Nominal interest rate. Nominal interest rate The real interest rate is equal to the nominal rate

It is customary to estimate the interest rate in two projections: nominal and real values.

The nominal interest rate reflects the current position of the asset value. Its main difference from the real rate is its independence from market conditions. The nominal rate in monetary terms reflects the cost of capital excluding inflationary processes. The real rate, in contrast to the nominal, demonstrates the value of the cost of financial resources, taking into account the value of inflation.

Based on the definition of this concept, it is clear that the nominal interest rate does not take into account changes in price increases and other financial risks. The nominal rate can be taken into account by market participants only as an indicative value.

Mathematical effect

The relationship between the nominal and real rates was mathematically reflected in the Fisher equation. This mathematical model looks like this:

Real rate + Expected inflation rate = Nominal rate

The Fisher effect is mathematically described as follows: The nominal rate changes by the amount at which the real rate remains unchanged.

It is the future rate of inflation, taking into account the maturity of the debt claim, and not the actual rate that was in the past that matters in the formation of the market rate.

Equality of the nominal and real rates is possible only in the complete absence of deflation or inflation. This state of affairs is practically unrealistic and is considered in science only in the form of ideal conditions for the functioning of the capital market.

Nominal compound interest rate

Most often, the nominal interest rate is applied for lending. This is due to the dynamic and competitive loan market. Determination of the cost of capital within the framework of credit lines is estimated based on the loan term, currency and legal characteristics of borrowing. Banks, trying to minimize their risks, prefer to lend to clients, with long-term cooperation, in foreign currency, and in the short-term - in domestic.

In order to correctly assess the expected income from the use of funds for a long period of time, economists advise taking into account the compound interest scheme. When calculating profit using the compound interest method, at the beginning of each new regulatory period, profit is accrued in the amount received at the end of the previous period.

Any market mechanism in a changeable environment, especially such as the domestic economy, is always associated with high risks. Whether it is a loan agreement or investment in securities, opening a new business or depository cooperation with a bank. Always assessing the potential profit, it is necessary to pay attention to external factors and the real state of the market. Based only on the nominal profitability, you can make a wrong, obviously unprofitable or even potentially disastrous financial decision.

The nominal interest rate is the market interest rate, excluding inflation, and reflects the current valuation of monetary assets.

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 one makes sense only in conditions of inflation or deflation. The American economist Irving Fisher put forward an assumption about the relationship between the nominal, real interest rate and inflation, called the Fisher effect, which states: the nominal interest rate changes by the amount at which the real interest rate remains unchanged.

As a formula, 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, then 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 process of making investment decisions by economic agents without taking into account the difference between nominal and real interest rates.

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

Nominal income gain of CU100 will be "eaten up" by 10% inflation. Thus, the distinction between the nominal interest rate and the real one is important for understanding how contracts are made in an economy with an unstable general price level (inflation and deflation).

Interest rate concept

The interest rate should be understood as the most important economic category, reflecting the profitability of an 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 managerial decisions, because any subject of the economy 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. individually, 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, the owners of capital assets often agree to work only under the condition of the extremely high level of interest rates, and borrowers are likely to acquire capital only if the interest rate is low. The examples considered are vivid proof of the fact 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, of course, the rate of return) into nominal and real. This allows you to fully assess the effectiveness of financial transactions. If the inflation rate is exceeded in relation to the interest rate received by the investor on investments, the result from the corresponding operation will be negative. Of course, in terms of the absolute value of his funds, he will significantly increase, that is, for example, in rubles he will have more money, but the purchasing power, which is characteristic of them, will significantly fall. This will lead to the opportunity for a new amount to buy only a certain amount of goods (services), less than it 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 terms of inflation or deflation. Inflation should be understood as a significant and sharp decline, while deflation should be understood as their significant drop. Thus, the nominal rate is the rate that is assigned 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 rate, which is adjusted for the inflationary process.

Irving Fisher, an American economist, has formed a hypothesis explaining how depends on the nominal. 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 remains "fixed": 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 inflationary processes, expressed in percentage terms.

The real interest rate is ...

A striking example of the Fisher effect, discussed in the previous chapter, is a picture where the expected rate of inflation is equated to one percent per annum. Then the nominal interest rate will also rise by one percent. But the real percentage will remain unchanged. This proves that the real interest rate is the same nominal interest rate, but minus the estimated or actual inflation rate. This rate is fully cleared 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 inflationary processes. In this way, real interest rate is the following relation: r (p) = (1 + r (n)) / (1 + i) - 1, where the calculated indicator corresponds to the real interest rate, the second unknown term of the ratio determines the nominal interest rate, and the third element characterizes the inflation rate.

