Interest rates

Interest rate(interest rate) - an amount indicated as a percentage of the loan amount that the recipient of the loan pays for using it for a certain period (month, quarter, year).

From the perspective of the theory of money, the interest rate is the price of money as .

Interest income is income from the provision of capital in debt in various forms (,), or it is income from investments in.

Interest rate is a fixed rate at which the interest amount is paid on time. Typically, the interest rate characterizes the ratio of the annual amount of interest (interest income) to the amount of the principal debt. The interest rate is also used in the process of increasing value.

Interest rate is the fee charged by banks for loans provided. The interest rate is the basis of banks' cost accounting. The interest rate depends on the size of the loan, its repayment period, the ratio of supply and demand for the loan, as well as the degree of risk that the credit institution bears when lending a certain amount to the debtor.

History of interest rates

Over the past two centuries, benchmark interest rates have been set either by national governments or central banks. For example, the US Federal Reserve federal funds rate ranged from 0.25% to 19% between 1954 and 2008, while base rates ranged from 0.5% to 15% between 1989 and 2009, and the spread base rates in Germany ranged from close to 90% in the 1920s to around 2% in the 2000s. During an attempt to overcome the spiral of hyperinflation in 2007, the Reserve Bank of Zimbabwe raised borrowing rates to 800%.

Central bank interest rates

Interest rate is the rate of the central bank for transactions with other credit institutions. Through the central bank it has the ability to influence interest rates of commercial banks in the country and.

When interest rates decrease, business activity increases and inflation increases. An increase in interest rates leads to a decrease in business activity, a decrease in inflation and an appreciation of the national currency.

The main interest rate in the United States: the Federal funds rate is the interest rate at which banks place available funds in accounts in the United States to other banks on .

The rate in the Eurozone is the refinancing tender rate - an interest rate that is the minimum possible for applications to raise funds in a tender.

Japan's Prime Interest Rate: The target interest rate for overnight borrowings is the level of interest that the average in the short-term deposit market wants to see.

The main interest rate in the UK, the so-called Repo rate, is the rate at which the Bank of England issues short-term loans secured by securities.

Canada's Prime Rate: The Overnight Rate Target is the level of interest that the Bank of Canada wants to see as the average for the short-term deposit market. To control the level of interest rates in the overnight market, the Bank of Canada sets a so-called operational band of 0.50%, the middle of which is always the target overnight interest rate.

Australia: Australian dollar overnight interest rate (Cash rate) is an interest rate determined as a result of supply and demand in the money market. The Reserve Bank of Australia sets the required level of this rate and maintains it by controlling.

Interest rates on loans can be greater than zero, equal to zero (“interest-free loan”) and less than zero (“negative” interest). If interest rates reach high levels, this leads to the emergence of usury.

Types of interest rates

There are several types of interest rates.

Fixed and floating rates

Depending on whether the rate changes over time, fixed and floating interest rates are distinguished:

- permanent, established for a certain period and does not depend on any circumstances.
subject to periodic review. The rate is changed based on fluctuations in certain indicators. A classic example of such indicators is (LIBOR, the weighted average rate on the London interbank market for credit resources). Accordingly, a floating rate of LIBOR+5% will mean that the nominal interest rate is 5% higher than the LIBOR rate.

Decursive and anticipatory bets

Depending on the timing of interest payments, there are two types of interest rates:

decursive rate- interest is paid at the end along with the principal amount of the loan;
anticipatory rate- interest is paid at the time the loan is granted (in advance) and is determined based on the final amount of the debt.

For the lender, an anticipatory rate is more profitable, and for a borrower, a decursive rate is more profitable. So, if the interest rate is 10%, then with a decursive rate for a loan of $1000, the lender will receive $1100 at the end of the term. With an anticipatory rate, he will give the borrower $900 and at the end of the term he will receive $1000.

Real and nominal rates

There is a distinction between nominal and real interest rates.

