### Mortgage Puzzle

I have recently acquired a 30 year mortgage.

Today I’ve received a letter offering to let me make payments on a biweekly basis instead of a monthly basis. If I accept this offer, I will make a biweekly payment equally to exactly half my current monthly payment — and my mortgage will paid off in 23.6 years instead of 30.

Question: How can such a small change in the timing of my payments shave a full 6.4 years off the life of my mortgage?

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#### 34 Responses to “Mortgage Puzzle”

1. 1 1 Alex Ignatiev

Thirteen payments.

2. 2 2 Manfred

Steve, I will venture an incomplete answer, because I did not work out the math.
Here is how I understand it: when you make monthly payments, as your mortgage payment schedule says, you make 12 payments per year. If you make bi-weekly payments, you make 26 payments per year, which works out to be 13 “monthly” payments. In other words, because of how the Gregorian Calendar is set up, you “add one month”, by making bi-weekly payments, and that month goes entirely to principal (which saves you the related future interest that you would pay on it).
To confirm your numbers of 23.6 years, one would have to run the math with the interest you pay.

There are other schemes: for example, instead of bi-weekly payments, you could add 1/12 (one-twelveth) of your monthly note to your monthly payment. In this way, after one year, you actually paid 13 months, instead of twelve. I am not sure this works out the same as the one you propose, but it is probably close.

3. 3 3 Walter Sales

26 bi-weekly payments per year will add roughly a month of payments to each year. That, combined with the early payment of some principal and voila you have a 23.6 year mortgage instead of a 30 year mortgage.

4. 4 4 Neil S

With a biweekly mortgage you are making 26 half payments per year rather than 12 full payments. This amounts to making paymnets that are 8 1/3% greater than the traditional schedule each year. Timing effects and Compounding interest takes care of the rest.

5. 5 5 Ken B

I think Manfred has identified the biggest factor, 13 vs 12. AS well they get an extra couple weeks interest on some of the half payments. To see this, pay not bi-weekly but on the 14th and last day of each month. They are collecting interest from the 15th to the end of the month on half the monthly amount. Thus they can afford to provide an incentive.

6. 6 6 Justin Ross

Ken B is correct. Instead of paying 12 times year, you pay every other week for a total of 26 times a year. You could get pretty much the same savings by doubling one of your mortgage payments once a year, and save yourself the switching costs.

7. 7 7 Ben

Both answers above are right; this puzzle mystified me as well a few months back, but simply increasing your monthly payment by 8.3% will accomplish the same thing.

8. 8 8 Jose Silva

An additional factor may be that people who represent a lower risk of default self-select into the biweekly payments. And since a lot of risk-segmentation approaches run afoul of the law, and this one presumably doesn’t, that might be a significant part of the change.

(How is the bi-weekly system implemented? I assumed that it would be something like the 14th and 30th of the month, which would make it easy to do. If it’s really every two weeks, then the people who select this will probably rely on some sort of automated payment, which is an additional screening device.)

9. 9 9 Alan Wexelblat

I suspect that at least part of the answer is that interest is computed on a daily basis so an ‘early’ payment reduces the principle. So each 2 weeks you are not paying 2 weeks worth of (compounded) interest on the amount of the principal you paid off with that check.

That said, one of the major effects of this plan is to reduce the amount of mortgage interest you deduct on your taxes. As someone with a high enough income to be able to take full advantage of this deduction (despite your dislike of it) the end result may not be financially sensible for you. This is doubly true if you do not expect to be in this house for the full 23 years.

I’m also puzzled by the logic of acquiring a 30-year fixed at this point rather than a 5-1 ARM with a conversion option. I think it’s statistically very likely that interest rates will remain at near-historic lows for the next 5 years and last time I looked 5-1 ARMs were being offered at a couple points below the 30-year fixed rate.

10. 10 10 AaronG

A large effect is coming from paying 13/12 as much per year with bi-weekly payments. Someone else noted the effect of half of each payment coming 2 weeks earlier, but this is pretty negligible. I suspect they are also giving you a rate differential while you are signed up for bi-weekly payments. To get a reduction of 6.4 years without a rate differential, your interest rate would have to be ~7%. If that is what you got, you need a better mortgage broker!

