6.2 Application of limits in economic calculations
Compound interest
In practical calculations, discrete percentages are mainly used, i.e. interest accrued for fixed equal time intervals (year, half year, quarter, etc.). Time is a discrete variable. In some cases, in proofs and calculations related to continuous processes, it becomes necessary to use continuous percentages. Consider the compound interest formula:
S = P(1 + i) n . (6.16)
Here P is the initial amount, i is the interest rate (as a decimal fraction), S is the amount formed by the end of the loan term at the end of the nth year. Compound interest growth is a process that develops exponentially. The addition of accrued interest to the amount that served as the basis for their determination is often called interest capitalization. In financial practice, one often encounters a problem that is the opposite of determining the accumulated amount: for a given amount S, which should be paid after some time n, it is necessary to determine the amount of the loan received P. In this case, they say that the amount S is discounted, and the interest in the form of the difference S - P are called the discount. The value of P, found by discounting S, is called the modern, or reduced, value of S. We have:
P = z P = = 0.
Thus, with very long payment terms, the present value of the latter will be extremely insignificant.
In practical financial and credit operations, continuous processes of accruing money, that is, accruing over infinitely small periods of time, are rarely used. Continuous growth is of much greater importance in the quantitative financial and economic analysis of complex industrial and economic objects and phenomena, for example, in the selection and justification of investment decisions. The need to use continuous accruals (or continuous percentages) is determined primarily by the fact that many economic phenomena are continuous in nature, therefore, an analytical description in the form of continuous processes is more adequate than based on discrete ones. Let's generalize the compound interest formula for the case when interest is accrued m times a year:
S = P (1 + i/m) mn .
The accumulated amount in discrete processes is found by this formula, here m is the number of accrual periods in a year, i is the annual or nominal rate. The larger m, the shorter the time intervals between the moments of interest calculation. In the limit as m ®¥ we have:
`S = P (1 + i/m) mn = P ((1 + i/m) m) n .
Since (1 + i/m) m = e i , then `S = P e in .
With a continuous increase in interest, a special type of interest rate is used - the force of growth, which characterizes the relative increase in the accumulated amount in an infinitely small period of time. With continuous capitalization of interest, the accrued amount is equal to the final amount, which depends on the initial amount, the accrual period and the nominal interest rate. In order to distinguish between continuous interest rates and discrete interest rates, we denote the former by d, then `S = Pe .
The growth force d is the nominal interest rate at m®¥. The multiplier is calculated using a computer or according to function tables.
Payment streams. financial rent
Contracts, transactions, commercial and production and business operations often provide not for separate one-time payments, but for many payments and receipts distributed over time. Individual elements of such a series, and sometimes the series of payments as a whole, is called a stream of payments. Payment stream members can be either positive (receipts) or negative (payments) values. The flow of payments, all members of which are positive values, and the time intervals between two successive payments are constant, is called financial rent. Annuities are divided into annual and p-urgent, where p characterizes the number of payments during the year. These are discrete rents. In financial and economic practice, there are also sequences of payments that are made so often that in practice they can be considered as continuous. Such payments are described by continuous annuities.
Example 3.13. Suppose that at the end of each year for four years, 1 million rubles are deposited in the bank, interest is accrued at the end of the year, the rate is 5% per annum. In this case, the first installment will turn to the amount of 10 6 ´ 1.05 3 by the end of the annuity period, since the corresponding amount has been on the account for 3 years, the second installment will increase to 10 6 ´ 1.05 2, since it has been on the account for 2 years . The last installment does not pay interest. Thus, at the end of the annuity period, contributions with accrued interest represent a series of numbers: 10 6 ´ 1.05 3 ; 10 6 ´ 1.05 2 ; 10 6 ´ 1.05; 10 6. The value accumulated by the end of the annuity period will be equal to the sum of the members of this series. To summarize what has been said, we derive the corresponding formula for the accumulated amount of the annual annuity. Let's designate: S - the accumulated amount of the annuity, R - the size of the annuity term, i - the interest rate (decimal fraction), n - the annuity term (number of years). The annuity members will bear interest for n - 1, n - 2,..., 2, 1 and 0 years, and the accumulated value of the annuity members will be
R (1 + i) n - 1 , R (1 + i) n - 2 ,..., R (1 + i), R.
