Introduction. 6
One-time payments .. 7
1.1 BASIC CONCEPTS .. 7
1.2 EASY INTEREST ... 8
1.3 COMPLEX INTEREST ... 10
1.3.1 Compound Interest Formula. ten
1.3.2 Determination of the future amount .. 10
1.3.3 Determination of the present value. Discounting. eleven
1.3.4 Determination of the term of the loan (deposit) 12
1.3.5 Sizing interest rate. 12
1.3.6 Rated and effective rate. 13
1.4 CALCULATION OF TAXES AND INTEREST ... 14
1.5 PERCENTAGE AND INFLATION .. 15
1.5.1 Basic concepts. 15
1.5.2 Accounting for inflation. 16
Tasks. eighteen
Chapter 2.20
PERMANENT REGULAR PAYMENT FLOWS .. 20
2.1 BASIC CONCEPTS .. 20
2.2 FUTURE AMOUNT OF PRESUMERANDO AND POSTNUMERANDO WITHOUT INITIAL AMOUNT ... 21
2.2.1 Pre-numberando rent. 21
2.2.2 Post-numerando rent. 21
2.3 EQUIVALENCE EQUATION IN GENERAL FORM .. 23
2.3.1 Determining the Future Amount .. 23
2.3.2 Determination of the current amount .. 24
2.3.3 Definition of recurring payments. 24
2.3.4 Calculation of the term of the annuity. 25
2.3.5 Determination of the size of the interest rate. 25
2.4 SOLVING FINANCIAL CHALLENGES WITH FINANCIAL FUNCTIONS Excel 26
2.4.2 Calling financial functions. 26
2.4.3 Calculation of the future value. 26
2.4.4 Calculation of the running amount .. 27
2.4.5 Definition of recurring payments. 27
2.4.6 Calculation of the term of the annuity .. 28
2.4.7 Determining the size of the interest rate. 28
2.5 SELECTING A LOAN BANK AND DRAWING UP A LOAN REPAYMENT PLAN 29
2.5.1 Problem statement. 29
2.5.2 Choosing a credit bank. 29
2.5.3 Loan repayment plan. thirty
2.6 PAYMENTS p ONCE A YEAR, AND CALCULATION OF INTEREST m ONCE A YEAR .. 32
2.7 CHOICE OF A MORTGAGE LOAN ... 34
Tasks. 36
Chapter 3.39
TOTAL PAYMENT FLOW .. 39
3.1 EVALUATION OF THE EFFECTIVENESS OF INVESTMENT PROJECTS .. 39
3.2 REGULAR NON-REGULAR PAYMENTS .. 39
3.2.1 Problem statement. 39
3.2.2 Accumulated amount of non-permanent annuity. 39
3.2.3 The discounted amount of non-permanent annuity .. 40
3.2.4 Internal rate of return. 41
3.2.5 Discount payback period of the investment project. 42
3.2.7 Comparison of the effectiveness of the two investment projects for payments m times a year 43
3.3 IRREGULAR AND IRREGULAR FLOWS ... 46
The amount of payments reduced to the moment t 0 46
3.4 FUTURE VALUE AT FLOATING PERCENTAGE RATE .. 47
Tasks. 48
Chapter 4.40
OPERATIONS WITH VEXELS .. 50
4.1 BASIC CONCEPTS ... 50
4.2 DISCOUNTING AT SIMPLE ACCOUNTING RATE .. 50
4.3 ACCOUNTING VEKSELS AT A COMPLEX RATE .. 52
4.4 VEXELS AND INFLATION .. 53
4.4.1 Simple discount rate and inflation. 53
4.4.2 Compound discount rate and inflation. 54
4.5 COMBINING VEXELS .. 55
4.5.1 Determination of the value of the combined bill. 55
4.5.2 Determining the maturity of the combined vector. 56
4.5.3 Consolidation of promissory notes for inflation. 57
4.6 EFFICIENCY OF TRANSACTIONS WITH VEXELS .. 58
4.6.1 The effectiveness of transactions at simple interest .. 58
4.6.2 Effectiveness of deals on compound interest .. 59
Tasks. 60
Chapter 5.62
DAMPING OF FIXED ASSETS AND INTANGIBLE ASSETS .. 62
5.1 BASIC CONCEPTS .. 62
5.2 LINEAR CUSHIONING METHOD ... 62
5.3 NONLINEAR, GEOMETRICALLY-DEGRESSIVE METHOD OF ACCOUNTING FOR DAMPING 64
5.4 Excel FUNCTIONS FOR CALCULATION OF CUSHIONING ... 65
5.4.1 Linear method accounting for depreciation. AMP functions. 65
5.4.2 Decreasing residue method (geometrically - degressive method). DDOB 66 function
5.5 COMPARISON OF THE LINEAR METHOD OF ACCOUNTING DEPRECIATION WITH THE METHOD OF REDUCING RESIDUE (Calculation in Excel) 66
Tasks. 68
Chapter 6 69
LEASING. 69
6.1 BASIC CONCEPTS .. 69
6.1.1 Financial (capital) leasing. 70
6.1.2 Operative leasing. 70
6.2 PAYMENT SCHEME UNDER A LEASING CONTRACT .. 70
6.3 CALCULATION OF LEASING PAYMENTS UNDER THE FIRST SCHEME .. 71
6.3.1 Lease payments under the linear law of depreciation. 71
