53.9
9a. Let's talk about the importance of investing early. You graduate and get a good job. By the time you're 25, you're in a position to start saving $200 per month for retirement. You plan to retire in 40 years, at the age of 65. You expect to earn 12% per year, compounded monthly, on your investments and you plan to put your first $200 away immediately. How much will you have when you retire? [8 pts]
What I'm asking for is the FV of an annuity due (an annuity whose first payment is today). We do this in 2 steps. First, compute the PV of the annuity and then compute the FV of that PV.
\( \large PV = 200 + 200 \left [ \frac{1}{\frac{.12}{12}} - \frac{1}{\frac{.12}{12} \left(1+\frac{.12}{12} \right )^{480} } \right ] = 20{,}031.43 \).
\( \large 20{,}031.43 \left ( 1 + \frac{.12}{12} \right)^{480} = 2{,}376{,}683.60 \)
9b. Let's say you procrastinate and wait until you're 35 to start saving. You still plan to retire at the age of 65, but now you've only got 30 years to save. How much more will you have to save per month in order to retire with the same amount of money as in part a? Assume that you will make your first payment in a month and make a total of 360 payments. [8 pts]
Here, we need to figure out what payment it would take to reach the FV of $2,376,683.60 that we found in part a. In order to do that, we have to figure out what the PV of $2376683.60 is when we're 35 at the start of this new annuity. Then, we can use that PV to solve for the necessary payments.
\( \large PV = \frac{2{,}376{,}683.60}{ \left ( 1 + \frac{.12}{12} \right)^{360}} = 66{,}111.469 \)
\( \Large CF = \frac{66{,}111.469} { \left [ \frac{1}{\frac{.12}{12}} - \frac{1}{\frac{.12}{12} \left (1+\frac{.12}{12} \right )^{360}} \right ] } = 680.03 \)
So we have to make payments that are $680.03-$200.00=$480.03 more than we would if we just started saving earlier.