essentially depend on expert knowledge on the system behavior. Later there have been several objective function based fuzzy clustering approach proposed in the literature that help us to identify the cluster partitions of the data without expert knowledge. From the most widely used Bezdek’s FCM [3], Gustafson and Kessel’s algorithm [18], Gath and Geva Algorithm [14] and as a possibilistic example Krishnapuram and Keller”s algorithm [24] can be considered as the other most widely used algorithms. In this paper Bezdek’s FCM clustering algorithm is used. The clusters generated by this algorithm are assumed to be approximately equal in size and shape. Therefore it is usually applied for the recognition of the positions of the clusters instead of shapes. We are interested in the position of the clusters since the crises are rare events. Each cluster is represented by its center. Also a center is often regarded as a representative of all data assigned to that cluster. Therefore centers represent the patterns associated with clusters. If we assume that a data set is representative of a system, one can neglect the slight modifications that additional data may cause. Hence the FCM membership function can be used as a continuous function. Thus it can be used to determine the memberships of all possible data points. Membership functions can often be assigned linguistic labels such as ‘‘low”, ‘‘medium” or ‘‘high”. Such labeling provides linguistic meaning representation for understanding.3 In particular, we want to capture the interactive input–output relationship and thus we do not separate the input and output clusters from each other. The use of all available information during the clustering seems advantageous for complex systems. Therefore we use the whole input output product space for clustering. FSM with FCM is the medium that is used in this paper as perception based inference method where fuzzy clusters are the dictionary catalogs that serve as the basis of objects. According to this approach any object can be defined as a pattern that is generated by experience. Clustering the similar patterns provide us the definition of translation catalogs that are used in approximate reasoning. This approach is a process which has four properties: (i) clustering, (ii) similarity, (iii) flexibility, and (iv) resolution of uncertainty. In

3

See [19, Chapter 8], rule generation with clustering for a clear exposition.

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this manner, we embed in our model gestalt perceptual prototypes by its properties of similarity, grouping, proximity, and continuity. Currency crises are rare events. Therefore one may not find sufficient experience (observations) to calculate objective probabilities. Just before crisis, investors face a decision problem under uncertainty and it can be analyzed in a fuzzy environment [6] by transforming this problem into a decision problem. In such decision problems, one can define the states of nature as the policies of a central bank to defend its currency. In conjunction, the actions can be defined as the portfolio decisions of the investors whereas the payoffs are the expected returns. In general it may be more realistic to consider the states of nature and actions available to investors as fuzzy sets with infinitely many elements. A payoff can be calculated for every nature of state but in this case the dimension of the payoff matrix will be too high to be useful. Therefore we design the payoff matrix to cover only the extremes of a central bank’s policies in order to reduce the problem to a manageable level. There are two states of nature which are ‘‘floating regime” or ‘‘crawling peg regime”4 and there are two actions which are ‘‘buy foreign assets” or ‘‘buy domestic assets”.5 In other words we design the payoff matrix so that only payoffs are fuzzy. 3. Structure of the proposed model In this section, the proposed model and its structural details are explained under five major headings which are: 1. 2. 3. 4. 5.

Background Fuzzy rule base with FCM Fuzzy system model Results Perception based inference

3.1. Background From 1990 to 2004 Turkish economy experienced two currency crises: One in 1994 and the other in 2001. In the spring of 1994, government’s attempt to lower the cost of interest rate on public debt, by changing its borrowing strategy, triggered the events that eventually led to a speculative attack on domestic currency and resulted in a large devaluation of TRL. Several banks failed, and the government signed a stand-by agreement with the International Monetary Fund (IMF). In 1999 the central bank adopted a crawling peg regime. Later, in the beginning of 2000, the government began to apply an exchange rate based stabilization program supported by the IMF in order to lower inflation and to keep expanding public debt at sustainable levels. When rising current account deficit is perceived to be unsustainable, a speculative attack following a political crisis resulted in the collapse of the stabilization program. The central bank moved to floating exchange rate regime in spite of its explicit commitment to defend the domestic currency (TRL) and TRL depreciated sharply. Our aim is to identify large changes in TRL/USD exchange rate and analyze the investor’s decision problem just before the 2001 crisis. Fig. 1 shows the daily TRL/USD return series for the period of 1990–2002. The daily return series DF(t) are defined as DF ðtÞ ¼

