How to calculate compound interest. Compound Interest Formula

Daria Nikitina

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Compound interest It is customary to call the effect when interest on profits is added to the principal amount and subsequently participates in the creation of new profits.
Compound interest formula- this is the formula by which the total amount is calculated taking into account capitalization (interest).

In this article:

Simple calculation of compound interest

To better understand the calculation of compound interest, let's look at an example.
Let's imagine that you deposited 10,000 rubles in the bank at 10 percent per annum.
A year later on your bank account the amount will be SUM = 10,000 + 10,000*10% = 11,000 rubles.
Your profit is 1000 rubles.
You decide to leave 11,000 rubles in the bank for the second year at the same 10 percent interest.
After 2 years, the bank will have accumulated 11,000 + 11,000*10% = 12,100 rubles.

The profit for the first year (1000 rubles) was added to the main amount (10000 rubles) and in the second year it was already generating new profit. Then, in the 3rd year, the profit for the 2nd year will be added to the principal amount and will itself generate new profit. And so on.

This effect is called compound interest.

When all the profit is added to the principal amount and then produces new profit itself.

Compound interest formula:

SUM = X * (1 + %) n

Where
SUM— final amount;
X - initial amount;
% — interest rate, percent per annum /100;
n — number of periods, years (months, quarters).

Calculation of compound interest: Example 1.
You deposited 50,000 rubles in the bank at 10% per annum for 5 years. How much money will you have in 5 years? Let's calculate using the compound interest formula:

SUM = 50,000 * (1 + 10/100) 5 = 80,525.5 rub.

Compound interest can be used when you open time deposit in the bank. According to the terms banking agreement Interest can be accrued, for example, quarterly or monthly.

Calculation of compound interest: Example 2.
Let's calculate what the final amount will be if you put 10,000 rubles for 12 months at 10% per annum with monthly interest accrual.

SUM = 10000 * (1+10/100/12) 12 = 11047.13 rub.

Profit amounted to:

PROFIT = 11047.13 - 10000 = 1047.13 rubles

The profitability was (in percent per annum):

% = 1047,13 / 10000 = 10,47 %

That is, when monthly accrual interest yield turns out to be greater than when interest is calculated once for the entire period.

If you do not withdraw your profits, then compound interest comes into play.

Compound interest formula for bank deposits

In fact, the compound interest formula in relation to bank deposits is somewhat more complicated than described above. The interest rate for the deposit (%) is calculated as follows:

% = p * d / y

Where
p— interest rate (percent per annum / 100) on the deposit,
for example, if the rate is 10.5%, then p = 10.5 / 100 = 0.105;
d— period (number of days) based on the results of which capitalization occurs (interest is accrued),
for example, if capitalization is monthly, then d = 30 days
if capitalization is once every 3 months, then d = 90 days;
y— the number of days in a calendar year (365 or 366).

That is, you can count interest rate for different deposit periods.

Compound interest formula for bank deposits looks like that:

SUM = X * (1 + p*d/y)n

When calculating compound interest, you need to take into account the fact that over time, the accumulation of money turns into an avalanche. That's the appeal of compound interest. Imagine a small fist-sized ball of snow that began to roll down a snowy mountain. While the lump is rolling, snow sticks to it from all sides and a huge snow stone will fly to the foot. Same with compound interest. At first, the increase created by compound interest is almost invisible. But after some time she shows herself in all her glory. This can be clearly seen in the example below.

Calculation of compound interest: Example 3.
Let's consider 2 options:
1. Simple interest. You invested 50,000 rubles for 15 years at 20%. Additional contributions No. You withdraw all profits.
2. Compound interest. You invested 50,000 rubles for 15 years at 20%. There are no additional fees. Each year, interest profits are added to the principal amount.

Starting amount: 50,000 rubles
Interest rate: 20% per annum
	Simple interest	Compound interest
	Sum	Profit in a year	Sum	Profit in a year
After 1 year	60,000 rub.	10,000 rub.	60,000 rub.	10,000 rub.
After 2 years	70,000 rub.	10,000 rub.	72,000 rub.	12,000 rub.
3 years later	80,000 rub.	10,000 rub.	86,400 rub.	14,400 rub.
After 4 years	90,000 rub.	10,000 rub.	RUB 103,680	RUB 17,280
After 5 years	100,000 rub.	10,000 rub.	RUB 124,416	RUB 20,736
After 6 years	110,000 rub.	10,000 rub.	RUB 149,299	RUR 24,883
After 7 years	120,000 rub.	10,000 rub.	RUB 179,159	RUB 29,860
After 8 years	130,000 rub.	10,000 rub.	RUR 214,991	RUB 35,832
After 9 years	140,000 rub.	10,000 rub.	RUB 257,989	RUB 42,998
After 10 years	150,000 rub.	10,000 rub.	RUB 309,587	RUB 51,598
After 11 years	160,000 rub.	10,000 rub.	RUB 371,504	RUB 61,917
After 12 years	170,000 rub.	10,000 rub.	RUB 445,805	RUB 74,301
After 13 years	180,000 rub.	10,000 rub.	RUB 534,966	RUB 89,161
After 14 years	190,000 rub.	10,000 rub.	RUB 641,959	RUB 106,993
After 15 years	200,000 rub.	10,000 rub.	RUB 770,351	RUB 128,392
Total profit:	150,000 rub.		RUB 720,351

. The base for calculating compound interest, unlike simple ones, does not remain constant Noah – it increases with each step in time. The absolute amount of accrued interest increases, and the process The amount of debt is increasing at an accelerating rate. Increment by compound interest can be represented as a follower new reinvestment of funds invested in simple projectscents for one accrual period ( running period ). JoinThe reduction of accrued interest to the amount that served as the basis for their calculation 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 peryear (annual interest). For this purpose it is used complex becoming kaextensions. To write the growth formula, we apply thosethe same notation as in the formula for increasing by simple pro cents:

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

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

P - term, number of years of accumulation,

i - the level of the annual interest rate, presented by dedecimal fraction.