6.3.2 Lease payments from accelerated depreciation(diminishing balance method) 73
6.4 CALCULATION OF LEASING PAYMENTS UNDER THE SECOND SCHEME. 74
Hence, the income of the leasing company. 75
6.5 CALCULATION OF LEASING PAYMENTS UNDER THE SECOND SCHEME C HELP Excel 76
6.6 DETERMINING THE FINANCIAL PERFORMANCE OF LEASING OPERATIONS .. 77
Tasks. 77
References .. 79
Introduction
Financial mathematics is the basis for banking operations and commercial transactions. The proposed guide deals with the calculation of simple and compound interest in one-time payments and payment streams, with constant and variable annuities and rates. It sets out a unified approach to solving a wide range of problems of determining various financial values: the future amount of the transaction, the current (discounted) amount, interest rate, payments, term of the transaction, its effectiveness, etc. The effect of inflation on the parameters of financial transactions is taken into account. Formulas of financial mathematics are used in the manual for calculating credit, deposit, mortgage transactions, bills accounting, to compare the effectiveness of financial transactions. To make leasing transactions clear, the guide outlines various methods for accounting for depreciation.
To study the manual, knowledge of school mathematics is enough. The output of all formulas is given.
By their nature, financial formulas, especially for non-constant and non-uniform payments, are cumbersome, which complicates direct calculations on them. Values such as the interest rate or the term of a financial transaction are generally not explicitly expressed. To determine them, it is necessary to solve a nonlinear equation, for example, by the iteration method.
Excel has built-in financial functions, allowing you to easily calculate all financial values in many practical cases using a personal computer. Therefore, the tutorial details the methods of using Excel to solve financial tasks... The author strongly recommends that students master these methods in order to further apply them in their practice to analyze the effectiveness of financial transactions and the work of their company.
The manual contains a large number of examples, many of which are of independent cognitive value. In order to consolidate theoretical knowledge at the end of each chapter, tasks are given for independent study.
The financial mathematics manual is intended for part-time students of distance education, but it can also be recommended for full-time students in financial and economic specialties. The manual is of practical interest for bank employees, financial companies, industrial enterprises and commercial structures.
The terminology adopted in the manual may seem unusual for economists brought up on the books of E. M. Chetyrkin and his followers. For example, the interest rate is denoted by the letter i (interest). However, in mathematics, the letter i is usually used to denote integer values. Therefore, in the manual "Financial Mathematics" introduced the designations used in Excel and in.
Chapter 1
One-time payments
BASIC CONCEPTS
All financial calculations are based on the principle of the temporary value of money ... Money is a measure of the value of goods and services. Purchasing power money falls as inflation rises. It means that sums of money received today (denote them PV-present value- present, current value), more, more valuable than the same amounts received in the future. In order for money to retain or even increase its value, it is necessary to provide an investment of money that brings a certain income. It is customary to denote income by a letter I(interest), in financial and household jargon, it is called interest.
There are many ways to nest ( investments ) of money.
