## Demystifying APR (Annual Percentage Rate) Annual Percentage Rate, or APR, represents the yearly interest rate you can expect to earn on an investment or pay on a loan. Crucially, APR is calculated as simple interest and **does not** factor in the effect of compounding interest within that year. It provides a straightforward, annualized view of the interest rate. Let's illustrate APR with an example: **Scenario:** * You borrow an initial principal amount of $5. * The daily interest rate is 0.07%, which translates to 0.0007 in decimal form. **Calculating Debt After 1 Year (without compounding):** With APR, the interest is calculated solely on the original principal amount ($5) each day. * After 1 day, your debt would be the principal plus one day's interest: `$5 (principal) + ($5 * 0.0007) (interest for day 1)` * After 2 days, you add another day's interest, again calculated on the original $5: `$5 (principal) + ($5 * 0.0007) (interest for day 1) + ($5 * 0.0007) (interest for day 2)` This pattern of adding interest calculated on the initial principal continues for all 365 days of the year. So, the total debt after 1 year can be expressed as: `Total Debt = $5 + ($5 * 0.0007) + ($5 * 0.0007) + ... (365 times) ... + ($5 * 0.0007)` This can be simplified by factoring out the common terms: `Total Debt after 1 year = $5 + ($5 * 0.0007 * 365)` `Total Debt after 1 year = $5 * (1 + 0.0007 * 365)` **Calculating the APR:** The APR is the total interest accrued over the year divided by the initial principal amount. `APR = (Total Debt after 1 year - Initial Borrowed Amount) / Initial Borrowed Amount` Substituting our values: `APR = ($5 * (1 + 0.0007 * 365) - $5) / $5` We can cancel out the $5 terms from the numerator and denominator: `APR = (1 + 0.0007 * 365) - 1` `APR = 0.0007 * 365` `APR = 0.2555` To express this as a percentage, we multiply by 100: `APR = 25.55%` **General Formula for APR:** A more general formula for APR is: `APR = R * N` Where: * `R` is the periodic interest rate (e.g., the daily interest rate of 0.0007 in our example). * `N` is the number of periods in a year (e.g., 365 for daily interest). Using our example values: `APR = 0.0007 * 365 = 0.2555`, or 25.55%. ## Unpacking APY (Annual Percentage Yield) Annual Percentage Yield, or APY, also represents the yearly interest rate, but with a critical difference: APY **includes** the effect of compounding interest. Compounding means that you earn interest not only on your initial principal but also on the accumulated interest from previous periods. This is often referred to as "interest on interest." Let's use the same scenario to see how APY differs: **Scenario:** * You borrow an initial principal amount of $5. * The daily interest rate is 0.07% (0.0007 as a decimal). **Calculating Debt After 1 Year (with daily compounding):** With APY, the interest earned each day is added to the principal, and the next day's interest is calculated on this new, slightly larger amount. * Debt on day 0 (when borrowed): `$5` * Debt after 1 day: The principal plus one day's interest. `$5 * (1 + 0.0007)` * Debt after 2 days: The previous day's balance is multiplied by (1 + daily interest rate). `[$5 * (1 + 0.0007)] * (1 + 0.0007) = $5 * (1 + 0.0007)^2` This compounding effect continues each day for the entire year. * Debt after 365 days: `Total Debt = $5 * (1 + 0.0007)^365` **Calculating the APY:** APY is the total interest earned (or paid), including compounded interest, over the year, divided by the initial principal. `APY = (Total Debt after 1 year - Initial Borrowed Amount) / Initial Borrowed Amount` Substituting our values: `APY = ($5 * (1 + 0.0007)^365 - $5) / $5` Again, we can cancel out the $5 terms: `APY = (1 + 0.0007)^365 - 1` Calculating the value: `APY ≈ (1.0007)^365 - 1` `APY ≈ 1.2909905 - 1` `APY ≈ 0.2909905` To express this as a percentage: `APY ≈ 29.099%` **General Formula for APY:** The general formula for APY is: `APY = (1 + R)^N - 1` Where: * `R` is the periodic interest rate (e.g., the daily interest rate of 0.0007). * `N` is the number of compounding periods in a year (e.g., 365 for daily compounding). Using our example values: `APY = (1 + 0.0007)^365 - 1 ≈ 0.2909905`, or approximately 29.099%. ## APR vs. APY: The Core Distinctions Understanding the difference between APR and APY is crucial for accurately assessing returns on investments or the true cost of borrowing in Web3 and traditional finance. * **APR (Annual Percentage Rate):** * Represents the simple interest rate calculated annually. * It does **not** account for the effects of compounding within the year. * The formula is `APR = R * N` (where `R` is the periodic rate, and `N` is the number of periods per year). * In our example with a 0.07% daily rate, the APR is 25.55%. * **APY (Annual Percentage Yield):** * Represents the effective annual rate of return, taking compounding into account. * It **includes** the effect of interest earning interest (compounding). * The formula is `APY = (1 + R)^N - 1` (where `R` is the periodic rate, and `N` is the number of compounding periods per year). * In our example with a 0.07% daily rate compounded daily, the APY is approximately 29.099%. **The Relationship:** Because APY incorporates the power of compounding, it will generally be higher than APR if interest is compounded more than once per year. The more frequently the interest is compounded (e.g., daily vs. monthly vs. annually), the greater the difference between APY and APR will be for the same nominal periodic interest rate. As clearly demonstrated with the 0.07% daily interest rate, the APY (29.099%) is notably higher than the APR (25.55%). This difference is entirely attributable to the effect of daily compounding included in the APY calculation. When evaluating financial products, especially in DeFi where compounding can be frequent, paying close attention to whether APR or APY is quoted will give you a more accurate picture of potential earnings or costs.