Approximate monthly loan payments

This post presents a simple method of estimating monthly payments on a loan. According to [1] this is a traditional Persian method and still commonly used in Iran.

A monthly payment amount is

(principal + interest)/months

but the total amount of interest over the course of a loan is complicated to compute.

Initially you owe all the principal, and the end you owe none of it, and so roughly on average you owe half of it. You could approximate the total interest as the simple interest on half the principal over the course of the loan. This is the Persian approximation. It's not exactly correct, but it makes a surprisingly good approximation.

Motivation

Why are approximations important? If you're just going to call some function in a spreadsheet, you might as well call the exact formula rather than an approximation.

However, the approximation is easier to understand. The exact formula is a nonlinear function of the interest rate, whereas the approximation is an affine function as we'll show below. It's easier, for example, to see the effect of a change in interest rates in the approximation.

Evaluating accuracy

Let P be the principal, N the number of months, and r the monthly interest rate. Then the exact loan payment is

whereas the Persian approximation is

A first-order Taylor series approximation for the exact formula gives

which is the Persian approximation with the N in the numerator replaced by N + 1. When N is large, the difference between N and N+1 doesn't matter so much, and the Taylor approximation is better when r is small, so we should expect the Persian approximation to be most accurate when N is large and r is small.

Let's see how the exact method and the two approximations compare for a five-year loan of $10,000 with 6% annual interest.

 P = 10_000 r = 0.06/12 N = 5*12 t = (1 + r)**N C = r*t*P/(t - 1) C1 = (P + 0.5*P*N*r)/N C2 = (P + 0.5*P*(N+1)*r)/N print(C, C1, C2)

The exact payment in this case is $193.33, the Persian approximation is $191.67, and the Taylor approximation is $192.08. The Persian approximation is a little simpler but also a little less accurate than the Taylor approximation.

[1] Peyman Milanfar. A Persian Folk Method of Figuring Interest. Mathematics Magazine. December 1996.

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