You can open an account in savings bank but the percentage must exceed the inflation rate. You can borrow money in the form of a loan for the purpose of obtaining in the future, the so-called, accrued amount FV(future value - future value). And you can invest in production.
The simplest financial transaction is a one-time grant or receipt of the PV amount with the condition of a refund over time. t the accumulated (future) amount of FV. The amount received by the debtor (for example, we are with you or the company) will be considered positive, and the amount that the creditor gives (again, we are with you or the bank) - negative.
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The effectiveness of such an operation is characterized by the growth rate Money, attitude r(rate-ratio) of income I to the basic value of PV, taken in absolute value.
. (1.1)
Capital growth rate r during t expressed as a decimal fraction or as a percentage and called interest rate , rate of return or cash turnover rate During this time.
Since PV and FV have opposite signs, the present and future values are related by the relation (let's call it the equivalence equation)
FV + PV (1 + r) = 0, (1.2)
where r is the interest rate over time t.
The value of K, showing how many times the future amount has increased in absolute value in relation to the current
K = FV / PV = (1 + r), (1.3)
are called capital growth ratio .
In calculations, as a rule, for r accept annual interest rate , they call her nominal rate.
There are two schemes for increasing capital:
· Simple interest scheme;
· Compound interest scheme.
SIMPLE INTEREST
Simple interest scheme assumes the invariability of the amount on which interest is accrued... Simple interest is used in short-term financial transactions(with a maturity less than the interest accrual period) or when interest is paid periodically and is not added to the capital stock.
Consider two types of deposits: standby and time-based.
1) By simple contribution(money for such a deposit can be withdrawn at any time) for t days will be credited
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FV + PV (1+ r) = 0 (1.4)
where T is the number of days in a year. The build-up ratio is
Depending on the determination of T and t, the following techniques are used.
1. Exact percentages ... In Russia, the USA, Great Britain and in many other countries, it is customary to consider T = 365 in a regular year and T = 366 in a leap year, and t is the number of days between the date of issue (receipt) of the loan and the date of its repayment. The date of issue and the date of redemption are counted as one day.
2. Banking method ... In this method, t is defined as the exact number of days, and the number of days in a year is taken as 360. The method is beneficial for banks, especially when issuing loans for more than 360 days, and is widely used by commercial banks.
3. Ordinary interest with approximate number of days ... In some countries, for example, France, Belgium, Switzerland, T = 360 is taken, and t is approximate, since it is considered that there are 30 days in a month.
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2) By term deposit
(money is deposited in the bank for a certain period: six months, a year, or another) interest is calculated after certain periods. We denote
m is the number of periods in a year.
m = 12 - with monthly interest;
m = 4 - with a quarterly charge;
m = 2 - when charged once every six months;
m = 1 - when charged once a year.
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FV + PV (1+) = 0 (1.5)
Build ratio
Determine the accrued amount
By formulas (1.2) - (1.5), one can solve inverse problem: what the initial amount of PV should be lent or deposited in the bank in order to receive the amount FV at the end of the term at a given annual interest rate r.
Receivers of receipts estimate their income as a total value for the full term of the payment, of course, taking into account the temporary disparity of money.
Accumulated amount- the sum of all payments with interest accrued on them by the end of the annuity period. This can be the generalized amount of debt, the total amount of investments, etc.
The logic of the financial operation of increasing financial rent
Accumulated individual payments are members of an exponential progression with the first member being equal R and a factor equal to (1 + i).
Let us consider the definition of the accrued amount using the example of the simplest case, the annual constant ordinary rent:
where FVA- the accrued rent amount;
R- the size of the rent term, i.e. the size of the next payment;
i- the annual interest rate at which compound interest is accrued on payments;
n- the term of the rent in years,
sn; i- the coefficient of rent increase.
Example. For five years, at the end of each year, an amount of 500 rubles will be deposited into a bank account, on which interest will be accrued at a rate of 30%. Determine the amount of interest that the bank will pay to the account holder.
Solution:
Since the annuity period is equal to one year, then this annual rent; interest is calculated once a year; contributions will be at the end of the annuity period, post-numerando, so this common rent; the payment amount is constant throughout the entire period of the annuity, which is typical for permanent rent; the number of members of the rent is five, i.e. of course, therefore limited rent; and the payments are unconditional, so this faithful rent.
