Making 13 Mortgage Payments A Year
Copyright 2010 by Morris Rosenthal - - contact info
Mortgage Math Workbook
Copyright 2010 by Morris Rosenthal
All Rights Reserved
How to Reduce Mortgage Principal with Bi-Weekly or Larger Payments
One of the subjects I didn't cover in my ebook about fixed rate mortgages is paying the mortgage off early by accelerating the payments and reducing the pricipal owed. The following discussion is about the math behind this process, it doesn't include any penalties or fees your bank might charge for early payments. If you have been making payments on your mortgage for some time and want to figure out how much principal you owe, you need to check or calculate an amortization table for the loan.
Years ago, a friend of mine wanted to accelerate the payments he was making on a piece of land he'd bought to build a house. I ran the calculations through the compound interest formula used for mortgages and gave him the wrong answer. The mistake wasn't intentional. If your mortgage allows you to prepay in random amounts at any given time, you can throw extra money at the mortgage when you feel like it and trust the bank to continually recompute your principal and amortization table. But people who are interested in paying off a 30 year mortgage in less than thirty years often go for the 13 payments a year idea, because they like the concept of making one extra payment by paying every four weeks instead of once a month. But it rarely works out that way, and making 13 payments a year is not the same as a bi-weekly mortgage, where you actually make 26 payments a year, with each payment being half of what a normal monthly payment would have been. If you get a bi-weekly mortgage where the payments are less than half of what they would have been on a standard 30 year mortgage, you aren't going to cut any years off the mortgage and will save a really trivial amount of interest by compunding the first payment two weeks early. The most sensible option, providing the bank goes along with it, is to divide a monthly payment by 12 and add that twelfth to the regular monthly payments, which amounts to making thirteen payments a year.
The basic formula for calculating mortgage interest is:
Monthly Payment = Principal [ i(1 + i)n ] / [ (1 + i)n - 1]
which is explained in depth with detailed examples in the ebook. The "i" in the formula is the interest per compounding period, or the annual interest divided by the number of payments per year. It works out to this because mortgages are structured so that the payments are made per compounding period, normally a month, which makes it easy to produce a meaningful amortization table. The "n" in the formula is the number of compounding periods in the life of the loan. For a normal 30 year mortgage, n=360.
So you go to a bank and get approved for a 15 year or a 30 year fixed rate mortgage, and then you decide you want to pay it down a little more quickly by making 13 payments a year. The banks aren't normally set up to take a payment every 4 weeks, so the normal way to go about it, assuming again no penalties or fees, is to add one twelfth to each of your monthly payments. The question then becomes, how much time will you shave off the mortgage? The answer is, it depends on the interest rate, which will be a known for the mortgage you are accelerating, as will the principal and the period. So if you start prepaying at the very beginning of the loan, the formula can be simplified so we can solve for n, the number of payments. We'll substitute "x" for the messy part of the equation as we move things around, so it reads,
M(onthly payment) = P(rincipal) [ix / (x-1)]
multiply both sides by (x-1)
Mx - M = Pix
divide by the monthly payment
x-1 = Pix/M
divide both by x
1-1/x = Pi/M
1-Pi/M = 1/x
x = 1 / (1-Pi/M)
substitute back in for x
(1 + i)n = 1 / (1-Pi/M)
which we might have reached faster if I was smarter about it, but that's as close to n as we're getting.
Now let's take some real numbers, like one of the examples from the ebook with a 30 year mortgage at 4.5% for $187,000 with a computed monthly payment of $947.51. If we were able to make 13 payments a year, we would use 13 compounding periods for:
i=0.045/13 = .003462
1+i = 1.003462
(1.003462)n = 1/[1-$187,000(.003462)/$947.51]
(1.003462)n = 1/(1-.6833) = 1/0.3167 = 3.16
So to solve for the amount of time lopped off the mortgage, we need to find an "n" that make
1.003462n = 3.16
The way to do this is to take a few guesses, let's start with 300 four week "months", and use the ^ for "to the power of"
1.003462 ^ 300 = 2.82, way too low, so let's try 312 "months"
1.003462 ^ 312 = 2.94, still too low, so let;s try 324 "months"
1.003462 ^ 324 = 3.06 getting close, so lets try 336 "months"
1.003462 ^ 336 = 3.19, too high, so try 333 "months"
1.003462 ^ 336 = 3.161, so that's what we're going with.
333 four week "months" has to be divided by 13 to see how many normal 12 month years we get
333/13 = 25.62 years.
Of course, it's not easy to find a bank that operates on 13 month years, but the difference between actually compounding 13 times a year and compounding 12 times a year with an extra /12 on each payment or a bi-weekly payment of $947.51 divided by two for the equivalent of 13 weekly payments will all come out around the same. It's easy enough to check the 12 payments with extra twelfths using the standard mortgage formula. Instead of
$947.51 per month, we'll pay $78.96 + $947.51 = $1026.47 per month giving:
i=0.045/12 = .00375
1+i = 1.00375
(1.00375)n = 1/[1-$187,000(.00375)/$1026.47]
(1.00375)n = 1/(1-.683) = 1/0.3168= 3.16 (notice that the 3.16 is the same factor we arrived at using the smaller payment and 13 four week "months"
We know the answer we expect to see is 25.62 years, so lets compute our first guess at n from 12 months times 25.62 years:
12 x 25.62 = 307.44
1.00375 ^ 307.44 = 3.16
So unless I made a math error somewhere, prepaying a the 1/12 extra payment each month is equivalent to making the payment 13 times a year. Now let's check bi-weekly for the same mortgage principal, using half the monthly payment on the 30 yr as the bi-weekly payment. Note that the interest being compunded bi-weekly means dividing the annual interest rate by 26:
$947.51 per month, we'll pay $947.51/2 = $473.76 per month giving:
i=0.045/26 = .001731
1+i = 1.001731
(1.001731)n = 1/[1-$187,000(.001731)/$473.76]
(1.001731)n = 1/(1-.683) = 1/0.31675 = 3.16 (notice that the 3.16 is the same factor we arrived at using the smaller payment and 13 four week "months" and the higher payment 12 times a year to the hundreds place, though there was a small difference, ie, 1 / 0.3168 vs 1/0.3156.
We know the answer we expect to see is 25.62 years, so lets compute our first guess at n from 26 biweekly payments times 25.62 years:
26 x 25.62 = 666.12
1.001731 ^ 666.12 = 3.16
The main point is that the savings in time and interest are all relative to a 30 year mortgage that you would have paid off otherwise. There's no winning or losing here. Taking out a 30 year mortgage that allows you to prepay and making the equivalent of 13 payments a year will cut around five years off the mortgage, but those payments are no different than if you had taken out a 25 year mortgage to start with. The math is the same. The bi-weekly mortgages don't offer any savings over adding to the monthly payment, they were simply set up to help people who were bad at saving to manage their money. Since many salary employees get paid every two weeks, a the bank could automatically debit their checking account biweekly for the mortgage amount, and may insist on your enrolling in automatic payroll deposit to set up the loan. The only equivalency between bi-weekly and 13 payment mortgage schemes and paying off your mortgage "early" is if you arrange them so as to pay more on an annual basis than you would have done with a 30 year mortgage, as we did above.