Dave wrote:
A simple example:a $30,000, 20 year mortgage at 7.499% yields monthly
payments (using the standard Excel PMT function, as well as just about any
other mortgage calculator around) of $241.66. Looking at the resulting
amort schedule at say 36 months, shows the principle reduced to +/-
$27,887.50.

That is the balance after 35 payments. After 36 payments, the balance
is 27820.11:

fv(7.499%/12, 36, pmt(7.499%/12, 20*12, -30000), -30000)

The Rembrandt calc yields monthly Payments of $241.77, and with the
principle reduced to only $28,059.74.

The large difference in the balance after 36 months might be explained
by the lender using a method that accelerates the percentage of total
interest (finance charge) paid per period.

The Rule of 78 is one such method. For example, if the payment is
241.66 per month over 240 months for a loan of 30000, the Rule of 78
would apply 232.52 (240/28920 of the 27998.31, the total interest) of
the first payment toward interest, instead of 187.48 that would apply
toward interest in a "normal" amortization schedule, assuming a nominal
interest rate of 7.499%.

I would explain the Rule of 78 further, but it seems moot because after
36 payments, the Rule of 78 would leave a balance of 29054.98 instead
of 28059.74. I think that difference is too great to believe that
Rembrandt uses the Rule of 78. (Besides, the Rule of 78 is usually not
used for long-term loans.)

But my point is: I believe the lender can accelerate interest payment
and reduce principal repayment in any arbitrary way. Look at the
disclosure agreement for the explanation.

But I am still hopeful that there is a more sensible explanation.

As for the small difference in the payment amount, I have explored
different compounding frequencies (e.g, typically daily for mortgages),
different payment frequencies (e.g, semimonthly), varying assumptions
about rounding -- even unsound mathematical practices that I believe
lenders follow, such as computing the daily compound rate as
7.499%/365, but determining the payment based on the monthly compound
rate of 7.499%/12 -- all to no avail so far.

My best guess is that the Rembrandt model presumes that the initial
loan accumulates interest for a few days before the monthly payment
window begins, for example to align payments with some "nice" time of
the month like the 1st, 15th, or end of the month. This is quite
common -- and often an unstated assumption -- in some mortgage
calculators that I have seen. However, whereas it is common for a
mortgage calculator to assume 15 days accumulation, that would work out
to about 2 days in your case, which seems odd unless, by coincidence,
you ran your numbers 2 days before one of those "nice" breakpoints.

I also want to explore the differences between the "actuarial method"
and the "US method" of computation described in Appendix J of the Truth
in Lending Act to see if that could explain some of the differences. I
doubt it; but I have never been clear on the differences between the
two methods.

This probably does not help much. Just thinking out loud in the hopes
that it might stimulate discussion.

I haven't got a Rembrandt amort schedule yet, but my brother-in-law banker
is getting me one. I thought I'd plot the data in Excel and let it tell me
what the equation was (is). Probably a mess, but what the heck.

I would be very interested in seeing the amortization schedule myself.
I would not try to infer the equation through any sort of regression
analysis. Instead, I would apply various predictable mathematical
models to analyze and reverse-engineer the numbers in the schedule.