Solving the TVM equation for the interest rate

[Albert Chan]

(02-12-2020 05:25 PM)Albert Chan Wrote:  This formula is based on Pade[1,1], I centered 0, of NFV = F + P + ((1+I)^N-1)*(P+M/I)

Solving Pade[1,1] approximated NFV = 0, for 1/I :

\( \Large {1\over I} ≈
{\binom{N}{3}M + \binom{N}{2}P \over \binom{N}{2}M + \binom{N}{1}P} -
{\binom{N}{2}M + \binom{N}{1}P \over F + P + M N}\)

Doing the same example, N=11, F=40000, P=0, M=-2564

1/I ≈ (-423060/-141020) - (-141020/11796) = 44102/2949

I ≈ 2949/44102 ≈ 6.687%, which under-estimated true rate (6.780%) by tiny 0.093%

We can reuse the formula, getting a closer estimate.

With above I=6.687%, calculated F = 39804.11, error = 40000 - 39804.11 = 195.89
Tried the formula again, with F = 40000 + 195.89, to compensate this error

I ≈ 1/(3 - (-141020/(11796 + 195.89))) ≈ 6.775%

Again, using this new I, calculated F = 39989.83, error = 40000 - 39989.83 = 10.17

Interpolate for 0 error, I = 6.775% - (6.687% - 6.775%) * 10.17/(195.89 - 10.17) = 6.7798%