[net.consumers] Semi-monthly mortgage repayments

marks@yogi.DEC (11/12/85)

I have recently read about a new type of real estate mortgage agreement,
in which the bank gives the buyer a mortgage at the normal insterest
rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
to pay off the loan semi-monthly rather than once a month.  The total
monthly payments are the same, they are just split in half and paid
twice a month.

The incredible part of this is that you manage to reduce your interest
so much by paying the loan semi-monthly, that you end up paying only a
fraction of the total interest you would otherwise have paid, while
expending the same amount of money per month, and you pay off what would
have been a 30 year mortgage in something like 12 years.

Clearly, this system benefits the consumer greatly, but banks are not
exactly rushing to sign people up under this system.  The only advantage
I can see from the bank's point of view is that the bank has the loan
repaid in a much shorter period of time than it otherwise would.  Of course,
it is losing lots of interest money on the deal.

My question is this:  does anyone know of any banks that are doing this?
I live in the Boston area, but the closest bank I have found that does 
this type of mortgage is the Hudson City National Bank in Chatham, N.Y.
(upstate).  Seems like a great deal to me.  Wonder how the banks could
be convinced.

R.M.

cc-06@ucbcory.BERKELEY.EDU (Ilya Goldberg) (11/13/85)

In article <1389@decwrl.UUCP> marks@yogi.DEC writes:
>
>I have recently read about a new type of real estate mortgage agreement,
>in which the bank gives the buyer a mortgage at the normal insterest
>rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
>to pay off the loan semi-monthly rather than once a month.  The total
>monthly payments are the same, they are just split in half and paid
>twice a month.
>
>The incredible part of this is that you manage to reduce your interest
>[...]  and you pay off what would
>have been a 30 year mortgage in something like 12 years.
>

12 years seems a much smaller time period than 30 and is not very believable.
Could someone post a formula either verifying '12 years' or giving the
correct period of time based on the above pay off arrangement?

						Ilya	(...!ucbvax!ilya)

ark@alice.UucP (Andrew Koenig) (11/13/85)

>>
>>I have recently read about a new type of real estate mortgage agreement,
>>in which the bank gives the buyer a mortgage at the normal insterest
>>rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
>>to pay off the loan semi-monthly rather than once a month.  The total
>>monthly payments are the same, they are just split in half and paid
>>twice a month.
>>
>>The incredible part of this is that you manage to reduce your interest
>>[...]  and you pay off what would
>>have been a 30 year mortgage in something like 12 years.
>>

>12 years seems a much smaller time period than 30 and is not very believable.
>Could someone post a formula either verifying '12 years' or giving the
>correct period of time based on the above pay off arrangement?

If you pay twice a month instead of once, and your payments are
half what they would be otherwise, you pay off the loan in half
the time (believe it or not)!  So the number above should have
been 15 years, not 12.

jimb@tekcbi.UUCP (Jim Boland) (11/13/85)

> 
> I have recently read about a new type of real estate mortgage agreement,
> in which the bank gives the buyer a mortgage at the normal insterest
> rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> to pay off the loan semi-monthly rather than once a month.  The total
> monthly payments are the same, they are just split in half and paid
> twice a month.
> 
> The incredible part of this is that you manage to reduce your interest
> so much by paying the loan semi-monthly, that you end up paying only a
> fraction of the total interest you would otherwise have paid, while
> expending the same amount of money per month, and you pay off what would
> have been a 30 year mortgage in something like 12 years.

Are we missing something here??? Somehow, the logic of all this escapes me.
It doesn't make a lot of sense.
Obviously, you need to pay off the principal plus the interest on the 
amount left owing (usual method).  That is why the initial payments are
primarily interest and the final payments primarily principal.
You said that you are paying the same amount per month, only twice
a month instead of once.  I fail to see where that would affect the
total amount payed by much.  I also fail to see how the term would be
shortened significantly - certainly not 18 years.  

