Accretion and discounting operations. Financial transactions in a market economy

The interest means should be understood as the absolute amount of profit received as a result of the provision of money. They can be transmitted in any form. It can be various financial transactions. For example, the issuance of a loan, the placement of funds on a deposit account, the sale of products on credit, the purchase of a savings certificate, bonds, the accounting of a bill and so on. Particular importance in this case is the relationship between the rate of increase and the discount rate. Let's consider these elements in more detail.

Specificity

The interest rate is the relative amount of the profit received for a certain (fixed) time interval. It is formed by the ratio of income to the amount of debt. It is measured in ordinary or decimal fractions or in percent. When analyzing financial transactions, specialists use this relative amount as an indicator of the degree of efficiency (profitability) of any commercial, economic, investment, credit activity. It does not matter whether the fact of investing funds and the process of increasing their volume, or it did not take place. The time interval to which the interest rate is associated is called the accrual period. They can be a year, a quarter, a half-year, a month and even a day in some cases. As a rule, in practice, annual amounts are used.

The logic of operations of discounting (accumulation) of capital

By agreement between the borrower and the creditor, interest is paid as they accrue, or they are included in the principal amount of the debt. The increase in the amount of funds over time due to accession is the increase in capital. He is also called an increase in the amount. The discount rate is the inverse of the increase rate. This is due to the fact that, with a reduction, the amount that relates to the upcoming period is reduced by the corresponding discount. In such cases, it is said that accounting (discounted) rates are applied. Interest earned on them is called antipattern, and those that arise from the sum of the increase are called decursive. This is the logic of operations of discounting (accumulation) of capital.

Accrual Features

In most cases, the interest is called interest rates. For their accrual, a fixed base is used. When the amount that was received at the previous stage of reduction or increase is accepted as it, compound interest is used. The increment and discounting in such cases passes according to certain schemes. Relative amounts can be fixed. In this case, the agreement determines their size. They can also be floating. In this case, the contract does not indicate a rate, but a base that changes in the time interval, as well as the amount of the margin-margin. The size of the latter is determined by the term of the loan, the borrower's solvency and other conditions. Throughout the entire period of the loan operation, it can be variable or permanent. In the case of consistent debt repayment, two options for calculating interest are allowed. In the first case, the interest rate (complex or simple) is applied to the actual amount of the debt. The second option is used for consumer lending. In this case, the accrual is carried out for the entire amount of the obligation without taking into account its consecutive repayment. In practice, discrete sums are used. They are charged for certain time intervals (half-year, year, etc.). The accretion and discounting operations can be carried out continuously, for infinitely small periods. In this case, apply the appropriate percentages (continuous).

Accretion and discounting formulas

An increased amount of debt (loan, deposit, other loans or invested funds) should be understood as the initial amount of money with interest at the end of the accrual period. Thus, we can designate:

• Interest for the whole period - I;
• The original amount of the debt is P;
• Increased amount of funds (at the end of the period) - S;
• Interest rate - i;
• The loan time is n.

For the whole period interest will be:

I = Pni.

The increase in the amount is determined by the addition of the initial funds and interest:

P + I = P + Pni = P (1 + ni) = S.

In practice, specialists often have to face the opposite problem. On the amount of S, which is payable through some time interval n, you need to determine the size of the loan that was received - P. In such cases, there is a discount. The calculation is carried out when interest from the amount of S will be withheld forward, immediately when the loan is issued. The process of accrual of interest and their cancellation is called accounting. The interest itself is called a discount or a discount. To calculate, we need to use the equality S = P (1 + ni). We obtain P = S / (1 + ni). Thus, P will be the modern size of S, paid after n years. The above calculations show simple types of discounting (build-up). In the latter case, a variant of the mathematical definition of the sum is considered. As you can see, the calculations use indicators that are used in both the build-up operation and discounting.

Period duration

The build-up and discounting operations can be calculated using two time bases. If K is 360 days, then commercial or ordinary interest is obtained. When applying the actual duration of a calendar year in 365 or 366 days, exact interest is charged. The number of loan days is taken exactly and approximately. In the latter case, the month will be 30 days. The exact number of days can be determined by calculating their number between the dates when the loan was issued and when it should be repaid. According to Art. 839, item 1 of the Civil Code, the days in which the deposit was opened and closed are not included in the general term for accrual.

Available options

In practice, there are three ways of calculating interest:

1. Exact amounts with a specific number of days. The designations ACT / ACT or 365/365 are used. This option is used by central and large commercial banking institutions in the US and the UK. This method of calculation allows you to get the most accurate amount.
2. Normal interest with the exact number of loan days. In this case, the designations AST / 360 or 365/360 are used. This method is sometimes referred to as banking. It is used in transactions between banks of different countries or one state. This method, in particular, is common in Switzerland, Belgium, France. With this calculation, a somewhat larger amount is obtained than when applying the exact percentages.
3. Normal interest with an approximate number of days (360/360). This method is practiced in commercial banks in Denmark, Germany, Sweden. This option is used in cases where the exact result is not needed (for example, for intermediate calculations).

