23

Most mortgages use an amortization schedule that have you pay more interest than principal at the beginning and then less interest and more principal as time goes one.

You can see this using this online schedule calculator: https://www.bankrate.com/calculators/mortgages/amortization-calculator.aspx

example of a portion of an amortization schedule

The example above is a 30 years fix mortgage in the US with a loan amount of $165,000 and a rate of 4.5%. (I put US to be specific, but I think this question is not specific to the US)

Let's say I pay off the mortgage at the end of year 1, I would have paid a total of $7,979.31 of interest. Let's say there is no fee at all for paying off early to make things easier. Then the annual rate I would have paid for this loan would be 7,979.31/165,000 = 4.84%

But if I pay off the loan at end of year 20, I would have paid $116,617.93 of interest, so the annual rate of the loan would be (1+116,617.93/165,000)^(1/20)-1 = 2.71%

Does that mean that paying a mortgage early causes you to effectively pay a higher interest rate? Or am I miscalculating something?

  • 49
    The way you calculate "annual rate" is not how you calculate annual rate. The formula is wrong and so the result is meaningless. – void_ptr Oct 31 at 3:35
  • 23
    The fact that you end up at a different annual rate than the annual rate, should tell you something ;-) You are not calculating that correctly. – Stian Yttervik Oct 31 at 7:32
  • 3
    7,979.31/165,000 = 4.84% this is wrong. You can't use the original balance in the denominator. The interest is calculated monthly, you need to either calculate monthly or take the average balance for the year. – dwizum Oct 31 at 17:06
  • 9
    Even if you were right, you're not accounting for all the years that you're not paying any interest. So it's high interest for 1 year, zero for 19 years, versus lower interest for 20 years. – Barmar Oct 31 at 19:43
  • 2
    @DavidSchwartz I thought it was normal, yes, but at least some 10 years ago banks here wanted you to pay them part of interest even with early finishing, until they were prohibited to do so by regulator. – Gnudiff Nov 1 at 9:58
65

I see your confusion:

You're looking at the interest as if it applied to the whole loan balance. That's not what actually happens--the interest rate remains the same but as you pay down the loan the amount that you are paying interest on drops.

That's what that chart really is showing--as the amount owed drops the interest drops and the amount that goes to the principal goes up. This is why financial advice strongly stresses paying extra on your mortgage. One extra payment at the start can (depending on the interest rate) cut as much as a year off the duration of the mortgage.

  • 44
    If you're going to make extra payments on your mortgage, it's best to do it early, since you'll reduce the interest paid over the rest of the loan period. But since mortgage interest rates (~4%) are generally lower than 401k or stock market return rates (~5-8%), you're often better off taking the money you would have put toward the mortgage and investing it instead. Of course, the mortgage payment is a guaranteed 4% return, and there's something to be said for owning your home outright, but it's something to consider. – Nuclear Wang Oct 31 at 15:44
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    There is also the tax treatment - in the UK at least the reduced mortgage payments is in effect a 4% tax free first free investment vs the taxable stock market returns. The main issue is liquidity. – Duke Bouvier Oct 31 at 18:42
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    @NuclearWang it also changes your risk profile. In a lot of places (but certainly not all) mortgages are an obligation of the property, not the borrower. If I owe $10,000,000 on a property in California, and it's suddenly destroyed (slides into the ocean, whatever), I call the bank and tell them they now have a ten million dollar problem - but I don't. (I have a note on my credit report saying I failed to pay a large mortgage, but given a choice between that and owing $10 million, I know which one I'd pick.) – James Moore Oct 31 at 19:09
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    @JamesMoore free legal advice from non-lawyers like me is worth what you pay for it. The difference is value is called a deficiency and the bank can get a judgement against you to cover the amount not covered by the sale of the foreclosed property. Even if you have a non-recourse loan you might have signed papers as part of getting the loan where you agreed to pay any deficiencies that arise in the future. This is the same deal as people walking away from underwater mortgages. You can still owe lots of money on top of the bad credit. – Ukko Nov 1 at 1:46
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    The extent to which foreclosing on a mortgage is being trivialized in this comment thread is deeply troubling. – dwizum Nov 1 at 13:30
48

Your first sentence is written in a way that highlights a common misconception about how term loans work:

Most mortgages use an amortization schedule that have you pay more interest than principal at the beginning

