Mortgage Calculator - Algebra II

Monthly mortgage payments are determined by the principal and interest rate. In fact, a good Algebra II student should be able to understand the derivation of the monthly payment for a standard 30 year mortgage.

Here is an example of a mortgage payment calculator:



The goal of this posting is to show how one could teach Algebra II students the math necessary for creating this widget. Prerequisites:

• Algebraic sophistication (very comfortable with manipulating equations)
  
• Basic understanding of sequences and series
• Understand the telescope method for deriving sequence formulas
• Know the formula for a geometric series

The Proof:

Let

• P_n = principle amount after n months
• r = annual interest rate (compounding monthly)
• m = monthly payment (constant - doesn't change month by month)
• N = the number of monthly payments until the principle is completely payed off (for a 30 year mortgage N = 360)

First derive a recursive definition for the sequence P_n . To get this, notice that the interest that needs to be payed is the principle times the interest percentage for one month. The interest percentage for one month is simply r / 12. Therefore the interest portion of payment n is:

• interest owed on month n = I = P_n • (r / 12)

The principle payed off is the amount of the monthly payment left over after paying the necessary interest. In other words

• Principle Payed on month n = m - I = m - P_n • (r / 12)

Therefore, the principle remaining is given by:

• Priciple Remaining = P_n+1 = P_n - ( m - P_n • (r / 12) )

Simplify this to:

P_n+1 = P_n• ( 1 + r / 12 ) - m

Finally, letting α = ( 1 + r / 12 ) the formula becomes

P_n+1 = α P_n - m

Applying the telescoping method, together with the formula for geometric series we get:

• P_n= αⁿ P₀ - m (αⁿ - 1) / (α- 1)

Finally, since the mortgage is payed off after the N'th payment we have

• P_N = 0
  

or plugging N into the 2nd to last formula:

• 0 = α^N P₀ - m (α^N - 1) / (α- 1)

or solving for m:

• m = α^N (α- 1) / (α^N - 1) P₀