Pay extra mortgage vs Invest: The Definitive Calculator

[FIPurpose]

Investing or paying your mortgage is one of the most contentious and popular topics on this forum. I am not trying to play arbiter as to whether you should pay off your mortgage or invest. But I want to attempt to provide a formula so that people can do their own cost benefit analysis.

Assumptions:
1. You already have a mortgage. But this formula may help you decide between buying or renting.
2. This is your primary home.
3. Using US tax system... (cause that's all I know, but can likely be easily translated to other countries
4. Houses appreciate at about the rate of inflation. We'll go ahead and work with inflation adjusted numbers.

So here is the info that you'll need to calculate:

Variables:
1. Interest Rate (IR)
2. PMI (PMI)
3. Estimated Tax Bracket (TAX)

* Note on TAX - Since we're making trade-offs, use your top bracket. If your on the edge of 2 brackets, use the lower of the 2.


Constants:
1. Inflation adjusted Stock Returns = 7% (SR)
2. Annual mortgage payment as percentage of mortgage = 8% (AMP)

* MPP is to estimate your needed cash flow to pay your mortgage. At the beginning of your mortgage it will be slightly lower, and will increase with time as your payment remains constant, while your mortgage decreases.


Formulas:

We'll split this into different scenarios where we subtract stock returns to understand what the long term trade-off of this decision is.

Scenario 1: I put 20% down on my house, I still work and am saving for retirement, I don't itemize:

SR - IR

Example: 7% - 4.5% = 2.5% (You are losing a 2.5% opportunity cost on the value of your mortgage)


Scenario 2: I itemize:

If you regularly itemize, you're likely in a high tax state, so you'll likely itemize anyways with or without a mortgage. If you only just barely itemize, (IF you only avoid using standard deduction by a small amount, use a non-itemize scenario)

SR - IR + (IR * TAX)

Example: 7% - 4.5% + (4.5% * 20%) = 3.4%


Scenario 3: I am retired/ plan on retiring soon, I plan on withdrawing IRA money to pay mortgage

In this scenario, you will be withdrawing from your retirement account, to pay for your mortgage. So you have to bump your tax bracket to pay for your mortgage. Even if you're doing a pipeline, consider how much additional money you'll need to withdraw.
   
SR - IR - (APP * TAX)

Example: 7% - 4.5% - (8% * 12%) = 1.54%


Should I pay extra till I get rid of PMI?
I would say usually that answer is yes. I hear most PMI's add an equivalent .5% to your payment, but since they are removed after 20% equity, the payoff is usually worth it especially the closer you are to 20%. But if you just want a quick formula. For PMI add .5% to the interest in any above formula.


What does the result mean?:

You'll notice that all the examples show stocks always win. That's because we are comparing 2 different types of assets. But they do not have the same risk. To compare risk/reward trade-offs in the market, we can look a couple of values. The one I'll do here is called the Sharpe ratio, which compares the risk-reward of an investment compared to a risk-free investment (ie our mortgage).

To do this we need to know the standard deviation of the market. The best number I could find for our comparison is about 12%. Which means that most years, the stock market returns 7%±12%.

So to calculate the sharpe ratio you take your answer from above and divide by 12%.

Examples:
4% / 12% = .33
2.5% / 12% = .21
1.5% / 12% = .125

Conclusions:

There are no hard a fast rules about risk-reward trade-offs. A good rule of thumb is to make sure that your portfolio has an average Sharpe of 1. So for the question of buy investment or pay off mortgage: invest in the correct proportion.

Example:
VTSAX Sharpe - 1.2
Mortgage Payment Sharpe - .3

So invest 80% - VTSAX
Pay mortgage 20% - (Make sure to account of what you already pay in your regular payment not just extra)

[Boofinator]

Should I pay extra till I get rid of PMI?
I would say usually that answer is yes. I hear most PMI's add an equivalent .5% to your payment, but since they are removed after 20% equity, the payoff is usually worth it especially the closer you are to 20%. But if you just want a quick formula. For PMI add .5% to the interest in any above formula.

