# The formula and calculation of the annuity loan payment

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By default, the Annuity Calculator calculates payments by loan amount, by loan term in months and annual interest rate. Using the appropriate form fields, you can change the calculator settings.

Using Advanced settings, you can take into account in the calculation of annuity payments the commission when issuing a loan, the monthly commission and the amount of monthly insurance.

If you plan to get a loan with the Last installment, enter the amount of the last installment in the corresponding field.

For the convenience of constructing a schedule of annuity payments, you can change the date of issue of the loan and the date of the first payment.

To calculate a mortgage loan, use a mortgage calculator.

## The formula for calculating annuity payments

Are you sure you want to see the annuity payment formula? Ok, here she is: P - monthly payment on an annuity loan (the same annuity payment that does not change during the entire loan repayment period),
S - credit amount,
i - monthly interest rate (calculated according to the following formula: annual interest rate / 100/12),
n - the period for which the loan is taken (the number of months is indicated).

At first glance, this formula may seem scary and incomprehensible. On the other hand, is it necessary to understand it? You just need to calculate the amount of the annuity payment, right? And what is needed for this? That's right, you just need to substitute your values ​​in the formula and make calculations. Let’s do it now!

## Calculation of the annuity loan payment

Let's say you decide to take a loan 50 000 rubles on 12 months under 22% per annum Naturally, the type of repayment will be annuity. You need to calculate the amount of monthly installments on the loan.

Let's start by beautifully formatting our initial data (we will need them not only in this, but also in further calculations):

Credit amount: 50 000 rub.
Annual interest rate: 22%.
Loan terms: 12 months.

So, before proceeding with the calculation of the annuity payment, you need to calculate the monthly interest rate (in the formula, it is hidden under the symbol i and calculated as follows: annual interest rate / 100/12). In our case, we get the following:

Now that we have found the value of i, you can begin to calculate the size of the annuity payment on our loan:

Using simple mathematical calculations, it turned out that the amount of monthly deductions on our loan will be equal to 4680 rubles.

In principle, this could end our article, but you probably want to know more. True? Tell me, do you want to know how much of the interest on the loan is paid, and what is the body of the loan? Anyway, how much will you overpay on a loan? If so, then we continue!

## Annuity payment

So, with an annuity loan repayment schedule, you pay the same amount every month, regardless of the outstanding balance. Another way to make monthly payments is to use a differentiated payment method. For comparison, with a differentiated loan repayment scheme, the principal amount is paid monthly in equal installments, and interest is calculated on the outstanding balance. In this case, the amount of the monthly payment decreases in the process of repaying the loan.

For example, the amount of interest for the first month of using the loan is:

S% 1 = S * i

• where S% 1 is the amount of interest for the first month,
• S - loan amount.
• i - interest rate on the loan per month (calculated as annual, divided by 12 months).

For the second and next months:

S% n = (S - ∆S) * i

• where ∆S is the amount of principal repaid.

## Loan repayment schedule by annuity payments

First, we will show you the annuity payment schedule itself, analyze it together with you, and then we will tell you in detail about how and by what formulas we calculated it.

Here is the annuity repayment schedule of our loan: And this is a diagram (for clarity): Both the schedule and the chart confirm what is written in the publication: What are annuity payments. If for some reason you did not read it, then be sure to do it - you will not regret it. And those who read can be sure that in the annuity schedule of loan repayments, payments are made in equal amounts, at the initial stage, the percentage of interest on the loan is the highest, and towards the end of the term it is significantly reduced.

Please note that the loan body is repaid from the first month of lending. It’s just that on some sites you can read something like this: “With an annuity loan repayment scheme, interest is paid first, and only then the loan body itself.” As you can see, this statement is not true. It would be more correct to say this:

Annuity payments contain at the initial stage a high proportion of interest on the loan.

The body of the loan is also repaid from the first month of lending. Thus, the amount of debt is reduced and, accordingly, the amount of interest payments on the loan.

Now let's take a closer look at our annuity payment schedule. As you can see, our monthly payment is 4680 rubles. It is this amount that we will pay to the bank every month throughout the entire loan term (in our case, throughout 12 months) As a result, the total amount of payments will be 56 157 rubles. On credit, we took 50 000 rubles (in the chart, this is the fourth column, which is called “Repayment of the loan body”). It turns out that the overpayment for this loan will be 6157 rubles. Actually, this is the interest on the loan, which are indicated in the third column of our annuity payment schedule. It turns out that the effective interest rate (or the full cost of the loan) with us will be - 12,31%. Let's make this information “beautiful”:

Monthly annuity payment: 4680 rub.
Loan body: 50 000 rub.
Total Payout: 56 157 rub.
Overpayment (interest) on the loan: 6157 rub.
Effective interest rate: 12,31%.

