# On the Adequacy of Mathematical Models of the Exchange Rate

Currently, there are many quantitative approaches based on the search for nonlinear and chaotic dynamics in behavior financial markets. Among these approaches, fractal and multifractal methods play a significant role, which are also used to predict stock market crises. Among them we can single out a class based on the study of local regularity of financial time series. For this purpose, various indicators are usually constructed, analyzing which one can draw conclusions regarding the regularity of the series exchange rate. To obtain an effective forecasting model, it is necessary to thoroughly analyze the nature of the impact of factors affecting the formation of the exchange rate. An important role here is played by the availability of statistical data related to the factors under consideration and the dynamics of the exchange rate. Market development is determined by fundamental factors, but the opposite is also true - fundamental factors are determined by the market, i.e. by the behavior of market participants, their assessments and expectations. At the same time, the ability to correctly assess the development of market situations depends on the ability to anticipate the prevailing expectations of market participants, rather than on the ability to predict changes in the real world.

The most important requirement for a mathematical model of the exchange rate is the requirement for its adequacy - the correct correspondence of the studied real object in relation to the selected system of its properties. This is understood, first of all, as:

1) proper qualitative description the properties of the object in question. For example, the possibility, based on the study of the model, to make a correct conclusion about the direction of change of any quantitative characteristics of these properties, about their interrelation, about the nature of the object's fluctuations, about the stability of its state or evolution.
2) a correct quantitative description of these properties with some reasonable accuracy. According to whether the second condition is set or not, we speak respectively about quantitative or qualitative models. Instead of quantitative adequacy, one also speaks of model accuracy.

It is natural to speak not just about the adequacy of the model, but also about greater or lesser adequacy. It should be emphasized that it should be considered only according to certain attributes - properties taken as the main ones in this study. If they are not explicitly stated, they should be implied or clarified in the course of the study. Therefore, the term "mathematical modeling" should be applied to the field of applied mathematics, which includes both the construction and study of mathematical models and the creation of computational algorithms and programs that implement these algorithms.

Earlier in the magazine ForTrader.org mathematical modeling of the system of trend indicators of technical analysis and heuristic methods based on the use of expert assessments; mathematical modeling of exchange rate forecasting based on analytical dependencies and factors affecting the exchange rate were considered. Based on all of the above, we can draw the main conclusion: any predictive model of exchange rate movements must include a huge number of variables, and in all cases a robust statistical test that takes into account factor displacement and randomness is required.

Academic institutions have made attempts to build an adequate mathematical model of exchange rate forecasting, but each time these attempts were unsuccessful. According to specialists, such a model should be based on purchasing power parity theoryHowever, the forecasts of exchange rates based on the data on money supply and incomes give significant discrepancies with reality. However, it is possible to forecast the trend component of the exchange rate and in the creation of a decision support system in the tasks of fundamental analysis and forecasting of macroeconomic dynamics and exchange rates along with the development of its functional structure, and information representation of the processes of international capital flows, the development of national economies and financial markets. The most important point is the choice of mathematical methods and models for solving such important tasks included in the functional structure of the system under consideration as the assessment and forecasting of processes of macroeconomic dynamics and exchange rates.

As noted above, the complexity of the development of macroeconomic processes of international capital flows, national economies and financial markets is such that it is advisable to use adaptive statistical models to solve the problems of identification and medium-term forecasting of these processes and exchange rates [4,5].

## Economic models

Mathematical models of the market economy have long been developed and relatively well studied, which cannot be said about the models of planned and, even more so, transition economy. The latter cannot be (even in principle) reduced to models of the classical type, because it must reflect the main features of both economic systems. An effective methodological approach to the construction of models possessing this synthetic property is to first build models of balances of material and financial flows, which in a certain sense are universal, i.e. suitable for describing the economy of any type. They are "deliberately" not closed, and the way of their closure directly depends on the behavior of economic agents, government policy, etc. When setting different types of production relations (scenarios) and thus setting different ways of closure, models for different types of economies are obtained (see Fig. 1)[4].