Now, if you had said that you make two full payments per month, then
I might buy that argument.  Primarily because one of those payments
would be primarily principal.  Then you would decrease the payback period
by over half.  In other words, if your payment were $500 per month for
30 years, you would not gain much by making two $250 payments per month.
You are still paying $500 per month for 30 years.  However, If you make
the $500 (of which $480 is interest - initially) payment on the first
and then make another $500 payment (primarily principal only) on the
fifteenth, you can see that the balance will go down rather quickly -
thus shortening the loan period significantly.

Or maybe what you're proposing is part of the new math that I don't
understand.):

This is one that someone should put to Bruce Williams.

mark@cbosgd.UUCP (Mark Horton) (11/13/85)

When I first saw this, the claim was that the bank having the
extra half payment for half the month was what was cutting down
on the interest.  But this doesn't make any sense because if you
were to just prepay half a payment extra when you took out the
loan, it would give the bank that same amount of money all the
time instead of half the time, and obviously it would just apply
to the principle and make almost no difference.

Finally I read something in the Sunday paper that explained it.
It isn't a semi-monthly payment, it's a bi-weekly payment.  Since
there are 52 weeks in a year, you in effect make 13 months worth
of payments every year instead of 12.  This extra payment is what
is bringing down the mortgage that much more quickly.

singhal@hound.UUCP (S.SINGHAL) (11/13/85)

In the semi monthly mortgage plan you pay every second week instead of twice a
month.  Since there are 52 weeks in a year, you make 26 payments each equal to
half the monthly payment.  Consequently you make 13 monthly payments a year
instead of just 12.  This extra payment, along with some interest savings,
enables you to pay off a 30 year mortgage in just 18 years or so.

Don't be misled by the banks trying to convince you that the early liquidation
of the mortgage is simply due to interest savings !

---Sharad Singhal

dennis@rlgvax.UUCP (Dennis Bednar) (11/14/85)

> In article <1389@decwrl.UUCP> marks@yogi.DEC writes:
> >
> >I have recently read about a new type of real estate mortgage agreement,
> >in which the bank gives the buyer a mortgage at the normal insterest
> >rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> >to pay off the loan semi-monthly rather than once a month.  The total
> >monthly payments are the same, they are just split in half and paid
> >twice a month.
> >
> >The incredible part of this is that you manage to reduce your interest
> >[...]  and you pay off what would
> >have been a 30 year mortgage in something like 12 years.
> >
> 
> 12 years seems a much smaller time period than 30 and is not very believable.
> Could someone post a formula either verifying '12 years' or giving the
> correct period of time based on the above pay off arrangement?
> 
> 						Ilya	(...!ucbvax!ilya)

Pure hogwash!!  I have a personal mortgage program, and I ran it with 4 tests
in which the amount borrowed was constant ($100,000), and the interest
rate was constant @ 12% per year, but I varied the number of years, and
I varied the number of payments per year.  The results were:

Test	Years		Payments Per Year	Avg Monthly Payment
1.	30		12			$1028.61
2.	12		24			$1311.96
3.	12		500			$1310.56
4.	30		24			$1028.35

As you can see, the average monthly payment went up ~$200 between
(1) and (2).  I then thought that maybe bi-monthly payments were
too many days apart, so in (3) I jacked the number up to 500,
(interest computed more than once per day), BUT it was
about the same amount as (2), unfortunately.  Last, I wanted to
see how much the avg monthly payment went down just by going from
monthly to bi-monthly payments (compare (1) and (4)), and it didn't
seem to matter that much, either.

I will post mortgage.c to net.sources.  Also, I remember seeing
a loan.c program about 2 years ago, which I think was posted to
the net.  It would be interesting to compare the output of the
2 programs.
-- 
Dennis Bednar	Computer Consoles Inc.	Reston VA	703-648-3300
{decvax,ihnp4,harpo,allegra}!seismo!rlgvax!dennis

suze@terak.UUCP (Suzanne Barnett) (11/14/85)

> 
> I have recently read about a new type of real estate mortgage agreement,
> in which the bank gives the buyer a mortgage at the normal insterest
> rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> to pay off the loan semi-monthly rather than once a month.  The total
> monthly payments are the same, they are just split in half and paid
> twice a month.
> 
What I have heard is that the payments are bi-weekly, not
semi-monthly, which means you pay approximately 13 months
worth of mortgage payments each year. For example, if your
payment is $1000 per month, and you pay $500 every two weeks,
you pay a total of $13000 in the year, not $12000.

