Fun math algorithms

[Albert Chan]

(10-17-2020 11:27 AM)Albert Chan Wrote:  C-1 ≈ IN*(6+IN)/12 + I/2

Let's rename C as \(C_+\), to signal compounding effect in the forward direction.

Knowing FV=0, we have \(N |PMT_+| = C_+ |PV|\)

For reverse direction, knowing PV=0, we have \(N |PMT_-| = C_- |FV|\)

\(C_- = \Large{C_+ \over (1+I)^N} = \frac{\left(I N \over 1-(1+I)^-N\right)}{(1+I)^N} = {I(-N) \over 1-(1+I)^N}\)

So, run the clock backwards (N → -N), \(C_+ \text{ turns into } C_-\)
Naturally, C₊ estimate formula apply:

\(C_± - 1 ≈ \large{IN (IN\;±\;6) \over 12} + {I\over2}\)

---
Lets build a car leasing formula !

\(N·PMT = N(PMT_+ - PMT_-) \)

PMT+ is car payments if we *buy* the car (not gives it back).
PMT− is credit of car payments, if we do give it back (turning buying, back to leasing)

\(N·PMT_+ = CAP · C_+ ≈ CAP \left(1 + \large{IN (IN\;+\;6) \over 12} + {I\over2}\right)\)
\(N·PMT_- = RES · C_- ≈ RES \left(1 + \large{IN (IN\;-\;6) \over 12} + {I\over2}\right)\)

Substract the 2, and divide by N, we have

\(PMT = \large \left(1 + {I\over2} + {(IN)^2 \over 12} \right) ({CAP - RES \over N})
+ ({I\over2}) \normalsize (CAP+RES) \)