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Posted by joeu2004 on February 16, 2006, 2:31 am
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What is the formula or process for computing the advertised
ARM APR given the following information? I am looking for a
mathematical explanation. 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)
where 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 bal1
is $192,168.14.
Both web calculators compute the same pmt1 and pmt2.
One calculator also displays the same bal1. The other
calculator does not display the outstanding balance after
the first term; but based on the fact that pmt2 is the same,
I infer that its bal1 is the same.
Since we all agree on pmt1, pmt2 and bal1, I do not see
how we compute different APRs.
Extra credit: Where is the process for computing the ARM
APR defined in law?
Web source cite Reg Z and Reg Z Appendix J. But as I read
those regulations, I do not find enough detail to define the
mathematical formulation.
-----
[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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Posted by Mstr Jack on February 22, 2006, 9:47 pm
Please log in for more thread options
Hey Joe,
The problem might lie in what you consider interest, compared to what the
mortgage calculator considers interest, I'm not sure. But the final
interest disclosure for mortgages in Florida does consider part of your loan
fee as "interest". You might try working the formula without the $1500
deduction, and see what you get. Also, there might be a difference in how
the rule of 22nd's applys when the interest rate becomes variable. It looks
to me as if your formula considers it as a "new" loan at that point. As far
as the "legal way" to compute interest on an ARM, I'm sure it is defined
with a formula, and I would say the two different on-line calculators use
that formula, since they gave you identical answers, and that's what they're
designed to do. Besides, calculating the exact interest would be impossible
at this point, if that's what you're trying to do, since no one can really
know what the index rate will be in 3 years (I see you did list it as
"estimated").
Right now, due to an inversion in the bond market, fixed rate mortgages are
actually a better deal, for the most part, and you might want to consider
one, if this calculation is for your own mortgage. Although rates are
higher at banks for a fixed rate, the margin is smaller than normal, and a
Mortgage Broker with access to wholesale lenders can probably find you a
program with a rate close to, or as good, as the banks ARM rate, but it
would be fixed. Although you would pay a fee for the brokers service, if
you're planning on keeping your home more than 5 years, you'd be past the
break even point (having saved over $5000 by not moving to a variable rate,
per your example). After that, you could conceivably save over $2600 a year
for the life of the loan, which would save you over $70,000.00 if you kept
the loan until maturity. Well worth the few thousand the Broker would
charge to find and negotiate the best deal for you.
If this is just a math exercise, try going to the sites you mentioned and
click VIEW and then SOURCE CODE on your IE bar, you then may be able to see
what formula they use, and how yours differs.
But, if this is more than just a math exercise, get in touch with a Licensed
Mortgage Broker and look at your options before you go with a 3/1 ARM. Feel
free to give me a call at 813.882.8878, if you'd like to find out more.
Karen Pooley, President
Star Mortgage, Inc.
813-882-8878
http://www.starmortgagebroker.com
> What is the formula or process for computing the advertised
> ARM APR given the following information? I am looking for a
> mathematical explanation. 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)
>
> where 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 bal1
> is $192,168.14.
>
> Both web calculators compute the same pmt1 and pmt2.
> One calculator also displays the same bal1. The other
> calculator does not display the outstanding balance after
> the first term; but based on the fact that pmt2 is the same,
> I infer that its bal1 is the same.
>
> Since we all agree on pmt1, pmt2 and bal1, I do not see
> how we compute different APRs.
>
> Extra credit: Where is the process for computing the ARM
> APR defined in law?
>
> Web source cite Reg Z and Reg Z Appendix J. But as I read
> those regulations, I do not find enough detail to define the
> mathematical formulation.
>
>
> -----
> [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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Posted by Jeff Strickland on February 26, 2006, 1:48 pm
Please log in for more thread options <top post>
Karen offers some good suggestions (view source code, and others) to see how
the APR is calculated, but I suspect the APR is calculated elsewhere and
plugged into the code on demand. If so, her idea won't work, but it's still
a nice try. Give it a shot and see what happens.
APR is merely the cost of money in terms of a percentage rate. Let's say you
applied for a fixed rate mortgage at 6.00% on a loan of $200,000, and the
non-recurring closing costs were $5,000. The payment on the loan would be
(rounded) $1,200, but since you had to come up with the closing costs --
either from the loan or out of pocket -- the actual loan is only $195,000.
Since the payment remains at $1,200, the APR goes to 6.24%. The affect of
the fees on the loan costs 0.24%. This number is ONLY useful to compare the
costs of Lender A's loan charges against the loan charges of Lenders B & C,
and so on. THERE IS NO AFFECT of the APR on the monthly payment, it only
describes the fees.
