Questions: If 1078.00 accumulates to 1521.77 in five years, eight months compounded monthly, what is the effective annual rate of interest? The effective annual rate of interest is %. (Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as needed.)

Solution Steps

To find the effective annual rate of interest, we first need to determine the monthly interest rate using the compound interest formula. Then, we convert this monthly rate to an effective annual rate. The compound interest formula is given by \( A = P(1 + r/n)^{nt} \), where \( A \) is the final amount, \( P \) is the principal, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years. We solve for the monthly rate and then convert it to the effective annual rate.

We start with the principal amount \( P = 1078.00 \), the accumulated amount \( A = 1521.77 \), the number of compounding periods per year \( n = 12 \), and the total time in years \( t = 5 + \frac{8}{12} = \frac{68}{12} = 5.6667 \).

Using the compound interest formula:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

we rearrange to find the monthly interest rate \( r \):

\[ \left(1 + \frac{r}{n}\right)^{nt} = \frac{A}{P} \]

Substituting the known values:

\[ \left(1 + \frac{r}{12}\right)^{12 \cdot 5.6667} = \frac{1521.77}{1078.00} \]

Calculating the right side gives:

\[ \frac{1521.77}{1078.00} \approx 1.409 \]

Taking the 68th root:

\[ 1 + \frac{r}{12} \approx 1.005083 \]

Thus, the monthly interest rate \( r \) is:

\[ \frac{r}{12} \approx 0.005083 \implies r \approx 0.060996 \]

The effective annual rate \( R \) is calculated from the monthly rate:

\[ R = \left(1 + \frac{r}{12}\right)^{12} - 1 \]

Substituting the monthly rate:

\[ R = \left(1 + 0.005083\right)^{12} - 1 \approx 0.062730 \]

Converting to percentage:

\[ R \approx 6.2730\% \]

Final Answer