Simple interest calculations. Determine the accrued amount Accrued debt
Calculations using compound interest assume that interest accrued on the original amount is added to this amount, and interest accrual in subsequent periods is made on the already accrued amount. The amount received as a result of the accumulation of interest is called the accrued, or future value of the deposit amount after the expiration of the period for which the calculation is carried out. The initial amount of the deposit is also called the present value.
The mechanism for increasing the initial amount (capital) for compound interest is also called capitalization.
The calculation of the accrued amount for compound interest is carried out using the formula:
n
FV = PV * (1 + i) , (1.1)
- FV- the increased (future) amount;
- PV- the initial (current) amount on which the interest is charged;
- i - rate compound interest as a decimal:
- P - the number of years during which interest is calculated.
Example # 1. The bank's client made a term deposit of 30 thousand rubles at 10% per annum. Interest is accrued once a year. Determine the amount of the accrued amount in four years.
FV = PV * (1 + i) = 30,000 * (1 + 0, 1) = 30,000 * 1, 4641 = 43923 p.
In accordance with the agreement between the client and the bank, interest can be accrued much more often than once a year - half-yearly, quarterly, monthly, ten-day or even daily. In these cases, to determine the accrued amount, you can use the accrual formula (1.1), where the value P would mean total number periods of interest, and the rate i- interest rate but already for corresponding period(half year, quarter, month, etc.).
In most cases, not a quarterly or monthly rate is indicated, but an annual rate, also called nominal. In addition, the number of periods is indicated. (m) accruals per year. In this case, the formula can be used to calculate the accrued amount:
n * m
FV = PV * (1 + j/ m) , (1.2)
- J - nominal interest rate;
- m - the number of periods of interest accrual per year;
- n - number of years.
Example # 2. The bank's client made a term deposit of 30 thousand rubles for three years at a nominal rate of 10% per annum. Interest is accrued on a quarterly basis. Determine the amount of the accrued amount.
n * m 3 * 4
FV = PV * (1 + j/ m) = 30,000 * (1 + 0, 1/4) = 30,000 * 1, 3449 = 40347 p.
When solving such problems, the question may arise: what annual interest rate must be set to get the same financial results, as in the case of m- one-time (monthly or quarterly) interest accrual per year at a rate j/ m.
In this regard, in addition to the nominal rate, there is the concept of the effective, or real, interest rate ie, which is determined by the formula:
m
ie= (1 + j/ m) - 1 , (1.3)
ie - effective compound interest rate.
Example No. 3. The client contacted the bank regarding the placement of his own free resources, after which he became aware that the bank, when using a nominal rate of 10%, makes a quarterly interest calculation. What is the effective compound interest rate, assuming the same accrued amount is received as when using the nominal rate j = 10%?
ie= (1 + j/ m) - 1 = (1 + 0,1 / 4) - 1 = 1.1038 – 1 = 0,1038 (10,38%)
Capitalization process invested funds- a powerful means of maintaining and increasing the real cost of factors of production. To illustrate this statement, you can use the mnemonic rule of magnitude 70 (in some publications of financial and economic orientation it is marked as the “Rule of 72 - x”), which allows you to approximately determine the period of doubling (only doubling) the initial amount at given interest rates.
n = 70 /i , (1.4)
-i - compound interest rate (in%)
- n - period (for the interest rate under the given conditions).
Note. The rule of magnitude 70 is recommended for bets in the range of 3 - 17 %%. The magnitude rule of 70 can also be used to estimate inflation.
Example No. 4. The owner of the money capital placed in the bank wants to know how many years it will take to double the capital with the accrued annual interest rate at a rate of 10%?
n = 70 /i = 70/10 = 7 years.
Inverse problems can also be solved using the rule of magnitude 70. So for a given period, which corresponds to doubling the capital, you can calculate the required level of interest rate.
Task number 1
BUT) In order to ensure mutual interests, the bank and the production and commercial company agreed to establish a minimum balance on the current account for a period of ____ year (s) in the amount of ____ thousand rubles. At the same time, the bank undertakes to periodically charge ______ percent per annum. Define:
Accumulated amount;
Effective compound interest rate.
The values of the quantities in table 1.1.
Table 1.1
Initial data
B) Define:
Average annual rate price increases;
Average annual price index,
if for ____ year (s) prices have doubled.