F ðt þ 1Þ  1: F ðtÞ

4 An exchange rate is the rate at which one currency can be exchanged for another. There are two ways the price of a currency can be determined against another; (i) fixed, or pegged, rate is a rate the government (central bank) sets and maintains as the official exchange rate and (ii) floating exchange rate is determined by the private market through supply and demand. In case of the crawling peg, government reassesses the value of the peg periodically and then changes the peg rate accordingly. 5 Buy foreign assets means buy foreign currency and buy domestic assets means invest in money market in our formulation.

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Fig. 1. Points of interest and the possible volatility clusters are indicated for the TRL/USD return series.

Fig. 1 suggests that there are volatility clusters (observed variance of returns of financial asset is high for certain extended periods and then low for certain other periods) and they can be modeled by ARMAX/GARCH [7], ANFIS [21] or rule based fuzzy system modeling (RBFSM) methodology. All procedures are explained in [29,30] for input selection, knowledge discovery and comparison of the alternative modeling techniques.6 As it is explained in Section 3.3, supervised learning results in close to one in value of the fuzzifier (m = 1.3) which results in near crisp clusters. Since the effect of the other cluster centers are quite low for the crises periods (high volatility periods) FCM finds at least one prototype near the center of the volatility clusters (since DFDT and DRs are positive high). Generally after crises periods show very sharp declining of interest rates or in some cases sharp appreciation of the domestic currency (as a result of overshooting, some speculators bet that domestic currency deeply undervalued), it is expected and also found another cluster center representing after crises development of the selected variables (since DFDT and DRs are negative high or they are not positive high). The normal probability plot and the histogram of daily returns is given in Fig. 2. Points of interest are those that have 0.001 or less probability of occurrence which represent the daily returns during the days of crises. It is well known that the crises are rare events with very low probabilities of occurrence. Since our analysis is based on the perceptions of market participants, we are constrained to use only the publicly available data that can be accessed easily via the Turkish governmental web sites.7 3.2. Fuzzy rule base with FCM FCM provides a researcher the use of unsupervised learning as well as supervised learning based on performance criteria during system identification. Once the clusters are identified based on a priori number of clusters, c, and level of fuzziness, m, and calculation of the membership values for each vector can be determined in a straightforward manner as it is shown in [3]. If we define kth input vector Xk = {x1,k, x2,k, . . . , xnv,k} where nv is the number of variables, k = 1, . . . , nd, where nd is the number of data vectors, and the kth output as Yk, then a rule in general can be written as IF x1 2 X k is I 1;j AND x2 2 X k is I 2;j AND    AND xnv 2 X k is I nv;j THEN y 2 Y k is Oj

6

The important macroeconomic variables were found to be the lagged values of the change in interest rate (DRti) and the change in dollar denominated bank deposits (DFDTti). Explanatory variables were selected as, DRt3, DRt2, DFDTt2, DFDTt1, See [30]. 7 In our analysis, selected macroeconomic variables are obtained from (i) http://www.tcmb.gov.tr/, (ii) www.die.gov.tr, (iii) www.dpt.gov.tr, and (iv) www.hazine.gov.tr.

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Histogram of Change in TRL/USD, Monthly Between March 1990 and April 2002 15

Mean: -0.0014245, standard dev: 0.048608

10

Interest Points 5

0 -0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Change in TRL/USD

Fig. 2. (a) Normal probability plot of TRL/USD daily return series. (b) Histogram of change in TRL/USD.

where, Ii,j and Oj are linguistic assignments for input and output variables respectively, for the jth rule (j = 1, . . . , c*), where c* is the number of rules in fuzzy rule base. The number of rules is equal to the number of clusters, but determination of the optimum number of clusters is not straightforward. In this study we adopted supervised learning approach using minimum root mean square error (RMSE) criterion in a rule base determination for the selection of the optimum (c, m) pair.8 In addition to knowledge discovery with FCM, the other important part of FSM is inference module. As an FSM inference module, we use Takagi–Sugeno approach [35], with the local least squares estimator (LSE). 3.3. Fuzzy system model The data set is split into two subsets. The first data set represents training data and includes the first crisis (from 1990 to 1997). The second data set represents the test data and includes the second crisis (1997–2002). 8

For various cluster validation indices see for instance [4,5,13].