The sum of all contributions with accrued interest will be equal to:
Calculation of the current value of the constant annual annuity POSTNUMERANDO when calculating% once a year.
In addition to the accumulated amount, the current value is a generalizing characteristic of the flow of payments. Current (current) value of the flow of payments(capitalized or discounted amount) is the amount of payments discounted at the commencement of the annuity at the rate of accrued compound interest. This is the most important characteristic of financial analysis, since is the basis for measuring the effectiveness of various financial and credit operations, comparing the terms of contracts, etc. This characteristic shows what amount should have been initially, so that, by dividing it into equal installments, on which the established interest would be charged throughout the entire period, it would be possible to obtain the specified accrued amount.
The logic of the financial transaction for determining the current value of the flow of payments
In this case, a discounting scheme is implemented: all elements are reduced to one point in time using discount multipliers, which allows them to be summed up.
In the simplest case, for an annual ordinary annuity with payments at the end of each year, when the moment of assessment coincides with the beginning of annuity, the present value of the financial annuity is:
Fraction in the formula - rent reduction factor (an; i), the values of which are tabulated for a wide range of values, since they depend on the interest rate ( i) and on the number of years ( n) (Appendix 5).
Example. Determine the current value of the rent based on the example data.
Solution:
The current value of the rent will be:
Thus, all payments made in the future are currently estimated at 1 "217.78 rubles.
16. Calculation of the accumulated amount of constantp- urgent rent POSTNUMERANDO when calculating%monce a year (p= m)
There are cases when rental payments are made several times a year in equal amounts (term rent), and interest is accrued only once a year. Then the accrued amount of rent will be determined by the formula:
It is also not uncommon for rental payments to be made several times a year and interest is also calculated several times a year, but the number of rental payments is not equal to the number of interest calculation periods, i.e. p ≠ m... Then the formula by which you can determine the accrued amount of financial rent will take the form:
In practice, the post-numerando flow has become more widespread, since according to general principles accounting, it is customary to summarize and evaluate the financial result of an operation or other action at the end of the next reporting period. As for the receipt of funds as payment, in practice they are most often distributed unevenly over time and therefore, for convenience, all receipts are attributed to the end of the period, which allows the use of formalized assessment algorithms.
The flow of prenumerando is important when analyzing various schemes for accumulating funds for their subsequent investment.
A prenumerando annuity differs from a regular annuity in the number of interest periods. Therefore, the accrued sum of the prenumerando rent will be greater than the accrued sum of the regular rent in (1 + i) once.
For a prenumerando annual annuity with interest accrued once a year, the formula will take the form:
For a prenumerando annual annuity with interest accrued several times a year:
Calculation of the current value of the constant p-term rent POSTNUMERANDO when calculating% m once a year (p = m).
Consider the calculation of the current value of rent for its various types:
annual rent with interest accrued several times a year:
term annuity when calculating interest once a year:
term rent with repeated accrual of interest throughout the year, provided that the number of payments is not equal to the number of accruals, i.e. p ≠ m :
17. Determination of the size of the next payment of permanent financial annuity POSTNUMERANDO (p= m=1)
Consecutive payments in the form of constant regular annual annuity are determined by the main parameters:
R- the amount of the payment;
n- the term of the rent in years;
i- annual interest rate.
However, when developing the conditions for a financial transaction, situations may arise when a given value is one of two generalizing characteristics and an incomplete set of rent parameters. In such cases, the missing parameter is found.
In determining annuity member two options are possible, depending on which value is the initial one:
a) accrued amount... If the amount of debt is determined at some point in the future ( FVA), then the amount of subsequent contributions during n years when interest is accrued on them at rate i can be determined by the formula:
Example. To buy a car in 5 years, 50 thousand rubles will be required. Determine the amount of annual contributions paid at the end of each year to the bank, which charges interest at a rate of 40%.
