1.1.2 Calculate the Effective Annual Rate Flashcards
List three ways to quote interest rates for investments paying interest more than once a year.
Periodic interest rate
Stated annual interest rate (quoted interest rate)
Effective annual rate (EAR)
What is Periodic interest rate?
Periodic interest rate (PIR) is the rate of interest earned over a single compounding period.
PIR = Annual Percentage Rate (APR) / Number of periods in a year (n)
PIR = APR/n
Example 1
Let’s say the annual interest rate is 10%, and one period is of a month. In such a situation, the total number for periods in a year would be 12, and the calculation will be as follows:-
10/12 = 0.83%
In the above example, the monthly Periodic Interest Rate will be 0.83%.
Example 2
Let’s say the annual interest rate is 12%, and one period is for a day. In such a situation, the total number of periods in a year would be 365, and the calculation will be as follows:-
12/365 = 0.0329%
In the above example, the daily Periodic Interest Rate will be 0.0329%.
Example 3
Let’s say the annual interest rate is 15%, and the interest is compounded semi-annually, with a total of two periods in a year.
15/2 = 7.5%
In the above example, the semi-annual Periodic Interest Rate will be 7.5%.
What is Stated annual interest rate (quoted interest rate)?
The stated annual interest rate (quoted interest rate), is the annual rate of interest that 【 does not account for compounding within the year. 】
It is the annual interest rate quoted by financial institutions and【 is equal to the periodic interest rate multiplied by the number of compounding periods per year. 】
For example, the stated annual interest rate of the above CD is 3% x 4 = 12%. It is strictly a quoting convention, and it does not give a future value directly.
What is Effective Annual Rate (EAR)?
Effective annual rate (EAR) is the annual rate of interest that takes full account of compounding within the year.
The periodic interest rate is the stated annual interest rate divided by m, where m is the number of compounding periods in one year:
Formula : EAR = ((1+(PIR/m))^m) - 1
Note that the higher the compounding frequency, the higher the EAR.
The difference between Periodic Interest Rate Vs Annual Percentage Rate (APR)?
Periodic Interest Rate
This rate of interest is quoted periodically.
This interest rate takes the compounding effect into consideration.
Mostly, Periodic Interest Rate is more accurate in comparison to APR. It gives the actual cost of borrowing or actual returns on investments.
This is mostly for a shorter period of time.
Annual Percentage Rate (APR)
APR, as the name suggests, is quoted annually.
APR does not take the compounding effect into consideration.
APR is comparatively less accurate as there are high chances of understating the cost of the loan or overstating the returns because of its long-term schedule.
This is mostly for a longer period of time
For example, a $1 investment earning 8% compounded semi-annually actually earns 8.16%: (1 + 0.08/2)^2 - 1 = 8.16. The annual interest rate is 8%, and the effective annual rate is 8.16%.
Calculate the Effective Annual Rate (EAR) if the nominal interest rate is 8%, find the effective annual rate with quarterly compounding.
Method 1: By Formula
m = 4, EAR = (1 + 0.08/4)4 - 1 = 0.0824
The effective interest rate with quarterly compounding is 8.24%.
- The effective annual interest rate is ______.
A. always equal to the annual percentage rate
B. greater than the annual percentage rate when the number of compounding periods per year is greater than one
C. less than the annual percentage rate when the number of compounding periods per year is greater than one
Correct Answer: B
- You borrow $10,000 at 8% to finance a new car. The loan will be paid back over 3 years of monthly payments. Your effective rate of interest would be closest to ______.
A. 8.00%
B. 8.24%
C. 8.30%
Correct Answer: C. With monthly payments, interest will be charged on a monthly basis. The effective interest rate is the stated rate compounded monthly: r = (1 + 0.08/12)12 - 1
- You borrow $10,000 at 8% to finance a new car. The loan will be paid back over 3 years of monthly payments. Your effective rate of interest would be closest to ______.
A. 8.00%
B. 8.24%
C. 8.30%
Correct Answer: C. With monthly payments, interest will be charged on a monthly basis. The effective interest rate is the stated rate compounded monthly: r = (1 + 0.08/12)12 - 1
- The new products officer for the Strong-n-Safe Bank is trying to determine the stated rate for a new 4 year CD. The effective rate will be 6.95% and the CD will offer continuous compounding. The advertised stated rate would be closest to ______.
A. 6.72%
B. 6.82%
C. 7.20%
A. 6.72%
- Which of the following statements is true?
I. If the compounding period is one year, stated annual interest is the same as effective annual interest.
II. If the compounding period is less than one year, effective annual interest is greater than stated annual interest.
III. If the compounding period is less than one year, stated annual interest is greater than effective annual interest.
Correct Answer: I and II
- Which of the following amounts is closest to the end value of investing $100,000 for 5 years at an effective annual interest rate of 12% compounded annually.
A. $182,212
B. $172,000
C. $176,234
Correct Answer: C End value = 100,000(1+0.12)5 = $176,234.17
- A money manager has $20 million to invest for one year. She has identified three alternative one-year certificates of deposit (CD) (shown below):
CD | Compounding frequency | Annual interest rate
CD1 | Monthly | 7.82%
CD2 | Quarterly | 8.00%
CD3 | Continuously | 7.95%
Which CD has the highest effective annual rate (EAR)?
A. CD 1
B. CD 2
C. CD 3
Correct Answer: C
CD1: (1 + 0.0782/12)12 - 1 = 8.1065%
CD2: (1 + 0.08/4)4 - 1 = 8.2432%
CD3: e0.0795 - 1 = 8.2746%
