Question

Betty and Bob secure a $1 million, 30 year mortgage based on a rate of 12% per annum compounded monthly.

            a. Compute the monthly payment.

            b. Compute the total interest paid over the 30 years.

            c. At what period will the interest payment per period and the principal payment

                per period be equal?

            d. Generalize your work in part c by finding a formula for the period in terms of

                 the inputs L, i , and n.

            e. At what period will Betty and Bob own half of their home; that is at what

                 period will their total contribution to equity be half of the loan amount?

            f. Generalize your work in part e by finding a formula for the period in terms of

                 the inputs L, i , and n.

            g. Further generalize your work in part f by finding a formula for the period at

                 which they own k percent (0<k<1) of their home in terms of the inputs L, i, n,

                 and k.

Answer 1

L=1000000 ,

i=12%/12 =0.01

n= 30 x 12 =360

Let Monthly payment =P

a) Monthly payment formula (P) = [L*i*(1+i)^n] / [ (1+i)^n -1]

   Hence P = [1000000*0.01*(1.01)^360] /(1.01^360 -1) = $ 10,286.13

b) total interest paid over 30 years = $ 10,286.13 x 360 -$ 1,000,000 =$ 2,703,006.8

c) Monthly payment (P) = 10,286.13

If principal and interest have to be equal then interest has to be 10,286.13/2 = 5143.07

If interest =5143.07 then principal outstanding should 5143.07/0.01 = $ 514,307

We have to find period t when principal = $ 514,307

t = ln[1- 0.01*1000000/10286.13] / ln(1.01) - ln[1- 0.01*514307/10286.13]/ln(1.01)

   = ln(0.027817)/ln(1.01) - ln(0.499999)/ln1.01

   Hence t = -360 + 69.6609 = 290.3391 months when both principal and interest will be equal

d) Generalized formula for calculation in part c

t = ln[1- i*L/P] / ln(1+i) - ln0.5/ln(1+i)

e) Let t be the period when they will own half of the home ie 500,000

   t = ln[1- 0.01*1000000/10286.13] / ln(1.01) - ln[1- 0.01*500000/10286.13]/ln(1.01)

   = ln(0.027817)/ln(1.01) - ln(0.513909)/ln1.01

   Hence t = -360 + 66.9032 = 293.097 months

f. Generalized formula for calculation in e. ie when they own 50%

t = ln[1- i*L/P] / ln(1+ i ) - ln[1- i*0.5L/P]/ln(1+i)

g. Generalized formula for calculation when they k%

   t = ln[1- i*L/P] / ln(1+ i ) - ln[1- i*k/100*L/P]/ln(1+i)