# regular payments formula

Jill has received $175000. She is going to deposit this into an account with an annual interest rate of 12% where the interest is compounded semi annually. She will make equal withdrawals every six months. Find the size of the withdrawal so that all money has been withdrawn after 8 years. I know the formula for compound interest but how I would do this question? ## 1 Answer Let$R$be the semiannual withdrawal. The present value of the$T=16$semiannual withdrawals must equal the initial capital$C =175,000$. Using the semiannual rate$i = \sqrt{1.12}-1$(please check if this what you mean by "compounded semi-annually"), you find the equality $$C = \sum_{t=1}^{T} R (1+i)^{-t}$$ Solving for$R$yields $$R = \frac{Ci}{1 - (1+i)^{-T}}$$ (The last step requires computing the sum of a geometric progression.) • I don't understand your formula for i ? Isn't i = 0.12 – user342624 Aug 28 '16 at 17:00 • Your question says that 0.12 is the annual interest rate. Since interest is compounded semi-annually, you need the semi-annual interest rate for the formula to apply. – mlc Aug 28 '16 at 20:45 • Normally$12\%$annual interest compounded semi-annually means$6\%$is paid twice a year, resulting in an effective annual interest of$1.06^2-1 =12.36\%$The$i$in the formula should then be$0.06\$ – Ross Millikan Aug 28 '16 at 21:12
• There are different conventions around the world, and I am not sure which one was implicitly meant. Ross Millikan's suggestion makes sense. (Instead, I assumed that the semi-annually compounded interest rate was meant to yield an effective annual interest of 1.12.) – mlc Aug 28 '16 at 21:18