From financial news. Interest Rate Forecasting

Predicting the level of interest rates is a time-honored profession. Economists are hired (sometimes at very high fees) to predict interest rate movements, because firms need to know how to plan for their future spending, while banks and investors need interest rate projections to know which assets to buy. Interest rate forecasters assume what will happen to the factors that affect the supply and demand for bonds and money. These are factors such as the state of the economy, the profitability of investment opportunities, the expected rate of inflation, the size of the government budget deficit, the receipt of loans, and the like. Forecasters then use the supply and demand toolkit outlined in this section to make interest rate forecasts.

The Wall Street Journal reports interest rate forecasts by leading forecasters twice a year (early January and July) in the Economy or Credit Markets column, which provide information on the state of the bond market on a daily basis. Interest rate forecasts are an uncertain matter. Unfortunately, even the predictions of the best forecasters are often far from the real development of events.

Let us assume that there is a one-time increase in the money supply today, which leads to an increase in prices, that is, their highest level next year. Since the price level rises during a given year, interest rates will increase due to the price level effect Only at the end of this year, when price growth peaked, will the price level effect be greatest.

The rising price level will also raise interest rates through the "expected inflation effect" because people will assume that inflation will be higher during that year. However, when the price level stops rising next year, the inflation rate and expected inflation will fall to zero. Any increase in interest rates that is the result of a previous increase in expected inflation will then be cancelled. We therefore see that, in contrast to the price level effect reaching its greatest impact next year, the expected inflation effect will be the next year's least impact (i.e. zero). The main difference between these two effects is that the price level effect remains even after the price increase has stopped, while the expected inflation effect does not.

The important point is that the effect of expected inflation will continue as long as prices rise. As we shall see in the analysis of monetary theory in the following sections, a one-time increase in the money supply is not inducible by a constantly rising price level. This level of induction only induces a higher money supply growth rate. So, a higher money supply growth rate is required for the "expected inflation effect" to continue.

Or does a higher growth rate in the money supply lower interest rates?

We can now put together all the effects we have analyzed that will help us resolve the issue, our analysis will support policy makers who advocate higher money supply growth when they think interest rates are too high. Of all the effects, only the liquidity effect shows that the higher the growth rate of money cause interest rates to fall. In contrast, the effects of income, the price level, and expected inflation suggest that interest rates will rise when the growth in the money supply becomes higher. Which of these effects has the strongest impact, and how quickly do they work? The answer to this question is critical in determining whether interest rates will rise or fall when the growth rate of the money supply increases.

The liquidity effect of a higher rate of growth in the quantity of money generally has an immediate effect, a rising money supply leads to an immediate decrease in the equilibrium interest rate. Effects

Chart 6.13.

Income and price levels take time to trigger because a rising money supply takes time for price and income levels to rise, which in turn raise interest rates. The expected inflation effect, which also raises interest rates, can work slowly or quickly depending on how slowly or quickly people adjust their inflation rate forecasts when the money supply growth rate rises.

Chart 6.13 outlines three possibilities, each of which shows how interest rates respond over time to an increased rate of growth in the money supply starting at time T. Part (a) of the chart shows a case in which the liquidity effect dominates the other effects, so the interest rate the rate falls from u1 in time T to the final level r2. The liquidity effect works quickly, lowering interest rates, but over time, other factors begin to work in the opposite direction, which stimulates the fall. And although the influence of the liquidity effect is stronger than other effects, still the interest rate never returns to its original level.

Part (b) of the graph has a weak other liquidity effect, with an expected inflation effect, and is slow to work because inflation forecasts are adjusted slowly. Initially, the liquidity effect lowers the interest rate. So the effects of income, the price level, and expected inflation will start raising this rate. As these effects dominate, the interest rate eventually rises over its output level to u2. In the short run, lower interest rates are a consequence of the increased rate of growth in the quantity of money, but in fact they cease to rise above their original level.

Part (c) of the graph shows the effect of expected inflation, which prevails over others, also acts quickly, because people's expectations of inflation quickly rise when the growth rate of the quantity of money rises The effect of expected inflation starts immediately to overpower the liquidity effect, so the interest rate immediately starts crawl up. Over time, as income and price level effects kick in, the interest rate rises even faster, and final result will be such that the interest rate will be significantly higher than the original. This result clearly shows that an increase in the rate of growth of the money supply is not a response to a decrease in interest rates, but rather the growth in the quantity of money should be reduced in order to reduce interest rates.

An important question for policy makers is whether, of the three scenarios, the closest real situation of things. If interest rates are to be lowered, then the growth rate of the money supply must be increased, because the liquidity effect dominates other effects (part a). Reducing the growth rate of the money supply is appropriate if other effects dominate the liquidity effect and inflation

Graph 6.14.

New hopes correct quickly (part c). If other effects dominate the liquidity effect, but inflation expectations adjust slowly (part b), then your desire to increase or decrease the growth in the money supply depends on whether you care more about what will happen in the short term or what will happen in the long term.

Is the scenario supported by the evidence? The relationship between interest rates and the growth in the quantity of money from 1951 to 1990 is depicted in Chart 6.14. When the growth rate of the money supply became faster in the mid-1960s, interest rates rose, indicating that the liquidity effect dominated the effects of prices, income, and expected inflation. Until the 1970s, interest rates reached levels unprecedented in the post-World War II period, when the pace of the money supply was rising.