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By using the system behavior knowledge extracted from the training data we predict the values of the exchange rate for test period. The following steps are applied for rule based fuzzy system model (RBFSM). – Set search space for number of clusters and level of fuzziness: cluster list C = {c1, c2, . . ., cnc}, and level of fuzziness list M = {m1, m2,. . ., mnm}, where nc and nm denote the maximum number of clusters and the maximum value for the level of fuzziness. acut is set to zero. Learning Phase: Training data set is used for the learning exercise of the system parameters (clusters and LSE of cluster regression coefficients). – For each cp, p = 1, . . . , nc, and ml, l = 1, . . . , nm of search space:  Do FCM clustering by using (n + 1) dimension (input + output, {XjY}, space9), where Mahalanobis distance is implemented as the similarity measure. Mahalanobis distance of the vector k, to the cluster center j is calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T Distðjxk jy k j; vxjy;j Þ ¼ ðjxk jy k j  vxjy;j Þ R1 ðjxk jy k j  vxjy;j Þ where R is the covariance matrix and it is calculated by using Training data set. vXjY,j is the jth cluster centers (j = 1, . . . , cp) that is written explicitly as vX jY ;j ¼ ðx1;j ; x2;j ; . . . ; xnv;j ; y j Þ The values of the cluster centers projected to the input space are vx;c ¼ ðx1;c ; x2;c ; . . . ; xn;c Þ  Re-calculate the membership values for each input vector by using projected input space cluster centers, vx,j. Separate the clusters by using acut. Estimate the coefficients of the linear regression functions by least square estimation for each cluster as in fuzzy modeling of Takagi–Sugeno [35]. In this way, complexity and nonlinearity in the system are decomposed into linear sub-systems. As a result, the model generates the rules in the form of; if X is Aj then y j ¼ bTj X þ b0j or in case of small clusters where number of data vector is smaller than the number of variables, the rules have constant consequent and they are of the form of if X is Aj then y j ¼ b0j . As a mathematical representation, for each cluster we obtain a regression function of the form either fj ðx1 ; . . . ; xnv Þ ¼ b0j þ b1j x1 þ    þ bnv j xnv

or f j ðx1 ; . . . ; xnv Þ ¼ b0j

where j = 1, . . . , cp, i.e., there is a regression equation for each fuzzy cluster.  Estimate the output value for each training data vector and calculate the root mean square error RMSE as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u cp nd nd u1 X X X _ Y ¼ lk;j fj ðX k Þ and RMSE ¼ t ð^y k  y k Þ2 nd j¼1 k¼1 k¼1 – Select, {c*, m*} pair based on minimum RMSE, for c* 2 C and m* 2 M. Once the number of clusters and the level of fuzziness are selected, the test data set is used to assess and validate the model performance. 9 The interaction of factors behind the crises is assumed to be more complicated. In addition, it is assumed that there are always unobserved indicators/variables that affect the economic system. Number of variables is reduced (hence uncertainty is increased) to increase the interpretability of the system during input selection. The relations found in the economic variables are unstable and they are prone to change time to time. Hence there is no ‘deterministic’ relationships exist between them. Therefore clustering in input + output space may result in ‘one to many mapping’ could do serve better for our purpose.

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I. Ozkan et al. / Information Sciences 178 (2008) 1923–1934 Change in TRL/USD Predictions and Actual Values for Test Data

TRL/USD Return

0.4 Actual Values Fuzzy Est

0.3

Crisis-2001

0.2 0.1 0 -0.1

0

10

20

30 Months Error

40

50

60

0.2 Error

Error

0.1 0 -0.1 -0.2

0

10

20

30

40

50

60

Months

Fig. 3. Predictions for 2001 currency crisis.

