Compound Interest
The final value (FV) of an investment P after t years is calculated with the formula:
[tex]FV=P(1+\frac{r}{m})^{m\cdot t}[/tex]
Where r is the annual interest rate and m is the number of compounding periods per year.
We are given the following data:
P = $7000
t = 13 years
r = 7% = 7 / 100 = 0.07
The compounding period varies from part to part.
a) Annually
In this case, m = 1 because the money compounds once per year. Applying the formula:
[tex]FV=7000(1+\frac{0.07}{1})^{1\cdot13}[/tex]
Calculating:
[tex]\begin{gathered} FV=7000(1.07)^{13} \\ \\ FV=16868.92 \end{gathered}[/tex]
The final value is $16868.92
b) Semiannually. The money compounds twice a year, so m = 2. Applying the formula:
[tex]\begin{gathered} FV=7000(1+\frac{0.07}{2})^{2\cdot13} \\ \\ FV=7,000(1+0.035)^{26} \\ \\ FV=17121.71 \end{gathered}[/tex]
The final value is $17121.71
c) Quarterly. The money compounds 4 times a year, so m = 4. Applying the formula again:
[tex]\begin{gathered} FV=7000(1+\frac{0.07}{4})^{4\cdot13} \\ \\ FV=7,000(1+0.0175)^{52} \\ \\ FV=17253.92 \end{gathered}[/tex]
The final value is $17253.92
d) Daily (calendar year). In this case, we use m = 365:
[tex]\begin{gathered} FV=7000(1+\frac{0.07}{365})^{365\cdot13} \\ \\ FV=7,000(1+0.0001917808)^{4745} \\ \\ FV=17388.74 \end{gathered}[/tex]
The final value is $17388.74
e) Continuously. We use a slightly different formula here:
[tex]FV=P\cdot e^{rt}[/tex]
Applying the formula:
[tex]\begin{gathered} FV=7000\cdot e^{0.07\cdot13} \\ \\ FV=17390.26 \end{gathered}[/tex]
The final value is $17390.26