Let's rewrite this series in reverse order. It is a geometric progression with the denominator (1+i) and the first term R. Let's find the sum of the terms of the progression. We get: S = R´((1 + i) n - 1)/((1 + i) - 1) = R´((1 + i) n - 1)/ i. Denote S n; i =((1 + i) n - 1)/ i and we will call it the rent accumulation factor. If interest is calculated m times a year, then S = R´((1 + i/m) mn - 1)/((1 + i/m) m - 1), where i is the nominal interest rate.
The value a n; i =(1 - (1 + i) - n)/ i is called the annuity reduction coefficient. The annuity reduction coefficient at n ®¥ shows how many times the present value of the annuity is greater than its term:
An; i \u003d (1 - (1 + i) - n) / i \u003d 1 / i.
Example 3.14. Perpetual annuity is understood as a sequence of payments, the number of members of which is not limited - it is paid for an infinite number of years. Perpetual annuity is not a pure abstraction - in practice it is some type of bonded loans, an assessment of the ability pension funds meet its obligations. Based on the essence of perpetual annuity, we can assume that its accumulated amount is equal to an infinitely large value, which is easy to prove by the formula: R´((1 + i) n - 1)/ i ® ¥ as n ® ¥.
Reduction coefficient for perpetual annuity a n; i ® 1/i, whence A = R/i, i.e. the present value depends only on the value of the annuity term and the accepted interest rate.
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The issues of calculation and forecasting of financial and economic indicators are becoming increasingly important. In modern conditions, financial mathematical models are an integral and very important part of statistical analysis in order to develop and make decisions.
In financial and economic calculations, cash flows (the amount of money) are always associated with specific time intervals. In this regard, in financial transactions (agreements, contracts), fixed terms, dates, frequency of payments (or receipt of Money). In financial mathematics, the time factor is taken into account by calculating (applying) an interest rate that takes into account the intensity of interest (interest money). The interest rate is the ratio of the amount of interest money paid for a strictly fixed period of time to the amount of credit, loans, etc. The time interval to which the interest rate is timed is called the accrual (accumulation) period.
Interest rates may apply to the same initial amount throughout the life of the loan, loan. This kind of interest is called simple interest rates. In this case, the distribution of the accumulation amount is described by a uniform linear distribution law, and the accumulation process itself can be expressed as an arithmetic profession:
FV=PV( 1 +n * i) or FV=PV + I,
where FV - accrued amount;
PV - current (initial) amount;
n is the number of accrual periods;
i - interest rate;
i= PV * n * i - interest income for the entire period.
In some cases, it is possible to apply discrete time-varying interest rates. For example, the rate of simple interest in the first year is 10%, in the second - 15%, in the third - 20%.
When the periods of accrual (for example, by years) are equal, then the accrual formula for simple interest is: FV=PV (1+n-i) m ,
where m is the total number of reinvestment operations.
In domestic practice, as a rule, they do not distinguish between the concepts of loan (credit) interest and the discount rate. Usually, a collective term is used - the interest rate. At the same time, the term discount rate is found in relation to the refinancing rate of the Central Bank of the Russian Federation, as well as to bill transactions.
It should be emphasized that in most cases interest is calculated at the end of each period (interval) of accrual. This method of determining and calculating interest is called the decursive method. In some cases, in accordance with the concluded agreements, an antisipative (preliminary) method is used, i.e. interest is calculated at the beginning of each accrual period.
In financial calculations, the most common tasks are to determine the accumulated amount of FV for a given (initial) value present value loan (credit) PV, as well as the current amount (received) PV for a given accumulated amount FV. The first type of tasks is called compounding (accumulation process), the second type of tasks is called discounting. The difference in the values of the current value PV of the accumulated amount FV is called the discount D k , i.e. D K = FV - PV.
Simple interest can be exact, when the year is taken to be equal to its actual length in days, or ordinary, when the length of the year is taken to be 360 days. The accepted number of days in a year is called the time base.
There are also concepts such as commercial (or bank) accounting, accounting of bills, discounting at a discount rate (for simple interest). In the practice of financial and credit relations, simple discount rates are used when accounting for bills of exchange and other monetary obligations. Depending on the form of representation of capital and the method of paying income, securities are divided into two groups: debt (coupon bonds, certificates, bills of exchange - having a fixed interest rate) and equity (shares), representing the holder's share in real ownership and providing dividends in unlimited time . All other types valuable papers are derivatives of debt and equity: these are options, futures contracts, privatization checks.
