Or does a higher rate of money supply growth lower interest rates? Methods for forecasting exchange rates.

To model interest rate levels in statistics, use Various types equations, including polynomials of various degrees, exponentials, logical curves and other types of functions.

When modeling interest rate levels, the main task is to select the type of function that most accurately describes the development trend of the indicator being studied. The mechanism for determining the function is similar to choosing the type of equation when constructing trend models. In practice, the following rules are used to solve this problem.

1) If the dynamics series tends to monotonic increase or decrease, then it is advisable to use the following functions: linear, parabolic, power, exponential, hyperbolic or a combination of these types.

2) If the series tends to rapid development indicator at the beginning of the period and decline by the end of the period, it is advisable to use logistic curves.

3) If a series of dynamics is characterized by the presence of extreme values, then it is advisable to choose one of the variants of the Gompertz curve as a model.

In the process of modeling interest rate levels great importance is given to careful selection of the type of analytical function. This is explained by exact specification The pattern of development of an indicator identified in the past determines the reliability of the forecast for its development in the future.

Theoretical basis statistical methods used in forecasting is the property of inertia of indicators, which is based on the assumption that the pattern of development that existed in the past will continue in the predicted future. The main statistical forecasting method is data extrapolation. There are two types of extrapolation: prospective, carried out into the future, and retrospective, carried out into the past.

Extrapolation should be assessed as the first step in making final forecasts. When applying it, it is necessary to take into account all known factors and hypotheses regarding the indicator being studied. In addition, it should be noted that the shorter the extrapolation period, the more accurate the forecast can be obtained.

In general, extrapolation can be described next function:

y i + T = ƒ (y i , T, a n), (26)

where y i + T – predicted level;

y i – current level of the predicted series;

T – extrapolation period;

and n is the parameter of the trend equation.

Example 3´´. Based on the data in example 3, we will extrapolate to the first half of 2001. The trend equation is as follows: y^ t = 10.1-1.04t.

y 8 = 10.1-1.04*8 = 1.78;

y 9 = 10.1-1.04*9 = 0.78.

As a result of data extrapolation, we obtain point forecast values. The coincidence of actual data for future periods and data obtained by extrapolation is unlikely for the following reasons: the function used in forecasting is not the only one to describe the development of the phenomenon; the forecast is carried out using a limited information base, and random components inherent in the levels of the initial data influenced the result of the forecast; unforeseen events in the political and economic life societies in the future can significantly change the predicted development trend of the indicator being studied.

Due to the fact that any forecast is relative and approximate, when extrapolating interest rate levels, it is advisable to determine the boundaries of the confidence intervals of the forecast for each value y i + T. The boundaries of the confidence interval will show the amplitude of fluctuations in the actual data of the future period from the predicted ones. In general, the boundaries of confidence intervals can be determined by the following formula:

y t ±t α *σ yt , (27)

where y t is the predicted level value;

t α – confidence value determined based on Student’s t-test;

σ yt – root mean square trend error.

In addition to extrapolation based on the alignment of series according to the analytical function, the forecast can be carried out using the extrapolation method based on the average absolute increase and the average growth rate.

The first method is based on the assumption that The general trend development of interest rate levels is expressed by a linear function, i.e. there is a uniform change in the indicator. To determine the predicted level of loan interest for any date t, the average absolute increase should be calculated and sequentially summed up by the last level of the dynamics series as many times as the time periods for which the series is extrapolated.

y i + T = y i + ∆¯*t, (28)

where i is the last level of the period under study for which ∆¯ is calculated;

t – forecast period;

∆¯ - average absolute increase.

The second method is used if it is assumed that the general development trend is determined by an exponential function. Forecasting is carried out by calculating the average growth rate raised to a power equal to the extrapolation period.

y i + T = y i * К t ¯. (29)

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In order to predict the further dynamics of a currency pair, it has been developed great amount techniques. However, quantity has not translated into quality, and getting a fairly effective forecast is not the most simple task. Let's look at the four most common methods for forecasting currency pair rates.

Purchasing power parity (PPP) theory

Purchasing power parity (PPP) is perhaps the most popular method. It is mentioned more often than others in economics textbooks. PPP theory is based on the principle of the “law of one price,” which states that the cost of identical goods in different countries should be the same.

For example, the price of a cabinet in Canada should be similar to the price of the same cabinet in the United States, taking into account the exchange rate and excluding shipping and exchange costs. That is, there should be no reason for speculation, to buy cheap in one country and sell more expensive in another.

According to PPP theory, changes in the exchange rate should compensate for . For example, this year prices in the United States should increase by 4%, in Canada over the same period - by 2%. Thus, the inflation differential is: 4% - 2% = 2%.

Accordingly, prices in the US will rise faster than in Canada. According to PPP theory, the US dollar must lose about 2% in value for the price of the same good in two countries to remain approximately the same. For example, if the exchange rate was 1 CAD = 0.9 USD, then according to PPP theory the predicted rate is calculated as follows:

(1 + 0.02) x (0.90 USD per 1 CAD) = 0.918 USD per 1 CAD

That is, to comply with PPP, the Canadian dollar must rise in price to 91.8 American cents.

The most common example of the use of the PPP principle is the Big Mac index, which is based on a comparison of its price in different countries, and which demonstrates the level of undervaluation and overvaluation of a currency.

The principle of relative economic stability

The method of this hike is described in the name itself. The economic growth rates of different countries are taken as a basis, which makes it possible to predict the dynamics of the exchange rate. It is logical to assume that stable economic growth and a healthy business climate will attract more foreign investment. To invest, it is necessary to purchase national currency, which, accordingly, leads to an increase in demand for national currency and its subsequent strengthening.

This method is suitable not only for comparing the state of the economy of two countries. With its help, you can form an opinion about the presence and intensity of investment flows. For example, investors are attracted to higher interest rates to maximize returns on their investments. Accordingly, the demand for the national currency is growing again and it is strengthening.

Low interest rates can reduce the flow of foreign investment and stimulate domestic lending. This is the case in Japan, where interest rates have been cut to record lows. There is a trading strategy based on the difference in interest rates.

The difference between the principle of relative economic stability and the PPP theory is that with its help it is impossible to predict the size of the exchange rate. It gives the investor only a general idea of the prospects for a currency to strengthen or weaken and the strength of the momentum. To get more full picture, the principle of relative economic stability is combined with other forecasting methods.

Construction of an econometric model

A very popular method for forecasting exchange rates is the method of creating a model that describes the relationship between a currency exchange rate and factors that, in the opinion of an investor or trader, influence its movement. When compiling an econometric model, as a rule, values from economic theory are used, but any other variables that have a significant impact on the exchange rate can be used in the calculations.

Let's take, for example, making a forecast for the coming year for the USD/CAD pair. We choose the key factors for the dynamics of the pair: the difference (differential) between the interest rates of the US and Canada (INT), the difference in and the difference between the growth rates of personal income of the US and Canada (IGR). The econometric model in this case will have the following form:

USD/CAD (1 year) = z + a(INT) + b(GDP) + c(IGR)

The coefficients a, b and c can be either negative or positive and show how strong the influence of the corresponding factor is. It is worth noting that the method is quite complex, however, if you have a ready-made model, to obtain a forecast it is enough to simply substitute new data.

