Long First Month Mortgage Payment Calculation

Date: 06/22/2001 at 00:15:28
From: Genevieve Chan
Subject: Long 1st month mortgage payment calculation

Dr. Math, 

All the mortgage payment formulae seem to deal with level pay whole 
monthly payments, given interest, loan principal, and length of the 
loan. But in the real world, if one applies for a loan, the first 
payment generally is not exactly one month from the time the loan is 
taken. 

For example, imagine a loan is taken out on Jan. 15, 2001. The first 
payment is due Jan. 25, 2001. Thereafter payment is due on the 25th of 
each month. The principal is 120,000, interest is 8%. The length of 
the loan is 15 years (180 payments). Can this problem be solved with 
one formula? If not how would you calculate the monthly payment 
amount, given that the payments have to be a uniform amount for the 
entire loan life? That is, the monthly payment amount is the same from 
payment 1 to 180. 

Thank you in advance.
Regards.

Date: 06/22/2001 at 06:55:07
From: Doctor Mitteldorf
Subject: Re: Long 1st month mortgage payment calculation

Dear Genevieve, 

In the real world, most banks and lenders duck this mathematical 
problem by treating the first month differently: the borrower gets his 
full loan, but then immediately pays back the partial month's interest 
in advance. This is called "prepaid interest." The remaining loan is 
an even number of payment periods, and is calculated in the standard 
way. 

The bank actually reaps a small benefit from this practice, because 
for the partial month the payment is in advance, whereas any other way 
of treating the partial month would follow the standard practice of 
charging interest in arrears.

If you want to do the calculation with level payments including the 
partial months, it is not any more difficult than the calculation for 
an even number of months. The trick is to think in terms of present 
values, and to use the principle that 

=> the present value of all payments as of the loan date is equal to 
the amount of the loan.
 
Let's take your example. The nominal interest rate is 8%, so the 
monthly interest is 8/12 of 1% or 0.006667. So each month, the 
present value of your payment is discounted by another factor of 
1/(1+0.006667) = 0.993377. For the partial month, there are varying, 
slightly different ways to compute the discounting factor; for now, 
I'll use the simplest and calculate it as 10/30 of a full month's 
interest, so 8/36 of 1% is 0.002222, and 1/(1+0.002222) = 0.997782.

Now let's call the unknown payment amount p. The first payment is made 
after 10/30 of a month, so its present value is discounted by the 
factor 0.99777. The second payment is discounted by 0.99777*0.993377, 
and the third by 0.99778*0.993377*0.993377, the fourth by 
0.99777*0.993377*0.993377*0.993377, etc. So the sum of the present 
values of the payments is

   p*0.99778*(1 + 0.993377 + 0.993377^2 + 0.993377^3 + ... + 
     0.993377^179)

We can find the sum of the series in parentheses using the same 
standard technique we always use. Call the sum S. Multiply S times 
0.99333, and notice that the product is almost the same, just missing 
the first term and with an extra term tacked on at the end:

         S = 1 + 0.993377 + 0.993377^2 + 0.993377^3 + ... + 
             0.993377^179

0.993378*S =     0.993377 + 0.993377^2 + 0.993377^3 + ... + 
             0.993377^180   

Therefore 0.993378*S =  S - 1 + 0.993378^180. Solve this for S to find

   S = (1-0.993378^180) / 0.0066225 = 105.338

Going back to our original equation with p, we have

   p * 0.997782 * 105.338 = $120,000

and this gives p = $1141.72

The upshot is that we do the same calculation as for a standard 
180-payment loan, but that the payment amount is reduced by a factor 
of 0.993378/0.997782. The numerator of this factor is the discounting 
factor you would have used for a full month; the denominator is the 
discounting factor you actually use for the partial month.

Read more about loan calculations at our FAQ page, 

   Loans and Interest
   http://mathforum.org/dr.math/faq/faq.interest.html   

and in the High School Interest area of our Dr. Math archives:

   http://mathforum.org/dr.math/tocs/interest.high.html   

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/