In transactions involving compound interest, the stated annual rate of interest in called the nominal rate of interest. Thus if an investment is made at 6% converted semiannually, the nominal rate of interest on this investment is 6% . It may be noted that the actual interest earned on the given investment will be more than 6% that year. For example, Rs. 100 invested at 6% converted semiannually amounts in one year to 100(1.03)² = Rs. 106.09. Thus the interest actually earned on this principal of Rs. 100 is Rs. 6.08 which represents an annual return of 6.09%. We say that the effective rate in this case is 6.09%. When the conversion period is a year he effective rate is the same as the stated annual rate.

Relationship Between the Effective Rate and the Nominal Rate. Let r_e denote the effective rate corresponding to the nominal rate r, converted m times a year. We shall use I exclusively for rate per conversion period. Thus i = r/m. At the rate i, the principal p amounts in one tear to p (1+ i)^m. Since an effective rate is the actual rate compounded annually, therefore at the effective rate re, the principal P amounts in one year to P (1+ r_e). Thus

 P (1+ r_e) = P (1+ i)^m

or,   1+ r_e = (1+ i)^m

r_e = (1 + i)^m-1

Hence

Effective Rate r_e  = (1+ i )^m -1 = (1 + r/m)^m -1

By use of formula, we can find the effective rate equivalent to the nominal rate r convert m times a year, that is, equivalent to the rate i = r/m per conversion period. However, if the nominal rate r is compounded continuously, the effective rate re is given by

re  = lim [ (1 + r/m) ^m -1]
     m→∞

= lim (1+r/m)^m -1
  m→∞

= lim [ (1 + r/m)^{m/ r} ] ^r -1
  m→∞

= [ lim (1+ r/m)^{m/ r} ] ^r -1
  m→∞

Let r/m = x, then as m→∞, x→0. Thus

r_e = [ lim (1+ x) ^1/x ] ^r -1 = e^r -1

Hence in the case of nominal rate r compounded continuously, we see that

Effective Rare r_e = e^r -1

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