Time series analysis

The time series analysis method is purely technical and does not take into account economic theory. The most popular model in time series analysis is the autoregressive moving average (ARMA) model. The method is based on the principle of predicting price patterns of a currency pair based on past dynamics. The calculation is carried out by a special computer program based on the entered parameters of the time series, the result of which is the creation of an individual price model for a specific currency pair.

Undoubtedly, forecasting exchange rates is an extremely difficult task. Many investors simply prefer to hedge currency risks. Other investors recognize the importance of forecasting exchange rates and seek to understand the factors influencing them. The above methods can be a good help for such market participants.

Rice. 9.6. Interest rate parity band

ability to predict changes in legislation and in the conditions for concluding and executing contracts.

The following objectives of forecasting the exchange rate can be distinguished;

a) currency risk management.

This goal is the leading one, but not the only one; b) short-term financing decisions. The currency in which we borrow must have a desirably low interest rate and a tendency to weaken during the financing period; c) short-term investment decisions. The currency in which we place deposits or provide loans should have the highest interest rate possible and tend to appreciate during the investment period; d) assessment of long-term investment projects. If we are going to invest money in another country, then the corresponding currency should ideally weaken. But if we invest funds within our country for subsequent export, then it will be desirable to strengthen the corresponding currency;

e) assessment of long-term borrowings. In principle, the approach is the same as for short-term financing, but the implementation of this forecasting goal is much more difficult; f) management of the flow of income received* abroad. If the currency in which the income is received strengthens, then this income most likely should be repatriated, that is, taken “home.” But if the opposite trend in the exchange rate is predicted, then it is best to reinvest them abroad.

The above list of goals shows the very significant impact that can have effective methods forecasting exchange rates on the profitability of international transactions. This determined the efforts and resources spent on solving the problem of their development. This task became extremely relevant in connection with the introduction of a system of floating exchange rates into world practice*3 in the mid-1970s In the period since then, an impressive arsenal of various forecasting methods has been created and extensive experience in their application has been accumulated.

The developed methods were based on theoretical studies on the movement of exchange rates, carried out in world financial science for last decades, which were discussed above. Over the past twenty to thirty years, a large number of methods for assessing the future movement of exchange rates have been developed and practically tested. They are based on four basic approaches: 1) technical forecasting; 2) fundamental forecasting; 3) forecasting based on market expectations; 4) forecasting based on expert assessments.

The first two approaches come from two generally accepted forecasting methods, applied not only to exchange rates, but also to the prediction of many other socio-economic parameters. The features of their application in foreign exchange markets are discussed in this section. The third approach is specific to forecasting exchange rates, so special attention will be paid to it. Finally, the fourth approach, using the intuitive opinions of experts, is quite obvious, and only some comments on the appropriateness of its use will be given below.

An approach based on technical forecasting can be formally presented as follows:

e(= a0 + a( x et_, + a2x ec_2 + + a„x ec_„, (9.17)

where e is the change in the exchange rate in the forecast period t\

e, -2, ???, e, -„ - changes in the exchange rate of the same currency in periods t - 1, t- 2, ..., t - n)

ak - statistical (weighting) coefficients obtained by correlation-regression or other methods (k from 0 to n);

n is the number of past periods on the basis of which the forecast is built.

Technical forecasting has another name in Russian, namely forecasting based on time series. Currently, quite a lot of new sophisticated methods for such forecasting have appeared, using various non-linear functions of past and future data, graphical analysis of exchange rate fluctuations, expert assessment of the possibility of transferring from previous periods some patterns of movement of this exchange rate, the so-called time series models. etc. Often this actually allows one to obtain satisfactory results. Nevertheless, at its core, this approach assumes the permissibility of extrapolation, the extension of development trends of a particular phenomenon that have developed in the past into the future. Both its possibilities and its limitations follow from this premise. The economic interpretation of the forecast is quite simple, but any significant change in the existing trends turns out to be detrimental to the quality of forecasting the future value of the exchange rate.

Fundamental forecasting, unlike technical forecasting, is based not on extrapolation of the past trend of changes in the exchange rate itself, but on the study of its dependence on certain factors outside the foreign exchange market. In this regard, in Russian-language literature, it is often called factorial. In formal form, this approach can be written as follows:

es =aa + akhhi + ... + apxP"(+ap^ +ui.x +- +ap+tut1_i (9.18)

where xx, ..., xn" are factors affecting the foreign exchange rate, the values of which are also predicted for the period

y„, _ „ ..., ut," _, - factors affecting the foreign exchange rate, the values of which can be calculated on the basis of actual data for period t -

n, t - the number of factors of the first and second groups.

The identification of these two groups of factors is necessary, since it reflects the essence of the approach to forecasting the exchange rate. Indeed, the construction factor models in the area under consideration should be based, first of all, on generally accepted theoretical considerations of the influence of certain parameters on the exchange rate.

The theory of the international Fisher effect discussed above defines a two-factor model in which the future value of the exchange rate depends on both the comparative rates of inflation and the comparative level of interest rates in the two countries between whose currencies the desired exchange rate is predicted. In this case, the inflation rates are taken for the period for which the forecast is made, i.e. they themselves must be forecast. You can also take the inflation rates for the previous period for which they are already known. However, this requires appropriate justification, i.e., determining what is more statistically significant: the relationship between the movement of the exchange rate and the accompanying inflation rates or those prevailing in the past period, and whether such a replacement will lead to a loss in the quality of the forecast.

As for interest rates, considered as a factor in this theory, at first glance they are valid for the forecasting period and in this sense are clearly determined already at the beginning of the period, and therefore can be interpreted as a factor of the previous period. However, this is not quite true. The fact is that we usually make a forecast in advance, with some anticipation, which means that interest rates for the forecast period are not yet known and themselves must be the subject of the forecast. As in the previous case, the rates of the previous period can also be considered as a factor, but the same additional justification is required here. Thus, the use of fundamental forecasting is associated with a number of problems, the degree of resolution of which directly affects the quality of the forecast.

Among these problems, it is first necessary to pay attention to the following. The first is to find the periods for which the factors are taken. In this case, we are talking not only about the forecasting period and the period immediately preceding it. It is possible that the quality of the forecast may be higher if earlier periods are taken or if the model includes values of the same factor for several periods: 4 ^ - 1, ? - 2, etc. This may be especially appropriate when building a short-term forecast, for example, a month or a week in advance.

If it turns out to be justified to use the values of factors in the forecast period, then naturally the problem arises of how to obtain these values.

After resolving issues related to determining the required set of factors, problems arise in constructing a correlation-regression or some other relationship between the factors under consideration and the desired value. At the same time, there are traditional dangers in the process of constructing regression equations and, above all, the possibility of missing unaccounted but significant factors, which makes the model as a whole not entirely adequate.

Finally, another very significant problem is the stability of the regression coefficients obtained as a result of calculating the regression equation. The instability and variability of these coefficients can stem from two main reasons. The first is that if the set of factors used or the method of calculating their values changes (for example, calculating given value for period t or? - 1) regression coefficients may change, and therefore reflect the unequal elasticity of changes in the exchange rate for the same factor.