Solution:
In this case, the accrued amount of constant financial rent is known, so the amount of annual contributions will be equal to:
Thus, in order to accumulate the necessary amount on the account for buying a car, you should set aside 4 "568 rubles at the end of each year for five years.
b) the modern value of the financial rent, then, based on the interest rate and the term of the rent, a one-time payment is found by the formula:
Example. The amount of 10 thousand dollars was lent for 5 years at 8% per annum. Determine the annual amount of debt repayment.
Solution:
The current amount of debt is known, hence:
Thus, it will be necessary to return the amount of 2,504.56 rubles annually.
You can check: the amount of debt with interest accrued on it by the end of the fifth year will be:
FV= 10 "000 (1 + 0.08) 5 = 14" 693.28 rubles.
The accrued amount for a stream of payments in the size of 2 "504.56 rubles. Will be:
Consequently, the value of the member of the financial rent is determined correctly. The slight discrepancy is due to rounding of calculations.
The present value of the prenumerando rent is calculated by multiplying the present value of the ordinary rent by the corresponding increment factor.
Accumulated amount formulas
Consider the build-up for various cases of rent accrual.
1. Ordinary annual annuity.
May at the end of each year during NS years, the current account is paid byRrubles, interest is charged once a year at the ratei. In this case, the first installment by the end of the annuity term will increase to the value as in the amount R interest accrued during ( n - 1) of the year. The second installment will increase to, etc. No interest is charged on the last installment.
Thus, at the end of the annuity period, its accrued 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 NS. This amount is
(1)
where
(2)
called rate of increase of rent. It only depends on the term of the rent NS and interest rate leveli.
Accrued amount of rent prenumerando in (1 + i) times more postnumerando and at m =p = 1
(3)
Example 1.
To create a pension fund, a postnumerando annuity in the amount of 10 million rubles is paid annually to the bank. Interest is charged on incoming payments at a complex annual rate of 18%. Determine the size of the fund in 6 years.
Solution.
By formula (1) we have:
million rubles
Answer. Pension Fund in 6 years it will be 99.42 million rubles.
2. Annual rent, 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 the rate is applied every timej/ m, where j - nominal interest rate. Then the members of the rent with the interest accrued to the end of the term have the form
If we read the previous line from right to left, then we get a geometric progression, the first term of which R, denominator (1+ j/ m) m, number of members NS. The sum of the members of this progression will be the accrued sum of rent. She is equal
(4)
The accrued amount of rent prenumerando is calculated by the formula
(5)
Example 2.
Under 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 lender benefits from example 2.2, so that interest is accrued on the rent on a quarterly basis, while the size of the fund will be 97.45 million rubles.
3. Rentp - urgent,m = 1.
Find the accrued amount, provided that the rent is paid R once a year in equal installments, and interest is calculated once at the end of the year.
If R - the annual amount of payments, then the amount of a separate payment isR/ p. Then the sequence of payments with interest accrued to 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 NS. Then the accrued sum of the considered rent is equal to the sum of the members of this geometric progression
(6)
where
(7)
p-term rent increase rate at m = 1.
The accrued amount of pre-numberand rent is calculated by the formula:
(8)
Example 3.
Mr. Ivanov contributes 500 rubles to the bank at the end of each month. Compound interest is charged on the received payments at an annual interest rate of 22%. Determine the amount of the accrued amount in 8 years.
Solution.
Using formula (6), we find the size 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, interest accrual and payment receipt often coincide in time. Thus, the number of payments R per year and the number of interest accruals T coincide, i.e. p = t... Then, to obtain the formula for calculating the accrued amount, we will use the analogy with the annual rent and one-time interest accrual 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 accrued amount of pre-numberand rent is calculated by the formula:
(10)
Example 4.
Mr. Petrov must repay a debt in the amount of 200 thousand rubles. In order to collect this amount, he plans to deposit the same amount into the bank at the end of every six months within 3 years, and compound interest is charged 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 to the bank once at the end of each year and interest is accrued 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 compounding interest:
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 deposited into the bank every year with the annual compounding 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 pay the bank every six months and half-year compounding of interest an amount equal to 55.228 thousand rubles. and the amount of 57.692 thousand rubles. with an annual contribution and an annual compounding interest. The first option is more profitable for him.
5. Rent R- urgent, p ³ 1 , m ³ 1.
This is the most common case R-time annuity with interest T once a year, and perhaps R ¹ T.
