Compound Interest
The final value of an investment of P dollars at an annual rate r for t years is given by:
[tex]FV=P\mleft(1+\frac{r}{m}\mright)^{t\cdot m}[/tex]
Where m is the number of compounding periods per year.
Ann invested P = $5400 in an account that pays an annual interest rate of r = 3.9%. Converting to decimal, r = 3.9 / 100 = 0.039.
The interest compounds daily, so m = 365.
(a) The final value after t = 1 year is calculated below:
[tex]\begin{gathered} FV=\$5400\mleft(1+\frac{0.039}{365}\mright)^{1\cdot365} \\ FV=\$5400(1.000106849)^{365} \\ FV=\$5400\cdot1.039768 \\ FV=\$5614.75 \end{gathered}[/tex]
(b) The effective annual interest rate is calculated as:
[tex]r_e=\frac{FV}{P}-1[/tex]
Substituting:
[tex]\begin{gathered} r_e=\frac{5614.75}{5400}-1 \\ r_e=1.03977-1 \\ r_e=0.03977 \end{gathered}[/tex]
Expressed as a percent: re = 3.98%