The second reason stems from the need to use predictive values of factors in certain cases. Such a forecast cannot be absolutely accurate and, moreover, in most cases it is inappropriate to refine it, for example, to average it, since this leads to an artificial smoothing of the resulting forecast values of the exchange rate, which does not reflect the full complexity of the relationship under study.

In order to better understand the last position and, in general, more clearly imagine the ways of using the economic interpretation of the results of fundamental forecasting, we will give an example.

Consider a two-factor fundamental forecasting model of the following type:

ec=a0+axxc+a2y1L, (9.19)

Where x1 is the difference in interest rates in two countries predicted for period I;

g/(_, - actual for the period? - 1 value of the difference in inflation rates between countries.

Let us assume that a statistically significant regression equation has been obtained for this model

e(=0(2-0^c(+O5y(_1. (9.20)

This equation can be interpreted in accordance with formulas (9.6) and (9.11) as follows.

Each percentage increase in the inflation rate in some conditional “our” country compared to the inflation rate in “another” country in the past period leads to a 0.5% increase in the direct exchange rate of “our” currency against the “other” currency in the forecasts. - In the period under review. An increase in the direct exchange rate of “our” currency, i.e. an increase in the price of a foreign currency, means a reduction in price, a weakening of “our” currency.

On the other hand, every percentage increase in the interest rate in “another” country compared to the interest rate in “our” country in the forecast period leads to a 0.6% depreciation of “our” currency in the same period and a corresponding increase in the price of foreign currency .

Let us pay special attention to the conclusion obtained in financial theory* and confirmed by the practice of countries with developed market economies. He ss| The point is that an increase in the interest rate in any country compared to other countries in a certain period (year, month) leads, other things being equal, to upward pressure, i.e., to an increase in the price of the currencies of this country in the same period. However, we note that the same increase| could lead, on the contrary, to downward pressure, to a depreciation of the given currency in the next period? + 1.

After the necessary explanations, we will enter some initial values taken! into the factor model. Let us assume that the actual value of the difference between | the inflation rate in the two countries under consideration in period 1 was! 1%. This means that the inflation rate in our country was higher. Let's do this! the same assumptions regarding the differences in interest rates obtained as a result of some calculations for the forecast period. These values are entered| are determined not by one number, but by a certain set of them, a distribution with an indication! We define for each of them the probability of implementation. The corresponding data are given in table. 9.4. |

Table 9.І Forecast values of the difference in interest rates Number of forecast option Forecast value for option, % Probability of option implementation, % 1 -4 10 2 -5 60 3 -6 30 As can be seen from table. 9.4, in all options the interest rate in “our* country is lower than in the “other” one, but the possible magnitude of the difference is not the same. In addition, the likelihood of each option being realized is also different. We emphasize that this principle of presenting forecast information is quite common, and moreover, it corresponds with modern ideas about financial risk as an objectively existing uncertainty of future results and many other economic parameters.

The results of the exchange rate forecast will also be presented in three options, which are shown in table. 9.5.

As can be seen from table. 9.5, both higher inflation rates and lower interest rates in “our” country lead to a weakening of “our” currency, which, depending on the possible size of the fall in interest rates, or, more precisely, on the projected degree of their lag from the level of interest rates in “other” country, can with a 60% probability be 3.7%,

Table 9.5

Forecast values of the exchange rate Option number “„ + ”L,., “A e, Probability of option implementation, % 1 0.7 2.4 z 10 2 0.7 3.0 3.7 60 3 0.7 3.6 4.3 30 and also 4.3% - with a probability of 30% and 3.1% - with a probability of 10%. Some average value can also be calculated ( expected value) exchange rate changes 3.1

x 0.10 + 3.7 x 0.60 + 4.3 x 0.30 = 3.82.

This value will occur if the average, mathematically expected forecast value of the gap in interest rates is realized, equal to 5.2%.

Let us now move on to consider the third approach in the field of forecasting the exchange rate, which is very different from the first two, since it uses a fundamentally different methodology and technique for forecast calculations. This approach is based on the use of the interest rate parity theory. The leading problem of its application to forecasting is the degree of correspondence between the forward rate and the future spot rate. The fundamental possibility of coincidence or sufficient proximity for forecasting of these rates is determined by the following two circumstances.

The first is that the forward rate is a value derived from market expectations about the future current rate of banks and other firms providing forward services. The specialists of these banks and firms have best knowledge the relevant foreign exchange markets, since they work professionally in them, and, in addition, are interested in minimizing the difference between calculated forward and actual spot rates that arise in the future, since this reduces the risk of providing forward services.

The second circumstance is that the convergence of forward and future current rates is ensured by processes of market self-regulation. The latter is based on currency-interest arbitrage: from a theoretical point of view, zero profitability of arbitrage operations can be achieved, which means the equilibrium state of the market segment under consideration. Of course, complete equilibrium or, as is commonly said, the state of a perfect financial market is achievable only ideally. Nevertheless, the measure of achieving equilibrium determines the measure of justification for using the method of forecasting the exchange rate based on market expectations.

Let's turn now to the problems practical use characterized methods taking into account the real restrictions existing in the economic system.

In a number of countries, extensive research has been carried out on the quality of forecasts obtained using various methods. Assessing the results of these studies in aggregate, two main conclusions should be pointed out. Firstly, none of the metros gives completely accurate forecasts in a statistical sense. There is almost always a statistically significant bias in the prognostic assessment relative to the actual one. Second, forecasting based on market expectations produced the least bias in most studies.

Highlighting this method as giving on average a minimal forecast error, it must be emphasized that this does not deny the advisability of using other methods in certain circumstances. At short periods forecasting (day, week), the method of technical forecasting becomes preferable, if only for the reason that in the future! kah developed countries There are simply no representative interest rate quotes available for such short periods. As the duration of these periods increases (a year or more), macroeconomic factors of exchange rate movements become more apparent and, accordingly, the method of fundamental forecasting becomes more important.

It should also be borne in mind that for the practical application of the forecasting method based on market expectations, three fundamental conditions under which it works must be met: 1) there are no sufficiently significant restrictions on the movement of money between the markets in question; 2) the vast majority of foreign exchange transactions are purely financial in nature and do not serve the processes of movement of goods or the provision of non-financial services; 3) commercial banks play a decisive role in the market; in any case, their total financial positions are not inferior to the positions of the central banks of those countries for the markets of which this approach is applied. The above conditions are met for countries with developed market economies, and this determines the fundamental possibility of forecasting based on this method.

To model interest rate levels in statistics, various types of equations are used, including polynomials of various degrees, exponentials, logical curves and other types of functions.

2) If the series tends to rapidly develop the indicator at the beginning of the period and decline towards the end of the period, then it is advisable to use logistic curves.

3) If a series of dynamics is characterized by the presence of extreme values, then it is advisable to choose one of the variants of the Gompertz curve as a model.

In the process of modeling interest rate levels, great importance is given to careful selection of the type of analytical function. This is explained by the fact that an accurate description of the pattern of development of an indicator identified in the past determines the reliability of the forecast for its development in the future.