The values of the quantities in table 1.2.
Table 1.2
Initial data
Financial math homework test
1. Determine the accrued deposit amount of 3 thousand rubles. with a deposit term of 2 years at a nominal interest rate of 40% per annum. Interest is accrued: a) once a year, b) half-yearly, c) quarterly, d) monthly
The accrued 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).
Banks' accepted interest rate for the accrual interval.
a) once a year:
(thousand roubles.)
b) half-yearly
- (thousand roubles.)
- c) quarterly,
- (thousand roubles.)
- d) monthly.
- (thousand roubles.)
- 2. The bank accepts deposits from the population at a nominal interest rate of 12% per annum. Interest accrual monthly. The $ 1200 deposit was withdrawn after 102 days. Determine the client's 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 rate of increase, the amount of accrued interest and the cost of the loan at the end of each year
Simple interest:
The ratio of simple interest accrual is determined by the formula:
Where
where S 0 is the loan amount;
n - interest accrual period;
i is the nominal interest rate.
S 0 - loan amount;
n - interest accrual period;
i is the nominal interest rate.
Table 1 shows data on the value of the accrual factor, the amount of interest and the cost of the loan at the end of each year (calculations were carried out in Microsoft Excel - Appendix A, task 3).
Table 1. Estimated data of the rate of increase, the amount of interest and the cost of the loan.
build-up ratio |
loan cost, $ |
percentage, $ |
|
Compound interest:
The build-up ratio is determined by the formula:
i is the nominal interest rate.
The interest amount is calculated using the formula:
where S is the loan amount;
n - interest accrual period;
i is the nominal interest rate.
Loan cost at the end of the period:
where S n is the cost of the loan (accrued cost);
S 0 - loan amount;
n - interest accrual period;
i is the nominal interest rate.
Table 2 shows data on the value of the accrual ratio, the amount of interest and the cost of the loan at the end of each year (calculations were carried out in Microsoft Excel).
Table 2. Estimated data of the rate of increase, the amount of interest and the cost of the loan.
build-up ratio |
loan cost, $ |
percentage, $ |
|
4. The firm is provided preferential loan$ 50 thousand for 3 years at 12% per annum. Loan interest is calculated once a year. Under the terms of the agreement, the company has the right to pay the loan and interest in a single payment at the end of the three-year period. How much should a firm pay when calculating simple and compound interest?
Simple interest:
The amount of simple interest is calculated using the formula:
where S is the loan amount;
n - interest accrual period;
i is the nominal interest rate.
The loan amount will be:
The amount of accrued compound interest is calculated using the formula:
where S is the loan amount,
n - interest accrual period,
i is the nominal interest rate.
The loan amount will be:
5. The industrial and commercial company received a loan of 900 thousand rubles. for a period of 3 years. Interest is complex. The interest rate for the first year is 40% and each subsequent year is increased by 5%. Determine the loan repayment amount
The loan repayment amount is determined by the formula:
where S n is the amount of the loan repayment at the end of the period;
S 0 - loan amount;
n - interest accrual period;
i is the nominal interest rate.
By condition, the interest rate grows by 5%:
The loan repayment amount for the 3rd year will be:
6. Determine the period of time required to double the capital at simple and compound interest at an interest rate of 12% per annum. In the latter case, interest is charged monthly
"Rule 70" and "Rule 100" allow you to answer the question of how many years capital will double at the 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 to triple the capital at simple and compound interest at an interest rate of 48% per annum. In the latter case, interest is accrued quarterly
Simple interest when trebling capital:
Compound interest when trebling capital:
8. How long does it take to keep a deposit in the bank at 84% per annum with monthly, quarterly and semi-annual interest accruals to double the deposit amount. Banking calculation method
Compound interest ("rule 70"):
where T is the period for which the capital will double;
m is the frequency of interest accrual;
i - interest rate.
- - monthly accrual: years.
- - Quarterly accrual: years.
- - semi-annual accrual: years.
- 9. The client made a deposit for a period of 4 months $ 1600. Interest accrual monthly. After the end of the term, he received $ 1,732. Determine the bank's interest rate
To determine the bank's interest rate, the accrual formula is applied Money by the method of compound interest:
j is the actual number of interest accrual periods;
n - term of the deposit (in years);
S0 - the amount of the deposit at the moment of opening the deposit;
Bank interest rate.