Real interest rate is the interest rate taking into account .

The relationship between real, nominal rates and inflation is generally described by the following (approximate) formula:

I r = I n - I i

Where I r- real interest rate;
I n- nominal interest rate;
I i- expected or planned level of inflation.

Irving Fisher proposed a more precise formula for the relationship between real, nominal rates and inflation, expressed by the Fisher formula named after him:

I r = (1 + I n)/(1 + I i) - 1 = (I n - I i)/(1 + I i)

At I i = 0 And I i = I n both formulas give the same value. It is easy to see that for small values of the inflation rate I i the results differ little, but if inflation is high, then the Fisher formula should be applied.

According to Fisher, the real interest rate must be numerically equal to the marginal productivity of capital.

Monetary policy is a set of activities and government in the field of money circulation and credit.

Central bank monetary policy (monetary policy)- this is a set of government measures that regulate the activities of the monetary system, the loan capital market, order in order to achieve a number of general economic goals: stabilization of prices, rates, strengthening of the monetary unit.

Monetary policy is the most important element.

All impacts are reflected in the value of the total social product and.

The main goals of the state's monetary policy:

Containment
Security
Tempo regulation
Mitigation of cyclical fluctuations in the economy
Ensuring the stability of the balance of payments

Principles of monetary and credit regulation of the economy

Monetary regulation of the economy is carried out on the basis of the principle compensation regulation, which assumes the following:

monetary policy restrictions, which involves limiting credit transactions by increasing the norms for reserving funds for participants in ; level up; restrictions on the growth rate in circulation compared to the commodity mass;
monetary policy expansion, which involves stimulating credit operations; reduction of reserve standards for subjects of the credit system; falling lending rates; acceleration of currency turnover.

Monetary Policy Instruments

The development and implementation of monetary policy is the most important function. It has the ability to influence the volume of money supply in the country, which in turn allows it to regulate the level of production and employment.

The main instruments of the central bank in implementing monetary policy:

Regulation of official reserve requirements
It is a powerful means of influencing the money supply. The amount of reserves (part of the banking assets that any commercial bank is required to keep in the accounts of the central bank) largely determines its lending capabilities. Lending is possible if the bank has enough funds in excess of the reserve. Thus, increasing or decreasing reserve requirements can regulate the lending activity of banks and accordingly affect the money supply.

Open Market Operations
The main instrument for regulating the supply of money is the purchase and sale of government securities by the Central Bank. When selling and purchasing securities, the Central Bank tries to influence the volume of liquid funds of commercial banks by offering favorable interest rates. By purchasing securities on the open market, he increases the reserves of commercial banks, thereby contributing to an increase in lending and, accordingly, an increase in the money supply. The sale of securities by the Central Bank leads to the opposite consequences.
Regulation of the discount rate (discount policy)
Traditionally, the Central Bank provides loans to commercial banks. The interest rate at which these loans are issued is called the discount rate. By changing the discount interest rate, the central bank influences banks' reserves, expanding or reducing their ability to lend to the population and enterprises.

Factors that influence demand, supply and interest rates can be collectively called “monetary policy instruments.” These include:

Interest rate policy of the Bank of Russia

The Central Bank sets minimum interest rates for transactions it carries out. The refinancing rate is the rate at which loans are provided by commercial banks, or it is the rate at which bills of exchange are rediscounted from them.

The Bank of Russia may establish one or more for various types of transactions or pursue an interest rate policy without fixing the interest rate. Bank of Russia uses interest rate policy to influence market interest rates in order to strengthen the ruble.

Bank of Russia regulates the total volume of loans issued to them in accordance with the accepted guidelines of the unified state monetary policy, using the discount rate as an instrument. Bank of Russia interest rates represent the minimum rates at which the Bank of Russia carries out its operations.

