PC-1211, PC-1250, etc. TVM

[Albert Chan]

(06-02-2024 06:37 PM)Albert Chan Wrote:  rate = pmt/(x-pv)
f = x + (PV+FV) / ((1+rate)^N-1) = 0      // flatten curve to make Newton's method efficient
...
Interestingly, converge rate seems to match NPMT formula with Newton's method.

I just checked. Barring rounding errors, both ways are equivalent!

I reuse variables from HP71B code. Let I = initial rate, I2 the next iteration.
H is newton correction for X (i.e. X --> X+H)

I = M/(X-P)
I2 = M/((X-P)+H) = I/(1+H/(X-P)) = 1/(1/I+H/M)

From I, we have X = M/I+P

Y = FNF(X)
Y*I = X*I + (P+F)/(S-1)*I = (P+F) * I/(S-1) + P*I + M

This is just NPMT formula, time-reversed!
Let Z = Y*I = NPMT formula

(dZ/dI)/Z
= d(ln(Y*I))/dI
= 1/I + (dY/dI)/Y                      -- H = -Y/(dY/dX)
= 1/I + (-1/H) * (dX/dI)            -- (dY/dI) = (dY/dX) * (dX/dI)
= 1/I + (-1/H) * (-M/I^2)          -- (dX/dI) = d(M/I+P)/dI
= (I+M/H)/I^2

I - Z/(dZ/dI) = 1/(1/I+H/M)       -- = I2. Doing dY/dX vs dZ/dI are equivalent!