Number of Payments Funded by an Annuity

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

Number of Payments Funded by an Annuity:

annuity formula jeff
Q. Jeff has $400,000 invested in a savings account at 4% a year. If he takes an annual income of $30,000 from this account, how long will it be before he runs out of money?

A. Jeff will run out of money in 19.4 years.

This could be calculated by inserting the numbers in the following annuity formula and rearranging it to find n:


We've done this for you and here is the equation in terms of n:


Using the annuity formula to answer the question:

• We wish to calculate n, the number of payments •

PV, the present value, is $400,000 •

RP, the regular payment, is $30,000 •

i, the interest rate, is 4%, which we write as 0.04 •

We now put these numbers into the equation and find:

n = [-log(1 - 0.04 x 400000/30000)]/[log(1 + 0.04)]

Therefore n = [-log(1 - 0.533333333)]/[log(1.04)]

And so n = 0.3309932/0.0170333

Hence n = 19.4


Now here's a practice question for you:

Q. John has $1,000,000 invested and obtains an annual return of 8%. If he draws down and spends $100,000 a year, when will he run out of money?

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























A. John will run out of money after 20.9 payments - in other words his money will last for almost 21 years.