Formula for mortgage monthly payment by few71840

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```									         Formula for mortgage monthly payment

September 25, 2009

P - principal;
r - annual rate of interest (ﬁxed, compounded monthly)
t - time (in years)
term - 30 years
r 12t
A = P 1 + 12

Let x denote the ﬁxed monthly payment. Let Ik and Bk denote the interest
acumulated during the k th month and the ballance remaining at the end of the
k th month, respectively.
The interest after the ﬁrst month is
r   12( 12 )
1
Pr
I1 = P 1 +                        −P =      .
12                     12
Thus, the ballance after the ﬁrst month is
Pr          r
B1 = P − (x − I1 ) = P +              −x=P 1+    − x.
12         12
For the second month we then have
B1 r         r                          r
I2 =        = P 1+    −x
12          12                         12
and so
r             r                              r
B2 = B1 − (x − I2 ) = P 1 +              −x−x+ P 1+    −x                             =
12            12                             12
r         2                r
=P 1+                  −x 1+          − x.
12                         12
It is not diﬃcult to show that
r   k             r       k−1                r   k−2                    r
Bk = P 1 +             −x 1+                   −x 1+                  − ... − x 1 +      − x.
12                12                         12                         12
Then, by the geometric progression formula,
k                                                r k
r     k           r
1 + 12 − 1        r                    k        12x   1+   12     −1
Bk = P 1 +            −x       r    =P 1+                           −                           .
12            1 + 12 − 1       12                                    r

1
After 30 years the ballance should be zero, so, in order to ﬁnd x, we set
B360 = 0. Then

r 360
r     360       12x     1+   12          −1
P 1+              =
12                            r
so
r 360
Pr 1 +      12
x=                                  .
r 360
12     1+      12     −    1

2

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