$title Repayment Factors for Loans (REPAY,SEQ=72)

$onText
This sample problem computes the monthly payments for a 1000 dollar
loan to be repaid in equal installments. The monthly payment is derived
as follows.

GAMS Development Corporation, Formulation and Language Example.

p(0) = 1000
p(1) = p(0)  *(1 + rate)/periods - pay
...
p(t) = p(t-1)*(1 + rate)/periods - pay = 0

where "p" is the principal outstanding at the beginning of a period and
"rate" is the annual interest rate. "periods" are the payment periods
per year. "pay" is the payment per period. The loan has to be repaid
completely after "t" payments within "years" years. The number of
payments "t" is "periods"*"years".

with sum(n, q**(ord(n) - 1)) = (1 - q**card(n))/(1 - q)
find
pay = 1000*rate/periods/(1 - (1 + rates/periods)**(-periods*years)).

Keywords: repayment factors, finance
$offText

Set
   r 'list of rates'       / 1*41 /
   m 'list of maturities'  / 12-years, 15-years, 30-years /;

Parameter
   rates(r) 'annual rates'
   years(m) 'maturities'   / 12-years 12, 15-years 15, 30-years 30 /
   periods  'payments per year'
   pay      'monthly payments for 1000 dollar loan';

rates(r) = .08 + .00125*(ord(r) - 1);
periods  = 12;

pay(r,"rate") = 100*rates(r);
pay(r,m)      = 1000*rates(r)/periods/(1-power(1+rates(r)/periods,-periods*years(m)));

display pay;