The real interest rate is. Nominal and real deposit rate

The most important characteristic of the modern economy is the depreciation of investments through inflationary processes. This fact makes it advisable to use not only a nominal, but also a real interest rate when making some decisions in the market. What is an interest rate? What does it depend on? How ?

Interest rate concept

The interest rate should be understood as the most important economic category that reflects the profitability of any asset in real terms. It is important to note that it is the interest rate that plays a decisive role in the process of making management decisions, because any economic entity is very interested in obtaining the maximum level of revenue at minimum costs in the course of its activities. In addition, each entrepreneur, as a rule, reacts to the dynamics of the interest rate in an individual way, because in this case the determining factor is the type of activity and the industry in which, for example, the production of a particular company is concentrated.

Thus, owners of capital assets often agree to work only if the interest rate is extremely high, and borrowers are likely to acquire capital only if the interest rate is low. The examples discussed are clear evidence that today it is very difficult to find equilibrium in the capital market.

Interest rates and inflation

The most important characteristic of a market economy is the presence of inflation, which determines the classification of interest rates (and, naturally, the rate of return) into nominal and real. This allows you to fully assess the effectiveness of financial transactions. If the inflation rate exceeds the interest rate received by the investor on investments, the result of the corresponding operation will be negative. Of course, in terms of absolute value, his funds will increase significantly, that is, for example, he will have more money in rubles, but the purchasing power that is characteristic of them will drop significantly. This will lead to the opportunity to buy only a certain amount of goods (services) with the new amount, less than would have been possible before the start of this operation.

Distinctive features of nominal and real rates

As it turned out, they differ only in conditions of inflation or deflation. Inflation should be understood as a significant and sharp decline, while deflation should be understood as a significant drop. Thus, the nominal rate is considered to be the rate set by the bank, and the purchasing power inherent in income and denoted as interest. In other words, the real interest rate can be defined as the nominal interest rate, which is adjusted for inflation.

Irving Fisher, an American economist, formed a hypothesis explaining how it depends on nominal values. The main idea of the Fisher effect (this is the name of the hypothesis) is that the nominal interest rate tends to change in such a way that the real one remains “stationary”: r(n) = r(p) + i. The first indicator of this formula reflects the nominal interest rate, the second - the real interest rate, and the third element is equal to the expected rate of inflation processes, expressed as a percentage.

The real interest rate is...

A striking example of the Fisher effect, discussed in the previous chapter, is the picture when the expected rate of the inflation process is equal to one percent on an annual basis. Then the nominal interest rate will also increase by one percent. But the real percentage will remain unchanged. This proves that the real interest rate is the same as the nominal interest rate minus the expected or actual inflation rate. This rate is completely free of inflation.

Calculation of the indicator

The real interest rate can be calculated as the difference between the nominal interest rate and the level of inflation processes. Thus, the real interest rate is to the following relation: r(р) = (1 + r(н)) / (1 + i) - 1, where the calculated indicator corresponds to the real interest rate, the second unknown member of the relationship determines the nominal interest rate, and the third element characterizes the inflation rate.

Nominal interest rate

When talking about lending rates, as a rule, we are talking about real rates ( the real interest rate is purchasing power of income). But the fact is that they cannot be observed directly. Thus, when concluding a loan agreement, an economic entity is provided with information about nominal interest rates.

The nominal interest rate should be understood as a practical characteristic of interest in quantitative terms, taking into account current prices. The loan is issued at this rate. It should be noted that it cannot be greater than zero or equal to it. The only exception is a loan on a free basis. Nominal interest rate is nothing more than interest expressed in monetary terms.

Calculation of the nominal interest rate

Suppose an annual loan of ten thousand monetary units pays 1,200 monetary units as interest. Then the nominal interest rate is equal to twelve percent per annum. After receiving 1200 monetary units on a loan, will the lender become rich? This question can be answered correctly only by knowing exactly how prices will change over the course of an annual period. Thus, with annual inflation equal to eight percent, the lender's income will increase by only four percent.

