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My brother has a 30 year fixed mortgage. He pays monthly. Every month my brother doubles his principal payment (so every month, he pays a little bit more, according to how much more principal he's paying).

He told me he'd pay his mortgage off in 15 years this way. I told him I though it'd take more than 15 years. Who's right? If I'm right (it'll take more than 15 years) how would I explain this to him?

CLARIFICATION: He doubles his principal by looking at his statement and doubling the "amount applied to principal this payment" field.

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@three_cups: Sorry, I'm not clear yet. Say the amount applied to principal in his last statement says "5". Does he add 5 to the payment for next month, then? –  Arturo Magidin Feb 20 '11 at 6:13
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@three_cups: "Princip a l". Principles is what people should have. (-; –  Arturo Magidin Feb 20 '11 at 7:21
    
Another issue is that the plan of doubling the principal payments seems easier initially than it really is, because the extra payments are comparatively small at the beginning of the loan, when most of the payment is going to interest, but become much larger at the end of the loan, when most of the payment is going to principal. –  JDH Feb 21 '11 at 12:10
    
Arturo: The best way I can describe it is in terms of looking at a mortgage statement/bill. When paying a mortgage, one has the option to pay more than required. The amount that is overpaid is applied to the principal. The mortgage statement also has a field that tells how much of the payment is going towards interest and how much is going towards principal. My brother doubles this field (the amount going towards principal) and overpays by that amount. –  three-cups Feb 21 '11 at 17:14
    
JDH - Agreed. I'm still deliberating whether to show my brother this question. Not sure if I want to burst his bubble... –  three-cups Feb 21 '11 at 17:14
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5 Answers

up vote 7 down vote accepted

Suppose you have a $\$ $100 debt at 10% monthly interest, and you pay $\$ $15 a month. The amortization for this payment is:

Month     Principal   Interest    Payment   Applied to Principal
  1         100          10         15              5
  2         95            9.5       15              5.50
  3         89.50         8.95      15              6.05
  4         83.45         8.35      15              6.65
  5         76.80         7.68      15              7.32
  6         69.48         6.95      15              8.05
  7         62.43         6.24      15              8.76
  8         53.67         5.37      15              9.63
  9         44.04         4.40      15             10.60
 10         33.44         3.34      15             11.66
 11         21.78         2.18      15             12.82
 12          8.96         0.90       9.86

So you pay it off in one year.

Now, I'm not clear from your description if your brother is planning to pay off double what the original amortization table would indicate, or double what that month's principal would have been (given that he already paid off more than he was expected to), so let me do both.

Suppose first that he simply pays pays his original 15, plus the column "applied to principal" from the original amortization: so the first month, instead of paying 15 with 5 towards principal, he pays 20 (10 towards principal). Next month, instead of 15, he pays 20.50 (so, whatever it will be towards principal, plus an extra 5.50 corresponding to the original amortization table "applied to principal" column; etc.) We have:

Month     Principal   Interest    Payment   Applied to Principal
  1         100          10        15+5            10
  2          90           9        15+5.50         11.50
  3          78.5         7.85     15+6.05         13.20
  4          65.3         6.53     15+6.65         15.12
  5          50.18        5.02     15+7.32         17.30
  6          32.88        3.29     15+8.05         19.76
  7          13.12        1.31     14.43

so it takes 7 months, more than half. Here, what I add each month to the payment is what the "Applied to Principal" column indicated for that month in the original amortization table.

However, if each month he pays each again the amount that would now apply to the principal, we have:

Month     Principal   Interest    Payment   Applied to Principal
  1         100          10         15+5          10
  2          90           9         15+6          12
  3          78           7.80      15+7.20       14.40
  4          63.60        6.36      15+8.64       17.38
  5          45.62        4.56      15+10.44      20.88
  6          24.74        2.47      15+9.74       24.74

so you pay it off in six month (the last month because 15-2.47 = 12.53, and 15+12.53 is more than what is owed). Here, what I add to the payment each month is equal to the difference between the basic payment of $\$ $15 and the interest that is being paid off. So for example, in month 4, you have to pay down $\$ $4.56 interest; that means your $\$ $15 payment will pay down 15-4.56=10.44 principal, so you add another $\$ $10.44 to the payment.

