The process of increasing money in connection with the addition of interest to the amount of debt is called accumulation.

The accrued amount of a loan (debt, deposit, etc.) is its initial amount, together with the interest accrued on it by the end of the term.

The process of changing the amount of debt with accrued simple interest can be represented as an arithmetic progression, the members of which are the values

R; P + Pi =P(1 +i); R(1 + i) +Pi =P( 1+ 2i) etc. up to R(1 +ni).

The first term of this progression is R, difference - Pi, then the last term is the accumulated sum

S = P (1 + ni),

where S- extended amount of money;

R- the initial amount of money,

i- simple interest rate;

Pi- accrued interest for one period;

n- the number of interest periods;

Pni- accrued interest P periods .

This formula is the accrual formula for simple interest, or the simple interest formula.

Multiplier (1 + ni) is called the accrual multiplier. It shows how many times the accrued amount is greater than the initial amount.

The accumulated amount can be represented in the form of two terms: the initial amount and the amount of interest

S=P+I,

where I = Pni- amount of interest.

The calculation of simple interest is usually used in two cases:

1) when concluding short-term contracts (provision of short-term loans, etc.), the term of which does not exceed one year;

2) when interest is not added to the amount of debt, but is paid periodically.

The interest rate is usually set on an annual basis, so if the transaction lasts less than a year, it is necessary to find out what part of the interest is paid to the creditor. For this, the value P expressed as a fraction

where P - term of the financial transaction in fractions of a year;

Y- number of days or months in a year (time base) year- year);

t- duration of the operation (loan) in days or months time- time).

In this case, the accrued amount is calculated by the formula:

Several options for calculating interest are possible, differing in the choice of time base Y and how the duration of a financial transaction is measured.

Often, a year is taken as the base for measuring time, conditionally consisting of 360 days (12 months of 30 days each). In this case, they say that they are calculating ordinary, or commercial interest. In contrast, exact interest is obtained when the actual number of days in a year is taken as the base: 365 or 366 if the year is a leap year.

Determining the number of days of a financial transaction can also be exact or approximate. In the first case, the actual number of days between two dates is calculated, in the second, the duration of the financial transaction is determined by the number of months and days of the transaction, approximately considering all months to be equal and containing 30 days each. In both cases, the start date and end date of the operation are considered to be the same day.

The calculation of the exact number of days between two dates can be done by taking the difference between these dates, or using a special table that shows the serial numbers of dates in a year (Appendix 2, 3).

Various options for the time base and methods for calculating the days of a financial transaction lead to the following interest calculation schemes used in practice:

Exact interest with the exact number of days of the loan (the British scheme 365/365, when 365 days are counted in a year, half a year is equal to 182 days and the length of the months is exact);

Ordinary interest with the exact number of days of the loan (French scheme 365/360, 360 days are accepted per year and the exact length of the months);

Ordinary interest with an approximate number of loan days (German scheme 360/360, it is considered that there are 360 ​​days in a year and 30 days in each month).

Since the exact number of loan days is in most cases greater than the approximate one, the amount of interest with the exact number of days is usually greater than with the approximate one.

The calculation option with exact interest and an approximate measurement of the loan time is not applicable.

The exact and approximate number of days for ordinary interest are related by the following dependencies:

i 360 = 0.986301 i 365 ; i 365 = 1.013889 i 360 .

Interest rates do not remain constant over time; loan agreements sometimes provide for discretely changing interest rates. In this case, the formula for calculating the accumulated amount takes the following form:

where i t- the rate of simple interest in the period with the number t, t = 1,…, k;

n t- duration t accrual period at the rate i t,i = 1,…, k.

The deposit amount received at the end of the specified period, together with the interest accrued on it, can be reinvested at this or another interest rate. The reinvestment process is sometimes repeated several times within the settlement period N. In the case of multiple investments in short-term deposits and the application of a simple interest rate, the accrued amount for the entire period N is found according to the formula

where P 1 , P 2 , n t- the duration of successive periods of reinvestment

where i 1 ,i 2 , …, i t- rates at which reinvestment is made.

When servicing current accounts, banks are faced with a continuous chain of receipts and expenditures of funds, as well as the need to accrue interest on a constantly changing amount. In banking practice, in this situation, the rule is used - the total amount of interest accrued for the entire period is equal to the amount of interest accrued on each of the amounts that are constant over a certain period of time. This applies to the debit and credit parts of the account. The only difference is that the credit interest is deductible.

To calculate interest on such constant amounts, percentage numbers are used:

The percentages for each constant amount are added up and divided by the motto:

Therefore, the entire absolute amount of accrued interest is calculated as follows:

The calculation of the rate of return of short-term financial transactions in the form of a simple interest rate is carried out according to the formula:

In practice, it is often necessary to solve the problem that is the inverse of interest accumulation, when, for a given accumulated amount corresponding to the end of a financial transaction, it is required to find the initial amount . This calculation is called discounting the accrued amount. .

The value found by discounting is called the present value, or present value, of the accrued amount.

In most cases, the time factor is taken into account in financial contracts with the help of discounting. Modern value Money is equivalent to the accumulated amount in the sense that after a certain period of time and at a given interest rate, as a result of the accumulation, it will become equal to the accumulated amount . Therefore, the operation of discounting is also called reduction.

You can bring the value of money to any desired point in time, not necessarily to the beginning of a financial transaction.

There are two types of discounts:

1. Mathematical discounting, which is a solution to the problem, the reverse of the increase in the original loan. If in the direct problem S = P (1 + ni), then in reverse

Expression 1/(1 + ni) is called the discount factor. It shows what proportion of the initial amount of money in the final amount of debt.

The discount of the accrued amount is equal to

D = S - R,

where D- discount.

2. Banking (commercial) accounting. Accounting operation, including accounting of bills, consists in the fact that the bank, before the due date for payment on a bill or other payment obligation, buys it from the owner (who is a creditor) at a price lower than the amount that must be paid on it at the end of the term, t .e. acquires (takes into account) it at a discount.

In this case, the present value of cash is

P=S(1 - nd),

where d- accounting interest rate.

Multiplier (1 - nd) is called the discount factor.