Note the banks are not doing anything mischievous or playing with numbers in such a way to cause you to pay more interest up front than you otherwise would. The amount of interest charged per month is exactly the correct amount based on your loan balance at that time. If you take your interest rate, which is 4.5%, and divide it by 12 to get your interest rate per month, and then multiply that by your balance, you'll get your interest charge for that month:

0.045/12 * 165,000 = $618.75

Which is the exact amount in the amortization table. You can repeat for the second month:

0.045/12 * 164,782.72 = $617.935

And so on for every month. In other words, you are always paying exactly 4.5% per year or approximately 4.5%/12 every month. Note this means if you ever make an additional principal payment, then from that day on you'll have changed the amortization table- you'll pay less interest and gain more principal than what you see on the original table. This is why even small amounts of additional principal payments can reduce your total loan term by months or even years.

As others have pointed out, the reason you calculated higher interest than 4.5% in the first year is because you counted 13 months instead of 12. If you use 12 months you would get:

7370.55 / 165000 = 4.467%

The reason that's lower than your stated APR is because your balance decreases each month and so you pay slightly less interest each month. If you were to use the average balance over the course of the year as the denominator instead of $165K you'd get closer to exactly 4.5%. The same is true if you take the average balance over 20 years and calculate the interest paid- it will be close to exactly 4.5%.

  • 1
    Right, so if you paid off the whole mortage after one payment, you would have paid interest for one month on the whole original balance. But if you never prepaid and went all the way to end, some months you'd only be paying interest on a very tiny balance. So the interest per month is lower in the second case, but the total amount of interest is much much more. – Kate Gregory Nov 2 at 13:33
17

It is not so much that you are miscalculating, but that what you are calculating is not meaningful in the context you're trying to use it ("the annual rate of the loan"). You are essentially trying to compare apples with oranges.

You have taken the total interest paid (over one or 20 years), expressed this as a fraction of the original loan amount, and then "un-compunded" that rate to get a "yearly percentage".

(Your figures are 4.84% for 1 year; 2.71% for 20 years. As noted below, these are "slightly wrong" because (a) you took the interest after an extra month, and (b) you should have used the number of months, not years, in the un-compounding process. My corresponding figures are 4.38% and 2.67%.)

The question then is: what do these rates represent?

They are the rates of return necessary on a single, lump-sum deposit (where interest is compounded monthly) in order to earn the equivalent of the interest paid on the mortgage over the same period.

  • So, $165,000 deposited for one year at an annual rate of 4.38%, where interest is compounded monthly, would grow by $7,373 after one year.

  • Similarly, $165,000 deposited for 20 years at an annual rate of 2.67%, where interest is compounded monthly, would grow by $116,280 after twenty years.

(These figures would also apply to an extreme form of a "balloon mortgage" where no repayments were made over the term of the loan, and the entire amount owing (accumulated interest + original principal) was repaid as a lump-sum at the end of the period.)

Yes, the 2.67% is a lot lower than 4.38%, but this just shows the effect of compound-interest over a longer period.

Using the same repayment calculator as you1 and a fixed-deposit calculator I found2, I prepared the following table (which also shows the figures for the full 30-year term):

4.5% Loan repaid after           1 Year      20 Years      30 Years
                                -------      --------      --------
Months:                              12           240           360
Loan amount:                    165,000       165,000       165,000
Interest paid (a):                7,370       116,315       135,971
Total paid:                     172,370       281,315       300,971
Interest as % of loan (b):         4.47%        70.49%        82.41%
Effective annual rate (c):         4.38%         2.67%         2.01%
Return on investment (d):       172,373       281,280       301,400

Notes:

(a) Interest paid is taken from the loan calculator page, using the October 20xx line in each case. Note: I originally mis-transcribed the 1-year figure as 7,360 (giving very slightly different % rates).

(b) This is the interest divided by the loan amount and expressed as a percentage.

(c) This is the figure from (b) converted to an effective annual rate, using

          ( ( 1 + interest/loan ) ^ ( 1 / months ) - 1 ) * 12

This is similar to the calculation you made to get 2.71%, except (i) I use the number of months when taking the nth-root, and then multiply by 12; and (ii) I used the October figures where you used the December figures.

(d) These figures are from the fixed-deposit calculator, using the rate as shown in the line above (e.g. "2.67"). If the exact, calculated EAR is used, the calculator's figures agree to the dollar with the interest paid figures from (a).

