## Too Good to be True – Math in the Popular Press I

I came across this statement in a  recent Atlantic Monthly.  “Further, if instead of paying a $2,000 mortgage once a month, you pay$1,000 every two weeks, you can pay off a 25-year mortgage five to seven years early!”   This was attributed to author David Bach.  I thought that this was too good to be true.

My first question is what would a mortgage company do if you paid them a half payment every two weeks instead of a full payment every month.  I think that they would just apply the money to the next monthly payment.  In that case nothing would change and no time would be saved.  If instead the company applied the $1000 to the principal, you would be in arrears at the end of the month since the next$1000 would not cover the interest owed.  So for the rest of this post, I will assume that we are changing the number of payments per year from 12 to 24.

I build a basic spreadsheet mortgage table with principal $359,000 and interest rate 4.5%. Here are the results.  Payment$2000 and frequency 12 and rate 4.5% takes 300 periods Payment $1000 and frequency 24 and rate 4.5% takes 598 periods Payment$1000 and frequency 26 and rate 4.5% takes 568 periods

You save one month using the twice the frequency – half the payment method.  I originally said”You save about one year and three months if you make 26 payments that is  you paid an extra \$2000 per year.”  Thanks to Sherry Ettlich, I now know this was wrong and it would save approximately 3 years 4 months.  I suspect that with a higher interest rate more time could be save.

The relationship between payment($p$), rate per period($r$), principal($P$) and number of periods($N$) is $p =\frac{rP}{1-(1+r)^(-N)}$.  If you halve the payment and double the frequency it looks like this $\frac{p}{2} =\frac{\frac{r}{2}P}{1-(1+\frac{r}{2})^(-M)}$ where M is the number of periods.  It is an easy matter to eliminate variables and get $(1+r)^N = (1+\frac{r}{2})^M$.  Now take the natural logarithm of both sides to get $M = N\frac{\ln (1+r)}{\ln (1+\frac{r}{2})}$. The first three terms of Wolfram|Alpha’s series approximation for the ratio $\frac{\ln (1+r)}{\ln (1+\frac{r}{2})}$ is $2 - \frac{r}{2}+\frac{3r^2}{8}$.  This comes out to 1.98 if $r = .045$.  Thus at best seven months are saved since $1.98*300 = 593$.

The upshot is that I didn’t believe what I read.  So I made some assumptions and confirmed my skepticism up to a point.  But probably all I did was confirm that I didn’t understand premises of that particular statement in the Atlantic Monthly.