Yes, this pays your interest faster, but, you could pay off the
loan even sooner if you have that extra $1000 applied directly to
the principal, not the interest first, as are most mortgage payments.
-- 
**************************************************************
Suzanne Barnett-Scott

uucp:	 ...{decvax,ihnp4,noao,savax,seismo}!terak!suze
phone:	 (602) 998-4800
us mail: CalComp/Sanders Display Products Division
	 (Formerly Terak Corporation)
	 14151 N 76th street, Scottsdale, AZ 85260

mkg@lzaz.UUCP (Marsh Gosnell) (11/15/85)

In article <1389@decwrl.UUCP>, marks@yogi.DEC writes:
> I have recently read about a new type of real estate mortgage agreement,
> in which the bank gives the buyer a mortgage at the normal insterest
> rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> to pay off the loan semi-monthly rather than once a month.  The total
> monthly payments are the same, they are just split in half and paid
> twice a month.

Check the fine print.  All of the half pay programs I've seen call for
you to pay half the amount every two weeks (not twice a month).  Having
recently purchased a house, I carefully investigated this kind of mortgage
and concluded that they aren't worth it.

The programs I investigated called for payments every two weeks to be
deducted automatically from a checking account you MUST have with the
bank.  One bank I checked with will drop you from the half pay program
if you don't have sufficient funds in your checking account to cover a
payment (e.g., NO late payments allowed).

Almost ALL of the interest savings come not from the fact you are paying
more often but because you are paying the equivalent of 13 "monthly"
payments each year.  If you were to take out a conventional 15 or 30 year
mortgage and pay 1/12 extra each month you pay off the mortgage within a
month or two of the half pay mortgage.  With the conventional mortgage,
you aren't forced to maintain a checking account with the bank, you have
a grace period for late payments, etc.

I wouldn't bother with a half pay mortgage.
   Marsh Gosnell   lzma!mkg

partha@gitpyr.UUCP (Partha Dasgupta) (11/15/85)

In article <4555@alice.UUCP>, ark@alice.UucP (Andrew Koenig) writes:

> >>The incredible part of this is that you manage to reduce your interest
> >>[...]  and you pay off what would
> >>have been a 30 year mortgage in something like 12 years.

> >Could someone post a formula either verifying '12 years' or giving the
> >correct period of time based on the above pay off arrangement?
> 
> If you pay twice a month instead of once, and your payments are
> half what they would be otherwise, you pay off the loan in half
> the time (believe it or not)!  So the number above should have
> been 15 years, not 12.

No Way! By your logic, if I made 4 payments a month, at one-fourth the monthly 
rate, I would pay off the mortgage in 7.5 years.

If you beleive that, I would love to borrow $100,000 from you, at those terms,
any time!

Partha

-- 
Partha Dasgupta
Georgia Insitute of Technology, Atlanta Georgia, 30332
...!{akgua,allegra,amd,hplabs,ihnp4,masscomp,ut-ngp}!gatech!gitpyr!partha

rs55611@ihuxk.UUCP (Robert E. Schleicher) (11/15/85)

> 
> I have recently read about a new type of real estate mortgage agreement,
> in which the bank gives the buyer a mortgage at the normal insterest
> rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> to pay off the loan semi-monthly rather than once a month.  The total
> monthly payments are the same, they are just split in half and paid
> twice a month.
> 
> The incredible part of this is that you manage to reduce your interest
> so much by paying the loan semi-monthly, that you end up paying only a
> fraction of the total interest you would otherwise have paid, while
> expending the same amount of money per month, and you pay off what would
> have been a 30 year mortgage in something like 12 years.
> 
There are two versions of this type of plan, which I think are being
mixed-up a little in the above discussion.