Interest rate policy of credit institutions, being part of the national monetary policy, has a significant impact on the development and its stability. are usually free to choose specific rates on loans and deposits and use certain indicators reflecting the state of the short-term money market as guidelines when implementing interest rate policy. On the other hand, the central bank, in the targeting process, sets intermediate monetary policy goals that it can influence, as well as specific tools for achieving them. This may be the refinancing rate or interest rates on central bank operations, on the basis of which the short-term interbank lending rate is formed, etc.

The problems of identifying factors influencing the interest rate policy of commercial banks have worried specialists since the formation of economic theory. However, answers to many questions have not yet been found. Modern research aimed at identifying optimal rules for implementing national monetary policy is largely based on.

Methods of direct and indirect regulation of national monetary policy are considered in theory and practice. From the point of view of interest policy in the narrow sense (rates on credit and deposit operations, the spread between them), the instrument of its direct regulation is establishment by the central bank of interest rates on loans and deposits of commercial banks, indirect instruments - establishing the refinancing rate and the rate for central bank operations in the money and open markets.

Interest rates on loans and deposits as instruments of direct regulation are not often used in world practice. For example, the People's Bank of China sets rates that are considered indicative for the banking system. At the same time, the bank's policy is aimed at reducing the spread, which in the first half of 2006 was 3.65%, and by the end of 2009 - 3.06%, which indicates sufficient liquidity of the Chinese banking system.

In many countries, including Russia, the refinancing rate has become more of an indicative indicator, giving the economy only an approximate benchmark for the value of the national currency in the medium term, because it remains unchanged for a long time, while real rates in the money market change every day.

Required reserve standards

According to existing legislation, commercial banks are required to transfer part of the raised funds to special accounts in.

Since January 2004 established by the Central Bank following amounts of contributions to the mandatory reserve fund Bank of Russia: for ruble accounts of legal entities and foreign currency of citizens and legal entities, as well as for ruble accounts of citizens - 3.5%.

The maximum amount of deductions, i.e., required reserve standards, is 20% and cannot change by more than 5% at a time.

This standard allows the Bank of Russia to regulate the liquidity of the banking sector.

Reserves serve as a current regulation of liquidity in the money market, on the one hand, and as a limiter on the emission of credit money, on the other.

In case of violation of required reserve standards, the Bank of Russia has the right to indisputably collect from the credit institution the amount of funds not deposited, as well as a fine in the established amount, but not more than double.

Open market operations

Open market operations, which mean the purchase and sale by the Bank of Russia of corporate securities, short-term transactions with securities with the completion of a reverse transaction later. The limit on open market operations is approved by the board of directors.

In accordance with the law of July 10, 2002 No. 86-FZ (as amended on October 27, 2008) “On the Central Bank of the Russian Federation (Bank of Russia),” the Bank of Russia has the right to buy and sell products of commercial origin with a maturity date of not more than 6 months, buy and sell bonds, certificates of deposit and other securities with a maturity of no more than 1 year.

Refinancing

Refinancing means lending by the Bank of Russia to banks, including accounting and rediscounting of bills. The forms, procedure and conditions of refinancing are established by the Bank of Russia.

Refinancing of banks is carried out by providing intraday loans, overnight loans and holding pawnshop credit auctions for a period of up to 7 calendar days.

Currency regulation

It should be considered from two sides. On the one hand, the Central Bank must monitor the legality of foreign exchange transactions, and on the other hand, monitor changes in the national monetary unit in relation to other currencies, avoiding significant fluctuations.

One of the methods of influencing the exchange rate is through central banks carrying out foreign exchange interventions or motto policies.

Currency intervention- this is the sale or purchase by the Central Bank of foreign currency for the purpose of influencing the exchange rate and the total demand and supply of money. These obviously include transactions for the purchase and sale of precious metals on the domestic market of the Russian Federation, the procedure for which is regulated by letter of the Central Bank of the Russian Federation dated December 30, 1996 No. 390.

The main objectives of exchange rate policy in Russia are strengthening confidence in the national currency and replenishing gold and foreign exchange reserves. Currently, the monetary base is fully backed by gold and foreign exchange reserves.