3.4. Results Our dependent variable is the change in US dollar (DFt) and independent variables are the 2nd and 3rd lagged values of the interest rate (DRt2, and DRt3), and 1st and 2nd lagged values of change in dollar denominated bank deposits10 (DFDTt1, DFDTt2). The optimum number of clusters, and the level of fuzziness are found as c = 5 and, m = 1.3. The cluster regression functions of the model are as follows: DF 1 ðtÞ ¼ 0:0213 þ 0:046DRt3 þ 0:0077DRt2 þ 0:0558DFDT t2 þ 0:21DFDT t1 DF 2 ðtÞ ¼ 0:0987 DF 3 ðtÞ ¼ 0:5819 DF 4 ðtÞ ¼ 0:0194 þ 0:0015DRt3 þ 0:05DRt2 þ 0:279DFDT t2  0:008DFDT t1 DF 5 ðtÞ ¼ 0:052  0:041DRt3 þ 0:007DRt2 þ 0:767DFDT t2 þ 0:67DFDT t1 Clusters 2 and 3 do not have enough data vectors to estimate regression coefficients and they are used as constant functions. Cluster 3 represents the crisis and cluster 2 represents the right after crisis period. Fig. 3 shows the predictions of RBFSM for the test data set which includes the month of crisis in 2001. The solid line represents the actual values of the change in TRL/USD rate and dashed line represents the RBFSM predictions. RBFSM model captured currency crisis successfully experienced in 2001 based on the knowledge extracted from the data set that covers 1994 currency crisis. The rules that were generated by fuzzy clusters are given in [30]. As an illustrative example, the rule that represents the crisis period is given in Fig. 4. This rule is obtained by cluster projection. We label the rule associated with the month of crisis as ‘‘If the change in interest rate, 3 months ago, was high and the change in interest rate, 2 months ago, was high and the change in foreign dominated assets, 2 months ago, was high and the change in foreign dominated assets, last month, was moderately high then the change in TRL/USD exchange rate is to be high” 10

All the bank accounts denominated in foreign currencies and converted into US dollar equivalent.

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R3 ΔR(t-3)

ΔR(t-2)

ΔFDT(t-2)

ΔFDT(t-1)

ΔF(t)

Fig. 4. Classification rule for cluster 3.

The linguistic labeling given above is in contrast with the standard risk management procedure. First the money market is disturbed and interest rates start to increase. Investors start to increase their foreign currency dominated assets when they perceive that the increase in interest rate is permanent (it is not vanished in the second month). If the investors continue to increase their foreign currency nominated assets then speculators find a good opportunity to attack the currency. The supply of foreign currency dries suddenly since everybody wants to buy foreign assets. This is an easily understandable situation and it increases our trust in our model findings. Since the results are easily perceivable, one can use the fuzzy inference system as perception based inference and use the result as expectation. 3.5. Perception based inference The fuzzy inference system is used as perception based inference that is applied to support a decision by creating an expectation. Every new pattern is assessed by its similarity to fuzzy clusters found during clustering. Each cluster has its own action or more precisely its regression function. All estimates from clusters are weighted by membership values of a new case. The defuzzified result is used as the expectation for each new case. Assume that d is the decision problem, s, is the state of nature, f is the inference function, p is payoff, X ¼ ðx1 ; x2 ; . . . ; xn Þ is an information vector (input vector), vx ¼ ðvx;1 ; vx;2 ; . . . ; vx;c Þ is the cluster center matrix T and vx;j ¼ ðv1;j ; v2;j ; . . . ; xn;j Þ is the jth cluster center projected to input space. The decision problem can be presented as d ¼ ðX ; s; fp ¼ f ðX ; vx ÞgÞ Inference function can be written as f ðX ; vx Þ ¼

c X j¼1

lj ðX ; vx Þ gj ðX Þ Pc vx Þ i¼1 li ðX ; 

!