Obviously, at the end of the first year the interest is equal to R i , and the increased amount will be. Towards the endin the second year it will reach the value IN end n of the year the accrued amount will be equal to

(4.1)

The interest rates for the same period are generally as follows:

(4.2)

Some of them are learned by accruing interest on interest. It amounts to

(4.3)

As shown above, compound interest growth is represented byis a process corresponding to geometric progress siya, the first term of which is equal to R , and the denominator is .The last term of the progression is equal to the accumulated amount at the end loan term.

Size called increment multiplier at compound interest. The meanings of thismultiplier for integers P are given in complex tables percent.Accuracy of multiplier calculations in practical calculationsdetermined by the permissible degree of rounding of the accruedamounts (down to the last penny, ruble, etc.).

The time when growing at a complex rate usually measures sya like AST/ A ST.

As you can see, the value of the growth multiplier depends on two parameters - iAnd P. It should be noted that for a long timeincrease, even a small change in the rate has a noticeable effectby the value of the multiplier. In turn, a very long timeleads to terrifying results even with smallinterest rate.

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

Formulas (4.1) - (4.3) assume that interest onCents are calculated at the same rate as when applied to the principal amount of the debt. Let's complicate the terms of interest calculationsComrade Let interest on the principal be calculated at the rateiand interest on interest - at the rate In this case

The series in square brackets represents the geometrica logical 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 Previously, when calculating interest, the location of the interest accrual 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 accrued for the entire period, interest cannot be attributed only to the lasthim period. In accounting, taxation,finally, in the analysis of the financial activities of the WHO enterprise The task of distributing accrued interest across periods arises.

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 entire interest accrual period. The instability of the monetary market forces us to modernize the “classical” scheme, for example, using example opinions floating rates ( floating rate). Naturally, the calculationfor the future at such rates is very conditional. Another thing -calculation after the fact. In this case, as well as when betrayalbet sizes are fixed in the contract, the total multiplier The growth factor is defined as the product of the quotients, i.e.

(4.5)

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

· Example 4.3

4. Calculation of interest for a fractional number of years. Often the deadline is dah for interest calculation is not an integer. In the rules of a number of commercial banks for some operations interest is calculated 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, calculation is carried out according to the formula:

(4.6)

Second, sme shady,The method involves calculating interest on the wholenumber of years using the compound interest formula and the fractional part term according to the formula simple interest:

,(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 periodThe accrual house is half-yearly, quarterly or monthly.

When choosing a calculation method, you should keep in mind that manythe growth rate using the mixed method turns out to be slightly larger than using the general method, since for P < 1 is fairin ratio

The biggest difference I see given at b = 1/2.

· Example 4.4

5. Comparison of growth using 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 greater than compound interest

2) for a period of more than a year

3) for a period of 1 year, the growth factors are equal to each other

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

1) for simple interest

2) for compound interest

Formulas for doubling capital are:

Undoubtedly, the profitability of a bank deposit is primarily determined by the interest rate. After all, this is what every potential client focuses on. But, in fact, the investor needs, in particular, to pay attention not to the annual interest rate, but to the method of calculating profit. After all, in financial system Bank there are two concepts: simple and compound interest. And for each investor you need to know exactly what simple and compound interest are, concepts and formulas in order to determine which deposit will be most profitable for him.

What is simple interest

First of all, simple interest is the accrual of remuneration for placing a deposit in a bank account for the entire period of storage of funds. If we talk in simple words, then simple interest is accrued only upon expiration of the deposit agreement; it is determined at the annual interest rate. Moreover, if the contract is automatically extended for the next term, then the remuneration for the previous period is not added to the body of the deposit.

To understand as accurately as possible what a simple profit calculation system is, let’s look at an example. You placed 50,000 rubles in the bank at 7% per annum for one year. At the end of the contract, your profit will be 50,000 × 0.07 = 3,500 rubles. If the contract is automatically extended for the next term, your profit will again be 3,500 rubles. That is, after 2 years you will be able to receive 50,000+3500+3500=57,000 rubles from the bank.

Important! The formula for calculating simple interest is as follows: K=D×p. Where K is the amount of profit, D is the body of the deposit, p is the annual interest rate (in the formula you need to indicate not the annual rate, but the rate divided by 100).

If you place funds for a period of less than one year, then the annual interest rate is divided by 12 and multiplied by the number of months during which the funds were in the bank account. For example, if the deposit period is 3 months and the interest rate is 10% per year, then the total profit is calculated as follows: 0.1/12×3 = 0.025. For example, if you placed 50,000 rubles for a period of 3 months, then the profit at the end of the contract will be as follows: 50,000 × 0.025 = 1,250 rubles.

Formulas for simple and compound interest

Compound interest on deposits

The difference between simple interest and compound interest is actually quite large. When choosing a deposit product, everyone has probably heard about such a concept as capitalization. That is, this is a profit accrual scheme in which the accrued profit is added to the body of the deposit, and income is again accrued on it in the future.