So, we have analyzed the annuity payment schedule. It remains to understand how the percentage and the share of the loan body in monthly payments are calculated. That's why in the first month, interest is exactly 917 rubles, in the second - 848 rubles, in the third - 777 rubles etc.? Do you want to know? Then read on!

## Calculation of interest on annuity payments

This formula will help you calculate the percentage of interest in annuity payments:

In - the amount in the annuity payment, which is used to repay interest on the loan,
Sn - the amount of the remaining loan debt (loan balance),
i - the monthly interest rate already familiar to you (in our case, it is - 0.018333).

For clarity, let's calculate the percentage of interest in the first payment on our loan:

Since this is the first payment, the amount of the remaining loan debt is the entire loan - 50 000 rub. Multiplying this amount by the monthly interest rate - 0.018333we will get 917 rub - the amount indicated in our schedule.

When calculating the amount of interest in the next annuity payment, the monthly interest rate multiplies the debt that formed at the end of the previous month (in our case, 46 237 rub.) The result is 848 rub - the percentage of interest in the second annuity payment. According to the same principle, interest is calculated in other payments. Next, let's calculate the component in annuity payments, which will be used to repay the loan body.

## Calculation of the share of the loan body in annuity payments

Knowing the percentage of interest in an annuity payment, you can easily calculate the share of the loan body. The calculation formula is simple and clear:

S - the amount in the annuity payment, which is used to repay the loan body,
P - monthly annuity payment,
In - the amount in the annuity payment, which is used to repay interest on the loan.

As you can see, there is nothing complicated. In fact, an annuity payment contains two components:

1. 1. Percentage of the loan.
2. 2. Body loan share.

If we know the size of the annuity payment itself and the size of the percentage share, then the repayment of the loan body in this payment will go what remains after the deduction of the interest amount from it.

The calculation of the share of the loan body in our first payment looks like this:

We hope that now everyone understands where in the column “Repayment of the loan body” of our schedule of annuity payments in payments for the first month the amount came from 3,763 rub. Yes, yes, this is exactly what remains after we are out of the amount of the annuity payment (4680 rub.) deducted the amount of interest on the loan (917 rub) The values ​​of this column for the following months are calculated in a similar way.

So, we sorted out the loan body. Now it remains to find out how the debt is calculated at the end of the month (in the annuity payment schedule this is our last column).

## How to calculate debt at the end of the month in the annuity payment schedule

First of all, you need to understand what exactly is your debt on the loan, and what payments contribute to its reduction. In our example, you take a loan 50 000 rubles - this is your duty. Interest overpaid on a loan (6157 rubles) Your debt is not, it is just a reward to the bank for the loan. Thus, we can conclude:

Repayment of interest on a loan does not help to reduce your debt to the bank.

In times of crisis, banks often "meet" their debtors. They say something like this: “We understand that you are in trouble now! Okay, our bank is ready to make concessions to you - you can simply repay the interest to us, but you do not need to repay the loan body itself. All the same people are brothers and must help each other! Blah blah blah…"

At first glance, such an offer may seem advantageous, and the bank itself - “white and fluffy lapuli”. Yeah, no matter how! If you pick up a calculator and carry out simple arithmetic calculations, it immediately becomes clear that the real offer of the bank looks something like this:

“Guys, you got money! There is nothing to be done, this is life! We offer you to become our slave for a while (and maybe forever) - you will pay interest on a monthly loan, and you don’t need to pay off the debt itself (well, so that the amount of interest payments does not decrease). Nothing personal - it's just business, friends! ”

Now remember the main point:

It is the repayment of the loan body that pulls you out of the debt hole. Not interest, namely the body of the loan.

Surely you have already guessed how the debt is calculated at the end of the month in our payment schedule. In general, the formula looks like this:

Sn2 - debt at the end of the month on an annuity loan,
Sn1 - the amount of current debt on the loan,
S - the amount in the annuity payment, which is used to repay the loan body.

Note! When calculating the debt at the end of the month, only the part of the payment that is used to repay the loan body (interest paid is not included) is subtracted from the total amount of current debt.

For clarity, let's calculate what the debt will be at the end of the month on our loan after making the first payment:

So, at the first payment, the current loan debt with us is equal to the entire loan amount (50 000 rub.) To calculate the debt at the end of the month, we do not subtract from this amount the entire monthly payment (4680 rub.), but only the part that went to repay the loan body (3,763 rub.) As a result, our debt at the end of the month will be 46 237 rub., it is on this amount that interest will be charged in the next month. Naturally, they will be less, since the amount of debt has decreased. Now you understand why it is important to repay the loan body?

