TITLE: Paying off a mortgage twice as fast?
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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.