Discount Rate

[Albert Chan]

(04-10-2022 12:19 PM)Thomas Klemm Wrote:  Newton Iteration
\[
\begin{align}
Z &= (1 + i)^{-n} \\
\\
i_\text{next} &= i \left[ 1 + \frac{P + \frac{Z - 1}{i}}{\frac{Z - 1}{i} + \frac{nZ}{1 + i}} \right]
\end{align}
\]

When interest rate converging to true rate, numerator goes 0, or (Z-1)/i = -P

I was reading https://brownmath.com/bsci/loan.htm#Newton.
It's rate newton formula actually replaced denominator (Z-1)/i by -P

The formula quoted above is good when guess rate is under-estimated.
This explained why it has i0 = 2*(n-p)/(n*(n+1)) instead of 2*(n-p)/(p*(n+1))

Too high a guess can make Newton's step diverge.
Example, with N=36, P=30, guess rate of 0.08, it barely able to converge.

0.08 → -0.071887 → -0.041499 → -0.014129 → 0.003911 → 0.009742 → 0.010205 → 0.010207

"-P" version had the opposite problem, and preferred over-estimated guess.
Again, with N=36, P=30, guess rate of 0.004, it totally diverged

0.004 → -0.009120 → -0.003414 → -0.000780 → -0.000059 ...

A better option is to solve NPMT=0 instead (now used for rate search in Plus42, see tvm_eq())
From NPMT=0 based loan_rate(n, pv, pmt, fv, i), simplified with FV=0, we get this:

\( \displaystyle \large
i_\text{next} = i \left[
1 -
\frac{(\frac{Z - 1}{i})/P + 1}
{(\frac{nZ}{1 + i})/(\frac{Z - 1}{i}) +1} \right] \)

This version, convergence is slightly faster, and *very* stable for guess i
Again, with N=36, P=30, but ridiculous rate guess:

10⁺¹⁰ → 0.033333 → 0.012232 → 0.010228 → 0.010207
10⁻¹⁰ → 0.000231 → 0.010782 → 0.010209 → 0.010207