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Here is how a normal fixed-rate monthly amortization schedule is calculated:

First we determine the total monthly payment:

Payment = [P*(r / n)(1 + r / n)^nt] / [(1 + r / n)^n*t - 1]

where n = 12, P is the loan, t is the loan term, and r is the interest rate.

Then the interest required each month is simply calculated:

Interest = r*B/n

where B is the current remaining balance

And the principal paid each month is:

principal = Payment - interest

How do lenders change these calculations when we do biweekly payments? For example do they divide by 24 or by 26? Or perhaps do the extra two payments each year go entirely towards interest so it's equivalent to normal monthly payments + 1 annual principal only payment?

Spreadsheet showing amortization schedule, feel free to copy it. I showed two different amortization approaches the bank may take with a biweekly payment schedule: https://docs.google.com/spreadsheets/d/19HSKLYnqsSY3M6a5h5xs18u8PoILS21773j5JPYS_zc/edit?usp=sharing

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  • it doesn't, it's all just marketing
    – njzk2
    Jan 22 at 21:58
  • Please clarify - "bi-weekly" is twice a week or every two weeks (ie fortnightly) ? We had this discussion at work recently and different cultures read it in one of those ways.
    – Criggie
    Jan 23 at 22:39
  • Typically the way I've had it sold to me is they take the monthly payment, cut it in half and you pay that biweekly. In that case you're making 26 payments of 1/24 the yearly payment amount - which amounts to an additional 8% payment over the year and cuts the loan duration from 25 years to 20.5, thereby reducing interest paid. Is this what your bank is proposing?
    – Chris
    Jan 24 at 21:38

4 Answers 4

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It depends on how the loan is set up and the lender's policy towards early payments. If the loan is set up with a monthly payment then the payment amount can be computed based on that formula where n=12. If the loan is set up as a biweekly loan (which would be unusual), they would use 26 and the biweekly payment would be slightly smaller.

There are two ways that I have seen lenders apply biweekly payments towards a loan with a fixed monthly payment - both used half of the fixed monthly payment as the biweekly payment amount:

  1. They "hold" the payment and apply the full monthly payment once the remainder of the payment is received, so you don't really save anything. Any amount over the normal monthly payment is applied fully toward principal, reducing interest going forward.
  2. They calculate the accrued interest up do the date of the payment using a daily rate (typically annual rate/365), and the rest of that payment goes towards principal. The next biweekly payment will have slightly less interest than the first since the principal amount is slightly lower.

There may be other methods, but in either case the big benefit to you is not the interest savings, but the "Extra" monthly payment each year (26 half payments per year instead of 12 full payments). I have done biweekly payment before because it fit my income stream better, and the bank used the first method so there was no interest savings - the big benefit was the "extra" 13th payment. I could have simply done the extra payment manually, but it was nice to have the half payment line up with my income frequency.

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  • Could you take a look at Sheet 1 and 2 in the spreadsheet? I took two approaches to how you could potentially put together an amortization schedule for a biweekly pay structure. First approach is where you calculate a new payment using n = 26, the second is when the monthly payment is just the monthly payment divided by 2. The second leads to huge gains in interest over the life of the loan, while the first approach saves just a few hundred dollars.
    – Nova
    Jan 22 at 22:30
  • 1
    My experience has always been that the monthly payment does not change, you just pay half of it bi-weekly. I have never seen a bank bank reduce the payment for a bi-weekly amortization. Since you make 1 extra half-payments a year, the total length of the loan and the interest is cut significantly versus reducing the payment amount and keeping the length of the loan the same. You could do it by just making extra half-payments yourself, so it depends on how the bank applies those payments.
    – D Stanley
    Jan 22 at 22:47
  • 4
    @Nelson Your interest is what it is - principal balance multiplied by interest rate. You pay exactly the interest that you accrued, no more and no less.
    – void_ptr
    Jan 23 at 4:44
  • 2
    @StrangerToKindness If the loan is reamortized to biweekly, meaning that the payment is recomputed so you pay it off over the same time but make 26 payments per year, then the payment will be slightly lower and you'll save very little interest. If instead you pay half of the monthly payment biweekly you pay it off faster because you pay 26 half payments instead of 12 full payments. In my experience the latter is much more common,
    – D Stanley
    Jan 23 at 14:24
  • 1
    @Nelson If your half-payment is less than the total interest for the month, then yes it usually goes to the interest part first (the bank gets the money you owe them before they let you pay down principal). It doesn't make much of a difference in the end. If your half payment was more than the monthly interest then part of it would go to principal.
    – D Stanley
    Jan 23 at 14:27
5

Payment = [P*(r / n)(1 + r / n)^nt] / [(1 + r / n)^n*t - 1]

where n = 12, P is the loan, and r is the interest rate.

If the periodic rate is given as r / n with n = 12 months, this implies that the quoted rate is a nominal annual rate compounded monthly.