The amount of discount or accounting held by the bank is equal to

D=Snd.

The simple annual discount rate is

Discounting at a discount rate is carried out in most cases, provided that the year is equal to 360 days.

A special case is the banking accounting process, when the operation period is expressed in days or months:

The discount rate can be used to increase:

The increment and discount operations are opposites, but they can be used to solve both problems. In this case, depending on the applied rate, one can distinguish between direct and inverse problems (Table 2.1).

Table 2.1 - Direct and inverse problems

In a market economy, any interaction between individuals, firms and enterprises for the purpose of making a profit is called a transaction. In credit transactions, profit is the amount of income from lending money, which in practice is realized through the accrual of interest (interest rate - i). Interest depends on the amount provided, the term of the loan, the terms of accrual, etc.

The most important place in financial transactions is the time factor (t). The principle of non-equivalence and non-equivalence of investments is connected with the time factor. In order to determine the changes that occur with the initial amount of money (P), it is necessary to calculate the amount of income from lending money, investing it in the form of a contribution (deposit), investing it in securities, etc.

The process of increasing the amount of money in connection with the calculation of interest (i) is called accumulation, or the growth of the initial amount (P). Thus, the change in the initial cost under the influence of two factors: interest rate and time is called accrued value (S).

Accrued value can be determined by the scheme of simple and compound interest. Simple interest is used when the accrued amount is determined in relation to a constant base, that is, the accrued interest is repaid (paid) immediately after accrual (thus, the initial amount does not change); in the case when the initial amount (initial) changes in the time interval, they deal with compound interest.

When calculating simple interest, the accrued amount is determined by the formula

S = P (1 + i t), (1)

where S is the accumulated amount (cost), rub.; P - initial amount (cost), rub.; i – interest rate expressed as a coefficient; t is the interest calculation period.

S \u003d 10,000 (1 + 0.13 1) \u003d 11,300, rub. (loan repayment amount);

ΔР \u003d 11,300 - 10,000 \u003d 1,300, rub. (the amount of accrued interest).

Determine the amount of repayment of the debt subject to the annual payment of interest, if the bank issued a loan in the amount of 50,000 rubles. for 2 years, at a rate of 16% per annum.

S \u003d 50,000 (1 + 0.16 2) \u003d 66,000, rub.

Thus, the calculation of simple interest is carried out in the case when the accrued interest does not accumulate on the amount of the principal debt, but is paid periodically, for example, once a year, half a year, a quarter, a month, etc., which is determined by the terms of the loan agreement. There are also cases in practice when settlements are made for shorter periods, in particular on a one-day basis.

In the case when the term of the loan (deposit, etc.) is less than one year, it is necessary to adjust the given interest rate in the calculations depending on the time interval. For example, you can represent the interest calculation period (t) as a ratio, where q is the number of days (months, quarters, six months, etc.) of the loan; k is the number of days (months, quarters, semesters, etc.) in a year.

Thus, formula (1) changes and has the following form:

S = P (1 + i ). (2)

The Bank accepts term deposits for a period of 3 months at 11% per annum. Calculate the client's income when investing 100,000 rubles. for the specified period.

S = 100,000 (1+ 0.11 ) = 102,749.9, rub.;

ΔР = 102,749.9 - 100,000 = 2,749.9, rub.

Depending on the number of days in a year, different calculation options are possible. In the case when a year, conventionally consisting of 360 days (12 months of 30 days), is taken as the base for measuring time, ordinary or commercial interest is calculated. When the actual number of days in a year is taken as the base (365 or 366 in a leap year), one speaks of exact percentages.

When determining the number of days of using the loan, two approaches are also used: exact and ordinary. In the first case, the actual number of days between two dates is calculated, in the second, the month is taken equal to 30 days. Both in the first and in the second case, the day of issue and the day of repayment are considered as one day. There are also cases when the calculation uses the number of settlement or working banking days, the number of which per month is 24 days.

Thus, there are four calculation options:

1) ordinary interest with the exact number of loan days;

2) ordinary interest with an approximate number of loan days;

3) exact interest with an approximate number of loan days;

4) exact interest with bank number of working days.

At the same time, it should be taken into account that in practice the day of issue and the day of repayment of the loan (deposit) are taken as one day.

The loan was issued in the amount of 20,000 rubles. for the period from 10.01.06 to 15.06.06 at 14% per annum. Determine the loan repayment amount.

1. Ordinary interest with the exact number of days of the loan:

156=21+28+31+30+31+15;

S \u003d 20,000 (1 + 0.14 ) \u003d 21,213.3, rub.

2. Ordinary interest with an approximate number of loan days:

S \u003d 20,000 (1 + 0.14 ) \u003d 21,205.6, rub.

3. Exact interest with an approximate number of loan days:

S \u003d 20,000 (1 + 0.14 ) \u003d 21,189.0, rub.

4. Exact interest with bank number of working days:

S \u003d 20,000 (1 + 0.14 ) \u003d 21,516.7, rub.

Data for calculating the number of days in the period are presented in Appendix. 12.

As mentioned above, in addition to the calculation of simple interest, complex accrual is used, in which interest is accrued several times over the period and is not paid, but accumulated on the amount of the principal debt. This mechanism is especially effective for medium-term and long-term loans.

After the first year (period), the accumulated amount is determined by the formula (1), where i will be the annual compound interest rate. After two years (periods), the accumulated amount S 2 will be:

S 2 \u003d S 1 (1 + it) \u003d P (1 + it) (1 + it) \u003d P (1 + it) 2.

Thus, when calculating compound interest (after n years (periods) of accrual), the accrued amount is determined by the formula

S = P (1 + i t) n , (3)

where i is the compound interest rate, expressed as a coefficient; n is the number of compound interest accruals for the entire period.

The accumulation coefficient in this case is calculated by the formula

Kn = (1 + i t) n , (4)

where Kn is the coefficient of accumulation of the initial cost, units.

The investor has the opportunity to place funds in the amount of 75,000 rubles. on a deposit in a commercial bank for 3 years at 10% per annum.

Determine the amount of accrued interest by the end of the deposit term, when calculating compound interest.

