CHAOS THEORY AND EXCHANGE RATE PROBLEM

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1 CHAOS THEORY AND EXCHANGE RATE PROBLEM Yrd. Doç. Dr TURHAN KARAGULER Beykent Universitesi, Yönetim Bilişim Sistemleri Bölümü Büyükçekmece- Istanbul Tel.: (212) Fax: (212) Key Words: chaos, modelling, exchange-rate Abstract Chaos theory has been well around in the world of science. In recent years, also the economists and management scientists have begun approaching the theory. It is well known fact that financial markets, particularly capital markets and market economy applications exhibit some chaotic behaviours. In addition to that, it has experimentally been proved that business cycles of both national and global economies also show chaotic characteristics. In this work, the application of the theory in a specific area, the prediction of exchange rates, is examined. 1-Introduction In science, for a long time, it has been assumed that regularity therefore predictability has been the centre of approaches to explain the behaviours of systems. Whereas in real life, it is a well known fact that systems exhibit unexpected behaviours which lead to irregular and unpredictable outcomes. This approach, named as non-linear dynamics, produces much closer representation of real happenings. The chaos theory which is one of methods of non-linear dynamics, has recently attracted many scientist from all different fields. In this work, the chaos theory is briefly introduced. A simple model of exchange rate problem by Ellis (1) is selected as an example to show a typical chaotic behaviour. 2-Non-linear systems and chaos In many problems, the behaviour of systems are accepted as linear types although their true characteristics are non-linear. The reason for the assumption is that modelling non-linear behaviours were highly difficult. This is why using some assumptions and making some simplifications for transferring non-linear relations into linear ones have been very practical and useful. This trend has been changing since powerful computers are available and usable for tackling complex calculations. Unlike a linear relationship in which a given cause has only one effect, in a non-linear relationship, a cause may have more than one outcomes. Thus non-linear equations can have more than one solution. Furthermore the additive property satisfied in linear systems do not 527

2 exist in non-linear systems. Because of existence of synergy within a non-linear system, understanding the behaviours of systems requires not a reductionist approach but a systemic approach in which all the patterns of behaviour are considered together as a whole. Non-linearity originates mainly from the presence of feedback systems representing the interaction between the various parameters of the system. The research work on non-linear feedback systems is relatively new. In 60 s, Lorenz s (2) and Feigenbaum s (3) works on atmospheric turbulence and bifurcations respectively are described as the first serious studies of the subject. Lorenz in his work on climate showed that atmospheric events are strongly unpredictable (all forecasts are valid for only a few days) and also very much sensitive to the changes in conditions (tiny changes that could not even be detected might lead to extensively different states of behaviours). However this unpredictability is restricted within specific boundaries. (in February, a temperature in Siberia can be anything but not as high as 40 0 C). This phenomena, in fact, is described as instability within stability or a mixture of order and disorder and represents the main characteristic of chaos. A chaotic nature can be highly complex and seemingly unstable. Yet they remain constrained due to the existence of attractors within the system. There are three types of attractors named as normal, periodic and strange. A normal or a point attractor leads the system to a steady state therefore to a predictable outcome. The periodic attractor shows itself in cases of regular and periodic motions such that a clock pendulum repeats the motion continuously. The rest of attractors which are neither point nor cyclic types are called as strange attractors and exhibit complex oscillations. The strange attractors are the ones lead to chaos as the path is non-periodic and non-stable but is not completely unstable either. The motions are constrained within the region of the attractors. This type of complex behaviours neither purely stable nor instable take place at the borders between stability and instability and describe a third state named as bounded instability or in other words chaos. Non-linear feedback systems or chaotic systems produce forms of behaviour that are neither stable nor unstable but continuously new and creative. 3-Chaos in exchange-rate The chaotic behaviours are commonly encountered in different fields of economics and management. For example in financial markets, particularly in capital markets, the chaotic characteristics are often observed. The business cycles of global and national economies also exhibit chaotic patterns. A detailed analysis of chaotic behaviours met in economics and management is given in Hobart paper by Parker and Stacey (4). In this section, the non-linear dynamics and the chaos are examined for the exchange-rate problem, in accordance with Li-Yorke theorem (5). In the model, the exchange rate is assumed to be determined by the interactions of speculators and trades. Non-linearity shows up in the model through the speculators demand for foreign currency. This nonlinearity is enough to generate chaotic dynamics at some values of parameters of the model. The speculators net demand for foreign currency is determined by the percentage deviation of the current exchange rate from the expected future exchange rate; 528 S t = α ( e f / e t - 1) (1) where α : sensitivity factor and its value is equal or greater than zero, e t : current domestic price of foreign currency, e f : expected future exchange rate.

