Banks offer their potential investors different types deposits, but all of them can be divided into two groups according to the methods of calculating profits. This is the accrual of interest on a deposit without capitalization, and the accrual using compound interest. To calculate the profit in the second case, you will need the compound interest formula for bank deposits.

We will tell you how to calculate compound interest on your own, and use this formula for competent capital investment. You will understand how banks charge you interest. This will help you easily navigate among the mass of different offers on deposits.

How to Calculate Compound Interest: Formula and Examples

Let's start from simple to complex. Typical bank deposit with simple interest does not provide for the possibility of capitalization of profits. You receive interest payments monthly, quarterly or at the end along with the principal amount, depending on the conditions of the bank. You can withdraw money and use it at your own discretion.

Here is an example of a classic simple deposit. You put 100,000 in the bank at 12% per annum. The bank pays you interest every month. Your total profit will be:

100,000 * 0.12 = 12,000 rubles

At the end of each period, you will receive approximately 1000 rubles. The calculation formula in the bank is more complicated, it takes into account the number of days in each month and the number of days in a year. Therefore, in February you will receive less than in April, and in April you will receive less than in May. But in total, the profit will be 12,000 rubles *.

* For those who love precision in everything. In fact, you won't even get 12,000 rubles, because banks use a more complex formula for accruing deposits. The amount of profit is calculated as follows: % \u003d p / (Dnper. / Dnyear.). Banks, as a rule, do not take into account the day the deposit was made, so in reality you will receive 100,000 * 0.12 / (364/365) = 100,000 * 0.119671232 = 11,967, 1232 rubles per year.

Compound interest on a deposit provides for the accrual of interest for the period specified in the agreement (month, year, quarter), and the subsequent addition of this amount to the total amount of the deposit. Interest for the next period will no longer be charged on the initial amount, but on the amount + interest. Therefore, the income for the new period will be higher.

The financial term "compound interest" means the total profit received for a deposit, subject to the addition of profit for each period. Adding interest to the original amount is called capitalization.

C profit \u003d C beg * (1 + %) w - C beg

Explanations for the compound interest formula:

• With profit - the amount that you will receive after the end of the contract, not including the initial deposit;
• From start - the amount for which the deposit was made (initial amount);
• % - designation of the interest rate. It is indicated as a decimal fraction. p(10% per annum is 0.1;
• 14.5% per annum - 0.145, and is calculated for each period according to the formula:% = R* (Ndn.per. / Nyr.);
• w is the number of capitalization periods. If the addition to the principal amount of the deposit is made every month, then w = 12. A simplified % formula for an approximate profit calculation would be: % = R / 12.

Using this simple version, compound interest can be calculated very quickly without additional programs and calculators.

Example. You put the same 100,000 rubles at 12% per annum, but with capitalization every month. Your profit will be: 100,000 * (1 + 0.12 / 12) 12 - 100,000 = 100,000 * (1 + 0.01) 12 - 100,000 = 112,682.503 - 100,000 = 12,682 rubles.

The actual amount will be different because the exact % formula for each month will be different due to the different number of days. Also, the first day of the first credit period is not taken into account (as in the case of calculating simple interest).

Most bank deposit products offer compound interest capitalized on a monthly or quarterly basis. The more periods of capitalization, the higher the profit will be. This is easy to check on the first example by changing the number of periods from 12 to 4: 100,000 * (1 + 0.12/4) 4 - 100,000 = 100,000 * (1.03) 4 - 100,000 = 100,000 * 1, 1255088 - 100,000 = 12,550.88 rubles.

Why do bank customers often have difficulties with compound bank interest? Most often, because they use a simplified formula for the calculation, and do not take into account the different rate for each period. But then the general formula cannot be applied either: after all, if in one quarter we get % = p * (90/365) = p * 0.2466, then already in the second quarter % = p * (91/365) = p * 0, 2493.

How does such a deposit differ from a standard deposit with interest capitalization? In this case, at the end of the first period (month), not interest for this period is added to the initial amount, but a certain fixed amount. In order to calculate compound interest with monthly replenishment, we will use a different formula.

