Appendix A: Derivation of Dividend Discount Model D1 D2 D3 A.1 Summation of Infinite Geometric Series P0 ¼ þ þ þÁÁÁ (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 þÁÁÁþARnÀ1 (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 nÀ1 n RS ¼ AR þ AR þÁÁÁþAR þ AR (A.2) 2 D1 D1ðÞ1 þ g D1ðÞ1 þ g P0 ¼ þ 2 þ 3 þÁÁÁ 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 À RS ¼ A À ARn 1 þ k ðÞ1 þ k ðÞ1 þ k ðÞ1 þ k ðÞ1 þ g 2 It can be shown  2 þÁÁÁ (A.7) ð1 þ kÞ AðÞ1 À Rn S ¼ (A.3) Comparing Equation A.7 with Equation A.4, i.e., 1 R À ; 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 nÀ1 S1 ¼ A þ AR þ AR þÁÁÁþAR þÁÁÁ 1; D1=ðÞ1 þ k þ AR (A.4) P ¼ 0 1 À ½ðÞ1 þ g =ðÞ1 þ k then, D1=ðÞ1 þ k ¼ ½1 þ k À ðÞ1 þ g =ðÞ1 þ k A D1=ðÞ1 þ k S1 ¼ (A.5) ¼ 1 À R ðÞk À g =ðÞ1 þ k D D ðÞ1 þ g ¼ 1 ¼ 0 k À g k À 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 ½QPðÞÀÀ V F À iD ðÞ1 À 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 À10,500 10,000 1,000 1,000 10 168.67 251.31 Project B À10,500 1,000 1,000 12,000 15 À390.69 À984.10 Cash flows of B – A 0 À9,000 0 11,000 20 À893.52 À2027.78 Based upon the information the table above we can cal- ; ; ; ; 1 000 10 000 culate the NPV of Project A and Project B under different 0 ¼½þ10 500 ÀÀðÞ10 500 À "#1"#þ Rc 1 þ Rc discount rates. The results are presented in Table A.1. 1; 000 1; 000 12; 000 1; 000 NPV(B) is higher with low discount rates and NPV(A) þ 2 À 2 þ 3 À 3 is higher with high discount rates. This is because the cash ðÞ1 þ Rc ðÞ1 þ Rc ðÞ1 þ Rc ðÞ1 þ Rc flows of project A occur early and those of project B occur (A.10) later. If we assume a high discount rate, we would favor project A; if a low discount rate is expected, project B will be Solving Equation A.10 by trial and error method for Rc,Rc chosen. In order to make the right choice, we can calculate equals 10.55%. the crossover rate. If the discount rate is higher than the Using the procedure of calculating internal rate of return crossover rate, we should choose project A; if otherwise, (IRR) as discussed in Equations A.8, A.9, and A.10,we we should go for project B. The crossover rate, Rc, is the calculate the IRR for both Project A and Project B. The rate such that NPV(A) equals to NPV(B). IRR for Project A and B are 11.45% and 10.95% respec- Suppose the crossover rate is Rc, then tively. From this information, we have concluded that Project A will perform better than Project B without consideration NPVðAÞ¼ À10; 500 þ 10; 000=ð1 þ RcÞþ1; 000= for change of discount rate. Therefore, the IRR decision rule 2 3 cannot be used for capital budgeting decisions when there ðÞ1 þ Rc þ 1; 000=ðÞ1 þ Rc (A.8) exists an increasing or decreasing net cash inflow. This is so called “The Timing Problem” for using the IRR method for NPVðBÞ¼ À10; 500 þ 1; 000=ð1 þ RcÞþ1; 000= 2 3 capital budgeting decisions. ðÞ1 þ Rc þ 12; 000=ð1 þ RcÞ NPVðAÞ¼NPVðBÞ (A.9) Therefore, Rearranging the above equation (moving all terms on the LHS to the RHS), we obtain Equation A.10 C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 915 # Springer Science+Business Media New York 2013 Appendix D: Capital Budgeting Decisions with Different Lives D.1 Mutually Exclusive Investment Projects Subtracting Equation A.12 from Equation A.11 gives: with Different Lives ÀÁ NPVðÞÀð N; t HÞNPVðÞ¼ N; t NPVðNÞ 1 À Htþ1 ÀÁ The traditional NPV technique may not be the appropriate NPVðNÞ 1 À Htþ1 criterion to select a project from mutually exclusive invest- NPVðÞ¼ N; t 1 À H ment projects, if these projects have different lives. The underlying reason is that, compared with a long-life project, Taking the limit as the number of replications, t, a short-life project can be replicated more quickly in the long approaches infinity gives: run. In order to compare projects with different lives, we compute the NPV of an infinite replication of the investment lim NPVðÞ¼ N; t NPVðÞ N; 1 project. For example, let Projects A and B be two mutually x!1 2 3 exclusive investment projects with the following cash flows. 1 ¼ NPV4 hi5 Year Project A Project B 1 À 1=ðÞ1 þ K N 0 100 100 "# N 17050 ðÞ1 þ K ¼ NPVðNÞ (A.13) 27050 ðÞ1 þ K N À 1 350 Equation A.13 is the NPV of an N-year project replicated By assuming a discount rate of 12%, the traditional NPV at constant scale an infinite number of times. We can use it to of Project A is 18.30 and the NPV of Project B is 20.09. This compare projects with different lives because when their shows that Project B is a better choice than Project A. cash-flow streams are replicated forever, it is as if they had However, the NPV with infinite replications for Project A the same (infinite) life. and B should be adjusted into a comparable basis. Based upon Equation A.13, we can calculate the NPV of In order to compare Projects A and B, we compute the Projects A and B as follows: NPV of an infinite stream of constant scale replications. Let NPV (N, 1) be the NPV of an N-year project with NPV (N), replicated forever. This is exactly the same as an annuity For Project A For Project B paid at the beginning of the first period and at the end of NPVðÞ2; 1 NPVðÞ3; 1 "#"# every N years from that time on. The NPV of the annuity is: ðÞ1 þ 0:12 2 ðÞ1 þ 0:12 3 ¼ NPVð2Þ ¼ NPVð3Þ : 2 : 3 ; NPVðNÞ NPVðNÞ ðÞ1 þ 0 12 À 1 ðÞ1 þ 0 12 À 1 NPVðÞ¼ N 1 NPVðNÞþ N þ 2N þÁÁ ðÞ1 þ K ðÞ1 þ K 1:2544 1:4049 ¼ ðÞ18:30 ¼ 20:09 In order to obtain a closed-form formula, let 0:2544 0:4049 (1/[(1 + K)N]) ¼ H.Thenwehave: ¼ 90:23 ¼ 69:71 ÀÁ NPV N; t NPV N 1 H H2 Ht (A.11) ðÞ¼ ð Þ þ þ þÁÁþ Consequently, we would choose to accept Project A over Multiplying both sides by H, this becomes Project B, because, when the cash flows are adjusted for different lives, A provides the greater cash flow. À Alternatively, Equation A.13 can be rewritten as an equiv- H½¼ NPVðÞ N; t NPVðNÞ H þ H2 Á alent annual NPV version as: þþHt þ Htþ1 (A.12) C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 917 # Springer Science+Business Media New York 2013 918 Appendix D NPVðNÞ Assume company A buys a machine that costs $1,000 and K  NPVðÞ¼ N; 1 (A.14) Annuity factor the maintenance expense of $250 is to be paid at the end of each of the 4 years.