Pay Off Mortgage Or Invest?
A person online compared two strategies for dealing with one’s mortgage:
- Pay off as much of the principal as possible every month, and then start investing only once the mortgage is completely paid off. This makes intuitive sense because delaying paying off the principal means larger interest payments for longer, which is money down the drain.
- Pay off nothing, invest everything, and then pay off the mortgage in full at the end of a 20-year period. This also makes intuitive sense because a twenty-year investment portfolio has generally earned higher return than the interest on a mortgage, historically speaking.1 Mortgages specifically have low interest because they are insured with a valuable collateral: the home.
This person had made rough estimations based on past returns in Excel and concluded very confidently that the second alternative is better, because it has higher expected return. Some other people online tried to point out that it obviously has higher expected return because it has higher risk, but the original author did not seem to understand that nuance: they quipped back that
Paying off the loan comes with very high risk of missed return on investment.
which sounds clever but that’s not what risk means in finance.
No, higher risk means that an investment has a broader range of outcomes. It might go up more, but it also might go down much more. Paying off a loan has virtually no risk, in this sense, because we know exactly how much the loan is reduced by for each payment.
We can see this if we plot net worth (savings minus mortgage principal remaining) over a 20-year horizon for the person who uses all their excess money to pay off the loan.2 This ignores the value of the home in order to focus on the effect purely on the loan and investments. Including the value of the home (or any other savings a person might have) would result in the line shifting upwards. It may also interact with the outcomes of the investments due to diversification effects, but this simple system is enough to get the point across.
This is based on very primitive models for interest rate movements and stock returns3 Vasicek and normal fee-adjusted log returns for a diversified portfolio, respectively., but they ought to be sufficient for the illustration. This represents someone who has borrowed $200,000 and has a monthly budget of $1,300 to service the loan.4 These figures are taken from the same Excel sheet that kicked this article off. The models are also calibrated against the same assumptions as the Excel sheet. This budget is used first to pay off the interest, and whatever remains is used to pay off the principal. Once the loan is fully paid off, investments start.5 This can be seen as a slight kink upwards in the last two years.
The reason this line looks … so much like a line, is that paying off a mortgage is a financially low risk strategy. The amount of the principal that can be paid off varies a little with how the interest rates change, but we know rather accurately how much we will increase our wealth each month.
This is in contrast to the investor, who has the same $1,300 budget, but only pays the interest from it, and then uses whatever remains to invest in a sensible financial portfolio. This investor does not know how much their wealth will increase each month, because it depends on the performance of their investment portfolio. Their corresponding plot might look like the following.
This person arrives at nearly the same wealth after twenty years as the first person, but it wobbles around a lot more on the way there. This is the definition of financial risk.
These are just two outcomes out of many possible. To visualise the final net wealth after twenty years, we’ll plot these numbers from many simulations into a histogram. These are the outcomes for the investor type.
The wiggling means, when considering all possible futures, that we cannot be very sure of what we will have at the end of 20 years. There’s nearly a quarter of a chance that we still aren’t able to fully pay off the loan. If we cannot, the mean we come up short is $40,000. The Russian roulette case6 The 1/6th quantile. I find this to be a useful intuition for what one could consider an unlucky but still not unreasonable outcome. is being -$20,000 in the red.
On the other hand, the median outcome is that we’re $73,000 in the black. The inverse Russian roulette case7 The 5/6th quantile. This is the lucky-but-not-unreasonable outcome. gives us almost $200,000 invested, even after having paid off the $200,000 mortgage! Let’s put these figures into a table for easier comparison.
| Outcome | Investor |
|---|---|
| Mean excess short | -$40,000 |
| Probability of short | 23 % |
| Russian roulette | -$20,000 |
| Median outcome | +$73,000 |
| Inverse Russian roulette | +$200,000 |
Now if we go for the low-risk strategy of paying off the principal as much as we can before we invest, we get these outcomes instead:
This is a much narrower range of outcomes (lower risk indeed) but also not with the same potential for outsize returns as the investment strategy. Comparing by table, we find
| Outcome | Investor | Repayer |
|---|---|---|
| Mean excess short | -$40,000 | -$30,000 |
| Probability of short | 23 % | 3 % |
| Russian roulette | -$20,000 | +$35,000 |
| Median outcome | +$73,000 | +$54,000 |
| Inverse Russian roulette | +$200,000 | +$70,000 |
Nothing of this says that it is better to follow one strategy or the other; to figure that out we would need to know total wealth, account for how the other savings are invested, the correlations between these investments, etc.
But it does hint that
- maybe if one has a primary goal of paying off the loan in full by year twenty, repaying as much as possible each month is vastly superior to investing: it gives us a slim 3 % risk of not reaching our goal, compared to 23 %.
- if one does not mind being in debt and wants a high-risk investment strategy for this money8 Because one has shown that to be optimal through analysis, not because one feels like it, to be clear., then investing it is better than paying off the debt: the inverse Russian roulette case is nearly 3× higher with that strategy.
- If one cares about the median outcome9 Which perhaps most of us should do, unless otherwise indicated then it doesn’t matter all that much which of these strategies one chooses, and the optimal may well be a combination.
This complicated outcome reflects the reality of the decision, in contrast to the spreadsheet that only considered arithmetic mean return, and paid no attention to risk.