From here, the bank's interest rate is calculated using 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 calculating interest: 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 moment of opening the deposit;
Sm - the amount of the deposit at the moment of opening the deposit;
Bank interest rate.
a) quarterly interest accrual:
b) monthly accrual of interest:
11. "Priorbank" offered the population a monetary deposit for 1996. The income on it amounted to 72% per annum for the first 2 months, 84% for the next 2 months, 96% for 5 months, and 108% per annum for 6 months. Determine the effective interest rate when placing money for 6 months at the indicated simple and compound interest. In the latter case, interest is charged monthly
The effective interest rate is the rate that reflects the actual income from the commercial transaction).
The effective interest rate, calculated using 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 using compound interest, is determined by the formula:
where m is the number of interest accruals;
n - term of the deposit (in years).
12. Advertising of one commercial bank offers 84% per annum with monthly interest. Another commercial bank offers 88% per annum at quarterly interest rates. Deposit storage period is 12 months. Which bank to give preference to?
The choice between commercial banks will depend on the rate of increase.
The compound interest accrual ratio is determined by the formula:
where n is the interest accrual period,
i is the nominal interest rate.
Bank preference 1.
13. Compare the conditions of the four banks: a) simple interest and an interest rate of 48%; b) nominal interest rate - 46% per annum, interest is accrued on half-yearly basis; c) nominal interest rate - 45%, interest accrual on a quarterly basis; d) nominal interest rate -44%, interest accrual monthly
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) interest is simple and the interest rate is 48%.
Simple interest accrual ratio:.
b) nominal interest rate - 46% per annum, interest is accrued on half-yearly basis.
c) nominal interest rate - 45%, interest accrual on a quarterly basis.
Compound interest accrual ratio:
d) nominal interest rate -44%, interest accrual monthly.
Compound interest accrual ratio:
Table 3 compares the conditions for the depositor, the borrower and the bank (lender).
Table 3
14. The client has placed a deposit of 100 thousand rubles. for a term deposit for a period of 8 months. Accrual of interest monthly, at a nominal interest rate of 36% per annum. Determine Accrued Amount and Effective Interest Rate
The accrued deposit amount is determined by the compound interest formula:
S 0 - the initial amount of the deposit;
n - interest accrual period;
i is the nominal interest rate.
15. The company 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 the future values excluding and taking into account commissions:
where m is the number of interest accruals;
n - loan term (in years);
S is the amount of the loan;
Bank nominal interest rate;
The amount of the commission paid to the bank.
where h is the bank's commission.
The effective rate is calculated using the formula:
- - once a year: ;
- - by quarter:;
- - 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 at simple interest, b) the loan was received at compound interest with interest accrued once a year, c) with monthly interest
a) the loan was received at simple interest
b) the loan was received at compound interest with interest accrued once a year:
c) the loan was received at compound interest with monthly interest accrual:
17. The firm received a loan of 40 thousand rubles. for one month at an annual interest rate of 12%. The interest is simple. The monthly inflation rate is 5.9%. Determine the monthly interest rate adjusted for inflation, accrued amount and interest money
The bank's interest rate per month is:
Bank interest rate per month, taking into account inflation:
where i p is the bank's real rate, taking into account inflation;
i is the nominal rate of the bank;
n is the number of years;
p - inflation rate.
The accrued loan amount is determined using the simple interest formula:
deposit credit bank income
18. The firm 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 months indicated. Determine the bank's interest rate, taking into account inflation, the return amount, the bank's discount
Inflation rate for three months:
Average inflation rate per month:
Accumulated refund amount:
Interest payments will be: 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 bank's simple interest rate, the accrued amount
Inflation rate for the year:
The inflation rate will be: or 53%.
Loan interest rate adjusted for inflation:
r is the real rate of return;
p - inflation rate.