In order to avoid mistakes and losses in the context of inflation (decrease in the purchasing power of money), it is necessary to take into account the mechanism of the influence of inflation on the result of financial transactions. When calculating, use relative value inflation rate, i.e. inflation rate α : α=(PV α – PV)/PV or α= PV/PV*100
where α - inflation rate;
PV α - the amount reflecting the actual purchasing power (the actual cost of goods over a period of time /);
PV - amount in the absence of inflation;
РV= PV α - PV - the amount of inflationary money.
The essence of simple interest is in that they are charged on the same amount of capital during the entire term of the loan (credit).
In the practice of conducting financial calculations, the date of issue and the date of repayment of the loan are always considered to be one day. In this case, one of two options is used
1)exact percentage obtained when the time base is taken as the actual number of days in a year (365 or 366) and the exact number of loan days:
where Nd is the duration of accrual in years;
D is the duration of the accrual period in days;
K is the length of the year in days.
The exact number of loan days D is determined by a special table, which shows the serial numbers of each day of the year (the number of the first day is subtracted from the number corresponding to the day the loan (loan) ends);
2)ordinary interest is obtained when the approximate number of days of the loan is applied, and the length of the full month is assumed to be 30 days. This method is used when redeeming bonds (loans). The accumulated amount FV in these cases is determined from the expression
Let's determine the interest rate, taking into account inflation Iα, according to the formula of I. Fischer.
Bespalova Ekaterina
The content of the work corresponds to the stated topic and is presented in accordance with a well-planned plan. In the "Introduction" section, the topic, goals and objectives of the work are defined, as well as research methods are listed. The goals and objectives of the work are quite competently and convincingly confirmed by the materials of the work. The authors successfully used such methods as analysis, synthesis, comparison. The materials of the work indicate that the researchers carefully studied the theoretical material on this topic, carried out calculations and made their own conclusions. The applied value of this topic is very high and affects the financial, economic, demographic and other spheres of our life. An understanding of percentages and the ability to perform percentage calculations and calculations is necessary for every person, since we encounter percentages in Everyday life. In the theoretical part design work everything you need to know about simple and compound interest is presented: formulas, explanations and calculations using these formulas. A good addition to the work is the research part, which is devoted to comparative analysis compound and simple interest, which shows the suitability of compound interest in banking system. The student independently conducted a study on deposits individuals in various banks, making a reasonable conclusion that compound interest plays a large role in the economy and the banking system. The material may be useful to teachers of mathematics, economics, students of educational organizations.
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State budgetary vocational educational institution of the Republic of Khakassia "Tekhnikum public utilities and service"
Project theme:
« The use of compound interest in economic calculations
Scientific adviser: Cherdyntseva L.A.
Student: Bespalova Ekaterina Andreevna
Group: TT-11
Abakan, 2016
Introduction
Every day we do the same thing - we live, work, eat and sleep, for us it is everyday life. We don't even notice that many terms are related to everyday life. For example, the economy is part of everyday life. People take part in daily economic activity live in an economic environment. In turn, no economy can do without interest. Interest is all around us.
But interest appeared in ancient times among the Babylonians. Cash settlements with interest were common in ancient Rome. The Romans called interest the money that the debtor paid to the lender for every hundred. From the Romans, interest passed to other peoples.
At present, interest is applied in all economic spheres activities: in enterprises, in statistics, in the banking system, etc. We will show our work on the example of banks.
Why banks? Banks are in the center economic life serve the interests of manufacturers by linking cash flow industry and trade, Agriculture and population. All over the world, banks have significant power and influence, they manage the huge money capital flowing to them from enterprises and firms, from merchants and farmers, from the state and private individuals.
Why does a person bring his savings to the bank? Of course, to ensure their safety, and most importantly - to receive income. And here, knowledge of the formula for simple or compound interest, as well as the ability to make a preliminary calculation of interest on a deposit, will come in handy more than ever. After all, forecasting interest on deposits or interest on loans is one of the components of a reasonable management of your finances.
This is the relevance of the topic.
Objective:
Study of simple and compound interest in economic calculations.
Tasks:
Compare simple and compound interest on deposits of individuals.
Compare income on deposits of individuals using compound interest formulas depending on the time period.
Conduct an analysis of income on deposits of individuals in various banks.
