A mortgage of $200,000 at 6% for 15 years is paid monthly. Find the… Step by Step

Step by step

The question

Question: A mortgage of $200000 at 6% for 15 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $200000.

r is the yearly rate: 6% is 0.06.

P = $200,000 loan, r = 6% = 0.06

n is how many times a year: 12.

t is the time: 15 years.

n = 12 a year, t = 15 years

2 Per payment

Interest is added each month.

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

0.06 divided by 12 is 0.005.

i = rn = 0.0612 = 0.005

i = the rate each month

And N is how many payments: 12 times 15 is 180.

N = n × t = 12 × 15 = 180

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 0.005 is 1.005.

PMT = 200000 × 0.0051 − 1.005⁻¹⁸⁰

4 Work it out

The top: 1000.

With a calculator, 1.005 to the power negative 180 is about 0.407482.

1 minus that is 0.592518.

200000 × 0.005 = 1000

1.005⁻¹⁸⁰ ≈ 0.407482

1 − 0.407482 = 0.592518

Now the top over the bottom.

PMT ≈ 10000.592518

1000 divided by 0.592518 is 1687.7137 and a bit.

PMT ≈ 1687.7137…

Lenders round up to the next cent: 1687 dollars and 72 cents a month.

PMT = $1,687.72

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $200,000 ÷ 180 ✗

5 Check

Let's check with the first month.

The interest that month is 200000 times 0.005: $1000.

month 1 interest: $200,000 × 0.005 ≈ $1,000.00

The rest of the payment, 687 dollars and 72 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$1,687.72 − $1,000.00 = $687.72 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