Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 4 THE DISCOUNT FACTOR inner product. The the Riesz representation theorem extends the proof to infinite-dimensional payoff spaces. See Hansen and Richard (1987). Proof 2: (Algebraic.) We can prove the theorem by construction when the payoff space X is generated by portfolios of a N basis payoffs (for example, N stocks). This is a common situation, so the formulas are also useful in practice. Organize the basis payoffs into a vector x = £ x1 x2 ... xN ¤ 0 and similarly their prices p. The payoff space is then X = {c 0 x}. We want a discount factor that is in the payoff space, as the theorem requires. Thus, it must be of the form x ∗ = c 0 x. Construct c so that x ∗ prices the basis assets. We want p =E(x ∗ x)= E(xx 0 c). Thus we need c = E(xx 0 ) −1 p.IfE(xx 0 ) is nonsingular, this c exists and is unique. A2 implies that E(xx 0 ) is nonsingular (after pruning redundant rows of x). Thus, x ∗ = p 0 E(xx 0 ) −1 x (51) is our discount factor. It is a linear combination of x so it is in X. It prices the basis assets x by construction. It prices every x ∈ X : E[x ∗ (x 0 c)] = E[p 0 E(xx 0 ) −1 xx 0 c]=p 0 c. By linearity, p(c 0 x)=c 0 p. What the theorem does and does not say The theorem says there is a unique x∗ in X. There may be many other discount factors m not in X. In fact, unless markets are complete, there are an infinite number of random variables that satisfy p = E(mx). Ifp = E(mx) then p = E [(m + ε)x] for any ε orthogonal to x, E(εx) =0. Not only does this construction generate some additional discount factors, it generates all of them: Any discount factor m (any random variable that satisfies p = E(mx)) can be represented as m = x∗ +ε with E(εx) =0. Figure 10 gives an example of a one-dimensional X in a two-dimensional state space, in which case there is a whole line of possible discount factors m. If markets are complete, there is nowhere to go orthogonal to the payoff space X, so x∗ is the only possible discount factor. Reversing the argument, x∗ is the projection of any stochastic discount factor m on the space X of payoffs. This is a very important fact: the pricing implications of any discount factor m forasetofpayoffsXarethe same as those of the projection of m on X. This discount factor is known as the mimicking portfolio for m. Algebraically, p = E(mx) =E [(proj(m|X)+ε)x] =E [proj(m|X) x] Let me repeat and emphasize the logic. Above, we started with investors or a contingent claim market, and derived a discount factor. p = E(mx) implies the linearity of the pricing function and hence the law of one price, a pretty obvious statement in those contexts. Here we work backwards. Markets are incomplete in that contingent claims to lots of states of nature are not available. We found that the law of one price implies a linear pricing function, and a linear pricing function implies that there exists at least one and usually many discount factors. 68
SECTION 4.2 NO-ARBITRAGE AND POSITIVE DISCOUNT FACTORS x * Payoff space X m = x * + ε space of discount factors Figure 10. Many discount facotors m can price a given set of assets in incomplete markets. We do allow arbitrary portfolio formation, and that sort of “completeness” is important to the result. If investors cannot form a portfolio ax + by, they cannot force the price of this portfolio to equal the price of its constituents. The law of one price is not innocuous; it is an assumption about preferences albeit a weak one. The point of the theorem is that this is just enough information about preferences to deduce the existence of a discount factor. 4.2 No-Arbitrage and positive discount factors The definition of arbitrage: positive payoff implies positive price. There is a strictly positive discount factor m such that p = E(mx) ifandonlyifthereare no arbitrage opportunities. No arbitrage is another, slightly stronger, implication of marginal utility, that can be reversed to show that there is a positive discount factor. We need to start with the definition of arbitrage: Definition (Absence of arbitrage): A payoff space X and pricing function p(x) leave no arbitrage opportunities if every payoff x that is always non-negative, x ≥ 0 (almost surely), and positive, x>0, with some positive probability, has positive 69
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Asset Pricing John H. Cochrane June
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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
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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 and 66: SECTION 4.1 LAW OF ONE PRICE AND EX
Page 67: SECTION 4.1 LAW OF ONE PRICE AND EX
Page 71 and 72: SECTION 4.2 NO-ARBITRAGE AND POSITI
Page 73 and 74: 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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