The LOAN Procedure

Computational Details

These terms are used in the formulas that follow:

p: periodic payment
a: principal amount
$r_{a}$: nominal annual rate
f: compounding frequency (per year)
$f’$: payment frequency (per year)
r: periodic rate
$r_{e}$: effective interest rate
n: total number of payments

The periodic rate, or the simple interest applied during a payment period, is given by

$r = \left( 1 + \frac{r_{a}}{f }\right)^{f/f'} - 1$

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 ${f = {f’}}$ (that is, when the payment and compounding frequency coincide), the preceding expression reduces to the familiar form:

$r = \frac{r_{a}}{f }$

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

$r = \mr{exp}{\left(\frac{r_{a}}{f'}\right)} - 1$

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,

$(1 + r_{e}) = (1 + r)^{f} = \left(1 + \frac{r_{a}}{f}\right)^{f }$

$r_{e} = \left(1 + \frac{r_{a}}{f}\right)^{f } - 1$

For continuous compounding, the effective interest rate is given by

$r_{e} = \mr{exp}\left(r_{a}\right) - 1$

See Muksian (1984) for details.

The payment is calculated as

$p = \frac{a r}{1 - \frac{1}{(1 + r)^{n}}}$

The amount is calculated as

$a ={\frac{p}{r}} \left(1 - \frac{1}{(1 + r)^{n}} \right)$

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

$n = \frac{- \ln \left(1 - \frac{ar}{p}\right)}{\ln (1 + r)}$

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 ${8*10^{-6}}$ and 0.1 with an initial midpoint 0.01 and successive midpoints bisecting.