Let's say Lender A has its list of fees, and Lender B has another list of
fees. You get a headache trying to figure out which fee is which when
comparing the different lenders. The bottom line is that you don't really
care what is what, you only care what is how much. The APR gives you a
bottomline summary of what is how much. Assuming the rate and loan amount is
the same, the different APRs will describe the different fee schedules as a
bottom line number. The monthly payment IS NOT calculated from the APR.
</top post>
> Hey Joe,
>
> The problem might lie in what you consider interest, compared to what the
> mortgage calculator considers interest, I'm not sure. But the final
> interest disclosure for mortgages in Florida does consider part of your
> loan fee as "interest". You might try working the formula without the
> $1500 deduction, and see what you get. Also, there might be a difference
> in how the rule of 22nd's applys when the interest rate becomes variable.
> It looks to me as if your formula considers it as a "new" loan at that
> point. As far as the "legal way" to compute interest on an ARM, I'm sure
> it is defined with a formula, and I would say the two different on-line
> calculators use that formula, since they gave you identical answers, and
> that's what they're designed to do. Besides, calculating the exact
> interest would be impossible at this point, if that's what you're trying
> to do, since no one can really know what the index rate will be in 3 years
> (I see you did list it as "estimated").
>
> Right now, due to an inversion in the bond market, fixed rate mortgages
> are actually a better deal, for the most part, and you might want to
> consider one, if this calculation is for your own mortgage. Although
> rates are higher at banks for a fixed rate, the margin is smaller than
> normal, and a Mortgage Broker with access to wholesale lenders can
> probably find you a program with a rate close to, or as good, as the banks
> ARM rate, but it would be fixed. Although you would pay a fee for the
> brokers service, if you're planning on keeping your home more than 5
> years, you'd be past the break even point (having saved over $5000 by not
> moving to a variable rate, per your example). After that, you could
> conceivably save over $2600 a year for the life of the loan, which would
> save you over $70,000.00 if you kept the loan until maturity. Well worth
> the few thousand the Broker would charge to find and negotiate the best
> deal for you.
>
> If this is just a math exercise, try going to the sites you mentioned and
> click VIEW and then SOURCE CODE on your IE bar, you then may be able to
> see what formula they use, and how yours differs.
>
> But, if this is more than just a math exercise, get in touch with a
> Licensed Mortgage Broker and look at your options before you go with a 3/1
> ARM. Feel free to give me a call at 813.882.8878, if you'd like to find
> out more.
>
> Karen Pooley, President
> Star Mortgage, Inc.
> 813-882-8878
> http://www.starmortgagebroker.com
>
>> What is the formula or process for computing the advertised
>> ARM APR given the following information? I am looking for a
>> mathematical explanation. 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)
>>
>> where 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 bal1
>> is $192,168.14.
>>
>> Both web calculators compute the same pmt1 and pmt2.
>> One calculator also displays the same bal1. The other
>> calculator does not display the outstanding balance after
>> the first term; but based on the fact that pmt2 is the same,
>> I infer that its bal1 is the same.
>>
>> Since we all agree on pmt1, pmt2 and bal1, I do not see
>> how we compute different APRs.
>>
>> Extra credit: Where is the process for computing the ARM
>> APR defined in law?
>>
>> Web source cite Reg Z and Reg Z Appendix J. But as I read
>> those regulations, I do not find enough detail to define the
>> mathematical formulation.
>>
>>
>> -----
>> [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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Posted by joeu2004 on March 1, 2006, 10:23 am
Please log in for more thread options Jeff Strickland wrote:
> Karen offers some good suggestions (view source code, and
> others) to see how the APR is calculated, but I suspect the
> APR is calculated elsewhere and plugged into the code on
> demand. If so, her idea won't work
Right. That was actually the first thing I had tried before posting
my question here.
> APR is merely the cost of money in terms of a percentage rate.
> [....]
> THERE IS NO AFFECT of the APR on the monthly payment, it
> only describes the fees.
I am well aware of the general theory and approach. My point was:
apparently using exactly the same fees and everything else being
equal, I compute a different APR.
I mentioned the similarity of monthly payments only to demonstrate
that we are all making the same assumptions, I believe, about the
number of interest rate adjustments (2), the outstanding loan
balance after the initial fixed rate ("teaser") term, the rate used
during the second term, and the number of remaining payments.
So I am looking for the mathematical algorithm or formula used
either in general or by the calculators that I cited in footnotes.
(Preferrably the latter.)
An example of a potential difference might be: I annualize the
monthly interest rate by multiplying by 12; perhaps the calculators
annualize by compounding the monthly rate. But when I try the
latter, my compounded APR (7.611%) is larger than the calculator
APR (7.544%). Scratch that theory!
