In the above illustration we have consider how the future value modify along with the modification in frequency of compounding. So as to understand the relationship among effectual and nominal rate let us compute the future value of Rs.1000 on the interest rate of 12 percent while the compounding is done yearly, quarterly, monthly and semiannually.
FV = 1,000(1 + .12)^{1}
= 1120
FV = 1000 ( 1 + .12/2)^{2}
= 1000(1.06)^{2}
= 1000 (1.1236)
= 1123.6
FV = 1000 ( 1 + .12/4)^{4}
1000 = (1.03)^{4}
1000 = (1.1255)
= 1125.5
FV = 1000 ( 1 + .12/12)^{12}
= 1000 (1.01)^{12}
= 1000 (1.1268)
= 1126.8
BY the above computations we can notice that Rs.1000 grows to Rs.1120, Rs.1123.6 and Rs.1125.5 and Rs.1126.8 though the time period and rate of interest are similar. In the given case 12.36, 12.55 and 12.68 are termed as effectual rate of interest. The connection among the effectual and nominal rate of interest is specified by:
r = (1 + k/m)^{m} - 1 .........................Eq(4)
Here r = effective rate of interest;
k = nominal rate of interest;
m = frequency of compounding yearly
depands on the above stated illustration the effective interest rate is computed as follows:
1) Effective interest rate for monthly compounding
r = (1 + .12/12)^{12} - 1
= (1.01)^{12} -1
= 1.1268 - 1
= 12.68
2) Effective interest rate for quarterly compounding
r = (1 + .12/4)^{4} - 1
= (1.03)^{4} - 1
= 1.1255 - 1
= .1255
= 12.55%
3) As the same the effective interest rate for semi-annual compounding is
r = (1 + 12/2)^{2} - 1
= (1.06)^{2} - 1
= 1.1236 - 1
= .1236
= 12.36