S = 75,000 (1+ 0.1 1) 3 = 99,825, rub.

ΔР = 24 825, rub.

Thus, the growth rate will be:

Kn \u003d (1 + 0.1 1) 3 \u003d 1.331

Therefore, the accumulation coefficient shows how many times the initial amount has increased under given conditions.

The share of calculations using compound interest in financial practice is quite large. Calculations according to the compound interest rule are often referred to as accruing interest on interest, and the procedure for adding accrued interest is called their reinvestment or capitalization.

Rice. 1. The dynamics of the increase in cash in the calculation of simple and compound interest

Due to the constant growth of the base due to the reinvestment of interest, the growth of the initial amount of money is carried out with acceleration, which is clearly shown in Fig. one.

In financial practice, interest is usually calculated several times a year. If interest is accrued and added more often (m times a year), then m-fold interest accrual takes place. In such a situation, the terms of the financial transaction do not stipulate the rate for the period, therefore, the annual interest rate i is fixed in financial contracts, on the basis of which the interest rate for the period () is calculated. At the same time, the annual rate is called the nominal rate, it serves as the basis for determining the rate at which interest is charged in each period, and the rate actually applied in this case (() mn) is effective, which characterizes the full effect (income) of the operation, taking into account intra-annual capitalization .

The accrued amount under the effective compound interest scheme is determined by the formula

S = P (1+ ) mn , (5)

where i is the annual nominal rate, %; (1+ ) mn is the coefficient of the effective rate increase; m is the number of cases of accrual of interest per year; mn is the number of cases of interest accrual for the period.

S = 20,000 (1+) 4 1 = 22,950, rub.

It should be noted that for a period of 1 year, the number of interest accruals per year will correspond to the number of interest accruals for the entire period. If the period is more than 1 year, then n (see formula (3)) will correspond to this value.

S \u003d 20,000 (1+) 4 3 \u003d 31,279.1, rub.

The calculation of compound interest is also applied not only in cases of calculating the amount of debt increased by interest, but also in case of repeated accounting valuable papers, determining the rent for leasing services, determining the change in the value of money under the influence of inflation, etc.

As discussed above, the rate that measures the relative return received over the period as a whole is called the effective rate. The calculation of the effective interest rate is used to determine the real yield of financial transactions. This yield is determined by the respective effective interest rate.

I ef \u003d (1+) mn - 1. (6)

Credit organisation calculates interest on term deposit, based on the nominal rate of 10% per annum. Determine the effective rate for daily compound interest.

i \u003d (1+) 365 - 1 \u003d 0.115156, i.e. 11%.

The real income of the depositor for 1 rub. invested funds will not be 10 kopecks. (from the condition), and 11 kopecks. Thus, the effective interest rate on the deposit is higher than the nominal one.

The bank at the end of the year pays 10% per annum on deposits. What is the real return on deposits when interest is calculated: a) quarterly; b) every six months.

a) i \u003d (1+) 4 - 1 \u003d 0.1038, i.e. 10.38%;

b) i \u003d (1+) 2 - 1 \u003d 0.1025, i.e. 10.25%.

The calculation shows that the difference between the rates is insignificant, however, the accrual of 10% per annum on a quarterly basis is more profitable for the investor.

The calculation of the effective interest rate in financial practice allows the subjects of financial relations to navigate the offers of various banks and choose the most appropriate investment option.

Loan agreements sometimes provide for a change in the interest rate over time. This is due to changes in contractual conditions, the provision of benefits, the imposition of penalties, as well as a change general conditions transactions, in particular, the change in the interest rate over time (as a rule, upwards) is associated with the prevention banking risks, possible as a result of changes in the economic situation in the country, rising prices, depreciation national currency etc.

The calculation of the accrued amount when the interest rate changes over time can be carried out both by calculating simple interest and compound interest. The interest calculation scheme is specified in the financial agreement and depends on the term, amount and conditions of the transaction.

Let the interest rate vary from year to year. For the first n 1 years, it will be equal to i 1, n 2 - i 2, etc. When calculating simple interest on the initial amount, it is necessary to add the interest rates i 1, i 2, i n, and for complex ones, find their product.

The formula for calculating simple interest is

S = P (1+i 1 t 1 + i 2 t 2 + i 3 t 3 + i n t n) , (7)

where i n is the simple interest rate; t n is the duration of the accrual period.

In the first year in the amount of 10,000 rubles. 10% per annum are charged, in the second - 10.5% per annum, in the third - 11% per annum. Determine the repayment amount if the interest is paid annually.

S \u003d 10,000 (1 + 0.10 1 + 0.105 1 + 0.11 1) \u003d 13,150, rubles;

ΔР = 3 150, rub.

The formula used to calculate compound interest is

S = P(1+i 1 t 1) (1+ i 2 t 2) (1+ i 3 t 3) (1+ i n t n) (8)

where i n is the compound interest rate; t n - the duration of the period of its accrual.

In the first year in the amount of 10,000 rubles. 10% per annum are charged, in the second - 10.5% per annum, in the third - 11% per annum. Determine the repayment amount if interest is capitalized.

S \u003d 10,000 (1 + 0.10 1) (1 + 0.105 1) (1 + 0.11 1) \u003d 13 492.05, rub.

The examples given confirm the fact that the calculation of simple interest is associated with the determination of the accrued amount in relation to a constant base, i.e., each year (period) interest is charged on the same initial cost. If we consider example 10, then in this case the accrued value will be:

- for the first year: S 1 \u003d 10,000 (1 + 0.10 1) \u003d 11,000, rubles;

ΔР 1 \u003d 1,000, rub.;

- for the second year: S 2 \u003d 10,000 (1 + 0.105 1) \u003d 11,050, rubles;

ΔР 2 \u003d 1,050, rub.;

- for the third year: S 3 \u003d 10,000 (1 + 0.11 1) \u003d 11,100, rubles;

ΔР 3 \u003d 1 100, rub.

Thus, the amount of interest for 3 years will be:

ΔР \u003d 1,000 + 1,050 + 1,100 \u003d 3,150, rub. (see example 10).

