My HP 10bII+ is much faster than HP 12c!

[Albert Chan]

XCas easily handle this rate problem.

XCas> nfv(n, r, pv, pmt, fv) := fv + pv + ((1+r)^n - 1) * (pv + pmt/r)

XCas> solve(nfv(168,x,0,-45,10925.76) = 0)      // OP, Galuppo's example
      → [0.00417538330383]                               // ×12 → 5.01045996459%

XCas> solve(nfv(30,x,1000,-1000,50000) = 0)   // my example returned 2 rates
      → [0.0365197435259, 0.999999952503]

Timings suggest solve (without guess) actually call proots, then removes the complex roots.

Some real roots for rate are meaningless, say with r ≤ -100%.
Let x = 1+r, and consider only positive x as valid, we get:

\( NFV = FV + PV x^n + PMT\left({x^n-1 \over x-1}\right)\)

\(\large {NFV \over PMT} = \left({PV \over PMT}\right) x^n + x^{n-1} + x^{n-2} + \;... +\;x + \left(1 + {FV \over PMT}\right)\)

If above has one sign change, we have exactly one positive solution for x.

For 2 sign changes, x has 0 or 2 positive roots (see Descartes' sign rules)