# What's the maximum deviation from loan amortization

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?

• The greatest savings would accrue if you made every payment exactly one month early. If the original amortization schedule pays off $P$ with monthly payments of $m$, your early-payment scheme would be at best reducing the interest to what it would be if you paid off a loan of $P-m$ with monthly payments of $m$. – Steve Kass Feb 15 '14 at 15:45
• This is true, but this answer doesn't target the larger question. If I pay consistently early, then I have saved myself interest. But my amortization is spot on - just adjust the dates. The question here is about inconsistent payments. Whether or not I make an early payment, my amortization isn't easily predictable if my payment dates are inconsistent. I do think the actual deviation is from the amortization is limited to one months interest on the original principle. That's what I want to prove mathematically. – Charles Forrest Feb 15 '14 at 16:03
• You save more than one month's interest on the entire principal, it seems. Here is a simple mock-up comparing on-time payments with all-early payments: imgur.com/MDFmNOv . Note that you will pay the loan off early. I do think you are making the picture more complicated than it needs to be by not realizing that the upper limit of deviation is achieved by making all payments exactly one month early. That's not a complicated situation to evaluate, and making any payment closer to the due date will necessarily increase the total interest. – Steve Kass Feb 15 '14 at 17:19
• My original hypothesis was that max deviation from standard amortization schedule would be 1 months compounding interest. From your example, I observe the maximum to be 1 months interest then itself compounded for the remainder of the loan. 15 * (1.015)^(14) = 18.476. This is the delta between 124.16 and 105.68. Thanks for that. I mean to prove that this is the maximum possible deviation resulting from "on-time" payments. Steve, your example makes me realize this max value I'm looking for is a function of the loan term length. – Charles Forrest Feb 16 '14 at 0:21

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.

• I'm sorry. This has me totally confused. The deviation is less than the interest rate? Perhaps I need more context or examples to follow you here. I did observe from Steve's example that the max deviation may be a function of the loan term. – Charles Forrest Feb 16 '14 at 0:33
• I calculated the relative deviation. The absolute deviation is $iR$, which means that the maximum difference between the classical and this amortisation is the interest that you perceive in one month times the classical amortisation value. If you disregard the hypothesis of limiting $\tau$ to one, you have the general case of a time-dependent deviation. – 7raiden7 Feb 16 '14 at 8:59

The classical amortisation would write as: $$P\cdot\sum_{j=0}^M(1+i_{12})^{t_j},$$ where $i$ is the monthly interest rate and $t_j$'s are the maturities.

Let's denote $t'_j=t_j\pm\tau_j$ the new payment dates. So you have accrued an interest as high as: $$P\cdot\sum_{j=0}^M(1+i_{12})^{t'_j}=P\cdot\sum_{j=0}^M(1+i_{12})^{t_j}\cdot(1+i_{12})^{\pm\tau_j},$$ where $\tau_j$ is the deferral period of the real payment date. Denoting with $R$ the value of your portfolio, you have: $$R'=P\cdot\sum_{j=0}^M(1+i_{12})^{t_j}\cdot(1+i_{12})^{\pm\tau_j},$$

So, if you pay in advance at time $t_j$, you will benefit from a reduction of $P(1+i_{12})^{-\tau_j}$. If you pay later, you'll will be charged for $(1+i_{12})^{\tau_j}$.

Realistically, the interest rate would differ, and would be specified ex-ante in the contract, i.e.: $$R'=P\cdot\sum_{j=0}^M(1+i_{12})^{t_j}\cdot(1+i'_{12})^{\pm\tau_j},$$

If you want to find some maximum/minimum amount for $R'$ it would be tricky in this general case, as you cannot decompose the sum into two factor easily.

Hope it helps.

• This is a great formula interpretation of the problem. Thank you for that. Of course it isn't a resolution. If we didn't have any information about length of each compounding interval, I'd say you're right we're done. Can't be solved. But we do know that each payment is made prior to the due date, and can use that to show that the average interval is 365/12 days. Also, the we know that the maximum deferral interval is 30 days. I still think there is a solution to this problem somewhere. Anyone else care to take a stab? – Charles Forrest Feb 15 '14 at 15:26
• In response to 7raidens comment "Realistically, the interest rate would differ, and would be specified ex-ante in the contract" I felt it necessary to update the question slightly. You may assume a known fixed interest rate. I agree that not knowing the interest rate at each period makes this problem a lot more complicated. – Charles Forrest Feb 15 '14 at 15:45