Fast Formulas #2: Average Life of Mortgage (Amortizing with Balloon)

Fast Formulas #1: Average Life of Mortgage (as Scheduled) showed a quick way to calculate the weighted average life (“WAL”) of a fixed-rate mortgage that pays as originally scheduled.  I found this formula (which is probably not original) by substituting the standard  level payment formula into the general solution to Puzzle #1: Mortgage Average Life.

Puzzle #3: Mortgage with Balloon is similar to Puzzle #1 but asks for a quick way to find the average life of a mortgage that amortizes on schedule for some period of time, but is then paid off early with a balloon payment.  I thought I had a good compact solution, but Ashwani Singh found a better one, which is discussed in Puzzle #3: A Better Answer.

Here’s the general formula for an amortizing mortgage with a balloon, based on Ashwani’s solution.  Let:

be the original term of the fully amortizing mortgage, in months;

be the monthly level payment of that -month mortgage;

be the WAL for that mortgage, in months;

be the number of months until the balloon payment;

be the number of months in the original term past the balloon payment;

be the monthly level payment of a -month mortgage;

be the WAL for a -month mortgage; and

be the WAL with the balloon payment.

Then:

.                                                                                         (Formula 2.1)

(Note that here I measure time in months instead of in years, as I did in Puzzle #3: A Better Answer.)

But suppose we don’t already know the average lives and level payments needed for Formula 2.1.  Let’s see what happens if we substitute the respective formulas into Formula 2.1.

First, we need to define:

, the monthly interest rate (1/12 of the annual rate);

, the one month discount factor; and

.

We can use two formulas from Fast Formulas #1.  The formula for a level monthly payment is:

;                                                                                                               (Formula 1.3)

and the average life for a mortgage without an early payoff is:

.                                                                                                        (Formula 1.4)

Now substitute Formulas 1.4 and 1.3 into Formula 2.1.  With a little work you will find that:

                                                                           (Formula 2.2)

with

.

Note that Formula 2.2, like Formula 1.4, does not require the principal amount to calculate an average life.

Let’s use Formula 2.2 in an example.  Suppose we had a 30-year mortgage at 6%, with a balloon payoff at ten years.  Then , , , and . We calculate:

;

;

;

;

;

and

months.  This is only 9 months before the balloon payment, which dominates the average.

I think that Formula 2.2 might be an original way to find the average life of an amortizing mortgage with a balloon.  I want to thank Ashwani again for the elegant contribution which led to this formula.

Copyright 2011, 2013.  All rights reserved.

Revised February 15, 2013 with LaTeX math formatting and minor changes to the text.