In the case of calculating compound interest, the initial amount changes after each accrual, since interest is not paid, but accumulated on the principal amount, i.e., interest is accrued on interest. Consider example 11:

- in the first year: S 1 \u003d 10,000 (1 + 0.10 1) \u003d 11,000, rubles;

- in the second year: S 2 \u003d 11000 (1 + 0.105 1) \u003d 12,100, rubles;

- in the third year: S 3 \u003d 12100 (1 + 0.11 1) \u003d 13,431, rubles.

Thus, the amount of interest for 3 years will be: i 3 \u003d 3,431, rubles. (see example 10).

When developing the terms of contracts or analyzing them, sometimes it becomes necessary to solve inverse problems - determining the term of the operation or the level of the interest rate.

Formulas for calculating the duration of a loan in years, days, etc. can be calculated by transforming formulas (1) and (5).

Loan term (deposit):

t = · 365 . (nine)

Determine for how long the depositor should place 10,000 rubles. on a deposit when accruing simple interest at a rate of 10% per annum, in order to receive 12,000 rubles.

t = ( ) 365 = 730 days (2 years).

The client has the opportunity to invest 50,000 rubles in the bank. for half a year. Determine the interest rate that ensures the client's income in the amount of 2,000 rubles.

t = ( ) = 0.08 = 8% per annum

Similarly, the required term for the completion of a financial transaction and its length, or the amount of the required interest rate when calculating compound interest, is determined.

To simplify the calculations, the values ​​​​of the coefficient (multiplier) of the accumulation are presented in App. 3.

Accrued amount formulas

Consider the accumulation for various cases of accrual of rents.

1. Ordinary annuity.

Let at the end of each year for P years to the current account is deposited according toRrubles, interest is accrued once a year at the ratei. In this case, the first installment by the end of the annuity period will increase to a value since the amount R interest accrued over n - 1) of the year. The second installment will increase to and so on. No interest is charged on the last installment.

Thus, at the end of the annuity term, its accumulated amount will be equal to the sum of the members of the geometric progression

in which the first term isR, denominator (1+ i), number of members P. This amount is equal to

(1)

where

(2)

called rent accumulation factor. It depends on the term of the lease. P and the level of the interest ratei.

The accrued amount of annuity prenumerando in (1 + i) times more postnumerando and at m =p=1

(3)

Example 1

To create a pension fund, annuity postnumerando in the amount of 10 million rubles is annually paid to the bank. Interest is accrued on incoming payments at a compound annual rate of 18%. Determine the size of the fund after 6 years.

Solution.

According to formula (1) we have:

million rubles

Answer. Pension Fund in 6 years will be 99.42 million rubles.

2. Annuity, interest accrual m once a year.

Let payments be made once at the end of the year, and interest is charged T once a year. This means that every time the rate is appliedj/ m, where j - nominal interest rate. Then the terms of the annuity with interest accrued before the end of the term have the form

If we read the previous line from right to left, we get a geometric progression, the first member of which R, denominator (1+ j/ m) m, number of members P. The sum of the members of this progression will be the accrued sum of the annuity. She is equal

(4)

The accrued amount of annuity prenumerando is calculated by the formula

(5)

Example 2

In the conditions of example 1, assume that the bank accrues interest on a quarterly basis at a nominal rate of 18% per annum. Make a conclusion which option for calculating interest is beneficial to the lender.

Solution.

By formula (4) we have

= 97.45 million rubles

Answer.The option of example 2.2 is beneficial for the creditor, so that interest is accrued on the rent quarterly, while the size of the fund will be 97.45 million rubles.

3. Rentp - urgent,m = 1.

Find the accumulated amount, provided that the rent is paid R once a year in equal installments, and interest is accrued once at the end of the year.

If R- the annual amount of payments, then the amount of a separate payment is equal toR/ p. Then the sequence of payments with interest accrued before the end of the term is also a geometric progression, written in reverse order,

which has the first memberR/ p, denominator (1+ i) 1/ p, total number of members etc. Then the accumulated amount of the annuity under consideration is equal to the sum of the members of this geometric progression

(6)

where

(7)

p-term annuity accumulation factor at m = 1.

The accumulated annuity amount prenumerando is calculated by the formula:

(8)

Example 3

Mr. Ivanov pays 500 rubles to the bank at the end of each month. Compound interest is charged on incoming payments at an annual interest rate of 22%. Determine the amount of the accrued amount after 8 years.

Solution.

Using formula (6), we find the amount of the accrued amount:

S=500 [ (1 + 0,22) 8 - 1 ] / [ (1 + 0,22) 1/8 - 1 ] = 52.806 thousand rubles

Answer.The amount accrued by the bank to Mr. Ivanov in 8 years will be 52.806 thousand rubles.

4. Rent p -urgent, p = t.

In contracts, the calculation of interest and the receipt of payment often coincide in time. So the number of payments R per year and the number of interest accruals T match, i.e. p = t. Then, to obtain the formula for calculating the accrued amount, we use the analogy with the annual annuity and one-time interest at the end of the year, for which

The only difference will be that all parameters now characterize the rate and payment for the period, and not for the year. Thus, we get

(9)

The accumulated annuity amount prenumerando is calculated by the formula:

(10)

Example 4

Mr. Petrov must repay a debt of 200 thousand rubles. In order to collect this amount, he plans to deposit the same amount to the bank at the end of each six months for 3 years, and compound interest is accrued on it every six months at an annual rate of 15%. What should be the amount of semi-annual deposits made by Mr. Petrov with semi-annual interest accrual? Consider the case when the amount is deposited in the bank once at the end of each year and interest is calculated at the same compound interest rate.

Solution.

From (9) we find the sum ( R), which must be paid to the bank every six months with a six-month compound interest calculation:

R = S j /[ (1 + j/m)mn- 1 ] = 200 × 0,15 / [ (1 + 0,15/ 2) 2 × 3 - 1 ] = 55.228 thousand rubles

From formula (1) we find the amount that must be paid to the bank every year with annual compound interest:

R = S j / [ (1 + j) n - 1 ] = 200 × 0,15 / [ (1 + 0,15) 3 - 1 ] = 57.692 thousand rubles

Answer.Mr. Petrov needs to deposit in the bank every six months and six months of compound interest an amount equal to 55.228 thousand rubles. and the amount of 57.692 thousand rubles. with an annual contribution and annual compound interest. The first investment option is more profitable for him.

