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I see a lot of investment and personal finance resources that propose scenarios like the following: if you have a loan for a house with a 5% interest rate and additional income after the minimum mortgage payment that you want to use to pay down your loan or invest, you should choose to invest in something like the stock market which has a 10% return because 10% is bigger than 5%.

Is this really fair to say? What math goes into this decision? Is it only equal if you intend to keep the money in the stock market for exactly the same amount of time as you would be paying off the loan?

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What that line of thinking is missing is risk. Yes, on average the stock market makes somewhere around 10% annually, but from year to year that can range from -30% to +40% or more depending on what you're investing in. So it's possible that investing in the market will be more profitable that paying down your mortgage, but there is a decent chance that you will actually lose money. So from a risk/reward standpoint, you're trading a risk-free "investment" (you will always "earn" your interest rate by paying down your principal and reducing your interest expense) for a risky investment.

Additionally, if you are relying on those gains to pay your mortgage (or have to tap into other reserves to cover losses), then you could be doubling your financial risk and making foreclosure a possibility.

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There is a term that applies to such considerations: ceteris paribus, which means “other things equal” (wikipedia).

That is, the comparison assumes that the only difference considered is the percentage return. Tax, tenor, trading costs etc are ignored.

In reality, of course, all those considerations need to be taken into account. Paying off a portion of a 5% loan can be taxed quite differently from paying the 5% interest and investing the (same) capital into a 5% investment. If the interest is not tax deductible but the 5% return is taxable, then this isn’t a good starting point for thinking about the 10% investment.

On the other hand, if one has some feel for the other factors, it might be possible to use a rule of thumb, such as using half the expected return, the other half going to taxes and fees. I’m just pulling figures out of thin air for argument’s sake. You might adjust for risk and other unequal factors in a similar way.

So to answer your question: no, you can’t just take compare the headline percentages if you want to work out whether your spare cash is better spent as an extra mortgage payment or invested.

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The other answers are good but don't really address your question about time:

Is it only equal if you intend to keep the money in the stock market for exactly the same amount of time as you would be paying off the loan?

I suggest thinking of where to put your money not as a one-time decision but as something that will be reevaluated periodically, say once a year. There are complicating factors, but all else being equal, more money (higher return) now is better than less. Later, you can make a new decision based on circumstances at the time, but you will reach that point with more money if you maximize your return this year.

The rough analogy is that you are putting your money to work. If you can invest at X% return, then each of your dollars earns a "wage" of X cents per year. Importantly, a dollar can only earn one investment "wage" at a time, though of course you can split your dollars between investments or move them back and forth (just as you personally can have two jobs, but there are only so many hours in the year and you generally can't spend the same hour working for company A and company B).

However long your money remains in a given investment, when you take it out you will be looking for the best new investment at that time (just as when you leave a job you will typically have, or find, another job). But you can never get back the time that has elapsed. Since time is the ultimate scarce resource, it makes sense that returns are compared as rates (per unit time) just as wages are.

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