The scenario described in more (a) seems doubtful, and the case for interest rates falling due to an increase in the growth rate of the money supply is highly unlikely. Looking back at Chart 6.6, which shows the relationship between interest rates and expected inflation, this is not too strange. The increase in the money supply growth rate in the 1960s and 1970s is offset by a large increase in expected inflation, and this led us to predict that the effect of expected inflation was dominant. This is the most plausible explanation for why interest rates have risen in spite of the superior rate of growth in the quantity of money. However, it actually follows from Chart 6.11 which of these two scenarios on parts (b) and (c) of Chart 6.13 is accurate. It depends in critical on how quickly people's inflation expectations adjust. How are expectations formed and how quickly are they adjusted? This is an important issue that is now being actively studied by economists and is analyzed in section 29.

Almost all investors financial markets to one degree or another, the question of future interest rates worries. For example, for holders of treasury bonds, this is one of the key issues. If investors in the bond market believe that interest rates will rise in the future, then they should probably avoid long-term bonds in favor of bonds with more short term execution.

yield curve
In the United States, the Treasury yield curve is a key factor in all domestic interest rates and also influences global rates. Interest rates on all other categories of bonds rise and fall in the wake of Treasuries, which are debt securities issued by the US government. To attract investors, any debt securities that carry more risk than Treasuries must offer higher returns. For example, the rate on 30-year mortgages in normal conditions set at 1%-2% above the yield on 30-year Treasury bills.

Below is the Treasury yield curve since December 5, 2003 ( diagram 1). This is the "normal" shape of the curve, as it slopes upwards and curves accordingly:

Let's look at the three elements of this curve. First, it shows nominal interest rates. Inflation destroys the value of future coupon and principal payments; the real interest rate equals the yield minus inflation. Therefore, the yield curve combines expected inflation and real interest rates. Second, the Federal Reserve directly adjusts only the short-term interest rate at the very beginning of the curve. The Federal Reserve has three instruments of monetary policy, the most powerful of which is the federal funds rate, which is the overnight rate. Third, the rest of the curve is determined by supply and demand at bond auctions.

Chart 1. Treasury yield curve.

Sophisticated institutional buyers have yield requirements that, along with their appetite for government bonds, determine how these institutional buyers place their offers on government bonds. Since these buyers have their own opinions about inflation and interest rates, many believe that the yield curve is the "magic crystal" that predicts future interest rates. In this case, investors assume that only unforeseen events (such as an unexpected rise in inflation) will shift the yield curve up or down.

Long-term rates follow short-term rates Technically, the Treasury yield curve could change different ways it can move up or down (parallel changes), become flatter or steeper (change in slope), or become more or less arched in the middle (change in curvature).

Chart 2 compares the 10-year Treasury yield (red line) with the 1-year Treasury yield ( green Line) from June 1976 to December 2003. The blue line represents the differential between these two returns:

Chart 2. 10-year and 1-year bond yields.

Looking at Diagram 2, two observations can be made. First, the two returns moved up and down almost at the same time (correlation was approximately 88%). Therefore, parallel changes are quite common. Second, while long-term rates follow short-term rates in direction, they tend to lag in value. You can definitely see that when short-term rates rise, the differential between 10-year and 1-year returns tends to narrow (the differential curve flattens), and when short-term rates fall, the differential widens (the curve gets steeper). In particular, the increase in rates from 1977 to 1981 was accompanied by a flattening and inversion of the curve (negative differential); rate cuts from 1990 to 1993 resulted in a steeper differential curve; the last rate cut from March 2000 to the end of 2003 resulted in a very steep differential curve by historical standards.

Demand Offer
So what moves the yield curve up or down? Within the framework of this article, we cannot pay due attention to the complex dynamics of capital movements, under the interaction of which market interest rates are formed. But it should be understood that the Treasury yield curve reflects the value of US government debt, and therefore ultimately reflects supply and demand.

Supply Factors
monetary policy
If the Fed wants to increase the federal funds rate, it supplies more short-term securities for open market operations. An increase in the supply of short-term securities limits the amount of money in circulation as borrowers give money to the Federal Reserve. In turn, this decrease in the money supply increases the short-term interest rate because there is less money left in circulation available to borrowers. By increasing the supply of short-term securities, the Fed raises the left end of the curve, and near-term yields will quickly adjust accordingly.

Can we predict future short-term rates? According to expectations theory, long-term rates include forecasts of future short-term rates. Let's look at the actual yield curve for December 2003 shown above ( diagram 1) which is "normal" but very cool. The one-year yield is 1.38% and the two-year yield is 2.06%. If you wanted to invest for a two-year period and if interest rates were constant, then you should buy two-year bonds outright (which have higher yields) instead of buying one-year bonds and then rolling them over. However, according to the theory of expectations, the market predicts an increase in the short-term rate. Therefore, at the end of the first year, you will be able to switch to one-year bonds with more favorable yields and, as a result, will receive approximately the same yield as two-year bonds. In other words, expectations theory says that a steep yield curve predicts higher future short-term rates.

Unfortunately, the theory pure form doesn't work interest rates often stay the same during a normal (sloping up) yield curve. This is probably due to the fact that longer-term securities are associated with a certain uncertainty about the interest rate and imply, accordingly, additional yield. If we look at the yield curve from this point of view, the two-year return contains two elements of the forecast of the future short-term rate plus additional return for uncertainty (ie risk premium). Thus, we could say that a steeply sloping yield curve portends an increase in the short-term rate. On the other hand, a gently sloping curve does not portend any change in the short-term rate; the upward slope should only reflect additional returns for the uncertainty associated with long-term liabilities.