Direct quantitative restrictions

Direct quantitative restrictions of the Bank of Russia include the establishment of limits on the refinancing of banks and the conduct of certain banking operations by credit institutions. The Bank of Russia has the right to apply direct quantitative restrictions in exceptional cases in order to implement a unified state monetary policy only after consultations with the government of the Russian Federation.

Benchmarks for growth of money supply indicators

The Bank of Russia can set growth targets for one or more indicators based on the main directions of the unified state monetary policy. In Russia, the main aggregate is the monetary aggregate.

Today, the monetary policy of central banks is guided by monetarist principles, where the Central Bank is tasked with strictly controlling the money supply, ensuring a stable, constant and long-term growth rate of the amount of money in the economy, equal to the growth rate of GDP.

Other factors influencing demand, supply and interest rates include:

the situation in the real sector of the economy;
return on investment in production;
the situation in other sectors of the financial market;
economic expectations of business entities;
the need of banks and other business entities for funds to maintain their liquidity.

The politics of cheap and expensive money

Depending on the economic situation in the country, the central bank pursues a policy of cheap or expensive money.

Cheap money policy

Characteristic of a situation of economic recession and high level. Its goal is to make credit money cheaper, thereby increasing aggregate spending, investment, production and employment.

To implement a cheap money policy, the central bank can reduce the interest rate on loans to commercial banks or make purchases on the open market or reduce the reserve requirement ratio, which would increase the money supply multiplier.

Dear money policy

It is carried out with the aim of reducing the pace by reducing total expenditures and limiting the money supply.

Includes the following activities:

Increasing the interest rate. Commercial banks begin to take less loans from the Central Bank, therefore the supply of money is reduced.
Sale of government securities by the central bank.
Increase in reserve requirements. This will reduce excess reserves of commercial banks and reduce the money supply multiplier.

All of the above monetary policy instruments related to indirect (economic) methods of influence. In addition to these general methods of monetary regulation, the central bank also uses direct (administrative) methods designed to regulate specific types of credit. For example, a direct limitation on the size of bank loans for consumer needs.

Monetary policy has pros and cons. Strengths include speed and flexibility, less dependence on political pressure than fiscal policy. Problems in the implementation of monetary policy are created by cyclical asymmetry. The effectiveness of monetary policy may also decrease as a result of counter-directional changes in the velocity of money.

For a particular currency, the interest rate is the price of money in that currency, that is, the cost of borrowing in that currency.

The interest rate is determined in the money market under the influence of supply and demand: the more people who want to borrow money, the higher it is; the more people who are willing to borrow money, the lower it is.

National governments, through Central Banks, control interest rates in money markets by limiting the amount of money in circulation. The interest rate affects economic activity in the country. If interest rates rise, and therefore loans become more expensive, then debt-financed projects become less attractive because they must be more profitable to cover costs. In other words, high loan interest suppresses economic activity and makes it impossible to implement a number of projects. On the contrary, a decrease in interest rates contributes to the growth of economic activity, increasing the attractiveness of projects with debt financing.

Impact of interest rates on the global market

Most businesses globally are financed through loans of a similar nature. The “cost” of these loans is determined by the corresponding interest rates. The higher the rate, the more profitability the business must have to cover the interest on the loan.

Essentially, the interest rate is the cost of doing business that affects every individual. The interest rates that consumers and homeowners pay on their credit cards and mortgages originate in the money markets.

Interest rates determine how much it costs an organization or individual to borrow money over a certain period of time. The expression “time is money” means that as long as the borrower owes money, interest accrues on the principal amount borrowed.

