# Jeremy wishes to withdraw \$750 from his account every six months for the nest 5 years. If the account pays 3.8% per annum compounded semi-annually, how much must he deposit in the account now to provide these withdrawals?

Jeremy has to deposit \$7522.32 initially to be able to withdraw \$750 once every 6 months for the next 5 years. Jeremy would like to withdraw \$750 from his account every six months for the next 5 years. The account pays 3.8% per annum compounded semi-annually. The initial amount to be deposited to enable this has to be determined.

The present value of an amount P at an interest rate r and time period t is given by the equation `PV = P/(1+r)^t` .

Jeremy's account gives an interest rate of 3.8% per annum. As interest is compounded semi-annually, the interest rate for each time period is 3.8/2 = 1.9%. The total number of withdrawals is 2*5 = 10. A withdrawal after n time periods has a present value of `750/(1+0.019)^n`.

The sum of the present value of all the withdrawals by Jeremy is given by the expression

`750/(1+0.019)^10 + 750/(1+0.019)^9+ 750/(1+0.019)^8 + + ... +750/(1+0.019)^1`

= `750*(1/(1+0.019)^10 + 1/(1+0.019)^9+ 1/(1+0.019)^8 + ... + 1/(1+0.019)^1)`

The sum of n terms of a geometric series `a, ar, ar^2 , ... ar^(n-1)` is given by `a*(r^n-1)/(r - 1)` . Using this gives

`750*(1/(1+0.019)^10 + 1/(1+0.019)^9+ 1/(1+0.019)^8 + ... + 1/(1+0.019)^1)`

= `750*(1 - 1/(1+0.019)^11)/(1 - 1/(1+0.019))`

= 7522.32

Jeremy needs to deposit an amount of \$7522 to make a withdrawal of \$750 every 6 months for the next 5 years if he gets an interest of 3.8% compounded semi-annually.

Last Updated by eNotes Editorial on