I have the following problem.
Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years?
I can tell that this problem is an annuity-immediate. Since the interest compounding period and the withdrawal period are equal the interest rate used would stay at 6% and there would be 40 compounding periods over the 20 years. I tried to use the following formula.
P = 5000/(a 40|.06), => (a 40|.06) = (1-(1.06^-40))/.06
The answer that I get when I do all of my calculations is $332 however this is way off from the listed answer in the back of the book. But from my best understanding, I'm doing everything correctly. Would somebody be able to enlighten me as to what I'm doing wrong?
$\require{enclose}$ The rate is a nominal $i^{(2)} = 0.06$ convertible semiannually, which means that the effective semiannual rate is $j = \frac{i^{(2)}}{2} = 0.03$. Hence the equation of value is $$5000 = X a_{\enclose{actuarial}{40}j} = X \frac{1 - v^{40}}{j} = \frac{1 - (1.03)^{-40}}{0.03} X \approx 23.1148 X,$$ from which it follows that $X \approx 216.312$. This assumes that withdrawals from the investment occur at the end of each $6$-month period.