I did my calculations using an annuity formula for the present value of a series of level equally spaced payments:
PV = (1 – (1 – i) ^ -n) / i
where
n is the total number of payments
i is the rate for the period of time between payments (i.e. for monthly payments, divide the nominal rate quoted by the bank by 12)

11. 11 11 AaronG

Sorry got a sign wrong when I typed it out, should read:

PV = (1 – (1 + i) ^ -n ) / i

12. 12 12 Patrick

Be careful, Aaron. For sums as large as morgages, small lacks of precision can result in large swings in the numbers. To correctly calculate a monthly rate, it’s not i/12, but rather (1+i)^(1/12) or the 12th root of i. At 6%, i/12 = 0.5%, but the monthly interest is 0.487%

13. 13 13 AaronG

Patrick
Banks normally quote rates in nominal terms so that it is correct to divide by 12. It won’t always be the case, but it is the most common.

14. 14 14 Ken B

What should Steve do?

Theory 1. If the offer is being made then the bank sees a profit in Steve switching. Steve should stay.

Theory 2. If Steve is getting a form letter this is probably a routine thing. The offer could not persist in the marketplace unless it benefits borrowers. Steve should switch.

Theory 3. The market is efficient. Steve shouldn’t care.

Theory 4. Steve should consider his own cash flow and the tax implications for the period until he can refinance and whether he has better investments available.

I know, Theory 4 is no fun, but Will A is out there waiting a chance to pound on me if I omit a useful answer :)

15. 15 15 Phil

In Canada, banks quote rates on a semi-annual compound basis. So if they quote a 6% mortgage rate, it’s really 3% every six months compounded every six months, so the monthly interest is the 6th root of 1.03, minus 1 (which works out to about 0.4938622% per month).

Kind of annoying, really.

16. 16 16 Phil

Ken B/8: I support Theory 4. You should base your decisions on your own utility and/or consumer surplus, and not what’s better or worse for the other party.

(Of course, you’re interested in theories 1-3 to help you determine whether you gain or lose utility by switching.)

In life, I’ve found that if you’re not stupid, the transactions on which the other party makes the most profit tend to be the ones on which I get the most consumer surplus. The margins on the iPad are huge, but I’d gladly pay double for mine. This is not coincidence.

17. 17 17 Pat T

Are they giving you a discount figuring that anyone who’d pay biweekly is less likely to default?

Off topic – when my wife was pregnant, it drove me crazy how all the sites and literature equated 40 weeks with 10 months. 28 days is not a month!

18. 18 18 AaronG

Phil 9
I suspect that reducing the number of calls from customers who can’t do this math is precisely why American banks quote loans the way they do. I work in insurance and we quote effective annual rates on our annuity products. You would not believe how many phone calls we get from customers complaining about a couple of missing pennies in the interest we credited them yesterday…

19. 19 19 NitinB

It is primarily due to 26 payments per year instead of 24, as most have noted above. Plus, the company making you this offer is charging fees that would reduce the impact of the extra payments. You can pay down even faster than 23.6 years if you made the “13th monthly” payment yourself instead of accepting this offer.

http://nitin.wlkr.net/2007/04/blog-post_14.html

20. 20 20 Jeremy N

I believe it’s mostly due to the interest. Interest can accumulate very quickly. Think about what’s happening. Every two weeks you pay down your principal. After one year, you’ll have paid off more than twice the principal under the biweekly plan. That’s less you’ll have to pay in interest for the next 29 years!

There’s a comparison calculator at http://www.bankrate.com/calculators/mortgages/bi-weekly-mortgage-calculator.aspx.

It shows that at 5% interest, you’ll pay about 93% of the total loan as interest with a monthly mortgage, while only 75% for the biweekly. You save 18% of the loan amount by paying biweekly!

21. 21 21 Ken B

@Phil: Yes. I was just making a wee jest with my thesis/antithesis/synthesis in 1,2,3.

Good point about the high margins. I think that’s often the case with new innovation. I jsut paid through the nose for a new OTC pill that cures a medical problem. I would have paid twice as much. To prevent the store getting ideas I won’t say what I bought!

22. 22 22 Kirk

Banks assume that most mortgages will not be a 30 year profit maker for them. Most are moved out of, re-fied, sold or otherwise discharged early. Bi-weekly payments get the banks more money on the front end, thus reducing their risk of default with collateral that won’t support the remaining debt. It also frees money for current lending (they make much more money on loan origination and first year than they do on existing loans – this is why most are sold to bundlers.) The “loss” that they take on interest is negligible in the short term and usually the loss on the back end never takes place.

The over-riding factor, in my mind, on whether to do this is based on your pay schedule from work. If you are paid weekly or bi-weekly, you can transition this into your budget simply. If you are paid monthly or semi-monthly you should simply increase the amount you pay each month, since the bi-weekly payments will eventually screw up your budget unless you save in anticipation of that month, and if you’re saving to make up for a triple payment month, you might as well put it on the principal up front and let it work it’s magic faster.