The role of interest rates

Market participants are always looking for ways to increase profits. The key criterion for assessing the prospects of investments is the real rate of return on them. For the currency in which investments are made, this rate is determined as follows:

Real rate of return = Interest rate - Apparent inflation rate

Money flows to countries with the highest real income rates. Central banks sometimes raise interest rates to attract capital into the country. High rates attract capital, which allows us to hope for an increase in demand for the national currency and for an increase in the exchange rate. To obtain high income, market participants must invest capital in the country and buy its national currency.

When the central bank raises interest rates, real income rates in the national economy rise, which attracts capital to the country. On the contrary, a decrease in interest rates entails an outflow of capital from the country. Capital inflows or outflows strengthen or weaken the currency, respectively. Thus, by changing interest rates in the national money market, the Central Bank influences the exchange rate.

Money market interest rates are linked to rates in other sectors of the financial market, and the Central Bank discount rate, Treasury bill rate, and interbank overnight lending rates ("overnight money") are the basis for the entire interest rate system.

In different countries, the Central Bank interest rate is called differently:

- discount rate in the USA, Germany, Japan and Switzerland
- intervention course in France
- bank rate in Canada
- UK money market dealing rate
- refinancing rate in Russia

Surely everyone who has ever taken out a loan or become a bank depositor first came across the concept of “bank interest rate”:

An interest rate is an amount, expressed in percentage terms, that is set by the bank for using a loan and paid for a certain period - a year, quarter or month.

If money is deposited into a checking bank account or deposit, the depositor is the lender to the bank and the bank is the borrower.
If a client borrows money from a bank (takes out a loan), then the bank is now the lender, and the client is the borrower.

Knowledge of these simple truths will get rid of the complexes that banks instill in the population, explaining to them many kilometers of formulas for calculating interest with Newton's binomials, factorials, complex roots, powers and other mathematical crap of complexity.

The interest rate determines the price of money

In either of these two cases, the interest rate has a monetary measurement: what the depositor's or bank's savings will be in a month, a year or several years.

The interest rate on depositors' deposits is usually lower than the rate on bank loans. This is the main income of banking and financial institutions - to take money at a lower price and dispose of it by borrowing it at a higher price.

For depositors, a deposit is mainly a way to save money, rather than earn money, so deposit rates are now low, and in some European banks they are even negative.

The base interest rate is the lowest loan interest rate offered to large, reliable companies and clients. The BPS is usually set by central banks.

Historical information about rates

The historical ranges of rates are impressive:

In Germany, for example, the base interest rate fluctuated between 90% and 2% between 1920 and 2000.
In the UK - 0.5 - 15% in 1989 - 2009.
In the USA, the US Federal Reserve rate in 1954 - 2008 varied between 19% and 0.25%.
In Zimbabwe, during the period of hyperinflation in 2007, the lending rate reached 800%.

Types of bets

Fixed and floating rates

Interest rates are:

Fixed - unchangeable for a certain period of time.
Floating - changeable and periodically revised by the bank, depending on certain indicators.

Thus, the classic indicator is LIBOR - the average rate of the London Interbank Credit Exchange.

Many banks determine the floating rate using the formula: LIBOR + n, where n is the fixed rate of a particular bank.

Russian banks can focus on an independent indicative rate, for example, MosPrime Rate.

In a growing lending rate market, it is more profitable for a borrower to take out a loan at a fixed interest rate.

According to the time of payment, bets are:

decursive - paid at the end along with the repayment of the loan;
anticipatory - paid in advance when granting a loan.

Decursive rates are beneficial for borrowers, and anticipatory rates are beneficial for lenders, but banks usually act in their own interests:

interest on deposits is calculated using a decursive method,
credit - antiseptic: when issuing a loan, the total interest is immediately determined, which is then divided by the number of periods (usually months).

Decursive and anticipatory methods are used when calculating simple and compound interest, when the initial amount of capital changes in each reporting period.

The decursive method is convenient to use with floating rates.
The anticipatory method is convenient during periods of instability as a guarantor for the payment of compound interest.

The decursive rate is also called loan interest, since it determines the ratio of the income received (interest) to the initial amount of money.

