TVM solve for interest rate, revisited

[Albert Chan]

(06-10-2022 08:25 PM)Albert Chan Wrote:  Proof: f is either concave-up, or concave-down (f'' have same sign)

g = n*x / ((1+x)^n-1) = 1 - (n-1)/2*x + (n²-1)/12*x² - (n²-1)/24*x³ + ...

Previous proof showed g'' > 0

In order to do that, it also have to show g above x=0 tangent line.
I had wrongly assumed second part true, for my first proof attempt.

This proof is more direct, by showing g'' has no roots.
As long as g'' has the same sign, f'' also have same sign throughout.

f = (pv+fv)/n*g + pv*x + pmt
f' = (pv+fv)/n*g' + pv
f'' = (pv+fv)/n*g''

XCAS> g := x*n/((1+x)^n-1)
XCAS> g := g(x=x-1); // shifted. Now, domain is x > 0
XCAS> h := normal(n/g)

(x^n-1) / (x-1)      // = x^(n-1) + x^(n-2) + ... + x + 1 > 1, if n>1

h is a polynomial, so does all its derivatives.

g'' = n*(h^-1)''

(h^-1)'' = (-h^-2 * h')' = (-h^-2 * h'' + 2*h^-3 * h'^2)
(h^-1)'' * h^3 = 2*h'^2 - h*h''

RHS looks promising to be positive; it is Halley's correction denominator.
But, all we needed is to show RHS never touch 0.

XCAS> m := factor(simplify(2*h'^2 - h*h''))

\(\displaystyle \frac {n\;x^{n-2} \cdot \left[
(n\!-\!1)\; x^{n+1}
\;-\; (n\!+\!1)\; x^{n}
\;+\; (n\!+\!1)\; x
\;-\; (n\!-\!1)
\right]}{\left(x-1\right)^{3}}\)

Above expression touched up a bit, to show polynomial sign changes.
For n>1, numerator has 3 sign changes, thus 1 or 3 positive roots.
Denominator had factor (x-1)^3, cancelled all numerator roots.

For n>1, m has no roots ⇒ g'' has no roots ⇒ f'' has no roots. QED

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Not needed for the proof, but we might as well get sign(g'')

XCAS> factor(limit(m, x=1))      // (n>1) ⇒ (m>0) ⇒ (g''>0)

n^2*(n+1)*(n-1)/6

Or, we just imagine x ≫ n > 1, sign(m) = + / + = +