Please note that capitalization is carried out at a certain frequency, for example, once a week, a month, a quarter or a year.

From this we can conclude that capitalization allows you to get greater profits compared to simple interest. To clearly see this, let’s look at the formula for calculating compound interest, and it will look like this: B=(K×H×P/N)/100, Where:

• B – amount of accrued profit;
• K – deposit body;
• H – annual rate;
• P – number of days during which capitalization occurs;
• N – number of days in a year.

To clearly understand how exactly compound interest will be calculated. Let's look at a simple example. The deposit amount is 50,000 rubles, the interest rate per year is 7%, capitalization is carried out monthly, the contract period is one year. Let's calculate the profit for the first month of using the deposit: B=(50000×7×30/365)/100=287.6 rubles – this is the profit for the first month. In the next period, the calculation will look like this: B=(50287.6×7×31/365)/100=298.9 rubles.

From the above example we can conclude that capitalization allows you to receive greater profits every month compared to the previous one. But when choosing a deposit offer, be sure to pay attention to the frequency with which interest is capitalized; the more often, the more benefits the client receives.

What is the difference

In fact, the system for calculating interest on deposits varies greatly, primarily for the reason that with the capitalization of interest, the benefit of the deposit can be significantly higher than with a simple system. Because with a simple system, profit grows in arithmetic progression, and with a complex one, in geometric progression. To clearly see this, below is a diagram of compound interest compared to a simple interest scheme.

Compound interest scheme versus simple interest scheme

But there are also pitfalls in this matter. The conditions of bank deposits are strictly individual, so when choosing a deposit product, first of all, pay attention to the number of capitalization periods over the entire term of the agreement. For example, the bank indicates that your deposit agreement provides for capitalization of interest, but it is carried out once every 6 months, that is, you will receive your first income six months after concluding an agreement with the bank.

At the same time, you decided to place funds only for 3 months, accordingly, you will receive your funds earlier than the bank capitalizes the interest and in this case it is more advisable to choose a simple calculation of interest on the deposit. Important! Most banks offer the same deposit offer

to their clients to make a choice to receive profit with a certain frequency or to include themselves in the body of the deposit; accordingly, the client has the opportunity to choose which system, simple or complex, he would like to receive his income.

In fact, it is quite simple to understand what the fundamental difference between simple and compound interest is, but the nuance is that banks do not indicate such concepts as simple and compound interest in the agreement; each potential investor must pay attention to all the terms of the agreement . If the agreement states that interest is paid upon expiration of the agreement, accordingly, capitalization under such an agreement is not provided. Compound interest is used in long-term financial and credit operations if interest is not paid periodically immediately after it has been accrued over the past period of time, but is added to the amount of debt. Adding accrued interest to the amount that served as the basis for determining it is often called capitalization

percent.

Compound interest formulaPLet the initial amount of debt beP(1+ i) , then after one year the amount of debt with added interest will be P(1+ i)(1+ i)= P(1+ i) 2 , in 2 years n, through P(1+ i) years -. Thus, we obtain the compounding formula for compound interest

S=P(1+i)n, (19)

Where S- accrued amount,i- annual compound interest rate,n- loan term, (1+ i) years -- growth multiplier.

In practical calculations, discrete percentages are mainly used, i.e. interest accrued at equal time intervals (year, half-year, quarter, etc.). Compound interest growth is a growth according to the law of geometric progression, the first term of which is equal toP, and the denominator (1+ i).

Note that when the deadlinen<1 growth using simple interest gives greater results than compound interest, and whenn>1 - vice versa. This is easy to verify using specific numerical examples. The greatest excess of the amount accrued at simple interest over the amount accrued at complex interest (at the same interest rates) is achieved in the middle part of the period.

Compound interest formula
when the rate changes over time

In the case when the compound interest rate changes over time, the compounding formula has the following form

(20)

where i 1, i 2,..., i k - successive values ​​of interest rates in effect during the periods n 1, n 2,..., nk respectively.

Example 6.

The contract states variable rate compound interest, defined as 20% per annum plus a margin of 10% in the first two years, 8% in the third year, 5% in the fourth year. Determine the value of the growth multiplier for 4 years.

Solution.

(1+0,3) 2 (1+0,28)(1+0,25)=2,704

Formula for doubling the amount

In order to assess their prospects, a creditor or debtor may ask: in how many years will the loan amount increase inNtimes at a given interest rate. This is usually required when forecasting your investment opportunities in the future. We get the answer by equating the growth factor to the valueN:

A) for simple interest

(1+ nisimple) = N, where

. (21)

B) for compound interest

(1+ icomplex) years -= N, where

. (22)

Especially often usedN=2. Then formulas (21) and (22) are called doubling formulas and take the following form:

A) for simple interest

, (23)

B) for compound interest

. (24)

If formula (23) is easy to use for rough calculations, then formula (24) requires the use of a calculator. However, for small interest rates (say, less than 10%), a simpler approximation can be used instead. It's easy to get if you consider that ln 2  0.7, and ln (1+ i)  i. Then

n» 0.7/ i. (25)

Example 7.

Solution.

a) With simple interest:

years.

b) With compound interest and the exact formula:

Of the year.

c) With compound interest and an approximate formula:

n» 0.7/ i= 0.7/0.1 = 7 years.

Conclusions:

1) The same value of simple and compound interest rates leads to completely different results.