The given block diagram at the macro level reflects rather complex relationships of economic partners regulating the production, exchange and distribution of products and services, which had developed in the Russian economy by the end of the first third of the 1990s. It can be seen that the model corresponds to a mixed, transitional economy: in addition to the state (the main agent of the planned system), it includes, for example, commercial banks operating in a competitive environment with the purpose of profit-making.

Without fully describing all the assumptions about production relations embedded in the model, let us characterize some of them:

1) sectors experiencing import competition and export sectors are distinguished;
2) labor collectives and administration in the sectors are interested in increasing the wage fund and, despite the reduction in demand for products, achieve this through mutual non-payments and soft loans from the Central Bank (CB); net investment is absent, production capacity decreases;
3) changes in production conditions affect wages but not employment; there are no enterprise bankruptcies, nominal unemployment is low;
4) only raw materials are exported and only consumer goods are imported;
5) markets are controlled by an industrial-financial oligarchy, with exporters at the top;
6) the macroeconomic policy of the state is reduced to the determination of tax rates, volumes of favorable loans from the Central Bank, government purchases, payments to the population from the state budget, subsidies to enterprises, etc.

The formulated scenario is translated into a general model, resulting in a specific transition model. In mathematical terms, it represents a bulky and complex system of nonlinear ordinary differential equations (augmented by a large number of algebraic equations) These inputs are relative to a few dozen basic economic variables (e.g., outputs of various products) and contain many decision-determining characteristics and parameters (e.g., inflation expectations of the population). These inputs are located and refined, as are the scenarios for the current state of the system.

For example, in one of the variants of the model it was considered that the Central Bank does not conduct operations on the domestic foreign exchange market, then by the end of 1993 the dollar exchange rate should, according to the model, have risen to 4000 r./dollar. However, since the middle of 1993, the Central Bank started the corresponding actions, and in reality the exchange rate reached "only" 1300 r./dollar. The model was modified to take into account the new policy, and the time series from that moment coincide well with the actual ones (see Fig. 2) [ 4 ].

Computational experiments both with this and with other models of the transforming economy constructed in the same way allowed us to draw a number of rather important general conclusions. In particular, it was established that the transition from the planned Soviet economy, which almost collapsed in the late 80s and early 90s, to the effective equilibrium state of the new market economic system, even in the best case, will take at least ten years, will be accompanied by high structural unemployment and bankruptcies of many enterprises.

## "Quasi-equilibrium" of the Russian economy

Another, no less significant result of experimenting with models is that it was possible to establish the "hit" of the post-reform Russian economy in a special type of quasi-equilibrium state, different from the models studied in classical political economy.

The model is also used to conduct more detailed studies of various specific issues of current economic policy. This policy is subject to the natural requirement of "security", which is interpreted in the model as the inadmissibility of abrupt destruction of the established and actually existing economic relations and structures, even if not too effective. This is by no means a far-fetched problem, since we are not talking about someone's conscious desire for destruction, but about the "unprofessional" use of economic instruments in a very complex and unstable situation. A typical task is to determine the size of soft loans given by the state to producers at virtually negative interest rates. The model has shown that extremes are very dangerous. Absence of soft loans leads not only to a sharp (within weeks and months) suppression of inflation (and even to the deflation), but also to the destruction of production structures, most of which have already adapted to inflation. Their incomes are reduced to such an extent that a mass "flight" of employees of enterprises and intensification of production decline are inevitable. In the opposite case, with very large concessional loans and the hyperinflation generated by them, the system of commercial banks collapses. They plan their profits based on inflation rates. As long as its growth is not too great, their actions, based even on rough forecasts, ensure steady profits. With hyperinflation, the inevitable inaccuracy of forecasts leads to systematic losses of banks and the actual "disappearance" (in the relative sense) of their equity capital. Let us also mention two other specific current events that are significant for the Russian economy and were analyzed by means of computational experiments with the model.

