Nominal, Period and Effective Interest Rates Based on Discrete Compounding of Interest

·         Usually, financial agencies report the interest rate on a nominal annual basis with a specified compounding period that shows the number of times interest is compounded per year. This is called simple interest, nominal interest, or annual interest rate. If the interest rate is compounded annually, it means interest is compounded once per year and you receive the interest at the end of the year. For example, if you deposit 100 dollars in a bank account with an annual interest rate of 6% compounded annually, you will receive 100∗(1+0.06) = 106100∗(1+0.06) = 106 dollars at the end of the year.

·         But, the compounding period can be smaller than a year (it can be quarterly, monthly, or daily). In that case, the interest rate would be compounded more than once a year. For example, if the financial agency reports quarterly compounding interest, it means interest will be compounded four times per year and you would receive the interest at the end of each quarter. If the interest is compounding monthly, then the interest is compounded 12 times per year and you would receive the interest at the end of the month.

·         For example: assume you deposit 100 dollars in a bank account and the bank pays you 6% interest compounded monthly. This means the nominal annual interest rate is 6%, interest is compounded each month (12 times per year) with the rate of 6/12 = 0.005 per month, and you receive the interest at the end of each month. In this case, at the end of the year, you will receive 100∗(1+0.005)12= 106.17100∗(1+0.005)12= 106.17 dollars, which is larger than if it is compounded once per year: 100∗(1+0.06)1= 106100∗(1+0.06)1= 106 dollars. Consequently, the more compounding periods per year, the greater total amount of interest paid.

·         Please watch the following video, Nominal and Period Interest Rates (Time 3:52).

Nominal and Period Interest Rates

Click for the transcript of “Nominal and Period Interest Rates” video.

Credit: Farid Tayari

Period interest rate i = r/m
Where m = number of compounding periods per year
r = nominal interest rate = mi

“An effective interest rate is the interest rate that when applied once per year to a principal sum will give the same amount of interest equal to a nominal rate of r percent per year compounded m times per year. Annual Percentage Yield (APY) is the standard term used by the banking industry to identify an effective interest rate.”

The future value, F1, of investing P at i% per period for m period after one year:

hen:

F1=F2P(1+i)m= P(1+E)1F1=F2P(1+i)m= P(1+E)1

Since P the same in both sides: (1+i)m= E+1(1+i)m= E+1

Then:

Effective Annual Interest:E = (1+i)m−1Effective Annual Interest:E = (1+i)m−1

(Equation 2-1)

If the effective Annual Interest, E, is known and equivalent period interest rate i is unknown, the equation 2-1 can be written as:

i = (E +1)1/m −1i = (E +1)1/m −1

(Equation 2-2)

Going back to the previous example,

i=6/12 = 0.005so, E=(1+0.005)12−1 = 1.0617 − 1 = 0.0617 or 6.17%