23. 23 23 khodge

24. 24 24 Fonzy Shazam

Ken B and Manfred are on to it, but it is a little more elegant than what has been said. The annual interest rate is not changing in the proposal. But the number of periods per year is. So now we are looking at interest being assessed on the principal over 26 periods per year rather than 12. So every payment the relative interest burden is lower, and if the interest payment is lower, the principal payment must be higher. Viola, you are accelerating your principal contributions saving you the mortgage holder interest. In an amortizing product, that comes in the form of a quicker payoff–you sooner get to stop making mortgage payments.

25. 25 25 Ken

Steve,

How can such a small change in the timing of my payments shave a full 6.4 years off the life of my mortgage?

I don’t believe you don’t know the answer to this question. The first 5 to 10 years of a 30 year mortgage pays down so little of the principle. For 3.75%, only 1/3 of the mortgage payment goes towards principle. This means that the single extra mortgage payment made (the two week plan has 26 half payments a year, rather than 12 full payments) is equivalent to 3 months worth of payments, since it goes straight to principle, and none to interest. Additionally, interest is calculated on only outstanding principle, so after just a few years, there is substantially less principle for the half payment every two weeks, rather than one payment per month.

26. 26 26 hanmeng

“The BiWeekly Advantage Plan is a convenient mortgage budgeting plan that can help you save thousands of dollars in interest and pay off your mortgage sooner. Instead of you making the normal 12 payments a year, the Plan will draft one half of your regular monthly payment every 14 days. This allows you to make the equivalent of one extra payment, all of which is applied to principal each year. Some homeowners try to make extra principal payments themselves, but most aren’t able to keep a consistent schedule. Let The BiWeekly Advantage Plan do the work for you.”

27. 27 27 Scott H.

@Ken B et al: My take is that an intelligent decision usually revolves around the person’s bi-weekly cashflow. If you’re getting paid bi-weekly, why hang on to extra money at 3.75% interest when you are getting probably 0.00000% interest in your checking account.

Also, as an ex-biweekly paid person I can tell you that those months where the extra pay periods fall are usually considered bonus month. They are often not considered in normal monthly budgeting plans.

28. 28 28 hanmeng

Whoops. Here’s something else:
“For just a \$3.00 transaction fee, collected with each draft, you will be on your way to mortgage-free homeownership years sooner.”

29. 29 29 Mike H

Paying your loan is like trying to balance a pole while scuba diving (the equations are the same, to first order). Your repayments are the force you apply to the pole. Fully paid mortgage = fully balanced pole.

If you’ve carefully tuned the force so it takes a whole 30 years to balance the pole, it’s not surprising if pushing a little harder gets it balanced much faster.

However, I have heard that these “extra payment” schemes are often not all they are cracked up to be. Check carefully exactly when your extra payments end up saving you interest, and especially if there are any fees, consider whether it’s worth just going with the monthly plan and making extra payments on your own now and then, to the tune of 1 extra payment per year.

The bank is making the offer for a reason.

30. 30 30 jrptwo

AaronG’s formula is the precise answer, of course. To motivate it:
Why is an extra 1/12 annual payment for the first 24 years equal to 1 eliminated annual payment for the last 6? It must be that the payments in the first 24 years are worth about three times as much as the payments in the last 6 years.

The payment in the first 24 years is paid on average 12 years from now; the payment in the last 6 years is paid on average 27 years from now. The difference in timing is 15 years (on average). At 7% interest, one dollar now grows to 1.07^14 in 15 years = \$2.75, or about three times.

The precise formula accounts for such inaccuracies in this explanation as using the average difference in timing.

31. 31 31 jrptwo

(Typo: 1.07^15 in 15 years)

32. 32 32 Ken B

@Scott H: Back when I had a mortgage we did made just the choice you indicate. I still have no idea if it was right, but since our taxes were simple (and Canadian, so no deductions for interest) there were no tax implications. For Steve, whose income probably varies a lot year to year due to books etc, and whose taxes may be complicated, it might be very different.

33. 33 33 Paul T

Mortgage Puzzle, PhD level:

Take the idea to the limit: suppose you pay the mortgage continuously – a continuous stream of \$\$ from your bank account – such that the monthly total is the same as currently, and same interest rate on unpaid principal. How long will it take, to pay off completely?

34. 34 34 Ken B

@Paul T: Not PhD level. I remember doing that in high school. We were doing e and limits at the time of course.