The frequency of compounding is extremely important in calculating the present value and future value of the cash flows. It means that at what frequency the interest amount is added to the principal amount.
Suppose you invest $1,000 in a bank. The bank is providing you 8 percent per annum interest (stated annual interest rate) with annual compounding. We need to calculate the future value of this investment after 3 years. Since the compounding frequency is annual, the interest will be added to the principal amount after one year. Therefore, value of principal amount after one year = 1,000 + 1,000*0.08 (annual interest) = 1,000(1+0.08) = $1,080. Now, in the second year, we will get interest on this principal amount ($1,080) rather than $1,000. So, we will end up getting slightly more interest. Value of principal amount after two years = 1,080 + 1,080*0.08 = 1,080(1+0.08)= 1,000(1+0.08)2 = $1,166.40. In the third year, we will get interest on $1,166.40 and the value of investment after three years = 1,166.40 + 1,166.40*0.08 = 1,166.40(1+0.08) = 1,080(1+0.08)2 = 1,000(1+0.08)3 = $1,259.71.
Since the compounding was annual in the previous example, the effective annual rate was also equal to the stated annual interest rate i.e. 8 percent.
If we change the compounding frequency to semi-annual, then the interest amount for each period will be based on 4 percent (=8 percent divided by 2) and the total number of periods will also increase to 6 (=3*2). Then, the future value will be equal to 1,000(1+0.04)6=$1,265.32.
The effective annual rate (EAR) is that interest rate which makes the future value equal to actual future value with any compounding frequency assuming an annual compounding frequency.
The EAR with the above example of semi-annual compounding will be such so that FV = 1,000(1+EAR)3 = 1,000(1+0.04)6. So, EAR = [(1+0.04)6](1/3) - 1 = (1+0.04)2 - 1 = (1+rs/m)m - 1 = (1+Periodic interest rate)Number of periods in a year - 1.
where:
m = number of periods in a year.
rs = stated annual interest rate
When the compounding frequency is continuous, then there will be infinite periods in a year.
EAR (continuous compounding) = lim (1+ rs/m)m - 1 = exp(rs) - 1.
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