3 In the formula above, if α = 0 there is no speculative demand for the foreign currency, and α = any deviation of the exchange rate from its expected value leads to infinite net demand. The above formula is indeed non-linear as more the current price of currency is undervalued, proportionally the demand gets larger. At small α values, little undervalued current price of the currency does not lead to large demands. Since in the model, the trade balance T t is assumed as a linear function of current and previous exchange rates therefore it is described as below T t = β (e t e f ) + γ (e t-1 e f ) (2) where β, γ >0. In the equation above, It is normally expected that the second part (the previous exchange rate) may have a bigger effect on T t. Therefore at much larger values of γ than β if so the system could exhibit a chaotic behaviour. When speculators do not intend to buy or sell, the exchange rate is said to be at its steady state value and at this state, the expected future exchange rate is determined by the fundamental variables such as the interest rate differential and relative money supplies which are not considered in this model. This is why time scale of the model is relatively short and limited to only daily or weekly periods in which the fundamentals are not expected to change. If the exchange rate is at its steady state value that e* = e t = e t-1 the trade balance becomes zero and the value of e* is assumed to be 1 for the sake of simplicity in the calculations. In the model, within each period the exchange rate is calculated at!s t = T t. From equations (1) and (2) β e t-1 e 2 t - {(β + γ) e t-1 - γ e 2 t-1 - α} e t - α e t-1 = 0 (3) At parameter values of α = 2, β = 3, γ = 20 as seen in figure 1, chaos appears in the model in accordance with Li-Yorke sense which states that chaos may happen if a function s value falls following 2 consecutive increases. The time variation of the exchange rate for 200 iterations shown in figure 2, also displays a chaotic behaviour as the exchange rate sometimes moves towards the steady state value but departs away shortly from this stabile point towards another unpredictable path. This feature may represent an actual case for the rate-time variation such that there might be periods of a highly volatile situation on the market followed by relatively a calm situation. In the model, if the current exchange rate is known, the next period exchange rate can be determined easily. However the main concern in the model is the high degree of dependency to the initial conditions and to the sensitivity parameters which cause the chaos. If the actual problem is chaotic, prediction of the future will be extremely difficult. Figure 3 displays the effect of initial conditions. A slight change in the value of starting point from to led to a great difference on the curve of exchange rate-time curves. Similarly Figure 4 displays the effect of slight change in the value of α from to Although the difference between values is as small as %0,1 still enough to generate a vast change in the result. In the work, the variation of actual exchange rates with the model is shown in figure 5. Although very little amount of data (Sterling & Dollar exchange-rates for 10 consecutive days in April 2000) is used in the graph, the trend looks interestingly similar. Obviously to be able to make a better comparison, much larger time scaled data along with optimum parameter values need to be used in the model. 529

4 Figure 1 The exchange-rate variation e t = f(e t-1 ) Figure 2 The time series of exchange rate (200 iterations) Figure 3 Sensitivity to initial values 530

Figure 4 Sensitivity to α parameter values Figure 5 Actual Exchange-Rate & Model Results 4-Conclusion The chaos theory and non-linear dynamics of systems behaviours are briefly introduced.

5 Figure 4 Sensitivity to α parameter values Figure 5 Actual Exchange-Rate & Model Results 4-Conclusion The chaos theory and non-linear dynamics of systems behaviours are briefly introduced. A simple model of exchange rate-problem is used to display the chaotic behaviour. As the results taken from the model show that the main sources of chaos for this particular problem are the high sensitivity of the system output to the initial values of input and slight changes of system parameter values. When a system exhibits a chaotic behaviour, it will be almost impossible to predict long-term outcomes as the future of a chaotic system is open-ended and inherently unknowable. Currently, there is no alternative suggestion for the chaotic systems except change and see what happens policy. Only the very near future can be predictable if distortion or noise free input and parameter values are provided to the models. Thus the big task for understanding behaviours of these systems is to obtain reliable and highly precise massive data 531

6 5-References [1] Ellis, J. M. An Investigation of Nonlinearities and Chaos in Exchange-Rates, Doctoral dissertation, University of Oregon (1992) [2] Lorenz, E. N. Deterministic Non-period Flows, Journal of Atmospheric Sciences, 20, pp: (1963) [3] Feigenbaum, M. Quantitative Universality for a class of Non-linear Transformations, Journal of Statistical Physics, 19, (1978) [4] Parker, D. & Stacey, R. Chaos, Management and Economics published by Institute of Economics Affairs, (1994) [5] Li, T. Y. & Yorke, J. A. Period Three Implies Chaos, American Mathematical Monthly, 82, pp:

The Nonlinear Real Interest Rate Growth Model : USA

Advances in Management & Applied Economics, vol. 4, no.5, 014, 53-6 ISSN: 179-7544 (print version), 179-755(online) Scienpress Ltd, 014 The Nonlinear Real Interest Rate Growth Model : USA Vesna D. Jablanovic