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# lecture4hout - MATH 006 Calculus and Linear Algebra(Lecture...

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MATH 006 Calculus and Linear Algebra (Lecture 4) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 19

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Outline 1 Present Value of an Annuity 2 Amortization Albert Ku (HKUST) MATH 006 2 / 19
Present Value of an Annuity Present Value of an Annuity Example How much should you deposit in a bank account paying 6% compounded semi-annually in order to be able to withdraw \$100 every 6 months for the next 3 years? (After the last withdrawal is made, no money is to be left in the account.) Remark Obviously, you do not need to deposit \$600 because part of your future withdrawals can be covered by the interest provided. From the bank’s viewpoint, the initial deposit can be regarded as the present value of the annuity . Albert Ku (HKUST) MATH 006 3 / 19

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Present Value of an Annuity Solution Idea: We compute the present value of each withdrawal one by one and sum them up. Present value of the 1st withdrawal: 100(1 + 0 . 06 2 ) - 1 Present value of the 2nd withdrawal: 100(1 + 0 . 06 2 ) - 2 Present value of the 3rd withdrawal: 100(1 + 0 . 06 2 ) - 3 ······ Present value of the 6th withdrawal: 100(1 + 0 . 06 2 ) - 6 Albert Ku (HKUST) MATH 006 4 / 19
Present Value of an Annuity Therefore, the amount of inital deposit is 100(1 + 0 . 06 2 ) - 1 + 100(1 + 0 . 06 2 ) - 2 + ··· + 100(1 + 0 . 06 2 ) - 6 = (1 + 0 . 06 2 ) - 6 ± 100 + 100(1 + 0 . 06 2 ) + ··· + 100(1 + 0 . 06 2 ) 5 ² = (1 + 0 . 06 2 ) - 6 ³ 100 · (1 + 0 . 06 2 ) 6 - 1 0 . 06 2 ! = 100 · 1 - (1 + 0 . 06 2 ) - 6 0 . 06 2 = \$541 . 72 Albert Ku (HKUST) MATH 006 5 / 19

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Present Value of an Annuity Formula for the Present Value of an Annuity We now introduce the formula of the present value of an annuity in the term of the notations used in ﬁnance. Theorem
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lecture4hout - MATH 006 Calculus and Linear Algebra(Lecture...

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