And the final word: it really does not matter what rate you calculate: the bottom line is that by paying the mortgage off early, you're only paying a little over $7k instead of over $116k in interest!


1 https://www.bankrate.com/calculators/mortgages/loan-calculator.aspx.

2 https://www.calculator.com.my/fd-savings. The fixed-deposit calculator happens to show amounts in Malaysian Ringgits (RM), but that does not matter for our purposes: we're only interested in the numbers.

  • 1
    Are you agreeing with OP's conclusion or (remarkably) just showing how their math is wrong even though you've done a great job duplicating it? I see no disclaimer that the 2% result ignores the correct way to calculate time value of money, unless that was the apple/orange remark. – JTP - Apologise to Monica Oct 31 at 9:41
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    I was intending to point out that their 2.71 (my 2.67) is "correct", in a sense, but is measuring something completely different than "the annual rate of a loan". I'll see if I can reword the opening to explain better. – TripeHound Oct 31 at 9:52
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    Yes. The 2.01 is a deposit, lump sum, and a 30 year return. I get it, but a bit more detail in the answer would make it perfect. – JTP - Apologise to Monica Oct 31 at 10:10
5

As long as there is no prepayment penalty, and there are no tricks like counting the effect on the rate due to “points” being charged, the rate is the rate. 4.5% is the same through the day you pay it off, no matter the time. I don’t know how you are getting the other numbers.

Each month, the interest is calculated by multiplying the remaining balance by the interest rate (as a monthly number, or annual divided by 12). The math you went through was not correct. When I set up a spreadsheet to calculate my mortgage balance each month, I am correct, to the penny, as long as I respect rounding that fraction of a cent.

  • The rate only holds if you keep the mortgage for 30 years. How can the rate be the rate when the ratio of interest you pay each month changes over time? Or am I missing something? – qwertzguy Oct 31 at 6:08
  • @qwertzguy No, that rate holds every year. That is why it is the annual rate. – Stian Yttervik Oct 31 at 7:33
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    I thought mortgages (in the US) are only simple interest - no compounding and so the effective rate should be the same as the APR... – TTT Oct 31 at 16:04
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    @TTT: Splitting the payment into principal and interest components before adjusting the balance used to compute interest (B2 = B1 - (P - i*B1)) is exactly equivalent to deducting the whole payment and then adding the interest (B2 = B1 - P + i*B1). That you pay interest on the remaining principal not on the interest does not matter because any dollar is worth the same as any other dollar. – Ben Voigt Oct 31 at 18:46
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    @TTT - I removed the line about "compounding". I did some math to write up an explanation of why I was right to use it, but proved myself wrong. Interesting. – JTP - Apologise to Monica Oct 31 at 22:09
1

If the terms of the mortgage were that you didn't have to make any payments for the duration of the loan, except for a balloon payment of the interest plus principle ($165,000 + $116,617.93 = $281,717.93) after 20 years, then taking ($281,717.93/$165,000)^(1/20) to calculate the effective interest rate would be a valid calculation. But you don't get to put off making payments until the end, you have to make them throughout the course of the loan.

When you pay early, you pay less interest in absolute dollar amount. The larger absolute dollar amount of the interest for the full length loan is compensated for by a larger time period. You are treating that longer time period as being 20 years, but that is too long of a time period, and so you are overcompensating and calculating an effective interest rate that is too low. While the overall duration is indeed 20 years, each individual payment has a term less than that. So to calculate the effective interest rate, you need the standard formula that takes into account the fact that there is no fixed term that applies to all the payments.

-1

You are not comparing apples to apples. You are correct. The APR for year 1 might be 4.84 if you pay it off in one year. But you are comparing it to a 20 year loan. Therefore you have to average that out; 4.84 the first year and 0 the remaining 19 in order to have a meaningful comparison. = .24% APR on 20 year "loan"

  • 1
    The APR for year 1 might be 4.84 if you pay it off in one year. - but it's not. No matter how many years you take to pay it off, the APR for the first year is not that number. The OP did the math wrong. There's no point in basing an answer on an assumption that the OP might be correct. – dwizum Oct 31 at 20:13
  • 1
    @dwizum I understood the OP had the math wrong. But he also had the concept wrong and no one else brought that up. I was trying to explain that 19 years of no interest is better even if the first year his higher than OP believed it should be. – Jammin4CO Nov 1 at 14:11

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