Since the surveillance of the Federal Reserve is professional occupation, it is not enough to wait for the actual change in the federal funds rate. It is important for an investor to try to stay one step ahead of the decisions of the monetary authorities, waiting instead of watching changes in interest rates. Market participants around the world scrutinize the wording of every Federal Reserve statement (and speech by Fed officials) in an attempt to discern their future intentions. AT recent times The Federal Reserve is becoming more and more transparent in its decisions. For example, in August 2003, the Federal Reserve stated that it would keep the discount rate low for a significant period of time, so market participants in the following months simply waited for the Fed to drop this phrase and thus signal its intention to raise the federal funds rate. .

fiscal policy
When the US government makes up a budget deficit, it borrows money by issuing long-term Treasury bills. The more the government borrows, the more debt it issues. When borrowing increases, at some point the US government must increase the interest rate to secure more lending. However, foreign lenders are always happy to purchase debentures american government, so they have high liquidity, and the United States has never violated its obligations (in fact, at the end of 1995 they were close to default, but the Secretary of the Treasury at the time, Robert Rubin, averted the threat and called a bond default "unthinkable and something akin to nuclear war"). However, foreign creditors can easily find an alternative in the form of European bonds (Eurobonds), and therefore they can charge higher interest rates if the US tries to sell too much of its debt.

Demand factors
Inflation
If we assume that US debt holders expect to receive a given real yield, then an increase in inflationary expectations will raise the nominal interest rate (nominal yield = real yield + inflation). Inflation also explains why short-term rates move faster than long-term ones. When the Fed raises short-term rates, long-term rates also rise, reflecting the expectation of higher short-term rates in the future. However, this increase is mitigated by lower inflation expectations, as higher short-term rates also mean lower inflation (as the Fed delivers more short-term Treasuries, it collects money and limits the money supply).

Diagram 3. The impact of the increase discount rate on yield (in blue initial yield curve, green after Fed rate hike).

An increase in the federal funds rate tends to flatten the yield curve because the yield curve reflects nominal interest rates: higher nominal rate = higher real rate + lower inflation.

Economic forces
Factors that create demand for Treasuries include economic growth, currency competitiveness and hedging opportunities. Just remember: anything that increases demand for long-term Treasuries puts downward pressure on interest rates (higher demand = higher price = lower yields or interest rates), and less demand for bonds tends to put downward pressure on interest rates . A stronger economy tends to make corporate (private) debt more attractive than government debt, reducing demand and raising rates. A weaker economy, on the other hand, stimulates "demand for quality" by increasing demand for Treasuries, leading to lower yields. It is sometimes assumed that a strong economy will automatically force the Fed to raise short-term rates, but not necessarily. Only when there is a threat that growth will be transformed into more high prices, the Federal Reserve is likely to go on raising rates.

In the global economy, US Treasuries compete with the debt securities of other countries. From a global perspective, US bonds represent investments in both US real interest rates and the dollar.

Finally, Treasury bills play a huge role as a hedge (insurance) for market participants. In an environment of falling interest rates, many holders of mortgage-backed securities, for example, can hedge their risk by buying long-term bonds. These insurance purchases can play big role in demand, helping to keep rates low, but at the same time, they can contribute to market volatility.

Conclusion
In this article, we have covered the key factors associated with interest rate movements. On the supply side, monetary policy determines how much government debt and money to put into the economy. On the demand side, inflation expectations are the key factor. However, we discussed other important factors that affect interest rates, including: fiscal policy (i.e. how much the government needs to borrow), as well as demand-side factors like economic growth and currency competitiveness. We understand that these other factors are constantly changing, but there are two important issues that you should constantly be asking yourself: "is fiscal policy creating too much debt in the market?" and "will demand for US debt keep the same momentum in the global market?"

David Harper

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

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

1) If the series of dynamics tends to monotonously 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 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 the 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 analytic function. This is explained by the fact that the exact characteristics of the patterns of development of the indicator identified in the past determine the reliability of the forecast of 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 exists 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 in the future, and retrospective, carried out in the past.

Extrapolation should be evaluated 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 studied indicator. In addition, it should be noted that the shorter the extrapolation period, the more accurate forecast available.

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

y i + T = ƒ (y i , T, a n), (26)

where y i + T is the predicted level;

y i is the current level of the predicted series;

T is the period of extrapolation;

and n is the trend equation parameter.

Example 3´´. Based on the data of 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 \u003d 10.1-1.04 * 8 \u003d 1.78;

y 9 \u003d 10.1-1.04 * 9 \u003d 0.78.

As a result of data extrapolation, we get point values ​​of the forecast. The coincidence of the actual data of future periods and the data obtained by extrapolation is unlikely for the following reasons: the function used in forecasting is not the only one for describing the development of the phenomenon; the forecast is carried out using a limited information base, and the 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 trend in the development of the indicator under study.

Due to the fact that any forecast is relative and approximate, when extrapolating the levels of interest rates, 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 value of the level;

t α is a confidence value determined on the basis of Student's t-test;

σ yt is the standard error of the trend.

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

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

y i + T = y i + ∆¯*t, (28)

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

t is the forecast period;

∆¯ - average absolute increase.

