Interest Rates with The Mathematician

This week we have a guest post from none other than Mike Dabkowski, aka The Mathematician (to be clear, I assigned him this nickname). He is diving into some of the math and other factors behind interest rates and how those impact mortgage rates.

Don’t worry, though! I ruthlessly edited this article so even I could follow the content. I’m considering it payback for his dissatisfaction of my explanation of the term “median” back in this article on home buyers in 2021. You will see my comments in pink with a JD attached because I couldn’t go a whole week without adding in my own two cents. -Dabs

Finally, an article that will interest you. Sorry, I couldn’t resist. I’ve appeared on this blog in name only, and I am excited for the opportunity to write! Okay, here we go, let’s consider certain translation invariant probability measures on fields… Just kidding.

Aside from my research in partial differential equations, I teach courses in Mathematical Finance. The first topic we discuss in these classes is interest. Albert Einstein supposedly once said, 

“Compound Interest is the eighth wonder of the world. He who understands it, earns it. He who doesn’t, pays it.” 

Albert Einstein

Compound Interest

Compound interest apparently beats out general relativity, which is quite the feat. (JD: I don’t know what this means either, but keep reading.) The idea behind how interest works is really neat: you give me some money now and I will return the same amount plus a fraction of what you gave me at some time in the future. This fraction is called the interest rate (technically, the effective rate of interest for the time period in question).

If the interest rate is i and you start with an initial amount of P, then your yearly balances will be:

As we can see, the amount is growing, but the expression looks messy. This is where high school algebra saves the day! (JD: The Mathematician is the only one who wants to take us back there.)

When we factor this expression, we see that the amount you have in your account after n years is 

Amount in account after n years = P(1+i)ⁿ

In other words your balance is growing exponentially. It may not seem like your wealth is growing exponentially from day to day, but it does in fact grow exponentially over long periods of time.

Your growing balance is great news, but there is one catch – interest rates change. The interest rate your bank gives you is likely to change (although not much day to day) over the course of your life. Why is does this change happen? This is where economics comes in. (JD: Buckle up, guys.)

Disclaimer: I am not an economist, but I often play one in my course.

Business Cycles

Economies go through business cycles over historically twelve year periods: recession, recovery, expansion, repeat. No one likes periods of recession, so the Federal Reserve Bank and the Federal Government try to stimulate the economy out of recession.

We won’t debate the Austrian versus Keynesian schools economic thought here; cable news does a great job of that (sarcasm). We have seen over the course of the pandemic the government issue stimulus checks and the federal reserve keep interest rates low. The fiscal (government) and monetary (central bank) policies have given people more money and made it easier to borrow money; effectively people have more money to spend to push the economy into expansion. 

Making Bank with Interest Rates

What does this mean for mortgage lending? The answer isn’t completely simple, but let’s try to understand the basics. The Federal Reserve Bank controls the supply of money in the country. It lends money to banks at an interest rate (Federal Funds Rate) that it sets at the quarterly meetings that financiers watch carefully.

The Federal Funds Rate is the rate at which banks can borrow and lend money overnight between themselves. Banks want to make money (JP Morgan, Citigroup, and Bank of America want to make money?), so they take the money that they have and lend it to customers at a higher interest rate than the Federal Funds Rate. This is an example of exploiting arbitrage. (JD: For whatever reason, the word “arbitrage” conjures up images of men in camouflage planning a raid.)

Think about it this way:

There are two banks: Bank 1 and Bank 2. Bank 1 offers 1% interest and Bank 2 offers 2% interest. You could take out a loan from Bank 1, and invest in Bank 2. The interest from Bank 2 covers the interest from Bank 1 and more. You have made money from nothing. 

This method is how the Federal Reserve can increase the amount of money in the economy. If this is done in an overly aggressive manner, it can add too much money to the economy. Similarly federal stimulus programs can add too much money to the economy. If the money supply increases too rapidly, you can end up with price inflation, where goods and services cost more (there is more money to pay for them after all).

So, the compound interest in your bank account might not be able to keep up with the rate of inflation. If inflation gets out of control, the Federal Reserve will reverse course (sound familiar?) and raise the interest rate. 

Impact to Mortgage Rates

Now we see where the mortgage rates come into play. When the Federal Reserve raises the interbank interest rate, the banks won’t simply throw up their hands and say, “There goes the profit margin!”. (JD: I kinda wish that did happen, though.)

Banks will raise their rates in response and your 30-year mortgage will cost you more in a monthly payment because you have to pay more interest on the same amount of principal. Exactly how much the rates will increase is hard to say, but there are plenty of resources (this blog and its author in particular, but I may be biased) which will help you estimate the potential cost. 

It is important not to get too worked up about potential rises in the interest rates. Even though mortgage rates will increase, inflation will decrease, savings rates will increase and the economy will maintain its overall upwards trajectory as it goes through normal business cycles. Even more importantly, a thirty year fixed-rate mortgage guarantees you the same interest rate for thirty years.

A lot can change in thirty years, but your mortgage payment never will. This fact makes buying a home a tremendous way of building wealth. You might make a $1500 per month mortgage payment now, but you will also pay $1500 per month twenty years from now. Think about how far that $1500 will go in twenty years; probably not too far.

If the calculation follows the past twenty years, you would effectively be paying $915 in today’s dollars. You receive an effective future discount since you have a level mortgage payment and a fixed rate of interest. (You can play around with the numbers using this calculator.)

Thank you for reading my guest post. As you know, Dabs is here to help, and I am here to help Dabs! You can ask any math finance questions to Dabs, and she will forward them to me.

Disclaimer: Mike Dabkowski is not a certified financial advisor. This article is informational in nature and should not be construed as advice applying to any specific real-world situation.

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