Nominal interest rate. Real and nominal interest rates Nominal and real interest rates characteristics

Percentage is absolute value. For example, if 20,000 is borrowed and the debtor must return 21,000, then the interest is 21,000-20,000=1000.

The lending interest rate (norm) - the price for using money - is a certain percentage of the amount of money. Determined at the point of equilibrium between the supply and demand of money.

Very often in economic practice, for convenience, when they talk about loan interest, they mean the interest rate.

There are nominal and real interest rates. When people talk about interest rates, they mean real interest rates. However, actual rates cannot be directly observed. By concluding a loan agreement, we receive information about nominal interest rates.

Nominal rate (i)- quantitative expression of the interest rate taking into account current prices. The rate at which the loan is issued. The nominal rate is always greater than zero (except for a free loan).

Nominal interest rate- This is a percentage in monetary terms. For example, if for an annual loan of 10,000 monetary units, 1,200 monetary units are paid. as interest, the nominal interest rate will be 12% per annum. Having received an income of 1200 monetary units on a loan, will the lender become richer? This will depend on how prices have changed during the year. If annual inflation was 8%, then the lender’s income actually increased by only 4%.

Real rate(r)= nominal rate - inflation rate. The real bank interest rate can be zero or even negative.

Real interest rate is an increase in real wealth, expressed as an increase in the purchasing power of the investor or lender, or the exchange rate at which today's goods and services, real goods, are exchanged for future goods and services. The fact that the market rate of interest would be directly influenced by inflationary processes was the first to suggest I. Fischer, which determined the nominal interest rate and the expected inflation rate.

The relationship between the rates can be represented by the following expression:

i = r + e, where i is the nominal, or market, interest rate, r is the real interest rate,

e - inflation rate.

Only in special cases, when there is no price increase in the money market (e = 0), do real and nominal interest rates coincide. The equation shows that the nominal interest rate can change due to changes in the real interest rate or due to changes in inflation. Since the borrower and lender do not know what rate inflation will take, they proceed from the expected rate of inflation. The equation becomes:

i = r + e e, Where e e expected inflation rate.

This equation is known as the Fisher effect. Its essence is that the nominal interest rate is determined not by the actual rate of inflation, since it is unknown, but by the expected rate of inflation. The dynamics of the nominal interest rate repeats the movement of the expected inflation rate. It must be emphasized that when forming a market interest rate, it is the expected inflation rate in the future, taking into account the maturity of the debt obligation, that matters, and not the actual inflation rate in the past.

If unexpected inflation occurs, then borrowers benefit at the expense of lenders, since they repay the loan with depreciated money. In the event of deflation, the lender will benefit at the expense of the borrower.

Sometimes a situation may arise where real interest rates on loans are negative. This can happen if the inflation rate exceeds the growth rate of the nominal rate. Negative interest rates can be established during periods of runaway inflation or hyperinflation, as well as during an economic downturn, when demand for credit falls and nominal interest rates fall. Positive real interest rates mean higher income for lenders. This occurs if inflation reduces the real cost of borrowing (credit received).

Interest rates can be fixed or floating.

Fixed interest rate is established for the entire period of use of borrowed funds without the unilateral right to revise it.

Floating interest rate- this is the rate on medium- and long-term loans, which consists of two parts: a moving basis, which changes in accordance with the market market conditions and a fixed amount, usually unchanged throughout the entire period of lending or circulation of debt

When people talk about interest rates, they usually mean real interest rates as opposed to nominal interest rates. However, actual rates cannot be directly observed. When concluding a loan agreement or viewing financial bulletins, we receive information primarily about nominal interest rates.

The nominal interest rate is interest in monetary terms.

For example, if a $1,000 annual loan pays $120 in interest, the nominal interest rate would be 12% per annum.

Having received an income of $120 on a loan, will the lender become richer? It depends on how prices have changed during the year. If prices rose by 8%, then the lender's real income increased by only 4% (12%-8%=4%).

The real interest rate is the increase in real wealth, expressed as an increase in the purchasing power of the investor or lender, or the exchange rate at which today's goods and services, real goods, are exchanged for future goods and services. Essentially, the real interest rate is the nominal rate adjusted for price changes.

The above definitions enable us to consider the relationship between nominal and real interest rates and inflation.

It can be expressed by the formula

i = r + r,(1.1)

Where i- nominal interest rate;

r- real interest rate;

R- inflation rate.

This equation shows that the nominal interest rate can change for two reasons: due to changes in the real interest rate and (or) due to changes in the inflation rate.

Real interest rates change very slowly over time because changes in nominal interest rates are caused by changes in the inflation rate.

A 1% increase in the inflation rate causes a 1% increase in the nominal rate.

When the borrower and lender agree on a nominal rate, they do not know what rate inflation will take at the end of the contract. They are based on expected inflation rates. The equation becomes:

i = r + r e . (1.2)

This equation is known as the Fisher equation, or the Fisher effect. Its essence is that the nominal interest rate is determined not by the actual inflation rate, since it is not yet known, but by the expected inflation rate ( R e).

