Find the EAR in each of the following cases.
Stated Rate (APR) Number of Times
8 % Quarterly %
16 % Monthly %
12 % Daily %
15 % Infinite %
8% APR paid quarterly is (1.02)^4 -1 = 8.243%
16% APR paid monthly is (1.01333)^12 -1 = 17.22%
12% APR paid daily is (1.0003288)^365 = 12.75%
15% APR compounded with infinite frequency is
e^.15 -1 = 16.18%
Thank you, that infinite was was real confusing. Any advice on the other two questions I posted?
To find the Effective Annual Rate (EAR) in each of the given cases, follow the steps below:
1. Case 1: Stated Rate (APR) = 8%, Number of Times = Quarterly
The formula to calculate EAR is: EAR = (1 + (APR / n))^n - 1, where n represents the number of compounding periods.
In this case, APR = 8% and n = 4 (quarterly compounding).
EAR = (1 + (0.08 / 4))^4 - 1
EAR = (1.02)^4 - 1
EAR = 1.0824 - 1
EAR = 0.0824 or 8.24%
2. Case 2: Stated Rate (APR) = 16%, Number of Times = Monthly
Using the same formula, APR = 16% and n = 12 (monthly compounding).
EAR = (1 + (0.16 / 12))^12 - 1
EAR = (1.01)^12 - 1
EAR = 1.1268 - 1
EAR = 0.1268 or 12.68%
3. Case 3: Stated Rate (APR) = 12%, Number of Times = Daily
Again, using the formula, APR = 12% and n = 365 (daily compounding).
EAR = (1 + (0.12 / 365))^365 - 1
EAR = (1.000328)^365 - 1
EAR = 1.1275 - 1
EAR = 0.1275 or 12.75%
4. Case 4: Stated Rate (APR) = 15%, Number of Times = Infinite
In this case, the EAR is calculated using continuous compounding. The formula for continuous compounding is: EAR = e^(APR) - 1, where e is the base of natural logarithm.
EAR = e^(0.15) - 1
Using a scientific calculator or software, the value of e^(0.15) is approximately 1.1618.
EAR = 1.1618 - 1
EAR = 0.1618 or 16.18%
Therefore, in each of the given cases:
- Case 1: EAR = 8.24%
- Case 2: EAR = 12.68%
- Case 3: EAR = 12.75%
- Case 4: EAR = 16.18%
To find the Effective Annual Rate (EAR) in each of the given cases, we need to consider the compounding period and use the appropriate formula:
1. For the case of 8% interest rate compounded quarterly:
EAR = (1 + (APR / n))^n - 1
Here, n is the number of compounding periods per year.
EAR = (1 + (8% / 4))^4 - 1
EAR = (1 + 0.02)^4 - 1
EAR = 1.02^4 - 1
EAR = 1.0824 - 1
EAR = 0.0824 or 8.24%
2. For the case of 16% interest rate compounded monthly:
EAR = (1 + (APR / n))^n - 1
Here, n is the number of compounding periods per year.
EAR = (1 + (16% / 12))^12 - 1
EAR = (1 + 0.013333)^12 - 1
EAR = 1.013333^12 - 1
EAR = 1.1721 - 1
EAR = 0.1721 or 17.21%
3. For the case of 12% interest rate compounded daily:
EAR = (1 + (APR / n))^n - 1
Here, n is the number of compounding periods per year.
Since there are 365 days in a year:
EAR = (1 + (12% / 365))^365 - 1
EAR = (1 + 0.032877)^365 - 1
EAR = 1.032877^365 - 1
EAR = 1.1276 - 1
EAR = 0.1276 or 12.76%
4. For the case of 15% interest rate compounded infinitely:
In this case, the EAR can be calculated by using the following formula:
EAR = e^(APR / 100) - 1
Here, e is the mathematical constant approximately equal to 2.71828.
EAR = e^(15% / 100) - 1
EAR = e^(0.15) - 1
EAR ≈ 1.1618 - 1
EAR ≈ 0.1618 or 16.18%
So, the EAR in each case is as follows:
- 8% compounded quarterly: 8.24%
- 16% compounded monthly: 17.21%
- 12% compounded daily: 12.76%
- 15% compounded infinitely: 16.18%