Question

If r is the nominal rate and n is the number of times interest is compounded annually, then R=(1+r/n)^(n)-1 is the effective rate. Here, R represents the annual rate that the investment would earn if simple interest were paid. Use this formula to determine the effective rate for $1 invested for 1 year at 4.8% compounded semiannually.

Answer 1

Effective Rate in Compound Interest

Given r as the nominal rate of investment and n the number of times the interest is compounded annually, the formula for the effective rate is:

`R=\mleft(1+(r)/(n)\mright)^n-1`

We are required to find the effective rate for a rate of r=4.8% compounded semiannually. This means the value of n is 2 since there are two periods where interest is added to the principal per year.

Substituting the given values in the formula (recall r must be used as a decimal value, i.e. r=4.8/100=0.048):

`R=(1+(0.048)/(2))^2-1`

Calculating:

`R=(1.024)^2-1=0.048576`

The effective rate is 4.86%