13.5
5.You would like to have $4 million in your investment account when you retire. If you plan to retire in 45 years, how much must you contribute to your retirement account on a monthly basis if you are to have $4 million in 45 years? Assume that you can earn 15% APR, compounded monthly. (10 pts)
This question starts with the future value of an annuity. In order to solve this problem, we have to consider the things we know how to do: we know how to solve for the payments of an annuity if we’re given the present value and we know how to get the present value of a future value. If we combine the two, we can solve the problem.
We want $4 million when we retire in 45 years and the interest rate is 15% APR, compounded monthly. That is equivalent to saying that we want:
\( \Large PV = \frac{FV}{\left( 1+ r \right)^{n}} = \frac{4{,}000{,}000}{\left( 1+ \frac{.15}{12} \right)^{540}} = 4{,}883.63 \) today
So we need to figure out what payment if made equally every month for 45 years would give us the same present value.
\( \Large P3 = \frac{PV} { \left [ \frac{1}{r} - \frac{1}{r \left (1+r \right )^{n}} \right ] } = \frac{4{,}883.63} { \left [ \frac{1}{\frac{.15}{12}} - \frac{1}{\frac{.15}{12} \left (1+\frac{.15}{12} \right )^{540}} \right ] } = $61.12 \)