First member of the annuityR/ p, paid later 1 / p year after the beginning, by the end of the term, together with the interest accrued on it
Membership NS p. As a result, we get the accrued amount
(11)
The accrued amount of prenumerando rent is determined by the formula:
(12)
Example 5.
The company 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 company's insurance fund in 10 years will amount to 7790.86 thousand rubles.
. The basis for calculating compound interest, unlike simple interest, does not remain constant noah - it increases with each step in time. The absolute amount of interest charged increases and the process the increase in the amount of debt is accelerating. Compound interest build-up can be thought of as a follower reinvestment of funds invested in simple businesscents for one accrual period ( running period ). Jointhe increase of accrued interest to the amount that served as the basis for their accrual is often called capitalization of interest.
Let's find a formula for calculating the accrued amount under the condition vii that interest is accrued and capitalized once inyear (annual interest). To do this, apply difficult becoming kabuild-up. To write the buildup formula, we use thosethe same designations as in the formula for building up by simple cents:
P - the original amount of debt (loans, credit, capital la, etc.),
S - the accrued amount at the end of the loan term,
NS - term, number of years of increase,
i - the level of the annual interest rate represented by defractional fraction.
Obviously, at the end of the first year, the percentages are equal to R i , and the accrued amount will be. To the endin the second year it will reach the value V end n th year the accrued amount will be is equal to
(4.1)
The percentages for the same period are generally as follows:
(4.2)
Some of them are taught through the accrual of interest on interest. She makes
(4.3)
As shown above, growth in compound interest isis a process corresponding to geometric progress this, the first term of which is R , and the denominator is.The last term of the progression is equal to the accumulated amount at the end loan term.
The quantity are called build-up multiplier on compound interest. The values of thisfactor for integers NS are given in tables of complex percent.The accuracy of calculating the multiplier in practical calculationsis determined by the permissible degree of rounding of the accruedamounts (up to the last kopeck, ruble, etc.).
The time for building up at a complex rate usually measures Xia as AST / A ST.
As you can see, the magnitude of the build-up factor depends on two parameters - iand NS. It should be noted that for a long timebuild-up, even a small change in the rate significantly affectsby the value of the multiplier. In turn, a very long periodleads to frightening results even with smallinterest rate.
The formula for building up compound interest is obtainedfor the annual interest rate and the term, measured in years.However, it can also be applied for other periods of accrual.niya. In these casesimeans the rate for one accrual period (month, quarter, etc.), and n - the number of such periods. On example if i- rate for half a year, then NS – number of semesters etc.
Formulas (4.1) - (4.3) assume that the interest on thecents are charged at the same rate as when charged on the principal amount of the debt. Let's complicate the conditions for calculating interestComrade Let interest on the principal debt be calculated at the rateiand interest on interest - at the rate In this case
The series in square brackets represents a geometricprogression with the first term equal to 1 and the denominator. As a result, we have
(4.4)
· Example 4.1
2. Accrual of interest in adjacent calendar periods. You Moreover, when calculating interest, the location of the interest calculation period relative to calendar periods was not taken into account. However, often the start and end dates of a loan are in two periods. It is clear that the accrued for the entire period, interest cannot be attributed only to the lastto him period. In accounting, in taxation,finally, in the analysis of the financial activities of the company the task of distributing the accrued interest over periods is eliminated.
The total loan term is divided into two periodsn 1 and n 2 . Respectively ,
where
· Example 4.2
3. Variable rates. The formula assumes constantrate throughout the entire period of interest accrual. The instability of the monetary market makes it necessary to modernize the “classical” scheme, for example, using the example neniya floating rates ( floating rate). Naturally, the calculationfor the future at such rates is very conditional. It's a different matter -calculation after the fact. In this case, as well as when cheatingthe size of the bets are fixed in the contract, the total The increment is defined as the product of quotients, i.e.
(4.5)
where - consecutive values of the rates; - the periods during which the corresponding rates.