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

In order for the results of work in the bond market to be better than the market average, simply purchasing bonds with the highest yield to maturity is not enough. In order to outperform the market, it is necessary to know how the yield required by investors from a particular bond issue will change (the expected change in the level of liquidity and credit quality of the issue), and, more importantly, what will be the situation with the level of interest rates in the economy in in general.

This will make it possible to keep predominantly short securities in the portfolio in anticipation of an increase in interest rates (the decrease in their value will be less than that of long ones). In the event of an expected decrease in interest rates, the portfolio will mainly contain bonds with a longer duration (the growth in their value will be more significant than short ones).

In order to determine the vector of the level of interest rates in the economy as a whole, Arsagera Management Company uses 5 models. All these models are based on the arbitrage principle.

Interest rate level vector

To determine what the level of interest rates will be in the future, Arsagera Management Company uses several economic models, each of which describes the behavior of various groups of economic agents in certain economic conditions.

inflation model

The inflationary model takes into account the behavior of domestic investors. Within the framework of this model, the level of interest rates in the country is compared with the level of inflation in the same country (inflation forecast for Russia is based on the MEDT forecasts). The main premise of this model is that investors in different countries are guided by the same level of real return (return reduced by the inflation rate in the country) when investing in instruments with the same level of risk. Thus, knowing what real profitability investors in different countries expect from investments with a certain level of risk, we, when predicting the inflation rate in Russia, can say what the profitability of specific instruments should be, so that investors would be interested in investing funds within the country, and not abroad. outside of it.

Example. The average yield of the most reliable corporate bonds in Russia is 7.5%. The inflation rate is expected to be 9.9% over the next year. In the US, the average yield on the most reliable corporate bonds is 5%, and inflation is expected to be 2.2%. Thus, it turns out that in Russia the real return on investment will be -2.4%, and in the US - +2.8%. We see that it is more interesting for investors to invest in the US market until the real profitability of instruments with the same level of risk evens out. The vector of the level of interest rates in Russia according to this model is +520 p.p.

Cash rate parity model

This model takes into account the behavior of global players involved in cross-border capital investment. Since investing funds in foreign (in relation to such an investor) markets involves the transfer of funds into the currency of another country, the final return expected by such an investor is affected by the expected change exchange rates. Availability a large number of investors engaged in cross-border investments leads to equalization (globally) of the returns of instruments with the same level of risk.

Thus, having a forecast for the future exchange rate and knowing the level of interest rates in one of these countries, we can say what level of interest rates investors expect to see in the second country.

Example. Assume that the current exchange rate of the ruble against the US dollar is 50 rubles to the dollar. The rate expected in a year is 55. Therefore, if the current return on instruments with a certain level of risk in the United States is 10% per annum, then investors' expected return on Russian instruments with the same level of risk in a year is 21% per annum (to compensate for the expected depreciation of the ruble). Since the forecast values ​​of exchange rates are announced not only by the Ministry of Economic Development, but also by the leading investment institutions of the West, we can calculate what profitability they expect from Russian assets.

Credit and deposit model

The credit and deposit model consists of three submodels. These models take into account the behavior of different groups of domestic investors:

• Borrowers (legal entities) who choose the method of raising funds for the development of the enterprise.

The company chooses from two alternatives: either to raise funds by placing a bond issue, or to take a loan from a bank. The more “cheaper” method will be more in demand, and over time, rates (including all costs) in both markets - bond and credit - will even out.

• Banks who choose a method of investing funds that will bring them higher returns.

When placing funds, banks choose between issuing a loan to an enterprise and purchasing corporate bonds. The divergence of returns in these markets will inevitably lead to capital outflows and returns will even out. At the same time, liquidity for a loan bank and a bond is different, which is also taken into account in the model in the form of a liquidity premium.

• Enterprises and population who are trying to place temporarily free funds with the highest yield.

By placing temporarily free funds, enterprises and individuals choose between purchasing bonds and opening a deposit in a bank. As in the previous model, the actions of participants seeking to maximize their returns will even out returns in these markets.

The models described above make it possible to understand what instruments each of the considered groups will use to achieve their goals, and how this will affect the level of interest rates in various markets. The results of all the models described above are weighted depending on the significance of the group of economic agents that are guided by a particular model.

Having received the vector of interest rates, we can say at what yield investors will be ready to buy any of the bond issues currently circulating on the market in a year. Further, by discounting coupon payments and payments of the body of bonds at the rate that investors will require in a year from investments in such securities, we calculate the future value of bonds.

For example, the results of model calculations indicate that in the coming year, the average level of return required by investors will increase by 0.5% compared to the current level. In this case, we need to choose which of the two bond issues to purchase:

• Company-1 - duration 1 year, coupon rate 10%, payments are made quarterly;
• Company-5 - duration 5 years, coupon rate 10%, payments are made once a quarter.

If within five years interest rates and, as a result, the yield required by investors will remain at current levels, then you can buy either of the two bond issues. The return on both investments will be the same and will amount to 10% per annum.

In the case under consideration, when we expect an increase in interest rates by 0.5%, the wrong choice can significantly reduce the efficiency of investments.

In the case of the Company-1 issue, despite the fact that the required yield from these bonds will be 10.5% per annum, while the coupon payments on these bonds will be 10% per annum, the investor will receive its nominal value after the bond issue is redeemed. price. He will be able to invest the funds received in the company's bonds with the same credit quality and liquidity, but the coupon rate for them will already be 10.5%.