See Calculating effective and nominal interest rates (Wikipedia)

The effective annual rate can be calculated like so

effr = (1 + r/n)^n - 1

For a bi-weekly calculation using the OP's Payment formula a nominal rate compounded bi-weekly should be used. E.g. with n = 26 fortnights

r = 26 ((1 + effr)^(1/26) - 1)

Example: with nominal annual r = 10% compounded monthly

P = 1000   r = 0.10   t = 10   n = 12

Payment = (P*(r/n) (1 + r/n)^(n*t))/((1 + r/n)^(n*t) - 1) = 13.2151

total interest = Payment*n*t - P = 585.809

Converting the given rate to a nominal annual rate compounded bi-weekly

effr = (1 + r/n)^n - 1 = 10.4713% per annum

r = 26 ((1 + effr)^(1/26) - 1) = 9.97766% compounded bi-weekly

Bi-weekly calculation

P = 1000   r = 0.0997766   t = 10   n = 26

Payment = (P*(r/n) (1 + r/n)^(n*t))/((1 + r/n)^(n*t) - 1) = 6.08564

total interest = Payment*n*t - P = 582.266

Without conversion, incorrectly keeping r = 0.10, total interest would be 583.746

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A mortgage is normally calculated based on monthly payments, that is, 12 payments a year. If you get biweekly payments, that is 26 payments a year. Usually each payment is half the monthly payment.

So say your monthly payment comes out to, just to use round numbers, $1000. In a year you will pay 12 x $1000 or $12,000. If you get biweekly payments, you will make 26 payments x $500, or $13,000. That extra $1000 per year reduces your balance faster. Most mortgages in the US have interest calculated each month based on the balance that month, so reducing the balance faster means that your interest payment each month will be a bit smaller. Not a lot at first, but it will add up over time.

Mortgages I've had have all, I think, calculated interest based on the balance at the end of the month. But if you had a mortgage that calculated based on average daily balance, like a credit card, then paying half in the middle of the month would reduce your balance for the second half of the month, and thus your average daily balance. But I don't think most mortgages in the US work this way. And in any case it wouldn't make much difference even if they did. If you're paying, say, 6%, paying $500 two weeks early would reduce your average daily balance by $250, 6% per year is 1/2% per month, so you'd save $1.25 a month on interest.

But paying an extra $1,000 a year means that after 1 year you're interest would be reduced by 6% of $1000 = $60. The second year you'd save $120 from the extra principal, plus 6% of the $60 that now went to principal instead of interest for another $3.60, total $123.60. That $3.60 may not sound like a lot but it will grow exponentially year after year.

The real magic isn't that you are paying more often, it's that you are paying a little extra each month. The biweekly versus monthly is just a psychological tool to make the budgeting easier.

Big warning: I've had several mortgages where I got a letter from the bank offering to set up biweekly payments, with an up front payment to do this and a monthly service charge. For most US mortgages, there is absolutely no reason to pay such extra fees. You can accomplish exactly the same thing by just paying a little extra each month. If it helps, create a new account and every 2 weeks put half your mortgage payment into that account. Then when the mortgage is due pay from that account. This will accomplish exactly the same thing without incurring any extra fees. Or if the bank charges you to create another account, just use your regular account but write on a piece of paper that you are setting this money aside. (In either case, don't give in to any temptation to spend this money on something else because "I'll replace it later". You won't.)

1

Math guy here.

You've confused yourself with that complicated formula, because it is a derivation of the central formula, and the derivation requires certain assumptions to be made.

The core formula is that for each interest calculation period, the added interest in that period is (interest rate / # of periods per year +1) * remaining principal. We add the interest, subtract the payment that was made, and that is the new principal. Then we repeat for each period.

This is the operating principle of all loans. The period is monthly, daily, or whatever suits the lender.

But then, people want loans which definitely end at a particular date. This is purely an artifice: the above formula doesn't care when the loan ends. You can do this in Excel easily enough: just make a spreadsheet with 360 lines in it (30 years x 12 months). And then you could sit there trying numbers over and over until you find the monthly payment that results in the principal being zero on the month you want to be the last.

However, thanks to calculus, you can take the original formula and deriveone that will spit out the right payment for any given amount, term in years, and payments per year. That's exactly the formula you quote in your question. That's not the absolute truth of interest calculation; it's just a derivation based on those assumptions.

So you are changing the assumptions, which breaks the formula you quoted.

So you need to go back and look at the contract. People are emotionally attached to the idea of the loan hitting zero on the last month, so lenders might do a little fudging, and treat a payment that arrived on the 27th or the 3rd as if it arrived on the 1st. Otherwise, the slightly changed interest will throw the payment schedule off. So you need to read your contract and see how they actually do that.

If they are compounding daily or continuously, you can simply reapply your formula with (days/14) number of payment periods over the loan duration. "14 days" does not go evenly into 365.24 days per year, so saying 26 periods per year will not give a correct answer.

If they are compounding monthly, then you need to model that: on 2 months per most years (and 3 months in a few years), your payment will be 150% of the normal payment. And the formula you put in your question is absolutely not designed to calculate that!

All this to say, when you do weird and complicated stuff with loads, the "standard" payment formula goes out the window, and you must find an alternate way, such as that 360-row Excel spreadsheet.

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  • I agree with this answer the most. Read the fine print of loan agreements I'm sure you'll find statements to the effect of "interest is compounded daily" or that the monthly payment schedule is "an estimate". The benefit of biweekly payments is interest only compounds only 14 days instead of about 30 days. This 16 day difference is huge.
    – gns100
    Jan 24 at 16:52
  • @gns100 Mortgage interest accrues daily but does not compound daily. And, lenders hold incomplete payment in a non-interest bearing buffer account anyway. It is not applied to the mortgage early. The only savings come from the 2 months in which 3 half-payments are made. Source: I actually read the fine print.
    – void_ptr
    Jan 24 at 17:41
  • @void_ptr, I should have put that as a caveat that my experience is related to the 10 or so mortgages/refinances I've done myself. I am not a mortgage broker/lender so have no knowledge how common the documentation I've seen really are. Thanks!
    – gns100
    Jan 24 at 20:56

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