The nominal interest rate is calculated as follows: r = (1 + percentage of income received by the bank) * (1 + increase in inflation rate) - 1 or R = (1 + r) × (1 + a), where the main indicator is the nominal interest rate, the second is the real interest rate, and the third is the growth rate of the inflation rate in the country corresponding to the calculations .

conclusions

There is a close relationship between nominal and real interest rates, which for absolute understanding it is advisable to present as follows:

1 + nominal interest rate = (1 + real interest rate) * (price level at the end of the time period under consideration / at the beginning of the time period under consideration) or 1 + nominal interest rate = (1 + real interest rate) * (1 + rate of inflation processes).

It is important to note that the real effectiveness and efficiency of transactions performed by the investor is reflected only by the real interest rate. It talks about the increase in funds of a given economic entity. The nominal interest rate can only reflect the increase in funds in absolute terms. It does not take inflation into account. Increase in real interest rate speaks of an increase in the level of purchasing power of the monetary unit. And this equals the opportunity to increase consumption in future periods. This means that this situation can be interpreted as a reward for current savings.

The interest rate is one of the most important macroeconomic indicators. There are many different interest rates in the financial market. First of all, interest rates on deposits and loans differ. For example, at the end of June 2012, rates on ruble deposits of individuals in Sberbank of Russia were in the range of 0.01-8.75% per annum, and rates on loans for the purchase of real estate in the same bank were in the range of 11-16.5% per annum. Sberbank's interest rates differ from rates in other commercial banks and rates in the interbank lending market. Interest rates in the banking system as a whole may differ from interest rates (or similar amounts, such as annual stock returns) in other segments of the financial market, such as private or government securities markets. Additionally, the size of the rates may be influenced by different degrees of investment risk in different segments of the financial market (a higher risk corresponds to a higher percentage). However, interest rate movements in different segments of the financial market are driven by similar mechanisms, and in most cases the broad spectrum of interest rates in a country moves in the same direction (barring short-term fluctuations). Therefore, in the future, by interest rate we will understand a certain single, abstract, “average” interest rate.

The importance of the interest rate lies primarily in the fact that it characterizes the cost of using borrowed funds in the financial market. Rising interest rates mean that borrowing from the financial market will become more expensive and less accessible to potential borrowers - for example, firms wanting to expand their business and upgrade their equipment, or home buyers wanting to get a mortgage. If rising interest rates force them to abandon investments, this could have far-reaching undesirable consequences for the entire economic system of the country. What could cause interest rates to rise? One of the reasons is increased inflation (especially in modern Russia). To describe the relationship between interest rates and inflation, it is necessary to introduce the concepts of real and nominal interest rates.

The typical interest rate that you might see when you go to a bank or other financial institution is called nominal (g). The nominal rates on deposits and loans at Sberbank in June 2012, given above. It is interesting that in 1992 in the same bank the interest rate on deposits (in rubles) could reach 190% per annum. Thus, every ruble placed on this deposit at the beginning of 1992 turned into 2 rubles over the year. 90 kopecks (1 ruble of the original deposit plus 190%). But did the owner of the deposit become richer as a result? Suppose at the beginning of 1992 for 1 rub. you could buy one loaf of bread. According to official statistics, in 1992 the inflation rate in Russia was approximately 2540%. If bread rose in price at such a rate, then its price over the year increased 26.4 times (see mathematical commentary “Growth and Increment Rates”) and by the end of the year amounted to 26 rubles. 40 kopecks Thus, at the beginning of the year, 1 ruble deposited could buy one loaf of bread. At the end of the year, for the 2 rubles received from the bank. 90 kopecks it was possible to buy only approximately one tenth of this loaf (to be precise, 2 rubles 90 koi: 26 rubles 40 koi "0.11 loaves of bread). Due to the fact that the growth in the size of the deposit in the bank lagged behind the rise in prices, the depositor lost nine-tenths of a loaf of bread, or, in other words, nine-tenths of the purchasing power of his money (to be precise, he lost 89% of its purchasing power, i.e. from one whole loaf at the beginning of the year there was only 0.11 loaves left at the end of the year and) The value -89%, in the calculation of which the nominal interest rate was adjusted for the inflation rate, is called real interest rate. It is usually denoted by the small letter r . Given data on the nominal interest rate i and the inflation rate π, the real interest rate can always be calculated using the Fisher formula:

