     Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test UNIT 11  :  MATHEMATICS OF INVESTMENT

LESSON 6: MORTGAGES

Example 1:

a)  Camille has just purchased a new house near Brantford.   She needs a mortgage of \$150 000 after her down payment..  She will repay it in monthly instalments over 25 years.

The interest rate is 6.6%/a, compounded semi-annually.  Find the monthly payment.

b)  Determine the total interest paid over the 25 year period.

Solution:

Here the payment interval( monthly ) is different than the interest period ( semi-annual).  This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the monthly rate that is equivalent to 6.6%/a, compounded semi-annually.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 6.6%/a, compounded semi-annually after 1 year. Step 2:  Let the equivalent monthly rate be i %.  (Note the equivalent yearly rate would be 12i %.)

Now find the value of \$1 invested at i % per month after 1 year.

A = 1(1 + i)12                           ** n = 12, the number of times interest is compounded per year.

Step 3:  These two amounts must be equal.  Hence The money in question is borrowed now – at point 0 on the time line.  Hence this is a PV general annuity question        Interest Period   0          1         2          3                                                                                                                                        298   299   300     Payment                          R         R        R                                                                                                                                          R      R      R  R(1.005425865)-1 R(1.005425865)-2

.

.

R(1.005425865)-298  R(1.005425865)-299 R(1.005425865)-300

This forms the following geometric series:

R(1.005425865)-300 + R(1.005425865)-59 + . . . + R(1.005425865)-2 + R(1.005425865)-1 b) Determine the total interest paid over the 25 year period.

Total amount repaid = 1013.85 x 300 = \$304 095.00

Mortgage amount                               = \$150 000

Interest paid = \$304 095 - \$!50 000    =\$154 095

Hence the total interest paid over 25 years is \$154 095.