We have a principal of $3,000.
The annual interest rate is 6% (r=0.06).
We have to find the balance after 3 years (n=3), compounded in different ways.
The general formula, for a subperiod m, is:
[tex]FV=PV(1+\frac{r}{m})^{n\cdot m}[/tex]where m is the number of superiods in a year. For example, a monthly compounded interest will have m=12.
a) Annually (m=1)
[tex]\begin{gathered} FV=PV(1+\frac{r}{m})^{n\cdot m} \\ FV=3000(1+\frac{0.06}{1})^{3\cdot1} \\ FV=3000(1.06)^3 \\ FV=3000\cdot1.191016 \\ FV\approx3573.05 \end{gathered}[/tex]The final value is $3,573.05.
b) Semi-annually (m=2)
[tex]\begin{gathered} FV=PV(1+\frac{r}{m})^{n\cdot m} \\ FV=3000(1+\frac{0.06}{2})^{3\cdot2} \\ FV=3000(1.03)^6 \\ FV\approx3000\cdot1.19045 \\ FV\approx3582.16 \end{gathered}[/tex]The final value when compounded semi-annually is $3,582.16.
c) Quarterly (m=4)
[tex]\begin{gathered} FV=PV(1+\frac{r}{m})^{n\cdot m} \\ FV=3000(1+\frac{0.06}{4})^{3\cdot4} \\ FV=3000(1.015)^{12} \\ FV\approx3000\cdot1.19562 \\ FV\approx3586.85 \end{gathered}[/tex]The final value when compounded quarterly is $3,586.85.
d) Monthly (m=12)
[tex]\begin{gathered} FV=PV(1+\frac{r}{m})^{n\cdot m} \\ FV=3000\cdot(1+\frac{0.06}{12})^{3\cdot12} \\ FV=3000\cdot(1.005)^{36} \\ FV\approx3000\cdot1.19668 \\ FV\approx3590.04 \end{gathered}[/tex]The final value when compounded monthly is $3,590.04.
Answer:
Annually: $3,573.05.
Semi-annually: $3,582.16.
Quarterly: $3,586.85.
Monthly: $3,590.04.