In the process of investing funds in a short-term deposit, in some cases a repeated sequential repetition of the build-up at a simple interest is used within the general prescribed period. Thus, reinvestment of the sums received at each stage of increasing the volume of funds by means of a variable or a fixed base is performed.

Reduction

Discounting can be considered as a definition of any value indicator relating to the upcoming time, for an earlier period. Such a method is called the reduction of a quantity to a certain, as a rule, initial moment. The sum of P, obtained by means of a reduction, is called the present value or the present value of the future payment. Depending on the type of interest rate used, there are two options for discounting:

1. Mathematical method.
2. Commercial (bank) accounting.

In the first variant, considered above, the obtained fraction is called the discounting factor. It reflects the share that the original amount of the debt amounts to in the final amount. When using the method of commercial accounting, the financial institution buys it before the payment due date on the bill or other payment obligation from the owner at a cost less than that indicated in the paper. Thus, the acquisition is subject to discounts. At maturity, the bank, having received money, implements interest income in the form of a discount. The owner of a paper with the help of accounting has the ability to receive funds before the time specified in it.

Peculiarities of a bill

This security is represented in the form of a debt receipt. The bill is drawn up in accordance with legislative requirements. Norms provide for special forms in which there is a name, the date of payment, the place where it is to be produced, information about the subject to be paid, information about the date and place of drawing up the paper, the signature of the drawer. Such debt receipts can be transferable and simple. The latter are presented in the form of documents that certify the unconditional financial obligation of the drawer to pay a certain amount to the owner of the paper upon the maturity of the obligation. A transferable document is a document that the borrower writes out. Tratt is a form of a special order to a direct payer (a banking organization, as a rule) to pay a certain amount to a bill holder (third person) within a set time.

Accounting of a bill

For such securities, a commercial (banking) method is used. In accordance with it, interest for the use of a loan in the form of a discount will be charged to the amount that must be paid at the end of the period. The accounting indicator in this case is d. The amount of the sum will be Snd. N will be measured in years, if d is the annual rate. The calculations are as follows:

P = S - Snd = S (1 - nd),

Where n is the period from the moment of accounting to the date of settlement of the obligation;

(1 - nd) is the discount factor.

Accounting, as a rule, is carried out at a temporary base of K, equal to 360 days, the number of loan days is most often taken accurate.

Other options

The accretion and discounting operations are calculated not only by simple interest. For example, amounts are not paid immediately after accrual, but are included in the amount of debt. This connection is called the capitalization of interest. In the calculation, you can apply the same indicators that were used above.

At the end of the first year, the percentages are Pi. The incremented sum will then be P + Pi = P (1 + i). By the end of the second year, it will become P (1 + i) + P (1 + i) i = P (1 + i) 2 and so on. At the end of the year n the sum will be S = P (1 + i) n, and the interest for this period is I = S-P = P [(1 + i) n-1].

(1 + i) n is the multiplication factor for complex percentages. Time in such cases is measured as AST / AST. Often the time for calculating interest is not an integer.

Interest accrual with increasing funds

There are the following options for accrual during the expansion:

1. The calculation is carried out using an integer number of years. It is taken from the compound interest formula. From the ratio of simple interest take a fractional part of the period.
2. According to the rules of some commercial banks for a number of transactions, the interest amount is charged only for whole numbers of periods (years or other terms).

To compare the results of the increase in different percentages, it will suffice to compare the corresponding multipliers. With equal levels of interest rates, the ratios of these indicators will be significantly dependent on the period. If n> 1, the difference will increase with the lengthening of the term. Working with compound interest, rule 72 is used: if the interest rate is i, then the doubling of the amount occurs about 72 / i years. For example, at 12% this will happen after 6 years.

Nominal and effective indicator

In today's conditions, capitalization of interest is carried out, as a rule, not once but several times during the year. This can be done quarterly or half-yearly. In some foreign commercial banking structures, daily accrual is practiced. If we take for the annual rate j, the number of periods in the year is m, each time the determination of interest will be made by j / m. The rate j is called nominal. There is also a valid (effective) indicator. It represents the annual rate of compound interest. With its use receive the same result as in the application of m - a one-time accrual of interest on j / m. This rate measures that relative real income, which is obtained as a whole for the year.

Bank account

When calculating using the commercial method, a complex rate is used. In such cases, the process of reducing the amount passes with a certain slowdown. This is due to the fact that each time the discount rate is applied not to the original amount of funds. It is used for the amount discounted at the previous stage in the time interval. An effective accounting indicator characterizes the degree of reduction for the year. In all cases, this rate will be less than the nominal for m> 1.