Accumulated refund amount:
20. The firm took out 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 rate of increase
Inflation rate for the year:
Loan interest rate (Fisher's formula):
Compound interest accrual ratio:
Simple interest accrual ratio:
21. A loan of 500 thousand rubles, received for a period of one year at a nominal interest rate of 18% per annum. Interest accrual monthly. The expected average monthly inflation rate is 3%. Determine the bank's interest rate adjusted for inflation and the accrued amount
The inflation rate for the year is calculated using the formula:
Let's determine the bank's interest rate, taking into account inflation:
Accrued amount:
22. Monthly inflation rates are expected at 3%. Determine the true interest rate of the annual deposit if banks accept deposits at nominal interest rates of 40%, 50%, 60%. Interest is complex and is charged monthly.
Inflation rate for the year:
or 42.58% per year
True Interest Rate:
where i is the 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. Average monthly inflation rate from January to June 1997 - 5.9%. What should be the bank's annual interest rate on deposits to ensure a real return on deposits of 12% per annum. Interest is complex and is charged monthly
The nominal interest rate on the deposit is determined by the formula:
where i is the nominal interest rate;
r is the real profitability of the deposit;
Inflation rate.
24. The commercial bank accepted deposits from the population in the first half of 1997 at an interest rate of 54% per annum. Interest is calculated monthly. 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 is the nominal interest rate;
r is the real profitability of the deposit;
Inflation rate.
Depreciation of the deposit takes place by 14.77%.
25. Commercial banks accept deposits from the population "on demand" at 60% per annum with monthly capitalization of interest. Determine the true interest rate of the bank taking into account inflation, the accrued 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 is the nominal interest rate;
True interest rate;
Inflation rate;
m is the number of interest accruals.
True interest rate for a nominal interest rate of 60%:
The accrued amount of the deposit with monthly capitalization of interest is determined by the formula:
where S n - the amount of the deposit at the end of the period;
S 0 - the initial amount of the deposit;
n - interest accrual period;
True interest rate.
The depositor's income by the end of the term will be:
where I n is the depositor's income 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
Decision:
The net present value of an investment project is determined by the formula:
where CF t - cash inflow (outflow) for the period t;
r is the comparison rate;
n is the life cycle of the project.
Table 4 shows the calculations performed in Microsoft Excel.
Table 4
discount coefficient |
present value of the stream |
||
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)
Internal norm IRR yield is the discount rate at which the NPV of the project is zero.
Table 5 shows the calculations performed in Microsoft Excel.
Table 5
Cost 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 the 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, with negative payments preceding 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 stream |
discount coefficient |
present value of the stream |
discount coefficient |
present value of the stream |
|||||
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 stream |
discount coefficient |
present value of the stream |
discount coefficient |
present value of the stream |
|||||
At a rate of 10%, the most effective is an investment project (-60, -70, -50, -40, 110, 110, 110, 110), since 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), since 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), since NPV = 2.13, PI = 0.02, payback period 57.71.
Bibliography
- 1. Tasks in financial mathematics: tutorial/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 economic processes", 2016. - 124 p.
- 3. Kuznetsov S.B. Financial mathematics: textbook / S.B. Kuznetsov; RANEPA, Sib. Institute of Management - Novosibirsk: Publishing House of SibAGS - 2014 - 263p.
- 4. Pechenezhskaya I.A. Financial mathematics: 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.
Under accrued amount debt (loan, deposit, etc.) understand the original 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:
where Y is the accrued amount, rubles;
R- the initial amount, rubles; q is the build up multiplier.
Simple and compound interest will be different.
Accrual multiplier simple
q = (l + nxi),
and the accrued amount - according to the formula
P (1 + P x /),
Where P - extension period, period;
/" - interest rate.
If the interest rate annual, and interest is paid during the year, it is necessary to determine what part of the annual interest is paid to the lender for the period. For this, the term of the buildup is calculated by the formula
Where? - the number of days after which interest is calculated and paid;
TO- the number of days in a year.
Example. Credit in the amount of 1 million rubles. issued on January 20 to October 5 inclusive (for 258 days) at 18% per annum. The interest is simple. In this case, the accrued amount will be
IN 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. The loan agreement provides next order accrual of 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:
In practical tasks, sometimes it becomes necessary 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:
Px (1 + "xr).
From this equation we obtain
The term in days will be calculated using the formula
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 profitability of a borrowing operation and when comparing contracts by their profitability 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 the 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
Accrual multiplier complex percent is calculated by the formula
I = 0 + 0",
and the accrued amount - according to the formula The percentages (Y) will be equal to:
In the case of using "floating" compound interest rates, the accrued amount is calculated using the formula
where is the value of the rate for the period n.