If the investor's funds are invested in Company-5 bonds, the repayment of which will occur only after five years, then the profitability of his investments will be lower.

This example shows the importance of correctly predicting the level of interest rates when choosing bonds.

Coupon payments are 10% per annum, while the required yield on investments in bonds with the same credit quality and liquidity will be 10.5% per annum.

As a manuscript Galkin Dmitry Evgenievich FORECASTING OF INTEREST RATES BASED ON THE THEORY OF DETERMINISTIC CHAOS AS A METHOD OF INTEREST RISK MANAGEMENT IN COMMERCIAL BANKS Specialty 08.00.13 - mathematical and instrumental methods of economics ABSTRACT of the dissertation for the degree of candidate of economic sciences Perm 2012 The work was done at the Department of Applied Mathematics HPE "Perm National Research Polytechnic University" (PNRPU) Supervisor: Doctor of Technical Sciences, Professor Pervadchuk Vladimir Pavlovich Official opponents: Doctor of Physical and Mathematical Sciences, Professor Rumyantsev Alexander Nikolayevich Candidate of Economic Sciences, Associate Professor Ivliev Sergey Vladimirovich State Technical University, Izhevsk The defense will take place on March 29, 2012 at 2 pm 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 The abstract was distributed on February 28, 2012. Scientific Secretary of the Dissertation Council, Doctor of Economics, Associate Professor T.V. Mirolyubova 2 GENERAL DESCRIPTION OF THE WORK Relevance of the research topic. The task of risk management in the banking sector is not trivial throughout the course of banking activities. The problem of banking risks in modern times is becoming increasingly important in the light of the increasing influence of the financial sector on the world economy. For example, in the United States, 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 enormous 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, interest rate risk occupies a special place, yielding the leading position in terms of the degree of influence only to credit risk. 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 particular 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 a period of economic instability, interest rate risk management should be carried out in a balanced way, taking into account possible scenarios that affect the level of interest rate risk. The circumstances mentioned above determine the relevance of the study. The degree of scientific development of the topic. Such scientists as Macaulay F., Redhead K., Hughes S., Entrop O., Cade E., Helliar C., 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 which Sevruk V.T., Larionova I. V., Vinichenko I.N., Lavrushina O.I., Sokolinskaya N.E., Valentseva N.I., Khandrueva A.A. 3 One of the dynamically developing areas in the study of economic objects and systems is the use of mathematical methods . Among them, we should separately note the approaches that make it possible to widely use the concepts of synergetics, deterministic chaos, and fractal geometry in the study. The following scientists were involved in the development and development of such methods: Takens F., Sornette D., Peters E., Bachelier L., Mandelbrot B., 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, a significant contribution to the development of this direction was made by Kurdyumov S.P., Malinetsky G.G., Bezruchko B.P., Loskutov A.Yu., Shumsky S.A., Kuperin Yu.A. The aim of the dissertation research is to develop theoretical and methodological foundations for managing interest rate risk in commercial banks based on interest rate forecasting using the theory of deterministic chaos. To achieve this goal, the following tasks were solved: 1. Research of existing approaches for forecasting financial time series and assessing interest rate risk in order to use existing experience in the development of a new method. 2. The choice of effective tools for the study of nonlinear dynamic systems based on the generated time series. 3. Study of the connection between the market of interest rates and interest rate 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 predicting interest rates. 6. Creation of interest rate risk management methodology based on the developed forecasting models. The object of the study is commercial banks exposed to interest rate risk as a result of operations 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 associated with interest rate risk. 4 The field of study 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" in the following points: 1.1. Development and development of the mathematical apparatus for the 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. 1.6. Mathematical analysis and modeling of processes in the financial sector of the economy, development of the method of financial mathematics and actuarial calculations. 2.3. Development of decision support systems for rationalization of organizational structures and optimization of economic management at all levels. theoretical and methodological basis are the scientific works of domestic and foreign scientists in the field of assessing and managing interest rate risk in banks, the theory of deterministic chaos, nonlinear dynamics, mathematical methods and models of financial markets, fractal geometry, synergetics, published in Russian and foreign press, as well as on the Internet. Practical calculations in the framework of this study were carried out using such applied software tools as MS Excel, MathWorks Matlab, Fractan, Tisean. The information base of the study consisted of: - data from information and analytical materials on the problem under study, presented in the scientific literature, periodicals and the Internet; – statistical sources in the form of quotations of LIBOR and EURIBOR interbank lending rates for various periods. The most significant results obtained personally by the author, which have scientific novelty and are submitted for defense, are: 1. Non-linearity and determinism of the LIBOR and EURIBOR interest rate market established using statistical methods. 2. A modified mathematical model for predicting interest rates based on a one-dimensional time series, taking into account the determinism of the 5 systems under study, as well as a developed approach to determine the scope of this model. 3. A mathematical model for predicting interest rates based on a multidimensional time series, which takes into account the determinism of the systems under study and makes it possible to use the dynamics of several systems when making a forecast. 