2) For small values ​​of the compound interest rate, the exact and approximate formulas give almost identical results.

Calculation of annual interest for a fractional number of years

For a fractional number of years, interest is calculated in different ways:

1) Using the compound interest formula

S=P(1+i)n, (26)

2) Based on a mixed method, according to which compound interest is calculated for a whole number of years, and simple interest for a fractional number

S=P(1+i) a (1+bi), (27)

Where n= a+ b, a- an integer number of years,b-fractional part of the year.

3) A number of commercial banks apply a rule according to which interest is not accrued for periods of time shorter than the accrual period, i.e.

S=P(1+i)a. (28)

Nominal and effective interest rates

Nominal rate . Let the annual compound interest rate bej, and the number of accrual periods per yearm. Then each time interest is calculated at the rate j/m. Bid jcalled nominal. Interest is calculated at the nominal rate according to the formula:

S=P(1+j/m)N, (29)

Where N- number of accrual periods.

If the loan term is measured by a fractional number of accrual periods, then whenmIn a one-time calculation of interest per year, the accrued amount can be calculated in several ways, leading to different results:

1) Using the compound interest formula

S=P(1+j/m) N/t, (30)

Where N/ t- number (possibly fractional) of interest calculation periods,t- interest accrual period,

2) According to the mixed formula

, (31)

Where a- an integer number of accrual periods (i.e.a= [ N/ t] - the integer part of the division of the entire loan termNfor the accrual periodt),

b- the remaining fractional part of the accrual period ( b= N/ t- a).

Example 8.

Loan size 20 million rubles. Granted for 28 months. The nominal rate is 60% per annum. Interest is calculated quarterly. Calculate the accrued amount in three situations: 1) when compound interest is charged on the fractional part, 2) when simple interest is charged on the fractional part, 3) when the fractional part is ignored. Compare results.

Solution.

Interest is calculated quarterly. There are a total of quarters.

1) = 73.713 million rubles.

2) = 73.875 million rubles.

3) S=20(1+0.6/4) 9= 70,358 million rub .

From a comparison of the accumulated amounts we see that it reaches its greatest value in the second case, i.e. when calculating simple interest on the fractional part.

Effective rate shows what annual compound interest rate gives the same financial result asm- one-time increase per year at the ratej/ m.

If interest is capitalizedmonce a year, each time at a ratej/ m, then, by definition, we can write the equality for the corresponding increment factors:

(1+iuh) n =(1+j/m) mn, (32)

Where iuhis the effective rate, andj- nominal. From this we obtain that the relationship between the effective and nominal rates is expressed by the relation

(33)

The inverse relationship has the form

j=m[(1+iuh) 1/m -1].(34)

Example 9.

Calculate effective rate interest, if the bank charges interest quarterly, based on a nominal rate of 10% per annum.

Solution

iuh=(1+0.1/4) 4 -1=0.1038, i.e. 10.38%.

Example 10.

Determine what the nominal rate should be when interest is calculated quarterly to ensure an effective rate of 12% per annum.

Solution.

j=4[(1+0.12) 1/4 -1]=0.11495, i.e. 11.495%.

Accounting (discounting) at a compound interest rate

Here, as in the case of simple interest, two types of accounting will be considered - mathematical and banking.

Mathematical accounting . In this case, the inverse problem of compound interest accumulation is solved. Let's write down the initial formula for the increase

S=P(1+i)n

and solve it relativelyP

, (35)

Where

(36)

accounting or discount factor.

If interest is chargedmonce a year, we get

, (37)

Where

(38)

discount factor.

Size P, obtained by discountingS, called modern or current value or given size S. Amounts P And Sequivalent in the sense that the payment in the amountS through nyears is equivalent to the amountPcurrently being paid.

Difference D= S- Pcalled discount.

Bank Accounting. In this case, it is assumed that a complex discount rate will be used. Discounting at a complex discount rate is carried out according to the formula

P=S(1-dsl) n, (39)

Where dsl- complex annual discount rate.

The discount in this case is equal to

D=S-P=S-S(1-dsl) n =S.(40)

When using a complex discount rate, the discounting process occurs with a progressive slowdown, since the discount rate is each time applied to an amount reduced over the previous period by the discount amount.

Nominal and effective interest rates

Nominal discount rate . In cases where discounting is usedmonce a year, use nominal discount rate f. Then in each period equal to 1/ mpart of the year, discounting is carried out at a complex discount ratef/ m. The discounting process for this complex accountingmonce a year is described by the formula

P=S(1-f/m)N, (41)

Where N - total number discount periods (N= mn).

Discounting is not one but monce a year reduces the discount amount faster.

Effective discount rate. The effective discount rate is understood as a compound annual discount rate equivalent (by financial results) nominal, applied for a given number of discounts per yearm.

In accordance with the definition of the effective discount rate, we will find its relationship with the nominal rate from the equality of discount factors

(1-f/m) mn =(1-dsl) n,

from which it follows that

dsl=1-(1-f/m)m. (42)

Note that the effective discount rate is always less than the nominal rate.

Increase at a complex discount rate. The increase is the inverse problem for discount rates. Formulas for compounding at complex discount rates can be obtained by resolving the corresponding formulas for discounting (39 and 41) with respect toS. We get

from P=S(1-d sl) n

, (43)

and from P= S(1- f/ m) N

. (44)

Example 11.

What amount should be entered in the bill, if the actual amount issued is 20 million rubles, the repayment period is 2 years. The bill is calculated based on a compound annual discount rate of 10%.