The theoretical basis of statistical methods used in forecasting is the property of inertia of indicators, which is based on the assumption that the pattern of development that existed in the past will continue in the predicted future. The main statistical forecasting method is data extrapolation. There are two types of extrapolation: prospective, carried out into the future, and retrospective, carried out into the past.

In general, extrapolation can be described by the following function:

y i + T = ƒ (y i , T, a n), (26)

where y i + T – predicted level;

y i – current level of the predicted series;

T – extrapolation period;

and n is the parameter of the trend equation.

Example 3´´. Based on the data in example 3, we will extrapolate to the first half of 2001. The trend equation is as follows: y^ t = 10.1-1.04t.

y 8 = 10.1-1.04*8 = 1.78;

y 9 = 10.1-1.04*9 = 0.78.

As a result of data extrapolation, we obtain point forecast values. The coincidence of actual data for future periods and data obtained by extrapolation is unlikely for the following reasons: the function used in forecasting is not the only one to describe the development of the phenomenon; the forecast is carried out using a limited information base, and random components inherent in the levels of the initial data influenced the result of the forecast; unforeseen events in the political and economic life of society in the future can significantly change the predicted development trend of the indicator being studied.

y t ±t α *σ yt , (27)

where y t is the predicted level value;

t α – confidence value determined based on Student’s t-test;

σ yt – root mean square trend error.

The use of the first method is based on the assumption that the general trend in the development of interest rates is expressed by a linear function, i.e. there is a uniform change in the indicator. To determine the predicted level of loan interest for any date t, the average absolute increase should be calculated and sequentially summed up by the last level of the dynamics series as many times as the time periods for which the series is extrapolated.

y i + T = y i + ∆¯*t, (28)

where i is the last level of the period under study for which ∆¯ is calculated;

t – forecast period;

∆¯ - average absolute increase.

Abstract of the dissertation on the topic "Forecasting interest rates based on the theory of deterministic chaos as a method of managing interest rate risk in commercial banks"

Galkin Dmitry Evgenievich

FORECASTING INTEREST RATES BASED ON

THEORIES OF DETERMINISTIC CHAOS AS A METHOD OF INTEREST RISK MANAGEMENT IN COMMERCIAL ENTITIES

Specialty 08.00.13 - mathematical and instrumental methods of economics

The work was carried out at the Department of Applied Mathematics of the Federal State Budgetary Educational Institution of Higher Professional Education "Perm National Research Poly Technical University» (PNRPU)

Scientific adviser:

Doctor of Technical Sciences, Professor

Pervadchuk Vladimir Pavlovich

Official opponents:

Doctor of Physical and Mathematical Sciences, Professor

Rumyantsev Alexander Nikolaevich

candidate economic sciences, assistant professor

Ivliev Sergey Vladimirovich

Lead organization:

Federal State Budgetary Educational Institution of Higher Professional Education "Izhevsk State Technical University", Izhevsk

The defense will take place on March 29, 2012 at 14:00 at a meeting of the dissertation council DM 212.189.07 at the Perm State National Research University at the address: 614990, Perm, st. Bukireva, 15, building 1, meeting room of the Academic Council.

The dissertation can be found in the library of the Perm State National research university. The abstract is posted on the official website of the Higher Attestation Commission of the Ministry of Education and Science of the Russian Federation: http://vak.ed.gov.ru/ and on the website of the Perm State National Research University www.psu.ru

Scientific secretary of the dissertation council, Doctor of Economic Sciences, Associate Professor

T.V. Mirolyubova

GENERAL DESCRIPTION OF WORK

The above-mentioned circumstances determine the relevance of the study.

One of the dynamic developing areas in the study of economic objects and systems is the use mathematical methods. Among them, special mention should be made of approaches that make it possible to widely use the concepts of synergetics, deterministic chaos, and fractal geometry in research. The following scientists were involved in the development and development of such methods: Takens F., Sornette D., Peters E., Bachelier L., Mandelbrot V., Gilmore R., Kantz H., Grassberger P., Procaccia I., Fama E., Lorenz E., Ruelle D., Casdagli M., Cao L., Haken H., Lefranc M. In Russian science, significant contributions to the development of this direction were made by Kurdyumov S.P., Malinetsky G.G., Bezruchko B.P., Loskutov A.Yu., Shumsky S.A., Kuperin Yu.A.

1.1. Design and development mathematical apparatus analysis of economic systems: mathematical economics, econometrics, applied statistics, game theory, optimization, decision theory, discrete mathematics and other methods used in economic and mathematical modeling.

2.3. Development of decision support systems to rationalize organizational structures and optimize economic management at all levels.

The theoretical and methodological basis are scientific works domestic and foreign scientists in the field of assessment and management of interest rate risk in banks, the theory of deterministic chaos, nonlinear dynamics, mathematical methods and models financial markets, fractal geometry, synergetics, published in Russian and foreign press, as well as on the Internet.

Practical calculations within the framework of this study were carried out using such applied software as MS Excel, MathWorks Matlab, Fractan, Tisean.

2. A modified mathematical model for forecasting interest rates based on a one-dimensional time series, taking into account the determinism of the studied

systems, as well as the developed approach to determine the scope of applicability of this model.

Theoretical significance of the results. The provisions and conclusions formulated in the dissertation research develop the theoretical and methodological basis for analyzing and forecasting the interest rate market, as well as methods for managing interest rate risk.

Practical significance of the results. The developed methodological approach provides commercial banks with the correct tool that allows, in the task of managing interest rate risk, to move from hypothetical scenario modeling to scenario modeling based on more probable forecast data.

Approbation of research results. The main provisions of the dissertation work were presented at the scientific and technical conference of students and young scientists of Perm State Technical University (Perm, 2007), at the XV International scientific and technical conference “Information and computing technologies and their applications (Penza, 2011), at the XII International Scientific and Technical Conference “Cybernetics and high tech XXI century (Voronezh, 2011), at the seminar of the Laboratory of Constructive Methods for Researching Dynamic Models of Perm State National Research University (Perm, 2011).

Also, the materials, methods and results of the dissertation are used at the Department of Applied Mathematics of the Perm National Research Polytechnic University when teaching the course “Mathematical analysis of dynamic models in economics” in the direction of preparation 010500.68 “Applied

mathematics and computer science" within the framework of the master's program "Mathematical methods in the management of economic processes" and when reading the course "Mathematical analysis of dynamic processes in economics" in the direction of preparation 080100.68 - "Economics" within the master's program "Mathematical methods of economic analysis".

Scope and structure of the dissertation work. The work is presented on 147 pages of typewritten text. The main results of the study are illustrated in 26 tables and 77 figures. The list of used literature includes 108 titles.

The structure of the dissertation is determined by the purpose, objectives and logic of the research. The work consists of an introduction, four chapters, a conclusion, a list of references and applications.

The introduction substantiates the relevance of the topic, sets the goals and objectives of scientific research, highlights the most significant achievements in the field of research, and presents the novelty of the results obtained.

The first chapter, “Application of mathematical methods in the study of financial time series,” examines existing methods and approaches to forecasting financial time series, evaluates their effectiveness, and defines the prerequisites for the use of nonlinear methods for modeling financial time series.

