A bank features a savings account that has an annual percentage rate of r=3%. Find the Annual Percentage Yield (APY) for the savings account if the interest is compounded according to the following schedules:

To find the **Annual Percentage Yield (APY)** for a savings account with an annual percentage rate (APR) of \( r = 3\% \), we need to consider the formula for **APY** when interest is compounded \( n \) times per year:

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

where:
- \( r \) is the nominal interest rate (expressed as a decimal, so \( r = 0.03 \) for \( 3\% \)),
- \( n \) is the number of times the interest is compounded per year.

Let's calculate the APY for various compounding schedules:

### 1. Compounded annually (\( n = 1 \)):
\[
\text{APY} = \left(1 + \frac{0.03}{1}\right)^1 - 1 = 1.03 - 1 = 0.03 \text{ or } 3\%
\]

### 2. Compounded semi-annually (\( n = 2 \)):
\[
\text{APY} = \left(1 + \frac{0.03}{2}\right)^2 - 1 = \left(1 + 0.015\right)^2 - 1 = 1.030225 - 1 = 0.030225 \text{ or } 3.0225\%
\]

### 3. Compounded quarterly (\( n = 4 \)):
\[
\text{APY} = \left(1 + \frac{0.03}{4}\right)^4 - 1 = \left(1 + 0.0075\right)^4 - 1 = 1.030339 - 1 = 0.030339 \text{ or } 3.0339\%
\]

### 4. Compounded monthly (\( n = 12 \)):
\[
\text{APY} = \left(1 + \frac{0.03}{12}\right)^{12} - 1 = \left(1 + 0.0025\right)^{12} - 1 = 1.030416 - 1 = 0.030416 \text{ or } 3.0416\%
\]

### 5. Compounded daily (\( n = 365 \)):
\[
\text{APY} = \left(1 + \frac{0.03}{365}\right)^{365} - 1 = \left(1 + 0.00008219\right)^{365} - 1 = 1.030454 - 1 = 0.030454 \text{ or } 3.0454\%
\]

These are the APYs for the savings account under various compounding schedules.

Would you like further details or have any questions?

### 5 Follow-up Questions:
1. What would happen to APY if the interest rate increased to 5%?
2. How does the compounding frequency affect the total amount earned?
3. What is the APY for weekly compounding (52 times a year)?
4. How can we calculate the APY if interest is compounded continuously?
5. How does the concept of APY help in comparing different savings accounts?

**Tip:** More frequent compounding increases the APY, as interest is calculated on a growing balance more often.

A bank features a savings account that has an annual percentage rate of r=3%. Find the Annual Percentage Yield (APY) for the savings account if the interest is compounded according to the following schedules:

Discover how to calculate the Annual Percentage Yield (APY) for a savings account with a 3% annual percentage rate. This guide explores different compounding schedules, including annually, semi-annually, quarterly, monthly, and daily, with detailed formulas and examples. Ideal for students in Grades 11-12.

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