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Introduction, sequences & Series, future value of payments  (Page 4/5)

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You are considering purchasing a flat for R200 000 and the bank's mortgage rate is currently 9% per annum payable monthly. You have savings of R10 000 which you intend to use for a deposit. How much would your monthly mortgage payment be if you were considering a mortgage over 20 years.

  1. The following is given:

    • Deposit amount = R10 000
    • Price of flat = R200 000
    • interest rate, i = 9 %

    We are required to find the monthly repayment for a 20-year mortgage.

  2. We are considering monthly mortgage repayments, so it makes sense to use months as our time period.

    The interest rate was quoted as 9% per annum payable monthly, which means that the monthly effective rate = 9 % 12 = 0 , 75 % per month. Once we have converted 20 years into 240 months, we are ready to do the calculations!

    First we need to calculate M , the amount of the mortgage bond, which is the purchase price of property minus the deposit which Sam pays up-front.

    M = R 200 000 - R 10 000 = R 190 000

    The present value of our mortgage payments X (which includes interest), must equate to the present mortgage amount

    M = X × ( 1 + 0 , 75 % ) - 1 + X × ( 1 + 0 , 75 % ) - 2 + X × ( 1 + 0 , 75 % ) - 3 + X × ( 1 + 0 , 75 % ) - 4 + ... X × ( 1 + 0 , 75 % ) - 239 + X × ( 1 + 0 , 75 % ) - 240

    But it is clearly much easier to use our formula than work out 240 factors and add them all up!

  3. X × 1 - ( 1 + 0 , 75 % ) - 240 0 , 75 % = R 190 000 X × 111 , 14495 = R 190 000 X = R 1 709 , 48

  4. So to repay a R190 000 mortgage over 20 years, at 9% interest payable monthly, will cost you R1 709,48 per month for 240 months.

Show me the moneyâ…

Now that you've done the calculations for the worked example and know what the monthly repayments are, you can work out some surprising figures. For example, R1 709,48 per month for 240 month makes for a total of R410 275,20 (=R1 709,48 × 240). That is more than double the amount that you borrowed! This seems like a lot. However, now that you've studied the effects of time (and interest) on money, you should know that this amount is somewhat meaningless. The value of money is dependant on its timing.

Nonetheless, you might not be particularly happy to sit back for 20 years making your R1 709,48 mortgage payment every month knowing that half the money you are paying are going toward interest. But there is a way to avoid those heavy interest charges. It can be done for less than R300 extra every month...

So our payment is now R2 000. The interest rate is still 9% per annum payable monthly (0,75% per month), and our principal amount borrowed is R190 000. Making this higher repayment amount every month, how long will it take to pay off the mortgage?

The present value of the stream of payments must be equal to R190 000 (the present value of the borrowed amount). So we need to solve for n in:

R 2 000 × [ 1 - ( 1 + 0 , 75 % ) - n ] / 0 , 75 % = R 190 000 1 - ( 1 + 0 , 75 % ) - n = ( 190 000 × 0 , 75 % 2 000 ) log ( 1 + 0 , 75 % ) - n = log [ ( 1 - 190 000 × 0 , 0075 2 000 ] - n × log ( 1 + 0 , 75 % ) = log [ ( 1 - 190 000 × 0 , 0075 2 000 ] - n × 0 , 007472 = - 1 , 2465 n = 166 , 8 months = 13 , 9 years

So the mortgage will be completely repaid in less than 14 years, and you would have made a total payment of 166,8 × R2 000 = R333 600.

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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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