(here all three values are expressed as percentages). An example of using Fisher's formula for our 1992 data:

If the inflation rate in the country is insignificant,

a simpler, approximate formula can be used that relates nominal, real interest rates and the inflation rate:. For example, if the annual inflation rate π was 1%, and the nominal rate i was equal to 3%, then the real interest rate was approximately

Let's return to the previously asked question, slightly modifying it. Why do nominal interest rates change? From the formula we find the nominal rate: . We get an effect called Fisher effect. In accordance with this effect, two main components of the nominal interest rate are distinguished - real interest and the inflation rate and, accordingly, two reasons for its change. Typically, a financial institution (say, a bank), when determining the nominal interest rate for the next year, proceeds from some target value of the real rate and its expectations regarding the future rate of inflation. If the target value of the real rate is +2% per annum and the bank’s experts expect a price increase over the next year of 1.5%, then the nominal rate will be set at 3.5% per annum. Please note that in this example, the formation of the nominal interest rate was influenced not by actual, but by expected inflation, which can be formalized as , where is the expected inflation rate (e - from English expected, expected).

Thus, the nominal rate is determined by two components - the real rate and the expected inflation rate. Note that fluctuations in the real interest rate are usually less significant than fluctuations in the expected inflation rate. In this case, according to the Fisher effect the dynamics of the nominal interest rate is largely determined by the dynamics of the expected inflation rate(Fig. 2.13 is offered as an illustration).

In turn, expected inflation is largely determined by the past history of this economic indicator: if inflation was insignificant in the past, it is expected that it will be insignificant in the future. If a country has previously experienced severe inflation, this gives rise to pessimistic expectations for the future. If in Russia until recently the inflation rate, as a rule, was a double-digit value, this also had an impact on the average interest rates in our country, and increased inflation led to an increase in funeral interest rates, and weakening inflation slightly reduced them.

Rice. 2.13.

The interest rate is indicated for 3-month Treasury bills; inflation is calculated as the growth rate of the CPI for All Urban Consumers in a given month relative to the same month last year. Sources: According to the US Federal Reserve (federalreserve.gov) and the US Bureau of Labor Statistics (bls.gov).

Inflation has a direct impact on the level of interest rates. Obtaining loans in conditions of inflation is associated with an increasing rate of bank rates, which reflect inflation expectations. Therefore, a distinction is made between nominal and real interest rates.

The terms “nominal” and “real” are widely used in economics: nominal and real wages, nominal and real profit (profitability) and these terms always indicate which of the indicators is calculated: one that does not take into account the level of inflation (nominal) and one cleared of inflation (ral).

Nominal interest rate– this is the amount of payment in monetary terms for the loan received by the borrower. This is the price of the loan in monetary terms.

Real interest rate– this is the income on a loan, or the price of a loan, expressed in natural measures of goods and services.

The concepts of “nominal” and “real” apply to all indicators that are affected by inflation.

To convert the nominal interest rate to the real interest rate, we use the following notation:

i – nominal interest rate;

r – real interest rate;

f – inflation rate.

Then i = r + f + r f, (15)

In the test, it is necessary to calculate what the nominal annual profitability of the enterprise should be so that the real annual profitability is equal to the interest rate indicated in column 3 of the table. P.3 at a monthly inflation rate equal to the value indicated in column 5 of the table. P.3.

For example , to ensure a real profit of the enterprise in the amount of 20% per year with an inflation rate of 1.5% per month, it is necessary to achieve a nominal profitability in the amount of:

Rh = 0.196 + 0.2 + 0.196 · 0.2 = 0.435 = 43.5%.

The annual inflation rate is calculated using the effective interest rate formula (calculation No. 8 of this test).

