(11C) TVM for HP-11C

[Albert Chan]

We can estimate rate using simple formula, by "setting" PV=0, or FV=0

NPMT = (Ce+N*I/2)*PV + (Ce−N*I/2)*FV + N*PMT
= (Ce+N*I/2)*(PV+FV) + N*(PMT−FV*I)            // (PV,FV,PMT) → (PV+FV, 0, PMT−FV*I)
= (Ce−N*I/2)*(PV+FV) + N*(PMT+PV*I)            // (PV,FV,PMT) → (0, PV+FV, PMT+PV*I)

(05-10-2022 09:35 PM)Albert Chan Wrote:  lua> guess_i(36, 30000, -550, -15000) -- car lease example
0.005804845779805948

I = guess_i(36, 30000, -550, -15000) = guess_i(36, 15000, -550+15000*I, 0)      // "set" FV=0

N = 36
P = 15000 / (550-15000*I) = 1 / (11/300 - I)      → I = 11/300 - 1/P

(04-09-2022 12:50 PM)Albert Chan Wrote:  

(10-16-2020 04:02 PM)Albert Chan Wrote:  XCas> C := I*N / (1 - (1+I)^-N)       // C = |N*PMT/PV|, "compounding factor"
XCas> series(C,I,polynom)

\(1
+\frac{I(N+1)}{2}
+\frac{I^2 (N^2-1)}{12}
+\frac{I^3 (-N^2+1)}{24}
+\frac{I^4 (-N^4+20N^2-19)}{720}
+\frac{I^5 (N^4-10N^2+9)}{480}\)

This may be a better rate estimate, by dropping compounding factor O(I^2)
With previous defined P, solve for I, we have:

\(\displaystyle I ≈ \frac{2\;(N-P)}{P\;(N+1)}\)

We can solve I = 2*(N-P)/(P*(N+1)) for I, but there is no need.
For this formula, we had already done the work, when builiding guess_i()
guess_i() I coefs:

[(N^2-1)*(FV+PV)/12, (-FV*N+N*PV+FV+PV)/2 , N*PMT+FV+PV]

Drop I^2 terms, solve for I, we have Dieter's formula

I ≈ 2*(PV+FV + N*PMT) / (FV*(N-1) - PV*(N+1))
  = 2*(30000-15000 + 36*-550) / (-15000*35 - 30000*37)
  ≈ 0.005872

(04-09-2022 05:47 PM)Albert Chan Wrote:  \(\displaystyle I ≈ \frac{1}{P} - \frac{P}{N^2}\)

This is even better, and work well with big N.

I ≈ (11/300 - I) - 1/(11/300 - I) / 36^2
(2*I - 11/300)*36^2 * (I - 11/300) ≈ 1

2592*I^2 - 142.56*I + 0.7424 ≈ 0

Solve quadratics, for the small root, we have I ≈ 0.005824

Or, we can solve symbolically:

XCas> ICOEFS := [n^2*pv*fv, n^2*pmt*(-pv+fv), -(n*pmt-pv-fv)*(n*pmt+pv+fv)]
XCas> proot(ICOEFS(n=36, pv=30000, pmt=-550, fv=-15000))

[0.00582443202789,0.0491755679721] // keep the small root.