PC-1211, PC-1250, etc. TVM

[Albert Chan]

(06-02-2024 09:46 PM)robve Wrote:  We could improve this a bit by iterating one more time after detecting a sign change or when f does not get better. We could also compare f to f0 to use the best rate corresponding to f or f0 whichever is closer to zero.

Not necessarily.

NPMT formula is strictly concave or convex function.
see TVM solve for interest rate, revisited, post 7, 24, 25.

If we start rate from the edge, NPMT Newton convergence is always 1 sided.
Each iteration is an improvement over previously, with quadratic convergence.

Both situations you mentioned technically should not happen.
The fact that they do suggested f and f0 numbers are suspect.

If we overshoot with f having opposite sign, this is as good as we can get.

Regarding f or f0, both same sign, but rate seems to converge, we don't know which is better.
That's why Plus42 pick the center of 2 "best" rate, to keep error radius small.

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(05-31-2024 10:39 PM)Albert Chan Wrote:  10 B=1 @ N=40 @ P=900 @ M=-40 @ F=-1000
20 DEF FNF(I) @ S=(1+I)^(-N) @ U=(1-S)/I @ K=1+B*I
30 FNF=P+K*M*U+F*S @ END DEF ! NPV
40 V=.01 @ W=FNF(V) @ I=.02 @ Y=FNF(I)
50 H=I-V @ V=I @ I=I-H*Y/(Y-W) @ W=Y @ Y=FNF(I)
60 DISP 100*I,Y @ IF V/I+I/V<>2 THEN 50

>run
 3.55877069046      -223.49575008
 4.3407684864        -68.532154113
 4.68660450765      -10.546204073
 4.74950349013      -.604877167
 4.75333056057      -.005731135
 4.75336716844      -.000003149
 4.75336718857      -.000000001

Above example, NPV=0 for rate, we have 1 sided convergence.
But we may lose this nice feature, with time-reversed setup.

>15 VARSWAP P,F @ P=-P @ F=-F @ N=-N
>run
 11.3570289363      -36923.539379
 2.35993812666      1387.63023153
 2.68581267787      1287.38125489
 6.87064131886      -3585.77746844
 3.79135227821      769.04731268
 4.33514429757      379.62865086
 4.86526534953      -115.24756927
 4.7418099072        11.55575603
 4.75306056619      .30740914
 4.75336803852      -.00085219
 4.75336718851      .00000007

NPV=0 for rate does not handle well with negative N
For the same reason, NFV=0 for rate have problem with positive N.