# Derek wishes to retire in 15 years, at which time he wants to have accumulated enough money to...

## Question:

Derek wishes to retire in 15 years, at which time he wants to have accumulated enough money to receive an annual annuity of $31,000 for 20 years after retirement. During the period before retirement, he can earn 12% annually, while after retirement he can earn 14% on his money.

What annual contributions to the retirement fund will allow him to receive the $31,000 annuity?

## Future Value and Present Value of Annuity:

The future value of an annuity is the lump sum at a future date that is equivalent to the series of payments. The present value, on the other hand, is the equivalent lump sum amount measured at current day's dollars.

## Answer and Explanation:

The annual contribution is $ 5,507.47.

The annual contribution is such that the future value of the contributions after 15 years, is equal to the value of the future withdrawals measured at the end of 15 years. We can use the following formula to compute the present value of an annuity with periodic payment {eq}M {/eq} for {eq}T{/eq} periods, given periodic return {eq}r{/eq}:

- {eq}\displaystyle \frac{M(1 - (1 + r)^{-T})}{r} {/eq}

Applying this formula, we can find the annual contribution (denoted by {eq}M{/eq} ) by solving the following equation:

- {eq}\dfrac{M*((1 + 12\%)^{15} - 1 )}{12\%} = \dfrac{31,000*(1 - (1 + 14\%)^{-20})}{14\%} {/eq}

- {eq}37.27971466 * M = 205317.0471 {/eq}

- {eq}M = 5,507.47 {/eq}

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from Business 110: Business Math

Chapter 8 / Lesson 3