To calculate compound interest with replenishment, the formula looks like this:

C profit \u003d C initial * (1 + %) w + (C additional * (1 +%) w + 1 - C additional * (1 + %)) / % - C initial

Example: you put 100,000 rubles into your account at 12% per annum, and every month you add another 5,000 to this deposit. At the same time, we do not take into account interest: we believe that you receive them on a separate account and use them differently.

You get: 100,000 * (1 + 0.01) 12 - 100,000 + (5,000 * (1 + 0.01) 13 - 5,000 * 1.01) / 0.01 = 12,682 + 1,904 = 14,586 rubles.

The formula for calculating the first period: C1 \u003d C beginning * (1 +%). C1 is not only interest, but also plus the initial amount of the contribution. Calculation for the second period: С2 = С1 * (1 + %). Keep in mind that the % value will be different in each case.

Calculate complex bank interest for a deposit of 100,000 rubles at 12% per annum, with capitalization every quarter. January 1 will be considered the day when the contract is drawn up.

C1 \u003d C beginning * (1 +%) \u003d 100,000 * (1 + 0.12 * (30 + 28 + 31) / 365) \u003d 100,000 * (1 + 0.12 * 0.2438356) \u003d 100,000 * (1 + 0.0292603) = 102,926.03 rubles;

C2 \u003d 102,926.03 * (1 + 0.12 * (30 + 31 + 30) / 365) \u003d 102,926.03 * (1 + 0.0299178) \u003d 106,005.35 rubles, etc. Continuing these calculations, we get 112514.93 rubles. That is, the profit will be 12,514.93 rubles (when calculated using a simplified formula, the result was 12,550 rubles).

Use such complex formulas not necessarily, unless you like exact numbers and want to check your bank to see if your deposits are being charged correctly.

How to profitably use compound bank interest

At equal interest rates, a deposit with capitalization will bring more income. But often the bank offers a choice: a deposit with a lower rate, but with capitalization, or a regular deposit with high stake without capitalization. To find the best way, you will have to use the above formula to calculate compound interest on deposits.

The formula can also be used in reverse. For example, calculate the interest rate at which you will receive the desired profit in a certain time. The formula will look like this:% \u003d (Desired / Initial) 1 / n - 1. For example, you want to calculate at what interest rate, having invested 10,000 rubles for a year with a quarterly capitalization, you will receive 15,000 rubles as a result. Calculate the rate: % = (15,000 / 10,000) ¼ - 1 = 0.10668. The rate should be 10.668%.

When opening a bank deposit, you need to pay attention not only to the size of the interest rate, but also to the type of interest accrual. There is a simple calculation of interest and a complex one. In this article, we will analyze the difference between the type of interest rate calculation, and also determine the benefits of one or another method of calculation.

What is the difference between simple and compound interest?

Banks usually offer simple interest accrual. What does it mean? This means that interest will be charged to your deposit only at the end of the term. Those. let's say you opened a deposit at 10% per annum and invested 10,000 rubles. In a year, you will be credited in the form of interest of 1,000 rubles. If you leave a contribution for the second year, then after this period you will be credited with another 1,000 rubles.

For 2 years, with a simple interest calculation, your total amount will be: 12,000 rubles.

If there was a complex calculation of interest, the picture changes a little. After 1 year, your account would also have 11,000 rubles (10,000 - your contribution + 1,000 rubles in interest).

However, this accrued thousand, at the end of the first period, would join the main body of the deposit. And all interest would already accrue on that total amount. Those. you would receive 10% in the second year, only not from 10,000 rubles, but from 11 thousand. In money it turns out - 1,100 rubles.

In total, for 2 years with complex accrual, your amount will be: 12,100 rubles

I think it makes no sense to explain what you choose: 12,000 or 12,100 rubles. Besides added benefit compound interest is the fact that they are also included in . Those. if the bank's license is revoked, then all accrued interest is also subject to return to the depositor.

With simple accrual, money is paid only at the end of the term, i.e. in fact, they were not credited, even if there was only one day left before the end of your deposit! And in this case, you have the right to return only the basic capital.

