It distresses me that I have seen more than one person decide to not pay-off their mortgage early “Pay-off” because they think mathematically it is incorrect to do so. I have written this post to dispel the myth.

The typical argument is constructed as so, the average rate of return for the stock market (Nominal of 10.7% / Real return of 8.5%) is higher than the current interest rate of 4% on my 30-year mortgage. Therefore, I will have a higher net worth if I “Invest” extra money instead of making additional principle payments my mortgage. The problem is they consider risk a qualitative factor and dismiss it from the calculation.

When you include risk in the calculation, Pay-off or Invest are both shown to be reasonable and mathematically correct alternatives. That’s right, one is not a feel-good option that is mathematically incorrect. However, it’s really hard to measure, quantify, project, and display risk. But I decided to give it a try.

The Scenario:

Frank just bought a house with a 30-year, 4% interest rate, $165,000 mortgage. He has an extra $1,500 per month after maxing out all tax advantaged accounts. What would happen if he put that discretionary money towards Pay-off or Invest? Nobody on the planet knows that answer, however we can make some educated projections.

The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present. The red line is Frank’s net worth if he puts the $1,500 towards Pay-off. The gray lines are all the outcomes that would have occurred, in the historical period described above, if he had chosen to Invest. The black line is the average of all the gray lines.

facebook photos hostingThe expected or average of the historical data is $20K higher than the expected result of Pay-off. However, there is a large variance in the outcomes that would have happened if Frank, at some point in the past, decided to Invest instead; he could have ended up $60K worse off by not paying off his mortgage early.

If Frank had decided to Invest at a date in the past in the data set, he would be riding one of those gray lines at random, which could have worked out great for him or been a swift kick to the backside. However, if he had payed off the mortgage early he would have selected the red line and would know exactly the track he was on and where he would end up.

Another interest way to look at the data; Under invest Frank would have a net worth of $0 between year 4 and year 10 while under Pay-off he would reach this mark at 7 years.

This graph emphasizes the actual outcomes of Invest if employed in the past. However, it is not accurate to assume that Frank’s expected value would be greater by $20k than Pay-off if he started in 2017. This statement would assume that the next eight years of stock market returns will be no better or worse than the last 145 years and the probability of any given rate of return occurring is exactly proportion to what occurred during the last 145 years. While this is possible it is very unlikely.

Stock returns in theory are equal to the risk-free rate plus an equity risk premium. Given that the risk-free rate is 0.80%, hence the low mortgage rates, while the normal risk-free rate has been around 4% is it reasonable to assume stock prices are more likely to under-perform historical averages than out-perform? Vanguard seems to think so.

facebook photos hosting2017 Vanguard Market Outlook

I have not covered the other factors that influence this decision such as liquidity, asset protection, or psychology which I may in a future post. The purpose of this post was to evaluate the math. In conclusion, the choice between Pay-off and Invest is a risk-reward trade off.