So, friends, we have figured out the formulas and calculations of annuity payments. We hope that now you have no questions on this topic, and you can easily make all the necessary calculations, as well as draw up a schedule for annuity loan payments. The only thing you would probably like is to somehow automate the settlement process. You will not believe it, but it is possible! Want to know how? Then we proceed to publication: Calculation of annuity loan payments in Excel.

## Calculation of annuity payment

Calculate monthly annuity payment can be according to the following formula:

x - monthly payment, S - initial loan amount, P - (1/12) of the interest rate, N - number of months.

The formula for determining which part of the payment went to repay the loan, and which to pay interest is quite complicated and without special mathematical knowledge it will be difficult for the average man to use it. Therefore, we calculate these quantities in a simple way, giving the same result.

To calculate the percentage component of an annuity payment, you need to multiply the loan balance for the specified period by the annual interest rate and divide all this by 12 (the number of months in a year).

 where - accrued interest, - balance outstanding for the period, P - annual interest rate on the loan.

To determine the part that goes to repay the debt, it is necessary to deduct the accrued interest from the monthly payment.

 s = x -, where s - part of the payment going to pay off the debt, x - monthly payment, - accrued interest, at the time of the nth payment.

Since the part going to repay the main debt depends on previous payments, therefore, the calculation of the schedule, according to this technique, calculate sequentially, starting from the first payment.

### An example of calculating an annuity loan repayment schedule

For example, we calculate the loan payment schedule in the amount of 100,000 p. and an annual interest rate of 10%. The loan repayment term is 6 months.

First, let's calculate the monthly payment.

Then we calculate the monthly percentage and credit part of the annuity payment.

 1 month Interest: 100000 * 0.1 / 12 = 833.33 Principal debt: 17156.14 - 833, 33 = 16322.81 2 month Credit balance: 100000 - 16322.81 = 83,677.19 Interest: 83,677.19 * 0.1 / 12 = 697.31 Principal debt: 17156.14 - 697.31 = 16458.83 3 month Credit balance: 83,677.19 - 16,458.83 = 67,218.36 Interest: 67,218.36 * 0.1 / 12 = 560.15 Principal debt: 17156.14 - 560.15 = 16595.99 4 month Credit balance: 67218.36 - 16595.99 = 50622.38 Interest: 50622.38 * 0.1 / 12 = 421.85 Principal debt: 17156.14 - 421.85 = 16734.29 5 month Credit balance: 50622.38 - 16734.29 = 33888.09 Interest: 33888.09 * 0.1 / 12 = 282.40 Principal debt: 17156.14 - 282.40 = 16873.74 6 month Credit balance: 33888.09 - 16873.74 = 17014.35 Interest: 17014.35 * 0.1 / 12 = 141.79 Principal debt: 17156.14 - 141.79 = 17014.35

If you are interested in knowing the size of the overpayment on an annuity loan, you need to pay monthly, multiply by the number of periods and subtract the initial loan amount from the resulting number. In our case, the overpayment will be as follows:

 17156,14 * 6 – 100000 = 2936,84

The result of calculations according to our example on the website www.platesh.ru will look like this:

Data entry form for annuity payment calculation An example of an annuity payment schedule

Which confirms the correctness of our calculations.

## How to calculate the monthly payment?

The formula for calculating the amount of the monthly payment for an annuity repayment scheme is as follows:

A = K • S

• where A is the amount of the monthly annuity payment,
• K - annuity ratio,
• S - loan amount.

The loan amount is known. And for calculating K - annuity ratio, the following formula is used:

• where i is the interest rate on the loan per month (calculated as the annual divided by 12 months),
• n is the number of periods (months) of loan repayment.

## Calculation Example

Suppose that you need to calculate the monthly loan payment with an annuity repayment schedule at an interest rate of 48% per annum for a period of 4 years in the amount of 20,000,000 rubles. Using the above formula for calculating the monthly payment (A = K • S) and the coefficient K, we calculate the annuity payment.

We have:

1. i = 48% / 12 months = 4% or 0.04
2. n = 4 years * 12 months = 48 (months)
3. S = 20,000,000

We calculate K:

And now we substitute the obtained value in the monthly payment formula:

A = 0.0472 * 20,000,000 = 943,613 rubles.

### Who benefits from annuity?

This is explained by the fact that during the entire loan repayment period, interest is accrued on the initial loan amount. With a differentiated schedule, interest for 100% of the loan amount is paid only in the first month (in the absence of a deferment of payment of the main debt), then interest is charged on the balance, which will result in less loan overpayment.

In other words, among two loans with the same interest rates, maturity and additional fees, a loan with an annuity repayment scheme will always be more expensive.

On the other hand, repayment of debt and interest in equal shares is convenient for the borrower.

