Question

diamond would like to save $2,350 at the end of every three months for the next 8.5 years in a savings account earning 3.2% compounded quarterly.
a) what would be the accumulated value of the investment at the end of the term? $
b) what would be the amount of interest earned? $

Explanation:

Step1: Identify the type of annuity

This is an ordinary annuity (payments at the end of each period) with quarterly payments. The formula for the future value (accumulated value) of an ordinary annuity is $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods.

Step2: Calculate the interest rate per period and number of periods

• The annual interest rate is $3.2\%$ or $0.032$. Since it's compounded quarterly, the interest rate per period $r = \frac{0.032}{4} = 0.008$.
• The number of years is $8.5$, so the number of periods $n = 8.5 \times 4 = 34$.

Step3: Calculate the future value (part a)

The payment per period $P = 2350$. Plugging into the formula:

$$ FV = 2350 \times \frac{(1 + 0.008)^{34} - 1}{0.008} $$

First, calculate $(1 + 0.008)^{34}$. Using a calculator, $(1.008)^{34} \approx 1.3012$. Then, $(1.3012 - 1) = 0.3012$. Then, $\frac{0.3012}{0.008} = 37.65$. Then, $2350 \times 37.65 \approx 88477.5$.

Step4: Calculate the total amount paid and then the interest (part b)

• Total amount paid over the period: $P \times n = 2350 \times 34 = 79900$.
• Interest earned = Future Value - Total Amount Paid = $88477.5 - 79900 = 8577.5$.

Answer:

a) The accumulated value is approximately $\$88477.5$
b) The amount of interest earned is approximately $\$8577.5$