# a mortgage math question

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I am a professional mathematician. Today I was tempted by a fedex express delivery offer from my mortgagee offering a refinance of my 3.75% 10yr. mortgage to a 2.75% 10yr. one, lowering my monthly payment from \$3500 to \$3000. I almost acted on it until I noticed that I had already paid mine for 2 years and this one would extend for 10 more years. Who can tell me why this is a bad move? (independent of closing costs.)

This might be a good math problem for an 8th grader, which doesn't say much for my chops as a professional mathematician!

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If you have 8 more years to pay on your mortgage at \$3,500 per month, your cost would total \$336,000 (96 x \$3,500).

Even though the rate drops by 1% for the new offer, extending it to 10 years results in a total cost of \$360,000 because you start over again paying mostly interest with your initial payments. The easiest way to see it is by comparing amortization tables.

You would not see much of a difference between a 28-year loan and a 30-year loan, but you will see a significant difference when stretching out shorter-term loans since you start paying down principal so quickly.

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I am not a professional mathematician. If any organization spends \$10 to FedEx me a letter they could have spend 29 cents mailing, explaining how much money I can save, I am just going to go out on a limb and assume that my best interests are not in play.

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If closing cost is absorb by the bank, I would refinance and put the extra \$500 per month into buying down the principal. In one year I would have pay down my principal by \$6,000. It would be easy to finish the new 10 year loan in less than 8 years.

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We looked at refinancing. Ran the numbers a couple of ways. It just didn't make enough sense. Rates are so so low though.

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I calculated it the way besroma did. Later I thought of it as a fraction problem: new pmt is 6/7 of old, but no. of payments is 5/4 of old, so total paid is 30/28 of old, or over 7% more. Arcadia also poses a possible option, treat it like an 8 year loan at a lower rate. But of course the lower rate was also a partial scam, as there were two numbers, an "interest rate" of 2.75 and an actual APR of 3.1, which no doubt included closing costs.

I haven't run the full numbers, and I could be wrong, but since principal payments are currently about 3 times interest, and skipping an interest payment requires making an extra principal payment, it seems to me that using the \$6,000 saved the first year (even ignoring closing costs) to reduce principal would save only about \$2,000 of the \$24,000 extra interest. This principal to interest ratio would become more costly every year afterwards. And even if one saved \$2,000/year on interest every year, it wouldn't seem to recoup the extra \$24,000 interest (not to mention closing costs) very quickly.

This rough estimate discourages me from even making the full calculation, as GGardner deduced even more simply. Arcadia's point is a good one however, the moral is that the primary savings is from paying off sooner. And I could be wrong about the details here.

As everyone knows as well, lenders are not fully bound by the preliminary teaser numbers, and often show up at closing with higher ones.

I suggest that every child learning math should work problems like this one and learn about the time value of money, i.e. inflation. I would discuss with them whether the state should be allowed to advertise a lottery prize that pays 100,000 a year for 10 years as a million dollar prize. Does that mean one that pays a dollar a year for a million years is worth a million dollars?

By the way if my arithmetic/interest calculations seem naive, I'm a professional geometer, and that's a different specialty!

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I haven't run the full numbers, and I could be wrong, but since principal payments are currently about 3 times interest, and skipping an interest payment requires making an extra principal payment, it seems to me that using the \$6,000 saved the first year (even ignoring closing costs) to reduce principal would save only about \$2,000 of the \$24,000 extra interest. This principal to interest ratio would become more costly every year afterwards. And even if one saved \$2,000/year on interest every year, it wouldn't seem to recoup the extra \$24,000 interest (not to mention closing costs) very quickly.

current payment would be \$875 principal \$2,625 interest out of \$3,500 monthly at 3.75%

assuming no closing cost,

payment after refinancing would be \$1,925 interest (2.75/3.75 * \$2,625)

so if you keep total payment the same, you could use \$1,575 for principal.

