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### Principal or Interest?

```Date: 08/08/2002 at 07:18:31
From: S. Jackson
Subject: New Car Loan

If I have a new car loan for \$20,000 at 9% simple interest, how do I
calculate how much of that monthly payment is going to the principal
versus the interest? I plan on making additional principal payments
monthly and want to keep up with the balance of the principal for an
early payoff.

Let's say the loan is for 5 years (60 months). So we are working with
\$20,000 loan for 60 months at 9% interest.

Thank you.
Sue Jackson
```

```
Date: 08/08/2002 at 08:39:09
From: Doctor Jerry
Subject: Re: New Car Loan

Hi Sue,

If at the beginning of any month, after you've made whatever payment
you wish, you still owe the bank A dollars, then during the coming
month your debt will increase by the amount of the interest, which
will be A*9/(12*100)=0.0075*A.

The notation a_0 means a sub 0.

amount borrowed: a_0 (dollars)
interest rate: r (like 7%)
number of months the loan is for: N
payments are made at the end of each month, in the amount x
you borrow the money at the first day of some month

let p_0 be the amount you owe the bank at time t=0 (months)
let p_j be the amount you owe the bank at time t=j (months)

Now,

p_0 = a_0
p_1 = a_0 + a_0*r/(12*100)-x=a_0(1 + r/1200) - x

(amount owed one month ago, plus interest accrued on this amount,
minus payment)

For convenience, let  1+r/1200 = w.  So,

p_1 = a_0*w - x

p_2 = p_1 + p_1*r/(12*100) - x = p_1*w - x=a_0*w^2-x*w - x
p_3 = p_2 + p_2*r/(12*100) - x
= p_2*w - x = a_0*w^3 - x*w^2 - x*w - x

So,

p_j = a_0*w^j - x[w^{j-1} + w^{j-2}+...+ w + 1]

Now,  w^{j-1} + w^{j-2}+...+ w + 1 = (1-w^j)/(1-w), finite geometric
sum. So,

p_j = a_0*w^j -x*(1-w^j)/(1-w).

We want to choose x so that p_N=0. Solving for x we find

x = a_0*w^N(1 - w)/(1 - w^N)

For your situation, in which a_0 = 20,000, N = 60, and r = 9, we find

x = 415.17.

This is the monthly payment. Just below I list the time (1 month, 2
months, and 3 months), the amount owed, the amount paid on principal,
and the amount on interest. The sum of the latter two must be the
payment. So, we need only calculate the second of these.

1, 19734.83, 150.00, 265.17

2, 19467.68, 148.01, 267.16

3, 19198.52, 146.01, 269.16

To calculate the interest for the first month, do this: take the
amount owed at the beginning of the month (20000) and calculate the
interest:

20000*(9/100)*(1/12) = 150

For the interest during the second month, take the amount owed at the
first of this month (20000-265.17 = 19734.83) and calculate the
interest:

19734.83*(9/100)*(1/12) = 148.01

It goes on like this.

I hope that this has been of some help.

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

```
Date: 08/08/2002 at 17:07:38
From: S. Jackson
Subject: Thank you

Thank you. I really appreciate it. I will work it into my schedule on
the spreadsheet and go from there. Again, thanks a lot!

Sue
```

```
Date: 08/09/2002 at 07:32:21
From: Doctor Jerry
Subject: Re: Thank you

Hi Sue,

Just one more thing. If you make an extra payment with the regular
payment, and the extra payment goes to reduce the principle, I think
amount from the principal used on the next line (on which the interest
for the coming month is calculated).

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Interest

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