where gj is jth cluster’s regression function and lj is membership values to jth cluster for information vector X and the normalization term is equal to one. f ðX ; vx Þ is simply a smooth interpolation of the local regression models and the weights on each local model is the value of the membership function. Local regression function gj is nv X bj;i xi gj ðX Þ ¼ bj;0 þ i¼1

and membership function l() is given as "  2 #1 c  X DistðX ; vx;j Þ m1 lj ðX ; vx Þ ¼  i¼1 DistðX ; vx;i Þ States of the nature represent the central bank’s policy alternatives that investors may face. The obvious extreme policy alternatives for central bank are (i) continue with the crawling peg, or (ii) switch to floating exchange rate regime. For every state of the nature, payoffs are the expected gain/loss of the investors, or simply the results of expectation function f ðX ; vx Þ which updates itself in each period with incoming new information. In our example, if the central bank decides to abandon the crawling peg regime to switch floating exchange rate regime, investors may have the payoff as they expected. The other possible extreme state is that central bank may have or obtain necessary foreign reserves and decide to defend the currency. In this case, the change

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Table 1 Payoff matrix before crisis Investors’ action

Central bank’s policy decision

Buy foreign assets Buy domestic assets

Floating (%)

Crawling peg (%)

31 4

1 4

Table 2 Minimum and maximum predictions for USD returns Months

Minimum

Maximum

Actual

3 2 1 Crisis +1 +2 +3 +4

0.0150 0.1156 0.0282 0.1158 0.2015 0.0217 0.0449 0.0696

0.1156 0.2200 0.1424 0.5000 0.4710 0.0527 0.0683 0.1131

0.0080 0.0097 0.1048 0.2885 0.2626 0.0617 0.0722 0.0882

Table 3 Payoff matrix with minimum and maximum predictions Investors’ action

Buy foreign assets Buy domestic assets

Central bank’s policy decision Floating (%)

Crawling peg (%)

31 (11.6, 50) 4

1 4

in the foreign exchange rate is dictated by the crawling peg regime. Under these assumptions the payoff matrix can be constructed as shown in Table 1. In the calculations of payoffs, the change in the exchange rate is used as a proxy for return on foreign assets. For the return on domestic assets, monthly average return of ON REPO (a form of short term borrowing for dealers in government securities.) is used as a proxy. The payoffs calculated for the action ‘buy foreign assets’ are 31% and 1%. 31% is calculated by the expectation function f ðX ; vx Þ, and 1% is already decided and dictated in the economic stabilization program. Monthly average return of ON REPO was around 4% before the 2001 crisis. This is far less than the expected gain if the speculative attack ends successfully. Risk neutral investors would buy domestic assets only if the probability to continue the crawling peg regime is greater than 90%. FSM for perception based inference gives us a power to calculate the uncertainty associated with fuzziness by using different level of fuzziness. To explore this type of uncertainty, we use the levels of fuzziness between 1.01 and 2. For this purpose, m values were chosen as {1.01, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0}. The maximum and the minimum values of predictions are found by the application of the fuzzy inference schema for TRL/USD returns and they are given in Table 2. As it can be seen in Table 2, the predictions have the largest range for the month of crisis. Clearly it is the month where uncertainty is at the highest level. The minimum and maximum prediction (expectation) values can be introduced into the payoff matrix as support values. The payoff matrix is shown in Table 3 for the crisis month with minimum and maximum expectations. The uncertainty is another support signal for a decision maker. Predictions for the crisis month are spread under different levels of fuzziness. Knowledge extracted from the past experiences does not help to produce narrow predictions. Therefore it can be thought as one type of uncertainty, namely fuzziness, which is significantly high for the crisis month. The minimum and maximum values of expectations help us to calculate the critical probability values for the decision of investment choices. In our example, critical probability values for

