The formula sounds quite intuitive if you assume linear principal amortization, and non-compounded interest: the average principal is P/2 so the total interest is I = P/2 x T x r. Overall, the monthly payment is (P + I)/Nb months.
When T is quite small, the error is small. The example in the article is r = 7%, T = 4. The error is only 0.8%. But if you use T = 30 years (a typical mortgage in the US), then for the same r = 7% the error is 14%.
I decided to see if the formula can be improved by adding interest on interest. In a very unscientific way I decided that Interest on interest is I x r x T/6, because 6 is 3! and is the denominator in the third term of a Taylor series. The new formula is (P + I + I-on-I)/N. For the same r = 7% and T = 30y the error went down to less than 1%.
When T is quite small, the error is small. The example in the article is r = 7%, T = 4. The error is only 0.8%. But if you use T = 30 years (a typical mortgage in the US), then for the same r = 7% the error is 14%.
I decided to see if the formula can be improved by adding interest on interest. In a very unscientific way I decided that Interest on interest is I x r x T/6, because 6 is 3! and is the denominator in the third term of a Taylor series. The new formula is (P + I + I-on-I)/N. For the same r = 7% and T = 30y the error went down to less than 1%.