Added. And then there seems to be a third option; from what youo describe, ti seems to me that he would look at the "Applied to Principal" line in the previous month, and add that amount to his payment fo the next month. If we do that, we get the following table:

Month     Principal   Interest    Payment   Applied to Principal
  1         100          10         15              5
  2          95           9.50      15+5           10.50
  3          84.50        8.45      15+10.50       12.05
  4          72.45        7.25      15+12.05       19.80
  5          52.65        5.27      15+19.80       29.53
  6          23.12        2.31      15+10.24       25.24

so it's 6 months this way, with a slightly larger final payment than the previous version, because he is not really doubling the amount of principal he would have paid down that particular month, but rather something slightly smaller. Here, what I add to the payment is the previous month's "Applied to Principal" column; so, since in month 4 there was $\$ $19.80 applied to principal, that's how much is added to the payment of Month 5.

So, in the first scenario (double what he would have paid towards principal in the original amortization), it takes him more than half the time. In the second, where he pays again over what he would have paid off that month in principal, it takes him about half the time. In the final scenario, where he pays again over what he paid last time towards principal, it's a bit more than half the time, but by very little compared to the first method.

These are just examples, of course, but they give you an indication of how things go over 30 years. It would seem he is correct, and it will take him just a bit over 15 years.

This agrees with the results reported by Ross using Excel: my "first scenario" is what Ross reports as taking 20 years; the second what he reports as taking 15 years. My third scenario is very close to the second, but he's a bit off from "doubling" because he is applying the previous amount that was used to pay down principal, not the current one.

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I put together an Excel spreadsheet. It depends upon what you mean by doubling the principal. For a 5% loan the 30 year payment is 5.3682/1000. If you look at the amortization schedule of the 30 year loan and increase the payment by the principal amount (with the additional applied to principal) it will take 20 years, not 15. But if you look at the principal amount based on the current balance and double it, you do pay off in 15 years. I would suggest you make your own spreadsheet and play with it. Each month you charge interest (in my case .05/12 of the current balance) then subtract the payment made to find the new balance. Excel has the PMT function to determine the payment.

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The short answer to your question is if your brother pays double the monthly payment then he will pay off the mortgage in less than $15$ years.

The right way to analyze such problems from a mathematical frame-point is to look at the present value of the total amount he is paying.

Say your brother pays $\$x$ per month for $30$ years starting from the end of the first month. Let $r \%$ be the rate of interest per annum. We will compare the present value of the mortgage/loan.

(Note: that $\$1$ today is worth $\$\left(1+\frac{r}{12}\right)$ at the end of the first month).

The present value of the total amount he pays in $30$ years is $$\frac{x}{\left(1+\frac{r}{12}\right)} + \frac{x}{\left(1+\frac{r}{12}\right)^2} + \cdots + \frac{x}{\left(1+\frac{r}{12}\right)^{12 \times 30}} = \frac{12x}{r} \left (1- \left (\frac{12}{12+r}\right)^{360} \right )$$

Instead of $\$x$ if he were to pay $\$(2x)$ starting from the end of the first month, and let him pay for $n$ years, the present value of the total amount he pays in $n$ years is $$\frac{2x}{\left (1+\frac{r}{12} \right )} + \frac{2x}{\left (1+\frac{r}{12} \right)^2} + \cdots + \frac{2x}{\left (1+\frac{r}{12} \right)^{12 \times n}} = \frac{24x}{r} \left(1- \left (\frac{12}{12+r}\right)^{12n} \right)$$

For the mortgage to be fair, both the present values must be the same.

So find $n$ such that $$2 \left(1- \left (\frac{12}{12+r}\right)^{12n} \right) = \left(1- \left (\frac{12}{12+r}\right)^{360} \right)$$

Solving for $n$, we get $$n = \frac{\log \left (1+\left (\frac{12}{12+r} \right )^{360} \right)-\log \left(2 \right)}{12 \log \left(\frac{12}{12+r} \right)}$$.

So say plugging in $r = 6\%$ i.e. $r = 0.06$, we get $n = 9.01466$ years.

Whatever be the rate of interest, the total amount, the amount he is paying per month, if he decides to double the monthly payment he will pay it off in less than $15$ years.