11. Calculation of performance indicators for investment projects

In this block it is necessary calculate the economic efficiency indicators of two investment projects and compare their results. The amount of investment for two projects is the amount indicated in column 2 of table. P.3. The interest rate is accepted in accordance with the data in column 3 of table. Clause 3 (annual interest rate No. 1).

The only difference between the projects is that in the second investment project the costs are incurred not in one year, as in the first, but in two years (divide the amount of investment in column 2 of Table A.3 by two). In this case, it is expected to receive net income within 5 years in the amounts indicated in column 6 of table. P.3. In the second investment project, receiving annual income is possible from the second year for 5 years.

In Fig. 11.1, 11.2 presents a graphical interpretation of these projects.

1Project

Rice. 11.1. Graphic interpretation of investment project No. 1

2 Project

Rice. 11.2. Graphic interpretation of investment project No. 2

To assess the effectiveness of an investment project, the following indicators should be calculated:

net present value (NPV);

net capitalized value (EW);

internal rate of return (IRR);

investment return period (IRP);

profitability index (ARR);

profitability index (PI).

The economic efficiency of complex investment projects is assessed using dynamic modeling of real cash flows. With dynamic modeling, the cost of costs and results decreases as they move away in time, since investments made earlier will bring greater profits. To ensure comparability of current costs and results, their value is determined on a specific date.

In the practice of assessing the economic efficiency of investments, the value of current costs and results is usually found at the end or beginning of the billing period. The value at the end of the billing period is determined by capitalization, the value at the beginning of the billing period is determined by discounting. Accordingly, two dynamic assessments are formed: a capitalization system and a discounting system. Both dynamic systems require identical preparation of initial information and provide an identical assessment of economic efficiency.

The economic effect for the billing period represents the excess of the value of capitalized (discounted) net income over the cost of capitalized (discounted) investments for the billing period.

Example , after carrying out measures to reconstruct the enterprise, the costs of which amount to 1000 USD. it became possible to reduce production costs by 300 USD. annually. Failure-free operation of the equipment is guaranteed for 5 years. Calculate the efficiency of these investments, provided that the interest rate on alternative projects is 15%.

Assessment of economic efficiency in the discounting system

Net present value indicator (NPV) is calculated as the difference between discounted income (D d) and discounted investments (I d):

NPV = D d – I d (16)

We will formalize the solution in the table. 11.1.

Table 11.1 Indicators of investment activity in the discounting system

Year number	Interest rate	Discount factor	Discounted investments (-), income (+)

General information is entered in columns 1 and 2 of Table 12. Column 4 contains the discount factor, which is calculated using formula (17).

K d = 1/ (1 + i) t. (17)

t- number of years.

Column 5 reflects discounted investments and annual discounted income. They are found as a line-by-line product of the values of columns 2 and 4. Column 6 “Financial position of the investor” shows how gradually discounted net income compensates for discounted investments. In year zero, only investments take place and the values of columns 2, 5, and 6 are equal in magnitude. For a year of capital use, net income appears. Part of the investment is compensated. The uncompensated part of the investment, found as the algebraic sum of the values of the zero and first year of column 5, is entered in column 6.

The last value of column 6 is the value of the economic effect. It is positive and net present value (NPV) equal to 5.64 c.u. A positive net present value indicates that our project is preferable to an alternative investment. Investment in this project will bring us additional profit in the amount of 5.64 USD.

In the table, the discounted return does not compensate for the investment until the fifth year. That means more than 4 years. Its exact value can be determined by dividing the amount of discounted investments not returned to the owner for 4 years by the amount of discounted income for the fifth year. That is, 4 years + 143.51 / 149.15 = 4.96 years.

The payback period is shorter than the guaranteed operating life of the equipment; that is, by this indicator our project can be assessed positively.

Profitability Index (ARR) characterizes the ratio of net present value to the total value of discounted investments, that is:

ARR = NPV / I d (18)

For our example, 5.64 / 1000 = 0.0056 > 0. Investments are considered economically profitable if the profitability index is greater than zero.