A deposit with monthly or quarterly interest capitalization becomes especially attractive. The lower the capitalization period for the deposit, the more high income He gives. It's about the cumulative effect. When accrued interest in the form of profit also accrues profit. Sometimes compound interest is called interest. including reinvestment or capitalization. Pay attention to this when concluding an agreement with a bank. If the agreement states that interest is accrued at the end of the deposit term, then we are talking on simple interest.

Banks do not often offer. Even if interest is accrued monthly or quarterly, banks prefer not to use the profits to charge additional interest on them, but transfer them to a separate account. The point here, as mentioned above, is the effect of refinancing, when the effective interest rate due to capitalization will be higher than originally declared by the bank.

Example. At nominal rate at 9% per annum, real effective rate taking into account reinvestment would be 9.4% per annum. At 10%, this figure would rise to 10.5%, and at 11%, to 11.6%.

Banks usually quote the nominal interest rate, since the effective interest rate may not happen if the interest is withdrawn.

The formula for calculating compound interest on deposits in banks

For those who want to understand for themselves how much they will receive by investing money at compound interest, the bank has a special formula for reinvesting or capitalizing a deposit:

S=K * (1+r/t)™

K is your initial amount that you deposited in the bank,

r - the annual interest rate at which you deposited in the bank, for example, 10% per annum is 0.1, 12% per annum is 0.12

t is the number of interest payments per year, for example, if interest is accrued annually, then t=1, quarterly t=4, monthly t=12

TM is the number of interest accrual periods, i.e. if you opened a deposit for 2 years, then with a quarterly accrual of periods there will be 8, with a monthly TM it will be 24.

S is the amount that will be in your account at the end of the deposit period.

Example.

You opened a deposit for a period of 2 years, at 12% per annum, interest capitalization is quarterly. You deposited 10,000 rubles.

How much will you have at the end of the term?

K=10 000
r=0.12%
t=4
TM=8

We get, S=10,000 * (1+0.12/4)∧8 = 12,668 rubles.

In total, for 2 years, such a contribution will bring you 2,668 rubles or a 26.68% return.

If, for example, we take a simple calculation of interest at the same 12% per annum for 2 years, with annual accrual, but without capitalization, then at the end of the term the amount will be slightly less, namely 2,400 rubles or 24% yield.

Of course, the difference of 2.68% is not that big. But everything changes if the amount of the deposit changes upwards or the term of the deposit increases. It is at large time intervals that the difference between simple and complex interest calculation is most noticeable. Over long periods of time, the difference in the result achieved can vary significantly. No wonder the Rothschilds (the richest family on the planet) called compound interest "".

Compound interest differs from ordinary interest in that it is charged not only on the principal amount of the deposit, but also on the amount of interest accumulated on it. For this reason, amounts in savings accounts with a compound interest rate grow faster than in accounts with a simple interest rate. Moreover, savings will grow even faster if interest is capitalized many times a year. Compound interest is found in various types of investments, as well as in certain types of loans, for example, credit cards. Calculating the increase in the original amount at the rate of compound interest is quite simple if you know the correct formula.

Steps

Part 1

1. Determine the annual capitalization. The interest rate on investments or loan agreements is set for a year. For example, if your car loan rate is 6%, then you pay 6% of the loan amount annually. When capitalizing interest once a year, it is easiest to calculate compound interest.

   • Interest on debts and investments can be capitalized (added to the principal amount) annually, monthly and even daily.
   • The more capitalization occurs, the faster the amount of interest increases.
   • The compound interest rate can be viewed from both the investor's and debtor's points of view. Frequent capitalization suggests that the investor's interest income will grow faster. For the debtor, this means that he will have to pay more interest for the use of borrowed funds until the loan is repaid.
   • For example, capitalization deposit may be carried out once a year, and the capitalization of the loan may be monthly or even weekly.

2. Calculate the interest capitalization for the first year. Let's say you have $1,000 and invest it in US government bonds at 6% per annum. Interest on US government bonds is calculated annually based on the interest rate and present value. security.