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
if the extension period is more than a year, then
(1 + t)
This situation is graphically shown in Fig. 4.1.
Fig.
Interest can be accrued (capitalized) not once, but several times a year - by half-year, quarter, month, etc. Since contracts, as a rule, stipulate an annual rate, the formula for building up compound interest is as follows:
Where } - nominal annual rate;
t- the number of periods of interest accrual per year.
Example. Initial amount in 1 RUB million 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 conditions credit operations With the use of compound interest, it is often necessary to solve the opposite problem - to calculate 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
If it is necessary to determine the term of the loan, at which the initial amount increases in N times, then the calculation formula is displayed:
for compound interest - from the expression (1 + /) "= N:
for simple interest - from the expression (1 + them *) = LH:
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
Consider Compound Interest - the accrual of interest both on the principal amount of debt and on previously accrued interest.
A bit of theory
The owner of the capital, lending it for a certain time, expects to receive income from this transaction. The size of the expected income depends on three factors: on the amount of capital provided on a loan, on the period for which the loan is provided, and on the amount of the loan interest or otherwise the interest rate.
There are various methods of calculating interest. Their main difference comes down to the definition of the initial amount (base) on which interest is charged. This amount can remain constant throughout the entire period or change. Depending on this, a distinction is made between the accrual method and compound interest.
When compound interest rates are used, interest accrued after each accrual period is added to the amount owed. Thus, the basis for compounding, as opposed to use, changes in each accrual period. The addition of accrued interest to the amount that served as the basis for their accrual is called interest capitalization. This method is sometimes referred to as "percentage by percentage".
The example file provides a graph to compare the accrued amount using simple and compound interest.
In this article, we will consider the calculation of compound interest in the case of a constant rate. Variable rate in case of compound interest.
Interest accrual once a year
Let the initial amount of the deposit be equal to P, then after one year the amount of the deposit with the added interest will be = P * (1 + i), after 2 years = P * (1 + i) * (1 + i) = P * (1 + i ) ^ 2, after n years - P * (1 + i) ^ n. Thus, we get the formula for increasing compound interest:
S = P * (1 + i) ^ n
where S is the accrued amount,
i - annual rate,
n - loan term in years,
(1+ i) ^ n - build factor.
In the case discussed above, capitalization is carried out once a year.
With capitalization m times a year, the compounding formula for compound interest looks like this:
S = P * (1 + i / m) ^ (n * m)
i / m is the rate for the period.
In practice, discrete percentages are usually used (interest calculated over the same time intervals: year (m = 1), half year (m = 2), quarter (m = 4), month (m = 12)).
In MS EXCEL, you can calculate the accrued amount by the end of the term of the deposit for compound interest in different ways.
Consider the problem: Let the initial amount of the deposit be equal to 20t.r., the annual rate = 15%, the term of the deposit is 12 months. Capitalization is made monthly at the end of the period.
Method 1. Calculation using a table with formulas
This is the most time consuming method, but the most intuitive. It consists in sequentially calculating the amount of the contribution at the end of each period.
In the example file, this is implemented on a sheet Constant rate.
For the first period, interest will be accrued in the amount of = 20,000 * (15% / 12), since capitalization is carried out monthly, and, as you know, 12 months a year.
When calculating interest for the second period, it is necessary to take not the initial amount of the deposit, but the amount of the deposit at the end of the first period (or the beginning of the second) as the base on which the% is accrued. And so on for all 12 periods.
Method 2. Calculation using the Accrued interest formula
Let's substitute the values from the problem into the formula of the accumulated sum S = P * (1 + i) ^ n.
S = 20,000 * (1 + 15% / 12) ^ 12
It must be remembered that the rate for the period (capitalization period) must be indicated as the interest rate.
Another way to write a formula is through the DEGREE () function
= 20,000 * DEGREE (1 + 15% / 12; 12)
Method 3. Calculation using the BS () function.
The BS () function allows you to determine investments under the condition of periodic equal payments and a constant interest rate, i.e. it is intended primarily for settlement in the event. However, by omitting the 3rd parameter (PMT = 0), you can use it to calculate compound interest.
