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Simple & Compound Interest

Present Value

Ordinary Annuities

Present Value Annuities

General Annuities & Equivalent Rates

Mortgages

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 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.

 

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