Fun math algorithms

[Albert Chan]

How to estimate car payments ?

XCas> C := I*N / (1 - (1+I)^-N)       // C = |N*PMT/PV|, "compounding factor"
XCas> series(C,I,polynom)

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

For car payments, N is usually not too big, we can use: \(\large C ≈ {(IN + 3)^2 + 3\over 12}\)

This estimate does not require converting annual interest rate to monthly rate.
All calculations could be done on a 4-banger

Example, using U.S. 2020 numbers

Quote:The average loan amount for a new car in the first quarter of 2020 was $33,631,
with an average interest rate of 3.6% for a 60-month loan.

I*N = (12*I) * (N/12) = 3.6% * 5 = 18%
C-1 ≈ (IN)*(6+IN)/12 = 18% * 6.18/12 = 9.27%

Fixed 2 mode:
33631 Enter 60 /    → 560.52 (monthly payment, if no interest)
9.27 %              →  51.96 (monthly finance charge, estimated)
+                   → 612.48 (monthly payment, estimated)

For reference, car payments (exact) = $613.31