1.  As described above, you make two 1/2 payments a month.  This provides
    a fair savings in overall interest payments, but not enough
    to pay off the loan in 12 years (it saves several years, though.)

2.  The other variation is that you pay 1/2 of what would have been your 
    monthly bill, but every two weeks (or, alternatively, your old
    monthly bill every four weeks).  This amounts to making 13 months
    of payments every year, and DOES pay off the loan very quickly,
    in about 12 years.  Of course, you're really paying a higher effective
    monthly payment.  In that sense, its not much different that a 15 year
    conventional loan.  It's basically a psychological trick to make the
    payment seem smaller by making them more often.  The reason banks
    and S&Ls don't like these is that their computer systems are often
    geared to monthly operation, with fixed late payment dates, etc.
    Also, it is often inconvenient for the consumer, whose income is still
    on a monthly schedule.


So, before you leap, make sure the terms of this wonder-plan are
understood clearly.

Bob Schleicher
ihuxk!rs55611

msc@saber.UUCP (Mark Callow) (11/19/85)

From: ark@alice.UucP (Andrew Koenig)
> If you pay twice a month instead of once, and your payments are
> half what they would be otherwise, you pay off the loan in half
> the time (believe it or not)!  So the number above should have
> been 15 years, not 12.

I don't believe it.  I got out my handy-dandy TI business analyst
and worked out the numbers for my own mortgage.  I divided my payment
by two and divided the monthly interest by 2 (i.e. increased the
compunding period to twice a month.)  I then calculated the number
of payments required to pay off the loan.  The answer?  718 payments.
Dividing by 2 to get the number of months, we get 359 or just
one month less than 30 years.  Yup, that's all the time you save; just
one month.

The real answer, as someone else said, is that they want you to make
biweekly payments so you are actually making the equivalent of 13 payments
a year.
-- 
From the TARDIS of Mark Callow
msc@saber.UUCP,  sun!saber!msc@decwrl.dec.com
...{decwrl,ucbvax}!sun!saber!msc, ...{amdcad,ihnp4}!saber!msc

vsh@pixdoc.UUCP (Steve Harris) (11/19/85)

I was told by a friend that the reason semi-monthly mortgage payments work
is due to the fact that half your (monthly) payment is made two weeks earlier
than with a single monthly payment; consequently the amount of time over which 
intrest is paid is reduced.

I don't really believe it, but that's what he said.
-- 

Steve Harris            |  {allegra|ihnp4|cbosgd|ima|genrad|amd|harvard}\
Pixel Systems Inc.      |               !wjh12!pixel!pixdoc!vsh
300 Wildwood Street     |
Woburn, MA  01801       |  617-933-7735 x2314

daver@hp-pcd.UUCP (daver) (11/19/85)

I tried a sample problem on my calculator (an HP-41CX with the Advantage ROM
for Time-Value-of-Money calculations) and it seems to work.  The problem I
computed was a $50,000 loan at 12% APR with monthly payments for 30 years:

	    PV = 50,000  (present value of loan - money received at start)
	    FV =      0  (future value of loan - money to be received at end)
	    N  =    360  (number of payments)
	    I  =    1.0  (interest % per payment at 12 payments/year)

     yields PMT=-514.31  (money received per payment period - i.e. -payment)

If we go biweekly, with 26 payments per year, each half the present payment,
the new interest rate per payment is 12/26 or 0.462... (all calculations are
carried to 10 digits), and the new payment is -514.31/2 or -257.15:

	    I  =   0.46  
	    PMT=-257.15

     yields N  =    494  (total payments)
       with FV =-120.36  (final balance after 494th payment - negative means
			  money flows away from borrower)

494 payments at 26 payments/year works out to 19 years.  Total money paid on
the loan is:

     case 1:  514.31 * 360 =          $185,150.27
     case 2:  257.15 * 494 + 120.36 = $127,154.02
				      -----------
				      $ 57,996.25  total saved

Annual payments come out to 

     case 1:  514.31 * 12 = $6171.68
     case 2:  257.15 * 26 = $6685.98

A good calculator makes it easy to answer questions like this.

