## 4/03/2007

### The "Inverse Rule of 72" for Mortgages

Golbguru of The Tao of Making Money recently posted about the common "Rule of 72" for figuring out when your money will double, based on making an investment that makes a particular interest rate. If your rate is 7%, it will take roughly 10 years to double your money, etc.

Well, that got me to thinking, particularly when I had a conversation with my dad today about my parents' house. They have been in the house for about 10 years and have a typical 30-year mortgage (although they are making payments every 2 weeks in an accelerated payoff plan). We were wondering how much they still owed on the mortgage, and unfortunately such calculations don't come quickly in our heads, and I had no quick shortcut. I wondered: What if there is a quick "Inverse Rule of 72" out there? How long would it take to halve the principal when we're paying interest?

So tonight I used Bankrate's Mortgage Calculator to figure out a few tricks. In line with the "double your money" idea from the Rule of 72, I wanted to know how long it would take to pay off half of the mortgage. Taking it one step further, I wanted to see how long it would take to pay off just 25% of the mortgage. The formulae for the final approximations here are not exact, but they are quite easy to calculate and good enough for casual use. All of these figures are based on a standard 30-year mortgage. First we have a table of the actual values for the month/year when the mortgage will reach a certain paid-off point:

 1/4 paid off 0% 6/07 1% 6/08 2% 5/09 3% 5/10 4% 6/11 5% 7/12 6% 8/13 7% 9/14 8% 9/15 9% 8/16 10% 7/17 1/2 paid off 0% 1/15 1% 2/16 2% 4/17 3% 4/18 4% 4/19 5% 3/20 6% 1/21 7% 10/21 8% 6/22 9% 1/23 10% 7/23
The 0% part is fairly obvious, and it forms the basis of our estimation formula. A 30-year 0% loan will be paid off halfway at the halfway point of the loan, or in the first month of the 15th year. One-quarter of the initial principal will be paid off one-quarter of the way through, or 7 years and 6 months into the loan. These form the baselines, and all other dates will be calculated from there. Now if you look at the tables, as you move up a percentage point, the target dates are pushed back by about a year per percentage point. At higher interest rates, there is a bit of compression at the back end of the loan, but a year is still a good estimate. There is only a 6-month difference between 9% and 10% to reach the halfway point (both occur in the 23rd year of the loan), but it is still 9 months between 6% and 7%. So as your interest rate gets above the "normal" threshold, the approximation will get less accurate for the half-paid scenario. For paying off one-quarter of the principal amount, each percentage point between 0% and 10% makes a difference of between 11 and 13 months, so one year is a very good approximation in that case.

So our final formula is:
1/2 Paid off: 15 years + 1 year per percent interest
1/4 Paid off: 7.5 years + 1 year per percent interest

Remember, these are all for 30-year mortgages! So, it will take roughly 22 years to pay off half of a 30-year mortgage at 7% interest and roughly 18 years at 3% interest. It will take roughly 10.5 years to pay off 1/4 of a mortgage at 3% and roughly 14 years at 7%. If you are interested in 15-year mortgages then you can just use the above formulas and divide your final answer by half.