Approximate analytical results for instalment loans

The basic variables involved in the mathematical formulation of an instalment loan are the amount of the loan, the number of instalments in the term of the loan, the size of each payment and the interest rate on the loan (commonly referred to as the annual percentage rate or APR). Given any three of these variables, the fourth can be calculated with the aid of annuity tables. The underlying relationship is, however, quite complex and it is therefore difficult to gain analytical understanding of the interplay between the variables, especially in view of the fact that the functional dependence of the APR on the remaining variables cannot be explicitly stated.

An approximate formula for the APR is derived in terms of a lower bound which is shown to be quite sharp provided the finance charge per unit financed is less than one. On the basis of this formula, several analytical results are derived. In particular, the rates of change characteristics of the payment size with respect to the loan amount and to the interest rate, and of the term of the loan with respect to the payment size are discussed. It is also possible to give intuitively appealing interpretations of the APR and the finance charge per unit financed. The APR for an instalment loan is shown to be roughly twice as high as the annual nominal rate on a comparable loan payable in a lump sum assuming the finance charge per unit financed is not large.