Asset Pricing John H. Cochrane June 12, 2000

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State 2 CHAPTER 4 THE DISCOUNT FACTOR x Single Payoff in R 2 X State 1 State 2 x 2 State 3 (into page) Two Payoffs in R 3 Figure 8. Payoff spaces X generated by one (left) and two (right) basis payoffs. x 1 State 1 functions of a basis payoff x, such as call options on x with strike price K, which have payoff max [x(s) − K, 0] . The law of one price. A2: (Law of one price, linearity) p(ax1 + bx2) =ap(x1)+bp(x2) It doesn’t matter how one forms the payoff x. The price of a burger, shake and fries must be the same as the price of a happy meal. Graphically, if the iso-price curves were not planes, then one could buy two payoffs on the same iso-price curve, form a portfolio whose payoff is on the straight line connecting the two original payoffs, and sell the portfolio for a higher price than it cost to assemble it. The law of one price basically says that investors can’t make instantaneous profits by repackaging portfolios. If investors can sell securities, this is a very weak characterization of preferences. It says there is at least one investor for whom marketing doesn’t matter, who values a package by its contents. The law is meant to describe a market that has already reached equilibrium. If there are any violations of the law of one price, traders will quickly eliminate them so they can’t survive in equilibrium. A1 and A2 also mean that the 0 payoff must be available, and must have price 0. The Theorem The existence of a discount factor implies the law of one price. This is obvious to the point of triviality: if x = y + z then E(mx) =E[m(y + z)]. The hard, and interesting part of the theorem reverses this logic. We show that the law of one price implies the existence of a discount factor. 66
SECTION 4.1 LAW OF ONE PRICE AND EXISTENCE OF A DISCOUNT FACTOR Theorem: Given free portfolio formation A1, and the law of one price A2, there exists a unique payoff x ∗ ∈ X such that p(x) =E(x ∗ x) for all x ∈ X. x∗ is a discount factor. A1 and A2 imply that the price function on X looks like Figure 7: parallel hyperplanes marching out from the origin. The only difference is that X may be a subspace of the original state space, as shown in Figure 8. The essence of the proof, then, is that any linear function on a space X can be represented by inner products with a vector that lies in X. Proof 1: (Geometric.) We have established that the price is a linear function as shown in Figure 9. (Figure 9 can be interpreted as the plane X of a larger dimensional space as in the right hand panel of Figure 8, laid flat on the page for clarity.) Now we can draw a line from the origin perpendicular to the price planes. Choose a vector x∗on this line. Since the line is orthogonal to the price zero plane we have 0=p(x) =E(x∗x) for price zero payoffs x immediately. The inner product between any payoff x on the price = 1 plane and x∗ is |proj(x|x∗ )|×|x∗ | Thus, every payoff on the price = 1 plane has the same inner product with x∗ . All we have to do is pick x∗ to have the right length, and we obtain p(x) =1= E(x∗x) for every x on the price = 1 plane. Then, of course we have p(x) =E(x∗x) for payoffs x on the other planes as well. Thus, the linear pricing function implied by the Law of One Price can be represented by inner products with x∗ . ¤ x* Price = 1 (returns) Price = 0 (excess returns) Figure 9. Existence of a discount factor x ∗ . Price = 2 The basic mathematical point is just that any linear function can be represented by an 67
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Asset Pricing John H. Cochrane June
Page 3 and 4:
Contents Acknowledgments 2 Preface
Page 5 and 6:
9.2 Intertemporal Capital Asset Pri
Page 7 and 8:
19.2 Yield curve and expectations h
Page 9 and 10:
Asset pricing problems are solved b
Page 11 and 12:
try to focus on the basic ideas and
Page 13 and 14:
Chapter 1. Consumption-based model
Page 15 and 16: SECTION 1.1 BASIC PRICING EQUATION
Page 17 and 18: discount factors, SECTION 1.3 PRICE
Page 19 and 20: SECTION 1.4 CLASSIC ISSUES IN FINAN
Page 23 and 24: equation (1.2) as SECTION 1.4 CLASS
Page 25 and 26: ate! In equations, if then SECTION
Page 33 and 34: SECTION 1.5 DISCOUNT FACTORS IN CON
Page 39 and 40: SECTION 1.6 PROBLEMS function with
Page 41 and 42: Chapter 2. Applying the basic model
Page 43 and 44: SECTION 2.2 GENERAL EQUILIBRIUM mig
Page 45 and 46: C t+1 SECTION 2.2 GENERAL EQUILIBRI
Page 47 and 48: SECTION 2.3 CONSUMPTION-BASED MODEL
Page 49 and 50: SECTION 2.4 ALTERNATIVE ASSET PRICI
Page 51 and 52: SECTION 2.5 PROBLEMS Since the opti
Page 53 and 54: that function. From SECTION 2.5 PRO
Page 55 and 56: SECTION 3.2 RISK NEUTRAL PROBABILIT
Page 57 and 58: The discount factor prices the asse
Page 59 and 60: SECTION 3.4 RISK SHARING Asset pric
Page 61 and 62: State 2 Payoff SECTION 3.5 STATE DI
Page 63 and 64: SECTION 3.5 STATE DIAGRAM AND PRICE
Page 65: SECTION 4.1 LAW OF ONE PRICE AND EX
Page 69 and 70: SECTION 4.2 NO-ARBITRAGE AND POSITI
Page 75 and 76: so SECTION 4.3 AN ALTERNATIVE FORMU
Page 77 and 78: Chapter 5. Mean-variance frontier a
Page 79 and 80: SECTION 5.1 EXPECTED RETURN -BETA R
Page 81 and 82: SECTION 5.2 MEAN-VARIANCE FRONTIER:
Page 83 and 84: SECTION 5.3 AN ORTHOGONAL CHARACTER
Page 85 and 86: SECTION 5.3 AN ORTHOGONAL CHARACTER
Page 87 and 88: SECTION 5.4 SPANNING THE MEAN-VARIA
Page 89 and 90: SECTION 5.5 A COMPILATION OF PROPER
Page 91 and 92: SECTION 5.5 A COMPILATION OF PROPER
Page 93 and 94: SECTION 5.6 MEAN-VARIANCE FRONTIERS
Page 95 and 96: SECTION 5.6 MEAN-VARIANCE FRONTIERS
Page 97 and 98: SECTION 5.7 PROBLEMS Hansen-Jaganna
Page 99 and 100: SECTION 6.1 FROM DISCOUNT FACTORS T
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Asset Pricing John H. Cochrane June 12, 2000

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