5. Rent R- urgent, p ³ 1 , m ³ 1.

This is the most general case R- term annuity with interest accrual T once a year, and possibly R ¹ T.

The first member of the annuityR/ p, paid later 1/r year after the beginning, will be by the end of the term, together with the interest accrued on it

Number of members P p. As a result, we get the accumulated amount

(11)

The accrued amount of annuity prenumerando is determined by the formula:

(12)

Example 5

The enterprise creates an insurance fund, for which it sends payments to the bank in the amount of 100 thousand rubles. at the end of every 4 months, the bank calculates compound interest once every six months at an annual rate of 18%. Determine the size insurance fund after 10 years.

Solution.

By formula (11) we find:

thousand roubles.

Answer.The size of the insurance fund of the enterprise in 10 years will be 7790.86 thousand rubles.

Discounting

Present Value (Refundable Amount)

Interest rate

Rice. 6. The logic of financial transactions

Mathematical discounting

Mathematical discounting is a formal solution to the problem, the reverse of the increase in the original loan amount. The task in this case is formulated as follows: what is the initial amount of the loan that must be issued in debt in order to receive the amount at the end of the term? S provided that the debt bears interest at the rate i ? By solving the equation (1) relatively P, we find:

(12)

The value established in this way P is the present value of the sum S which will be paid through n years. Expression 1/(1 + n∙i) is called discount multiplier, which shows the present value of one currency unit.

Difference ( S – P) can be considered not only as interest accrued on P, but also as a discount of the amount S. Let's denote the latter by D. The discount, as a discount from the final amount of the debt, is not necessarily determined through the interest rate, it can be set by agreement of the parties and in the form of an absolute value for the entire period.

Consider examples.

Example 8

A year later, the owner of a bill issued by a commercial bank should receive 220 thousand rubles on it. What amount was deposited in the bank at the time of purchase of the promissory note, if the annual rate is 12%?

Given: Solution:

S= 220 tr. Let's represent the problem graphically

n= 1 year

i = 12%; n= 1 g

S= 120t.r.

discounting

Using the expression(12) we get:
thousand roubles.

Example 9

The loan must be repaid in a year in the amount of 200 thousand rubles. The lender asked to repay the loan 270 days after issuance at 10% per annum. How much will the lender receive?TO = 365 days

Given: Solution:

S= 200 thousand rubles. Let's represent the problem graphically:

n= 1g.

n1 = 270 days

i = 10%

n = 365-270

S= 200t.r.

discounting

n 1 = 270

n 0 = 95 days

n = 365

Find the number of days left until the loan is repaid:

n 0 = n – n 1 = 365 - 270 = 95 (days)

Using the expression (12) we find:

(thousand roubles.)

Banking or commercial accounting (accounting for bills of exchange)

When accounting for a bill, bank accounting is used. Under this method, interest on the use of a loan is calculated as a discount on the amount due at the end of the term. In doing so, it applies discount rated. (Fig. 7)

Р discounting (accounting) S

Rice. 7

Discounting using a simple discount rate

The calculation formula for calculating these percentages is derived based on the following reasoning.

Let from 1 rub. an annual accounting (discount, advance) rate is taken d, then the debtor receives the amount (1- d) and after the expiration of the period must return 1 rub. That is, if 1 rub. is the amount returned S, then the initial sum will be equal to: P = S – d(provided that the period is one year), or in our case, P = 1 – d. If the value S, R And n are arbitrary, then

P = S – S ∙ n ∙ d = S ∙ (1 – n ∙ d), (13)

where S∙n∙d is the amount of the discount, and n- the period from the moment of accounting to the date of repayment of the bill. Value (1 - n∙d) is called discount multiplier using the discount rate. Accounting through the discount rate is carried out most often with a time base K= 360 days, the exact number of loan days is taken (ordinary interest with the exact number of loan days).

To clarify the practical application, consider a discount bill. Using the denomination of the bill (S) , discount rate (d) , time remaining until maturity (t) , subtract the discount (D) – discount from face value, i.е. the difference between S And R.

Then calculate the redemption (invoice) value of the bill to maturity

(13a)

Consider an example:

Example 10

The owner of a bill with a par value of 100 thousand rubles. and circulation period 105 days., 15 days. before the due date, it is taken into account in the bank at a discount rate of 20%. Determine the amount received by the owner of the bill.

Given: Solution:

S= 100 thousand rubles. Let's represent the problem graphically:

Per. appeal - 105 days.

n= 15 days

R - ? S = 100

n= 15 days

Using the expression(13a) we get:

(thousand roubles.)

In some cases, a situation may arise when the accrual of interest at the accrual rate is combinedi and discounting at a discount rated . In this case, the amount received during accounting is determined as:

P` = P ∙ (1 + n ∙ i) ∙ (1 – n` ∙ d) (14)

S`

whereP ( S ) - nominal amount;n - the total term of the payment obligation; n ` - the period from the moment of accounting to the date of payment repayment;R` - the amount received when accounting for a liability.

Example 11.

A debt obligation providing for the payment of 400 thousand rubles. with 12% per annum accrued on them, repayable in 90 days. The owner of the obligation (creditor) took it into account in the bank in 15 days. before maturity at a discount rate of 13.5%. The amount received after accounting was:

Given: Solution:

S= 400 thousand rubles. In this problem, the nominal value

n= 90 days (refundable amount) is taken as

n` = 15 days initial:S = P (see graph).

d = 13,5%

P(S) =400 tr. S`

i = 12%; n= 90 days

d = 13,5%; n` = 15 days

discounting

P` -?

1. First, we determine the accumulated amount of the obligationS ` , taking its face value for the initial amount:

(thousand roubles.)

2. Find the amount received after accounting:

(thousand roubles.)

3. Using an expression (14) we get the same amount:

(thousand roubles.)

The need to use a simple discount rate to calculate the accumulated amount arises in the case of determining the nominal value of a bill when issuing a loan. In this case, the amount of the debt affixed to the bill will be equal to

(15)

Value 1/(1- n ∙ d ) in this case is incremental multiplier when using a simple discount rate.

