Question

1. İYou and your spouse have found your dream home. The selling price is $220,000; you will put $50,000 down and obtain a 30-year fixed-rate mortgage at 12% compounded monthly for the balance a) Assume that monthly payments begin in one month. What will each payment be? b) How much interest will you pay in dollars) over the lifetime of the loan? (Assume you make each of the required 360 payments on time.) c) Although you will get a 30-year mortgage, you plan to prepay the loan by making an additional payment each month along with your regular payment. How much extra must you pay each month if you wish to pay off the loan in 20 years?

Answer 1

Selling price = $ 220,000             Down payment = $ 50,000

Therefore, Amount of mortgage = $ 170,000

Rate of interest = 12% per annum (compounded monthly)

            Hence, rate of interest(i) = 12% / 12 months = 1 % per month.

Time period = 30 years = 360 months

SOLUTIONS:

[A]

To find the monthly instalments, we use the formula of Present Value of Annuity,

Present Value = Annuity *

(1+i)"–1 i(1+i)n

Hence,                                  170,000 = A *

(1+0.01)360 –1 0.01(1+0.01)360

                                                170,000 = A *

34.94964 0.3594964

                                                A (Monthly payment) = $ 1,748.64

[B]

To find the total interest to be paid over the lifetime of the loan, we use compound interest formula,

                        Total Amount (A) = Principal (1 + i)ⁿ

                                                            A = 170,000 (1 + 0.01)³⁶⁰

                                    A = 170,000 (1.01)³⁶⁰

                                    A = $ 6,111,439.

Hence, Compound Interest (C) = Amount (A) – Principal (P)

                                                C = 6,111,439 – 170,000

                                                C = $ 5,941,439

[C]

Now, the time period of loan (n) = 20 years = 240 months.

To find the monthly instalments, we use the formula of Present Value of Annuity,

Present Value = Annuity *

(1+i)"–1 i(1+i)n

Hence,                                  170,000 = A *

(1+0.01)240 –1 0.01(1+0.01)240

                                                170,000 = A *

9.89255 0.1089255

                                                A (Revised Monthly payment) = $ 1,871.85

Therefore,                           Extra payment per month = 1871.85 – 1748.63 = $ 123.22