In the second chapter, “Selection and justification of methods for studying nonlinear dynamic systems based on time series" the main approaches to the study of dynamic systems using the theory of deterministic chaos are determined, Critical Assessment and the most optimal and correct tools for studying systems based on time series are identified.

In the third chapter, “Assessment and Research,” interest rate risk in Banking" examines the role of interest rate risk for commercial banks. The classification of interest rate risk and the main factors generating interest rate risk is studied in order to

identify the nature of the relationship between the interest rate market and interest rate risk.

In the fourth chapter, “Development of a method for managing interest rate risk based on forecasting interest rates,” the interest rate market is studied for nonlinearity and determinism. A forecasting model based on a one-dimensional time series is being adapted to the interest rate market; Forecasting models based on multivariate time series are being developed. Based on the obtained models, a methodology for managing interest rate risk in a commercial bank is created.

BASIC PROVISIONS AND RESEARCH RESULTS FOR DEFENSE

1. The nonlinearity and determinism of the interest rate market, established using statistical methods LIBOR rates and EURIBOR.

This provision is based on research into the 3-month LIBOR and 1-, 3- and 6-month EURIBOR interest rates, which are the most popular reference floating rates and to which the pricing of floating rate loans in US dollars and euros is tied. These rates reflect the cost Money in the interbank lending market for first-class borrowers with a credit rating of AA and higher for the appropriate period and in a certain currency.

The dissertation established a qualitative connection between the interest rate market and the level of interest rate risk for commercial banks. As a result, interest rates LIBOR and EURIBOR, as the most popular rates for pricing in global financial markets, were examined for nonlinearity and determinism.

Previously, to obtain quasi-stationarity, the time series under study were transformed based on the transformation

У, = log(x,) - bg(x,_!) = log(-) ,t = 2ji (1)

To study the signs of nonlinearity of systems, we used the BDS test proposed by Brock, Dechert and Shenkman, the idea of which is to calculate statistics based on the difference in correlation integrals (2) for embedding dimensions m and 1.

Сд,(/,Г)= 2 Y/,(*,",*?,/) (2)

"NV N ~H l

Where X^ = (X,XI+1,...,X,+N_1) And Xs = (^i^+lvi^j+.V-l) represent historical data, TN = 71 - jV +1, and

f 1, for Lf -x^ II

Oh, with be, - xs > /

The resulting statistics (3) should have a normal distribution N(0,1) if the process under study is white noise.

If the value of statistics for various values of / exceeds the critical value, then the hypothesis that the process is white noise is rejected.

BDS statistics were calculated for each process under study for various values of I and embedding dimensions m. The results obtained allowed us to reject the null hypothesis for each process, i.e. the samples are not independent and uniformly distributed. In addition, BDS statistics were calculated for the residuals of the autoregressive AR(1) model, as a result of which the null hypothesis for each process was also rejected, which in turn allowed us to conclude that the processes under study were nonlinear.

Another stage in the study of systems for determinism was the calculation of the Hurst exponent for the systems under study in order to determine the extent to which the objects under study have long-term memory. The assessment was made based on the calculation of the normalized range of the time series:

where R = max(x") - wn(dg") - range of the time series, N - number of observations, R - Hurst exponent, S - standard deviation of the series x".

Based on the log-log graph of the dependence of the normalized range R/S on the number of observations N, the value of the Hurst exponent is determined as the slope of the approximating straight line. For the systems under study, the calculation results are given in Table. 1 (3mLIBOR - LIBOR rates for a period of 3 months, lmEURIBOR - EURIBOR rates for a period of 1 month, 3mEURIBOR - EURIBOR rates for a period of 3 months, 6mEURIBOR - EURIBOR rates for a period of 6 months):

Table 1

The value of the Hurst exponents for the systems under study_

System 3mLIBOR lmEURIBOR 3mEURIBOR 6mEURIBOR

I 0.7007 0.7493 0.7863 0.7791

The results obtained (H > 0.5) indicate that the systems under study are persistent, i.e. have long-term memory and strive to maintain the trend. Based on this, as well as the results of the BDS test for these systems, we can conclude that the processes under study are deterministic.

When studying a time series of interest rates, it can be considered as an implementation of a more complex process of higher dimension. In this case, it is possible to reconstruct the attractor and, thereby, investigate the process itself that generates the time series.

The attractor reconstruction is carried out using the coordinate delay method:

*(/) = (s(t),s(t + r),..-At + (t- 1Yu) (5)

where m is the dimension of the embedding, and m > 2d +1, d is the Minkowski dimension.

The projection of the reconstructed attractor of the 3mLIBOR system into the R2 space is shown in Fig. 1, where diagonal structures are confirmation of the determinism of the system.

0.04 -0.03 -O 02 -0.01 0 0.01 0.02 O.OE 0.04

Rice. 1. Reconstructed 3mLIBOR attractor

Let us consider a discrete dynamic deterministic system, the dynamics of which are defined as

= /(*,) (6) Let s(t) = h(x,) - time series, which is the implementation of the dynamic system (6); in relation to the objects of study, the time series is a transformed series of interest rate values. It can be noted that the value of the time series generated by a deterministic system at a certain point in time can be represented as

This representation is valid for any point in the time series s(t) in any period of time, with the only difference being the number of impacts of the system / on the initial condition. Those. Having considered t consecutive values of a time series, they can be expressed as

2) = /,(/(*,+1) = h(J(Mx,m = F2 (x,)

As a result, all m values of a time series can be expressed

through the value x, using the set of functions F1.....Fm. Having produced

changing the variables zt+x =(s(t + l),i(i + 2),...,j(i + m)) and introducing the vector function A, which depends on / and on / (8) we can rewrite as

In accordance with Takens' theorem, if A: Md -> Rm

diffeomorphic, then it is possible to embed Md into Rm without self-intersections. Because A has a smooth inverse function, equality (9) can be written in the form

X,=h~\z„x) (10)

Substituting (10) into s(t + m +1) = Fm+l(x,), we obtain that s(t + m +1) = Fm+l(A~\zl+l) = ^(L" 1 (s(t + l),s(r + 2),..., s(t + u)))

= ®(j(/ +1), s(t + 2),..., sit + m)) (11)

Thus, the next value of a time series is determined through m of its previous values, where m has the topological meaning of the embedding dimension.

Due to the fact that the function Ф is not specified analytically, its approximation was carried out using a three-layer neural network, where the number of neurons in the input layer is equal to m, and in the output layer - 1.

To increase the efficiency of this model, the maximum Lyapunov exponent L, which determines the predictability of the system, and the Hurst exponent I, which determines the determinism of the system, were considered as functions of time. For this, window iv was used, the length of which was selected individually for each time series under study, and with the movement of the window, the specified characteristics were calculated. Based on this, the region where A > 0 and R > 0.5 was selected for application of the model.

In Fig. Figure 2 shows the time series of the 3mLIBOR interest rate together with the maximum Lyapunov exponent and the Hurst exponent as a function of time, on the basis of which the range of applicability of the model was determined.

Iterative forecast next value was built on the basis of previous historical data.

