To determine which investment yields the greater return, we will compare the final amounts obtained from [tex]$14000$[/tex] invested at [tex]$9 \%$[/tex] compounded quarterly to [tex]$14000$[/tex] invested at [tex]$8.85 \%$[/tex] compounded continuously over a period of [tex]$5$[/tex] years. We will use the compound interest formulas for both scenarios.
Scenario 1: [tex]$9 \%$[/tex] Compounded Quarterly
The formula for compound interest when interest is compounded periodically is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount ([tex]$14000$[/tex]),
- [tex]\( r \)[/tex] is the annual interest rate ([tex]$0.09$[/tex]),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year ([tex]$4$[/tex] for quarterly),
- [tex]\( t \)[/tex] is the number of years the money is invested ([tex]$5$[/tex]).
Let’s calculate the amount:
[tex]\[ A_{\text{quarterly}} = 14000 \left(1 + \frac{0.09}{4}\right)^{4 \times 5} \][/tex]
[tex]\[ A_{\text{quarterly}} = 14000 \left(1 + 0.0225\right)^{20} \][/tex]
[tex]\[ A_{\text{quarterly}} = 14000 (1.0225)^{20} \][/tex]
Substituting the values, we get:
[tex]\[ A_{\text{quarterly}} = 21847.128809586015 \][/tex]
Scenario 2: [tex]$8.85 \%$[/tex] Compounded Continuously
The formula for continuously compounded interest is given by:
[tex]\[ A = Pe^{rt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount ([tex]$14000$[/tex]),
- [tex]\( r \)[/tex] is the annual interest rate ([tex]$0.0885$[/tex]),
- [tex]\( t \)[/tex] is the number of years the money is invested ([tex]$5$[/tex]),
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately [tex]$2.71828$[/tex]).
Let’s calculate the amount:
[tex]\[ A_{\text{continuous}} = 14000 e^{0.0885 \times 5} \][/tex]
[tex]\[ A_{\text{continuous}} = 14000 e^{0.4425} \][/tex]
Substituting the values, we get:
[tex]\[ A_{\text{continuous}} = 21792.31379939193 \][/tex]
Comparison:
Now, let’s compare the results for both investment options:
- [tex]$9 \%$[/tex] compounded quarterly yields: \[tex]$21847.13 (approximately)
- $[/tex]8.85 \%[tex]$ compounded continuously yields: \$[/tex]21792.31 (approximately)
Clearly, [tex]$21847.13$[/tex] is greater than [tex]$21792.31$[/tex].
Thus, the investment that yields the greater return over 5 years is [tex]$9 \%$[/tex] compounded quarterly.
