Lecture Notes for Finance 1 (and More).

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Gordon’s growth formula capital budgeting 28 CHAPTER 3. PAYMENT STREAMS UNDER CERTAINTY Another classical formula concerns the present value of a geometrically growing payment stream (c, c(1 + g), . . ., c(1 + g) n−1 ) as = = n� i=1 (1 + g)i−1 c (1 + r) i c �n−1 1 + r c r − g (1 + g) i=0 i (1 + r) i � � �n� 1 + g 1 − . 1 + r Although we have not taken into account the possibility of infinite payment streams, we note for future reference, that for 0 ≤ g < r we have what is known as Gordon’s growth formula: ∞� i=1 c(1 + g) i−1 c = (1 + r) i r − g . 3.4.1 Some rules that are inconsistent with the NPV rule. Corresponding to our definition of internal rate of return in Chapter 3, we define an internal rate of return on a project c with initial cost c0 > 0, denoted IRR(c0; c), as a solution to the equation c0 = T� i=1 ci (1 + x) i, x > −1 As we have noted earlier such a solution need not be unique unless c > 0 and c0 > 0. Note that an internal rate of return is defined without referring to the underlying term structure. The internal rate of return describes the level of a flat term structure at which the NPV of the project is 0. The idea behind its use in capital budgeting would t hen be to say that the higher the level of the interest rate, the better the project (and some sort of comparison with the existing term structure would then be appropriate when deciding whether to accept the project at all). But as we will see in the following example, IRR and NPV may disagree on which project is better: Consider the projects shown in the table below (whose last column shows a discount function d):
3.4. IRR, NPV AND CAPITAL BUDGETING UNDER CERTAINTY. 29 date proj 1 proj 2 d 0 -100 -100 1 1 50 50 0.95 2 5 80 0.85 3 90 4 0.75 IRR 0.184 0.197 - NPV 19.3 18.5 - Project 2 has a higher IRR than project 1, but 1 has a larger NPV than 2. Using the same argument as in the previous section it is easy to check, that even if a cash flow similar to that of project 2 is desired by an investor, he would be better off investing in project 1 and then reforming the flow of payments using the capital market. Another problem with trying to use IRR as a decision variable arises when the IRR is not uniquely defined - something which typically happens when the cash flows exhibit sign changes. Which IRR should we then choose? One might also contemplate using the payback method and count the number of years it takes to recover the initial cash outlay - possibly after discounting appropriately the future cash flows. Project 2 in the table has a payback of 2 years whereas project 1 has a payback of three years. The example above therefore also shows that choosing projects with the shortest payback time may be inconsistent with the NPV method. 3.4.2 Several projects Consider someone with c0 > 0 available at date 0 who wishes to allocate this capital over the T + 1 dates, and who considers a project c with initial cost c0. We have seen that precisely when NPV (c0; c) > 0 this person will be able to obtain better cash flows by adopting c and trading in the capital market than by trading in the capital market alone. When there are several projects available the situation really does not change much: Think of the i ′ th project (p i 0 , p) as an element of a set Pi ⊂ R T+1 . Assume that 0 ∈ Pi all i representing the choice of not starting the i’th project. For a non-scalable project this set will consist of one point in addition to 0. Given a collection of projects represented by (Pi) i∈I . Situations where there is a limited amount of money to invest at the beginning (and borrowing is not permitted), where projects are mutually exclusive etc. may then be described abstractly by the requirement that the collection of selected projects (p i 0 , pi )i∈I are chosen from a feasible subset P of the Cartesian product ×i∈IPi. The NPV of the chosen collection of projects is then just the sum
Page 1: Lecture Notes for Finance 1 (and Mo
Page 4 and 5: 4 CHAPTER 1. PREFACE
Page 6 and 7: 6 CHAPTER 2. INTRODUCTION A key rol
Page 8 and 9: 8 CHAPTER 2. INTRODUCTION will have
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Page 12 and 13: financial market security price sys
Page 14 and 15: 14 CHAPTER 3. PAYMENT STREAMS UNDER
Page 16 and 17: yield to maturity term structure of
Page 18 and 19: annuity serial loan bullet bond ann
Page 20 and 21: ullet bond serial loan 20 CHAPTER 3
Page 26 and 27: NPV criterion 26 CHAPTER 3. PAYMENT
Page 32 and 33: duration, Macaulay convexity 32 CHA
Page 40 and 41: inomial model, one-period stock mon
Page 42 and 43: 42 CHAPTER 4. ARBITRAGE PRICING IN
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Page 48 and 49: utility function equilibrium state-
Page 52 and 53: call option, European 52CHAPTER 5.
Page 54 and 55: eplication hedging derivative secur
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78 CHAPTER 6. OPTION PRICING Typica
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80 CHAPTER 6. OPTION PRICING p uS0
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hedging call option, European 82 CH
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84 CHAPTER 6. OPTION PRICING delta
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86 CHAPTER 6. OPTION PRICING S0�
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put option, American 88 CHAPTER 6.
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90 CHAPTER 6. OPTION PRICING In oth
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volatility smile firm debt equity 9
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94 CHAPTER 6. OPTION PRICING clear
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96 CHAPTER 7. THE BLACK-SCHOLES FOR
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Central Limit Theorem Black-Scholes
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100 CHAPTER 7. THE BLACK-SCHOLES FO
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zero coupon bond, ZCB short rate pr
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110 CHAPTER 8. STOCHASTIC INTEREST
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term structure of interest rates, p
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forward contract futures contract 1
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ate of return relative portfolio we
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mean/variance analysis Markowitz an
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minimum variance portfolio minimum
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136 CHAPTER 9. PORTFOLIO THEORY Exp
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138 CHAPTER 9. PORTFOLIO THEORY var
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capital market line, CML tangent po
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142 CHAPTER 9. PORTFOLIO THEORY in
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144 CHAPTER 9. PORTFOLIO THEORY wil
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146 CHAPTER 9. PORTFOLIO THEORY Rol
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148 CHAPTER 9. PORTFOLIO THEORY tha
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150 CHAPTER 9. PORTFOLIO THEORY in
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152 CHAPTER 9. PORTFOLIO THEORY
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portfolio arbitrage opportunity APT
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156CHAPTER 10. FACTOR MODELS OF RET
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Arrow-Debreu prices debt equity bon
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166CHAPTER 11. CORPORATE FINANCE: F
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ankruptcy costs NPV criterion 168CH
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174 CHAPTER 12. EFFICIENT CAPITAL M
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Index (·) + -notation, 41 geometri
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188 INDEX stock holders, 164 swap c
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