Original temporary rad

2000 3000 4000 5000

Dynamics of the maximum Lyapunov exponent

2000 3000 4000 5000

Dynamics of the Hurst exponent

О 1000 2000 3000 4000 5000

Fig.2. Identification of the scope of applicability of the model for HSE

The results of predicting the next value of the temporary rad ZtYVSZH are presented in Fig. 3. This forecasting approach was more effective 25% of the time than the method using the current value as the forecast value (the most optimal forecasting method for a random walk).

Rice. 3. Original (solid line) and forecast ( dotted line) time series ZtYVSZH

3. A mathematical model for forecasting interest rates based on a multidimensional time series, taking into account the determinism of the systems under study and allowing the dynamics of several systems to be used when constructing a forecast.

If you have information about interest rates in one currency for different periods, you can consider these time series as implementations of one process, i.e. as projections of one process onto three coordinate axes. However, in in this case The difficulty lies in the correct reconstruction of the attractor: each time series has different metric characteristics. To overcome this problem, the creation of an expanded nesting space is provided:

( */!>■*/>-!-, >hp-2-t1 >->hp-(t,-1)-t, >

gp ~ Up>Ul-tg>Ul-2t 7i-(t2_1)G2> (12)

gp"2p-t1 >2i-2-g, " >2„_(I)-1).g, )

where r is the coordinate delay parameter defined for the /th system; t, is the dimension of the embedding of the /"-th system; xt y„, r„ - reports of the corresponding time series.

When considering an attractor embedded in a space of dimension £) = mx + m2 + m3, Takens’ theorem will also be valid, since compliance with the requirements for the minimum dimension of the embedding will be met in advance by “subembeddings”, the dimension of which initially ensured the fulfillment of Takens’ theorem. In this form, the artificially increased dimension of the embedding at the expense of other time series will make it possible to take into account additional information about the system, incl. on the term structure of interest rates.

In this mathematical model, a nonparametric model is used for prediction in the form of kernel smoothing of the coordinates of the next points for the ¿-nearest neighbors of the trajectory point in the reconstructed phase space. Then the forecast trajectory point will look like:

*/+!= T,(Uy-Uk + 2>k^,Uk) (13)

where N„(2,) is the number of neighbors for point r, and u"¿(r,y*) are weighting coefficients.

According to the Nadaraya-Watson formula, the weights ^k(r„uk) can be determined as

where kernel function Kk (x) = -K(-) = - e

Generally speaking, the type of kernel in (13), as well as the width of the window of the kernel function, is determined experimentally. In this case, the kernel function is a Gaussian function, and the window width A = 0.5.

According to H. Kantz and T. Schreiberg, this approach to modeling chaotic time series is quite robust to noisy data and effective for experimental systems.

In addition, this model is a representative of the class of mixed models, i.e. in a certain way combines the features of local and global models, which is reflected in its features: on the one hand, it takes into account the global behavior and direction of the system, on the other, it successfully models local dynamics.

In Fig. Figure 4 presents a long-term forecast of the interest rate 1tEiShV(Zh values from 1703 to 1751 as a result of applying this mathematical model to a set of interest rates E1Zh1VSZh for a period of 1, 3 and 6 months. The previous values were used as the initial data for the forecast.

Rice. 4. Original (solid line) and forecast (dashed line) ISHESHIVOYA time series

The proposed mathematical forecasting model carries out correct forecasting with a forecast horizon not

more than 15 values, and forecasting can be carried out for any component of the set of interest rates. This approach to time series forecasting was compared with other popular forecasting methods: the AMMA, AIMA-vaYASN models and the radial basis neural network. In Fig. Figure 5 shows the results of forecasting using the indicated models for a certain section of the interest rate 1tEiGShY11.

0.52 0.51 0.50 0.49 0.48 0.47

Original series

*" * "Model based on TDH

AMMA-OAYASN

VLR-Network

1 2 3 4 5 6 7 8 9 10 II 12 13 14 15

Rice. 5.1тЭ1ЛШ (Zhi its predicted values based on various models

In table Figure 2 presents the results of a numerical comparison of forecasting efficiency based on normalized standard deviation (NSDE)

where d2 is the dispersion of the test set and the average absolute error (D x):

table 2

Model based on TDH! ASHMA-AIMA 1 SAISN YAVR-network

NSC 0.375 1.262 i 0.808 0.699

A, 0.006 0.021 i 0.013 0.011

From the presented set of models and the summary table of the effectiveness of forecasting results, we can conclude that the proposed model based on the theory of deterministic chaos (DCT) is the most effective.

4. Methodology for managing interest rate risk in commercial banks, which is based on a mathematical model for forecasting interest rates based on methods of the theory of deterministic chaos, which allows for scenario modeling using predictive data.

Based on the proposed mathematical models, a methodology was developed for managing interest rate risk in a commercial bank (Fig. 6).

Rice. 6. Methodology for managing interest rate risk

Thus, the first stage consists of analyzing the current position exposed to interest rate risk using gap analysis and assessing the sensitivity of profitability to changes in interest rates in the context of repricing intervals. Thanks to this, the interest rates that most determine the change in profitability are identified. Based on the selected set of interest rates, the attractor is reconstructed and the invariants are calculated, then forecasting is carried out. The forecast results are interpreted in terms of risk acceptance or risk reduction. When reducing risk, depending on the predicted dynamics and the current risk position, actions are taken: in the case of forecasting upward dynamics in the interest rate market

rates with a positive risk position on them or downward dynamics with a negative risk position, assets sensitive to interest rate risk increase, which is carried out through the following actions:

Purchase of floating rate securities; -conversion of loan rates from fixed to floating;

Replacement of funding for loans with a floating interest rate to funding with a fixed interest rate; Otherwise, liabilities sensitive to interest rate risk increase.

1. The existing set of tools of the theory of deterministic chaos for studying systems based on time series is critically assessed and based on this, as well as a comparative approach, the most effective methods for reconstructing the attractor, calculating the correlation dimension and characteristic Lyapunov exponents are determined.

2. A qualitative relationship was identified between interest rate risk and the interest rate market, and the latter object was identified as one of the main causal factors in the occurrence of interest rate risk in commercial banks.

3. The non-linearity and determinism of interest rates LIBOR for a period of 3 months and EURIBOR for a period of 1, 3 and 6 months has been established. Dynamic systems were reconstructed based on time series, metric and dynamic invariants were assessed, the results of which once again confirmed the hypothesis about the determinism of the systems under study.

4. A mathematical forecasting model based on a one-dimensional time series has been adapted to the interest rate market; Criteria for its applicability have been developed based on determining the area of determinism and predictability.

5. For the interest rate market, a new mathematical forecasting model has been developed based on a multidimensional time series of interest rates using an expanded investment space and kernel smoothing of neighboring trajectory points, the efficiency of which exceeds the efficiency of classical approaches to forecasting financial markets.

6. A methodology has been created for managing interest rate risk in commercial banks based on the developed model for forecasting the interest rate market.

PUBLICATIONS ON THE TOPIC OF RESEARCH

1. Pervadchuk V.P., Galkin D.E. Application of methods of the theory of deterministic chaos to forecast the dynamics of the interbank lending rate LIBOR // Vestnik Izhevsk, state. tech. un-ta. -No. 2 (46). - Izhevsk, 2010. - p.45-49.