4. Methodology for managing interest rate risk in commercial banks, which is based on a mathematical model for predicting interest rates based on the methods of the theory of deterministic chaos, which allows scenario modeling using predictive data. Theoretical significance of the results. The provisions and conclusions formulated in the dissertation research develop the theoretical and methodological basis for the analysis and forecasting of 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 a correct tool that allows them to switch from hypothetical scenario modeling to scenario modeling based on more probable forecast data in the problem of interest rate risk management. Approbation of the research results. The main provisions of the dissertation work were reported at the scientific and technical conference of students and young scientists of PSTU (St. 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 the Study of Dynamic Models of PSNIU (Perm, 2011). The results of the study have found practical application in CJSC UniCredit Bank. The work of this organization uses the interest rate risk management methodology, as well as the interest rate forecasting model described in the study. 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 reading the course "Mathematical Analysis of Dynamic Models in Economics" in the direction of preparation 010500.68 "Applied 6 Mathematics and Informatics" as part of the master's program "Mathematical Methods in the Management of Economic Processes ”and when reading the course “Mathematical Analysis of Dynamic Processes in the Economy” in the direction of preparation 080100.68 - “Economics” within the framework of the master's program “Mathematical Methods for Analyzing the Economy”. The implementation of the research results in these organizations is confirmed by relevant documents. Publications. On the topic of the dissertation, the author published eight papers with a total volume of 3.72 pp, including two papers in publications recommended by the Higher Attestation Commission for publication of the results of the dissertation (1.16 pp). Volume and structure of the dissertation work. The work is presented on 147 typewritten pages. The main results of the study are illustrated in 26 tables and 77 figures. The list of used literature is 108 titles. The structure of the dissertation is determined by the purpose, objectives and logic of the study. 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 goals and objectives scientific research, highlights the most significant achievements in the field of research, and presents the novelty of the results. The first chapter "Application of mathematical methods in the study of financial time series" considers the existing methods and approaches to forecasting financial time series, evaluates their effectiveness, determines the prerequisites for using nonlinear methods for modeling financial time series. In the second chapter "Selection and justification of methods for studying nonlinear dynamical systems based on time series", the main approaches to the study of dynamical systems using the theory of deterministic chaos are defined, a critical assessment is made, and the most optimal and correct tools for studying systems based on time series are identified. 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 that give rise to interest rate risk is studied in order to reveal the nature of the relationship between the interest rate market and interest rate risk. In the fourth chapter, "Development of an interest rate risk management method based on interest rate forecasting," the interest rate market is studied for non-linearity and determinism. A forecasting model based on a one-dimensional time series is being adapted to the interest rate market; forecasting models based on a multivariate time series are being developed. On the basis of the models obtained, 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, an assessment of the practical significance of the work. MAIN PROVISIONS AND RESULTS OF THE RESEARCH TO BE DEFENDED 1. Non-linearity and determinism of the LIBOR and EURIBOR interest rate market established by statistical methods. This provision is based on a study of interest rates LIBOR for 3 months and EURIBOR for 1, 3 and 6 months, which are the most popular reference floating rates and to which the pricing of floating rate loans in US dollars and euros is linked. These rates reflect the cost Money in the interbank lending market for first-class borrowers with a credit rating of AA and above for the appropriate period and in a specific currency. The dissertation established a qualitative relationship between the interest rate market and the level of interest rate risk for commercial banks. As a result, the interest rates LIBOR and EURIBOR, as the most popular rates for pricing in the global financial markets, were examined for non-linearity and determinism. Previously, to obtain quasi-stationarity, the studied time series were transformed based on the transformation x (1) y t log(xt) log(x t 1) log(t) , t 2, n x t 1 8 To study the signs of nonlinearity of systems, the BDS test proposed by Brock, Dechert was used and Shenkman, whose idea is to calculate statistics based on the difference of correlation integrals (2) for embedding dimensions m and 1. 2 (2) C N (l , T) I t (xtN , xsN , l) TN (TN 1) t s where and xtN (xt , xt 1 ,..., xt N 1) xsN (x s , x s 1 ,..., x s N 1) are historical data, TN T N 1 , and 1, at x N x N l , t s where is the supremum norm. , l) N N 0, for xt x s l The resulting statistics (3) should have a normal distribution N (0,1) if the process under study is white noise. T (C N (l , T) C1 (l , T) N) (3) wN (l , T) N (l , T) different meanings l 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 different values ​​of l and embedding dimensions m. The results obtained made it possible to reject the null hypothesis for each process, i.e. the readings are not independent and uniformly distributed. In addition, BDS statistics were calculated for the residuals of the AR(1) autoregressive model, as a result of which the null hypothesis for each process was also rejected, which in turn made it possible to conclude that the studied processes were non-linear. 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 how much the studied objects have a long-term memory. The estimate was made on the basis of calculating the normalized range of the time series: R / S cN H (4) I t (xtN , x sN 9 where R max(x tn) min(x tn) is the range of the time series, N is the number of observations, H is Hurst exponent, S is the standard deviation of xtn series Based on the log-log plot of the dependence of the normalized range R / S on the number of observations N, 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 - rates LIBOR for a period of 3 months, 1mEURIBOR - 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): 0.7863 0.7791 The obtained results (H 0.5) indicate that the studied systems are persistent, i.e. they have a long-term memory and strive to maintain the trend. From the results of the BDS test for these systems, we can conclude that the processes under study are deterministic. 