Billy Bob, who is 22, won a prize of $5000 at McDonalds. He invests the money at
8% compounded quarterly for 43 years until he retires. When he retires, he then
reinvests the money at 7% compounded monthly and makes equal monthly
withdrawals for a further 25 years at which time the money would run out. How
much money would he get each month? Show all work
This is how I answered:
FV = R (1+i)^n = 5000(1+0.02)^172 where i=0.08/4 and n=43*4
FV = $150729.9473
Then:
PV= P {[(1 + i)^n]- 1 / [1 + i]^n}/i where i=0.07/12 and n = 24*12
$150729.9473 = P {[(1 + 0.07/12)^24*12]-1 / [1+0.07/1]^24*12}/0.07/12
P= $150729.9473 / 141.48690338
P = $1,065.33
However the answer should be 1,044.74 . I think I am doing something wrong with the interest. I know my answer is very close to the right answer but it is still incorrect.
Best Answer
If I´m right in total your equality was more or less
$$5000\cdot 1.02^{43\cdot 4}\cdot (1+\frac{0.07}{12})^{25\cdot 12}=P\cdot \frac{(1+0.07/12)^{25\cdot 12}-1}{\frac{0.07}{12}}$$
As lulu has already mentioned, it has to be $25\cdot 12$. But nevertheless I get the same result, $P=1065.33$. See here the result of the calculator w.a. And I agree with your calculation. I don´t see any mistake.
After testing other variations I´ve found out that they have used the $\texttt{effective interest rate}$ (aka equivalent interest rate) for the $7\%$.
To get the equivalent interest rate one has to solve the fowing equation
$\left(1+\frac{i}{12} \right)^{12}=1.07 \Rightarrow i=(1.07^{1/12}-1)\cdot 12=0.06784974465$
If you replace $0.07$ by the equivalent interest rate you´ll get the desired solution: calculator w.a. I think a little less number of digits is sufficient.