   • Interest for the first year of the investment will be $60 ($1000*6% = $60).
   • To calculate the interest for the second year, you first need to add the previously accrued interest to the initial investment amount. In the example above, that would be $1060 (or $1000 + $60 = $1060). That is, the current value of the government bond is $1060, and further interest is calculated from this value.

3. Calculate the interest capitalization for subsequent years. To more clearly see the difference between compound interest and ordinary interest, calculate their value for subsequent years. From year to year, the amount of interest will increase.

   • For the second year, multiply the present value of the $1060 bond by the interest rate ($1060*6% = $63.60). The amount of interest for the year will increase by $3.60 (or $63.60 - $60.00=$3.60). This is because the principal amount of the investment has grown from $1,000 to $1,060.
   • In the third year, the present value of the investment is $1123.60 ($1060 + $63.60 = $1123.60). Interest for this year will already be equal to $67.42. And this amount will be added to the current value of the security to calculate interest for the 4th year.
   • The longer the term of the loan/investment, the greater the impact of compound interest on the total amount. The term of a loan is the length of time the borrower has still not repaid their debts.
   • Without capitalization, the interest for the second year will be $60 ($1000 * 6% = $60). In effect, the interest for each year will be $60 if it is not included in the principal amount. In other words, this simple interest.

4. Create a spreadsheet in Excel to calculate compound interest in full. It is helpful to visualize compound interest as a simple spreadsheet in Excel that will show you the growth of your investment. Open the document and label the top cells in columns A, B, and C as "Year", "Cost" and "Acrued Interest".

   • Enter years 0 through 5 in cells A2-A7.
   • Enter the original investment amount in cell B2. Let's say if you started with an investment of $1000. Enter 1000 here.
   • Enter the formula "=B2*1.06" (without quotes) in cell B3 and press enter. This formula says that every year your interest is capitalized at a rate of 6% (0.06). Click on the bottom right corner of cell B3 and drag the formula to cell B7. The amounts in the cells will be calculated automatically.
   • Put a zero in cell C2. In cell C3, enter the formula "=B3-B$2" and press enter. This will give you the difference between the current and original cost of the investment (cells B3 and B2), which is the total amount of interest accrued. Click on the bottom right corner of cell C3 and drag the formula down to cell C7. The amounts will be calculated automatically.
   • In the same way, you can make calculations for as many years ahead as you like. You can also easily change the initial amount and interest rate by changing the formula for calculating interest and the contents of the corresponding cells.

5. Perform mathematical operations on the formula. Simplify the expression by calculating the individual parts, starting with brackets and the fraction located there.

   • Divide the fraction first. The result will be the following: F V = $ 5000 (1 + 0 , 00288) 2 ∗ 12 (\displaystyle FV=\$5000(1+0.00288)^(2*12)).
   • Add up the amounts in brackets. You will get: F V = $ 5000 (1 , 00288) 2 ∗ 12 (\displaystyle FV=\$5000(1.00288)^(2*12)).
   • Calculate the degree itself (expression above in brackets). The result will be like this: F V = $5000 (1, 00288) 24 (\displaystyle FV=\$5000(1.00288)^(24)).
   • Raise the number in brackets to the appropriate power. This can be done on a calculator: first enter the amount in brackets (1.00288 in our example), click on the exponentiation button x y (\displaystyle x^(y)), and then enter the exponent value (24) and press enter. The result will look like this: F V = $5000 (1, 0715) (\displaystyle FV=\$5000(1.0715)).
   • Finally, multiply the original amount by the number in brackets. In the example above, multiply $5000 by 1.0715 to get $5357.50. This is the future value of your investment in two years.

6. Subtract the original amount from the result. The difference will represent the amount of accrued interest.

   • Subtract the original $5,000 from the future value of the $5,357.50 deposit and you have $357.50 ($5,375.50-$5,000=$357.50).
   • That is, after two years, you will earn $357.50 in interest.

