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.
To determine the difference in APY (Annual Percentage Yield) when the rate is compounded monthly versus quarterly, we need to understand the impact of compounding on the overall interest earned.
APR (Annual Percentage Rate) represents the interest rate applied annually to the loan amount. In this case, the bank is advertising an APR of 5.5% on personal loans.
When the rate is compounded monthly, the interest is calculated and added to the initial loan amount every month, leading to more frequent compounding throughout the year. On the other hand, when the rate is compounded quarterly, the interest is calculated and added to the initial loan amount every quarter (every three months).
In order to calculate the APY, we need to account for the compounding frequency by using the formula:
APY = (1 + r/n)^n - 1
where r is the APR as a decimal (5.5% = 0.055), and n is the number of compounding periods in one year.
For monthly compounding:
APY_monthly = (1 + 0.055/12)^12 - 1
For quarterly compounding:
APY_quarterly = (1 + 0.055/4)^4 - 1
Now we can calculate the difference in APY:
APY_difference = APY_monthly - APY_quarterly
To simplify:
APY_difference = [(1 + 0.055/12)^12 - 1] - [(1 + 0.055/4)^4 - 1]
By calculating this expression, we can find the exact difference in APY between the monthly and quarterly compounding methods.