Question

1) Consider a $126,714 35-year mortgage with an interest rate of 8% compounded monthly.

a) Calculate the monthly payment.

b) How much of the principal is paid the first, 25th, and last year?

c) How much interest is paid the first, 25th, and last year?

d) What is the total amount of money paid during the 35 years?

e)What is the total amount of interest paid during the 35 years?

f) What is the unpaid balance after 25 years?

g)How much has to be deposited now into a savings account with an interest rate of 5% compounded quarterly in order to pay the unpaid balance of the mortgage after 25 years?

h)How much has to be deposited each quarter year from now in a fund with an interest rate of 6% compounded quarterly in order to cover the unpaid balance after 25 years?

Answer 1

a.

Formula to compute EMI is:

EMI = [P x r x (1+r) ⁿ]/[(1+r) ⁿ – 1]

P = Principal of loan = $ 126,714

r = Periodic rate = 0.08 /12 = 0.006666667 p.m.

n = Number of periods = 35 years x 12 months = 420 periods

EMI = [$ 126,714 x 0.006666667 x (1+0.006666667) ⁴²⁰]/ [(1+0.006666667) ⁴²⁰ – 1]

     = [$ 126,714 x 0.006666667 x (1.006666667) ⁴²⁰]/ [(1.006666667) ⁴²⁰ – 1]

     = ($ 126,714 x 0.006666667 x 16.2925521636058)/ (16.2925521636058– 1)

     = $ 13763.2970538924/ 15.2925521636058

     = $ 900.000007 or $ 900

Monthly payment is $ 900

b.

Principal paid in month n = EMI/[(1+r) ^{(N – n +1)}]

N = Total number of payments = 420

r = Monthly rate = 0.006666667

n = 1;

Principal paid in 1^st month = $ 900/ [(1+0.006666667) ^{(420 – 1 +1)}]

                                               = $ 900/ [(1.006666667)⁴²⁰]

                                               = $ 900/ 16.2925521636058

                                               = $ 55.2399643077661 or $ 55.24

n = 25;

Principal paid in 25^th month = $ 900/ [(1+0.006666667) ^{(420 – 25 +1)}]

                                               = $ 900/ [(1.006666667)³⁹⁵⁺¹]

                                               = $ 900/ [(1.006666667)³⁹⁶]

                                               = $ 900/ 13.8909708192533

                                               = $ 64.7902880015103 or $ 64.79

n = 420;

Principal paid in last month = $ 900/ [(1+0.006666667) ^{(420 – 420 +1)}]

                                               = $ 900/1.006666667

                                               = $ 894.039734803298 or $ 894.04

Principal paid in 1^st, 25^th and last payments are $ 55.24, $ 64.79 and $ 894.04 respectively.

c)

Interest paid in each payment = EMI – Principal portion

Interest paid in 1^st month = $ 900 - $ 55.24 = $ 844.76

Interest paid in 25^th month = $ 900 - $ 64.79 = $ 835.21

Interest paid in last month = $ 900 - $ 894.04 = $ 5.96

Interest paid in 1^st, 25^th and last payments are $ 844.76, $ 835.21 and $ 5.96 respectively.

d)

Total amount of money paid = Total number of EMI x EMI = 420 x $ 900 = $ 378,000

e)

Total amount of interest paid = Total amount of money paid – Loan principal amount

                                                    = $ 378,000 - $ 126,714 = $ 251,286