Solution.

million rubles

Example 12.

Solve the previous problem, provided that the increase at a complex discount rate is carried out not once, but 4 times a year.

Solution.

million rubles

Accumulation and discounting

The accrued amount at discrete percentages is determined by the formula

S= P(1+ j/ m) mn,

Where jis the nominal interest rate, andm- number of interest periods per year.

The more m, the shorter the time intervals between the points of interest accrual. In the limit atm® ¥ we have

S= lim P(1+j/m) mn =P lim [(1+j/m) m ] n . (45)

m ® ¥ m ® ¥

It is known that

lim (1+j/m) m =lim [(1+j/m) m/j ] j =e j ,

m ® ¥ m ® ¥

Where e- the base of natural logarithms.

Using this limit in expression (45), we finally obtain that the accumulated amount in the case of continuous interest accrual at the ratej equal to

S= Pe jn. (46)

In order to distinguish the continuous interest rate from discrete interest rates, it is called the growth rate and is denoted by the symbol d. Then

S=Pedyears -. (47)

The power of growth d represents nominal rate percent atm® ¥ .

Discounting based on continuous interest rates is carried out using the formula

P=Se-dn. (48)

Relationship between discrete and continuous interest rates

Discrete and continuous interest rates are in a functional relationship, thanks to which it is possible to transition from the calculation of continuous to discrete interest and vice versa. The formula for the equivalent transition from one bet to another can be obtained by equating the corresponding increase multipliers

(1+i) n =edyears -. (49)

From the written equality it follows that

d = ln(1+ i) , (50)

i= ed-1 . (51)

Example 13.

The annual compound interest rate is 15%, which is the equivalent growth rate,

Solution.

Let's use formula (50)

d = ln(1+ i)= ln(1+0,15)=0,13976,

those. the equivalent growth force is 13.976%.

Calculation of loan term and interest rates

In a number of practical problems the initial ( P ) and final (S ) the amounts are specified by the contract, and it is necessary to determine either the payment period or the interest rate, which in this case can serve as a measure of comparison with market indicators and the characteristics of the profitability of the operation for the lender. The indicated values ​​can be easily found from the initial formulas for accumulation or discounting. In fact, in both cases the inverse problem is solved in a certain sense.

Loan term

When developing the parameters of the agreement and assessing the time frame for achieving the desired result, it is necessary to determine the duration of the transaction (loan term) through the remaining parameters of the transaction. Let's consider this issue in more detail.

i.

S=P(1+i)n

follows that

(52)

where the logarithm can be taken to any base, since it is present in both the numerator and the denominator.

monce a year from the formula

S=P(1+j/m) mn

we get

(53)

d. From the formula

P=S(1-d)n

we have (54)

m once a year. From

P=S(1-f/m) mn

we arrive at the formula

(55)

When building up by constant growth force. Based

S= Pedyears -

we get

ln( S/ P)= d n. (56)

Interest Rate Calculation

From the same initial formulas as above, we obtain expressions for interest rates.

A) When increasing at a complex annual ratei. From the original growth formula

S=P(1+i)n

follows that

(57)

B) When increasing at a nominal interest ratemonce a year from the formula

S=P(1+j/m) mn

we get (58)

B) When discounted at a complex annual discount rated. From the formula

P=S(1-d)n

we have (59)

D) When discounting at a nominal discount ratem once a year. From

P=S(1-f/m) mn

we arrive at the formula

(60)

D) When increasing by constant growth force. Based

S= Pedyears -

we get

(61)

Interest and inflation

The consequence of inflation is a fall purchasing power money, which for the periodncharacterized by indexJn. Purchasing Power Index equal to the reciprocal of the price indexJp, i.e.

Jn=1/ Jp. (62)

Price indexshows how many times prices have increased over a specified period of time.

Increase at simple interest

If extended over n years the amount of money isS, and the price index is equal toJp, then the actually increased amount of money, taking into account its purchasing power, is equal to

C=S/Jp. (63)

Let the expected average annual inflation rate (characterizing the increase in prices over the year) be equal to h . Then the annual price index will be (1+ h).

If the increase is made at a simple rate duringnyears, then the real increase at the inflation rate h will be

(64)

where in general

(65)

and, in particular, with a constant rate of price growthh,

J p =(1+h)n. (66)

The interest rate that compensates for inflation when calculating simple interest is equal to

(67)

One way to compensate for the depreciation of money is to increase the interest rate by the so-called inflation premium. The rate adjusted in this way is called gross rate. Gross rate, which we will denote by the symbolr, is found from the equality of the inflation-adjusted increase multiplier at the gross rate to the increase multiplier at the real interest rate

(68)

where

(69)

Compound interest compounding

Extended at compound interest the amount by the end of the loan term, taking into account the fall in the purchasing power of money (i.e. in constant rubles) will be

(70)

where the price index is determined by expression (65) or (66), depending on the volatility or constancy of the inflation rate.

In this case, the fall in the purchasing power of money is compensated by the ratei= hensuring equalityC= P.

Apply two ways to compensate for losses from a decrease in the purchasing power of money when compound interest is calculated.

A) Interest rate adjustment, according to which the increase is made, by the amount inflation premium. The interest rate increased by the inflation premium is called the gross rate. We will denote it by the symbolr. Assuming that the annual inflation rate is equal toh, we can write the equality of the corresponding increment factors

(71)

Where i - real rate.

