As we have seen earlier, under the compound interest scheme, interest during a period is earned on the amount available at the end of the previous period.  And if the previous period is not period zero, the amount at the end of the previous period will comprise the principal as well as the interest earned on the principal.  Thus, interest on interest as well as interest on principal is earned under the compound interest scheme.  It follows then that more interest is earned when you compound more frequently.  Lending agencies are in the business of making money.  So they favor compounding at more frequent intervals than annual.  When that happens the effective interest rate paid on a loan will be larger than the nominal interest rate for the year.  Nominal interest rate is also known as the annual percentage rate (APR).

Effective interest rates are computed by first determining the interest rate per interest period.  If r is the annual interest rate and if we have M interest periods in a year, then the rate per interest period is simply r/M.    Effective interest rates are associated with the period that they apply to.    In other words, if we want to compute the effective interest rate for a year on a loan in which the interest is compounded at a more frequent interval than annually, we would denote the effective interest rate as i_a.  The subscript a is to denote that the interest rate is computed annually.   If we wanted to compute the effective interest rate for some other time interval, we would have to change the subscript a to some other notation that refers to the time period more accurately.  For example, if we wanted to compute the effective interest rate for a quarter, we would denote that effective interest rate by i_q.  We don't have to use the symbol q to denote the effective interest for a quarter; any other notation would suffice, as long as it is unique and denotes the effective time period unambiguously. 

The derivation of the formula for effective interest rate is based on the fact that interest is the difference in worth of the amount in question at the beginning and the end of the time period.  If you are finding the effective interest rate for a one-year period, you would compute the difference in value of some amount at the beginning of the year and at the end of the year.  The derivation is simplified, if you do this computation for a dollar.  So at the beginning of the period the value of the dollar is one dollar.  To compute the value of the dollar at the end of the year you have to know the number of compounding/interest periods in the year.  Once that is done, you    compute the interest rate per interest period by using the r/M formula.   This is the interest rate you will use in the compound interest formula to compute the value of the dollar at the end of the year.  The effective interest paid is then the difference between the future value and the present value.  This computation yields the effective interest rate in decimal units.  If you want the rate in percentage units, you just need to multiply the decimal value by 100.  Equation 5.1 is the end result of the derivation procedure described above.  And Examples 5.1 and 5.2 illustrate the use of the formula.

Equation 5.2 extends the effective interest formula to time periods other than a year.    If your payment period is different from your interest period, you compute the effective interest for the payment period.  So we no longer do the compounding for M periods but for the number of interest periods in the payment period.  This is denoted by the variable C in formula 5.2.  As before we start with the nominal interest rate of r, divide it by M and use this interest rate to compound the present value to C periods.  Taking the difference between the value computed and the present value will yield the effective interest rate i for the payment period as shown in Equation 5.2.  For clarity, you may want to subscript the i in Equation 5.2 by p, for payment period,  and denote the effective interest rate for the period by i_p.