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Discounting Notes 2

# Discounting Notes 2 - Discounting Future Cash Flows Daniel...

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Discounting Future Cash Flows Daniel K. Saunders 1 Geometric Series 1.1 Perpetuity (Infinitely Many Terms) For a geometric series , let 0 < x < 1. Then the following is true X n =0 x n = 1 1 - x With this fact we can show that . 9999 ... = 1 . First, notice that 0 . 9999 .... = 9 10 + 9 100 + 9 1000 + ... = 9 10 1 + 1 10 + 1 100 + ... Then the series inside of the paranthesis is a geometric series , and we know that X n =0 1 10 n = 1 1 - 1 10 = 1 9 10 = 10 9 Therefore, we are left with . 9999 ... = 9 10 × 10 9 = 1 1.2 Annuity (Finitely Many Terms) With finitely many terms (0 < x < 1) the adjusted result of a geometric series is as follows N - 1 X n =0 x n = 1 - x N 1 - x Notice that this result is consistent with the infinite case, since lim N →∞ N - 1 X n =0 x n = lim N →∞ 1 - x N 1 - x = 1 1 - x for 0 < x < 1 1

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2 Application to Finance 2.1 A Technical Note For this course, we consider the current period ( t = 0) as a separate issue from discounting future values. For example, if you look at the first set of lecture notes regarding the present value of an annuity , you will notice that the index, n , begins with n = 1.
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