Dave Rabinowitz
hplabs!hp-pcd!daver

jimb@tekcbi.UUCP (Jim Boland) (11/19/85)

I have read so many different answers to the original question, that I
am totally confused about the whole thing.  Am I the only one???? or did
I just happen to fall off a turnip truck |-).??????

Let's summarize what we have got here --------:

R.M. posts the original in article 975
> I have recently read about a new type of real estate mortgage agreement,
> in which the bank gives the buyer a mortgage at the normal insterest
> rate for, say, a 30-year, fixed-rate mortgage, but allows the mortgagor
> to pay off the loan semi-monthly rather than once a month.  The total
> monthly payments are the same, they are just split in half and paid
> twice a month.
> 
> The incredible part of this is that you manage to reduce your interest
> so much by paying the loan semi-monthly, that you end up paying only a
> fraction of the total interest you would otherwise have paid, while
> expending the same amount of money per month, and you pay off what would
> have been a 30 year mortgage in something like 12 years.

Well, I couldn't believe any of this.  Then we heard from:

>Ilya Goldberg
>12 years seems a much smaller time period than 30 and is not very believable.
>Could someone post a formula either verifying '12 years' or giving the
>correct period of time based on the above pay off arrangement?

Right On.  Let's see some formulas and proofs.  How about some examples???
Next we heard from:

>S.SINGHAL
>In the semi monthly mortgage plan you pay every second week instead of twice a
>month.  Since there are 52 weeks in a year, you make 26 payments each equal to
>half the monthly payment.  Consequently you make 13 monthly payments a year
>instead of just 12.  This extra payment, along with some interest savings,
>enables you to pay off a 30 year mortgage in just 18 years or so.

Oh, now we're making one extra payment per year.  OK, I can buy that.  I can
see where it will shorten the length of time somewhat.  But not by 12 years!
And then along comes:

>Andrew Koenig
>If you pay twice a month instead of once, and your payments are
>half what they would be otherwise, you pay off the loan in half
>the time (believe it or not)!  So the number above should have
>been 15 years, not 12.

Oh, now we cut it in half.  Let's see, now.  Some say 12 years, some
say 18 years, some say 15 years.  Where are the facts to support
these claims??
And how about the statement from:

>Partha Dasgupta
>No Way! By your logic, if I made 4 payments a month, at one-fourth the monthly 
>rate, I would pay off the mortgage in 7.5 years.
>If you beleive that, I would love to borrow $100,000 from you, at those terms,
>any time!

Boy, count me in too.  Why just $100K.  At rates like that, how about $1M?
Then we heard from:

>Mark Horton
>When I first saw this, the claim was that the bank having the
>extra half payment for half the month was what was cutting down
>on the interest.  But this doesn't make any sense because if you
>were to just prepay half a payment extra when you took out the
>loan, it would give the bank that same amount of money all the
>time instead of half the time, and obviously it would just apply
>to the principle and make almost no difference.

>Finally I read something in the Sunday paper that explained it.
>It isn't a semi-monthly payment, it's a bi-weekly payment.  Since
>there are 52 weeks in a year, you in effect make 13 months worth
>of payments every year instead of 12.  This extra payment is what
>is bringing down the mortgage that much more quickly.

Oh, OK, I think.  I guess we have a consensus(sp) on one point, anyway.
It is a bi-weekly payment, not semi-monthly.

Then someone with experience wrote:

>Marsh Gosnell
>Check the fine print.  All of the half pay programs I've seen call for
>you to pay half the amount every two weeks (not twice a month).  Having
>recently purchased a house, I carefully investigated this kind of mortgage
>and concluded that they aren't worth it.

>Almost ALL of the interest savings come not from the fact you are paying
>more often but because you are paying the equivalent of 13 "monthly"
>payments each year.  If you were to take out a conventional 15 or 30 year
>mortgage and pay 1/12 extra each month you pay off the mortgage within a
>month or two of the half pay mortgage.  

And then, Robert is going to help explain it:

>Robert E. Schleicher
>There are two versions of this type of plan, which I think are being
>mixed-up a little in the above discussion.