· Example 4.3
4. Accrual of interest with a fractional number of years. Often a term in th qax for interest calculation is not an integer. In the rules of a number of commercial banks for some transactions interest is calculated only for a whole number of years or other periods of calculation. The fractional part of the period is discarded. In most cases, the full term is taken into account. Whereintwo methods are used. According to the first, let's call it general, the calculation is carried out according to the formula:
(4.6)
Second, sm shany,the method assumes the accrual of interest for the wholethe number of years according to the formula of compound interest and for a fractional part term using the simple interest formula:
,(4.7)
where - loan term, a- an integer number of years,b - fractional part of the year.
A similar method is used in cases where the periodhouse accrual is half a year, quarter or month.
When choosing a calculation method, it should be borne in mind that manythe resident of the building up according to the mixed method turns out to be somewhat more than according to the general one, since for NS
<
1 fairin relation 
The largest difference is observed given when b = 1/2.
· Example 4.4
5. Comparison of growth in compound and simple interest. Let the time base for accrual be the same, the level of interest rates is the same, then:
1) for a term of less than a year, simple interest is more complex
2) for more than a year
3) for a period of 1 year, the multipliers of the build-up are equal to each other
Using the increment ratio for simple compound interest, you can determine the time required to increase the initial amount in n once. For this, it is necessary that the growth rates were equal to n:
1) for simple interest
2) for compound interest
The formulas for doubling the capital are as follows:
The accrued amount of debt (loans, deposits, etc.) is understood as the initial amount with accrued interest at the end of the term. The accrued amount is determined by multiplying the original amount by the accrual factor, which shows how many times the accrued amount is greater than the original:
V credit agreements sometimes interest rates vary over time - “floating” rates. If these are simple bets, then the amount accumulated at the end of the term is determined from the expression
Example. Loan agreement provides the following procedure for calculating interest: the first year - the rate is 16%, in each subsequent half of the year the rate is increased by 1%. It is necessary to determine the build-up multiplier for 2.5 years:
V practical tasks sometimes there is a need to solve secondary problems - to determine the term of the increase or the size of the interest rate in one form or another, with all other given conditions.
The length of the build-up period in years or days can be determined by solving the equation:
Example. Let us determine the duration of the loan in days, so that the debt equal to 1 million rubles would increase to 1.2 million rubles, provided that simple interest is charged at a rate of 25% per annum (K = 365 days).
The value of the interest rate can be determined in a similar way. Such a need for calculating the interest rate arises when determining the yield borrowing operation and when comparing contracts by their yield in cases where interest rates are not explicitly indicated. Similarly to the first case, we obtain
Example. The loan agreement provides for the repayment of an obligation in the amount of RUB 110 million. after 120 days. The initial amount of the debt was RUB 90 million. It is necessary to determine the profitability of the loan operation for the lender in the form of the annual interest rate. We get
In the case of using "floating" compound interest rates, the accrued amount is calculated using the formula
Since the accumulation multiplier for simple and complex bets is different, the following pattern is observed.
If the extension period is less than a year, then
Interest can be accrued (capitalized) not once, but several times a year - by half-year, quarter, month and
This situation is graphically shown in Fig.
etc. Since contracts, as a rule, stipulate an annual rate, the formula for building up compound interest is as follows:
Example. The initial amount is 1 million rubles. placed on deposit for 5 years at compound interest at an annual rate of 20%. Interest is charged on a quarterly basis. Let's calculate the accrued amount:
Obviously, the more often the interest is charged, the faster the build-up process goes.
When developing the terms of credit transactions using compound interest, it is often necessary to solve the opposite problem - calculating the duration of a loan or credit (the term of the increase) or the interest rate.
When increasing at a complex annual rate and at a nominal rate, we get
Example. Let us determine for what period (in years) the amount equal to 75 million rubles will reach 200 million when interest is charged at a complex rate of 15% once a year and quarterly:
The value of the interest rate when increasing on compound interest will be determined by the equations
Example. The bill was bought for 100 thousand rubles, the redemption amount is 300 thousand rubles, the term is 2.5 years. Determine the level of profitability. We get
Example. Let us determine the number of years required to increase initial capital 5 times, applying simple and compound interest at a rate of 15% per annum: for simple interest we get
More on the topic 4.3. Accrued amount:
- Section 1 "The amount of tax (the amount of advance tax payment) payable to the budget according to the taxpayer's data"