The dynamics of the nominal interest rate follows the movement of the expected inflation rate.

Since it is impossible to accurately determine the future rate of inflation, rates are adjusted according to the actual level of inflation. Expectations match current experience.

If the inflation rate changes in the future, there will be deviations in the actual rate from the expected rate.

These are called the inflation surprise rate and can be expressed as the difference between the future actual rate and the expected inflation rate ( r - r e).

If the unexpected inflation rate is zero ( p = p"), then neither the lender nor the borrower loses or gains anything from inflation.

If unexpected inflation occurs ( r - r" > 0 ), then borrowers benefit at the expense of lenders, since they repay the loan with depreciated money.

In case of unexpected deflation, the situation will be reversed: the lender will benefit at the expense of the borrower.

From the above, three important points can be highlighted: 1) nominal interest rates include a markup or premium on expected inflation; 2) due to unforeseen inflation, this premium may turn out to be insufficient; 3) as a result, there will be an effect of redistribution of income between lenders and borrowers.

This problem can be looked at from the other side - from the point of view of real interest rates. In this regard, two new concepts arise:

• - expected real interest rate - the real interest rate that the borrower and lender expect when granting a loan. It is determined by the expected inflation rate ( r = i - r e);
• - actual real interest rate. It is determined by the actual inflation rate ( r = i - r).

Since the lender expects to earn income, the nominal interest rate on new borrowings must be at a level that will provide good prospects for real income consistent with current estimates of future inflation.

Deviations of the actual real rate from the expected one will depend on the accuracy of the forecast of future inflation rates.

At the same time, along with the accuracy of forecasts, there is difficulty in measuring the real rate. It consists of measuring inflation and choosing a price index. In this matter, one must proceed from how the funds received will ultimately be used. If loan proceeds are intended to finance future consumption, then the appropriate measure of income is the consumer price index. If a company needs to estimate the real cost of borrowed funds to finance working capital, then the wholesale price index will be adequate.

When the rate of inflation exceeds the rate of increase in the nominal rate, the real interest rate will be negative (less than zero). Although nominal rates typically rise when inflation rises, real interest rates have been known to fall below zero.

Negative real rates are holding back lending. At the same time, they encourage borrowing because the borrower gains what the lender loses.

Under what conditions and why does a negative real rate exist in financial markets? Negative real rates may be established for some time:

• - during periods of runaway inflation or hyperinflation, lenders provide loans even if real rates are negative, since receiving some nominal income is better than holding cash;
• - during an economic downturn, when demand for loans falls and nominal interest rates decrease;
• - at high inflation, to provide income to creditors. Borrowers won't be able to borrow at such high rates, especially if they expect inflation to slow soon. At the same time, rates on long-term loans may be lower than the inflation rate, since financial markets will expect a fall in short-term rates;
• - if inflation is not sustainable. Under the gold standard, the actual rate of inflation may be higher than expected, and nominal interest rates will not be high enough: “inflation takes merchants by surprise.”

Positive real interest rates mean higher income for lenders. However, if interest rates rise or fall in line with inflation, then the lender suffers a potential capital gain loss. This happens in the following cases:

• 1) inflation reduces the real cost of a loan (loan received). A homeowner who takes out a mortgage loan will find that the amount of debt they owe decreases in real terms. If the market value of his home rises but the face value of his mortgage remains the same, the homeowner benefits from the decreasing real real value of his debt. The lender will suffer a capital loss;
• 2) the market value of securities, such as government bonds, falls if the market nominal interest rate rises, and, conversely, rises if the interest rate falls.

1. r + i[*. (2)

• expected real interest rate - the real interest rate that the borrower and lender expect when granting a loan. It is determined by the expected level of inflation (r- i - ts*);
• actual real interest rate. It is determined by the actual level of inflation (r = g - l).

• during periods of runaway inflation or hyperinflation, lenders provide loans even if real rates are negative, since earning some nominal income is better than holding cash;
• during an economic downturn, when demand for loans falls and nominal interest rates fall;

• at high inflation, to provide income to creditors. Borrowers will not be able to borrow at such high rates, especially if they expect inflation to slow soon. At the same time, rates on long-term loans may be lower than the inflation rate, since financial markets will expect a fall in short-term rates;
• if inflation is not sustainable. Under the gold standard, the actual rate of inflation may be higher than expected, and nominal interest rates will not be high enough: “inflation takes merchants by surprise.”

More on the topic Nominal and real interest rates:

1. Difference between real and nominal interest rates
2. 13.2. The economic basis for the formation of the level of loan interest
3. 13.2. The economic basis for the formation of the level of loan interest
4. 11.3. Loan interest rate, its types, relationship and differences from loan interest and profit rate\r\n
5. Investments and reinvestments. Formation of market interest rate
6. Loan, deposit, discount interest, their determining factors
7. 8.6. ROLE OF INTEREST RATE IN ENSURING INVESTMENT EFFICIENCY

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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

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

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.