2. A modified mathematical model for predicting 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 this model. When studying the time series of interest rates, it can be considered as the implementation of a more complex process of higher dimension. In this case, it is possible to carry out the reconstruction of the attractor and, thereby, to investigate the process that generates the time series itself. The attractor is reconstructed using the coordinate delay method: x(t) (s (t), s (t),. .., s (t (m 1))) (5) where m is the embedding dimension, and m 2d 1 , d is the Minkowski dimension. The projection of the reconstructed attractor of the 3mLIBOR system into the space R2 is shown in fig. 1, where the diagonal structures are a confirmation of the determinism of the system. 10 s (t m) h(f (m) (x t) Fm (x t) (8) As a result, all m values ​​of the time series can be expressed in terms of the xt value using the set of functions F1,…,Fm. By changing the variables z t 1 0.04 0.03 (s (t 1), s (t 2),..., s (t m)) and introducing a vector function that depends on t and f, (8) can be rewritten as z t 1 (x t) (9) 0.02 0.01 In accordance with the Takens theorem, if: M d R m is diffeomorphic, then it is possible to embed M d into R m without self-intersections Since it has a smooth inverse function, equality (9) can be written as x t 1 (z t 1) ( 10) Substituting (10) into s (t m 1) Fm 1 (x t) , we get that 0 -0.01 -0.02 -0.03 s (t m 1) Fm 1 (1 (z t 1) Fm 1 (1 (s (t 1 ), s (t 2),..., s (t m))) -0.04 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Fig. 1. Reconstructed attractor 3mLIBOR Consider a discrete dynamical deterministic system whose dynamics is defined as xt 1 f (xt) (6) For 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 s (t) h(f (t) (x 0)) (7) This representation is valid for any point of the time series s (t) in any period of time, with the only difference being the number of system actions f on the initial condition. Those. considering m successive values ​​of the time series, we can express them as s (t 1) h(f (x t) F1 (xt) s (t 2) h(f (xt 1) h(f (f ((x t))) ) F2 (xt) … 11 (s (t 1), s (t 2),..., s (t m)) (11) meaning of the embedding dimension.Due to the fact that the function 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 m, and in the output layer is 1. To increase the efficiency of this model, the maximum Lyapunov exponent, which determines the predictability of the system , and the Hurst exponent H, which determines the determinism of the system, were considered as functions of time. For this, a window w was used, the length of which was chosen individually for each time series under study, and with the movement of the window, the indicated characteristics were calculated. Based on this, an area was allocated for the application of the model, where 0 and H 0.5 . On fig. Figure 2 shows the time series of the interest rate 3mLIBOR together with the maximum Lyapunov exponent and the Hurst exponent as a function of time, on the basis of which the model applicability area was determined. An iterative forecast of the next value was built on the basis of previous historical data. 12 Original time series 10 5 0 0 1000 2000 3000 4000 5000 6000 7000 6000 7000 Dynamics of the maximum Lyapunov exponent 0.1 0.05 0 -0.05 -0.1 0 1000 2000 3000 4000 5000 , which takes into account the determinism of the systems under study and makes it possible to use the dynamics of several systems when constructing a forecast. If information is available on interest rates in one currency for various terms it is possible to consider these time series as realizations of one process, i.e. as projections of one process onto three coordinate axes. However, in this case, the complexity lies in the correct restoration of the attractor: each time series has different metric characteristics. To overcome this problem, the creation of an extended nesting space is provided: (xn , xn , xn 2 ,..., xn (m 1) , (12) z n yn , yn , yn 2 ,..., yn (m 1) , z n , z n , z n 2 ,..., z n (m 1) ) 1 1 2 1 2 3 0.5 0 0 1000 2000 3000 4000 5000 6000 7000 Identification of the model applicability area for 3mLIBOR The results of predicting the next value of the 3mLIBOR time series are shown in Fig. 3. This forecasting approach was 25% more efficient than the method using the current value as the predictor (the best random walk forecasting method). 13 3 1 2 3 where i is the coordinate delay parameter defined for the i-th system; mi is the embedding dimension of the i-th system; xn, yn, zn - reports of the corresponding time series. When considering an attractor embedded in a space of dimension D m1 m2 m3 , the Takens theorem will also be valid, since the requirements for the minimum embedding dimension will be met in advance by “subembeddings”, the dimension of which initially ensured the fulfillment of the Takens theorem. In this form, an artificially increased embedding dimension due to other time series will allow taking into account additional information about the system, incl. on the term structure of interest rates. In this mathematical model, a non-parametric model in the form of kernel smoothing of the coordinates of the next points for the k-nearest neighbors of the trajectory point in the reconstructed phase space is used for the prediction. Then the forecast point of the trajectory will look like: zt 1 Fig. 3. Original ( solid line) and forecast (dashed line) time series 3mLIBOR 3 1 2 N n (zt) (yk 1 yk zt)wk (zt , yk) (13) k 1 where N n (z t) is the number of neighbors for point zt , and wk (zt , yk) are weight coefficients. 14 According to the Nadarai-Watson formula, the weights wk (zt , yk) can be defined as K h (zt y k) (14) wk (z t , y k) N (z) p1 K h (zt y p) n t x2 () x 1 1 where kernel function K h (x) K () e 2h . h h 2 h Generally speaking, the form of the 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 is h 0.5 . According to Kantz H. and Shreiberg T., this approach to modeling chaotic time series is quite robust to noisy data and effective for experimental systems. Besides, 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 orientation of the system, on the other hand, it successfully models local dynamics. On fig. 4 shows a long-term forecast of the interest rate 1mEURIBOR for the values ​​from 1703 to 1751 as a result of applying this mathematical model to a set of EURIBOR interest rates for a period of 1, 3 and 6 months. The previous values ​​acted as the initial data for the forecast. more than 15 values, while 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: with ARIMA, ARIMA-GARCH models and a radial basis neural network. On fig. Figure 5 shows the results of forecasting using these models for a certain section of the interest rate 1mEURIBOR. 