From here we get the Fisher formula

r=i+h+ih. (72)

That is, the inflation premium is equal toh+ ih.

B) Indexation of the original amount P . In this case the amountPadjusted according to the movement of a pre-agreed index. Then

S=PJ p (1+i) n. (73)

It is easy to see that in both case A) and case B) we ultimately arrive at the same growth formula (73). In it, the first two factors on the right side reflect the indexation of the original amount, and the last two reflect the adjustment of the interest rate.

Measuring the real interest rate

In practice, we also have to solve the inverse problem - finding the real interest rate in conditions of inflation. From the same relationships between the increment multipliers, it is easy to derive formulas that determine the real rateiat a given (or announced) gross rate r.

When calculating simple interest, the annual real interest rate is equal to

(74)

When calculating compound interest, the real interest rate is determined by the following expression

(75)

Practical applications of the theory

Let's look at some practical applications of the theory we've discussed. Let us show how the formulas obtained above are applied when solving real problems of calculating the efficiency of some financial transactions, compare different calculation methods.

Currency conversion and interest calculation

Let's consider combining currency conversion (exchange) and increasing simple interest, compare the results from direct placement of existing Money into deposits or after preliminary exchange for another currency. There are a total of 4 options for increasing interest:

1. No conversion. Currency funds are placed as a foreign currency deposit, and the initial amount is increased at the foreign exchange rate by directly applying the simple interest formula.

2. With conversion. Original currency are converted into rubles, the increase is at the ruble rate, at the end of the operation the ruble amount is converted back to the original currency.

3. No conversion. The ruble amount is placed in the form of a ruble deposit, on which interest is accrued at the ruble rate using the simple interest formula.

4. With conversion. The ruble amount is converted into any specific currency, which is invested in a foreign currency deposit. Interest is calculated at the foreign exchange rate. The accrued amount is converted back into rubles at the end of the operation.

Transactions without conversion are not difficult. In an accrual operation with double conversion, there are two sources of income: interest accrual and exchange rate changes. Moreover, interest accrual is an unconditional source (the rate is fixed, we are not considering inflation yet). A change in the exchange rate can be either in one direction or the other, and it can be both a source additional income, and lead to losses. Next, we will specifically focus on two options (2 and 4), which provide for double conversion.

Let us first introduce the following NOTATION:

Pv- deposit amount in foreign currency,

P r- deposit amount in rubles,

S v- accrued amount in foreign currency,

Sr- accrued amount in rubles,

K 0 - exchange rate at the beginning of the operation (currency rate in rubles)

K 1 - exchange rate at the end of the transaction,

n- deposit term,

i- accrual rate for ruble amounts (in the form of a decimal fraction),

j- accrual rate for a specific currency.

OPTION:CURRENCY ® RUBLES ® RUBLES ® CURRENCY

The operation consists of three stages: exchanging currency for rubles, increasing the ruble amount, and converting the ruble amount back into the original currency. The accrued amount received at the end of the transaction in foreign currency will be

.

As you can see, the three stages of the operation are reflected in this formula in the form of three factors.

The growth multiplier taking into account double conversion is equal to

,

Where k= K 1 / K 0 - rate of growth of the exchange rate over the period of the operation.

We see that the growth factormis linearly related to the rateiand inverse with the exchange rate at the end of the transactionK 1 (or with the growth rate of the exchange ratek).

Let us theoretically study the dependence of the total profitability of an operation with double conversion according to the CURRENCY scheme® RUBLES ® RUBLES ® CURRENCY from the ratio of final and initial exchange ratesk .

The simple annual interest rate, characterizing the profitability of the operation as a whole, is equal to

.

Let us substitute into this formula the previously written expression forS v

.

Thus, with increasingk profitabilityi eff falls along a hyperbola with asymptote -1 / n . See fig. 2.

Rice. 2.

Let us examine the singular points of this curve. Note that whenk =1 the profitability of the operation is equal to the ruble rate, i.e.i eff = i . Atk >1 i eff < i , and whenk <1 i eff > i . In Fig. 1 can be seen, at some critical valuek , which we denote ask * , the profitability (efficiency) of the operation turns out to be zero. From equalityi eff =0 we find thatk * =1+ ni , which in turn meansK * 1 = K 0 (1+ ni ).

CONCLUSION 1: If the expected valuesk orK 1 exceed their critical values, then the operation is clearly unprofitable (i eff <0 ).

Now let's define maximum allowed exchange rate at the end of the transaction K 1 , in which the efficiency will be equal to the existing rate on deposits in foreign currency, and the use of double conversion does not provide any additional benefit. To do this, let’s equate the growth factors for two alternative operations

.

From the written equality it follows that

or

.

CONCLUSION 2: A currency deposit through conversion into rubles is more profitable than a foreign currency deposit if the exchange rate at the end of the transaction is expected to be lowermaxK 1 .

OPTION:RUBLES® CURRENCY® CURRENCY® RUBLES

Let us now consider the option with double conversion, when the original amount is in rubles. In this case, the three stages of the operation correspond to three factors of the following expression for the accumulated amount

.

Here, too, the increase multiplier linearly depends on the rate, but now on the foreign exchange interest rate. It also depends linearly on the final exchange rate.

Let us conduct a theoretical analysis of the effectiveness of this double conversion operation and determine the critical points.

.

From here, substituting the expression forSr , we get

.

Performance Indicator Dependencei eff fromk linear, it is shown in Fig. 3

Rice . 3.

At k=1 i eff =j , at k>1 i eff >j , at k<1 i eff .

