Formula for mortgage monthly payment

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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




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