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Press Ctrl-A ©G Dear2008 – Not to be sold/Free to use 1 Present Value of an Annuity Stage 6 - Year 12 General Mathematic (HSC)

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22 Present Value of Annuity (1/6) This is the single amount of money which invested, as compound interest, under the same terms as the annuity that will produce the same financial result. N = A (1+r) n A A is the future value of the investment. N N is the present value of an annuity. r r is the rate of interest per period as a decimal. n n is the number of periods.

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33 Present Value of Annuity (2/6) Mary has an annuity that has a Future Value of $200 000 on her retirement in 30 years. The annuity is invested at 4.5% compounding annually. Calculate the NPV. N = 200 000 (1+0.045) 30 = 200 000 1.045 30 = $53 400.00 N = A (1+r) n

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44 Present Value of Annuity (3/6) Mary has an annuity that has a Future Value of $200 000 on her retirement in 30 years. The annuity is invested at 4.5% compounding annually. Calculate the NPV. P = N = $53 400.00 A = P (1 + r) n A = 53 400 (1 + 0.045) 30 A ≈ $ 1 999 999.99 = 53 400 x 1.045 30

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55 Present Value of Annuity (4/6) This is the single amount of money which invested, as compound interest, under the same terms as the annuity that will produce the same financial result. N =M (1+r) n -1 r(1+r) n M M is the amount of the equal periodical investments. N N is the present value of an annuity. r r is the rate of interest per period as a decimal. n n is the number of periods. { }

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6 6 Present Value of Annuity (5/6) N =M (1+r) n -1 r(1+r) n { } Mark has an annuity that has annual deposits of $2 000 for his retirement in 30 years. The annuity is invested at 4.5% compounding annually. Calculate the NPV. N = 2000 (1+0.045) 30 -1 0.045(1+0.045) 30 = 2000 x (1.045 30 -1) (0.045x1.045 30 ) = $32 577.78

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77 Present Value of Annuity (6/6) Mark has an annuity that has annual deposits of $20 000 for his retirement in 30 years. The annuity is invested at 4.5% compounding annually. Calculate the NPV. N = P = $32 577.78 A = P (1 + r) n A = 32 577.78 (1 + 0.045) 30 ≈ $ 122 014.15 = 32 577.78 x 1.045 30 { } A = MA = M (1+r) n -1 r { } A = 2000 (1 + 0.045) 30 - 1 0.045 = 2000 x (1.045 30 - 1) 0.045 = $122 014.14 Future Value of the Annuity

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