# The Present Value of Regular Payments

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

Q. Mother Goose breaks into her piggy bank. With the $250,000 she finds there (it's a big piggy bank) she buys a lifetime annuity from an insurance company in return for an annual income of $18,000.
The company invests the money and she receives an annual return of 5%.
If she lives for another 20 years, will her annual income of $18,000 be fully funded by the $250,000?

A. The present value of all the payments Mother Goose will receive is $224,319.79. Therefore, her annual income is fully funded and the insurance company will make a profit.

This was calculated using the following annuity formula:

• We wish to calculate

•

•

•

We now put these numbers into the equation and find:

Therefore

Now here's a practice question for you:

Q. Jack buys a lifetime annuity from an insurance company for $180,000 in return for an annual income of $12,000. The company invests the money and receives an annual return of 6%. If Jim lives for another 40 years, will his annual income of $12,000 be fully funded by the $180,000?

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

A. The present value of Jack's 40 payments of $12,000 is $180,555.56. Hence his payments were not fully funded by the inital annuity of $180,000. The insurance company makes a loss.

**The Present Value of Regular Payments**:

A. The present value of all the payments Mother Goose will receive is $224,319.79. Therefore, her annual income is fully funded and the insurance company will make a profit.

This was calculated using the following annuity formula:

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

• We wish to calculate

**PV**, the present value •

•

**RP**, the regular payment, is $18,000 •

•

**i**, the interest rate, is 5%, which we write as 0.05 •

•

**n**, the number of payments, is 20 •

We now put these numbers into the equation and find:

**PV**= 18000 x [1 - (1 + 0.05)

^{-20}]/0.05

Therefore

**PV**= 18000 x [1 - 0.376889482]/0.05

**Hence PV = $224,319.79**

Now here's a practice question for you:

Q. Jack buys a lifetime annuity from an insurance company for $180,000 in return for an annual income of $12,000. The company invests the money and receives an annual return of 6%. If Jim lives for another 40 years, will his annual income of $12,000 be fully funded by the $180,000?

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

A. The present value of Jack's 40 payments of $12,000 is $180,555.56. Hence his payments were not fully funded by the inital annuity of $180,000. The insurance company makes a loss.