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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 (7A1) For a partial sum up to N, Equation 7A1 may be rewritten as follows: " Vps Dps 1 1 1 ... 1 2 1 rps 1 rps 1 rps N # (7A2) Multiply both sides of Equation 7A2 by (1 + rps), which yields " Vps 1 rps Dps 1 1 1 ... 1 1 2 1 rps 1 rps 1 rps N  1 # (7A3) Subtract Equation 7A2 from Equation 7A3 to obtain " Vps 1 rps  1 Dps 1  1 1 rps N # (7A4) As N , 1/(1 + rps)N 0 and Equation 7A4 approaches Vps rps Dps Thus we obtain Equation 78 from the chapter: Vps Dps rps 1 2 Web Extension 7A: Derivation of Valuation Equations Although we used preferred stock notation, Equations 73 and 78 are valid for any perpetuity. VALUES OF A CONSTANT GROWTH STOCK The proof of Equation 72, the chapter's formula for the value of a constant growth ^ stock, P0 = D1/(rs  g), is developed as follows. Rewrite Equation 7A1 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 (7A5) Multiply both sides of Equation 7A5 by (1 + rs)/(1 + g): " # ^ 1 rs P 1 g1 1 g2 ... 1 gN1 0 D0 1 1 g 1 rs 1 1 rs 2 1 rs N1 (7A6) Subtracting Equation 7A5 from Equation 7A6, we obtain " # ^ 1 rs 1 gN 1 P0 D0 1 1 g 1 rs N (7A6) 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 72 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.
 Spring '10
 MR.Wroshr

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