Question

if an eight-day single payment loan has a periodic interest rate of 11.1%, what is the approximate effective interest rate of the loan?

121,815.9%
12,181.6%
12,081.6%
120,815.9%

Answer 1

To calculate the effective interest rate of a loan for a period shorter than one year, we can use the formula for converting a periodic interest rate to an effective interest rate:

\[
\text{Effective Interest Rate} = (1 + r)^n - 1
\]

where \( r \) is the periodic interest rate (expressed as a decimal), and \( n \) is the number of periods in a year.

In this case, the periodic interest rate (\( r \)) is 11.1%, or 0.111 in decimal form. The loan duration is 8 days. To find \( n \), we need to determine how many periods of 8 days fit into a year (which has approximately 365 days):

\[
n = \frac{365}{8} \approx 45.625
\]

Now, we can substitute these values into the formula:

\[
\text{Effective Interest Rate} = (1 + 0.111)^{45.625} - 1
\]

Calculating \( (1 + 0.111)^{45.625} \):

1. Calculate \( 1 + 0.111 = 1.111 \).
2. Raise it to the power of 45.625:

Using a calculator,

\[
(1.111)^{45.625} \approx 12.1816
\]

Now, subtract 1:

\[
\text{Effective Interest Rate} \approx 12.1816 - 1 = 11.1816 \\
\text{Effective Interest Rate} \approx 12.1816
\]

Thus, if we convert it back to a percentage, we have approximately:

\[
\text{Effective Interest Rate} \approx 1218.16\%
\]

Among the provided options:
- 121,815.9%
- 12,181.6%
- 12,081.6%
- 120,815.9%

The correct answer is **12,181.6%**.