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This will be basic for you pros, please forgive me.

I have been paying the minimum balance on my student loan all year, but the balance shown on the site has barely gone down. I assumed that this was because they make you pay the interest first, then the principal. However, when I looked at the payoff amount, it was about the same as the balance! How is this possible?

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36  
Sad... nobody should be allowed to graduate without knowing enough math to understand amortization tables. Further, never make "minimum payments" on anything unless you like giving away money. Figure out how long you want to pay, how much per month you can afford, and pay that much. – Carl Witthoft Mar 2 at 14:08
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@CarlWitthoft I don't know that I'd go so far as to say that for anything, but it is definitely good advice for stuff with higher interest rates. For a low-interest mortgage, though, you're likely better off paying the normal ("minimum") payment and investing the rest of what you could have afforded to pay against it. In addition to likely being able to out-earn the interest rate on the mortgage (especially since said interest is deductible,) this also leaves you with more liquid assets in case some emergency expense arises. – reirab Mar 2 at 17:54
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If they called it what it is -- the interest maximizing payment -- then you wouldn't see it as a good thing at all. That's why its called the minimum payment. – Eric Lippert Mar 3 at 0:20
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@horsehair The interest amortization problem does not quite fit in a differential equation, because changes happen at at discrete intervals rather than continuously, but you should be able to enhance your feeling for what is happening by studying the corresponding continuous problem. You have an outstanding balance that increases at a rate proportionate to its value, but has another term that decreases it at a fixed rate. The boundary conditions are that it is equal to the principal at t=0, and will be zero at the end of the loan term. – Patricia Shanahan Mar 3 at 12:36
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@CarlWitthoft: It's not a question of knowing how to do arithmetic; it's a question of knowing what arithmetic to do. If students who can do math are being graduated without understanding the most basic thing about their student loans - and they are - then how that situation came about is a good question. Why on earth would lenders, governments and for-profit schools collude to produce a system that makes financially ignorant, deeply-indebted wage slaves out of the smartest segment of the population? That's a tough one right there. – Eric Lippert Mar 3 at 15:48
up vote 118 down vote accepted

While it's common to think of it that way - pay off the interest first, then the principal - that's not actually how your payments work over time. It's true of any one payment, though.

Interest is earned over time. It might be added on daily, weekly, monthly, or any other frequency. For simplicity's sake, let's assume it is added 1/12*(apr) once per month.

So you have a 6% loan, starting at $10,000 principal balance. That's 0.5% per month (6%/12). So each month, you owe .005*(principal) in interest.

Your first month, then, you owe:

$10,000 Principal
$    50 Interest
-------
$10,050 Total

You will always pay at least the interest every month. Some payment plans are called "interest-only"; in those, you pay only that $50 per month, and the $10,000 never goes down for the interest-only period. (Of course, eventually you have to start paying principal...)

Any amount you pay over $50 per month, either as part of your payment or as extra (and extra designated to go to principal - an important distinction), will lower that principal. That's what actually pays off the loan.

Since payments to principal reduce the total amount you owe, they also reduce the interest due. So a $100 monthly payment, with $50 going to interest and $50 to principal, would then leave you with, next month:

$ 9,950.00 Principal
$    49.75 Interest
----------
$  9999.75 Total

You'd then pay $100 again, with now $49.75 paying interest and $50.25 paying principal.

The reason people think of this as paying interest first, is in particular with some mortgages and longer term repayment plans the far majority (commonly 80%, but in some cases higher) of each payment is allocated to paying the interest on the loan. The way these plans work is that you have a fixed monthly payment for, say, 30 years - but that's at first nearly entirely interest, because you don't have to pay much principal off to eventually get the loan paid off.

For example, to pay off that loan in 30 years, you'd only have to pay $60 per month - $50 for interest initially and $10 for principal. Since that principal will slowly rise over time (as interest slowly drops), you end up paying it off. (30 years is 360 payments, or about $21,600 - so you're going to pay a lot of interest this way, of course, over 100% interest over the life of the loan).

Since you're only paying $10 to principal each month to start with, if you add even a small amount to that payment, you pay it off far faster and pay far less interest. Add $15 to your payment - $75 instead of $60 - and now you're paying $25 to principal instead of $10, meaning you now pay it off in 18 years at that payment structure and pay only a bit over $16,000 in total - saving nearly half of the interest. Add $40 ($100 per month) and you now pay it off in closer to 10 years, and pay $14,000 in total. Even just add that for the first few years, and you'll dramatically increase your payoff rate.

I recommend using a mortgage payoff calculator, like this one which I set up with the above loan, to see how things work out. It shows you the amount going to principal and interest each year, and lets you alter the payments to see how they affect things.

Here are charts of the three options, so you can see visually how your payments break down. Like above, this assumes interest is calculated and capitalized monthly (which may be an oversimplification for your loan, but it doesn't change the numbers much).

$60  per month payment chart $75  per month payment chart $100 per month payment chart

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+1 for a lot of work. Great answer Joe. – Pete B. Mar 1 at 16:20
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@horsehair Well, they certainly go down more if you pay it off in one payment :) But even adding half again your monthly payment is enough to cut the length of the loan down drastically (by around half). Like I said, go to the mortgage calc and play around with payment amounts to see what different things do. – Joe Mar 1 at 16:36
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Gotta love how kids can graduate college and not understand this... so basic yet so powerful. – WernerCD Mar 1 at 19:51
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I will admit to not understanding this very well either when I had just graduated college. It wasn't for several years until I understood it well. We just don't teach practical home finances well in this country. And I graduated with an economics degree from arguably the top Econ school in the country... – Joe Mar 1 at 19:53
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@CarlWitthoft That is not what at least my economics classes were about. Much more theoretical, except in Econometrics (which is much more technical). That would be covered in finance, not economics. – Joe Mar 2 at 14:20

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