HP12c Credit Card Payment Calculation

[Csaba Tizedes]

OK, first thing: you'll never will pay back that 2000.- USD, because (without interest (18% / year)) you'll pay 2.7%, but always remain 97.3% in every period.

Your real question is: if I can pay (in the future) an affordable PV \( (PV_a) \) in one payment, how many periods I must to pay until \( PV_{actual}<PV_a \)?

This is easy to calculate, because the change of the present value for one period is \( \displaystyle \Delta PV=PV·(d+r) \), where \( d=-0.027 \) and \( r=+0.0139 \) (this is the annual 18% converted to one period = 1.39%).

You can rearrange the above equation: \( \displaystyle \frac{\Delta PV}{PV}=(d+r) \), which is equal \( \displaystyle \frac{1}{PV}·\frac{dPV}{dt}=(d+r) \). Then simply integrate this equation from the initial PV \( (PV_0) \) to affordable PV \( (PV_a) \) and you get the number of periods \( (N) \):

\( \displaystyle \int_{PV_0}^{PV_a} \frac{dPV}{PV} = \int_{0}^{N} d+r \,dt \)

\( \bbox[yellow,5px] { \displaystyle N= \frac {ln \frac{PV_a}{PV_0}}{d+r} } \)

For example, if you can pay in the future 100.- USD \( (PV_a=100) \), then you need pay payments for \( N=228 \) periods, or if \( PV_a=50 \) you will pay for \( N=281 \) periods.

Of course I have checked in EXCEL and OK.

Csaba