Apologies for revising a somewhat old thread, but there seems to be a large misconception of the financial hit that is PMI. The most common approach I see for modelling the cost of PMI is the one given by FIPurpose, which is to simply add the PMI percentage to the mortgage percentage, in which case the expected return of mortgage plus PMI would be significantly less than the historical returns from stock (and hence a sound financial decision). Unfortunately, this is a severe under-representation of the true cost. The correct formula to represent PMI cost is

Effective Interest Rate (PMI) = Interest Rate (Mortgage) + Interest Rate (PMI) * (Mortgage-to-Loan-%) / (Mortgage-to-Loan-% - 80%)

So to use an example, if someone has the option for a 4% mortgage and is debating whether to put down 20% versus 10% with 0.5% PMI, the effective interest rate for the extra 10% will be

4%+0.5%(90%/(90%-80%))=8.5%

Since this result is below the generally agreed upon historical expected return for stocks (let's say 7%), PMI in this case is not good debt (unlike the rest of the mortgage). In fact, in almost all cases, PMI is a bad bet (not that some people haven't gotten ahead with PMI and some luck).

Edit to add:

For those less mathematically inclined, let's use a specific example. Say we are in a high cost-of-living area and are purchasing a $1M house (with a 4% interest rate offer). We are offered 0.5% PMI up to 90% loan-to-value. In this case, if we take the PMI, for the option of borrowing an extra $100k we get to pay $8500 per year (8.5%). This can be broken up to the mortgage rate cost on the extra $100k ($100k*4%=$4k) plus the cost of PMI on the full loan value ($900k*0.5%=$4.5k).

What makes PMI even worse than the numbers indicated is that as the loan gets paid off, the PMI doesn't decrease. So for the same example, say you took the PMI route and are now down to the last $1000 to get under an 80% loan-to-value amount. At this point you are effectively paying $4540 per year (PMI plus mortgage rate on the remaining $1000) to borrow $1000, resulting in an effective interest rate of 454%! But wait, it gets worse. The bank isn't required to automatically shut off PMI until you hit 78% loan-to-value, at which point you've passed an infinite interest rate on borrowed money and are now simply paying the bank an ignorance fee.

To determine whether PMI is a good bet, you really need to take the expected returns from stock versus the expected savings and returns from the lower mortgage payment and no PMI over the duration of the 30 year mortgage. In most cases, no PMI wins handily, and this is before taking risk into account.

[Viking Thor]

That's a great point and often overlooked, avoiding PMI is more valuable than a lot of people think.

[FIPurpose]

You are right that PMI math can get very crazy, though I've personally never purchased a $1,000,000 home so it didn't really work this way for me.

I bought a condo sometime 4-5 years ago and only did 6.5% down. If I recall correctly, the PMI came out to .5% over the whole loan (which was ~125k).

So we were paying $625 per year for the benefit of not putting down an extra 17k or ~3.5% based on your calculation. But personally, I had to compare that with having to sell stock and the capital gains that would cause to raise the cash to cover.

 But as I said, that number rapidly increases since PMI doesn't decline the closer you get to 80% LMV. I think we waited for a year, saved up an extra 10k then paid it off all at once.

So when the actual numbers are smaller, the additional costs can be pretty insignificant, but as interest rates continue to rise and the more expensive of a house you buy, the more likely you'll want to pay PMI off as quickly as possible.

[Viking Thor]

The numbers work the same for large or small mortgages though, in your case the $625 was reasonable if looking at 17k additional  borrowed, but when the balance was down to 3k less than 20% and you were still paying 625 that would not be good. I know you said you paid it early to get rid of it but if not it would have been expensive by the end.

It's not related to size of mortgage as much as it is keeping the PMI until it would naturally expire, which makes it expensive.

[ditheca]

I got lucky with PMI by paying attention.  Bought a home two years ago with 3.5% down.  Booming market, the house has appreciated 27% since I purchased it.

Paid Wells Fargo $110 to reappraise it, and just got a letter saying I no longer have to pay PMI.  Think I saved about $4000 just by asking for this reassessment.