The first of which is. "Black Tuesday" October 11, 1994, when there was a catastrophic fall of the ruble exchange rate against the dollar, which returned to approximately the same level a few days later. Sufficient adequacy of the model made it possible not only to describe (ex post facto) the dynamics of the main economic macroindicators after "Tuesday", but also to reasonably identify the economic agents that, albeit unwittingly, won (basic industries, state budget revenues) and lost (the bulk of the population, importers) as a result of this event.

The second is. army operation in ChechnyaThe Chechen crisis, which began at the end of 1994 and required significant additional state expenditures for its implementation and for measures to restore the republic's economy and social sphere (according to various estimates, from several trillion to tens of trillions of rubles). The main conclusion from the modeling results is that although the "Chechen crisis" cannot cause hyperinflation, even with a strict anti-inflationary policy of the state it makes a significant contribution to inflation and contributes to the decline in real incomes of the majority of the population [4].

## Economic efficiency and exchange rates

However, the model does not take into account the uncertainty of the value of exports in national currency if the invoice is in foreign currency, which may deter exports because there is doubt that the exported goods can ultimately be realized at a profit. Uncertainty in the domestic currency value of imports priced in foreign currency increases the risk of import losses because the price may not be competitive in terms of domestic currency. Thus, uncertainty in the exchange rate may hinder the development of foreign economic activity. In addition, the depreciation of foreign currency, which undermines the income from exports of goods in terms of national currency, is accompanied by an appreciation of the national currency and leads to an increase in the price of exports in foreign currency, which reduces the competitiveness of the enterprise. This effect will be particularly negative in conditions of price-sensitive demand.

Economic risk arises from the unfavorable impact of the exchange rate on changes in commodity prices for manufactured or purchased products, which in turn affects the economic position of the enterprise. For example, in conditions of decrease in the foreign currency exchange rate and the corresponding increase in the level of commodity prices, an exporting enterprise may reduce the level of its turnover and lose part of its market for finished products. The similar situation is faced by importing enterprises, which receive invoices in foreign currency in conditions of its exchange rate growth, which negatively affects the sales volumes of imported products, when, for example, their competitors are domestic manufacturers. An importing enterprise comes to the same situation when it discovers that a foreign supplier changes prices for its products in accordance with the growth of the exchange rate of its national currency or inflation rate.

Economic risk also arises when an enterprise that sells its products exclusively in the domestic market and has costs paid only in local currency incurs losses due to the appreciation of the local currency because competitive imported goods may be cheaper.

The exchange rate, which is the dollar exchange rate, and commodity markets are thought to move in opposite directions. An appreciation of the dollar counteracts inflation and ultimately causes commodity prices to fall. In turn, a fall in commodity prices causes the lower interest rates and higher bond prices. And rising bond prices contribute to the growth of the stock market.

A falling dollar causes the exact opposite effect - rising inflation (rising commodity prices), falling bond and stock prices. The peak of the bond market against the background of economic recovery serves as a signal of the economy's transition from a state of normal inflation-free growth to a phase of "unhealthy" growth. Investors sell bonds due to increased inflationary pressures and fears of a subsequent increase in interest rates. After a while, rising interest rates begin to put bearish pressure on the stock market and it also turns downward. When rising inflationary pressures cause interest rates to peak, investors' desire to buy dollars begins to reverse. Commodity markets also begin to turn downward due to a possible subsequent slowdown in production. Then, due to the slowdown in economic growth, the need for goods and money decreases, inflationary pressures subside, and commodity prices begin to fall.

As commodity prices and interest rates fall, the bond market begins to rise. Gradually, the stock market turns after it. After that, the commodity market also enters the growth phase and inflationary pressure begins to form. Investors have a renewed desire to buy the dollar. This example shows the close connection of exchange rates with the processes of macroeconomic dynamics, which acts as an external environment of foreign economic activity, which necessitates their joint consideration when solving the tasks of forecasting the development of the situation on the currency markets.

