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.

Question

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.