Interest
Interest is the amount paid for the use of money.
Interest is divided into simple and compound
1) Simple interest - interest that is charged on the initial amount.
S - the amount of funds due to be returned to the depositor at the end of the deposit (ie deposit).
I - annual interest rate
t - the number of days of accrual of interest on the attracted deposit
K - number of days in a calendar year (365 or 366)
P - the initial amount of funds attracted to the deposit
We came up with a problem so that you can see how simple interest is applied in bank calculations.
Task 1.
The bank made a contribution in the amount of 100,000 rubles, and after 5 years the account had 168,000 rubles. Determine the bank's interest rate using simple interest.
Solution:
I= (168000-100000)*(365*100%)/100000*1825=13.6%
Answer: 13.6% rate.
2) Compound interest - interest earned on accrued interest.
I - annual interest rate;
j - quantity calendar days in the period following the results of which the bank capitalizes accrued interest;
K is the number of days in a calendar year (365 or 366);
P is the initial amount of funds attracted to the deposit;
n - the number of operations for capitalization of accrued interest during general term attracting funds;
S - the amount of funds due to be returned to the depositor at the end of the deposit term. It consists of the amount of the deposit plus interest.
And now we will solve the problem in the same way, but with compound interest
Task 2.
The bank made a deposit of 100,000 rubles. at 13.6%, for 5 years. Interest is charged once a year. How much money will the depositor withdraw from the account at the end of 5 years?
Solution:
S= 100000* (1+ (13.6%*365)/ 365*100%) 5 \u003d 100000 * 1, 1365 \u003d 189187, 2 rubles.
Answer: 189187.2 rubles.
Let's compare simple and compound interest to understand the difference between them:
Task 3. A deposit of 100,000 rubles was made to the bank. at 12% for 10 years. Determine how much money will be through each year, using simple and compound interest.
In the table we see that it is more profitable to use compound interest:
Graph of capital growth using simple and compound interest:
And now let's compare the compound interest on the deposit, depending on the time period.
Task 4. A deposit of 100,000 rubles was made to the bank. for 1 year at an interest rate of 12% per annum. Compare the amounts that will be due back to the depositor when interest is calculated: daily, weekly, monthly, quarterly, semiannually and annually.
In the table, we see that the more often the interest calculation interval, the more income we receive.
Studying simple and compound interest, we analyzed in which bank it is better to invest money at the moment and why.
We took three banks as a basis - these are Binbank, Alfa-Bank and VTB 24.
VTB 24 - Profitable deposit
Alfa-Bank - Pobeda deposit
Binbank - deposit "Maximum income"
Task 5. We have 500,000 rubles. and choose which bank to put this amount to get the highest income for 1 year.
At the moment, it is best to make a deposit in Alfa-Bank
Output:
Conducted a study of simple and compound interest in economic calculations.
We compared simple and compound interest on deposits of individuals.
We compared the income on deposits of individuals using compound interest formulas depending on the time period.
Conducted an analysis of income on deposits of individuals in various banks
. REFERENCES AND INTERNET RESOURCES
1. Chetyrkin, E. M. Financial mathematics / E. M. Chetyrkin,
textbook. - 6th ed., corrected. - M.: Delo, 2006. - 399 p.2. Samarov, K. L. Financial Mathematics: Prakt. course: study guide / K. L Samarov. - M.: Alfa-M; INFRA-M, 2006. - 78 p.
3. Financial mathematics: a textbook for universities / P. P. Bocharov. - 2nd ed. - M.: Fizmatlit, 2005. - 574 p.
4 Financial mathematics: textbook.-method. complex / S. G. Valeev. - Ulyanovsk: UlGTU, 2005. - 106 p.
5. Financial mathematics. V. Malykhin: http://www.finansmat.ru/.
6. Financial mathematics. A. Fedorov (lectures): http://wdw2005.narod.ru/FM_lec.htm#_Toc179997391.
7. Mathematical Bureau: http://www.matburo.ru/index.php.
8. Financial mathematics (lectures):
http://treadwelltechnologies.com/index.html.
9. The financial analysis: http://www.finances-analysis.ru/financial-maths/.
10. Knowledge - to the masses: http://www.finmath.ru/.
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2 slide
INTRODUCTION 1. Relevance 2. History of origin. 3. Origin of the designation. 4. Set rules. 5. Comparison of values in percent 6. Types of percent. 7. Factors taken into account in financial and economic calculations. 8. Conclusion.
