Question

A bank advertises an APR of 5.5% on personal loans. How much more is the APY when the rate is compounded monthly as compared to when it's compounded quarterly? Explain your answer.
use this to answer the question:
Using the Convert APR to APY calculator:
APR as a decimal (a) =
n = 12 (compounded monthly = 12 times per year)
APY = %
APR as a decimal (a) =
n = 4 (compounded quarterly = 4 times per year)
APY = %
Find the difference: % - % = %

Answer 1

To determine how much more the Annual Percentage Yield (APY) is when the APR of 5.5% is compounded monthly compared to being compounded quarterly, we first need to convert the APR to APY using the formula for APY:

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

where:
- \( a \) is the APR expressed as a decimal (0.055 for 5.5%)
- \( n \) is the number of compounding periods per year.

### Step 1: Calculate APY for Monthly Compounding
1. **Convert APR to Decimal**:
\[
a = 5.5\% = 0.055
\]
2. **Set n for Monthly Compounding**:
\[
n = 12
\]
3. **Calculate APY**:
\[
APY_{monthly} = \left(1 + \frac{0.055}{12}\right)^{12} - 1
\]
\[
APY_{monthly} = \left(1 + 0.0045833\right)^{12} - 1
\]
\[
APY_{monthly} = (1.0045833)^{12} - 1 \approx 0.056875
\]
\[
APY_{monthly} \approx 0.056875 \times 100 \approx 5.69\%
\]

### Step 2: Calculate APY for Quarterly Compounding
1. **Set n for Quarterly Compounding**:
\[
n = 4
\]
2. **Calculate APY**:
\[
APY_{quarterly} = \left(1 + \frac{0.055}{4}\right)^{4} - 1
\]
\[
APY_{quarterly} = \left(1 + 0.01375\right)^{4} - 1
\]
\[
APY_{quarterly} = (1.01375)^{4} - 1 \approx 0.056370
\]
\[
APY_{quarterly} \approx 0.056370 \times 100 \approx 5.64\%
\]

### Step 3: Find the Difference
Now we subtract the APY for quarterly compounding from the APY for monthly compounding:

\[
\text{Difference} = APY_{monthly} - APY_{quarterly}
\]
\[
\text{Difference} = 5.69\% - 5.64\% \approx 0.05\%
\]

### Final Answer:
The APY is approximately **0.05% higher** when the APR is compounded monthly as compared to being compounded quarterly.