That is an increased downpayment of \$700 more towards principal without you being out any more money than now.

the next month your remaining principal would have already been down \$700 further so your interest payment portion of the loan would again drop. So by compounding the interest save each month by the extra drop in principal, it is easily more than \$6,000 saved per annum. It is a lot easier to calculate and compare on an excel spreadsheet just for curiousity sake.

ETA:

The calculations are based on interest being three times the amount of payment, I mis-read the question and obviously need another cup of coffee :)

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I am not a professional mathematician. If any organization spends \$10 to FedEx me a letter they could have spend 29 cents mailing, explaining how much money I can save, I am just going to go out on a limb and assume that my best interests are not in play.

LOL :D

Bill

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If closing cost is absorb by the bank, I would refinance and put the extra \$500 per month into buying down the principal. In one year I would have pay down my principal by \$6,000. It would be easy to finish the new 10 year loan in less than 8 years.

Assuming no "early payment" penalty. Look at the fine print, as many (shady) mortgages exclude principal pay down without penalty.

Bill

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Thanks for the details. Some of these numbers are not quite realistic though, since there are apparently closing costs, which is why the scam come on "interest rate", displayed two numbers, the fake 2.75% and the real 3.1%. Also I made a typo in stating the current rate as 3.75, instead of 3.375.

Moreover the current payment is (principal/interest) = 3, not 1/3. So it is the interest payment that is now \$875, and the principal payment is \$2625.

So the new interest payment would be something like (3.1)/(3.375)x875 = \$803 (figuring in closing costs as added interest), instead of \$875, leaving only a little for added principal.

I may still be confused, but you have inspired me to think about adding extra principal to the loan I already have! Even if I only add \$100/month to principal, I will make some progress.

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Assuming no "early payment" penalty. Look at the fine print, as many (shady) mortgages exclude principal pay down without penalty.

My mortgage has a fine print of penalty if prepaid before 3 years. But we could keep a dollar left in the loan and still not get hit by the penalty. That is what we did with our HELOC before the time period was up. We paid interest on the \$1 left in the HELOC for a few months.

Moreover the current payment is (principal/interest) = 3, not 1/3. So it is the interest payment that is now \$875, and the principal payment is \$2625.

So the new interest payment would be something like (3.1)/(3.375)x875 = \$803 (figuring in closing costs as added interest), instead of \$875, leaving only a little for added principal.

LOL. I need another cup of coffee. Mis-read it to be principal is 1/3 of interest. At your current rate of 3.375%, it would be hard to break even by just reducing to 3.1% even without closing cost.

Mine went down by 1.875% due to Libor rates and snowballing the savings from interest into principal downpay makes sense to us.

ETA:

Amortization depending on types of home loan and compound interest might make for an interesting consumer maths assignment though.

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Interesting. Refinancing on a 10 year loan would have to have a HUGE drop in interest imo. We recently refinanced our 30 year loan. Unfortunately they were not able to give us a 15 year loan like we wanted. It rolled into another 30 year loan. However the rate dropped so much for us that we could pay it off in 15 years by paying just a bit more than we were before. Which is why we went for it. FWIW I would have never considered it without making an amortization schedule in excel to really see how it all plays out. Now it is fun to pay extra every month and see the payments at the end disappear. And thus the interest we would have to pay. I color them just to make myself feel better.

I dont feel like the mortgage companies will give you a clear enough picture by talking to them and the interest rate and terms will often change from your original offer til closing. Being an educated consumer will keep you from getting bamboozled and if I want to teach anything to my kids it will be to figure those kind of things out for themselves and not rely on others to provide them all the sketchy details

Christina

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Use bankrate.com to take a look at their loan amoritization. It'll show you monthly payments, total payments and what happens if you add extra principal if you wanna do that.

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you have mind-bogglingly low mortgage rates (just so you know) :-)

Eta. There are a lot of great things about NZ but mortgage rates have never been one of them.

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