3 slide
Modern life makes interest tasks relevant, as the scope of practical application of interest calculations is expanding. Relevance.
4 slide
The word "percent" comes from the Latin word pro centum, which literally translates as "per hundred", or "from a hundred". Percentages are very convenient to use in practice, since they express parts of whole numbers in the same hundredths. History of origin.
5 slide
The % sign is due to a typo. In manuscripts, pro centum was often replaced by the word "cento" (one hundred) and abbreviated - cto. In 1685, a book was printed in Paris - a guide to commercial arithmetic, where by mistake the typesetter instead of cto scored%. The origin of the designation.
6 slide
In the text, the percent sign is used only for numbers in digital form, from which, when typing, it is separated by a non-breaking space (67% income), except when the percent sign is used for abbreviated notation compound words, formed using the numeral and adjective percent. Set rules.
7 slide
Sometimes it is convenient to compare two quantities not by the difference between their values, but by percentage. Percent comparison
8 slide
Distinguish between simple and compound interest. When using simple interest, interest is charged on the initial amount of the deposit (loan) throughout the entire period of accrual. Interest types
9 slide
Methods of financial mathematics are used in calculating the parameters, characteristics and properties of investment operations and strategies, parameters of state and non-state loans, loans, credits, in calculating depreciation, insurance premiums and premiums, pension accruals and payments, in drawing up debt repayment plans, assessing the profitability of financial transactions . Factors taken into account in financial and economic calculations.
It is a well-known situation that the same amount of money is not equivalent in different periods of time. Accounting for the time factor in financial transactions carried out through the calculation of interest.
Interest money (interest) is the amount of income from lending money in any form (loans, opening deposit accounts, buying bonds, renting equipment, etc.).
The amount of interest money depends on the amount of debt, the term of its payment and the interest rate that characterizes the intensity
interest calculation. The amount of debt with accrued interest is called the accrued amount. The ratio of the accrued amount to the initial amount of debt is called the accrual multiplier (coefficient). The time interval for which interest is calculated is called the accrual period.
When using simple interest rates, the amount of interest money is determined based on the initial amount of debt, regardless of the number of accrual periods and their duration according to the formula:
The above formula is used to determine the value of the accumulated cost of capital for short-term financial investments.
If the term of the debt is given in days, the following expression must be inserted into the above formula:
where 5 is the duration of the accrual period in days;
The number of days in a year can be taken exactly - 365 or 366 (exact interest) or approximately - 360 days (ordinary interest). The number of days in each whole month during the term of the debt can also be taken exactly or approximately (30 days). In the world banking practice usage:
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approximate number of days in each whole month and ordinary interest is called "German practice";
the exact number of days in each month and ordinary interest - "French practice";
the exact number of days and exact percentages - "English practice".
Depending on the use of a particular practice of calculating interest, their amount will vary.
Consider examples of financial and economic calculations for securities.
Example 7.1.
Savings certificate with a face value of 200 thousand rubles. issued on 20.01.2005 due on 05.10.2005 at 7.5% per annum.
Determine the amount of accrued interest and the redemption price of the certificate when using various ways interest calculation.
Let's determine the exact and approximate number of days until the certificate is redeemed.
tT04H = 11 days of January + 28 days of February + 31 days of March + 30 days of April + 31 days of May + 30 days of June + 31 days of July + 31 days of August + 30 days of September + 5 days of October = 258 days.
Iapprox \u003d 11 days of January + 30 x 8 days (February - September) + 5 days of October \u003d 256 days.
According to certificates, income is accrued according to interest rate. There are three ways to calculate interest:
1) exact interest, loan term - the exact number of days:
Іfinal \u003d 0.075 x 200 x 258/365 \u003d 10.6 thousand rubles; certificate redemption price:
51 \u003d 200 + 10.6 \u003d 210.6 thousand rubles;
2) ordinary interest, loan term - the exact number of days, the redemption price of the certificate:
52 \u003d 200 + 10.8 \u003d 210.8 thousand rubles;
3) ordinary interest, loan term - an approximate number
Іbіkn = 0.075 х 200 х 256/360 = 10.7 thousand rubles, certificate redemption price:
53 \u003d 200 + 10.7 \u003d 210.7 thousand rubles.
Example 7.2.