How to calculate loan interest and increase amount

Formula for determining loan interest:

i = I/P (1), where:

i (income) - loan interest;
I - the sum of all interest accrued during the reporting period;
P is the initial amount of money (present value).

The amount of increase F (future value) is determined by the formula:

F = P + i*n*P = P*(1 + i*n). (2)

Here n is the number of billing periods.

The F/P ratio is the build-up factor k n .

k n = 1 + i*n. (3)

Calculating the amount of build-upF is called compounding.

Compounding using calculation example

We will compound a bank loan of 1 million rubles, issued at 12% per annum (simple rate), for a period of 10 years according to formula (2)

F = 1,000,000 *(1 + 0.12 *10) = 2,200,000 rub.

The initial amount of money issued by the bank on a long-term ten-year loan, often used in mortgages, increased by 1,200,000 rubles, that is, more than doubled.

You can also calculate the amount of increase over a short period (less than a year). In this case, the formula for determining F (2) is transformed:

F = P * (1 + i * d/K). (4)

d — the number of calendar days for which the loan was taken out;
K is the number of days in a year, i.e. 365 or 366.

Let's calculate the increased loan amount in the amount of 50,000 rubles issued by an MFO at the annual simple rate of 15% specified in the agreement for a period of 91 days.

By inserting the values into formula (4), we get:

F = 50,000 * (1 + 0.15 *91/365) = 51,870 rub.

Often banks and microfinance organizations require the return of amounts greater than the estimated amount - this means that additional hidden interest has been calculated in the form of all kinds of commissions. Before concluding an agreement, you should carefully read all its clauses in search of illegal ways to increase capital.

Discounting

The reverse operation - calculating the initial amount P based on the accumulated F - is called discounting.

Discounting is calculated using the formula:

P = F/ (1 + i*n). (5)

Let's make the calculation using formula (5):

P = 100,000/(1 + 0.1*3) = 76,923 rubles.

Calculations with a floating rate

If the rate is floating, then the increased amount is calculated by summing the rates for each period of their change, and the formula is converted into some abstract one:

F = P *(1 + ∑(1…N) n*i) (6), where:

n—period from one to N;
i- variable bet;
∑(1…N) is the sum of the products n*i for all billing periods.

It looks scary at first glance, but how this happens is very easy to understand using the example:

We use formula (6) for calculation:

F = 500,000 *(1 + 0.11 + 0.5 (0.125 + 0.14 + 0.155 + 0.17)) = 500,000 * 1.405 = 702,500 rub.

Please note that the growth coefficient k, calculated for a fixed percentage using formula (3), for a floating percentage is determined by the expression in brackets of formula (6):

K = 1 + ∑(1…N) n*i. (7)

In this example, its value is 1.405.

Compound interest calculations

This calculation method in the banking industry is used when calculating interest on long-term deposits, when interest is accrued on the amount accumulated by previous interest.

The formula for calculating compound interest is shown in the figure below.

Rate size and inflation

The interest rate can be nominal and real:

Nominal - set by the bank.
Real - adjusted for inflation.

The real rate i real is less than the nominal rate i nom by the inflation rate π.

i real = i nom - π.

This formula is usually used when inflation is low. At a high inflation level, calculations are made using the more complex Fisher formula:

i real = (i nom - π)/(1 + π).

Real value of money

To determine the real value of money taking into account inflation after some time, use the formula:

R= N/(1+i)ª.

R is the real value of money;

N—nominal value;

i- inflation rate;

a is the number of periods (years, months, etc.).

Banks usually increase interest rates during periods of high inflation, factoring the increase into the nominal rate. Such a step, in addition to combating the decline in the price of money, gives them the opportunity to raise the interest rate on deposits so as not to lose depositors.

Financial illiteracy of the population is beneficial for bankers

Sometimes lending rates, especially fast ones, contradict common sense and are a veiled scam. Therefore, everyone who wants to take out a loan should have an understanding of what bank interest is and how to calculate the amount of the increase.

