78106_07_Web_Ch07A_p01-02

78106_07_Web_Ch07A_p01-02 - WEB EXTENSION Derivation of...

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Unformatted text preview: WEB EXTENSION Derivation of Valuation Equations 7A T he derivation of the formulas for the values of a perpetual preferred stock and the values of a constant growth stock are presented in this extension. VALUES OF A PERPETUAL PREFERRED STOCK Dps 1 rps 1 The value of a perpetual preferred stock is given by Vps Dps 1 rps 2 ... Dps 1 rps (7A-1) For a partial sum up to N, Equation 7A-1 may be rewritten as follows: " Vps Dps 1 1 1 ... 1 2 1 rps 1 rps 1 rps N # (7A-2) Multiply both sides of Equation 7A-2 by (1 + rps), which yields " Vps 1 rps Dps 1 1 1 ... 1 1 2 1 rps 1 rps 1 rps N - 1 # (7A-3) Subtract Equation 7A-2 from Equation 7A-3 to obtain " Vps 1 rps - 1 Dps 1 - 1 1 rps N # (7A-4) As N , 1/(1 + rps)N 0 and Equation 7A-4 approaches Vps rps Dps Thus we obtain Equation 7-8 from the chapter: Vps Dps rps 1 2 Web Extension 7A: Derivation of Valuation Equations Although we used preferred stock notation, Equations 7-3 and 7-8 are valid for any perpetuity. VALUES OF A CONSTANT GROWTH STOCK The proof of Equation 7-2, the chapter's formula for the value of a constant growth ^ stock, P0 = D1/(rs - g), is developed as follows. Rewrite Equation 7A-1 as 1 2 3 N ^ P0 D0 1 g D0 1 g D0 1 g ... D0 1 g 1 rs 1 1 rs 2 1 rs 3 1 rs N " # 1 g1 1 g2 1 g3 ... 1 gN D0 1 rs 1 1 rs 2 1 rs 3 1 rs N (7A-5) Multiply both sides of Equation 7A-5 by (1 + rs)/(1 + g): " # ^ 1 rs P 1 g1 1 g2 ... 1 gN-1 0 D0 1 1 g 1 rs 1 1 rs 2 1 rs N-1 (7A-6) Subtracting Equation 7A-5 from Equation 7A-6, we obtain " # ^ 1 rs 1 gN -1 P0 D0 1- 1 g 1 rs N (7A-6) Assume that rs > g. Then, as N approaches infinity, the term in brackets on the righthand side of the equation approaches unity, leaving ^ 1 rs -1 P0 D0 1 g This can be rearranged as ^ 1 rs - 1 g P 0 D0 1 g which simplifies to Equation 7-2 in the chapter: ^ rs - gP0 D0 1 g D1 ^ P0 D1 rs - g ...
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This note was uploaded on 05/03/2010 for the course FRR 3032 taught by Professor Mr.wroshr during the Spring '10 term at Crafton Hills College.

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