Final answer:
To calculate the effective annual rate (EAR), we account for compounding by using the formula EAR = (1 + periodic rate)^number of periods per year - 1. The EAR for this loan is approximately 1.057^52 - 1. The APR, traditionally calculated as the nominal rate without compounding, would equal the weekly rate of 5.7% multiplied by 52, resulting in 296.4%.
Step-by-step explanation:
Calculating Effective Annual Rate and APR for a Payday Loan
To calculate the effective annual rate (EAR) for the payday loan, we need to account for the frequency of compounding. Since the payday loan is a 7-day loan, we can assume there are approximately 52 weeks in a year, hence 52 compounding periods.
The discount rate of 5.7% per week translates into the borrower receiving 94.3% of the principal. After receiving $94.3 on a $100 loan, the borrower must pay back the full $100 at the end of the week. The EAR can be calculated using the formula for compound interest:
EAR = (1 + periodic rate)number of periods per year - 1
But we need to convert the discount rate to a periodic interest rate first. Since $94.3 is 94.3% of the loan, the interest rate for that period is (100% - 94.3%) = 5.7%.
Let's calculate the EAR:
EAR = (1 + 0.057)52 - 1 ≈ 1.05752 - 1
To calculate APR, which is the annual percentage rate, we would generally consider the nominal rate without compounding. APR is often calculated by multiplying the periodic rate by the number of periods in a year:
APR = 5.7% × 52 = 296.4%
However, these rates are hypothetical and are likely to exceed legal limits for payday loans in many jurisdictions.