Discount Rate

[Albert Chan]

(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:

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

For previous 2 examples, formula give slightly better guess rate.
Newton steps (denoted by →) converge faster.

I(N= 36, P= 3000/100) ≈ 0.010811 → 0.010203 → 0.010207
I(N=-36, P=-5000/100) ≈ 0.016000 → 0.018003 → 0.017958