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When you take out a traditional mortgage the first payment you make to a bank pays off mostly interest and then later payments start paying off principal.

What is the reason for this?

Is it just a historical accounting/industry convention? Or is there some reason why it happens this way?

63

In other words, math. All the other answers are great, but I thought I might add something concrete to clarify slightly.

Consider a counterexample. Suppose I borrow $120000 at 1%/month interest (I know mortgages are usually priced with annual rates, but this will make the math simpler). Further suppose that I want to pay a fixed amount of principal each month, rather than a fixed payment. Let's say we want to pay off the loan in 10 years (120 months), so we have a fixed principal payment of $1000/month.

So what's the interest for month 1? One percent of $120K is $1200, so your total payment will be $2200. The second month, the interest will be on $119K, so your payment will be $2190. And so on, until the last month you will be paying $1010. So, the amount of interest you pay each month declines, as does your monthly payment.

But for most people, paying big payments at the beginning and smaller ones toward the end is completely backwards, since most of us earn more as we progress in our careers. Sixteen years after I took out a mortgage with a $1300/month payment, I find it fairly easy to pay, although it was a bit challenging to our cash flow initially.

The standard amortization requires a fixed payment each month, but the interest amount still has to decline as the principal declines. That means the amount of principal paid must increase as you go along.

  • 6
    I like the reversal of the assumption here. It's a nice counter-demonstration. – Chris Cudmore Jun 10 '13 at 20:57
  • Paying a big payment at the beginning isn't as counter-intuitive as you'd think - it's just usually done right at the start, and called a 'down payment'. ;) – Zibbobz May 10 '17 at 17:31
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    And with most loans, you can make these larger payments if you wish. All the extra will go to principal. – David Schwartz May 11 '17 at 13:55
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It's not correct. You pay both principal and interest on amortized loans. What happens is that you pay the interest accumulated on that principal during the period. As the time passes - some of the principal is paid off, allowing you to leave more for the principal because the interest becomes less. Thus the longer in the term - the quicker the growth of the principle payout portion out of the fixed payments.

That is how amortized loans work. Balloon loans work differently.

  • but why that particular amortization schedule? en.wikipedia.org/wiki/Amortization_schedule – bernie2436 May 23 '13 at 19:22
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    @akh2103 because that ensures that you pay the interest on the balance owed at any given time, while keeping the total payments fixed. – littleadv May 23 '13 at 19:24
  • I remember making car payments every 14 days. I recall it saved me a few months of payments. I take it I was making more against the principal this way? I read some contracts don't allow this type of payment scheme if you are to pay monthly. – Sun Sep 10 '14 at 3:46
16

Assume a month to month mortgage. This is a simplification, but it will illustrate the point.

Borrow $100,000 at .5% a month. Make a payment of $1000 each month.

So, for the first month, it will cost you $500 in interest to borrow the entire balance for one month. When you make your payment, $500 goes to interest, and 500 goes to principal.

Your new balance is $99,500. Now forget about the past, forget about the future.
What does it cost you to borrow this amount for one month? $497.5 -- Leaving $502.50 towards principal.

On month three, we want to borrow $98,997.50 for a month at a cost of $494.99.

And so on...

Nearer the end of the loan, when you have only 10,000 remaining, the interest portion will be nearer $100 a month, meaning you're paying principle much faster.

In essence, the interest portion of the mortgage payment is the cost of borrowing the outstanding balance for 1 month. As the balance is (should be!) decreasing, so will the interest portion of the payment.

In reality, the interest is calculated on the opening balance every six months. (Canadian Banks - Fixed Rate)

  • 8
    This is the actual, accurate answer to the question. A loan balance accumulates interest, and you should see every period (say, a month) as if you closed the previous period's loan and opened a new one with the new balance. In essence, there is no such pay to "principal" and "interest". There's just pay for the remaining loan balance, and you should make it bigger than the accrued interest, or you will pay forever! – ysap Dec 1 '13 at 23:49
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    Great explanation! this is easy to understand, without diving into math. The part "forget about the past, forget about the future" made it click for me – verboze Aug 4 '16 at 3:18
  • Good explanation, I upvoted, but I question your last sentence. Any mortgage I've ever had, interest is calculated based on the balance for that month. The amount of the mortgage that goes to interest goes down every month and the amount that goes to principle goes up every month. (Maybe I'm misunderstanding what you were trying to say.) – Jay Jun 23 '17 at 18:47
  • I'm Canadian, so that's how it works here. Unless you go variable, in which case it's done monthly. YMMV depending on location. – Chris Cudmore Jun 23 '17 at 19:14
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Banks don't make you pay different amount of principal at different stages of the mortgage. It's a consequence of how much principal is left.