Taking advantage of the financial illiteracy of the population, banks today offer such sophisticated and complex calculation formulas that an engineer or programmer’s calculator is required. Meanwhile, calculating the total amount of loan payments (also known as the build-up amount), as can be seen from the examples, is quite simple on a regular calculator and even on a piece of paper. You can calculate payments based on the loan body and interest using different formulas, but the deviations between your final calculations and the bank ones should still not be too large.

Banks today practically do not use a differentiated method of loan repayment, in which the remaining amount of debt is taken into account when calculating interest, and not the original amount. This is supposedly motivated by “concern for clients”: why, they say, should they strain their brains and make complex calculations every month? Thus, it turns out that our lending is one of the most unprofitable in the world.

And calculation of the parameters of this transaction.

The financial mathematics course consists of two sections: one-time payments and payment flows. One-time payments- these are financial transactions in which each party, when implementing the terms of the contract, pays the amount of money only once (either lends or repays the debt). Payment flows- these are financial transactions in which each party, when implementing the terms of the contract, makes at least one payment.

There are two parties involved in a financial transaction - the lender and the borrower. Each party can be either a bank or a client. The basic financial transaction is the lending of a certain amount of money. Money is not equivalent in relation to time. Modern money is usually more valuable than future money. The value of money over time is reflected in the amount of accrued interest money and the pattern of its accrual and payment.

The mathematical apparatus for solving such problems is the concept of “percentage” and and .

Percentage - basic concepts

Percent- one hundredth of the pre-agreed base (that is, the base corresponds to 100%).

Examples:

Answer: more

	original debt amount
(days)	a fixed period of time to which the interest (discount) rate is tied (usually one year - 365, sometimes 360 days)
	interest (discount) rate for the period
	debt term in days
	debt term in shares of the period
	amount of debt at the end of the term

Interest rate

Interest rate- the relative amount of income for a fixed period of time. The ratio of income (interest money - the absolute amount of income from lending money) to the amount of debt.

Accrual period- this is the time interval to which the interest rate is dated; it should not be confused with the accrual period. Usually I take a year, half a year, quarter, month as such a period, but most often we deal with annual rates.

Interest capitalization- adding interest to the principal amount of debt.

Extension- the process of increasing the amount of money over time due to the addition of interest.

Discounting- inverse to the increase, in which the amount of money related to the future is reduced by the amount corresponding to the discount (discount).

The quantity is called the accumulation factor, and the quantity is called the discounting factor under the corresponding schemes.

Interest Rate Interpretation

With the scheme " simple interest"The initial basis for calculating interest throughout the entire term of the debt in each period of application of the interest rate is the original amount of the debt.

With the scheme " compound interest"(for integers) the initial base for calculating interest throughout the entire period in each period of application of the interest rate is the amount of debt accrued over the previous period.

Adding accrued interest money to the amount that serves as the basis for calculating it is called capitalization of interest (or reinvestment of the deposit). When applying the “compound interest” scheme, interest capitalization occurs at each period.

Interpretation of the discount rate

With the "simple interest" scheme ( simple discount) - the initial basis for calculating interest throughout the entire term of the debt at each period of application of the discount rate is the amount to be paid at the end of the deposit term.

With the “compound interest” scheme (for integers) ( complex discount) - the initial basis for calculating interest throughout the entire period in each period of application of the discount rate is the amount of debt at the end of each period.

Simple and complex interest rates

"Direct" formulas

	Simple interest	Compound interest
- interest rate			buildup
- interest rate			discounting (banking accounting)

"Reverse" formulas

	Simple interest	Compound interest
- interest rate			discounting (mathematical accounting)
- interest rate			buildup

Variable interest rate and reinvestment of deposits

Let the debt term have stages whose length is equal to , ,

- with a simple interest scheme

1 . The contract provides for the accrual of a) simple, b) compound interest in the following order: in the first half of the year at an annual interest rate of 0.09, then next year the rate decreased by 0.01, and in the next two half-years increased by 0.005 in each of them . Find the amount of the increased deposit at the end of the term if the amount of the initial deposit is $800.

