Final answer:
The future value of a $4000 loan at 4% interest compounded semiannually for 3 years is approximately $4504.65, and the total compound interest earned over this period is $504.65.
Step-by-step explanation:
To calculate the future value of a loan of $4000 at an annual interest rate of 4% compounded semiannually for 3 years, we must use the compound interest formula. The formula for finding the future value is Future Value = Principal × (1 + interest rate / number of compounds per year)^(total number of compounding periods).
For this loan:
- Principal = $4000
- Annual interest rate = 4% or 0.04
- Compounding frequency (semiannually) = 2 times per year
- Total time = 3 years
Therefore, the number of compounding periods is 3 years × 2 = 6. The interest rate per period is 0.04 / 2 = 0.02.
The future value is calculated as follows:
Future Value = $4000 × (1 + 0.02)^6
Future Value = $4000 × 1.126162
Future Value = $4504.65 (approximately)
The compound interest earned can be found by subtracting the principal from the future value:
Compound interest = Future Value - Principal
Compound interest = $4504.65 - $4000
Compound interest = $504.65
After three years, the total future value of the loan is $4504.65 and the total compound interest is $504.65. This is the extra amount earned due to interest being compounded semiannually.