In general, if a mortgage is for $N$ years and a person needs to pay $x$ per month, if the person decides to pay $k x$ per month instead of $x$ per month and $k>1$, then the person will pay the mortgage in less than $\frac{N}{k}$ years.

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But I confess that I did not understand the description as saying he would double his payment, but only he would double the portion of his payment that goes towards principal (in some unclear way)... That is, he is not paying $2x$ per month, but only $x+p$, where $p$ is the previous amount applied to principal. –  Arturo Magidin Feb 20 '11 at 6:29
    
@Arturo: I actually didn't read the question completely. I just looked at the words doubling the payment and hence wrote this answer. Now reading the question, I am actually confused. We need to wait till the OP confirms it. –  user17762 Feb 20 '11 at 6:32
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It seems your brother is essentially right.

In a standard amortization schedule, the amount applied to principal each month increases geometrically, at the interest rate. Doubling these amounts (or increasing them by any constant factor over the amounts in the original amortization schedule) corresponds to making payments at a higher constant level that amortizes the loan over a shorter total time.

Here's the math: For month $j=0,1,2\ldots$ of the loan, let $P_j$ be the principal remaining at the start of the month, and $Y$ the payment, paid at the end of the month. The amount paid toward interest is $I_j=rP_j$ with $r=0.05/12$, and the amount paid toward principal is $A_j=Y-I_j=Y-rP_j$. Then the new principal is $$ P_{j+1}= P_j-A_j = P_j(1+r)-Y, $$ so $$ I_{j+1}= rP_{j+1}= (1+r)I_j-rY = (1+r)(I_j-Y)+Y. $$ Hence $(1+r)A_j=A_{j+1}$ and therefore $A_j=(1+r)^jA_0$.

The standard payment $Y$ is rigged to make $P_N=I_N=0$ with $N=360$ months. A higher (constant) payment $\hat Y$ corresponds to principal payments $\hat A_j$ that are larger than $A_j$ by always the same proportion.

For a 30 year loan at 5 percent, the standard monthly payment is \$5.3692 per \$1000. Doubling the principal payment results in a monthly \$6.5697 per \$1000, which amortizes the loan over about 20 years. Increasing the principal payments by 200 percent (tripling them) amortizes the loan over a bit more than 15 years.

But from your description it seems your brother is doing something different, something that increases his payments each month. A spreadsheet calculation shows that he would indeed pay off the loan in 15 years, if he adds, to the standard 30-year payment $Y$, the amount of principal that the payment $Y$ would pay off this month. (This amount may be listed on his statement.) This means his payment at the end of month $j$ is $$ Y_j=Y+ (Y-rP_j). $$ As above, now his remaining principal satisfies $$ P_{j+1}= P_j+I_j - Y_j = P_j(1+2r)-2Y. $$ So effectively his principal is reduced as if he makes the constant payment $2Y$ on a loan with interest rate $2r$. As it happens, with $Y$ being the original 30-year standard payment, $2Y$ is almost just the right value to amortize this loan over 15 years.

Any way you do it, paying principal off early is a great way to save lots on interest later.

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Let's look at two scenarios: two months of payment $P$ vs. one month of payment $2P$. Start with the second scenario. Assume the total amount to be payed is $X$ and the rate is $r > 1$, the total amount to be payed after one month would be $$r(X-2P).$$ Under the first scheme, the total amount to be payed after two months would be $$r(r(X-P)-P) = r(rX - (1+r)P).$$ Most of the time, $X$ is much larger than $P$, and so $X-2P$ is significantly smaller than $rX - (1+r)P$ (remember $r \approx 1$). So it should take your brother less than 15 years.

Note I first subtract the payment and then take interest, but it shouldn't really matter.

This all assumes the payments are fixed, but looking at the other answers this is not really the case... I guess my banking skills are lacking. Too young to take loans.

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The payment is usually fixed, but each payment is divided into two parts: one pays interest that has accumulated over the past month, the other part pays down the principal. So each month, with the same fixed payment, you are paying down more of the principal (since less principal generates less interest). Anything you pay above the required payment pays down the principal more, so that you generate less interest for the next month. From the description, he is only increasing the payment by enough to double the portion going towards principal, not to double the total payment. –  Arturo Magidin Feb 20 '11 at 5:55
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