Final answer:
The future value of $100 at 10 percent interest is greater when compounded semiannually compared to annually, due to interest being applied more frequently.
Step-by-step explanation:
The future value of $100 at 10 percent compounded semiannually is greater than the future value of $100 at 10 percent compounded annually. This is because with semiannual compounding, interest is calculated and added to the principal twice a year, effectively applying interest to the interest more frequently within the same year. In comparison, annual compounding implies the interest is added only once at the end of the year.
Compound interest can be calculated using the formula A = P(1 + r/n)^nt, where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
To illustrate, for semiannual compounding: A = $100(1 + 0.10/2)^(2*1) after one year.
For annual compounding: A = $100(1 + 0.10/1)^(1*1) after one year.