Final answer:
The future value of a $7,500 biannual investment over 7 years at a 5.96% semiannual interest rate can be calculated using the future value of an annuity formula, resulting in approximately $376,502.48, where Option A is the closest to this value.
Step-by-step explanation:
To calculate the future value of an investment with a series of equal payments (annuities) and compound interest, we can use the future value of an annuity formula:
FV = P × rac{((1 + r)^n - 1)}{r}
Where:
- FV is the future value of the annuity.
- P is the payment amount per period.
- r is the interest rate per period.
- n is the total number of periods.
In this case:
- P = $7,500 (deposited every six months)
- r = 5.96% / 2 = 2.98% (since interest is compounded semiannually)
- n = 7 years × 2 = 14 periods (since there are two periods in a year)
Let's plug the values into the formula:
FV = $7,500 × rac{((1 + 0.0298)^14 - 1)}{0.0298}
Carrying out the calculations:
FV = $7,500 × rac{((1.0298)¹⁴ - 1)}{0.0298}
FV = $7,500 × 50.20033128 = $376,502.48
Option A is closest to this result. This example illustrates how compound interest can significantly increase the value of repeated investments over time as opposed to simple interest.