Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS The construction (5.79) can be used to derive the formula (5.78) for the Hansen-Jagannathan bound for the finite-dimensional cases discussed above. It’s more general, since it can be used in infinite-dimensional payoff spaces as well. Along with the corresponding return formula Rmv = R∗ + wRe∗ , we see in Chapter 8 that it extends more easily to the calculation of conditional vs. unconditional mean-variance frontiers (Gallant, Hansen and Tauchen 1995). It will make construction (5.79) come alive if we find equations for its components. We find x∗ as before, it is the portfolio c0x in X that prices x: x ∗ = p 0 E(xx 0 ) −1 x. Similarly, let’s find e ∗ . The projection of 1 on X is proj(1|X) =E(x) 0 E(xx 0 ) −1 x. (After a while you get used to the idea of running regressions with 1 on the left hand side and random variables on the right hand side!) Thus, e ∗ =1− E(x) 0 E(xx 0 ) −1 x. Again, you can construct time-series of x∗and e∗ from these definitions. Finally, we now can construct the variance-minimizing discount factors m ∗ = x ∗ + we ∗ = p 0 E(xx 0 ) −1 x + w £ 1 − E(x) 0 E(xx 0 ) −1 x ¤ or m ∗ = w +[p − wE(x)] 0 E(xx 0 ) −1 x (80) As w varies, we trace out discount factors m ∗ on the frontier with varying means and variances. It’s easiest to find mean and second moment: E(m ∗ )=w +[p − wE(x)] 0 E(xx 0 ) −1 E(x) E(m ∗2 )=[p − wE(x)] 0 E(xx 0 ) −1 [p − wE(x)] ; variance follows from σ2 (m) =E(m2 ) − E(m) 2 . With a little algebra one can also show that these formulas are equivalent to equation (5.78). As you can see, Hansen-Jagannathan frontiers are equivalent to mean-variance frontiers. For example, an obvious exercise is to see how much the addition of assets raises the Hansen- Jagannathan bound. This is exactly the same as asking how much those assets expand the mean-variance frontier. It was, in fact, this link between Hansen-Jagannathan bounds and mean-variance frontiers rather than the logic I described that inspired Knez and Chen (1996) and DeSantis (1994) to test for mean-variance efficiency using, essentially, Hansen- Jagannathan bounds. 96
SECTION 5.7 PROBLEMS Hansen-Jagannathan bounds have the potential to do more than mean-variance frontiers. Hansen and Jagannathan show how to solve the problem min σ 2 (m) s.t. p = E(mx),m>0. This is the “Hansen-Jagannathan bound with positivity,” and is strictly tighter than the Hansen- Jagannathan bound. It allows you to impose no-arbitrage conditions. In stock applications, this extra bound ended up not being that informative. However, in the option application of this idea of Chapter (18), positivity is really important. That chapter shows how to solve for a bound with positivity. Hansen, Heaton and Luttmer (1995) develop a distribution theory for the bounds. Luttmer (1996) develops bounds with market frictions such as short-sales constraints and bid-ask spreads, to account for ludicrously high apparent Sharpe ratios and bounds in short term bond data. <strong>Cochrane</strong> and Hansen (1992) survey a variety of bounds, including bounds that incorporate information that discount factors are poorly correlated with stock returns (the HJ bounds use the extreme ρ =1), bounds on conditional moments that illustrate how many models imply excessive interest rate variation, bounds with short-sales constraints and market frictions, etc. Chapter 21 discusses what the results of Hansen Jagannathan bound calculations and what they mean for discount factors that can price stock and bond return data. 5.7 Problems 1. Prove that Re∗ lies at right angles to planes (in Re )ofconstantmean return, as shown in figure 14. 2. Should we typically draw x∗ above, below or on the plane of returns? Must x∗always lie in this position? 3. Can you construct Re∗ from knowledge of m, x∗ ,orR∗ ? 4. What happens to R∗ ,Re∗ and the mean-variance frontier if investors are risk neutral? (a) If a riskfree rate is traded. (b) Ifnoriskfreerateistraded? 5. (Hint: make a drawing or think about the case that payoffs are generated by an N dimensional vector of basis assets x) x∗ = proj(m|X). IsR∗ = proj(m|R)? 6. We showed that all m are of the form x∗ + ε. What about R−1R? 7. Show that if there is a risk-free rate—if the unit payoff is in the payoff space X—then Re∗ =(Rf − R∗ )/Rf . 97
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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
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 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: SECTION 5.6 MEAN-VARIANCE FRONTIERS
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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