Let's start by analyzing the two different compounding methods: daily compounding and continuous compounding.
### 7% Compounded Daily
With daily compounding, the interest is applied to your investment every day. The formula for daily compounding is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (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,
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
For this investment:
- [tex]\( P = 13,000 \)[/tex]
- [tex]\( r = 0.07 \)[/tex]
- [tex]\( n = 365 \)[/tex] (since it is compounded daily)
- [tex]\( t = 5 \)[/tex]
So,
[tex]\[ A_{\text{daily}} = 13000 \left(1 + \frac{0.07}{365}\right)^{365 \times 5} \][/tex]
### 6.87% Compounded Continuously
With continuous compounding, the interest is compounded an infinite number of times per year. The formula for continuous compounding is given by:
[tex]\[ A = Pe^{rt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828),
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal),
- [tex]\( t \)[/tex] is the number of years,
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
For this investment:
- [tex]\( P = 13,000 \)[/tex]
- [tex]\( r = 0.0687 \)[/tex]
- [tex]\( t = 5 \)[/tex]
So,
[tex]\[ A_{\text{continuous}} = 13000 \cdot e^{0.0687 \times 5} \][/tex]
### Comparing the Two Methods
After calculating the amounts using the formulas above, we find:
- The amount after 5 years with 7% compounded daily is approximately [tex]$18,447.26.
- The amount after 5 years with 6.87% compounded continuously is approximately $[/tex]18,328.36.
### Conclusion
Since [tex]$18,447.26 (7% compounded daily) is greater than $[/tex]18,328.36 (6.87% compounded continuously), the 7% compounded daily rate yields the larger amount after 5 years. Therefore, investing at 7% compounded daily is the better option in this scenario.