Nominal interest rate

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

The nominal interest rate should be understood as the practical characteristics of interest in quantitative terms, taking into account the 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 free loan. The nominal interest rate is nothing more than the interest expressed in money terms.

Calculating the nominal interest rate

Suppose that on an annual loan of ten thousand currency units, 1200 currency units are paid as interest. Then the nominal interest rate is equal to twelve percent per annum. After receiving a loan of 1200 monetary units, will the lender get rich? Correctly answer this question you can only know exactly how prices will change over the course of the year. Thus, at an annual inflation rate of eight percent, the creditor's income would only increase by four percent.

The nominal interest rate is calculated as follows: r = (1 + percentage of income earned by the bank) * (1 + inflationary growth) - 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 an absolute understanding, it is advisable to represent 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 + inflation rate).

It is important to note that the real efficiency and productivity of the operations performed by the investor is reflected only by the real interest rate. It says about the increase in the funds of a given economic entity. The nominal interest rate can only reflect the magnitude of the increase in cash in absolute terms. It doesn't 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 is an equal opportunity to increase consumption in future periods. This means that this situation can be interpreted as a reward for current savings.

Quite often, you can see, at first glance, lucrative offers that promise financial independence. These can be bank deposits and opportunities for investment portfolios. But is everything as profitable as the advertisement says? We will talk about this within the framework of the article, having found out what the nominal rate and the real rate are.

Interest rate

But first, let's talk about the basis of the basics in this matter - the interest rate. It reflects nominally the benefit that a particular person can receive when investing in something. It should be noted that there are quite a few opportunities to lose your savings or the interest rate that a person should receive:

Therefore, it is necessary to study in great detail what you are going to invest in. It should be remembered that the interest rate is often a reflection of the riskiness of the project under study. So, the safest are those that offer a rate of return of up to 20%. The high-risk group includes assets that promise up to 70% per annum. And everything that is more than these indicators is a danger zone, into which one should not meddle without experience. Now that there is a theoretical basis, we can talk about what the nominal rate and the real rate are.

The concept of the nominal rate

Determining the nominal is very simple - it is understood as the value that is given to market assets and evaluates them without taking into account inflation. As an example, we can cite you, the reader, and a bank that offers a deposit at 20% per annum. For example, you have 100 thousand rubles and want to increase them. Therefore, they put it in the bank for one year. And at the end of the term, they took 120 thousand rubles. Your net profit is as much as 20,000.

But is it really so? Indeed, during this time, food, clothing, travel could have significantly increased in price - and, say, not by 20, but by 30 or 50 percent. What to do in this case to get real picture cases? What should you give preference to when you have a choice? What should be chosen as a benchmark for yourself: the nominal rate and the real rate, or something one of them?

Real rate

For such cases, there is such an indicator as the real rate of return. It is noteworthy that it is quite easy to calculate it. To do this, the expected inflation rate should be subtracted from the nominal rate. Continuing the example given earlier, we can say this: you put 100 thousand rubles in the bank at 20% per annum. Inflation was only 10%. As a result, the net nominal profit will be 10 thousand rubles. And if we adjust their cost, then 9,000 according to the purchase opportunity of the last year.

This option allows you to get, albeit insignificant, but profit. Now we can consider another situation in which inflation has already reached 50 percent. You don't need to be a genius of mathematics to understand that the state of affairs forces you to look for some other way to save and increase your funds. But so far, all this has been in the style of a simple description. In economics, the so-called Fisher equation is used to calculate all this. Let's talk about it.

Fisher's equation and its interpretation

It is possible to talk about the difference between the nominal rate and the real rate only in cases of inflation or deflation. Let's take a look at why. For the first time, the idea of the relationship between nominal and real rates and inflation was put forward by the economist Irving Fisher. In the form of a formula, everything looks like this:

NS = RS + OTI

NA is the nominal rate of return;

OTI is the expected inflation rate;

PC is the real bet.

The equation is used to describe the Fisher effect mathematically. It sounds like this: the nominal interest rate always changes by the amount at which the real remains unchanged.

It may seem difficult, but now let's figure it out in more detail. The fact is that when the expected is 1%, then the denomination also rises by 1%. Therefore, it is impossible to create a quality investment decision-making process without taking into account the differences between rates. Earlier you just read about the thesis, but now you have mathematical proof that everything described above is not a simple invention, but, alas, a sad reality.

