Imagine a local bank representative is assisting you with establishing an account. The account you two are discussing has an APR of 6.5%. Determine the APY with quarterly compounding and with monthly compounding. How does changing the compounding period affect the annual yield?
say $100 apr 6.5 simple = 6.50%
at 6.5 quarterly
(.065/4 +1)^4 = 1.0667 so 6.67 %
at 6.5 monthly
100 (.065/12 +1)^12 = 1.06697 so 6.70%
uncomplete need to know APY after find the value of A
To determine the APY (Annual Percentage Yield) with different compounding periods, we use the formula:
APY = (1 + r/n)^n - 1
Where:
- r is the annual interest rate (APR) expressed as a decimal (6.5% = 0.065)
- n is the number of compounding periods per year
For the given scenario, we need to determine the APY with quarterly and monthly compounding.
Quarterly Compounding:
In this case, n = 4 because there are 4 quarters in a year. Therefore, the formula becomes:
APY = (1 + 0.065/4)^4 - 1
To calculate this, we evaluate the expression inside the brackets first using the exponentiation (^) operator:
1 + 0.065/4 = 1.01625
Then, we raise the result to the power of 4:
1.01625^4 = 1.066987817
Finally, subtract 1 from this result:
1.066987817 - 1 = 0.066987817
So, the APY with quarterly compounding is approximately 0.06699 or 6.699%.
Monthly Compounding:
In this case, n = 12 because there are 12 months in a year. Therefore, the formula becomes:
APY = (1 + 0.065/12)^12 - 1
Similar to the previous calculation, we evaluate the expression inside the brackets first:
1 + 0.065/12 = 1.005416667
Then, we raise the result to the power of 12:
1.005416667^12 = 1.06750675
Finally, subtract 1 from this result:
1.06750675 - 1 = 0.06750675
So, the APY with monthly compounding is approximately 0.06751 or 6.751%.
Now, let's discuss how changing the compounding period affects the annual yield:
Increasing the compounding period leads to a higher APY. This is because more frequent compounding allows the interest to be calculated and added to the account balance more frequently. As a result, the interest earns interest more often, leading to a higher overall yield. In this case, the monthly compounding provides a higher APY compared to quarterly compounding, indicating that more frequent compounding is beneficial for account holders. However, it's important to note that the difference between the APYs may not always be significant, especially for smaller interest rates or shorter time periods.