A payday loan company charges a $60 fee for a $550 payday loan that will be repaid in 10 days.

Treating the fee as interest paid, what is the equivalent annual interest rate?

To determine the equivalent annual interest rate for a payday loan, we can use the concept of the effective annual rate (EAR). Since the loan period is shorter than a year, we need to annualize the interest rate. Here's the step-by-step process:

### Step 1: Calculate the Interest Rate for the Loan Period
The loan amount is $550, and the fee (treated as interest) is $60 for a 10-day loan period.

The interest rate for the 10-day period ($ r_{10} $) can be calculated as:

\[
r_{10} = \frac{\text{Interest Paid}}{\text{Loan Amount}} = \frac{60}{550} \approx 0.1091 \text{ or } 10.91\%
\]

### Step 2: Convert the 10-Day Interest Rate to an Annual Rate
To find the equivalent annual interest rate, we need to account for how many such 10-day periods there are in a year. Since there are 365 days in a year:

\[
\text{Number of 10-day periods in a year} = \frac{365}{10} = 36.5
\]

Now, we can annualize the 10-day interest rate using the formula for the effective annual rate (EAR):

\[
\text{EAR} = \left(1 + r_{10}\right)^{\text{Number of periods}} - 1
\]

Substituting the values:

\[
\text{EAR} = \left(1 + 0.1091\right)^{36.5} - 1
\]

### Step 3: Calculate the Effective Annual Rate (EAR)

Let's compute this:

\[
\text{EAR} = \left(1.1091\right)^{36.5} - 1 \approx 42.58 - 1 = 41.58
\]

\[
\text{EAR} \approx 4158\% \text{ annually}
\]

This result is surprisingly high, which is typical for payday loans due to their short-term nature and high fees.

**Summary:**

The equivalent annual interest rate for the payday loan is approximately **4158%**.

Would you like more details or have any questions?

Here are some related questions that might interest you:

1. How would the interest rate change if the loan term were longer, say 20 days?
2. What are the implications of such a high annual interest rate on borrowers?
3. How does this interest rate compare to more traditional loans like credit cards?
4. If the fee were $70 instead of $60, what would the new annual interest rate be?
5. What are alternative short-term financing options with lower interest rates?
6. How is the annual interest rate affected by compounding frequency?
7. What would be the interest rate if the loan amount were $1,000 with the same fee?
8. How much total interest would you pay over a year if you kept rolling over the loan?

**Tip:** Payday loans often carry extremely high annual interest rates, so it's important to explore other borrowing options whenever possible.

Learn how to calculate the equivalent annual interest rate (EAR) for a $550 payday loan with a $60 fee over a 10-day period. Understand the concept of effective annual rate and its application in financial mathematics.

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