Part 3

1. Learn the formula. Compound interest will grow even faster if you regularly increase the deposit amount, for example, deposit a certain amount into a deposit account every month. The formula applied in this case becomes larger, but is based on the same principles. It looks like this: FV = P (1 + ic) n ∗ c + R ((1 + ic) n ∗ c − 1) ic (\displaystyle FV=P(1+(\frac (i)(c)))^(n* c)+(\frac (R((1+(\frac (i)(c)))^(n*c)-1))(\frac (i)(c)))). All variables in the formula remain the same, but one more indicator is added to them:

   • "P" - initial amount;
   • "i" – annual interest rate;
   • "c" – frequency of capitalization (how many times per year interest is added to the principal amount);
   • "n" – duration of the period in years;
   • "R" - the amount of monthly replenishment of the deposit.

2. Determine the initial values ​​of the variables. To calculate the future value of a deposit, you need to know the initial (current) amount of the deposit, the annual interest rate, frequency of interest capitalization, the term of the deposit, and the amount of the monthly deposit replenishment. All this can be found in the agreement you signed with your bank.

   • Don't forget to translate annual percentage into a decimal. To do this, simply divide it by 100%. For example, the 3.45% rate mentioned above would be 0.0345 (or 3.45%/100%=0.0345) in decimal form.
   • As the frequency of capitalization, specify how many times a year interest is added to the total amount of the deposit. If this happens annually, enter one, monthly - 12, daily - 365 (don't worry about leap years).

3. Substitute the data in the formula. Continuing with the above example, let's say you decide to deposit $100 each month. At the same time, the initial deposit amount is $5,000, the rate is 3.45% per annum, and capitalization occurs monthly. Calculate the growth of the deposit for two years.

   • Plug in your data into the formula: FV = $ 5 , 000 (1 + 0.0345 12) 2 ∗ 12 + $ 100 ((1 + 0.0345 12) 2 ∗ 12 − 1) 0.0345 12 (\displaystyle FV=\$5,000(1+(\frac (0.0345)( 12)))^(2*12)+(\frac (\$100((1+(\frac (0.0345)(12)))^(2*12)-1))(\frac (0.0345)(12 ))))

4. Make a calculation. Again, remember the correct order of operations. This means that you need to start by doing the actions in parentheses.

   • First of all, calculate the fractions. That is, divide "i" by "c" in three places to get the same result of 0.00288 everywhere. Now the formula will look like this: FV = $ 5000 (1 + 0 , 00288) 2 ∗ 12 + $ 100 ((1 + 0 , 00288) 2 ∗ 12 − 1) 0 . 00288 (\displaystyle FV=\$5000(1+0.00288)^( 2*12)+(\frac (\$100((1+0.00288)^(2*12)-1))(0.00288))).
   • Add in parentheses. That is, add one to the result of previous calculations where required. You will get: FV = $ 5000 (1 , 00288) 2 ∗ 12 + $ 100 ((1 , 00288) 2 ∗ 12 − 1) 0 . 00288 (\displaystyle FV=\$5000(1.00288)^(2*12)+( \frac (\$100((1.00288)^(2*12)-1))(0.00288))).
   • Calculate degree. To do this, multiply the two numbers at the top outside the brackets. In our example, the degree value will be 24 (or 2*12). The formula will appear as follows: FV = $ 5000 (1 , 00288) 24 + $ 100 ((1 , 00288) 24 − 1) 0 . 00288 (\displaystyle FV=\$5000(1.00288)^(24)+(\frac (\$100( (1.00288)^(24)-1))(0.00288))).
   • Raise the required numbers to a power. You should raise the numbers in brackets to the power that you got at the previous stage of the calculations. To do this, on the calculator, enter the number from the brackets (in the example it is 1.00288), press the exponentiation button x y (\displaystyle x^(y)), and then enter a value for the degree (in this case, 24). You will get: FV = $ 5000 (1 , 0715) + $ 100 (1 , 0715 − 1) 0 . 00288 (\displaystyle FV=\$5000(1.0715)+(\frac (\$100(1.0715-1))( 0.00288))).
   • Subtract. Subtract one from the result of the previous calculation on the right side of the formula (in the example, subtract 1 from 1.0715). Now the formula looks like this: FV = $ 5000 (1 , 0715) + $ 100 (0 , 0715) 0 . 00288 (\displaystyle FV=\$5000(1.0715)+(\frac (\$100(0.0715))(0.00288) )).
   • Do the multiplication. Multiply the initial investment amount by the number in the first brackets, and the monthly top-up amount by the same amount in brackets. You will get: F V = $5357 , 50 + $ 7 , 15 0 , 00288 (\displaystyle FV=\$5357.50+(\frac (\$7.15)(0.00288)))
   • Do the division. You will get the following result: F V = $ 5 , 357.50 + $ 2 , 482.64 (\displaystyle FV=\$5,357.50+\$2,482.64)
   • Add up the numbers. Finally, add up the remaining two numbers to find out the future amount in the account. In other words, add $5357.50 and $2482.64 to get $7840.14. This is the future value of your investment in two years.