Example 12.

The entrepreneur applied to the bank for a loan in the amount of 200 thousand rubles. for a period of 55 days. The bank agrees to lend the specified amount subject to the accrual of interest on a simple discount rate of 20%. Find the refundable amount.

Given: Solution:

R= 200 thousand rubles. In this problem, the increment is made

n= 55 days at a simple discount rate.

R = 200 S - ?

buildup

d = 20; n= 55 days

Using the expression(15) we get:

thousand roubles.

If the amount was issued at a simple interest rate( i ) , then the accumulated amount would be equal to thousand roubles. , i.e. accretion at the discount rate is faster and it is less beneficial to the debtor 206,111< 206,304 т.е. возвращаемая сумма в первом случае будет больше.

Determination of the loan term when using the discount rate is made according to the formulas:

, (16)

, (17)

where n – the term of the loan in years; t – loan term in days; k - temporary base.

Consider an example:

Example 13

The company needs a loan of 500 thousand rubles. The bank agrees to issue a loan, provided that it is returned in the amount of 600 thousand rubles. The discount rate is 21% per annum. How long will the bank lend to the company?TO = 365 days

Given: Solution:

S= 600 thousand rubles. Graphical illustration of the task

R= 500 thousand rubles.

R= 500 tr. S= 600 tr.

d = 20%; n - ?

discounting

When solving problems of this kind, it is easier to use the expression(17) , then the loan term will immediately turn out in days (when using the expression(16) time will be expressed in fractions of a year):

(days)

The discount rate is calculated using the formulas:

, (18)

. (19)

Example 14

Contract for a loan of 500 thousand rubles. provides for the repayment of the debt in 300 days in the amount of 600 thousand rubles. Determine the discount rate applied by the bank.TO = 365 days.

Given: Solution:

R= 500 thousand rubles.

S= 600 thousand rubles.

t= 300 days

R= 500 tr. discounting S= 600 tr.

d = ? t= 300 days

According to the formula(19) we get:
ord = 20,27%

In transactions with discount financial instruments, the discount rate can sometimes be set implicitly: in the form of a total relative share of the denomination or as the ratio of the discounted amount to the face value ; thend located as or

(20)

whered ` - discount percentage;t – term before accounting (term of the bill).

Example 15

The amount of interest withheld when issuing a semi-annual loan is 20% of the loan amount. Let's define the mortgaged interest rate (discount rate).TO = 365

Given: Solution:

d` = 20%

t= 0.5 g (180 days)

TO= 365 days

d - ?

Example 16

State short-term three-month bills are quoted at the rate of 90. Let's calculate the discount rate.TO =360.

Given: Solution:

P / S = 0,9 discount in our case: 1 - 0.9 = 0.1

d - ? then:

Control homework in financial mathematics

1. Determine the accumulated deposit amount of 3 thousand rubles. with a deposit term of 2 years at a nominal interest rate of 40% per annum. Interest is calculated: a) once a year, b) semi-annually, c) quarterly, d) monthly

The accumulated amount by the end of the deposit term is determined by the formula:

where m is the number of interest accruals per year;

n - term of the deposit (in years);

The annual interest rate specified in the deposit agreement (nominal rate).

The interest rate accepted in banks for the accrual interval.

a) once a year:

(thousand roubles.)

b) by half a year

• (thousand roubles.)
• c) quarterly

• (thousand roubles.)
• d) monthly.

• (thousand roubles.)
• 2. The Bank accepts deposits from the public at a nominal interest rate of 12% per annum. Interest calculation monthly. The $1200 deposit was withdrawn after 102 days. Determine customer income

To calculate the duration of a financial transaction, we take the exact number of days in a year. The duration of a financial transaction is determined by the formula:

where t is the actual number of days for the financial transaction.

n - term of the deposit (in years).

3. For the construction of the plant, the bank provided the company with a loan of $ 200 thousand for a period of 10 years at the rate of 13% per annum. Calculate the accrual rate, the amount of accrued interest and the cost of the loan at the end of each year

Simple interest:

The coefficient of accrual of simple interest is determined by the formula:

where

where S 0 - loan amount;

n - interest calculation period;

i - nominal interest rate.

S 0 - loan amount;

n - interest calculation period;

i - nominal interest rate.

Table 1 shows data on the value of the accrual coefficient, the amount of interest and the cost of the loan at the end of each year (calculations were made in Microsoft Excel - Appendix A, task 3).

Table 1. Estimated data on the accumulation coefficient, the amount of interest and the cost of the loan.

	accumulation factor	loan cost, $	percentage, $

Compound interest:

The accumulation coefficient is determined by the formula:

i - nominal interest rate.

The interest amount is calculated by the formula:

where S is the loan amount;

n - interest calculation period;

i - nominal interest rate.

Loan value at the end of the period:

where S n - the cost of the loan (accumulated value);

S 0 - loan amount;

n - interest calculation period;

i - nominal interest rate.

Table 2 shows data on the value of the accrual coefficient, the amount of interest and the cost of the loan at the end of each year (calculations were made in Microsoft Excel).

Table 2. Estimated data on the accumulation coefficient, the amount of interest and the cost of the loan.

	accumulation factor	loan cost, $	percentage, $

4. The company was granted a preferential loan of $50,000 for 3 years at 12% per annum. Interest on the loan is calculated once a year. Under the terms of the agreement, the firm has the right to pay the loan and interest in a single payment at the end of the three-year period. How much should the firm pay when calculating simple and compound interest?

Simple interest:

The sum of simple interest is calculated by the formula:

where S is the loan amount;

n - interest calculation period;

i - nominal interest rate.

The loan amount will be:

The amount of accrued compound interest is calculated by the formula:

where S is the loan amount,

n - interest calculation period,

i - nominal interest rate.

The loan amount will be:

5. A manufacturing and commercial firm received a loan of 900 thousand rubles. for a period of 3 years. Interest is compound. The interest rate for the first year is 40% and each subsequent year increases by 5%. Determine the loan repayment amount

The loan repayment amount is determined by the formula:

where S n - the amount of the loan repayment at the end of the period;

S 0 - loan amount;

n - interest calculation period;

i - nominal interest rate.