Since the monthly payment is constant and does not require clarification of the required contribution amount at the bank.

While with a differentiated schedule, each month the amount of payment will be different.

## Calculation of annuity payment

A = P * (1 + P) ^ N / ((1 + P) ^ N - 1)

• where A is the annuity coefficient,
• P - monthly interest rate (annual interest rate divided by 12),
• N - the number of months for which a loan is issued.

Thus, we obtained the most important parameter of the loan calculation formula. Now you can get the size of the monthly payment by multiplying the loan amount by our ratio.

Annuity ratio:

• Ak = 0.01 * (1.01) ^ 10 / (1.01) ^ 10 - 1 = 0.01 * 1.105 / 1.105 - 1 = 0.01105 / 0.105 = 0.1052381
• Annuity payment: A * 100,000 = 0.1052381 * 100,000 = 10523.81

This annuity does not include any third-party payments (insurance, fees, etc.). But since additional payments usually do not change during the loan term, you can simply add them to the annuity after receiving your monthly payment.

Проще всего воспользоваться кредитными калькуляторами, которых полно на просторах интернета, чтобы не считать аннуитет самому. Да и банки при оформлении кредита обязаны предоставить вам график ежемесячных платежей. А вы можете проверить, посчитав аннуитет, насколько реальная ставка процента отличается от указанной в рекламе.

### Формула аннуитетного платежа

When you take a loan from a bank, you are obligated to pay the amount of the loan taken and interest on it within a certain time period. There are several ways to repay a loan, a common way is annuity payments. In this article, we will consider what annuity payments are, find out the annuity payment formula, and make a calculation.

That is, with an annuity payment, you pay the same amount every month (credit + interest on it) regardless of the remaining amount of debt.

Another way to repay the loan is through differential payment, that is, payment of interest on the remaining debt. With differentiated payments, your monthly repayment amount will decrease by the end of the loan term, since you will pay interest on the loan for the remaining amount of debt.

For example, paying off 80% of the loan, you will pay interest on the remaining amount (20%).

For the banks themselves, it is more profitable to use annuity payments, since in this case they receive more interest income.

For borrowers, annuity payments are more profitable in the sense that it is more convenient to pay the same amount every month than each time a different amount and specify how much he needs to pay in the next month.

In accordance with the annuity payment formula, the amount of periodic (monthly) payments will be:

A = K • S

• where A is the monthly annuity payment,
• K - annuity ratio,
• S - loan amount.

The annuity ratio is calculated by the following formula:

K = (i * (1 = i) n): ((1 + i) n-1)

• where i is the monthly interest rate on the loan (= annual rate / 12),
• n - the number of periods during which the loan is paid.

If the interest rate is 12% per annum, then the monthly rate:

### Payment calculation

Initial data:

• S = 30,000 rubles
• i = 1.5% (18% / 12 months) = 0.015
• n = 36 (3 years x 12 months)

We substitute these values ​​in the formula and determine the annuity coefficient:

K = (0.015 * (1 + 0.015) 36): ((1 + 0.015) 36-1) = 0.03615

Amount of monthly payments:

A = K * S = 0.03615 * 30000 = 1084.57 rubles.

### Annuity payments early repayment

I want to repay part of the loan ahead of schedule. The bank offers me two options: reducing the loan term and reducing monthly payments. Which option is more profitable? In fact, in addition to the two methods mentioned by the reader for early repayment of a loan, there are others. Below we will consider them in more detail. But first, you need to recall the bank loan options offered by banks, because possible options for early repayment depend on this.

## Different graphics

In the case of annuity, the client pays the bank the same amount every month, which includes repayment of the principal and interest. Most banks prefer annuity payments because they allow you to earn more on interest (the reason is that the body of the debt decreases more slowly than with differentiated payments).

However, this payment option has advantages. Firstly, the financial burden at the initial stage of loan payments in the case of annuity is less. Secondly, at the same income level, annuity payments allow you to take a larger loan amount compared to differentiated payments (this is a consequence of the lower financial burden at the initial stage of payments).

The monthly payment is gradually reduced as payments are made, because interest is accrued on the constantly contracting body of debt. We have already mentioned that differentiated payment is more beneficial in terms of minimizing interest payments.

However, there are also disadvantages, which, in fact, are the reverse side of the advantages of annuity: with differentiated payments, the maximum possible loan size is slightly less than with annuity payments, and at the beginning of lending there is an increased debt burden.

For clarity, we give a specific example of differentiated and annuity payments for loans in the amount of 100 thousand rubles at 11% per annum for a period of 6 months (see tables).

As can be seen from the example, in the case of a differentiated payment (without early repayment), the client will pay 24.4 rubles less than with an annuity.

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