Question

Maryann is planning a wedding anniversary gift of a trip to Hawaii for her husband at the end of 5 years. She will have enough to pay for the trip if she invests $5,000 per year until that anniversary and plans to make her first $5,000 investment on their first anniversary. Assume her investment earns a 4 percent interest rate, how much will she have saved for their trip if the interest is compounded in each of the following ways?

a. Annually b. Quarterly c. Monthly

Answer 1

Assuming that Maryann invests her $5,000 at the beginning of each year, and for 5 years. The investment is therefore an annuity due, and we need to find future value at the end of 5 years. For each year, we are provided:

PMT = $5,000

i = 4%

n = 5 years.

The future value $(FV)$ of an annuity due is found by the formula:

$FV = PMT \left[ \dfrac {\left(1+ \dfrac {i}{m}\right)^{mn}-1}{\left(\dfrac {i}{m}\right)} \right]\left(1 + \dfrac {i}{m}\right)$

Where, $i$ is the annual interest rate, $n$ is the number of years, $PMT$ is the annual payment, and $m$ is the number of compounding periods.

$\bold {(a) \space Annual \space Compounding}$

$\bold {Answer}$

$FV =\$28,164.88$

$\bold {Solution}$

$m = 1$

$FV = 5,000 \left[\dfrac {(1+0.04)^5- 1}{0.04}\right](1+0.04)$

$FV = 5,000 \left[\dfrac {1.04^5- 1}{0.04}\right](1.04)$

$= \$28,164.877312$

$=\bold {\$28,164.88}$

$\bold {(b) \space Quarterly \space Compounding}$

$\bold {Answer}$

$FV = \$111,195.97$

$\bold {Solution}$

$m = 4$

$FV = 5,000 \left[ \dfrac {\left(1+ \dfrac {0.04}{4}\right)^{(5×4)}-1}{\left(\dfrac {0.04}{4}\right)} \right]\left(1 + \dfrac {0.04}{4}\right)$

$FV = 5,000 \left[\dfrac {(1+0.01)^{20} - 1}{0.01}\right](1+0.01)$

$FV = 5,000 \left[\dfrac {1.01^{20}- 1}{0.01}\right](1.01)$

$= \$111,195.97015$

$= \bold {\$111,195.97}$

$\bold {(c) \space Monthly \space Compounding}$

$\bold {Answer}$

$FV = \$332,599.87$

$\bold {Solution}$

$m = 12$

$FV = 5,000 \left[ \dfrac {\left(1+ \dfrac {0.04}{12}\right)^{(5×12)}-1}{\left(\dfrac {0.04}{12}\right)} \right]\left(1 + \dfrac {0.04}{12}\right)$

$FV = 5,000 \left[ \dfrac {\left(1+ \dfrac {0.04}{12}\right)^{60}-1}{\left(\dfrac {0.04}{12}\right)} \right]\left(1 + \dfrac {0.04}{12}\right)$

$=\$332,599.873883$

$= \bold {\$322,599.87}$