Let us now find the critical valuek * , at whichi eff =0 . It turns out to be equal

or .

CONCLUSION 3: If the expected valuesk orK 1 is less than its critical values, then the operation is clearly unprofitable (i eff <0 ).

Minimum permissible valuek (the rate of growth of the exchange rate over the entire period of the operation), providing the same profitability as a direct deposit in rubles, is determined by equating the increase multipliers for alternative operations (or from the equalityi eff = i )

,

where min ormin .

CONCLUSION 4: Depositing ruble amounts through currency conversion is more profitable than a ruble deposit if the exchange rate at the end of the transaction is expected to be higherminK 1 .

Now let's look at combining currency conversion and increasing compound interest. Let's limit ourselves to one option.

OPTION:CURRENCY® RUBLES® RUBLES® CURRENCYk =1 i uh = i , atk >1 i uh < i , and whenk <1 i uh > i .

Critical valuek , at which the efficiency of the operation is zero, i.e.i uh =0 ,

is defined ask * =(1+ i ) n , which means that the average annual growth rate of the currency exchange rate is equal to the annual growth rate at the ruble rate: .

CONCLUSION 5: If the expected valuesk orK 1 more than its critical values, then the double conversion operation in question is clearly unprofitable (i uh <0 ).

Maximum allowed valuek , at which the profitability of the operation will be equal to the profitability of direct investment of foreign currency funds at the rate

Financial Transaction Outline

Financial or credit operations require a balance of investments and returns. The concept of balance can be explained in a graph.

Rice. 5.

Let the loan amountD 0 issued for a periodT . During this period, for example, two interim payments are made to repay the debtR 1 AndR 2 , and at the end of the term the balance of the debt is paidR 3 , bringing the balance of the operation.

Over time intervalt 1 debt increases toD 1 . In the momentt 1 debt is reduced toK 1 = D 1 - R 1 etc. The operation ends with the creditor receiving the balance of the debtR 3 . At this point, the debt is fully repaid.

Let's call it type b) outline of a financial transaction. A balanced operation necessarily has a closed loop, i.e. the last payment completely covers the balance of the debt. The transaction outline is usually used when repaying debt through partial interim payments.

Successive installment payments are sometimes used to pay off short-term obligations. In this case, there are two methods for calculating interest and determining the balance of debt. The first one is called actuarial and is mainly used in transactions with a deadline more than a year. The second method is called merchant's rule. It is usually used by commercial firms in transactions with a deadline no more than a year.

Comment: When calculating interest, as a rule, ordinary interest is used with an approximate number of days of time periods.

Actuarial method

The actuarial method involves the sequential calculation of interest on the actual amounts of debt. The partial payment goes primarily to repay the interest accrued on the payment date. If the payment amount exceeds the amount of accrued interest, then the difference goes to repay the principal amount of the debt. The outstanding balance of the debt serves as the basis for calculating interest for the next period, etc. If the partial payment is less than the accrued interest, then no offsets are made against the amount of the debt. This receipt is added to the next payment.

For the case shown in Fig. 5 b), we obtain the following calculation formulas for determining the debt balance:

K 1 =D 0 (1+t 1 i)-R 1; K2 =K 1 (1+t 2 i)-R 2; K2 (1+t 3 i)-R 3 =0,

where are the time periodst 1 , t 2 , t 3 - are given in years, and the interest ratei - annual.

Merchant Rule

The merchant rule is another approach to calculating installments. There are two possible situations here.

1) If the loan term does not exceed, the amount of the debt with interest accrued for the entire period remains unchanged until full repayment. At the same time, partial payments are accumulated with interest accrued on them until the end of the term.

2) In case the period exceeds a year, the above calculations are made for annual period of debt. At the end of the year, the accumulated amount of partial payments is subtracted from the debt amount. The balance is repaid next year.

With a total loan termT £ 1 algorithm can be written as follows

,

WhereS - balance of debt at the end of the term,

D - increased amount of debt,

K - increased amount of payments,

R j - amount of partial payment,

t j - time interval from the moment of payment to the end of the term,

m - number of partial (interim) payments.

Variable invoice amount and interest calculation

Let's consider a situation where a savings account is opened at a bank, and the account amount changes during the storage period: funds are withdrawn, additional contributions are made. Then in banking practice, when calculating interest, a calculation method is often used to calculate the so-called percentage numbers. Every time the amount in the account changes, a percentage number is calculatedCj over the past periodj , during which the amount in the account remained unchanged, according to the formula

,

Wheret j - durationj -th period in days.

To determine the amount of interest accrued for the entire period, all interest numbers are added up and their sum is divided by a constant divisorD :

,

WhereK - time base (number of days in a year, i.e. 360 or 365 or 366),i - annual simple interest rate (in%).

When closing the account, the owner will receive an amount equal to the last value of the amount in the account plus the amount of interest.

Example 14.

Let a demand account be opened on February 20 in the amountP 1 =3000 rubles, the interest rate on the deposit wasi =20% per annum. The additional contribution to the account wasR 1 =2000 rub. and was done on August 15th. Withdrawal from the account in the amountR 2 =-4000 rub. recorded on October 1, and the account was closed on November 21. It is required to determine the amount of interest and the total amount received by the depositor upon closing the account.

Solution.

