Final answer:
The future value of the ordinary annuity with payments of $9,400, at an interest rate of 6% compounded semiannually for 5 years, is approximately $107,760.18, with $94,000 from contributions and $13,760.18 from interest.
Step-by-step explanation:
To find the future value of an ordinary annuity, we use the formula:
FV = R \times \left(\frac{\left(1 + \frac{i}{n}\right)^{nt} - 1}{\frac{i}{n}}\right)
Where:
- R is the regular payment amount
- i is the annual interest rate (in decimal)
- n is the number of times interest is compounded per year
- t is the number of years
In this scenario, we have:
- R = $9,400
- i = 6% or 0.06
- n = 2 (since the interest is compounded semiannually)
- t = 5 years
Let's plug these values into the formula:
FV = $9,400 \times \left(\frac{(1 + \frac{0.06}{2})^{2 \times 5} - 1}{\frac{0.06}{2}}\right)
Calculating the terms within the parentheses first, we get:
(1 + 0.03)^{10} - 1
FV = $9,400 \times \left(\frac{(1.03)^{10} - 1}{0.03}
Now we compute the future value (FV) and determine the contributions and interest.
The total contributions are simply the payment amount times the number of periods:
Total contributions = R \times nt
Total contributions = $9,400 \times (2 \times 5) = $94,000
To find the total interest, we subtract the total contributions from the future value:
Total interest = FV - Total contributions
Using a calculator:
(1.03)^{10} - 1 \approx 0.343916
FV \approx $9,400 \times \left(\frac{0.343916}{0.03}\right) \approx $9,400 \times 11.46387 \approx $107,760.18
Total interest = $107,760.18 - $94,000 = $13,760.18
Therefore, the future value of the annuity is approximately $107,760.18, with $94,000 from contributions and $13,760.18 from interest (rounded to the nearest cent).