Question

An investor needs $16,000 in 17 years.(a) What amount should be deposited in a fund at the end of each quarter at 8% compounded quarterly so that there will be enough money in the fund?(b) Find the investor's quarterly deposit if the money is deposited at 6% compounded quarterly.The deposit should be $______(Do not round until the final answer. Then round to the nearest cent as needed.)

Answer 1

a) We have to find the amount that have to be deposited each quarter at 8% compounded quarterly to reach $16,000 in 17 years.

We can use the formula for the future value of an annuity as:

`FV=P\cdot((1+r\/m)^(nm)-1)/(r\/m)`

where the annual nominal rate is r = 0.08, the number of periods is n = 17, the number of subperiods per year is m = 4 and the future value is FV = 16000.

Replacing with the known values we can calculate P, the deposit, as:

`\begin{gathered} 16000=P((1+0.08\/4)^(17\cdot4)-1)/(0.08\/4) \\ \\ 16000=P((1.02)^(68)-1)/(0.02) \\ \\ 16000=P\cdot(3.84425-1)/(0.02) \\ \\ 16000=P\cdot(2.84425)/(0.02) \\ \\ P=(0.02)/(2.84425)\cdot16000 \\ \\ P\approx112.51 \end{gathered}`

b) If the nominal interest is 6% instead of 8% we can calculate the deposit needed as:

`\begin{gathered} 16000=P\cdot((1+0.06\/4)^(68)-1)/(0.06\/4) \\ \\ 16000=P\cdot((1.015)^(68)-1)/(0.015) \\ \\ 16000=P\cdot(2.752269-1)/(0.015) \\ \\ 16000=P\cdot(1.752269)/(0.015) \\ \\ P=(0.015)/(1.752269)\cdot16000 \\ \\ P\approx136.97 \end{gathered}`

Answer:

a) $ 112.51

b) $ 136.97