According to the condition, the interest rate increases by 5%:

The loan repayment amount for the 3rd year will be:

6. Determine the period of time required to double the capital on simple and compound interest at an interest rate of 12% per annum. In the latter case, monthly interest

"Rule 70" and "Rule 100" allow answering the question of how many years capital will double at interest rate i.

Simple interest (“rule of 100”):

i - interest rate.

where T is the period for which the capital will double;

i - interest rate.

7. Determine the period of time required for the capital to triple at simple and compound interest at an interest rate of 48% per annum. In the latter case, quarterly interest accrual

Simple interest on tripling capital:

Compound interest when capital is tripled:

8. How long does it take to keep a deposit in a bank at 84% per annum with monthly, quarterly and semi-annual interest accrual in order for the deposit amount to double. banking calculation method

Compound interest ("Rule of 70"):

where T is the period for which the capital will double;

m - frequency of interest calculation;

i - interest rate.

• - monthly accrual: years.
• - quarterly accrual: years.

• - semi-annual accrual: years.
• 9. The client deposited $1600 for a period of 4 months. Interest calculation monthly. After the end of the term, he received $1732. Determine the bank's interest rate

To determine the interest rate of the bank, the formula for accruing funds using the compound interest method is used:

j is the actual number of interest accrual periods;

n - term of the deposit (in years);

S0 - the amount of the deposit at the time of opening the deposit;

bank interest rate.

From here, the bank's interest rate is calculated by the formula:

The bank's interest rate will be:

10. What should be the minimum interest rate in order for the deposit to double in a year when interest is calculated: a) quarterly, b) monthly

The minimum interest rate is determined by the formula:

where m is the number of interest accruals;

n - term of the deposit (in years);

S0 - the amount of the deposit at the time of opening the deposit;

Sm - the amount of the deposit at the time of opening the deposit;

bank interest rate.

a) quarterly interest calculation:

b) monthly interest calculation:

11. "Priorbank" offered the population for 1996 a cash deposit. The income on it was 72% per annum for the first 2 months, 84% per annum for the next 2 months, 96% per annum for 5 months, and 108% per annum for 6 months. Determine the effective interest rate when placing money for 6 months at the specified simple and compound interest. In the latter case, monthly interest

The effective interest rate is the rate that reflects the real income from a commercial transaction).

The effective interest rate calculated on simple interest is determined by the formula:

where m is the number of interest accruals;

n - term of the deposit (in years).

The effective interest rate calculated on compound interest is determined by the formula:

where m is the number of interest accruals;

n - term of the deposit (in years).

12. An advertisement for one commercial bank offers 84% ​​per annum with monthly interest. Another commercial bank offers 88% per annum with quarterly interest. The deposit term is 12 months. Which bank do you prefer?

The choice between commercial banks will depend on the accrual rate.

The compound interest accrual coefficient is determined by the formula:

where n is the interest calculation period,

i - nominal interest rate.

Preference for Bank 1.

13. Compare the conditions of four banks: a) simple interest and an interest rate of 48%; b) nominal interest rate - 46% per annum, interest is accrued semi-annually; c) nominal interest rate - 45%, quarterly accrual of interest; d) nominal interest rate -44%, monthly interest accrual

To determine the most profitable option, it is necessary to compare the proposed conditions (all calculations are carried out for a period of 1 year).

a) simple interest and an interest rate of 48%.

The coefficient of accrual of simple interest: .

b) nominal interest rate - 46% per annum, interest is accrued semi-annually.

c) nominal interest rate - 45%, interest accrual quarterly.

Compound interest rate:

d) nominal interest rate -44%, monthly interest accrual.

Compound interest rate:

Table 3 compares the conditions for the depositor, the borrower and the bank (creditor).

Table 3

14. The client placed a deposit of 100 thousand rubles. on a term deposit for a period of 8 months. Interest is calculated monthly, at a nominal interest rate of 36% per annum. Determine the accumulated amount and the effective interest rate

The accumulated deposit amount is determined by the compound interest formula:

S 0 - initial deposit amount;

n - interest calculation period;

i - nominal interest rate.

15. The enterprise received a loan for 3 years at a nominal interest rate of 40% per annum. The commission is 5% of the loan amount. Determine the effective interest rate when calculating interest: a) once a year, b) quarterly, c) monthly

The effective rate is determined by equating future values ​​excluding and including commissions:

where m is the number of interest accruals;

n - loan term (in years);

S - the amount of the loan;

Nominal interest rate of the bank;

The amount to pay the commission to the bank.

where h - bank commission.

The effective rate is calculated using the formula:

• - once a year: ;
• - quarterly: ;

• - monthly: .
• 16. The company received a loan for 3 years at an annual interest rate of 48%. The commission is 5% of the loan amount. Determine the effective interest rate of the loan if: a) the loan was received under simple interest, b) a loan received at compound interest with interest accrued once a year, c) with monthly interest accrual

a) a loan received at simple interest

b) a loan received at compound interest with interest accrued once a year:

c) the loan was received at compound interest with monthly interest accrual:

17. The company received a loan of 40 thousand rubles. for one month at an annual interest rate of 12%. Interest is simple. The monthly inflation rate is 5.9%. Determine the inflation-adjusted monthly interest rate, accrued amount and interest money

The interest rate of the bank per month is:

Bank interest rate per month adjusted for inflation:

where i p - the real rate of the bank, taking into account inflation;

i - nominal rate of the bank;

n - number of years;

p is the rate of inflation.

The accrued loan amount is determined by the simple interest formula:

deposit credit bank income

18. The company applied to the bank for a loan of 100 thousand rubles. for a period of one month. The Bank provides such loans at a simple annual interest rate of 24%, excluding inflation. Monthly inflation rates for the previous three months: 1.8%; 2.4; 2.6%. The loan was allocated taking into account the average inflation rate for the three indicated months. Determine the bank's interest rate adjusted for inflation, return amount, bank discount

Inflation rate for three months:

Average inflation rate per month:

Accrued refund amount:

Interest payments will amount to: rub.

