Asset Pricing John H. Cochrane June 12, 2000

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Chapter 4. The discount factor Now we look more closely at the discount factor. Rather than derive a specific discount factor as with the consumption-based model in the last chapter, I work backwards. A discount factor is just some random variable that generates prices from payoffs, p = E(mx). What does this expression mean? Can one always find such a discount factor? Can we use this convenient representation without implicitly assuming all the structure of the investors, utility functions, complete markets, and so forth? The chapter focuses on two famous theorems. The law of one price states that if two portfolios have the same payoffs (in every state of nature), then they must have the same price. A violation of this law would give rise to an immediate kind of arbitrage profit, as you could sell the expensive version and buy the cheap version of the same portfolio. The first theorem states that there is a discount factor that prices all the payoffs by p = E(mx) if and only if this law of one price holds. In finance, we reserve the term absence of arbitrage for a stronger idea, that if payoff A is always at least as good as payoff B, and sometimes A is better, then the price of A must be greater than the price of B. The second theorem is that there is a positive discount factor that prices all the payoffs by p = E(mx) if and only if there are no arbitrage opportunities, so defined. These theorems are useful to show that we can use stochastic discount factors without implicitly assuming anything about utility functions, aggregation, complete markets and so on. All we need to know about investors in order to represent prices and payoffs via a discount factor is that they won’t leave law of one price violations or arbitrage opportunities on the table. These theorems can be used to describe aspects of a payoff space (such as law of one price, absence of arbitrage) by restrictions on the discount factor (such as it exists and it is positive). Chapter 18 shows how it can be more convenient to impose and check restrictions on a single discount factor than it is to check the corresponding restrictions on all possible portfolios. Chapter 7 discusses these and other implications of the existence theorems. The theorems are credited to Ross (1978), and Harrison and Kreps (1979). My presentation follows Hansen and Richard (1987). 4.1 Law of one price and existence of a discount factor Definition of law of one price; price is a linear function. p = E(mx) implies law of one price. The law of one price implies that a discount factor exists: There exists a unique x∗ in X such that p = E(x∗x) for all x ∈ X = space of all available payoffs. 64
SECTION 4.1 LAW OF ONE PRICE AND EXISTENCE OF A DISCOUNT FACTOR Furthermore, for any valid discount factor m, x ∗ = proj(m | X). So far we have derived the basic pricing relation p = E(mx) from environments with a lot of structure: either the consumption-based model or complete markets. Suppose we observe a set of prices p and payoffs x, and that markets — either the markets faced by investors or the markets under study in a particular application — are incomplete, meaning they do not span the entire set of contingencies. In what minimal set of circumstances does some discount factor exists which represents the observed prices by p = E(mx)? This section and the following answer this important question. This treatment is a simplified version of Hansen and Richard (1987), which contains rigorous proofs and some technical assumptions. Payoff space The payoff space X is the set (or a subset) of all the payoffs that investors can purchase, or it is a subset of the tradeable payoffs that is used in a particular study. For example, if there are complete contingent claims to S states of nature, then X = RS . But the whole point is that markets are (as in real life) incomplete, so we will generally think of X as a proper subset of complete markets RS . The payoff space will include some set of primitive assets, but investors can also form new payoffs by forming portfolios. I assume that investors can form any portfolio of traded assets: A1: (Portfolio formation) x1,x2 ∈ X ⇒ ax1 + bx2 ∈ X for any real a, b. Of course, X = RS for complete markets satisfies the portfolio formation assumption. If there is a single basic payoff x, then the payoff space must be at least the ray from the origin through x. If there are two basic payoffs in R3 , then the payoff space X must include the plane defined by these two payoffs and the origin. Figure 8 illustrates these possibilities. The payoff space is not the space of returns. The return space is a subset of the payoff space; if a return R is in the payoff space, then you can pay a price $2 to get a payoff 2R,so the payoff 2R with price 2 is also in the payoff space. Also, −R is in the payoff space. Free portfolio formation is in fact an important and restrictive simplifying assumption. It rules out short sales constraints, bid/ask spreads, leverage limitations and so on. The theory can be modified to incorporate these frictions, but it is a substantial modification. If investors can form portfolios of a vector of basic payoffs x (say, the returns on the NYSE stocks), then the payoff space consists of all portfolios or linear combinations of these original payoffs X = {c0x} where c is a vector of portfolio weights. We also can allow truly infinite-dimensional payoff spaces. For example, investors might be able to trade nonlinear 65
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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
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: SECTION 3.5 STATE DIAGRAM AND PRICE
Page 67 and 68: 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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