Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW the price is one and the payoff is a return. Start with the basic pricing equation for returns, 1=E(mR i ). I denote the return R i to emphasize that the point of the theory is to distinguish the behavior of one asset R i from another R j . The asset pricing model says that, although expected returns can vary across time and assets, expected discounted returns should always be the same, 1. Applying the covariance decomposition, and, using R f =1/E(m), or 1=E(m)E(R i )+cov(m, R i ) (11) E(R i ) − R f = −R f cov(m, R i ) (<strong>12</strong>) E(R i ) − R f = − cov[u0 (ct+1),Ri t+1] E[u0 . (13) (ct+1)] All assets have an expected return equal to the risk-free rate, plus a risk adjustment. <strong>Asset</strong>s whose returns covary positively with consumption make consumption more volatile, and so must promise higher expected returns to induce investors to hold them. Conversely, assets that covary negatively with consumption, such as insurance, can offer expected rates of return that are lower than the risk-free rate, or even negative (net) expected returns. Much of finance focuses on expected returns. We think of expected returns increasing or decreasing to clear markets; we offer intuition that “riskier” securities must offer higher expected returns to get investors to hold them, rather than saying “riskier” securities trade for lower prices so that investors will hold them. Of course, a low initial price for a given payoff corresponds to a high expected return, so this is no more than a different language for the same phenomenon. 1.4.3 Idiosyncratic risk does not affect prices Only the component of a payoff perfectly correlated with the discount factor generates an extra return. Idiosyncratic risk, uncorrelated with the discount factor, generates no premium. You might think that an asset with a high payoff variance is “risky” and thus should have a large risk correction. However, if the payoff is uncorrelated with the discount factor m,the asset receives no risk-correction to its price, and pays an expected return equal to the risk-free 24
ate! In equations, if then SECTION 1.4 CLASSIC ISSUES IN FINANCE cov(m, x) =0 p = E(x) . Rf This prediction holds even if the payoff x is highly volatile and investors are highly risk averse. The reason is simple: if you buy a little bit more of such an asset, it has no first-order effect on the variance of your consumption stream. More generally, one gets no compensation or risk adjustment for holding idiosyncratic risk. Only systematic risk generates a risk correction. To give meaning to these words, we can decompose any payoff x into a part correlated with the discount factor and an idiosyncratic part uncorrelated with the discount factor by running a regression, x = proj(x|m)+ε. Then, the price of the residual or idiosyncratic risk ε is zero, and the price of x is the same as the price of its projection on m. The projection of x on m is of course that part of x which is perfectly correlated with m. Theidiosyncratic component of any payoff is that part uncorrelated with m. Thus only the systematic part of a payoff accounts for its price. Projection means linear regression without a constant, proj(x|m) = E(mx) E(m 2 ) m. You can verify that regression residuals are orthogonal to right hand variables E(mε) =0 from this definition. E(mε) =0of course means that the price of ε is zero. µ E(mx) p (proj(x|m)) = p E(m2 ) m µ 2 E(mx) = E m E(m2 = E(mx) =p(x). ) The words “systematic” and “idiosyncratic” are defined differently in different contexts, which can lead to some confusion. In this decomposition, the residuals ε can be correlated with each other, though they are not correlated with the discount factor. The APT starts with a factor-analytic decomposition of the covariance of payoffs, and the word “idiosyncratic” there is reserved for the component of payoffs uncorrelated with all of the other payoffs. 1.4.4 Expected return-beta representation 25
Page 1 and 2: Asset Pricing John H. Cochrane June
Page 3 and 4: 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: equation (1.2) as SECTION 1.4 CLASS
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 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
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SECTION 5.3 AN ORTHOGONAL CHARACTER
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SECTION 5.4 SPANNING THE MEAN-VARIA
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SECTION 5.5 A COMPILATION OF PROPER
Page 91 and 92:
SECTION 5.5 A COMPILATION OF PROPER
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SECTION 5.6 MEAN-VARIANCE FRONTIERS
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SECTION 5.6 MEAN-VARIANCE FRONTIERS
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SECTION 5.7 PROBLEMS Hansen-Jaganna
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SECTION 6.1 FROM DISCOUNT FACTORS T
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Asset Pricing John H. Cochrane June 12, 2000

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