Final answer:
Using the future value of an annuity formula, it's calculated that with an annual deposit of $5,000 and a 4% interest rate compounded semiannually over 5 years, Company B's deposits will be worth approximately $27,082.
Step-by-step explanation:
To determine the value of Company B deposits at the end of 5 years with an interest rate of 4% compounded semiannually, we need to calculate the future value of an annuity. The formula for finding the future value of an annuity compounded semiannually is given by:
FV = P × {[(1 + r/n)^(nt) - 1] / (r/n)}
Where:
- FV is the future value of the annuity.
- P is the periodic payment amount.
- r is the annual interest rate (in decimal form).
- n is the number of times the interest is compounded per year.
- t is the number of years the money is deposited for.
For this problem:
- P = $5,000 (the annual deposit)
- r = 0.04 (4% annual interest rate)
- n = 2 (since the interest is compounded semiannually)
- t = 5 (the money is deposited over 5 years)
Substituting these values into the formula:
FV = 5000 × {[(1 + 0.04/2)^(2×5) - 1] / (0.04/2)}
Calculating the value inside the brackets:
FV = 5000 × {[(1 + 0.02)^(10) - 1] / 0.02} = 5000 × {[(1.02)^10 - 1] / 0.02}
Finally, calculating the future value gives:
FV ≈ $27,082
Therefore, the value of the deposits after 5 years is approximately $27,082, which corresponds to answer choice b.