## Nominal And Effective Rates Of Interest

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# Nominal And Effective Rates Of Interest

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)

P (1+ r

or, 1+ r

r

Hence

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→∞

= lim (1+r/m)

m→∞

= lim [ (1 + r/m)

m→∞

= [ lim (1+ r/m)

m→∞

Let r/m = x, then as m→∞, x→0. Thus

r

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

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^{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}). ThusP (1+ r

_{e}) = P (1+ i)^{m}or, 1+ r

_{e}= (1+ i)^{m}r

_{e }= (1 + i)^{m}-1Hence

**Effective Rate****r**= (1+ i )_{e}^{m}-1 = (1 + r/m)^{m}-1By 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}-1m→∞

= lim [ (1 + r/m)

^{m/ r}]^{ r}-1m→∞

= [ lim (1+ r/m)

^{m/ r}]^{r}-1m→∞

Let r/m = x, then as m→∞, x→0. Thus

r

_{e}= [ lim (1+ x)^{1/x}]^{r}-1 = e^{r}-1Hence in the case of nominal rate r compounded continuously, we see that

**Effective Rare r**_{e}= e^{r }-1For more help in

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