The way it works is that you always pay off interest first, and then any excess goes to pay off the principal. However early in the mortgage there is more interest, and so less of the payments go toward principal. Later in the mortgage there is less interest, so more of the payments go to principal.

If you didn't do that - say if more of your payments went to pay down principal early on - then you would find that the interest wasn't being all paid off. That interest would be added to the principal, which means your principal wouldn't be decreasing by the full amount you paid off. In fact the effect would be exactly the same as if you had paid off interest first.

6

Most of the initial payments pay more interest as a percentage because the payments are fixed. This and all discounted cash flows are variations of geometric series.

In the case of a mortgage, formed by this formula

P = L[c(1 + c)^n]/[(1 + c)^n - 1]

, or any other discounted cash flow where the flows are held constant over time, P, the temporal payment, L, the total loan, c, the interest rate, and n, the number of payments to be made to satisfy the loan, are all held constant; therefore, the only variable allowed to vary is the percentage of P that flows to interest.

This varying percentage of P that is paid to interest declines asymptotically.

This particular formula is used to simply the payment process for the benefit of the borrower. If P were allowed to vary, cash flows would become more complex and less predictable for the purposes of budgeting.

The bank benefits from charging the correct interest rate and receiving payment.

  • 1
    +1 for the word "asymptotically." I used it once over Christmas break and got a blank stare. – JoeTaxpayer Jan 6 '14 at 3:25
  • Please note using EU APR the calculation is P = L*r/(1 - (1 + r)^-n) where the monthly rate r = (1 + APR/100)^(1/12) - 1. You can check this here. – Chris Degnen Jan 6 '14 at 14:31
4

Banks make you pay accrued interest on the current outstanding balance of the loan each month. They want their cost of capital; that's why they gave you the loan in the first place. On top of this, you will want to pay some additional money to reduce the principal, otherwise you're paying interest forever (this is basically what large companies do by issuing coupon bonds, but I digress). At the beginning of the loan, the balance is large and therefore so is the interest accrued each month. As the remainder of your payment begins to whittle away at the principal amount, the accrued interest decreases, meaning that the same payment can now pay more principal, which further decreases the interest accrued on the lower balance, and so on.

The math behind this has been a staple of the financial industry for decades. The equation to calculate a periodic payment P for a loan of balance B at a periodic compounding rate R over a number of periods T is known as the "reverse annuity formula" (because it basically works the same for the bank as it would for you if you had the same balance B in a retirement account, earning R each period, and needed to take out P each period for T periods) and is as follows:

P = [ B(1 + R)TR ] / [ (1 + R)T - 1 ]

You can also play "what-ifs" using what's called an "amortization table". This is very easy to understand; take your balance, add the amount of interest accrued each month based on the rate (1/12 of the APR), then subtract your scheduled payment, and the result is your new balance, on which you repeat the process the next month. Plugging this basic series of operations into rows of a spreadsheet allows you to count the number of payments by simply watching for when the balance drops below zero (you can easily set most spreadsheets up to subtract the lesser of the payment amount or the current balance plus interest, in which case when the balance and interest is less than the scheduled payment it will drop to zero and stay there). You can then "goal seek" to find a payment, or a rate, that will pay off a particular balance in a set number of payments.

2

It will be easier to understand if you treat interest the following way: it's the sum you pay the bank to get permission to not return the whole principal right now. Like you borrowed one million and then the bank comes after that million and you're allowed to pay some relatively small sum (like one thousand) so that the bank doesn't bother you for one month.

Now with the scheme above you never pay the loan back, you only pay interest. You're perhaps interested in paying back the principal so that you're debt free at some point. You have two basic options: equal payments or equal amortization.

With equal payments you pay the same amount of money every month and some of that sum goes to paying interest and the remaining goes to paying off the principal. Since you're slowly paying off the principal the latter decreases and so your interest payment decreases - because the lower the principal the less money you have to pay to the bank so that the bank "goes away for one more month". Note that because the interest gets smaller every month the sum that goes towards the principal increases.

With equal amortization you pay different sums every month - the part that goes towards the principal is always the same and the interest decreases for the same reason as with equal payments.

So it's only natural - you pay off part of the principal with each payment and so your debt gets smaller and so you have to pay less interest every month because the sum which the bank would demand if it wanted the debt paid in full gets smaller.

protected by Community Aug 4 '14 at 22:43

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