The Bank accepts deposits for 3 months at a rate of 4% per annum, for 6 months at a rate of 10% per annum and for a year at a rate of 12% per annum. Determine the amount that the owner of the deposit will receive 50 thousand rubles. in all three cases.
The amount of the deposit with interest will be:
1) for a period of 3 months:
S \u003d 50 x (1 + 0.25 x 0.04) \u003d 50.5 thousand rubles;
2) for a period of 6 months:
S \u003d 50 x (1 + 0.5 x 0.1) \u003d 52.5 thousand rubles;
3) for a period of 1 year:
S \u003d 50 x (1 + 1 x 0.12) \u003d 56 thousand rubles.
When deciding on the placement of funds in a bank, an important factor is the ratio of the interest rate and the inflation rate. The inflation rate shows how many percent prices have increased over the period under review, and is defined as:
The inflation index shows how many times prices have risen over the period under review. The inflation rate and the inflation index for the same period are related by the ratio:
where Ju is the inflation index for the period;
N is the number of periods during the period under consideration.
The inflation rate for the period.
Example 7.3.
Determine the expected annual inflation rate at a monthly inflation rate of 6% and 12%.
Ju = (1 + 0.06)12 = 2.01.
Therefore, the expected annual inflation rate will be = 2.01 - 1 = 1.01, or 101%.
Ju = (1 + 0.12)12 = 3.9.
The expected inflation rate will be:
3.9 - 1 = 2.9, or 290%.
Inflation affects the profitability of financial transactions.
The real value of the accumulated amount with interest for the deadline, given by the time the money is loaned, will be:
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Example 7.4.
The bank accepts deposits for six months at a rate of 9% per annum. Determine the real results of the deposit operation for a deposit of 1000 thousand rubles. with a monthly inflation rate of 8%.
The amount of the deposit with interest will be:
S \u003d 1 x (1 + 0.5 x 0.09) \u003d 1045 thousand rubles.
The inflation index for the term of the deposit is equal to:
Ju = (1 + 0.08)6 = 1.59.
The accumulated amount, taking into account inflation, will correspond to the amount:
1045 / 1.59 \u003d 657 thousand rubles.
When using compound interest rates, interest accrued after the first accrual period, which is part of the total debt term, is added to the debt amount. In the second accrual period, interest will accrue based on the original amount of the debt, increased by the amount of interest accrued after the first accrual period, and so on for each subsequent accrual period. If compound interest is calculated at a constant rate and all accrual periods have the same duration, then the accrued amount will be equal to:
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where P is the initial amount of the debt;
in - interest rate in the accrual period;
n is the number of accrual periods during the term.
Example 7.5.
Deposit 50 thousand rubles. deposited in the bank for 3 years with compound interest at the rate of 8% per annum. Determine the amount of accrued interest.
The amount of the deposit with accrued interest will be equal to:
S \u003d 50 x (1 + 0.08) 3 \u003d 63 thousand rubles.
The amount of accrued interest will be:
I \u003d S - P \u003d 63 - 50 \u003d 13 thousand rubles.
If interest were accrued at a simple rate of 8% per annum, their amount would be:
I \u003d 3 x 0.08 x 50 \u003d 12 thousand rubles.
Thus, the calculation of interest at a compound rate gives a large amount of interest money.
Compound interest can be compounded several times a year. At the same time, the annual interest rate, on the basis of which the amount of interest in each accrual period is determined, is called
nominal annual interest rate. With a debt term of n years and compound interest accrual m times a year, the total number of accrual periods will be equal to:
The accumulated amount will be equal to:
1) term of debt:
Example 7.6.
The depositor makes a deposit of 40 thousand rubles. for 2 years at a nominal rate of 40% per annum with monthly accrual and interest capitalization. Determine the accumulated amount and the amount of accrued interest.
The number of accrual periods is equal to:
Therefore, the accumulated amount will be:
A bill of exchange or other monetary obligation before the maturity date on it can be bought by the bank at a price less than the amount that must be paid on them at the end of the term, or, as they say, discounted by the bank. In this case, the bearer of the obligation receives money earlier than the period specified in it, minus income
bank in the form of a discount. The Bank, upon the due date of payment of a bill or other obligation, receives the full amount indicated in it.
If the period from the moment of accounting to the moment of repayment of the obligation will be some part of the year, the discount will be equal.