Suppose you have a loan with principle P and fixed interest rate i compounded daily. Suppose you make fixed payments every month, but not on the same day. The only constraint is that you make every payment before the "due date", which is constant every month. A standard amortization would be inaccurate because the payment dates vary, but I think we can prove mathematically that the loan should not deviate from the regular amortization schedule by more than P(1+i/365)365*31 or 1 months compounding interest. Please show me how to prove/solve this problem.
This is a real-world problem. It's easy to see amortization of a loan when you make payments (payments are processed) in equal consistent payment periods. But realistically, they are made at various times of the month. A lender may use that complexity to hide overcharging. The above proof would allow a borrower to recognize errors over long payment schedules.
I suspect that the most interest a lender could charge is if the payment was made consistently on the payment "due date". The least amount would if payments were consistently paid on the first day of the payment cycle (at-most 31 days prior to due date). The length of term and variations in payment date shouldn't matter. For example: If in 2 consecutive, 30-day months you have a 40 day payment period, the next should be at most 2o days, so the interest charged should be a wash. How do we prove the above generally?
Best Answer
If you think about the relative deviation from the original amortisation process, you simply want to know the value $\frac{|R-R'|}{P}$, i.e.: $$ \sum_{j=0}^M(1+i)^{t_j}\cdot (1+i)^{-\tau_j}-(1+i)^{t_j}, $$ Thus: $$ \sum_{j=0}^M(1+i)^{t_j}((1+i)^{-\tau_j}-1). $$ Since you say that the maximum deferral date is one month, i.e. $0\leq\tau_j\leq1, \forall j>0$.
Hence: $$ \sum_{j=0}^M(1+i)^{t_j}((1+i)^{-\tau_j}-1)\geq0, $$ and $$ \sum_{j=0}^M(1+i)^{t_j}((1+i)^{-\tau_j}-1)\leq\sum_{j=0}^M(1+i)^{t_j}((1+i)^1-1)= $$ $$ =i\sum_{j=0}^M(1+i)^{t_j}=i\cdot\frac{R}{P}. $$ Thus: $$ \frac{R'-R}{P}\leq i\frac{R}{P}, $$ Therefore: $$ \frac{R'-R}{R}\leq i. $$ I guess this is what you can prove under the hypothesis you stated above.