Of course, one conjecture is that the calculators include some
hidden fees. That seems unlikely because (a) both calculators
display the total fees they use, and (b) two different calculators
are not likely to include the same hidden fees.
(Arguably, I cannot prove that the two calculators are indeed
different. They only appear to be different, and they are produced
for two different lenders, who might have different hidden fees, if
any.)
------ my previous posting ------
What is the formula or process for computing the advertised
ARM APR given the following information? I am looking for a
mathematical explanation. 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)
where 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 bal1
is $192,168.14.
Both web calculators compute the same pmt1 and pmt2.
One calculator also displays the same bal1. The other
calculator does not display the outstanding balance after
the first term; but based on the fact that pmt2 is the same,
I infer that its bal1 is the same.
Since we all agree on pmt1, pmt2 and bal1, I do not see
how we compute different APRs.
Extra credit: Where is the process for computing the ARM
APR defined in law?
Web source cite Reg Z and Reg Z Appendix J. But as I read
those regulations, I do not find enough detail to define the
mathematical formulation.
-----
[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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Posted by Jeff Strickland on March 5, 2006, 3:36 pm
Please log in for more thread options
> Jeff Strickland wrote:
>> Karen offers some good suggestions (view source code, and
>> others) to see how the APR is calculated, but I suspect the
>> APR is calculated elsewhere and plugged into the code on
>> demand. If so, her idea won't work
>
> Right. That was actually the first thing I had tried before posting
> my question here.
>
>> APR is merely the cost of money in terms of a percentage rate.
>> [....]
>> THERE IS NO AFFECT of the APR on the monthly payment, it
>> only describes the fees.
>
> I am well aware of the general theory and approach. My point was:
> apparently using exactly the same fees and everything else being
> equal, I compute a different APR.
>
> I mentioned the similarity of monthly payments only to demonstrate
> that we are all making the same assumptions, I believe, about the
> number of interest rate adjustments (2), the outstanding loan
> balance after the initial fixed rate ("teaser") term, the rate used
> during the second term, and the number of remaining payments.
>
> So I am looking for the mathematical algorithm or formula used
> either in general or by the calculators that I cited in footnotes.
> (Preferrably the latter.)
>
I am by no means an expert in Calyx Point (loan origination software), but
there is a worksheet that lets one enter the various parameters and Point
will attempt to give the APR of an ARM.
Sorry, I can't tell you how to calculate the APR of an ARM, but given two
loans for the same amount of money at the same terms, the one with the
lowest APR is the better deal for you. Beyond that, I know of no particular
value of knowing the APR.
> An example of a potential difference might be: I annualize the
> monthly interest rate by multiplying by 12; perhaps the calculators
> annualize by compounding the monthly rate. But when I try the
> latter, my compounded APR (7.611%) is larger than the calculator
> APR (7.544%). Scratch that theory!
>
> Of course, one conjecture is that the calculators include some
> hidden fees. That seems unlikely because (a) both calculators
> display the total fees they use, and (b) two different calculators
> are not likely to include the same hidden fees.
>
Given the fact that RESPA disallows any hidden fees whatsoever, I think the
variance you are finding is in the way they determine the compounding term.
As you found, the same interest rate compounded monthy or annually will give
different APRs. Another thing to remember is that a month is always 30 days,
and a year is always 360 days. They might be compounding daily by using the
Average Daily Balance method.
There is also a thing called a Start Rate which will change the APR. And,
I'll discuss this but you sound like you already know, there is the whole
Index + Margin thing, where the index changes throughout the loan term but
the margin remains a constant. Add the index and the margin to find the
Interest Rate.
I would assume that the APR would include any negative ammoritazation that
would occur by making the Minimum Payment. Of course, the negative would
have to be added to the outstanding balance, then compounded so it comes out
on the APR.
> (Arguably, I cannot prove that the two calculators are indeed
> different. They only appear to be different, and they are produced
> for two different lenders, who might have different hidden fees, if
> any.)
>
>
> ------ my previous posting ------
>
> What is the formula or process for computing the advertised
> ARM APR given the following information? I am looking for a
> mathematical explanation. 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)
>
> where 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 bal1
> is $192,168.14.
>
> Both web calculators compute the same pmt1 and pmt2.
> One calculator also displays the same bal1. The other
> calculator does not display the outstanding balance after
> the first term; but based on the fact that pmt2 is the same,
> I infer that its bal1 is the same.
>
> Since we all agree on pmt1, pmt2 and bal1, I do not see
> how we compute different APRs.
>
> Extra credit: Where is the process for computing the ARM
> APR defined in law?
>
> Web source cite Reg Z and Reg Z Appendix J. But as I read
> those regulations, I do not find enough detail to define the
> mathematical formulation.
>
>
> -----
> [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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