compound interest It is customary to call the effect that occurs when profits and interest accumulate, as a result of which interest payments increase exponentially. Most modern banks accept clients precisely at compound interest, which is undoubtedly beneficial for the depositor. Even Einstein himself appreciated the importance of the discovery of compound interest, calling it the main "driving force in the world."

In order to better understand what compound interest is, you need to go to examples with calculations.

How is compound interest calculated?

A simple formula is used to calculate:

In the formula, SUM means the final amount of the settlement with the client, X is the investment amount, n is the number of billing periods. On the graph you can see what is meant by the exponential increase in the sum:

For bank deposits, the formula is a little more complicated, since a new element of the equation is introduced -:

So, we need to know the frequency of capitalization. Capitalization refers to the recalculation of the amount on which interest is accrued - the base amount is added to the amount accrued for the last period. If the recalculation occurs monthly, the frequency of capitalization (in our formula it is D) is 30 days, if quarterly it is 90 days.

The remaining unfamiliar indicators in the formula for calculating bank compound interest are Y - the number of days in a year (365 or 366) and P - the interest rate. The entire block of values ​​after the unit under the bracket is called interest rate ratio.

Consider an example:

Citizen I invests 100,000 rubles at 15% per annum with a monthly capitalization. How much will he be able to earn in 8 years?

A) with simple interest?

B) with compound interest?

So, we calculate the simple percentage first. 15% of 100,000 rubles is 15,000 rubles. If 15 thousand rubles are multiplied by 8, then you get a profit from a deposit of 120 thousand rubles. Thus, after 8 years, citizen I will be able to withdraw 220 thousand rubles.

To calculate compound interest, we substitute the data in the formula:

The result of the calculations should be an unpleasant surprise - the profit will be the same 120 thousand rubles. Then let's try to calculate the amount for annual capitalization, and not for monthly:

We will get a result that will satisfy us much more - 306 thousand profits. We conclude: the less often capitalization occurs, the higher the profit will be. Interest is calculated annually as follows:

It can be seen that under compound interest they grow like a snowball. The longer the depositor does not withdraw them, the greater will be his profit from month to month.

Other Useful Formulas

Other formulas may be useful for calculating deposits:

1. Interest rate. The formula shows at what percentage you need to deposit funds in order to get the desired result.

We know all the indicators, so let's try to solve the example right away:

At what percentage should 10,000 rubles be put in order to receive 80,000 rubles in 15 years?

It is clear that you need to put money at 15% per annum.

1. Number of periods. The formula shows how many interest periods you need to deposit funds to achieve the desired result:

Again, we try to solve an example:

How long does it take to deposit money at 20% per annum in the amount of 150,000 thousand rubles in order to receive 1 million rubles?

Funding is required for 10 years.

. 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 accrued interest increases, and the process increase in the amount of debt is accelerating. Compound interest accumulation can be represented as a follower new reinvestment of funds invested under simple procents for one accrual period ( running period ). Jointhe addition of accrued interest to the amount that served as the basis for their calculation is often called interest capitalization.