## Unstable macrosystems

In most known developments in the field of fundamental analysis and forecasting of macroeconomic dynamics, medium-term forecast estimates of macroeconomic processes, including exchange rates, are obtained using multifactor econometric models based on the hypotheses of "efficient market" and "rational expectations", as well as the linear paradigm of market situations development. In particular, the model of the U.S. economy, developed at the CEMI of the USSR Academy of Sciences and designed to analyze the structure of the economy, medium-term forecasting of the main trends of economic development, as well as quantitative assessment of the effectiveness of government regulation. This model is built as a standard econometric module and includes 41 equations, including 27 linear regression equations and 13 algebraic equations. At the same time, the model equations are conventionally combined into five blocks: the block of final demand, production, income, public finance, and monetary policy. A similar model of econometric forecasting of the development of the American economy was developed in the NEMI of the USSR Academy of Sciences. The econometric model of US money circulation developed by M. Boton and T. Naylor contains 17 equations that explain the behavior of 6 intermediate benchmarks: means of circulation, perpetual deposits, time deposits, interest rates on government bonded loans and on mortgages on residential buildings, as well as the number of mortgages.

As applied to the macrosystem under consideration, this means that under certain conditions structural instability may occur in the ring structures of cause-effect relations of the system under study, leading to a significant change in the values of fundamental indicators included in this structure, including the exchange rate. Recognizing and modeling the development of such situations is an important point in the system of analyzing and forecasts exchange rates and macroeconomic dynamics in risk assessment. Since this ensures the consideration of trends in the development and interaction of factors affecting the exchange rate, which in turn strengthens the capabilities of the forecasting system.

Within the framework of forecasting economic processes, the development of the adaptive approach takes place along three directions: the first of them is focused on the complication of adaptive forecasting models; the idea of the second direction is to improve the adaptive mechanism of forecasting models; the third direction realizes the approach of joint use of adaptive principles and other forecasting methods.

Analysis of known developments in the field of economic forecasting at the macro level showed that in most of the works the authors limit themselves to the construction of linear econometric macroeconomic models based on the hypotheses of "efficient market" and "rational expectations". However, the existing contradictions and imperfections in these hypotheses are often the reason for obtaining erroneous results. Besides, these models do not provide for taking into account the changes in the structure of fundamental indicators' influence, as well as for taking into account the attitude of market participants to uncertainties and contradictions in the development of market situations, etc. This makes it expedient to develop other approaches and methods to the tasks of medium-term forecasting of exchange rates and fundamental analysis of macroeconomic dynamics. At the same time, it is important that these methods ensure the solution of the problem of forecasting macroeconomic processes taking into account the uncertainty in their development, associated with the complexity of the structure of relations of fundamental macroeconomic indicators, non-stationary nature of the influence of some fundamental indicators on others, as well as with the subjective perception and assessments of the participants of macroeconomic dynamics, as well as the impact on macrodynamics of the behavior of active elements of the system, which is still insufficiently reflected in the known times.

## Literature

1. Sobolev V. В. Currency dealing on the financial markets / Yuzh.-Ros. gos. tehn. un. (NPI). - Novocherkassk, 2009. - 442 с.
2. Sobolev V. V. Financiers/ Yuzh.-Ros. gos. tehn. un. (NPI).-Novocherkassk, 2009.-315 p..
3. Soros J. The Alchemy of Finance : Translated from English - M.: "Infra-M", 1996. - 416 с.
4. Samarskiy A.A., Mikhailov A.P. Mathematical Modeling: Ideas. Methods. Examples.- M.: Nauka, 1997 - 320 pp.
5. Lukashin Yu. P. Adaptive methods of short-term forecasting of time series: Textbook. - Moscow: Finance and Statistics, 2003. - 416 с.
6. Mishkin F. Economic Theory of Money, Banking and Financial Markets: Textbook for Universities / Translated from English by D.V. Vinogradov, ed. by M.E. Doroshenko. Doroshenko. - Moscow: Aspect Press, 1999. - 820 с.
7. Peters E. Fractal Analysis of Financial Markets: Application of Chaos Theory in Investments and Economics. - Moscow: Internet-trading, 2004. - 304 с.
8. Naylor T. G. Machine simulation experiments with models of economic systems. - M.: Mir, 1975 - 432 p.