These terms are used in the formulas that follow:
periodic payment
principal amount
nominal annual rate
compounding frequency (per year)
payment frequency (per year)
periodic rate
effective interest rate
total number of payments
The periodic rate, or the simple interest applied during a payment period, is given by
Note that the interest calculation is performed at each payment period rather than at the compound period. This is done by adjusting the nominal rate. See Muksian (1984) for details.
Note that when (that is, when the payment and compounding frequency coincide), the preceding expression reduces to the familiar form:
The periodic rate for continuous compounding can be obtained from this general expression by taking the limit as the compounding frequency f goes to infinity. The resulting expression is
The effective interest rate, or annualized percentage rate (APR), is that rate which, if compounded once per year, is equivalent to the nominal annual rate compounded f times per year. Thus,
or
For continuous compounding, the effective interest rate is given by
See Muksian (1984) for details.
The payment is calculated as
The amount is calculated as
Both the payment and amount are rounded to the nearest hundredth (cent) unless the ROUND= specification is different than the default, 2.
The total number of payments n is calculated as
The total number of payments is rounded up to the nearest integer.
The nominal annual rate is calculated using the bisection method, with a as the objective and r starting in the interval between and 0.1 with an initial midpoint 0.01 and successive midpoints bisecting.