#### Mortgage Calculations by Hand

First you must define some variables to make it easier to set up: **P** = principal, the initial amount of the loan I = the annual interest rate (from 1 to 100%) **L** = length, the length (in years) of the loan, or at least the length over which the loan is amortized.

The following assumes a typical conventional loan where the interest is compounded monthly. First we'll define two more variables to make the calculations easier: **J** = monthly interest in decimal form = **I / (12 x 100) N** = number of months over which loan is amortized = L x 12

Now for the big monthly payment (**M**) formula ... it is:

J M = P x ------------------------ 1 - ( 1 + J ) ^ -N where 1 is the number one (it does not appear too clearly on some browsers)

So to calculate it, you would first calculate **1 + J** then take that to the **-N** (minus N) power, subtract that from the number 1. Now take the inverse of that (if you have a **1/X** button on your calculator push that). Then multiply the result times **J** and then times **P**.

The one-liner for a program would be (adjust for your favorite language):

M = P * ( J / (1 - (1 + J) ** -N))

So now you should be able to calculate the monthly payment, **M**. To calculate the amortization table you need to do some iterations (i.e. a simple loop). Here are the simple steps :

**Step 1:** Calculate **H = P x J**, this is your current monthly interest

**Step 2:** Calculate **C = M - H**, this is your monthly payment minus your monthly interest, so it is the amount of principal you pay for that month

**Step 3:** Calculate **Q = P - C**, this is the new balance of your principal of your loan.

**Step 4:** Set **P** equal to **Q** and go back to **Step 1:** You thusly loop around until the value **Q** (and hence **P**) goes to zero.

Many people have asked how to find N (number of payments) given the payment, interest and loan amount. The answer to the actual formula is in the book: **The Vest Pocket Real Estate Advisor** by Martin Miles (Prentice Hall). Here's the formula:

**N = -1/Q * (LN(1-(B/M)*(R/Q)))/LN(1+(R/Q))**

Where:

- Q = amount of annual payment periods
- R = interest rate
- B = principle
- M = payment amount
- N = amount payment period
- LN = natural logarithm