Market interest rate as the most important macroeconomic indicator

The interest rate is important. The interest rate is the charge for money lent. There were times when the law did not allow remuneration for the fact that unspent, borrowed money was lent. In the modern world, loans are widely used, for the use of which interest is set. Since interest rates measure the cost of using money by entrepreneurs and the reward for non-use of money by the consumer sector, the level of interest rates plays a significant role in the economy of the country as a whole.

Very often in economic literature the term "interest rate" is used, although there are many interest rates. Interest rate differentiation is related to the risk taken by the lender. The risk increases as the loan term increases, since the likelihood that the lender may need the money before the established date for repayment of the loan becomes higher, and the interest rate increases accordingly. It increases when a little-known entrepreneur applies for a loan. A small firm pays a higher interest rate than a large one. For consumers, interest rates also vary.

However, no matter how different interest rates are, they are all influenced by: if the supply of money decreases, then interest rates increase, and vice versa. That is why the consideration of all interest rates can be reduced to the study of the patterns of one interest rate and in the future we will use the term “interest rate”

Distinguish between nominal and real interest rates

Real interest rate determined taking into account the level. It is equal to the nominal interest rate, which is set under the influence of supply and demand, minus the inflation rate:

If, for example, a bank makes a loan and charges 15%, and the inflation rate is 10%, then the real interest rate is 5% (15% - 10%).

Interest calculation methods:

Simple interest rate

Simple interest growth chart

Example

Determine the interest and the amount of accumulated debt if the simple interest rate is 20% per annum, the loan is equal to 700,000 rubles, the term is 4 years.

I = 700,000 * 4 * 0.2 = 560,000 rub.
S = 700,000 + 560,000 = 1,260,000 rub.

A situation where the loan term is less than the accrual period

The time base can be equal to:

360 days. In this case we get ordinary or commercial interest.
365 or 366 days. Used for calculation exact percentage.

Number of loan days

The exact number of days of the loan is determined by counting the number of days between the date of the loan and the date of its repayment. The day of issue and the day of repayment are considered one day. The exact number of days between two dates can be determined from the table of serial numbers of days in the year.
The approximate number of days of the loan is determined from the condition that any month is taken to be equal to 30 days.

In practice, there are three options for calculating simple interest:

Exact interest with exact number of loan days (365/365)
Ordinary interest with the exact number of days of loan (bank; 365/360). If the number of loan days exceeds 360, this method results in the amount of accrued interest being greater than the annual rate.
Ordinary interest with an approximate number of days of loan (360/360). It is used in intermediate calculations, as it is not very accurate.

Example

A loan in the amount of 1 million rubles was issued on January 20 until October 5 inclusive at 18% per annum. What amount must the debtor pay at the end of the term when simple interest is calculated? Calculate in three options for calculating simple interest.

First, let's determine the number of days of the loan: January 20 is the 20th day of the year, October 5 is the 278th day of the year. 278 - 20 = 258. Approximately - 255. January 30 - January 20 = 10. Month 8 multiplied by 30 days = 240. total: 240 + 10 + 5 = 255.

1. Exact interest with the exact number of days of loan (365/365)

S = 1,000,000 * (1 + (258/365)*0.18) = 1,127,233 rub.

2. Ordinary interest with the exact number of days of loan (360/365)

S = 1,000,000 * (1 + (258/360)*0.18 = 1,129,000 rub.

3. Ordinary interest with an approximate number of days of loan (360/360)

S = 1,000,000 (1 + (255/360)*0.18 = 1,127,500 rub.

Variable rates

Loan agreements sometimes provide for interest rates that vary over time. If these are simple bets, then the amount accrued at the end of the term is determined as follows.