How To Calculate Loan Payments and Amortization on the Back of an Envelope With a Cheap Calculator

In a previous article we presented a simple formula to calculate the amount of a monthly home mortgage loan payment. The formula applies to any compound interest loan. The only special equipment you need is a calculator with a power function key. That's the key with the y superscript x (y ^ x). If you have kids in school you probably already have one.

Here is a review of monthly payment formula.

The variables are:

N = loan period in months. i.e. 20 years = 240 months.

R = interest rate in whole numbers. i.e. 8% written as 8.

P = principal amount of the loan. The amount borrowed.

Q = the Q factor. An intermediate calculation.

M = monthly payment amount

Here's the entire formula for the monthly payment amount of a compound interest loan:

M = (P * R * Q) / (1200 * (Q -1))

Easy enough, but first you have to calculate the value of Q. Here is the formula:

Q = (1 + R/1200) ^N. Pretty simple, but you do need the power function key. N can get large.

In our earlier example we calculated a monthly payment of $418.22 on a $50,000 second mortgage at 8% for 20 years. You have paid the 2nd mortgage loan for 5 years (60 months). The pay off amount is $43,763 (rounded). This is how to calculate the pay off amount on any compound interest loan after N number of payments.

This is an easy three step process with a subtraction at the end. First calculate the growth value of the loan amount (P). P increases by a factor of (1 + R/1200) per month, so after N months the value of the principal amount of the loan would have inflated to P * (1 + R/1200) ^ N. For the current $50,000 second mortgage the calculation looks like this:

50000 * (1 +8/1200) ^60 = 74492.28 (step one)

The monthly payments have also inflated by a factor of (1 + R/1200) per month so in math talk we have a geometric series with n terms. The monthly payment part is a little more complicated and the formula looks like this:

1200 * M * ((1 + R/1200) ^N -1) / R

Plug in the actual values and it looks like this:

1200 * 418.22 * (1 + 8/1200) ^60 / 8 = 30729.49 (step two)

Now finish up by subtracting the inflated repayment value from the inflated loan amount value to get the pay off amount:

74492.28 - 30729.49 = 43762.79 (pay-off)

Once you know how to calculate the monthly payment and pay-off amount for any compound interest loan on the back of an envelope, you can noodle mortgage and car loan what-ifs from anywhere.

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