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From: schooler@inmet.UUCP
Newsgroups: net.consumers
Subject: Re: Semi-monthly mortgage repayments
Message-ID: <22100024@inmet.UUCP>
Date: Wed, 20-Nov-85 20:53:00 EST
Article-I.D.: inmet.22100024
Posted: Wed Nov 20 20:53:00 1985
Date-Received: Sat, 23-Nov-85 11:05:04 EST
References: <1389@decwrl.UUCP>
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Nf-ID: #R:decwrl:-138900:inmet:22100024:000:1673
Nf-From: inmet!schooler    Nov 20 20:53:00 1985


  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