To determine the future value of an investment with different compounding frequencies, we will use the compound interest formula:
[tex]\[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value of the investment.
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested.
### Given Data
- Principal Amount ([tex]\( P \)[/tex]): \[tex]$10,000.00
- Annual Interest Rate (\( r \)): 3.5% (or 0.035 as a decimal)
- Term (\( t \)): 3 years
- Compounding Frequencies:
- Annually: \( n = 1 \)
- Quarterly: \( n = 4 \)
- Monthly: \( n = 12 \)
- Weekly: \( n = 52 \)
- Daily: \( n = 365 \)
- Hourly: \( n = 8760 \)
### Step-by-Step Calculations
#### 1. Annually (n = 1)
\[ FV = 10000 \left(1 + \frac{0.035}{1}\right)^{1 \cdot 3} \]
\[ FV \approx \$[/tex]11087.18 \]
#### 2. Quarterly (n = 4)
[tex]\[ FV = 10000 \left(1 + \frac{0.035}{4}\right)^{4 \cdot 3} \][/tex]
[tex]\[ FV \approx \$11102.03 \][/tex]
#### 3. Monthly (n = 12)
[tex]\[ FV = 10000 \left(1 + \frac{0.035}{12}\right)^{12 \cdot 3} \][/tex]
[tex]\[ FV \approx \$11105.41 \][/tex]
#### 4. Weekly (n = 52)
[tex]\[ FV = 10000 \left(1 + \frac{0.035}{52}\right)^{52 \cdot 3} \][/tex]
[tex]\[ FV \approx \$11106.71 \][/tex]
#### 5. Daily (n = 365)
[tex]\[ FV = 10000 \left(1 + \frac{0.035}{365}\right)^{365 \cdot 3} \][/tex]
[tex]\[ FV \approx \$11107.05 \][/tex]
#### 6. Hourly (n = 8760)
[tex]\[ FV = 10000 \left(1 + \frac{0.035}{8760}\right)^{8760 \cdot 3} \][/tex]
[tex]\[ FV \approx \$11107.10 \][/tex]
### Summary of Results
- Annually: \[tex]$11087.18
- Quarterly: \$[/tex]11102.03
- Monthly: \[tex]$11105.41
- Weekly: \$[/tex]11106.71
- Daily: \[tex]$11107.05
- Hourly: \$[/tex]11107.10
These values represent the future value of a \$10,000 investment over 3 years at an annual interest rate of 3.5%, compounded at different frequencies. Notice that more frequent compounding results in slightly higher future values. Fill in the corresponding results in the table under the "Future Value" column.
