Mortgage Solution

The solution to yesterday’s mortgage puzzle:

Commenters pointed to several reasons why biweekly payments of (say) $500 will pay your mortgage off so much faster than monthly payments of $1000, but of these by far the most important is that biweekly payments of $500 add up to 500 x 26 = 13,000 dollars, whereas monthly payments of $1000 add up to 1000 x 12 = 12,000 dollars. With the biweekly payments, you make the equivalent of 13 monthly payments every year.

In other words, the key observation is that two weeks is not half a month.

My colleague Michael Wolkoff posed this puzzle to me many years ago, and I’m embarrassed to admit I failed to solve it before Michael gave me the solution. I was reminded of it yesterday when I got a biweekly-plan offer in the mail.

Click here to comment or read others’ comments.


16 Responses to “Mortgage Solution”

  1. 1 1 Andy

    Are the rates the same? If they are and you are allowed to make over payments, I see no reason why you should accept it since you can create the bi-weekly payment plan if you want to but you can also choose to stick with the monthly if for some reason you would prefer that. Not that you have said you would accept, just pointing it out.

  2. 2 2 Jeremy N

    This is not solved to my satisfaction. So, yes, you make one extra payment per year, but that should only shorten the length of the term by 360/12-360/13 = 2.3 years. Your term was shortened by 6.4 years. The remaining 4.1 years must come from interest reductions (some because of the earlier half-payments, some because of the extra payments).

    Have I erred?

  3. 3 3 Ken B

    @Jeremy N: I think you have erred. Payments are made to apply to interest and principal. You need to pay principle to reduce the loan. So an extra payment of principal each year, compounded, makes a big difference. The key word is compounded.
    As a secondary effect some of these payments come early each month, thus reducing accrued debt.
    I have not done the math to see if 6.4 is the reduction that the extra payments cause to pop out of the formula. Indeed I do not know what formula Steve’s bank uses.

    As I noted in comment 2, the bank is earning extra interest on roughly half the monthly payments so might have an incentive to further reduce the amortization period to make the offer more attractive. That would depend on a lot of things.

    The way to see is to compare twice monthly with biweekly payments. Then only the extra interest matters.

  4. 4 4 Ken

    Ken B,

    Indeed I do not know what formula Steve’s bank uses.

    They all use the same formula. You can use that same formula to compute the monthly payment of a standard 30 year fixed mortgage for any rate and any starting loan amount. Once you’ve figured what that is, you can use all sorts of tools (like excel, pen and paper, a simple program, etc.) to figure out how long it would take to pay off a mortgage by making an extra payment a year. You can even use the sequence in the above article to make it more general, noting that no interest is paid every 13th payment and noting to divide the ending payment, n, by 13, rather than twelve (because in the biweekly formula you make 13 payments per year, instead of the normal 12).

  5. 5 5 Andy

    The formula is =PMT(rate, nr of periods, loan amount) in excel (at least in the UK). It doesn’t break it out into how much is interest and how much is repayment as it is not constant as mentioned in #3. It is the same as listing the periods and solving for the amount paid per period (interest and repayment) such that the last payment fully repays the mortgage.

  6. 6 6 Dave

    Steve – can you please post or link something about how Hurricane Sandy is not good for the economy?

    I keep hearing people say that at least this will create a whole bunch of construction jobs that we desperately need.

  7. 7 7 Ken B

    @Ken: Thanks. I knew there was a formula, and understood it once, but haven’t looked at it in 20 years.

  8. 8 8 Ken B
  9. 9 9 Manfred

    The best source is Frederic Bastiat’s opus “That Which is Seen, and That Which is Unseen”. It is available free in several websites.

    Also look at Donald Boudreaux’s entry in Cafe Hayek:

  10. 10 10 Steve Landsburg


    Steve – can you please post or link something about how Hurricane Sandy is not good for the economy?

    I’ve written so many times on this topic that I fear I have nothing new to say.

