321.

223.4

4.You're planning to retire in 45 years. You figure you can comfortably save $500 per month.

  1. If you earn 12%, compounded monthly, what will you have when you retire? (7 pts.)

    2. There are two steps here: First calculate the present value of your annuity. r=.12/12, n=540, P3 = 500

    \( \Large PV = 500 \left [ \frac{1}{\frac{.12}{12}} - \frac{1}{\frac{.12}{12} \left(1+\frac{.12}{12} \right )^{540} } \right ] = 49{,}768.03 \).

    and then figure out what it will be when you retire, the FV of that PV.

    \( \Large FV = 49{,}768.03 \left( 1+ \frac{.12}{12} \right)^{540} = 10{,}727{,}346.52 \)

  2. Assume you plan to inherit $100,000 in 30 years. If you wait 10 years from now to start saving $500 per month, will you still have the same amount to retire? (6 pts.)

    3. If you take the $100,000 and invest it for the remaining 15 years, you will have:

    \( \Large 100{,}000 \left( 1+ \frac{.12}{12} \right)^{180} = 599{,}580.20 \) at retirement.

    This lowers the amount you need from your annuities: \( 10{,}727{,}346.52 - 599{,}580.20 = 10{,}127{,}766.32 \). If you wait 10 years, you will only save:

    \( \Large PV = 500 \left [ \frac{1}{\frac{.12}{12}} - \frac{1}{\frac{.12}{12} \left(1+\frac{.12}{12} \right )^{420} } \right ] = 49{,}234.03 \)

    \( \Large FV = 49{,}234.03 \left( 1+ \frac{.12}{12} \right)^{420} = 3{,}215{,}479.69 \)

    So you will have considerably less.

  3. If you want to retire in 25 years with the same amount of money as in (a), how much will you have to save each month? (5 pts.)

    4. Start by getting the PV of your retirement goal, but only 25 years prior (instead of 45 years prior). That would be only 300 months.

    \( \Large PV = \frac{10{,}727{,}346.52} {\left (1+\frac{.12}{12} \right )^{300}} = 542{,}100.96 \)

    Next use that present value as the PV of your savings annuity (also for 300 months), and solve for the casflow.

    \( \Large PV = \frac{542{,}100.96} { \left [ \frac{1}{\frac{.12}{12}} - \frac{1}{\frac{.12}{12} \left (1+\frac{.12}{12} \right )^{300}} \right ] } = 5709.54 \)

    So the moral is start saving early!

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