Analyzing Monthly Payments

Date: 05/13/98 at 18:56:15
From: Scott Holmes
Subject: Principal left after n months

I am trying to calculate how much interest versus principal has been 
paid on a loan at various points in the loan repayment. For example, 
if I borrow $10,000 at an APR of 7% (compounded daily), and I make 
monthly payments of $50, how much of each of my payments is going to 
interest? I know the first payment is almost all interest and my last 
payment is almost all principal, but what is the formula to calculate 
just how much goes to each?

So the questions I would like to answer are:

   How much principal have I paid after n payment cycles?

   What will be my payoff (principal only) after n payment cycles?

Thanks.

Date: 05/15/98 at 11:55:53
From: Doctor Schwa
Subject: Re: Principal left after n months

Great -- you have almost the whole answer in your statements already. 
What first springs to my mind is that 7% of 10,000 is $700 per year,
which is more than $50 per month, so you're going to end up going 
deeper and deeper in debt, not even managing to pay off the full
interest amount.

Dr. Anthony wrote a nice response about mortgage payments. His answer
doesn't exactly address your question, but is a good illustration of 
the techniques you need to apply to answer your question. You can find 
it at:

   Formula for Mortgage
   http://mathforum.org/library/drmath/view/54623.html   

I found it by searching the dr.math archives for the word "mortgage."

One approach to solving this problem, which I recommend trying, is
to work with a spreadsheet (like Excel) to calculate line-by-line
how much you owe each day, using a formula such as:

   A2 = A1*(1 + r) - B2

where r is the DAILY interest rate, (.07)/365 for your example. Then 
at the end of every month, on whatever day that is, put in column B 
the amount of your monthly payment. For instance, B30 = 50 might apply 
in your case.

Using the spreadsheet really helped me to get a feel for the numbers 
(and double-check my algebra!).

I'm going to simplify matters a bit by supposing that a year consists 
of 360 days, split into 12 30-day months. If you want to do 
calculations where you pay on the 1st of each real month, and compound 
the interest daily, you can, but it gets even more messy.

The amount you owe at the end of one month, in my simplified 
assumption, is 30 days of the daily interest, or

   (initial amount)*(1 + r/360)^30

with r this time being the ANNUAL interest rate, .07 in your case. I'll 
let "f" (for factor) stand for (1 + r/360)^30 (about 1.00585 in your case).

At the start of the loan, you owed some initial amount; let's call it
L, for loan amount. Then after one month, you owe L*f. Then you make a 
payment of (let's call it) P dollars per month. So now you owe 
(L*f - P). And then you pay interest on that for a month, which 
multiplies by f, giving:

   (L*f - P)*f = L*f^2 - P*f

Then you make another payment P, so you owe L*f^2 - P*f - P. Another 
month goes by, multiplying by f, and then subtracting P, so now you 
have:

   L*f^3 - P*f^2 - P*f - P

owed at the end of the third month.

I hope by now you can see the pattern; the amount you owe is:

   L*f^n - P*(f^(n-1) + f^(n-2) + ... + f^2 + f + 1)

at the end of the nth month.

Fortunately, there's a convenient formula for the sum of the geometric 
series in parentheses (feel free to ask if you don't know where this 
formula comes from):

   f^(n-1) + f^(n-2) + ... + f^2 + f + 1 = (f^n - 1)/(f - 1)

So at the end of n months, amount owed is:

   L*f^n - P*(f^n - 1)/(f - 1)

The amount of principal you've paid is the initial amount of the loan,
L, minus the amount you still owe, from the above formula. And the 
amount of interest you've paid is your total payment, P*n, minus the 
amount of principal you've paid.

If you wonder what's going to happen in just this one month, find the 
amount of interest paid in that month (amount owed, times the factor, 
will give you the new amount owed; so amount owed, times (f-1), is the 
increase in the amount owed, which is the interest). And the amount of 
principal paid is the payment P minus the amount that goes to take 
care of the interest.

This is a very practical kind of question -- I know, because with
interest rates so low these days, I recently had to decide which of
several options for refinancing my home mortgage would serve me best 
in the long run. The only extra complication when you try to turn this 
into a real-world problem is that mortgage interest is tax-deductible, 
while payments of principal are not.

-Doctor Schwa, The Math Forum
Check out our web site! http://mathforum.org/dr.math/