Conclusion

What can be said in conclusion? Whenever you have a choice, you need to have a high-quality approach to choosing an investment project for yourself. It doesn't matter what it is: a bank deposit, participation in a mutual investment fund, or something else. And to calculate future income or potential losses, always use economic tools. So, the nominal interest rate may promise you a pretty good profit now, but when evaluating all the parameters, it will turn out that not everything is so rosy. And the economic toolkit will help to calculate which decision will be the most profitable.

Inflation has a direct impact on the level of interest rates. Receiving loans in an inflationary environment 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 the economy: nominal and real wages, nominal and real profit (profitability) and these terms always indicate which of the indicators is calculated: not taking into account the inflation rate (nominal) and 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 monetary value of the loan.

Real interest rate Is the income for a loan, or the price of a loan, expressed in physical terms of goods and services.

The terms “nominal” and “real” apply to all indicators that are influenced by inflation.

To translate the nominal interest rate into the real interest rate, we use the following notation:

i is the nominal interest rate;

r is the 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. A.3 at the rate of inflation per month equal to the value indicated in column 5 of the table. A.3.

for instance , to ensure the real profit of the enterprise in the amount of 20% per year at 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 indicators of the effectiveness of investment projects

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

The difference between the projects is only in the fact 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). At the same time, it is expected to receive a net income within 5 years in the amounts indicated in column 6 of the table. A.3. In the second investment project, the receipt of annual income is possible from the second year within 5 years.

In fig. 11.1, 11.2 provide a graphical interpretation of these projects.

1Project

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

2 Project

Rice. 11.2. Graphic interpretation of the 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 (RVR);

profitability index (ARR);

profitability index (PI).

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

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

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

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

Evaluation of economic efficiency in the discounting system

The indicator of the net present value (NPV) calculated as the difference between discounted income (D d) and discounted investment (I d):

NPV = D d - I d (16)

We will draw up the decision in table. 11.1.

Table 11.1 Indicators of investment activity in the discounting system

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

General information is entered in columns 1 and 2 of table 12. In column 4, the discount factor is entered, which is calculated by the 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 the 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 the zero year, only investments take place and the values of columns 2, 5, and 6 are equal in magnitude. For a year of using capital, 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 years of column 5, is entered in column 6.

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

In the table, discounted income does not offset the investment until the fifth year. This means more than 4 years. Its exact value can be determined by dividing the value of the discounted investment not returned to the owner in 4 years by the value of the 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 life of the equipment; that is, according to this indicator, our project can be assessed positively.

Profitability index (ARR) characterizes the ratio of the net present value to the total value of the discounted investment, 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 (PI) characterizes the cost of net income for the settlement 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 q = 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 unit; accordingly, investments are considered economically efficient if the profitability index is greater than one. This is also true for our project.

Net capitalized value (EW) represents the excess of the value of capitalized income over the value of capitalized investments for the accounting period. Net capitalized value is determined as the difference between the capitalized net income (D k) and capitalized investments (I k):

EW = D k - I k (20)

A positive net capitalized value indicates the cost-effectiveness of an investment. The capitalization ratio is determined by the formula (21):

Кк = (1 + i) t. (2)

Let us formulate the solution of the proposed problem 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 ratio	Capitalized investments (-), income (+)	Investor's financial position

The net capitalized investment value (EW) is $ 11.35. To check it, we will recalculate it into the economic effect under the discounting system. For this you need:

Or multiply the effect size in the discounting system by the capitalization ratio for the 5th year (lead to the end point in time) 5.64 · 2.0113 = = 11.34 c.u .;

Or, multiply the effect size in the capitalization system by the discount rate 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, investment return period (RVR) will be:

4 years + 288.65 / 300 = 4.96 years.

Note that the periods of return in the system of capitalization and discounting are the same.

Profitability index (ARR) shows the value of the net cash received during the settlement period per unit of investment. For our example, the profitability index is: ARR = EW / I k = 11.35 / 2011.36 = 0.0056> 0.

Profitability indexPI in the capitalization system is defined 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) the owner's investment, it is necessary to find such a value of the interest rate at which the net present value and the net capitalized value are equal to zero. To do this, you need to change the interest rate by 1-2%. If the effect takes place (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 Graphical interpretation of changes in internal rate of return

When calculating the value of the internal rate of return, interpolation or extrapolation should be applied. Interpolating the values, we get the value of the indicator 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 the indicators calculated above characterize our project as profitable and economically viable. It should be noted that the project considered in the discounting system as positive is just as positive in the capitalization system. The net present value is equal to the net capitalized value reduced to one point in time. All other indicators in the systems of discounting and capitalization are equal in value. The choice of a specific system is determined by the requirements and qualifications of decision makers.