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ADVANCED PROGRAMME MATHEMATICS SECTION: FINANCE

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LOANS EXAMPLE 1 A home loan of R is amortized over a period of 20 10,5% p.a. compounded monthly. Determine (a)the monthly payment (b)the value of the loan plus interest after 20 years (c)the value of the repayments plus interest after 20 years

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ANSWER: R i (12) = 10,5% p.j. (a)

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(b) (c) R i (12) = 10,5% p.j.

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Note: At any stage:

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Balance = ? Balance = 0 ANSWER We determine the future value (F) of the loan after 6 years and subtract the future value (F v ) of the first 6 years' 72 paid payments. EXAMPLE 2 Determine the outstanding balance of the house loan mentioned above after 6 years.

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ALTERNATIVE METHOD Calculate the outstanding balance after 6 years by not using a future value but a present value annuity. (A shorter way!)

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Balance = ? Balance = 0 We regard the 6 th year as the present (the “now”) and then determine the present value of the following 14 years’ 168 unpaid instalments. (I.o.w. what must still be paid at this moment if it can be paid in cash. The future interest gets discounted.)

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Balance = ? Balance = 0 We regard the 6 th year as the present (the “now”) and then determine the present value of the following 14 years’ 168 unpaid instalments.

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THEREFORE ALTERNATIVE METHOD CONCLUSION To calculate the outstanding balance, we therefore calculate the present value of the unpaid instalments of the loan. SUMMARY:

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PROOF THAT the 2 methods produce the same result. Where k = the number of payments already made n = the total number of payments therefore n – k = the number of outstanding payments...And this is the present value of the unpaid / outstanding payments!!

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In more detail:

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Do Exercise 5.1 p. 170 Enrichment The notation a n ┐ i ( read: a angle n at i ) is used for the factor to be multiplied to a regular payment x to calculate the present value of the annuity.

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