Annuities and loans

Question

You want to be able to withdraw 40,000 each year for 15 years. Your account earns 8% interest. How much do you need in your account at the beginning?

Answer 1

Let P = the amount of money deposited at the beginning of the first year (unknown)

Let W = the amount that is withdrawn at the beginning of each year starting the second year. ($40,000)

Interest = 8% and 1.08 is the total yearly amount after interest has been applied. So, a table of the beginning and end of the year would look like this....

First of year End of Year

First year Deposit of W 1.08W

Second Year 1.08P - W 1.08²P - 1.08W

Third Year 1.08²P-1.08W-W 1.08³P-1.08²W-1.08W

Fourth Year 1.08³P-1.08²W-1.08W-W You can see the trend and we want the total money to

be zero at the beginning of the 16th year after the last $40,000 is removed!

So, the series up to the 16th year is as follows:

Sum = 1.08¹⁵P -1.08¹⁴W-1.08¹³W- ..................... - 1.08W-W

= 1.08¹⁵P -W{ 1.08¹⁴ + 1.08¹³ + ............. + 1.08 + 1 }

If you sum what is in the brackets, you obtain (1.08¹⁵ -1)/(.08) = 27.152 (Details not shown here)

Substituting, we get Sum = 1.08¹⁵P - 27.152W and this will be zero

So 1.08¹⁵P = 27.152(40,000) = 1,086,080

Solving for P we have P = $342,377.71

There are also standard formulas to make this calculation but I thought you may find it interesting to see where those formulas come from!