To simulate levels interest rates Various types of equations are used in statistics, including polynomials of various degrees, exponentials, 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 view 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 - last level 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 ax 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 return investors expect in various countries from investments with a certain level of risk, we, predicting the level of inflation in Russia, can say what the profitability of specific instruments should be, so that investors would be interested in investing in the country, and not abroad.

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 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 per dollar. The rate expected in a year is 55. Therefore, if the current yield of instruments with a certain level of risk in the United States is 10% per annum, then the return expected by investors 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.

In order to predict the further dynamics of the currency pair, a huge number of methods have been developed. However, the quantity has not turned into quality, and getting a fairly effective forecast is not an easy task. Let's consider the four most common methods for forecasting the rates of currency pairs.

Purchasing power parity (PPP) theory

Purchasing power parity (PPP) is perhaps the most popular method. It is mentioned more often than others in textbooks on economics. 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 must be the same as the price of the same cabinet in the United States, taking into account the exchange rate and excluding transport and exchange costs. That is, there should be no reason for speculation to buy cheaply in one country and sell more expensively in another.

According to PPP theory, changes in the exchange rate should compensate. For example, in current year prices in the US should rise 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 in order for the price of the same commodity 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/CAD) = $0.918/CAD

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

The most common example of using the PPP principle is the Big Mac index, which is based on comparing its price in different countries, and which shows the level of undervaluation and overvaluation of the currency.

The principle of relative economic stability

The method of this hike is described in the title itself. The growth rates of the economies of different countries are taken as a basis, which make 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. For investment, it is necessary to purchase the national currency, which, accordingly, leads to an increase in demand for national currency and its subsequent strengthening.

This method is suitable not only when comparing the state of the economy of two countries. It can be used to form an opinion on the presence and intensity of investment flows. For example, investors are attracted by higher interest rates, which allow them to get the maximum return 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 from the theory of PPP is that with its help it is impossible to make a forecast of the size of the exchange rate. It gives the investor only general idea about the prospects for strengthening or weakening of the currency and the strength of the momentum. To get more complete picture, the principle of relative economic stability is combined with other forecasting methods.

Building an econometric model

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

Take, for example, making a forecast for the coming year for the USD/CAD pair. We select the key factors for the dynamics of the pair: the difference (differential) in the interest rates of the USA and Canada (INT), the difference in and the difference between the growth rates of personal incomes of the population of the USA and Canada (IGR). The econometric model in this case will look like this:

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

The coefficients a, b and c can be both negative and positive and show how strong the influence of the corresponding factor is. It should be noted that the method is rather complicated, however, if there is a ready-made model, it is enough to simply substitute new data to obtain a forecast.

Time series analysis

The method of time series analysis 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 forecasting price models 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 particular currency pair.

Undoubtedly, forecasting exchange rates is an extremely difficult task. Many investors simply prefer to insure currency risks. Other investors are aware of the importance of forecasting exchange rates and seek to understand the factors that affect them. The above methods can become good help for such market participants.

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 for interest rate forecasts the supply and demand toolkit outlined in in general terms in this section.

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. As the price level rises during a given year, interest rates will rise due to the price level effect.

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 effect of the price level reaching its greatest influence next year, the effect of expected inflation will be the next year's least influence (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 short term lower interest rates is a consequence of the increased rate of growth in the quantity of money, but in fact they cease to rise above the initial 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.