We will carry out the calculation according to the scheme (360/360). There are three periods during which the amount in the account remained unchanged: from February 20 to August 15 (P 1 =3000, t 1 =10+5*30+15=175), from August 15 to October 1 (P 2 = P 1 + R 1 =3000+2000=5000 rub.,t 2

The amount payable upon account closure is

P 3 +I=1000+447.22=1447 rub. 22 cop.

Now we will show the connection of this technique with the simple interest formula. Let us consider the example presented above in algebraic form.

Cwe find the amount paid upon closing the account as follows:

Thus, we have obtained an expression from which it follows that for each amount added to or withdrawn from the account, interest is charged from the moment the corresponding transaction is completed until the account is closed. This scheme corresponds to the merchant rule discussed in Section 6.2.

Changing the terms of the contract

In practice, there is often a need to change the terms of the contract: for example, the debtor may ask for a deferment of the debt repayment period or, on the contrary, express a desire to repay it ahead of schedule; in some cases, there may be a need to combine (consolidate) several debt obligations into one, etc. In all these cases, the principle of financial equivalence of old (replaced) and new (replaced) obligations is applied. To solve the problems of changing the terms of the contract, the so-called equivalence equation, in which the amount of replaced payments reduced to any one point in time is equal to the amount of payments under the new obligation reduced to the same date. For short-term contracts, simple interest rates are applied, and for medium- and long-term contracts, compound rates are applied.

People have always thought about their future. They tried and are trying to protect themselves and their children and grandchildren from financial adversity, building at least a small island of confidence in the future. By starting to build it now with the help of small bank deposits, you can ensure stability and independence in the future.

The basic principle of banking operations is that funds can increase only when they are in constant circulation. In order for clients to confidently navigate the field of financial services and be able to correctly select conditions that are beneficial to them in a certain period of time, they need to know a number of simple rules. This article will focus on long-term investments that allow you to receive a significant profit from a relatively small amount of initial capital over a certain number of years or use the deposit further, withdrawing accruals for everyday needs.

To correctly calculate profit, you need to perform simple arithmetic operations based on the formulas below.

Compound interest formula (calculated in years)

For example, you decide to deposit 100,000.00 rubles. at 11% per annum, in order to take advantage of savings in 10 years, which have grown significantly as a result of capitalization. To calculate the total amount, you should use the compound interest calculation method.

The use of compound interest implies that at the end of each period (year, quarter, month) the accrued profit is summed up with the deposit. The amount received is the basis for a subsequent increase in profit.

To calculate compound interest, we use a simple formula:

• S – the total amount (“body” of the deposit + interest) due to be returned to the depositor upon expiration of the deposit;
• P – initial deposit amount;
• n is the total number of transactions for interest capitalization for the entire period of raising funds (in this case it corresponds to the number of years);
• I – annual interest rate.

Substituting the values ​​into this formula, we see that:

after 5 years the amount will be equal to rub.,

and in 10 years it will be rub.

If we were calculating over a short period, then it would be more convenient to calculate compound interest using the formula

• K – number of days in the current year,
• J – the number of days in the period following which the bank capitalizes accrued interest (the remaining designations are the same as in the previous formula).

But for those who find it more convenient to withdraw interest on their deposit monthly, it is better to become familiar with the concept “deposit capitalization”, implying the calculation of simple interest.

The graph shows how capital will grow with the capitalization of interest on the deposit if you invest 100,000.00 rubles. for 10 years at 10%, 15% and 20%

Compound interest formula (calculated in months)

There is another, more beneficial for the client, method of calculating and adding interest rates - monthly. The following formula is used for this:

where n also corresponds to the number of capitalization operations, but is already expressed in months. The interest rate here is additionally divided by 12 because there are 12 months in a year, and we need to calculate the monthly interest rate.

If this formula were used to calculate the deposit quarterly, then the annual interest would be divided by 4, and the indicator n would be equal to the number of quarters, and if the interest was calculated by half-year, then the interest rate would be divided by 2, and the designation n would correspond to the number of half-years.

So, if we made a contribution in the amount of 100,000.00 rubles. with monthly interest capitalization, then:

in 5 years (60 months) the deposit amount would increase to 172,891.57 rubles, which is approximately 10,000 rubles. more than in the case of annual capitalization of the deposit; rub.

and after 10 years (120 months) The “increased” amount would be 298,914.96 rubles, which is already a full 15,000 rubles. exceeds the figure calculated using the compound interest formula, which provides for calculation in years.

rub.

This means that the profitability when interest is calculated monthly is greater than when interest is calculated once a year. And if the profit is not withdrawn, then compound interest works to the benefit of the investor.

Compound interest formula for bank deposits

The compound interest formulas described above are most likely illustrative examples for clients to understand how compound interest is calculated. These calculations are somewhat simpler than formula applied by banks to actual bank deposits.

The unit used here is the interest rate coefficient for the deposit (p). It is calculated like this:

Compound interest (“accumulated” amount) for bank deposits is calculated using the following formula:

Based on it and taking the same data as an example, we will calculate compound interest using the banking method.

First, we determine the interest rate coefficient for the deposit:

Now we substitute the data into the main formula:

rub. – this is the amount of the deposit that has “grown” over 5 years*;

rub. – in 10 years*.

*The calculations given in the examples are approximate, since they do not take into account leap years and different numbers of days in a month.

If we compare the amounts from these two examples with the previous ones, they are somewhat smaller, but still the benefit from interest capitalization is obvious. Therefore, if you are determined to put money in the bank for a long period of time, then it is better to do a preliminary calculation of profit using the “banking” formula - this will help you avoid disappointments.