Question

A student desired to invest $1,540 into an investment at 9% compounded semiannually for 6 years. With all else equal, what is the future value of this investment after the six-year period?

Answer 1

To find the future value of an investment with compound interest, you can use the formula:

\[
FV = P \times \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \(FV\) = Future Value
- \(P\) = Principal amount (the initial investment)
- \(r\) = annual interest rate (in decimal)
- \(n\) = number of times interest is compounded per year
- \(t\) = number of years the money is invested for

In this case:
- \(P = 1540\)
- \(r = 0.09\) (9% expressed as a decimal)
- \(n = 2\) (since the interest is compounded semiannually)
- \(t = 6\)

Now, substitute the values into the formula:

\[
FV = 1540 \times \left(1 + \frac{0.09}{2}\right)^{2 \times 6}
\]

Calculating the values step-by-step:

1. Calculate \(\frac{r}{n}\):
\[
\frac{0.09}{2} = 0.045
\]

2. Calculate \(nt\):
\[
2 \times 6 = 12
\]

3. Now substitute back into the equation:
\[
FV = 1540 \times \left(1 + 0.045\right)^{12}
\]

4. Calculate inside the parentheses:
\[
1 + 0.045 = 1.045
\]

5. Raise to the power of 12:
\[
(1.045)^{12} \approx 1.601032
\]

6. Finally, multiply by the principal:
\[
FV \approx 1540 \times 1.601032 \approx 2461.59
\]

Thus, the future value of the investment after the six-year period is approximately **$2,461.59**.