How to Calculate a Loan Payment

No Gimmicks, No Pop Ups. Just type in your loan amount, interest rate and term and press compute to get an estimate of the corresponding monthly payment.

Click here for Loan Payment Estimator!!!

No Gimmicks, No Pop Ups. Just type in your loan amount, interest rate and term and press compute to get an estimate of the amortization schedule of your loan.

Click here for Loan Amortization!!!

Why an estimate? Different banks amortize loans differently. You may have a longer or shorter period in which to make your first payment than the subsequent payments. Your bank may accrue interest daily, monthly or another way. Banks usually round the numbers causing one payment to be more or less than others. These items make it impossible to do a schedule that would fit all loans. However, this schedule will be very close. Most likely within pennies.

Time Value of Money

The time value of money is the basis for the finance world. Simply put, it is better to have a dollar today than tomorrow. Although there are several reasons for this, we are going to focus on the fact that we can use that dollar to make more money.

Today the bank is the one with the dollar. You want that dollar from the bank to use today and return to it later. The bank would rather keep its dollar to use to make more money. In order to get the bank to lend you its dollar, you have to promise to give them more than a dollar in the future (ie the more money referred to earlier). The amount that the bank wants is usually stated as a percent of the amount you have borrowed from it which is called the interest rate.

Interest Rate and Discounting

We are going to start with a very simple loan. We are going to borrow a dollar from the bank and pay them back in one year with interest of 10 cents. This can be shown mathematically as:

Loan + Interest = Payment
$1 + $0.10 = $1.10

Given this loan is so simple, we can easily see a few things.

First, the interest rate is 10%. This is computed by dividing the interest amount by the loan amount (ie $0.10 divided by $1).

This leads to the concept of discounting. Discounting asks what is an amount of money a year from now worth today given a certain interest rate. From the example, you can see that $1.10 in a year is worth $1 today at 10% interest. This can be shown mathematically as:

Future Amount - Interest = Today's Equivalent
$1.10 - $0.10 = $1

How did we determine the interest? It was determined by multiplying the interest rate by the loan amount(ie $1 x 10% = $0.10).

Since mathematicians like a lot of variables, lets add some. For those of you not so good at math, a variable is just a letter, symbol or other form of representing something to make writing it easier. As an example, we can use the letter "i" to represent the interest rate. "i" is much shorter and easier to write than interest rate, especially if you need to write it 100 times.

Other variables to get started are:

Loan Amount variable = LA
Payment = P

We can now write the first equation shown earlier as:

LA + (i x LA) = P or
Loan Amount + (interest rate * Loan Amount) = Payment

Using a little algebra, we can change the formula as follows:

LA x (1 + i) = P or

LA = P / (1+i)

This last equation is the key to computing your loan payment.

Computing the Loan Payment

Now lets make a slightly more complicated loan. Our loan will now be 2 equal payments over a 2 year period. We can show this mathematically as follows:

LA = P / (1 + i) + P / ((1+i) x (1+i))

Notice the first payment is discounted one year for interest as before. However, the second payment is discounted twice. This is because it is 2 years into the future. Keeping with this logic a payment that is 3 years into the future would be discounted 3 times.

Now if you know a little bit of algebra, you can probably solve for the payment given the other variables.

ie
LA = $10,000
i= 10%

10,000 = P / (1 + 10%) + P / ((1 + 10%) x (1 + 10%))

P = $5,761.90

If your algebra is not so good, then just plug my answer into the equation where you see P and see what you get for the loan amount. I have rounded my answer to the nearest penny so you should find that your answer shouldbe a fraction of a penny from $10,000.

Now, not too many people take out a two payment loan and this calculation could get tedious for a 360 payment loan. Now it is a good thing that most mathematicians are lazy (unless trying to solve something new). Those mathematicians have developed a trick for simplifying the longer formula.

Simplifying the Formula

Lets start with the loan from above but for 3 years expressed as:

LA = P / (1 + i) + P / ((1+i) x (1+i)) + P / ((1+i) x (1+i) x (1+i))

I am going to use a little algebra to rewrite as:

LA = P x ( 1 / (1 + i) + 1 / ((1+i) x (1+i)) + 1 / ((1+i) x (1+i) x (1+i)) )

Now I can be a bit lazy myself so I am going to introduce a few more variables so I do not have to write so much. I am going to let:

1 / (1 + i) = v

and

v x v = v ^ 2, v x v x v = v^3 and so on

Now I can rewrite the loan from a bove as:

LA = P x ( v + v^2 + v^3)

Using a little more algebra:

LA = P x v x (1 + v + v^2)

Now for the trick. If you know that any number multiplied by one is the same thing and:

(1- v) / (1 - v) = 1

Then we can rewrite the equation without changing the answer as:

LA = P x v x (1- v) / (1 - v) x (1 + v + v^2)

Even more algebra:

LA = P x v / (1 - v) x (1 + v + v^2 - v - v^2 - v^3)

You can see from this that the formula simplifies to:

LA = P x v / (1 - v) x (1 - v^3)

If you have a lot of payments, lets say n payments, then the formula simplifies to:

LA = P x v / (1 - v) x (1 - v ^ n)

Using a little algebra you can rearrange and simply further to:

Payment = Loan Amount divided by (1 - v^n) / i

As a note, this article assumes annual payments. If you want to have monthly payments and are given an annual interest rate, your i would be the annual interest rate divided by 12.

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