(10C) Mortgage Loan Interest Rate

[Albert Chan]

(11-01-2018 03:22 AM)Albert Chan Wrote:  Let compounding factor C = nR/P, so C = 1 if i = 0

C = ni / (1 - (1+i)^-n)
C = 1 + (n+1)/2 i + (n²-1)/12 i² - (n²-1)/24 i³ + ...

if keep only linear terms, i1 = 2*(C-1) / (n+1)
if keep quadratics too, i2 = (2 i1) / (1 + sqrt(1 + (2 i1)*(n-1)/3))

For P=2500, R=86.67, n=36:
C = nR/P = 1.248048
i1 = 0.013408, or annual rate of 16.1%
i2 = 0.012497, or annual rate of 15.0%

Hi, Dieter

The formulas were meant for rough estimation, using "normal" calculators (no X^Y)

All formula were derived from C Taylor expansion at i = 0
The quadratic formula were using more accurate form: 2c / (-b - sqrt(b²-4ac))
where, a= (n²-1)/12, b=(n+1)/2, c=1-C, a non-positive number.

This is how the quick formula is derived:
C = 1 + (n+1)/2 i + (n²-1)/12 i² - (n²-1)/24 i³ + ...
~ 1 + (n+1)/2 i + (ni)² / 12
~ i/2 + (3 + 9 + 6(ni) + (ni)²) / 12
= i/2 + ((ni + 3)² + 3) / 12

To get more accurate interest rate, do interpolation (but X^Y is required):
i1 = 0.013408, C1 = n*i1 / (1 - (1+i1)^-n) = 1.267246
i2 = 0.012497, C2 = n*i2 / (1 - (1+i2)^-n) = 1.247888
Interpolate for C = nR/P = 1.248048, we get i = 0.0125045