>1.  As described above, you make two 1/2 payments a month.  This provides
>    a fair savings in overall interest payments, but not enough
>    to pay off the loan in 12 years (it saves several years, though.)

>2.  The other variation is that you pay 1/2 of what would have been your 
>    monthly bill, but every two weeks (or, alternatively, your old
>    monthly bill every four weeks).  This amounts to making 13 months
>    of payments every year, and DOES pay off the loan very quickly,
>    in about 12 years.  Of course, you're really paying a higher effective
>    monthly payment.  

Somehow, I can't see it.  How about taking a real life example and showing
it?  Considerations:
    Does it work at all interest levels?
    Does it work at all principal levels?
I'll give you one to work on--
  My loan is for $32K.  length is 30 years. rate is 8.75%.  monthly payment
  for principal and interest is $232.
If the payout is twelve years, then that means I have paid 12 * $232 extra
for principal = appx $2850.  That is less than 10% of the loan.  Somehow
I can't see where this is going to work out.  How about a month by month
listing of what the principal balance and interest are?
But read on:
>
>Suzanne Barnett-Scott
>What I have heard is that the payments are bi-weekly, not
>semi-monthly, which means you pay approximately 13 months
>worth of mortgage payments each year. For example, if your
>payment is $1000 per month, and you pay $500 every two weeks,
>you pay a total of $13000 in the year, not $12000.
>
>Yes, this pays your interest faster, but, you could pay off the
>loan even sooner if you have that extra $1000 applied directly to
>the principal, not the interest first, as are most mortgage payments.

Hmmmm. And finally, the first posting with some meat----- I mean substance:

>Andrew Koenig
>Whether or not you can pay additional principal (please note the spelling!)
>on your mortgage depends on whether the lender allows it.  This, in turn,
>depends at least in part on whether the laws of your state require the
>lender to allow it.  In New Jersey, for example, all mortgage loans
>must be prepayable in any part without penalty.  Such loans are sometimes
>called "simple interest" loans.
>
>Here's how it works.  On January 1, you closed on a house and got a
>$100,000 loan at 12% nominal rate to finance it.  Although the banks
>here say the loan is 12%/year, it is really 1%/month, or, if
>you like, 12%/year compounded monthly.
>
>Thus, at the end of January, you owe the bank:
>
>		100,000		your principal as of January 1
>		+ 1,000		1% interest on your principal.
>
>plus enough extra to make up the payment you agreed on (which, according
>to our assumptions, is $1028.61 for a 30-year term).  This extra is
>$28.61 for the first payment, so after you have made this payment,
>you owe the bank $99971.39.  Now, at the end of February, you owe the bank
>
>		99,971.39	your principal as of February 1
>		+  999.71	1% interest on your principal.
>
>plus enough extra to bring your payment up to $1028.61.  This extra is
>$28.90 this time around.  See, you're paying it off already!  :-)
>
>Here are the similar figures for the first year:
>
>period   balance   principal   interest
>     1 100000.00       28.61    1000.00
>     2  99971.39       28.90     999.71
>     3  99942.49       29.19     999.42
>     4  99913.30       29.48     999.13
>     5  99883.82       29.77     998.84
>     6  99854.05       30.07     998.54
>     7  99823.97       30.37     998.24
>     8  99793.60       30.68     997.94
>     9  99762.93       30.98     997.63
>    10  99731.94       31.29     997.32
>    11  99700.65       31.61     997.01
>    12  99669.04       31.92     996.69
>
>The important point to realize is that on a simple interest loan
>with a fixed payment and interest rate, the only thing that determines
>how much time you have left to pay it off is what your principal is.
>
>For instance, my $1028.61 payment at the end of January reduced my balance
>to $99971.39.  If I had paid $1086.70 instead, I would have reduced my
>balance to $99913.30, and would effectively now be beginning period 4.
>I would effectively have reduced the term of my loan by two months
>for the cost of $58.09.
>
>Sounds like an irresistable deal, right?  Well maybe.  If instead I had
>taken that $58.09 and invested it at 12% a year, compounded monthly,
>I would have exactly enough money 29 years and 10 months from now to
>pay off the balance of the loan, two months early.
>

Now it's 29yrs and 10 months.  
And I am still confused.  My gut tells me that you can pay it off
in less than 29 5/6 years but certainly more than 12-18 yrs.  I would
imagine it to be somewhere around 25 years.  But, I am not a financial
person.  
What we need now are no more "I Think"'s but some real examples of
how it works, itemized out so we can understand it.

