Question

The future value of $100 at 10 percent compounded semiannually is ______ the future value of $100 at 10 percent compounded annually.

Answer 1

To compare the future value of $100 at 10 percent compounded semiannually and 10 percent compounded annually, we can use the future value formula:

\[
FV = P \times (1 + r/n)^{nt}
\]

Where:
- \( FV \) = future value
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years

### 1. Future Value Compounded Semiannually

For semiannual compounding:
- \( P = 100 \)
- \( r = 0.10 \)
- \( n = 2 \) (as interest is compounded twice a year)
- Assume \( t = 1 \) year

Substituting the values into the formula:

\[
FV = 100 \times \left(1 + \frac{0.10}{2}\right)^{2 \times 1}
\]
\[
= 100 \times \left(1 + 0.05\right)^{2}
\]
\[
= 100 \times (1.05)^{2}
\]
\[
= 100 \times 1.1025
\]
\[
= 110.25
\]

### 2. Future Value Compounded Annually

For annual compounding:
- \( P = 100 \)
- \( r = 0.10 \)
- \( n = 1 \) (as interest is compounded once a year)

Substituting the values into the formula:

\[
FV = 100 \times (1 + 0.10)^{1}
\]
\[
= 100 \times (1.10)
\]
\[
= 100 \times 1.10
\]
\[
= 110.00
\]

### Conclusion

Now we can compare the two future values:

- Future value compounded semiannually: **$110.25**
- Future value compounded annually: **$110.00**

Thus, the future value of $100 at 10 percent compounded semiannually is **$0.25** more than the future value of $100 at 10 percent compounded annually.

So, the future value of $100 at 10 percent compounded semiannually is greater than the future value of $100 at 10 percent compounded annually.