Question

To offer scholarships to children of employees, a company invests $13,000 at the end of every three months in an annuity that pays 9% compounded quarterlya. How much will the company have in scholarship funds at the end of ten years?b. Find the interestClick the icon to view some finance formulasa. The company will have $ in scholarship funds,(Do not round until the final answer. Then round to the nearest dollar as needed)b. The interest is $(Use the answer from part (a) to find this answer. Round to the nearest dollar as needed.)h

Answer 1

Annuities

Suppose a fixed investment R is done every fixed number of periods m per year for t years at a constant rate r.

a.

The final value of the investments plus the interest is calculated as follows:

$FV=R\cdot\frac{(1+i)^n-1}{i}$

Where:

n = number of total periods of the investment.

n = m*t

$i=\frac{r}{m}$

The company invests R = $13,000 for t = 10 years at the end of every quarter (3 months), thus m = 4. The interest rate is r = 9% = 0.09.

The interest rate compounds quarterly.

Calculate:

n = 4*10 = 40

i = 0.09 / 4 = 0.0225

$\begin{gathered} FV=\ 13,000\cdot\frac{(1+0.00225)^{40}-1}{0.0225} \\ FV=\operatorname{\ }13,000\times\frac{(1.00225)^{40}-1}{0.0225} \\ FV=\operatorname{\ }13,000\times\frac{2.435188965-1}{0.0225} \\ FV=\operatorname{\ }13,000\cdot63.786176 \end{gathered}$

Calculating:

FV = $829,220

The company will have $829,220 in scholarship funds

b. The interest can be found by subtracting the final value and the initial value. We have to calculate the latter:

$\begin{gathered} A=FV(1+i)^{-n} \\ A=\ 829,220(1.0225)^{-40} \\ A=\ 340,516 \end{gathered}$

Thus, the interest is:

welcI = $829,220 - $340,516

I = $488,704

The interest is $488,704