2. Galkin D.E. Forecasting multidimensional financial time series based on methods of the theory of deterministic chaos // Inzhekon Bulletin. - 2011. - No. 3(46). - Ser. Economy. - St. Petersburg, 2011.-359-363 p.

In other publications:

3. Galkin D.E., Pervadchuk V.P. Fractal analysis of the dynamics of exchange rates // Abstracts of scientific and technical conference of students and young scientists Permsk. state tech. un-ta. - sir. Applied mathematics and mechanics, 2007. - p. 26-27.

4. Pervadchuk V.P., Galkin D.E. Rationale for the use of methods of the theory of deterministic chaos for forecasting economic systems // Vestnik Perm. state tech. un-ta. - sir. Mathematics and applied mathematics. - Perm, 2008. - p. 15-24.

5. Pervadchuk V.P., Galkin D.E. Application of fractals in the study of financial time series // Vestnik Perm. state tech. un-ta. - No. 14. - sir. Mathematics and applied mathematics. -Perm, 2008.-p. 8-15.

6. Pervadchuk V.P., Galkin D.E. Modeling of economic systems using methods of the theory of deterministic chaos // Cybernetics and high technologies of the XXI century: collection of reports of the XII international scientific and technical conference. - Volume 1. - Voronezh, 2011. - p. 277-282.

7. Galkin D.E. Features of reconstructing a phase attractor for forecasting economic systems // Information and computing technologies and their applications: collection of articles of the XV International Scientific and Technical Conference. - Penza: RIO PGSHA, 2011. - p.27-31

8. Pervadchuk V.P., Galkin D.E. The role of the interbank lending rate LIBOR in the global economy // Vestnik Perm. state tech. un-ta. - sir. Socio-economic sciences. - Perm, 2011. - p. 101105.

Signed for publication on February 20, 2012. Format 60x84/16 Conditional oven l. 1.45. Circulation 100 copies. Order 5O. Printing house of Perm State National Research University. 614990. Perm, st. Bukireva, 15

Dissertation: text in Economics, Candidate of Economic Sciences, Galkin, Dmitry Evgenievich, Perm

Perm National Research Polytechnic University

As a manuscript

Galkin Dmitry Evgenievich

Forecasting interest rates based on the theory of deterministic chaos as a method for managing interest rate risk in commercial banks

Specialty 08.00.13 - Mathematical and instrumental methods

economy

Dissertation for the degree of Candidate of Economic Sciences

Scientific supervisor, Doctor of Technical Sciences, Professor V.P. Pervadchuk

Perm, 2011

INTRODUCTION........................................................ ........................................................ ...............4

CHAPTER 1. APPLICATION OF MATHEMATICAL METHODS IN THE STUDY OF FINANCIAL TIME SERIES.........................................12

1.1. Analysis and forecasting of time series. Development of scientific thought 12

1.1.1. Linear models........................................................ ..................................14

1.1.2. Nonlinear models........................................................ ..............................18

1.2. Development of methods for analyzing financial time series based on

theory of deterministic chaos................................................................... .......................25

1.2.1. Local models........................................................ ................................25

1.2.2. Global methods................................................... ...............................26

1.2.3. Mixed methods................................................................... ...............................28

1.2.4. Topological approach................................................... .......................thirty

1.3. Brief conclusions........................................................ ...............................................34

CHAPTER 2. SELECTION AND JUSTIFICATION OF METHODS FOR RESEARCHING NONLINEAR DYNAMIC SYSTEMS BASED ON TIME SERIES................................................... ........................................................ ...................................35

2.1. The concept of complex systems and the theory of deterministic chaos..........35

2.2. Reconstruction of an attractor based on a time series....................................40

2.3. Calculation of the correlation dimension of a dynamic system.................................48

2.4. Characteristic Lyapunov exponents and entropy as a measure

predictability........................................................ ........................................................ .53

2.5. Brief conclusions........................................................ ...............................................57

CHAPTER 3. ASSESSMENT AND RESEARCH OF INTEREST RISK IN BANKING................................................... ................................58

3.1. Studying the role of interest rates in the economy and the interbank lending market for banks.................................................. ...................................................58

3.2. Research on the concept of interest rate risk, identification of the most significant types and main factors.................................................... ............................65

3.2.1. Interest rate risk and its place among banking risks.................................65

3.2.2. Classification of interest risks................................................................... ....70

3.2.2. The interest rate market as the main factor causing the emergence of interest rate risk.................................................. ...........................74

3.3. Study of approaches to assessing interest rate risk....................................76

3.4. Brief conclusions........................................................ ...............................................89

CHAPTER 4. DEVELOPMENT OF A METHOD FOR MANAGING INTEREST RISK BASED ON INTEREST RATES FORECASTING...................................91

4.1. Formulation of the problem................................................ ........................................91

4.2. Research of the interest rate market for stationarity, nonlinearity and determinism.................................................... ........................98

4.3. Estimation of metric and dynamic invariants....................................................107

4.4. Adaptation of a mathematical forecasting model based on a one-dimensional time series, taking into account limited determinism and predictability................................................... ........................................................ ..........113

4.5. Development of a mathematical forecasting model based on a multivariate time series.................................................... ................................120

4.6. Comparison of forecast results and development of a methodology for managing interest rate risk based on forecasting interest rates.........125

4.7. Brief conclusions........................................................ ........................................135

CONCLUSION................................................. ........................................................ ....136

LIST OF REFERENCES USED.................................................................... 138

APPLICATIONS........................................................ ........................................................ ....148

INTRODUCTION

Relevance of the research topic. The task of risk management in the banking sector is non-trivial throughout the entire banking activity. The problem of banking risks in modern times is becoming increasingly relevant in light of the increasing influence of the financial sector on world economy. So, for example, in the USA, the largest economy in the world, in the 1970s the share of financial sector income in total corporate income did not exceed 16%, and in the 2000s it reached 41%. Taking into account the colossal role of banks in the global financial crisis of 2008 and the growing crisis of 2011, the problem of risk management and control in the banking sector requires close attention and study.

Among all types of risk inherent in banking activities, interest rate risk occupies a special place, second only to credit risk in terms of influence. However, one of the significant differences between interest rate risk and credit risk is the fact that the area subject to its influence is much wider. As a result, the significance of interest rate risk is high not for one individual line of business, but for the bank as a whole.

In addition, taking into account the high volatility of financial markets, including the interest rate market, during periods of economic instability, interest rate risk management should be carried out carefully, taking into account possible options developments of events affecting the level of interest rate risk.

Taking into account the above circumstances, it should be recognized that the problem of managing interest rate risk in the banking sector is extremely relevant.

The degree of scientific development of the topic. By exploring the concept of interest rate risk and researching various aspects The problems of assessing and managing this type of risk were dealt with by such scientists as Macaulay F., Redhead K., Hughes S., Entrop O., Cade E., Helliar S., Fabozzi F., Gardener E., Mishkin F., van Greuning H. ., Patnaik I., Madura J., Amadou N.

