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The Annual Percentage Rate (APR) is the yearly interest expressed as a percentage. It helps consumers understand how much they will spend or earn in interest over a year.

In contrast to other interest calculations, APR is straightforward because it does not consider the effects of interest compounding. It's different from the nominal interest rate because APR can include fees or additional costs associated with a loan. However, for calculation simplicity in many examples, it often just represents the simple interest rate over a year.

Lenny Loanshark provides 1% interest per week. To find out his APR, multiply the weekly rate by the number of weeks in a year. Since there are 52 weeks, the formula to compute APR is:
\[ APR = 1\% \times 52 = 52\% \]

This calculation shows that Lenny's annual interest rate without considering any compounding effect is 52%.

The Effective Annual Rate (EAR) represents the true economic cost or yield of an investment over one year, taking compounding into account.

Compound interest means that interest in each period is calculated on the principal as well as all accumulated interest from previous periods.

EAR provides a more accurate measure of financial impact compared to APR because it includes the effect of compounding. It is critical when comparing financial products with different compounding periods.

To determine EAR for Lenny's loans, we use the formula: \[ EAR = \left(1 + \frac{i}{n}\right)^{nt} - 1 \] where:

• \(i\) is the nominal interest rate (1%)
• \(n\) is the number of compounding periods per year (52 for weekly compounding)
• \(t\) is the time the money is invested or borrowed for (1 year in this case)

By plugging in Lenny's values, we calculate: \[ EAR = \left(1 + \frac{0.01}{52}\right)^{52 \times 1} - 1 \approx 54.43\% \]

This means the effective annual interest rate, accounting for weekly compounding, is approximately 54.43%.

Compounding interest is the process by which interest earned is added to the principal, so interest in the next period is earned on both the original principal and the accumulated interest.

It has a significant effect on the growth of investments and the cost of loans. The frequency of compounding (such as annually, semi-annually, quarterly, monthly, or weekly) influences how much total interest will be paid or earned.

Understanding compounding requires awareness of the following:

• **Frequency of Compounding:** More frequent compounding results in a higher total amount of interest.
• **Impact on EAR:** The effective rate increases with more frequent compounding periods, even if the nominal rate remains unchanged.

For Lenny's weekly compounding scenario, compounding interest significantly increases the yearly cost of the loan compared to if interest were not compounded. This demonstrates why knowing both the APR and EAR is important for making informed financial decisions.