Consider an account with an APR of 5.9​%. Find the APY with quarterly​ compounding, monthly​ compounding, and daily compounding. Comment on how changing the compounding period affects the annual yield.


To calculate the **Annual Percentage Yield (APY)**, we use the following formula for different compounding periods:

\[
APY = \left( 1 + \frac{r}{n} \right)^n - 1
\]

Where:
- \( r \) is the annual interest rate (APR) as a decimal.
- \( n \) is the number of compounding periods per year.

Given that the **APR is 5.9%**, or \( r = 0.059 \), let's calculate the APY for **quarterly**, **monthly**, and **daily** compounding:

1. **Quarterly Compounding**:
   - \( n = 4 \) (since there are 4 quarters in a year).
   \[
   APY = \left( 1 + \frac{0.059}{4} \right)^4 - 1 = (1 + 0.01475)^4 - 1
   \]

2. **Monthly Compounding**:
   - \( n = 12 \) (since there are 12 months in a year).
   \[
   APY = \left( 1 + \frac{0.059}{12} \right)^{12} - 1 = (1 + 0.0049167)^{12} - 1
   \]

3. **Daily Compounding**:
   - \( n = 365 \) (since there are 365 days in a year).
   \[
   APY = \left( 1 + \frac{0.059}{365} \right)^{365} - 1 = (1 + 0.00016164)^{365} - 1
   \]

I'll compute the exact values for you.

The calculated **Annual Percentage Yields (APY)** are as follows:

1. **Quarterly Compounding**: \( 6.03\% \)
2. **Monthly Compounding**: \( 6.06\% \)
3. **Daily Compounding**: \( 6.08\% \)

### Effect of Changing the Compounding Period:
As the frequency of compounding increases (from quarterly to daily), the APY also increases slightly. This is because more frequent compounding allows interest to be applied more often, which leads to higher returns over time. However, the difference becomes smaller as the compounding frequency increases.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the APY change if the APR were higher or lower?
2. How do semi-annual and continuous compounding affect APY?
3. How does the time horizon affect the total returns with different compounding frequencies?
4. What would the APY be with a different APR (e.g., 3.5%)?
5. How is continuous compounding different from daily compounding?

**Tip**: For small APR values, the difference between daily and monthly compounding is often minimal, but it becomes more noticeable with larger APRs.

Consider an account with an APR of 5.9%. Find the APY with quarterly compounding, monthly compounding, and daily compounding. Comment on how changing the compounding period affects the annual yield.

This problem covers how to calculate the Annual Percentage Yield (APY) for a 5.9% APR using quarterly, monthly, and daily compounding. Learn how changing the compounding frequency affects the yield, with detailed formulas and step-by-step calculations. Suitable for high school students in Grades 10-12.

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