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choosing the action ‘buy domestic assets’ are {71.7%, 94%} in the risk neutral world.11 This means that if the probability that the central bank’s ability to continue with the crawling peg regime is greater than 94%, then risk neutral investors chose ‘buy domestic assets’ action. This probability value is clearly too high. The expectation of a large devaluation grows to be self-fulfilling and speculative attack on domestic currency becomes inevitable. 4. Conclusion Currency crises are the results of human actions which are the results of the human decisions. Perception is an essential part of the human decision process. Uncertainty is the inherent part of any decision problem. At any moment of a decision problem, economic agents have to form expectations about uncertain future events based on their perception which is built on the past experiences of the perceivers. One can relate some recent discussions on similarity based decision and case based reasoning in the theory of decision making under uncertainty [15,16] to perception based decision. In this paper, we build a decision model to analyze currency crises under perception based decision theory framework. Within this framework, we were able to link fuzzy modeling methodology to the problem of decision making under uncertainty. Our results suggest that speculative currency attack is inevitable when increased uncertainty is perceived, and the expectations are revised accordingly. References [1] K. Asai, H. Tanaka, T. Okuda, Decision making and its goal in a fuzzy environment, in: L.A. Zadeh et al. (Eds.), Fuzzy Sets and Their Application to Cognitive and Decision Process, Academic Press, 1973. [2] J. Baron, Thinking and Deciding, Cambridge University Press, 2000. [3] J.C. Bezdek, Fuzzy Mathematics in Pattern Recognition, Ph.D. Thesis, Applied Mathematics Center, Cornell University, 1973. [4] J.C. Bezdek, Cluster validity with fuzzy sets, J. Cybernet. 3 (1974) 58–72. [5] J.C. Bezdek, Mathematical models for systematics and taxonomy, in: G. Estabrook (Ed.), Proceedings of the 8th International Conference on Numerical Taxonomy, Freeman, San Francisco, CA, 1975, pp. 143–166. [6] R.E. Belman, L.A. Zadeh, Decision making in fuzzy environment, Manage. Sci. 17 (1970) 151–169. [7] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econ. 31 (1986) 307–327. [8] D. Dubois, H. Prade, Criteria aggregation and ranking alternatives in the framework of fuzzy set theory, in: Zimmerman et al. (Eds.), Fuzzy Sets and Decision Analysis, North Holland, New York, 1984. [9] F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets Syst. 115 (2000) 67–82. [10] F. Herrera, E. Herrera-Viedma, J.L. Verdegay, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets Syst. 78 (1996) 73–87. [11] M. Delgado, F. Herrera, E. Herrera-Viedma, L. Martinez, Combining numerical and linguistic information in group decision making, J. Inform. Sci. 107 (1998) 177–194. [12] R. Flood, N. Marion, Perspectives on the recent currency crisis literature, Int. J. Finance Econ. 4 (1999) 1–26. [13] Y. Fukuyama, M. Sugeno, A new method of choosing the number of clusters for the fuzzy c-means method, in: Proceedings of Fifth Fuzzy Systems Symposium (in Japanese), 1989, pp. 247–250. [14] I. Gath, A.B. Geva, Unsupervised optimal fuzzy clustering, IEEE Trans. Pattern Anal. Mach. Intell. 11 (1989) 773–781. [15] I. Gilboa, D. Schmeidler, Act-similarity in case-based decision theory, Econ. Theory 9 (1996) 47–62. [16] I. Gilboa, D. Schmeidler, Case based decision theory, Quart. J. Econ. 110 (1995) 605–639. [17] S. Guillaume, M. Cemagref, Designing fuzzy inference systems from data: an interpretability-oriented review, Fuzzy Syst. IEEE Trans. 9 (3) (2001) 426–443. [18] E.E. Gustafson, W.C. Kessel, Fuzzy clustering with a fuzzy covariance matrix, in: IEEE CDC, San Diego, California, 1979, pp. 761– 766. [19] F. Hoppner, F. Klawonn, R. Kruse, T. Runkler, Fuzzy Cluster Analysis, John Wiley and Sons, 1999. [20] R. Jain, Decision making in the presence of fuzzy variables, IEEE Trans. Syst. Man Cybernet. 6 (1976) 698–703. [21] J.-S.R. Jang, ANFIS: adaptive-network-based fuzzy inference systems, IEEE Trans. Syst. Man Cybernet. 23 (1993) 665–685. [22] G.L. Kaminsky, Varieties of currency crisis, NBER Working Paper No. 10193, 2005. P This is calculated by using expected return, EðrÞ ¼ i pi ri where pi is the probability of ith return ri. Critical probability values are calculated by setting both actions’ payoffs to be equal. Let p denotes the probability of ‘floating’ policy decision of central bank, then 4% = (1  p){min, max} + p% ) pj{min=11.6,max=50} = {71.7%, 93.9%}. 11

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