Profitability index (P.I.) characterizes the value of net income for the billing period per unit of investment. In the discounting system, the profitability index is determined by the formula:

PI = D d / I d = ARR + 1 (19)

For our project D d = 260.87 + 226.84 + 197.25 + 171.53 + 149.15 = 1005.645, then PI = 1005.64 / 1000 = 1.0056.

The profitability index is greater than the profitability index by one; Accordingly, investments are considered cost-effective if the profitability index is greater than one. This is also true for our project.

Net Capitalized Value (E.W.) represents the excess of the value of capitalized income over the value of capitalized investments for the accounting period. Net capitalized value is defined as the difference between capitalized net income (Dk) and capitalized investments (Ik):

EW = D to – I to (20)

A positive net capitalized value indicates the economic efficiency of the investment. The capitalization ratio is determined by formula (21):

Kk = (1 + i) t . (2)

The solution to the proposed problem will be presented in the form of a table. 11.2.

Table 11.2 Indicators of investment activity in the capitalization system

Year number	Current investments (-), income (+)	Interest rate	Capitalization rate	Capitalized investments (-), income (+)	Investor's financial situation

The net capitalized value of the investment (EW) is CU 11.35. To check, let’s recalculate it into an economic effect using the discounting system. To do this you need:

Or multiply the magnitude of the effect in the discounting system by the capitalization factor for the 5th year (bring to the final point in time) 5.64 · 2.0113 = = 11.34 USD;

Or multiply the magnitude of the effect in the capitalization system by the discount factor for the 5th year (bring the effect to the zero point in time) 11.35 × 0.4972 = 5.64 c.u.

The error in both calculations is insignificant, which is explained by the rounding of values in the calculations.

For the fifth year, it remains to return 288.65 USD. capitalized investments. Hence, return on investment period (RIR) will be:

4 years + 288.65 / 300 = 4.96 years.

Note that the return periods in the capitalization and discounting systems coincide.

Profitability Index (ARR) shows the value of net cash generated per unit of investment during an accounting period. For our example, the profitability index is equal to: ARR = EW / Iк = 11.35 / 2011.36 = 0.0056 > 0.

Profitability indexP.I. in the capitalization system is determined similarly to the discounting system. PI = D k / I k = 2022.71 / 2011.36 = 1.0056 > 1. Investments are economically justified.

To determine internal rate of return (IRR) owner's investment, it is necessary to find the interest rate at which the net present value and net capitalized value are equal to zero. To do this, you need to change the interest rate by 1-2%. If the effect occurs (NPV and EW > 0), it is necessary to increase the interest rate. Otherwise (NPV and EW< 0) необходимо понизить процентную ставку.

For this example, increasing the interest rate by 1% led to losses estimated in the discounting system NPV = - 16.46 c.u. (Fig. 3).

Rice. 11.3 Graphic interpretation of changes in internal rate of return

When calculating the internal rate of return, interpolation or extrapolation should be used. Having interpolated the values, we obtain the value of the internal rate of return in the amount of:

IRR = 15 + 5.64 / (5.64 + 16.46) = 15.226%.

Therefore, IRR = 15.226%.

Comparing the internal rate of return with the alternative interest rate, we come to the conclusion that the project in question offers a higher interest rate and, accordingly, can be successfully implemented.

All indicators calculated above characterize our project as profitable and economically feasible. It should be noted that a project considered positive in the discounting system is also positive in the capitalization system. Net present value equals net capitalized value at one point in time. All other indicators in the discounting and capitalization systems are equal in size. The choice of a specific system is determined by the requirements and qualifications of the persons implementing the decisions.

Inflationary processes depreciate investments, therefore decisions on the loan capital market are made taking into account not only the nominal, but also the real interest rate. Nominal interest rate - This is the current market rate and does not take inflation into account. Real interest rate - This is the nominal rate minus the expected (implied) inflation rate. The distinction between nominal and real interest rates only makes sense when inflation(increase in the general price level) or deflation(decrease in the general price level).