0.53 0.52 Original series Model based on TDH ARIMA ARIMA-GARCH RBF-Network 0.51 0.50 0.49 0.48 0.47 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. 1mEURIBOR and its forecast values ​​based on various models Table 2 presents the results of numerical comparison of forecasting efficiency based on the normalized standard deviation (NSSD) 1 N NSSD N (x pr x real) 2 i 1 2 (15) where 2 is the variance of the test set and absolute error (x): x 1 N N x pr x real (16) i 1 Fig. 4. Original (solid line) and predictive (dashed line) time series 1mEURIBOR The proposed mathematical forecasting model performs correct forecasting with a forecast horizon of not 15. theory of deterministic chaos (TDC) is the most effective. 16 4. Methodology for managing interest rate risk in commercial banks, which is based on a mathematical model for predicting interest rates based on the methods of the theory of deterministic chaos, which allows 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). start А Assessment of yield sensitivity to changes in interest rates Identification of the most risky positions and identification of determining interest rates Reconstruction and calculation of invariants for the main rate systems Risk acceptance Yes Gap analysis Yes Yes Upward dynamics Positive risk position Increase in assets Yes No No Negative risk position Increase liabilities Evaluation of model parameters Forecasting interest rates No No Increase in assets Evaluation of forecasting efficiency Correction of model parameters A end Fig. 6. Methodology for interest rate risk management Thus, the first stage consists in analyzing the current position subject to interest rate risk using gap analysis and assessing the sensitivity of yield to changes in interest rates in the context of repricing intervals. Thanks to this, the interest rates that determine the change in yield to the greatest extent are identified. Based on the selected set of interest rates, the attractor is reconstructed and the invariant is calculated, then forecasting is carried out. The results of the forecast are interpreted in terms of risk acceptance or risk reduction. When the risk is reduced, depending on the predicted dynamics and the current risk position, actions are taken: in case of forecasting upward dynamics in the interest rate market 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 due to the following actions: purchase of securities with a floating rate; conversion of interest rates on loans from fixed to floating; replacement of funding for loans with a floating interest rate for funding with a fixed interest rate; Otherwise, liabilities sensitive to interest rate risk increase. Conclusions 1. The existing set of tools of the theory of deterministic chaos for the study of systems based on time series was critically evaluated and based on this, as well as a comparative approach, the most effective methods for the reconstruction of the attractor, calculation of the correlation dimension and characteristic Lyapunov exponents. 2. A qualitative relationship between interest rate risk and the interest rate market was revealed, and the latter object was identified as one of the main causal factors in the occurrence of interest rate risk in commercial banks. non-linearity and determinism 3. LIBOR interest rates are set for a period of 3 months and EURIBOR for a period of 1, 3 and 6 months. The reconstruction of dynamic systems based on time series was carried out, the evaluation of metric and dynamic invariants was carried out, the results of which once again confirmed the hypothesis of 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 are developed on the basis of 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 extended embedding space and nuclear smoothing of neighboring trajectory points, the efficiency of which exceeds the efficiency of classical approaches to forecasting financial markets. 18 6. A methodology has been created for managing interest rate risk in commercial banks based on the developed model for predicting the interest rate market. Pervadchuk V.P., Galkin D.E. The role of the interbank lending rate 8. LIBOR in the global economy // Bulletin of Perm. state tech. university - ser. Socio-economic sciences. - Perm, 2011. - p. 101105. PUBLICATIONS ON THE TOPIC OF THE RESEARCH Publications in publications recommended by the HAC: 1. Pervadchuk V.P., Galkin D.E. Application of methods of the theory of deterministic chaos for forecasting the dynamics of the interbank lending rate LIBOR // Vestnik Izhevsk. state tech. university - No. 2 (46). - Izhevsk, 2010. - p.45-49. 2. Galkin D.E. Forecasting of multidimensional financial time series based on methods of the theory of deterministic chaos // Bulletin of Inzhekon. - 2011. - No. 3 (46). - Ser. Economy. - St. Petersburg, 2011. - 359-363 p. In other editions: 3. Galkin D.E., Pervadchuk V.P. Fractal analysis of the dynamics of exchange rates // Abstracts of the scientific and technical conference of students and young scientists Permsk. state tech. university - ser. Applied Mathematics and Mechanics, 2007. - p. 26-27. 4. Pervadchuk V.P., Galkin D.E. Substantiation of the application of methods of the theory of deterministic chaos for the forecast of economic systems. Vestnik Perm. state tech. university - ser. Mathematics and applied mathematics. - Perm, 2008. - p. 15-24. Pervadchuk V.P., Galkin D.E. The use of fractals in the 5. study of financial time series // Bulletin of Perm. state tech. university - No. 14. - ser. Mathematics and applied mathematics. - Perm, 2008. - p. 8-15. V.P., Galkin D.E. Modeling 6. Pervadchuk 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 phase attractor recovery for forecasting economic systems // Information and Computational Technologies and Their Applications: Collection of Articles of the XV International Scientific and Technical Conference. - Penza: RIO PGSKhA, 2011. - p.27-31 19 _______________________ Signed for publication on 20.02.2012. Format 60х84/16 Usl. oven l. 1.45. Circulation 100 copies. Order ___ . Printing house of PSNIU. 614990. Perm, st. Bukireva, 15 20