# Annuity Present Value - Interest and Capital Repayment

On the Annuity Formula homepage, we used the following example to introduce:

Q. Jack and Jill are considering using a $300,000 mortgage to buy a home.
They intend retiring in 20 years, so the mortgage must be paid completely in that time.
They are offered a mortgage of $300,000 with a fixed annual interest rate of 6%.
They can afford to pay $2500 a month.
Will Jack and Jill be able to afford the monthly mortgage payments?

A. This mortgage will cost $2149.29 each month, therefore it is affordable for them.

This was calculated using the following annuity formula:

The sharp eye among you may notice that this is the same equation as The Present Value of Regular Payments, which has been rearranged using a little algebra.

• We wish to calculate

•

•

•

We now put these numbers into the equation and find:

Therefore

and so

Now here's a practice question for you:

Q. Which mortgage will require the lower monthly repayments?

Option 1. $200,000 borrowed for 10 years at 6% per annum.

Option 2. $400,000 borrowed for 20 years at 6% per annum.

Try doing the calculation yourself, then scroll down to check your answer at the bottom of the page.

A. Option 1 requires a monthly payment of $2220.41. Option 2 costs $2865.72 a month, therefore Option 1 has lower monthly repayments.

**Annuity Present Value - Interest and Capital Repayment**:

A. This mortgage will cost $2149.29 each month, therefore it is affordable for them.

This was calculated using the following annuity formula:

The sharp eye among you may notice that this is the same equation as The Present Value of Regular Payments, which has been rearranged using a little algebra.

**Using the annuity formula to answer the question:**

• We wish to calculate

**RP**, the regular payments •

•

**PV**The present value is the capital borrowed, which is $300,000 •

•

**i**, is the interest rate. This is 6% a year. We divide this by 12 to get the rate per month. This is 0.5%, which we write as 0.005 •

•

**n**is the number of payments. This is 20 x 12 = 240 •

We now put these numbers into the equation and find:

**RP**= 300,000 x 0.005/[1 - (1 + 0.005)

^{-240}]

Therefore

**RP**= 300,000 x 0.005/0.06979039

and so

**RP**= 300,000 x 0.0071643

**Hence RP = $2149.29**

Now here's a practice question for you:

Q. Which mortgage will require the lower monthly repayments?

Option 1. $200,000 borrowed for 10 years at 6% per annum.

Option 2. $400,000 borrowed for 20 years at 6% per annum.

Try doing the calculation yourself, then scroll down to check your answer at the bottom of the page.

A. Option 1 requires a monthly payment of $2220.41. Option 2 costs $2865.72 a month, therefore Option 1 has lower monthly repayments.