Asset Pricing John H. Cochrane June 12, 2000

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Chapter 6. Relation between discount factors, betas, and mean-variance frontiers In this chapter, I draw the connection between discount factors, mean-variance frontiers, and beta representations. In the first chapter, I showed how mean-variance and a beta representation follow from p = E(mx) and (in the mean-variance case) complete markets. Here, I discuss the connections in both directions and in incomplete markets, drawing on the representations studied in the last chapter. The central theme of the chapter is that all three representations are equivalent. Figure 18 summarizes the ways one can go from one representation to another. A discount factor, a reference variable for betas – the thing you put on the right hand side in the regressions that give betas – and a return on the mean-variance frontier all carry the same information, and given any one of them, you can find the others. More specifically, 1. p = E(mx) ⇒ β: Givenm such that p = E(mx), m, x ∗ ,R ∗ ,orR ∗ + wR e∗ all can serve as reference variables for betas. 2. p = E(mx) ⇒ mean-variance frontier. You can construct R ∗ from x ∗ = proj(m|X), R ∗ = x ∗ /E(x ∗2 ),andthenR ∗ , R ∗ + wR e∗ are on the mean-variance frontier. 3. Mean-variance frontier ⇒ p = E(mx). IfR mv is on the mean-variance frontier, then m = a + bR mv linear in that return is a discount factor; it satisfies p = E(mx). 4. β ⇒ p = E(mx). If we have an expected return/beta model with factors f, then m = b 0 f linear in the factors satisfies p = E(mx) (and vice-versa). 5. If a return is on the mean-variance frontier, then there is an expected return/beta model with that return as reference variable. The following subsections discuss the mechanics of going from one representation to the other in detail. The last section of the chapter collects some special cases when there is no riskfree rate. The next chapter discusses some of the implications of these equivalence theorems, and why they are important. Roll (197x) pointed out the connection between mean-variance frontiers and beta pricing. Ross (1978) and Dybvig and Ingersoll (1982) pointed out the connection between linear discount factors and beta pricing. Hansen and Richard (1987) pointed out the connection between a discount factor and the mean-variance frontier. 6.1 From discount factors to beta representations m, x ∗ , and R ∗ can all be the single factor in a single beta representation. 98
SECTION 6.1 FROM DISCOUNT FACTORS TO BETA REPRESENTATIONS f = m, x*, R* E(R i ) = α + β i’λ m = b’f proj(f|R) on m.v.f. f = R mv LOOP m exists p = E(mx) m = a + bR R* is on m.v.f. mv R mv on m.v.f. E(RR’) nonsingular R mv exists Figure 18. Relation between three views of asset pricing. 6.1.1 Beta representation using m p = E(mx) implies E(R i )=α + β i,mλm. Startwith Thus, 1=E(mR i )=E(m)E(R i )+cov(m, R i ). E(R i )= 1 E(m) − cov(m, Ri ) . E(m) Multiply and divide by var(m),define α ≡ 1/E(m) to get E(R i µ µ i cov(m, R ) )=α + − var(m) var(m) = α + β E(m) i,mλm. As advertised, we have a single beta representation. For example, we can equivalently state the consumption-based model as: mean asset returns should be linear in the regression betas of asset returns on (ct+1/ct) −γ .Furthermore, the slope of this cross-sectional relationship λm is not a free parameter, though it is usually treated as such in empirical evaluation of factor pricing models. λm should equal the ratio of 99
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Asset Pricing John H. Cochrane June
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Contents Acknowledgments 2 Preface
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9.2 Intertemporal Capital Asset Pri
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19.2 Yield curve and expectations h
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Asset pricing problems are solved b
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try to focus on the basic ideas and
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Chapter 1. Consumption-based model
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SECTION 1.1 BASIC PRICING EQUATION
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discount factors, SECTION 1.3 PRICE
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SECTION 1.4 CLASSIC ISSUES IN FINAN
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equation (1.2) as SECTION 1.4 CLASS
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ate! In equations, if then SECTION
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SECTION 1.5 DISCOUNT FACTORS IN CON
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SECTION 1.6 PROBLEMS function with
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Chapter 2. Applying the basic model
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SECTION 2.2 GENERAL EQUILIBRIUM mig
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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 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: SECTION 5.7 PROBLEMS Hansen-Jaganna
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Asset Pricing John H. Cochrane June 12, 2000

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