A home loan of $150,000 at 4.5% for 30 years is paid monthly. Find… Step by Step

Step by step

The question

Question: A home loan of $150000 at 4.5% for 30 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $150000.

r is the yearly rate: 4.5% is 0.045.

P = $150,000 loan, r = 4.5% = 0.045

n is how many times a year: 12.

t is the time: 30 years.

n = 12 a year, t = 30 years

2 Per payment

Interest is added each month.

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

0.045 divided by 12 is 0.00375.

i = rn = 0.04512 = 0.00375

i = the rate each month

And N is how many payments: 12 times 30 is 360.

N = n × t = 12 × 30 = 360

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 0.00375 is 1.00375.

PMT = 150000 × 0.003751 − 1.00375⁻³⁶⁰

4 Work it out

The top: 562.5.

With a calculator, 1.00375 to the power negative 360 is about 0.259896.

1 minus that is 0.740104.

150000 × 0.00375 = 562.5

1.00375⁻³⁶⁰ ≈ 0.259896

1 − 0.259896 = 0.740104

Now the top over the bottom.

PMT ≈ 562.50.740104

562.5 divided by 0.740104 is 760.0280 and a bit.

PMT ≈ 760.0280…

Lenders round up to the next cent: 760 dollars and 3 cents a month.

PMT = $760.03

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $150,000 ÷ 360 ✗

5 Check

Let's check with the first month.

The interest that month is 150000 times 0.00375: 562 dollars and 50 cents.

month 1 interest: $150,000 × 0.00375 ≈ $562.50

The rest of the payment, 197 dollars and 53 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$760.03 − $562.50 = $197.53 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