I have gotten numerous requests from individuals wondering what the simple formula is for calculating the monthly payment and also how to generate the amortization table, including the accrued interest and extra principal payments. Instead of just showing some boring source code, I thought I would try to more fully explain it.

Spreadsheet (Excel, Lotus, Quattro) users should look here

**NEW!** Want to see how this is derived? Find a full derivation here! (Thanks goes to "Hans" Gurdip Singh.)

**NOTE:** This first part is for **United States** mortgages. Look here for the Canadian formula.

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 percent)
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. J = monthly interest in decimal form = I / (12 x 100)
N = number of months over which loan is
amortized = L x 12
Okay now for the big monthly payment ( 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**.
Sorry, for the long way of explaining it, but I just wanted to be clear for everybody.

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 iteration
(i.e. a simple loop). I will tell you 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.

Programmers will see how this makes a trivial little loop to code, but I have found that many people now surfing on the
Internet are **NOT** programmers and still want to calculate their mortgages! So this page was dedicated more to the latter. If you have
any further questions you can
contact me for more info.

Many people have asked me how to find N (number of payments) given the payment, interest and loan amount. I didn't know the answer and in my calculators I find it by doing a binary search over the payment formula above. However, Gary R. Walo ( nenonen5@southeast.net) found the answer to the actual formula in the book:

**n = - (LN(1-(B/m)*(r/q)))/LN(1+(r/q))**

**# years = - 1/q * (LN(1-(B/m)*(r/q)))/LN(1+(r/q))**

Where:

- q = amount of annual payment periods
- r = interest rate
- B = principal
- m = payment amount
- n = amount payment periods
- LN = natural logarithm

where:P = P * (1 - ((1 + J) ** t - 1) / ((1 + J) ** N - 1))

- P = principal, the initial amount of the loan
- I = the annual interest rate (from 1 to 100 percent)
- L = length, the length (in years) of the loan, or at least the length over which the loan is amortized.
- J = monthly interest in decimal form = I / (12 x 100)
- N = number of months over which loan is amortized = L x 12
- t=number of paid monthly loan payments

If you would like to calculate an outstanding loan balance but have not made regular fixed payments by the due date, you will have to fill out an accurate payment schedule within a spreadsheet, or similar application where you can account for missing and different payments.

min_rate = 0; max_rate = 100; # Set Maximum and minimum rate while (min_rate < max_rate - 0.0001) { mid_rate = (min_rate + max_rate) / 2; # Divide by 2 to find midpoint J = mid_rate / 1200; # Convert to monthly decimal percentage # calculate payment based on this interest, term of F and loan_amt guessed_pmt = loan_amt * ( J / (1 - (1 + J) ** -N )); if (guessed_pmt > actual_payment) { max_rate = mid_rate; # current rate is new maximum } else { min_rate = mid_rate; # current rate is new minimum } } print " The Rate is ", mid_rate;On any modern computer this runs pretty much instantaneously.

This was contributed to me by: Mike Morley (morleym@interlog.com)

Canadian mortgages are compounded semi-annually instead of monthly like US mortgages. Monthly Pmt = (P*(((1+i/200)^(1/6)-1))/(1-(((1+i/200)^(1/6)))^-(n*12))) Where: P = principal outstanding i = annual interest rate percentage n = number of years

Here is a easier to read representation:

i 1/6 ( 1 + --- ) - 1 200 Pmt = Principal x ------------------------ i 1/6 -12 x n 1 - [ (1 + --- ) ] 200 Or to convert canadian interest rates to US interest rates: Can. Rate 1/6 US Rate = 1200 x [ ( 1 + --------- ) - 1 ] 200 or as a formula, US Rate = 1200 * ((1 + Can.Rate/200)^(1/6) - 1)

You'll note if you plug this into the US formula you get the above formula for payment.

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