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Posted by joeu2004 on February 16, 2006, 10:53 pm
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What is the formula or algorithm for computing the advertised
ARM APR given the following information? I am looking for a
mathematical formulation. See my example below.
Loan amount: $200,000
Loan fees: 1,500 (includes any prepaid interest)
Loan term: 360 months
Initial interest rate: 6%
Initial term: 36 months
Estimated index rate: 5%
Margin: 2.75%
I compute 7.358%. But two web-based calculators [2] [3]
compute 7.554%. How was 7.554% computed?
According to [1], the ARM APR is 12 times the monthly IRR
of the cash flows. I infer that the monthly IRR is r derived from
the following:
0 = -(200000 - 1500) + SUM(pmt1 / (1+r)^t, t=1,...,36)
+ SUM(pmt2 / (1+r)^t, t=37,...,360)
In Excel terms, pmt1 is PMT(6%/12,360,-200000), pmt2 is
PMT((5%+2.75%)/12,(360-36),-bal1), and bal1 is
FV(6%/12,36,pmt1,-200000)). In other words, pmt1 is the
monthly payment during the initial term, bal1 is the outstanding
loan balance at the end of the initial term, and pmt2 is the
monthly payment for the remainder of the loan term based
on the index and margin.
I compute $1199.10 for pmt1, $1417.11 for pmt2 and $192,168.14
for bal1.
Both web calculators compute the same pmt1 and pmt2.
One calculator displays the same bal1. The other calculator
does not display the outstanding balance after the first term;
but I infer that it is the same bal1, based on the fact that
pmt2 is the same.
Since we all agree on pmt1, pmt2 and bal1, I do not see
how they compute a different APR. Well, I can imagine one
explanation, but it seems unlikely. I presume they use a
different algorithm -- but __that__ I cannot imagine what.
Extra credit: Where is the algorithm for computing the ARM
APR defined in law?
Web sources cite Reg Z and Reg Z Appendix J. But as I
read those regulations, they do not define the mathematical
formulation in sufficient detail.
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[1] http://web.mit.edu/11.431j/www/Fall91602/431_GMch17.ppt
notably slides 32, 38 and 41.
[2] http://nt.mortgage101.com/partner-scripts/1144.asp?p=buyerstrust
[3] http://www.lenderhomepage.com/calc/calc13.php
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