= -BS (15% / 12; 12 ;; 20,000)
Or so = -BS (15% / 12; 12; 0; 20000; 0)
Note. In the case of a variable rate, to find the Future value using the compound interest method BZRASPIS ().
Determine the amount of accrued interest
Consider the problem: The client of the bank put 150,000 rubles on the deposit. for 5 years with annual compounding interest at a rate of 12% per annum. Determine the amount of accrued interest.
The amount of accrued interest I is equal to the difference between the amount of the accrued amount S and the initial amount P. Using the formula to determine the accrued amount S = P * (1 + i) ^ n, we get:
I = S - P = P * (1 + i) ^ n - P = P * ((1 + i) ^ n –1) = 150,000 * ((1 + 12%) ^ 5-1)
Result: 114 351.25 rubles.
For comparison: the accrual at a simple rate will give the result of 90,000 rubles. (see example file).
Determine the term of the debt
Consider the problem: A bank client has put on a deposit a certain amount with an annual compounding interest at a rate of 12% per annum. After what time will the deposit amount double?
Taking the logarithm of both sides of the equation S = P * (1 + i) ^ n, we solve it with respect to the unknown parameter n.
The example file provides a solution, the answer is 6.12 years.
Calculating the compound interest rate
Consider the problem: The client of the bank put 150,000 rubles on the deposit. with annual compounding interest. At what annual rate will the deposit amount double in 5 years?
The example file provides a solution, the answer is 14.87%.
Note... Effective interest rate.
Accounting (discounting) at compound interest
Discounting is based on the concept of the value of money over time: the money currently available is worth more than the same amount in the future due to its potential to generate income.
Consider 2 types of accounting: mathematical and banking.
Mathematical accounting... In this case, the problem is solved inverse to the increase in compound interest, i.e. calculations are made according to the formula P = S / (1 + i) ^ n
The value of P, obtained by discounting S, is called the modern, or current value, or the reduced value of S.
The amounts P and S are equivalent in the sense that the payment of S in n years is equal to the amount P currently being paid. Here the difference D = S - P is called the discount.
Example... After 7 years, the policyholder will be paid the amount of 2,000,000 rubles. Determine the present value of the amount, provided that a compound interest rate of 15% per annum is applied.
In other words, it is known:
n = 7 years,
S = RUB 2,000,000,
i = 15%.
Decision. P = 2,000,000 / (1 + 15%) ^ 7
The value of the current value will be less, because opening Today a deposit in the amount of P with an annual capitalization at a rate of 15% we will receive in 7 years the amount of 2 million rubles.
The same result can be obtained using the formula = PS (15%; 7 ;; - 2,000,000; 1)
The PS () function returns the reduced (to the current moment) value of the investment and.
Bank accounting... In this case, the use of a complex discount rate is assumed. Discounting at a complex discount rate is carried out according to the formula:
P = S * (1- dsl) ^ n
where dc is the compound annual discount rate.
When using a complex discount rate, the discount process occurs with a progressive deceleration, since the discount rate is each time applied to the amount reduced for the previous period by the amount of the discount.
Comparing the formula for compound interest accrual S = P * (1 + i) ^ n and the discount formula for the compound discount rate P = S * (1- dsl) ^ n, we come to the conclusion that replacing the sign of the rate with the opposite one, we can for calculating the discounted value, use all three methods of calculating compound interest accrual, discussed in the section of the article Accrual of interest several times a year.
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. 10
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 Determination of the size of the 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. fifteen
1.5.2 Accounting for inflation. sixteen
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 Determination of 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 lending 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 ESTIMATES 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 NON-LINEAR, GEOMETRICALLY-DEGRESSIVE METHOD OF ACCOUNTING FOR DAMPING 64
5.4 Excel FUNCTIONS FOR CALCULATION OF CUSHIONING .. 65
5.4.1 Linear method of 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 ON THE SECOND SCHEME USING 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 is taken into account financial transactions... Formulas of financial mathematics are used in the manual for calculating credit, deposit, mortgage transactions, bills of exchange, to compare the effectiveness of financial transactions. To make leasing transactions clear, the guide outlines various methods for accounting for depreciation.
Knowledge of school mathematics is enough to study the manual. 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 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. The purchasing power of 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 ) 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 of funds, the ratio 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 over 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 equity.
Let's consider two types of deposit: 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 granting 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.