A home loan of $90,000 at 5.4% for 20 years is paid monthly. Find… Step by Step

Step by step

The question

Question: A home loan of $90000 at 5.4% for 20 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $90000.

r is the yearly rate: 5.4% is 0.054.

P = $90,000 loan, r = 5.4% = 0.054

n is how many times a year: 12.

t is the time: 20 years.

n = 12 a year, t = 20 years

2 Per payment

Interest is added each month.

The rate for one month is i: r over n.

0.054 divided by 12 is 0.0045.

i = rn = 0.05412 = 0.0045

i = the rate each month

And N is how many payments: 12 times 20 is 240.

N = n × t = 12 × 20 = 240

3 Use the formula

The payment each month: PMT = P times i, over 1 minus (1 + i) to the power negative N.

PMT = P × i1 − (1 + i)^−N

PMT = the payment each month

We put the numbers in: 1 plus 0.0045 is 1.0045.

PMT = 90000 × 0.00451 − 1.0045⁻²⁴⁰

4 Work it out

The top: 405.

With a calculator, 1.0045 to the power negative 240 is about 0.340419.

1 minus that is 0.659581.

90000 × 0.0045 = 405

1.0045⁻²⁴⁰ ≈ 0.340419

1 − 0.340419 = 0.659581

Now the top over the bottom.

PMT ≈ 4050.659581

405 divided by 0.659581 is 614.0264 and a bit.

PMT ≈ 614.0264…

Lenders round up to the next cent: 614 dollars and 3 cents a month.

PMT = $614.03

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $90,000 ÷ 240 ✗

5 Check

Let's check with the first month.

The interest that month is 90000 times 0.0045: $405.

month 1 interest: $90,000 × 0.0045 ≈ $405.00

The rest of the payment, 209 dollars and 3 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$614.03 − $405.00 = $209.03 pays the loan down

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Last: check.

Answer