Final answer:
The future value of a $10,000 investment at a 5% annual interest rate, compounded semiannually for five years, is $12,801 after rounding to the nearest dollar.
Step-by-step explanation:
To calculate the future value of a $10,000 investment with a 5% annual interest rate compounded semiannually for five years, we need to use the compound interest formula:
FV = P × (1 + r/n)^(nt)
Where:
- FV is the future value of the investment,
- P is the principal amount ($10,000),
- r is the annual interest rate (5% or 0.05),
- n is the number of times interest is compounded per year (2 for semiannually),
- t is the number of years the money is invested (5 years).
First, divide the annual interest rate by the number of compounding periods per year:
r/n = 0.05/2 = 0.025
Next, add 1 to the result and raise it to the power of the total number of compounding periods:
(1 + r/n)^(nt) = (1 + 0.025)^(2×5) = (1.025)^10
Calculate the power of 1.025 to 10:
(1.025)^10 = 1.280084145
Finally, multiply this result by the principal amount to get the future value:
FV = 10,000 × 1.280084145 = $12,800.84
Round the future value to the nearest dollar which gives us $12,801
In conclusion, the $10,000 investment is worth $12,801 after five years with compound interest.