Let's find a formula for calculating the accumulated amount under the condition that interest is accrued and capitalized once ayear (annual interest). For this, it is applied complex becoming kaextensions. To write the growth formula, we apply thosethe same notation as in the formula for increasing by simple pro cents:

P - the initial amount of debt (loans, credit, capital la, etc.),

S - accumulated amount at the end of the loan term,

P - term, number of years of accrual,

i - level annual rate percent submitted by decent fraction.

Obviously, at the end of the first year, the interest is equal to the value R i , and the accumulated amount will be K concin the second year it will reach the value IN end n -th year, the accumulated amount will be is equal to

(4.1)

The interest for the same period as a whole is as follows:

(4.2)

Some of them are learned by calculating interest on interest. She is

(4.3)

As shown above, compound interest growth isis a process corresponding to a geometric progression si, the first term of which is equal to R , and the denominator is .The last member of the progression is equal to the accumulated sum at the end loan term.

the value called incremental multiplier at compound interest. The meanings of thismultiplier for integers P are given in complex tables percent.Multiplier calculation accuracy in practical calculationsis determined by the allowable degree of rounding of the accumulatedamounts (down to the last penny, ruble, etc.).

Compound rate build time usually measures Xia as AST/ A ST.

As you can see, the value of the accumulation multiplier depends on two parameters - iAnd P. It should be noted that for a long timeeven a small change in the rate significantly affectsby the value of the multiplier. In turn, a very long timeleads to frightening results even with a smallinterest rate.

The compound interest accrual formula is obtainedfor an annual interest rate and a term measured in years.However, it can be applied to other accrual periods as well.niya. In these casesimeans the rate for one accrual period (month, quarter, etc.), and n is the number of such periods. On the example if i– half-year rate, then P – number of semesters etc.

Formulas (4.1) - (4.3) assume that the interest on procents are charged at the same rate as when charged on the principal amount of the debt. We will complicate the conditions for calculating interestcomrade Let the 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 the geometrica progression with the first term equal to 1 and the denominator. As a result, we have

(4.4)

· Example 4.1

2. Calculation of interest in adjacent calendar periods. You Earlier, 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 the loan are in two periods. It is clear that the accrued for the entire term, interest cannot be attributed only to the lasthis period. In accounting, taxation,Finally, in the analysis of the financial activity of the enterprise There is no problem of distributing accrued interest over periods.

The total loan term is divided into two periodsn 1 And n 2 . Respectively ,

where

· Example 4.2

3. Variable rates. The formula assumes a constantrate throughout the interest period. The instability of the monetary market makes it necessary to modernize the “classical” scheme, for example, using the opinions floating rates ( floating rate). Naturally, the calculationfor the future at such rates is very conditional. Another thing -post factum calculation. In this case, and also whenbet sizes are fixed in the contract, the total multiplier The extension agent is defined as the product of quotients, i.e.

(4.5)

where - consecutive values ​​of rates; - periods during which the corresponding rates.

· Example 4.3

4. Calculation of interest for a fractional number of years. Often time in th dax for interest calculation is not an integer. In the rules of a number of commercial banks for some operations interest is charged only for a whole number of years or other accrual periods. 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 crazy,method involves the calculation of interest on the wholenumber of years using the compound interest formula and for the fractional part term using the simple interest formula:

,(4.7)

where - loan term, but is an integer number of years,b - fractional part of the year.

A similar method is applied in cases wherehome accrual is semester, quarter or month.

When choosing a calculation method, it should be borne in mind that manythe resident of the growth according to the mixed method turns out to be somewhat larger than according to the general method, since for P < 1 is fairin relation

The largest difference is observed given at 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 interest rates matches, then:

1) for a period of less than a year, simple interest is greater than compound interest

2) for more than a year

3) for a period of 1 year, the accrual multipliers are equal to each other

Using the simple compound interest accumulation factor, you can determine the time required to increase the initial amount in n once. For this, it is necessary that the growth coefficients be equal to the value n:

1) for simple interest

2) for compound interest

The formulas for doubling the capital are: