Personal Finance Equations You Should Know: the Amortization Equation

Filed in Mortgages , Personal Finance 16 comments

Amortization is probably one of the most ubiquitous financial concepts that encompasses our daily lives, but yet, few of us know what it means, how it works, much less, how the monthly payments on all our stuff is calculated.

By definition, amortization is:

The reduction of a debt over time by making periodic payments (usually monthly), where depending upon the length of time the loan is held, a portion of each payment is paid to interest and a portion is paid to reduce loan principal.

Mathematically, you can use several different variations of the same equation, but I feel the easiest amortization equation is one that you might remind me of something you would have seen in your high school algebra class.

If you’re algebra skills need some refreshing, nothing works better than a piece of paper, a pencil and a scientific calculator. If you’re more experienced with spreadsheet program like Microsoft Excel or Open Office Calc (FYI – Calc is free), then you can simply plug and chug the equation whatever variables you need.

Putting this equation to work, let’s assume you’re applying for a 30 year mortgage at a 6% interest rate totaling $200,000. The equation would then look like:

The resulting answer would equal a monthly mortgage payment of $1199.10, not including insurance, taxes, PMI and other fees.

Examining this a bit further, here is a closer look at what your amortized payment schedule (broken down in 5 year increments) would look like:


Hat tip to Bret Whissel and for his easy to use Amortization Calculator.

Seems like pretty boring stuff, right? Probably so.

Until you realize that you’ve been in the same house for 10 years and you’re still paying more in interest (e.g. bank profits) than towards your loan principal.

Negative Observations

By taking a look at the amortization schedule, several things should pop out at you.

  1. In your first mortgage payment, you’re paying slightly more than 5 fold of your payment in interest rather than repaying principal.
  2. It takes 222 payments, or 18 and half years, of paying the minimum monthly payment, for the amount of interest paid and the amount of principal paid in your monthly payment to roughly equal one another (see yellow highlights in amortization schedule).
  3. On a $200,000 mortgage, the cumulative total of all your monthly payments will total $431,676.00 (assuming you pay minimum monthly payments).
  4. On a $200,000 mortgage, you will pay $231,677.03 in interest payments alone (assuming you pay minimum monthly payments). You will get a tax deduction on interest paid, but the figures are still worth noting.
  5. If you only pay the minimum monthly payment and you repay the mortgage in full, you paid roughly 116% of the original mortgage value. That’s more than double the loan amount.

Naturally, these are some of the negative observations when considering to taking out a mortgage. However, like most things in life, there are two sides to every coin.

Positive Observations

There are definite positives to taking out a mortgage.

  1. With a fixed interest rate, future inflation is definitely your friend. Meaning that the cost of your living expenses (your mortgage) will decrease over time as your salary increases and your native currency slowly loses value.
  2. You have a place to live that, one day, you may own free and clear.
  3. Since most people don’t have $200,000 just lying around, taking out a mortgage to reduce the high barrier to entry to home ownership is the next best thing.
  4. The interest paid may be deductible on your income taxes.
  5. If you’re a real estate investor, you can use amortization as leverage. For example, if your monthly rental income is higher than your the monthly mortgage payment on your investment property, you’re generating positively monthly cash flow and an annual profit on your investment.

Final Thoughts

I hope this basic personal finance equation gives you some insight into why you’re paying the amount of money you’re paying on your largest assets every month.

Obviously, using the example of a traditional 30 year mortgage is, in my opinion, the best way to show the positive side of amortization since it applies to a large percentage of the general public.

However, the amortization equation also applies to other large price tag items that impact your monthly budget. Such as, your car payment, student loan(s), and if you’re carrying a balance on your credit card, how much interest you’re paying every month.

Depending upon your point of view when it comes to your debt, you can use the amortization equation as a positive motivational tool for wealth building (by leveraging low interest debt to generate positive cash flow or on must have items) or as a negative motivational tool to keep yourself out of debt by avoiding high interest debt (paying 25% interest on credit card debt to buy more stuff).

Either way, when taking on new debt in the future, you should now know how much of your money is actually going to repay what you borrowed, and how much is ends up as bank profits. As this example shows, the answers might surprise you!

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Posted by Matt SF   @   3 February 2010 16 comments
Tags : , , ,

16 Comments

Comments
Feb 4, 2010
12:14 am

Just makes me sick when I look at how much interest costs you over a long-term loan. For us it is a great motivator to pay off our mortgage as soon as possible!

[Reply]

Matt SF Reply:

Yeah, same here. It’s even more sickening when you amortize credit card debt on 20% interest rates. I nearly included it, but as you said, it’s actually very disheartening. Maybe I’ll include it in a different post in the future.

[Reply]

Feb 4, 2010
8:18 am

I lust for equations and calculators like these.
Give me more…give me more!!!!

[Reply]

Matt SF Reply:

Thanks man. I’ll see what I can dig up!

[Reply]

Evan Reply:

Best website for this stuff

http://www.dinkytown.com

STUPID name, but amazing amount of calculators (I think over 300)

[Reply]

Matt SF Reply:

Yeah, lots of finance related calculators there. Almost overwhelming, but still, lots of good resources.

From my experience, it’s best to make the math as simple as possible, while showing your work as much as possible, so others can substitute in their own numbers. That way, they don’t really have to worry about understanding how to do the algebra, but still have a chance at coming up with the correct answers.

[Reply]

Feb 4, 2010
9:08 am

I think that the Amortization Schedule is one of my favorite tools! It helps to see the true costs of a mortgage (or even a car note), and helps people understand that a primary home is not as much of an “sure investment” as we are lead to believe.

[Reply]

Matt SF Reply:

Thanks Khaleef. What bugged me most from doing the background research on this was the 18.5 years for the Interest vs. Principal payments to break even. I mistakenly thought they would break even at 15 years (or reasonably close), but didn’t think it would have a 3 1/2 year difference.

That’s good if you’re the lender, but bad if you’re the paying the note. Probably why I like the idea of P2P Investing and Lending Club so much! (Which I almost included!)

[Reply]

Feb 4, 2010
2:49 pm
#4 Faisal :

Can you provide an equation in which the elapsed number of years is related to the amount of principal in a monthly payment? The equations you provided are time independent ( time as in the left most column in the table ). Not sure if there is a closed form for that. But great work nontheless. THanks a million.

[Reply]

Matt SF Reply:

Thanks Faisal! I’m not an expert mathematician by any means, so not exactly sure how one could rearrange the equation to your specifications.

Just curious, but why would you want an amortization equation that’s time dependent? The key point of amortization (at least in how I see it) is that time is independent so the lender can count on a scheduled method of repayment. If I’m wrong in my assumption, please let me know and maybe we can dig a little deeper.

[Reply]

Feb 4, 2010
6:52 pm

A couple thoughts.
First – I love the equation. Like the quadratic equation, I once spent the time to derive the amortization equation from scratch. Don’t know that I could do it so quickly now.

The 18.5 years is interesting, I’m sure it goes further out as the interest rate rises.

But the fear of mortgage interest is a bit overblown. Think about this – 6% after taxes will cost the borrower say 4.5%. Inflation has run 3% or so the past couple decades. So your cost is about 1.5% to live in that house. On a $200K house, that’s $250/mo. I am not talking about the house going up in value. I am only referring to the inflation making the remaining principal worth 3% less each year. Fewer hours worked to make the payment.

Last – remember, one can always pay extra principal each month, no bi-weekly, no crazy programs, just throw in the extra money and mark the payment as “principal” or “extra toward principal.” This gets you a return of a guaranteed 6% if that’s your rate. Of course you should first pay any higher interest debt, such as CC debt, and capture the matched 401(k) money, but after those I have no objection to retiring the mortgage. “mort” is the Latin root for “death” or “dying” and I’ll not argue that a dead mortgage is a good mortgage.

[Reply]

Matt SF Reply:

Ugh, just had a flashback to Algebra II when you mentioned the quadratic equation. Don’t miss those days. Although, advanced trig was my favorite math related course, probably b/c it was an applied math, but I never had an interest in mathematical derivation. Was more interested in what I could use the equations for, versus how the math was derived.

And you’re correct that the lower the interest rate, the sooner you would break even on Interest Paid vs Principal Paid in each monthly payment…

5.0% = 194 months
6.0% = 222 months
7.0% = 242 months
8.0% = 257 months

… so it definitely pays to keep that credit report as pristine as possible.

Maybe one day I’ll create a spreadsheet (sure there are some already out there) that shows the power of paying an extra $X per month on a 30 year mortgage and how it affects the amortization rate and time to pay off the note.

[Reply]

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