(17B/19B) GPM: initial payment

[Werner]

[Updated to correct an embarrassing error, that still made the first example work ;-)]
Actually, the summation can also be represented in terms of the 'standard' financial functions.
This is what I came up with, on a real machine should be a lot faster to execute:

PV of 1 year of PMT: PMTY = PMT*USPV(I%/12,12)
First N years: PVN = PMTY + PMTY*(G/F) + PMTY*(G/F)^2 + .. PMTY*(G/F)^(N-1)
  G = 1 + G%/100
  F = 1 + F%/100 effective annual rate = (1 + I%/1200)^12, F% = USFV(I%/12,12)*I%/12
  K = 1 + K%/100 = G/F -> K% = (G% - F%)/F
  then PVN = PMTY*USFV(K%,N)
Remaining M-N years PMTY*(G/F)^N*(1 + 1/F + 1/F^2 + .. 1/F^(M-N-1))
  (G/F)^N = SPFV(K%,N)
  1 + 1/F + .. 1/F^(M-N-1) = USPV(F%,M-N)*F

All combined:

T := USFV(I%/12,12);
F% := T*I%/12;
F := 1 + F%/100;
K% := (G% - F%)/F;
PV := PMT*USPV(I%/12,12)*(USFV(K%,N) + SPFV(K%,N)*USPV(F%,M-N)*F);

remark that

             (1+I%/100)^N - 1
USFV(I%,N) = ----------------
                I%/100

             1 - (1+I%/100)^-N
USPV(I%,N) = ----------------
                I%/100

so USPV(I%,N) = USFV(I%,N)/(1+I%/100)^N

given that F = (1+I%/1200)^12
then USPV(I%/12,12) = T/F
and

PV := PMT*T*(USFV(K%,N)/F + SPFV(K%,N)*USPV(F%,M-N));

or, as a 17BII/Plus42 equation, and using the sign convention:

Code:

GPM:0×L(F:1+L(F%:L(T:USFV(I%÷12:12))×I%÷12)÷100)×L(K%:(G%-G(F%))÷G(F))+PV+PMT×G(T)×(USFV(G(K%):N)÷G(F)+SPFV(G(K%):N)×USPV(G(F%):M-N))

Enter
12 I%
7.5 G%
60000 PV
5 N
30 M
PMT -> -474.83 is the initial payment

3 I%
2 G%
300000 PV
PMT -> -1161.50

Cheers, Werner