The current level of development of this problem in our country is reflected in the works of domestic scientists and specialists, among whom we should highlight Sevruk V.T., Larionova I.V., Vinichenko I.N., Lavrushin O.I., Sokolinskaya N.E., Valentseva N.I., Khandrueva A.A.

One of the dynamically developing areas in the study of economic objects and systems is the use of mathematical methods. Among them, special mention should be made of approaches that make it possible to widely use the concepts of synergetics, deterministic chaos, and fractal geometry in research. The following scientists were involved in the development and development of such methods: Takens F., Sornette D., Peters E., Bachelier L., Mandelbrot V., Gilmore R., Kantz H., Grassberger P., Procaccia I., Fama E., Lorenz E., Ruelle D., Casdagli M., Cao L., Haken H., Lefranc M. In Russian science, significant contributions to the development of this direction were made by Kurdyumov S.P., Malinetsky G.G., Bezruchko B.P., Loskutov A.Yu., Shumsky S.A., Kuperina Yu.A.

The purpose of the dissertation research is to develop theoretical and methodological foundations for managing interest rate risk in commercial banks based on forecasting interest rates using the theory of deterministic chaos.

To achieve this goal, the following tasks were solved:

1. Study of existing approaches for forecasting financial time series and assessing interest rate risk in order to use existing experience in developing a new method.

2. Selection of effective tools for studying nonlinear dynamic systems based on generated time series.

3. Study of the relationship between the interest rate market and interest risk in commercial banks.

4. Adaptation of a one-dimensional mathematical forecasting model to the interest rate market, taking into account limited determinism and predictability.

5. Development of a multidimensional mathematical model for forecasting interest rates.

6. Creation of a methodology for managing interest rate risk based on developed forecasting models.

The object of the study is commercial banks exposed to interest rate risk as a result of transactions with interest-bearing products.

The subject of the study is methods and tools for managing interest rate risk in commercial banks, as well as methods and algorithms that provide modeling of systems related to interest rate risk.

The area of research corresponds to the passport of the specialty of the Higher Attestation Commission of the Russian Federation 08.00.13 “Mathematical and instrumental methods of economics” on the following points:

1.1. Design and development of the mathematical apparatus for analyzing economic systems: mathematical economics, econometrics, applied statistics, game theory, optimization, decision theory, discrete

mathematics and other methods used in economic and mathematical modeling.

1.6. Mathematical analysis and modeling of processes in financial sector economics, development of the method of financial mathematics and actuarial calculations.

2.3. Development of decision support systems to rationalize organizational structures and optimize economic management at all levels.

The theoretical and methodological basis is research in the field of assessment and management of interest rate risk in banks by domestic and foreign scientists. Other areas of knowledge in which advances were used in the study are the theory of deterministic chaos, nonlinear dynamics, mathematical methods and models of financial markets, fractal geometry, and synergetics.

Practical calculations within the framework of this study were carried out using such applied software as MS Excel, Math Works Matlab, Fractan, Tisean.

The information base of the study consisted of:

Data from information and analytical materials on the problem under study, presented in scientific literature, periodicals and the Internet;

Statistical sources in the form of quotes for interbank lending rates LIBOR and EURIBOR for various periods.

The most significant results obtained personally by the author, having scientific novelty and submitted for defense are:

1. The nonlinearity and determinism of the LIBOR and EURIBOR interest rate market, established using statistical methods.

2. A modified mathematical model for forecasting interest rates based on a one-dimensional time series, taking into account the determinism of the systems under study, as well as a developed approach for determining the scope of applicability of this model.

Approbation of research results. The main provisions of the dissertation work were presented at the scientific and technical conference of students and young scientists of Perm State Technical University (Perm, 2007), at the XV International scientific and technical conference “Information and computing technologies and their applications (Penza, 2011), at the XII International Scientific and Technical Conference “Cybernetics and High Technologies of the 21st Century” (Voronezh, 2011).

The research results found practical use at CJSC UniCredit Bank. This organization uses the interest rate risk management methodology and also applies the interest rate forecasting model described in the study.

The materials, methods and results of the dissertation are used at the Department of Applied Mathematics of the Perm National Research Polytechnic University when teaching the course “Mathematical analysis of dynamic models in economics” in the direction of preparation 010500.68 “Applied mathematics and computer science” within the framework of the master’s program “Mathematical methods in the management of economic processes” and when reading the course “Mathematical analysis of dynamic processes in economics” in the direction of preparation 080100.68 - “Economics” within the framework of the master’s program “Mathematical methods of economic analysis”.

The implementation of the research results in these organizations is confirmed by relevant documents.

Structure of the dissertation work. The structure of the dissertation is determined by the purpose, objectives and logic of the research. The work consists of an introduction, four chapters, a conclusion, a list of references and an appendix.

The introduction substantiates the relevance of the topic, sets the goals and objectives of scientific research, highlights the most significant achievements, and presents the novelty of the results obtained.

The first chapter, “Application of mathematical methods in the study of financial time series,” examines existing methods and approaches to forecasting financial time series and evaluates them

efficiency, the prerequisites for the use of nonlinear methods for modeling financial time series are determined.

The second chapter, “Selection and justification of methods for studying nonlinear dynamic systems based on time series,” defines the main approaches to the study of dynamic systems using the theory of deterministic chaos, makes a critical assessment and identifies the most optimal and correct tools for studying systems based on time series.

The third chapter, “Assessment and study of interest rate risk in banking,” examines the role of interest rate risk for commercial banks. The classification of interest rate risk and the main factors generating interest rate risk is studied in order to identify the nature of the relationship between the interest rate market and interest rate risk.

In the fourth chapter, “Development of a method for managing interest rate risk based on forecasting interest rates,” the interest rate market is studied for nonlinearity and determinism. The forecasting model based on a one-dimensional time series is adapted to the interest rate market; A forecasting model is being developed based on a multivariate time series. Based on the obtained models, a methodology for managing interest rate risk in a commercial bank is created.

The conclusion contains the main results and conclusions of the dissertation research and an assessment of the practical significance of the work.

The list of references shows the main sources used in writing the dissertation.

The Appendix includes a description of the calculation results not included in the main text of the work.

The main results of the study are illustrated in tables and graphs. The dissertation includes 77 figures, 26 tables, 93 formulas. The list of used literature includes 108 titles. The total volume is 147 pages.

CHAPTER 1. APPLICATION OF MATHEMATICAL METHODS IN THE STUDY OF FINANCIAL TIME SERIES

1.1. Analysis and forecasting of time series.

Development of scientific thought

In this dissertation, forecasting of financial time series is considered as a method for managing interest rate risk of a commercial bank. In this regard, there is an obvious need to trace the development of scientific thought in relation to forecasting financial time series and time series in general, to consider the approaches and methods used to carry out forecasting, and to evaluate their advantages and disadvantages.

Generally speaking, interest in predicting the states of the objects under study appeared simultaneously with the definition of the object of study, which is quite understandable from the point of view of a scientist: when analyzing the essence of the object of study, the scientist always ends up trying to predict its future state, to model the “behavior” of the object. With the development of the mathematical apparatus, the methods of formulating also changed.