Appendix A: Derivation of Dividend Discount Model
D1 D2 D3 A.1 Summation of Infinite Geometric Series P0 ¼ þ þ þ (A.6) 1 þ k ðÞ1 þ k 2 ðÞ1 þ k 3 Summation of geometric series can be defined as: Where P0 ¼ present value of stock price per share D ¼ dividend per share in period t (t ¼ 1, 2,...,n) S ¼ A þ AR þ AR2 þ þARn 1 (A.1) t If dividends grow at a constant rate, say g, then, D ¼ D (1 + g), D ¼ D (1 + g) ¼ D (1 + g)2, and so on. Multiplying both sides of Equation A.1 by R, we obtain 2 1 3 2 1 Then, Equation A.6 can be rewritten as:
2 n 1 n RS ¼ AR þ AR þ þAR þ AR (A.2) 2 D1 D1ðÞ1 þ g D1ðÞ1 þ g P0 ¼ þ 2 þ 3 þ or, Subtracting Equation A.1 by Equation A.2, we obtain 1 þ k ðÞ1 þ k ðÞ1 þ k D1 D1 ðÞ1 þ g D1 P0 ¼ þ þ S RS ¼ A ARn 1 þ k ðÞ1 þ k ðÞ1 þ k ðÞ1 þ k ðÞ1 þ g 2 It can be shown 2 þ (A.7) ð1 þ kÞ AðÞ1 Rn S ¼ (A.3) Comparing Equation A.7 with Equation A.4, i.e., 1 R ; D1 1þg P0 ¼ S1 1þk ¼ A, and 1þk ¼ R as in the Equation A.4. n 1þg < > If R is smaller than 1, and n approaches to 1, then R Therefore, if 1þk 1orifk g, we can use Equation A.5 approaches to 0 i.e., to find out P0 i.e., 2 n 1 S1 ¼ A þ AR þ AR þ þAR þ 1; D1=ðÞ1 þ k þ AR (A.4) P ¼ 0 1 ½ ðÞ1 þ g =ðÞ1 þ k then, D1=ðÞ1 þ k ¼ ½ 1 þ k ðÞ1 þ g =ðÞ1 þ k
A D1=ðÞ1 þ k S1 ¼ (A.5) ¼ 1 R ðÞk g =ðÞ1 þ k D D ðÞ1 þ g ¼ 1 ¼ 0 k g k g A.2 Dividend Discount Model
Dividend Discount Model can be defined as:
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 911 # Springer Science+Business Media New York 2013 Appendix B: Derivation of DOL, DFL and DCL
B.1 DOL B.2 DFL
Let P ¼ price per unit 9 > V ¼ variable cost per unit Let i ¼ interest rate on = iD ¼ interest payment F total fixed cost outstanding debt on dept ¼ ;> Q ¼ quantity of goods sold D ¼ outstanding debt The definition of DOL can be defined as: N ¼ the total number of shares outstanding DOL (Degree of operating leverage) t ¼ corporate tax rate EAIT ¼ ½ QPðÞ V F iD ðÞ1 t
The definition of DFL can be defined as:
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 913 # Springer Science+Business Media New York 2013 914 Appendix B
B.3 DCL (Degree of Combined Leverage) Appendix C: Derivation of Crossover Rate
Suppose there are two projects under consideration. Cash Table A.1 NPV of Project A and B under different discount rates flows of project A, B and B – A are as follows: Discount rate (%) NPV (Project A) NPV (Project B) 0 1500.00 3500.00 Period 0 1 2 3 5 794.68 1725.46 Project A 10,500 10,000 1,000 1,000 10 168.67 251.31 Project B 10,500 1,000 1,000 12,000 15 390.69 984.10 Cash flows of B – A 0 9,000 0 11,000 20 893.52 2027.78