Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS E(R) R* + w i R e* R* σ(R) Figure 15. Orthogonal decomposition of a return R i in mean-standard deviation space. 5.4 Spanning the mean-variance frontier The characterization of the mean-variance frontier in terms of R ∗ and R e∗ is most natural in our setup. However, you can equivalently span the mean-variance frontier with any two portfolios that are on the frontier – any two distinct linear combinations of R ∗ and R e∗ .In particular, take any return UsingthisreturninplaceofR e∗ , R α = R ∗ + γR e∗ , γ 6= 0. (65) R e∗ = Rα − R ∗ γ you can express the mean variance frontier in terms of R ∗ and R α : where I have defined a new weight y = w/γ. R ∗ + wR e∗ = R ∗ + y (R α − R ∗ ) (5.66) = (1− y)R ∗ + yR α 88 n i R i
SECTION 5.5 A COMPILATION OF PROPERTIES OF R ∗ ,R e∗ AND x ∗ The most common alternative approach is to use a risk free rate or a risky rate that somehow behaves like the risk free rate in place of R e∗ to span the frontier. When there is a risk free rate, it is on the frontier with representation R f = R ∗ + R f R e∗ I derive this expression in equation (5.72) below. Therefore, we can use (5.66) with R a = R f . When there is no risk free rate, several risky returns that retain some properties of the risk free rate are often used. In section 5.3 below I present a “zero beta” return, which is uncorrelated with R ∗ , a “constant-mimicking portfolio” return, which is the return on the traded payoff closest to unity, ˆR = proj(1|X)/p[proj(1|X)] and the minimum variance return. Each of these returns is on the mean-variance frontier, with form 5.65, though different values of α. Therefore, we can span the mean-variance frontier with R ∗ and any of these risk-free rate proxies. 5.5 A compilation of properties of R ∗ ,R e∗ and x ∗ The special returns R∗ , Re∗ that generate the mean variance frontier have lots of interesting and useful properties. Some I derived above, some I will derive and discuss below in more detail, and some will be useful tricks later on. Most properties and derivations are extremely obscure if you don’t look at the picture! 1) E(R ∗2 1 )= E(x∗2 . (67) ) To derive this fact, multiply both sides of (5.62) by R∗ , take expectations, and remember R∗ is a return so 1=E(x∗R∗ ). 2) We can reverse the definition and recover x∗ from R∗ via x ∗ = R∗ E(R∗2 . (68) ) To derive this formula, start with the definition R∗ = x∗ /E(x∗2 ) and substitute from (5.67) for E(x∗2 ) 3) R∗ can be used to represent prices just like x∗ . This is not surprising, since they both point in the same direction, orthogonal to planes of constant price. Most obviously, from 5.68 p(x) =E(x ∗ x)= E (R∗x) E(R∗2 ∀x ∈ X ) For returns, we can nicely express this result as E(R ∗2 )=E(R ∗ R) ∀R ∈ R. (69) 89
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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 1.5 DISCOUNT FACTORS IN CON
Page 37 and 38: 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 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: SECTION 5.4 SPANNING THE MEAN-VARIA
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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