19. The bank issued a loan to the client for 3 months. Loan amount - 24 thousand rubles. The bank requires the real rate of return to be 12% per annum. The projected average monthly inflation rate is 3.6%. Determine the simple interest rate of the bank, the accumulated amount

Inflation rate for the year:

The inflation rate will be: or 53%.

Interest rate of the loan adjusted for inflation:

r - real rate of return;

p is the rate of inflation.

Accrued refund amount:

20. The firm took a loan from a commercial bank for two months at an interest rate of 30% per annum (excluding inflation). The estimated average monthly inflation rate is 2%. Determine the interest rate of the loan, taking into account inflation and the accrual rate

Inflation rate for the year:

Loan interest rate (Fischer formula):

Compound interest rate:

Simple Interest Accrual Factor:

21. A loan of 500 thousand rubles, received for a period of one year at a nominal interest rate of 18% per annum. Interest calculation monthly. The expected average monthly inflation rate is 3%. Determine the bank's interest rate adjusted for inflation and the accrued amount

The annual inflation rate is calculated using the formula:

Determine the bank's interest rate adjusted for inflation:

Accrued amount:

22. Monthly inflation rates are expected at 3%. Determine the true interest rate of return on an annual deposit if banks accept deposits at nominal interest rates of 40%, 50%, 60%. Interest is compound and accrues monthly.

Inflation rate for the year:

or 42.58% per year

True interest rate:

where i - nominal interest rate;

True interest rate;

inflation rate;

True interest rate for a nominal interest rate of 40%:

True interest rate for a nominal interest rate of 50%:

23. The average monthly inflation rate from January to June 1997 is 5.9%. What should be the bank's annual interest rate on deposits in order to ensure a real return on deposits of 12% per annum. Interest is compound and accrued monthly

The nominal interest rate on the deposit is determined by the formula:

where i - nominal interest rate;

r is the real return on the deposit;

inflation rate.

24. The commercial bank accepted deposits from the public in the first half of 1997 at an interest rate of 54% per annum. Interest is calculated monthly. The average monthly inflation rate is 5.9%. Determine the real interest rate of return

The real interest rate of return is determined by the formula:

where i - nominal interest rate;

r - real profitability of the deposit;

inflation rate.

There is a depreciation of the contribution by 14.77%.

25. Commercial banks accept deposits from the population "on demand" at 60% per annum with monthly interest capitalization. Determine the true interest rate of the bank, taking into account inflation, the accumulated amount and the client's profitability from a deposit of 3 thousand rubles. after 1 year if the average inflation rate is 3.5%.

Inflation rate for the year:

or 51.11% per year

True interest rate:

where i - nominal interest rate;

True interest rate;

inflation rate;

m - the number of interest accruals.

True interest rate for a nominal interest rate of 60%:

The accrued amount of the deposit with monthly interest capitalization is determined by the formula:

where S n - deposit amount at the end of the period;

S 0 - initial deposit amount;

n - interest calculation period;

true interest rate.

The investor's income at the end of the term will be:

where I n - income of the investor for the period n;

n - term of the deposit (in years).

26. Calculate NPV for an investment project with the following cash flow for a comparison rate of 15% per annum.

Table 3

Solution:

The net present value of an investment project is determined by the formula:

where CF t -- cash inflow (outflow) for period t;

r -- comparison rate;

n -- project life cycle.

Table 4 shows the calculations performed in Microsoft Excel.

Table 4

		discount coefficient	present value of the flow

The NPV value for the investment project was negative. So the project should be rejected.

27. Find the internal rate of return (IRR) for an investment project with the following regular cash flow (-200, -150, 50, 100, 150, 200, 200)

The IRR is the discount rate at which the project's NPV is zero.

Table 5 shows the calculations performed in Microsoft Excel.

Table 5

	Costs I

The internal rate of return is 19%.

28. Compare investment projects (-50, -50, -45, 65, 85, 85, 20, 20) and (-60, -70, -50, -40, 110, 110, 110, 110) if annual the interest rate is: a) 10% per annum; b) 15% per annum; c) 20% per annum.

The presented investment projects characterize a typical investment flow, in which negative payments precede positive ones.

Table 6 shows the calculations performed in Microsoft Excel.

Investment flow (-50, -50, -45, 65, 85, 85, 20, 20)

Table 6

		discount coefficient	present value of the flow		discount coefficient	present value of the flow		discount coefficient	present value of the flow

Table 7 shows the calculations performed in Microsoft Excel.

Investment flow (-60, -70, -50, -40, 110, 110, 110, 110)

Table 7

		discount coefficient	present value of the flow		discount coefficient	present value of the flow		discount coefficient	present value of the flow

At a rate of 10%, the most effective is investment project(-60, -70, -50, -40, 110, 110, 110, 110) NPV=66.96 PI=0.34, payback period is 2.91

At a rate of 15%, the most effective is an investment project (-50, -50, -45, 65, 85, 85, 20, 20), because NPV=22.26, PI=0.17, payback period is 5.73

At a rate of 20%, the most effective is an investment project (-50, -50, -45, 65, 85, 85, 20, 20), because NPV=2.13, PI=0.02, payback period 57.71.

Bibliography

• 1. Tasks in financial mathematics: textbook / P.N. Brusov, P.P. Brusov, N.P. Orekhov, S.V. Skorodulina - M.: KNORUS, 2016 - 286 p.
• 2. Katargin N.V. Methods of financial calculations: Texts of lectures / N.V. Katargin - M.: Financial University, Department of System Analysis and Modeling of Economic Processes, 2016. - 124 p.
• 3. Kuznetsov S.B. Financial mathematics: textbook / S.B. Kuznetsov; RANEPA, Sib. Institute of Management - Novosibirsk: SibAGS Publishing House - 2014 - 263p.
• 4. Pechenezhskaya I.A. Financial mathematics: a collection of problems / I.A. Pechenezhskaya - Rostov n / a: Phoenix, 2010 - 188 p.
• 5. Financial mathematics: textbook /P.N. Brusov, P.P. Brusov, N.P. Orekhov, S.V. Skorodulina - M.: KNORUS, 2012 - 224 p.