Housing Math And Interest Rates…


Any real-estate transaction we as normal earthlings enter into usually involves some kind of a bank loan (mortgage) secured against that property. And of course it would be foolish not to borrow even if you have the cash to buy that real estate outright in today’s interest rate environment where a 30-year fixed-rate mortgage could be had at 4% a year. To understand the implication of what happens to your mortgage payment or potentially the value of your home when interest rates change, we need to go through some annuity basics and that is where this write-up comes in.

So what is an annuity? You put up some capital and loan it to say an insurance company and they in turn guarantee a fixed monthly or annual payment for the life of that annuity. A type of annuity that has no end term is called a perpetuity where the cash flow continues indefinitely i.e., in perpetuity. Each cash payment by that insurance company has two components; return of principal as well as the interest earned on the amount put up.  Annuities usually get a bad rap not because of the structure of the product itself but the way it is packaged and sold by an insurance company but we will leave that discussion for another day.

Getting back to the mortgage thing, when a bank loans you money, they are in fact buying the right to a stream of cash flow over the term of that mortgage. That is, the bank is in fact buying an annuity from you. The cash flow the bank collects are the monthly mortgage payments you make. And this annuity of course, does not continue forever.

Picture below shows the mechanics of how the monthly payment for a mortgage or cash flow for an annuity is derived. Same exact math but two different applications. And sorry for the unartistic and real feel to it.


There are two points marked on the timeline; the first is a perpetuity at time t = 0 and the second, a perpetuity at time t = 180 months (corresponds to a 15-year mortgage).

The present value of a perpetuity at t=0 is basically a ratio of the cash flow you derive divided by the interest rate. So if you need an annual cash flow of say $60,000 and the going interest rate is 4% per year then the value of that perpetuity is $60,000 / 0.04 = $1,500,000. You put up that much money today with say an insurance company at 4% annual interest rate and you should expect to receive that $60,000 forever.

The present value of a 15-year annuity or the amount of money borrowed to buy a home for a loan term of 15 years is the difference between the present value of two perpetuities; the first bought at time t=0 and the other bought at time t = 180 months. We are looking at present value because you are borrowing today and hence there is a discount factor in the denominator for the second term in the mortgage equation above.

Now that you have this twisted math cleared, let’s run some numbers through that equation. Let’s assume that you can afford $5,000 monthly mortgage payment and the 15-year quoted APR is 4%. What is the maximum house you can afford assuming you can come into the transaction with $200,000 down payment?


With that out of the way, let’s use this math to see the impact of interest rate changes on the amount of house you can afford at three different monthly mortgage payment amounts. The type of mortgage used this time is a fixed-rate, 30-year loan. Down payment is not part of this calculation and can be added to the numbers below for the total house value you can afford.


As is evident, for someone who can afford $5,000 per month in mortgage payment, a jump from 4% APR to 7% APR can cause his affordability to drop by about $300,000. To put it another way, that jump can technically cause the value of that home to decline by that much if that person’s monthly affordability does not change. Similar calculations are shown for $7,500 and $10,000 monthly amounts.

And 7% is not that unreal. You could have had that rate if you had bought a home around 2006-2007 timeframe but hey, at that time you also had access to no-money down, interest only loans making your ability to afford that house easier. Those things hopefully are not around anymore unless you want a repeat of the housing crash.


Bay area housing could be an exception to this logical interest rate math because people routinely come into windfall profits a la the Box IPO yesterday that allows them to buy down that loan amount with a larger down payment. But even in the bay area, one can’t escape the fact that in the long term, that interest rate logic will hold.

So there you have it. Now go have yourself a nice rest of the weekend.

Image credit – Bryan Alexander, Flickr