American economist Irving Fisher put forward a hypothesis regarding the relationship between nominal and real rates. She got the name Fisher effect , which means the following: The nominal interest rate changes so that the real interest rate remains unchanged: i = r + π e ,

Where i– nominal interest rate, r- real interest rate, π e – expected inflation rate in percent.

The distinction between nominal and real interest rates is important for understanding how contracts are negotiated in an economy with an unstable general price level. Thus, it is impossible to understand the investment decision-making process by ignoring the difference between nominal and real interest rates.

6. Discounting and investment decision making

Fixed capital is a durable production factor; therefore, the time factor becomes particularly important in the functioning of the fixed capital market. From an economic point of view, identical amounts that have different time localizations differ in size.

What does it mean to receive $100 in 1 year? This (at a market rate of, say, 10%) is the same as putting $91 in a time deposit at the bank today. Over the course of a year, interest would accrue on this amount, and then in a year you could receive $100. In other words, the present value of future (received in 1 year) $100 is equal to $91. Under the same conditions, $100 received in 2 years is worth $83 today.

The discounting method developed by economists allows us to compare amounts of money received at different times. Discounting - this is a special technique for measuring the current (today's) and future values of money.

The future value of today's amount of money is calculated using the formula:

Where t - number of years, r - interest rate.

The present value of a future amount of money ( current discounted value) is calculated by the formula:

Example.

For example, if you invest Today 5 million dollars in fixed capital, then you can build a plant for the production of household utensils, and within future 5 years to receive 1200 thousand dollars annually. Is this a profitable investment project? (in 5 years will $6 million be received, will the profit be equal to $1 million?)

Let's calculate two options. The interest rate on risk-free assets, for example, in the first case is 2%. We use it as discount rates, or discount rates. In the second option, the discount rate taking into account risks is 4%.

At a discount rate of 2%, the current discounted value will be $5.434 million:

at a discount rate of 4% it is equal to $4.932 million.

Next, you need to compare two values: the amount of investment (WITH) and the sum of the current discounted value (PV), those. define net present value (NPV). It is the difference between the discounted amount of expected returns and the costs of the investment: NPV = PV- WITH.

Investing only makes sense when NPV > 0. In our example, the net present value at a rate of 2% will be: 5.434 million - 5 million = 0.434 million dollars, and at a rate of 4% - a negative value: 4.932 - 5 = -0.068 million dollars. Under such conditions, the net present value criterion shows the inexpediency of the project.

Thus, the discounting procedure helps business entities make rational economic choices.

The Nominal Interest Rate is the market interest rate, excluding inflation, that reflects the current valuation of monetary assets.

The Real interest rate is the nominal interest rate minus the expected inflation rate.

For example, the nominal interest rate is 10% per annum and the projected inflation rate is 8% per annum. Then the real interest rate will be: 10 - 8 = 2%.

Nominal and real inflation rate

The difference between the nominal rate and the real rate makes sense only in conditions of inflation or deflation. American economist Irving Fisher suggested a connection between the nominal and real interest rates and inflation, called the Fisher effect, which states: the nominal interest rate changes by the amount at which the real interest rate remains unchanged.

In formula form, the Fisher effect looks like this:

i = r + πe

where i is the nominal interest rate;
r is the real interest rate;
πe is the expected inflation rate.

For example, if the expected inflation rate is 1% per year, the nominal rate will increase by 1% for the same year, therefore, the real interest rate will remain unchanged. Therefore, it is impossible to understand the investment decision-making process of economic agents without taking into account the difference between the nominal and real interest rates.

Let's take a simple example: let's say you intend to give someone a loan for one year in an inflationary environment, what exact interest rate will you set? If the growth rate of the general price level is 10% per year, then by setting the nominal rate at 10% per annum for a loan of 1000 rubles, you will receive 1100 rubles in a year. But their real purchasing power will no longer be the same as it was a year ago.

Nominal increase in income amounting to CU 100. will be “eaten up” by 10% inflation. Thus, the distinction between the nominal interest rate and the real interest rate is important for understanding how exactly contracts are concluded in an economy with an unstable general price level (inflation and deflation).