We have an investment that went from a initial value of PV = 1400 to a value of FV = 2177.36 in 71 months.
As the time is expressed in months, we assumed a monthly compounded interest, with a number of subperiods per year of m = 12.
Then, we have a total number of subperiods of n*m = 71.
We then can write the relation between the initial and final value as:
[tex]FV=PV(1+\frac{r}{m})^{n\cdot m}[/tex]where r is the nominal rate.
We will calculate r as:
[tex]\begin{gathered} \frac{FV}{PV}=(1+\frac{r}{m})^{n\cdot m} \\ \sqrt[n\cdot m]{\frac{FV}{PV}}=1+\frac{r}{m} \\ \frac{r}{m}=\sqrt[n\cdot m]{\frac{FV}{PV}}-1 \\ r=m\cdot(\sqrt[n\cdot m]{\frac{FV}{PV}}-1) \end{gathered}[/tex]Replacing with the values we get:
[tex]\begin{gathered} r=m\cdot(\sqrt[n\cdot m]{\frac{FV}{PV}}-1) \\ r=12\cdot(\sqrt[71]{\frac{2177.36}{1400}}-1) \\ r\approx12\cdot(\sqrt[71]{1.555}-1) \\ r\approx12\cdot(1.00624-1) \\ r\approx12\cdot0.00624 \\ r\approx0.075 \end{gathered}[/tex]We can now transformed this rate to an equivalent annually compounded rate as:
[tex]\begin{gathered} 1+i=(1+\frac{r}{m})^m \\ i=(1.00624)^{12}-1 \\ i=1.0775-1 \\ i=0.0775=7.75\% \end{gathered}[/tex]Answer: The equivalente annually compounded rate is 7.75%
