Respuesta :
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:
[tex]FV=P\cdot\frac{(1+r\/m)^{nm}-1}{r\/m}[/tex]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:
[tex]\begin{gathered} 16000=P\frac{(1+0.08\/4)^{17\cdot4}-1}{0.08\/4} \\ \\ 16000=P\frac{(1.02)^{68}-1}{0.02} \\ \\ 16000=P\cdot\frac{3.84425-1}{0.02} \\ \\ 16000=P\cdot\frac{2.84425}{0.02} \\ \\ P=\frac{0.02}{2.84425}\cdot16000 \\ \\ P\approx112.51 \end{gathered}[/tex]b) If the nominal interest is 6% instead of 8% we can calculate the deposit needed as:
[tex]\begin{gathered} 16000=P\cdot\frac{(1+0.06\/4)^{68}-1}{0.06\/4} \\ \\ 16000=P\cdot\frac{(1.015)^{68}-1}{0.015} \\ \\ 16000=P\cdot\frac{2.752269-1}{0.015} \\ \\ 16000=P\cdot\frac{1.752269}{0.015} \\ \\ P=\frac{0.015}{1.752269}\cdot16000 \\ \\ P\approx136.97 \end{gathered}[/tex]Answer:
a) $ 112.51
b) $ 136.97
