Pay off or hold your mortgage

Let's say you win the lottery and you've got a net (after-tax) proceed of a cool $1 million. Would you pay off your mortgage?

For many of us, our mortgage is our biggest or second biggest debt (after medical school ± undergraduate tuition). Let’s assume that you’re only concerned about your mortgage obligation. How should you manage a cool $1 million coming your way?

Mortgage: $400,000 | Periods: 30 years | Periods remaining: 28 years | Compounding frequency: monthly |Stated annual interest rate: 2.75%

Nearly all of you would suggest investing the $1 million and paying down the mortgage liability as you normally would. In other words, maintain the status quo as it pertains to the mortgage. I concur.

What is more interesting, however, is this new scenario. Say you want to rid yourself of the mortgage liability. You want the financial freedom and security that comes with having no major debt. At the same time, you want to capitalize on any interest proceeds that you could accrue if you simply invested the $1 million (and did not pay off your mortgage). Can you do both? How can you do both?

These questions are what we will answer in this exposition.

The Scenarios

Let’s start by laying out the scenarios. We will have two:

• Scenario A: one in which you pay your mortgage over a 30 year period and you invest the $1 million dollars concurrently and

• Scenario A’: one in which you use the $1 million dollars to immediately pay off your mortgage and then invest the remaining money for 30 years

To make things simple, we will ignore that you’ve already paid 2 years of mortgage before receiving the $1 million dollars. We will start off at time = 0 where:

• you have a $400,000 mortgage and

• you just received a net $1 million dollars

Details of the mortgage

Let’s unpack the details of this mortgage. In Scenario A, you will pay the mortgage at the status-quo speed…that is, over 30 years. Since mortgages are compounded monthly*, we will perform some conversions to align the math.

*Mortgages are actually compounded daily, but for simplicity, we will stick with a monthly compounding period. The stated annual interest rate is 2.75% (as noted in all the way at the top of this post). An SAIR of 2.75% converts to a effective monthly interest rate of 0.229% over 360 (=30 x 12) compounding periods.

You’ll note that we have indicated (Line 3) that $1 million has been received at time = 0. This is a net amount received (again, that’s an assumption).

Cash inflow

Line 8 is where things start happening.

First, focus on Scenario A. Your monthly payments will be $1632.96 for 360 months. You begin your first payment at t=1 (which is what line 8 represents). At the same time, you have $1 million that you need to invest.

Things are different in Scenario A’. At t=1 (line 8), you pay off your mortgage completely using the $1 million you received. That leaves you with $600,000 remaining that needs to be invested.

Our goal in this exercise is the following: what type of investment must we select in Scenario A’ for our $600,000, over 30 years, in order to earn the same profit as we would in Scenario A (where we pay our mortgage over 30 years *and* invest $1 million in that same time period)?

We would realize 9.89% every year, on average, for the next 30 years. Let’s assume that we reinvest all our earnings back into the same instrument that returns 9.89% every year. And since we’ve determined that the compounding interval for the mortgage is monthly, let’s do the same for the investment instrument.

Cash outflows

Line 10 shows the effective monthly interest rate for the investment instrument that we will realize each month for 360 months (compounding periods).

End of mortgage period of scenario A

At t = 360 (30th year), your net profit (excluding everything else that happens in life) is $18,286,764 (= $19,198,656 - $911,891).

$18,286,764 is what you need to earn after 360 compounding periods if you invested $600,000 as in Scenario A’. What type of investment instrument would you need in order to make the profits in Scenario A equal to Scenario A’?

Such an effective monthly interest rate converts to a new stated annual interest rate (SAIR) of 11.44%.

Indifference point

Therefore, in order to *benefit* from paying off your mortgage immediately (i.e., in order to benefit from Scenario A’), you would need to find an instrument (or many instruments over time) that have an SAIR of greater than 11.44% each year for 30 consecutive years.

Is that doable? I suppose yes, when you consider that the lifetime average annual stock market return is near 10%. Since 10% is the average, there must be some investments in a given year that return more than 10%, and preferably more than 11.44% (which is your threshold to overcome a state of indifference).

Thanks and I hope this has been helpful. Leave your comments below.