    If you find yourself in an argument, try asking your opponent whether he thinks it’s good for the ants when you put a stick down an anthill and shake it all around to destroy their infrastructure. Go ahead and acknowledge that it will put a lot of ants to work.

  11. 11 11 Jeremy N

    @Ken B
    Thanks for the post, I agree with almost everything you said, but the way it’s configured, the loanee is not making an extra payment on principal every year. He’s making an extra payment, some of which is principal, some is interest, according to the amortization schedule.

    The extra principal he pays down reduces the total amount of interest he’ll pay and reduces the duration of the loan. That’s my point. Most of the reduction in the loan duration is because of reduced interest not (directly) because of the extra payment.

  12. 12 12 Ken


    Bastiat still has the clearest explanation as to why the destruction of wealth is just that: the destruction of wealth.

  13. 13 13 Al V.

    Using the Excel financial functions, an annual interest rate of 4.42% requires a monthly $1,000 payment on a $200,000 loan, paid monthly. The total amount paid is $360,000 (obviously). If I pay $500 bimonthly, Excel says the number of periods required to pay off the loan is 671, or 25 years and 40 weeks. The total amount paid would be $335,300.

    Of course, as noted by Andy, if you just make a monthly payment of $1,083. you accomplish the same thing. Which leads me to ask, what’s in it for Steve’s bank? Why are they making this suggestion, since he can accomplish the same thing on his own?

  14. 14 14 Paul T

    Dave: “can you please post or link something about how Hurricane Sandy is not good for the economy?”
    SL: “I’ve written so many times on this topic that I fear I have nothing new to say.”

    “Sir, I’ve given you an explanation. I am not obliged to find you an understanding.”
    - Samuel Johnson

    This is one of those things, if someone doesn’t get it right off, he never will.

    Ask, how can calamity possibly be positive? He’ll reply, “yeah, well, that’s true, I guess”, then he imagines the work that will be created, thanks to Sandy, and doesn’t really get it.

    Correctly and simply, one should note that those reconstruction resources would be allocated elsewhere, in a calamityless world, creating new wealth. But this demands a level of abstract thinking beyond the ken of Joe Sixpack. “Market allocation of scarce resources” – sure thing, Perfessor, works fine in the ivory tower, but we live in the real world!

    Bryaan Caplan wrote an insightful piece, “The 4 myths voters believe”. One of those is: markets don’t work. Which applies here. Hence, Joe Spud believes those construction workers will remain unemployed, they’ll be unable to pay their cable tv bill, so the kids can’t watch Discovery Channel, but the chinese kids will, they’ll pull ahead of us, steal our good manufacturing jobs, get rich, build their military, invade, take away our burgers and fries and Coors, force a diet of egg rolls and green tea, destroying our way of life!!!! Thank the Lord for hurricanes, Allah bless america! (actually, chinese is already the most popular fast food, but never mind that now)

    Notice this attitude is endemic; talk about cutting the military, the response is: what will all those people do? Markets don’t work!

    PS Obamney doesn’t get it either.

  15. 15 15 Paul T

    Ken: “Bastiat still has the clearest explanation as to why the destruction of wealth is just that: the destruction of wealth.”

    Ken, I suggest a bracer of Jack Daniels and Pepto Bismol before checking:

    Read carefully, what is he saying about global warming?
    Then check this birdbrain’s astonishing credentials.

  16. 16 16 Alecia

    There are more than 2 weeks in a month (except for most Febrarys) and 52 weeks in a year which adds up to 26 biweekly payments ($13,000) in a year.

    Why would mortgage payers choose to make payments more frequently in order to pay off their mortgages faster? It seems that consumers need to utilize more of their income for living, costs such as food, with the rising prices. Or are they so future oriented that the want to free up future income for future consumption? Furthermore, aren’t their penalties for paying off mortgages faster as banks don’t receive as much interest?

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