Anyone else feel this way????   Is there a net.financial?????

schooler@inmet.UUCP (11/21/85)

  I derived a mortgage formula which is evidently the same one
the banks use, since my results match their tables down to the
last cent.  Here goes:

  At month 0, I owe the principal (p).  After one month, I pay
my payment (x).  I now (at month one) owe the principal increased
by the interest rate minus the payment, or p * (1 + r) - x.  At
month two, I owe the previous amount, again increased by the
interest rate, again minus my payment, or (p * (1 + r) - x) * (1 + r) - x,
or p * (1 + r)^2 - x * (1 + r) - x.  In summary, at month n, I
owe p * (1 + r)^n - x * SUM ((1 + r)^n from 0 to n - 1).  After a
certain amount of time, I will owe 0, so I set the above formula
and solve for x.  The result is:

	x = p * r * (1 + 1 / ((1 + r)^n - 1)).

  Thus, if n = 360 (30 years), p = $100,000, and r = .01 (12% nominal
annual interest rate), then x = $1028.61.  To solve the problem at
hand, we instead solve for n to get:

	n = log (1 + (p * r) / (x - (p * r))) / log (1 + r).

  If we now set x and r to half the previous amounts to model paying
half a monthly payment twice a month, we get n = 718, or almost exactly
30 years, so we aren't saving much.  If we follow someone else's suggestion,
and pay the half-monthly amount every two weeks, then this is equivalent
to paying 13/12 ths the monthly amount every month, so x = $1114.33.
Keeping r = .01, we get n = 228 months, or 19 years.

  If the bank fiddles the interest rate a little, the payment term
could change drastically, as the above formulas are highly non-linear.
If I have missed something in the baove analysis, please let me know.

		-- Richard Schooler
		Intermetrics, Inc.
		{ihnp4,ima}!inmet!schooler

fred@mot.UUCP (Fred Christiansen) (11/22/85)

> I have read so many different answers to the original question, that I
> am totally confused about the whole thing.  Am I the only one???? or did
> I just happen to fall off a turnip truck |-).??????
>
> I am still confused.  My gut tells me that you can pay it off
> in less than 29 5/6 years but certainly more than 12-18 yrs.  I would
> imagine it to be somewhere around 25 years.  But, I am not a financial
> person.  
> What we need now are no more "I Think"'s but some real examples of
> how it works, itemized out so we can understand it.

ok.  here goes.  let's compare 1) a $95000 mortgage at 12% interest
with $1000/mo payments (these were easy numbers to pull out of my head)
versus 2) the same thing but with $500 bi-weekly payments.  since the
program i wrote deals only in months, let's convert those $500 bi-weeklies
into equivalent monthlies (note that this is being *very* conservative;
actual results are slightly better due to interest computation).
that computes to equivalent monthly payments of $1083.33.

per my results(*), the bi-weekly approach saves you $70,000+ and pays off
your mortgage in ~8 fewer years.

(*)
1) ((25 yrs * 12 months) + 1 month) * $1000) + $68.93 = $301,068.93
2) ((17 yrs * 12 months) + 6 months) * $1083.33) + $589.06 = $228,088.36

on a mortgage where the payments where computed to actually run 30 years,
the numbers you'd get would be even better than these.
-- 
<< Generic disclaimer >>
Fred Christiansen ("Canajun, eh?") @ Motorola Microsystems, Tempe, AZ
UUCP:  {seismo!terak, trwrb!flkvax, utzoo!mnetor, ihnp4, attunix}!mot!fred
ARPA:  oakhill!mot!fred@ut-sally.ARPA          "Families are Forever"