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 Find the size of the withdrawal so that all money has been withdrawn after 8 years.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?


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.)

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  • $\begingroup$ I don't understand your formula for i ? Isn't i = 0.12 $\endgroup$ – user342624 Aug 28 '16 at 17:00
  • $\begingroup$ 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. $\endgroup$ – mlc Aug 28 '16 at 20:45
  • $\begingroup$ 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 $0.06$ $\endgroup$ – Ross Millikan Aug 28 '16 at 21:12
  • $\begingroup$ 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.) $\endgroup$ – mlc Aug 28 '16 at 21:18

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