TVM solve for interest rate, revisited

[Albert Chan]

(07-11-2022 09:08 PM)Albert Chan Wrote:  (*) Another way is simply get a better guess.
For above example, we may ignore pmt, and directly solve for rate.

pv*(1+x)^n + fv = 0      → x = expm1(log(-fv/pv)/n)

We have x = 5.310, which is a great starting guess.

Had we put back pmt term, we get:

newx = function(x) return expm1(log1p(-(pv+fv)/(pmt/x+pv))/n) end

newx(x = ±Inf) reduced to quoted pmt=0 approximation.
x ← newx(x) work well for huge x; it is almost as good as Newton's method.

x - (newx - x) / (newx - x)' ≈ x - (newx - x) / (0 - 1) = newx

lua> require'fun'()
lua> genx = function(f,x) f=f(x); return f, f, f-x end
lua> n,pv,pmt,fv = 10, -100, 10, 1e10
lua> iter(genx,newx,Inf) :take(8) :each(print)

Code:

5.309573444801933       -Inf
5.321581577924096       0.012008133122162867
5.321554197006317       -2.738091777931828e-005
5.321554259298183       6.229186588768698e-008
5.321554259156466       -1.4171686046893228e-010
5.321554259156789       3.232969447708456e-013
5.3215542591567875      -1.7763568394002505e-015
5.3215542591567875      0

Some references, for how Lua iterator work

https://luafun.github.io/under_the_hood.html#iterators
http://lua-users.org/wiki/IteratorsTutorial
https://www.lua.org/pil/7.1.html