Demystifying How Mortgages Are Calculated

October 3, 2012 8 Comments »
Demystifying How Mortgages Are Calculated

Calculating a Mortgage

Mortgages can be complicated business – fortunately there are a ton of great calculators out there that take the legwork out of all the tricky math.

But as your teachers probably told you in school time and time again, it’s good to actually know how things work too.

And hey, it’s never smart to rely too heavily on technology in case something goes wrong. Oh, and you can impress your friends too. Well, probably not, but let’s move on.

That brings us to how mortgage interest works. Ready to do some light algebra? Neither am I, but let’s try it anyway.

Let’s say you’ve got a 30-year fixed mortgage with a loan amount of $200,000 and your mortgage rate is 3.5%. Fairly common scenario these days.

The Interest Part Is Easy

A simple way to determine how much your interest payment is each month is to multiple your loan amount by the interest rate, and then divide by 12.

$200,000 x 0.035 / 12= $583.33

So in the scenario above, we’d come up with $583.33. This would be the interest portion of your monthly mortgage payment. Pretty basic stuff here. No algebra required!

And hey, had this article been written back in 2006, we’d be done because most people held interest-only loans and didn’t even make principal payments.

But times have changed, and now everyone wants to pay off their mortgages. Funny how things change…

How Do You Calculate the Entire Payment, Including Principal?

Most people probably don’t care nor want to know this second part, but I figured I’d share just to cover all the bases and blow your mind.

If you want to calculate your entire mortgage payment, including both the principal and interest portion, you need to use the very complex monthly mortgage payment formula below.

And yes, it’s heavy on the algebra, real heavy for those of us not so thrilled with math. Warning: It’ll hurt your head.

P = L[c(1 + c) n]/[(1 + c) n – 1]

P= monthly payment
L = loan amount = $200,000
C = periodic interest rate = 0.002917 (3.5%/12 months)
N = number of payments = 360 (30 years)

Lost yet? Don’t worry; I won’t make you do the math. Heck, I used an algebraic calculator to come up with the answer.

Let’s break it down:

P = 200,000[0.002917(1.002917)^360]/ [(1.002917)^360-1]

P = 200,000 x 0.00449045

P = $898.09

Still with me? Phew. So the total monthly mortgage payment is $898.09. And because we know the interest portion already ($583.33), the principal portion of the payment must be $314.76.

Of course, it’s not that simple, nothing ever is. This calculation above is only good for the very first payment based on the $200,000 loan amount and a 30-year amortization schedule.

When calculating the following month’s payment, you would have you use the new loan balance, which falls to $199,685.24 thanks to that $314.76 principal payment.

Fortunately, we already know the total payment amount, which is fixed for the full loan term, so we can just calculate interest and then the rest must be principal.

So in month two, we calculate interest by doing the following:

$199,685.24 x 0.035 / 12 = $582.42

Taking our fixed total monthly payment amount of $898.09 and subtracting $582.42, we come up with $315.67, which is the second principal payment.

As you can see, the interest portion of the payment dropped slightly, while the principal portion increased.

Over time, the interest portion of the mortgage payment falls, thanks to the smaller outstanding balance, and the principal portion of the payment rises.

In fact, at the end of the loan term, assuming you don’t refinance or prepay, the interest portion will account for just a few bucks of the total payment.

However, the total payment amount doesn’t change. It’s just how your payment is allocated over time that changes.

I’ve probably confused more people than intended here, but it’s always good to know how things work, even if you don’t actually do the math yourself.

For the record, I recommend using a mortgage calculator as opposed to trying to do all this math by hand. It’s interesting to know how it is calculated, but way too much work.

Read more: Are mortgage calculators actually accurate?

(photo: Jorge Franganillo)

8 Comments

  1. Mark M. December 20, 2013 at 7:21 am -

    So that’s how they come up with my mortgage payment…that’s some serious math dude. Hurt my head.

  2. Cheri January 31, 2014 at 12:09 am -

    Thank god for calculators! Cool to know though, but yeah, I’ll be using a mortgage calculator, not complex equations I’ll probably flub in the process.

  3. raj June 5, 2014 at 5:57 pm -

    Colin this is really wonderful post, You’r explanation is outstanding and very detailed. Good work Keep it up!

  4. raj June 5, 2014 at 6:02 pm -

    Question Colin, my mortgage broker says if the APR increases, my monthly payment (Interest Rate) decreases, is this true? All the mortgage rate show in the websites (for example 3.5%, 4.5% etc ) are APR based or the Interest rate.
    If the market hit 5% then will my monthly payment reduce?

  5. Colin Robertson June 6, 2014 at 12:00 am -

    Raj,

    They typically advertise both the actual interest rate and the APR, with the latter usually in parentheses. The APR is usually a random number like 3.56% or 4.21%, whereas actual rate is 3.25% or 4%. Not sure what the broker is trying to tell you, but if the interest rate is fixed, it’s not going to go up or down ever. Can you give me more details so I can figure out what they’re getting at.

  6. Captain Ned March 24, 2015 at 9:15 pm -

    Just get a free HP-12C app for your smartphone of choice. Makes it simple.

  7. Steve Attar AFN May 20, 2015 at 9:51 am -

    That is a very interesting piece. I will recommend this post on my blog.

    While calculators are good, it is good to understand how things actually work. You might end up impressing some people.

  8. Gert April 30, 2016 at 1:01 am -

    Ummm, wow Colin. That was definitely more in depth then I was expecting.

    I gotta Agree with Steve that I feel like to many people now a days are just to lazy and not willing to take the time to understand how it all works. Instead they turn to the calculator.

    It’s unfortunate but I suppose that’s how our society is transitioning, my kids don’t even have to use long division/math in school anymore it’s all done by calculators.

    Regardless great post and a solid resource for anyone wanting to learn more about it.

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