Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW We can write p = E(mx) as E(R i )=R f + β i,mλm We can express the expected return equation (1.12), for a return Ri ,as E(R i )=R f µ µ i cov(R ,m) + − var(m) var(m) E(m) or E(R i )=R f + β i,mλm where βim is the regression coefficient of the return Ri on m. Thisisabeta pricing model. It says that expected returns on assets i =1, 2,...Nshould be proportional to their betas in a regression of returns on the discount factor. Notice that the coefficient λm is the same for all assets i,while the βi,m varies from asset to asset. The λm is often interpreted as the price of risk and the β as the quantity of risk in each asset. Obviously, there is nothing deep about saying that expected returns are proportional to betas rather than to covariances. There is a long historical tradition and some minor convenience in favor of betas. The betas refer to the projection of R on m that we studied above, so you see again a sense in which only the systematic component of risk matters. With m = β (ct+1/ct) −γ , we can take a Taylor approximation of equation (1.14) to express betas in terms of a more concrete variable, consumption growth, rather than marginal utility. The result, which I derive more explicitly and conveniently in the continuous time limit below, is (14) (15) E(R i ) = R f + βi,∆cλ∆c (1.16) λ∆c = γvar(∆c). Expected returns should increase linearly with their betas on consumption growth itself. In addition, though it is treated as a free parameter in many applications, the factor risk premium λ∆c is determined by risk aversion and the volatility of consumption. The more risk averse people are, or the riskier their environment, the larger an expected return premium one must pay to get investors to hold risky (high beta) assets. 1.4.5 Mean-variance frontier All asset returns lie inside a mean-variance frontier. Assets on the frontier are perfectly correlated with each other and the discount factor. Returns on the frontier can be generated as portfolios of any two frontier returns. We can construct a discount factor from any frontier 26
SECTION 1.4 CLASSIC ISSUES IN FINANCE return (except R f ), and an expected return-beta representation holds using any frontier return (except R f ) as the factor. Asset pricing theory has focused a lot on the means and variances of asset returns. Interestingly, the set of means and variances of returns is limited. All assets priced by the discount factor m must obey To derive (1.17) write for a given asset return R i and hence ¯ ¯E(R i ) − R f¯ ¯ ≤ σ(m) E(m) σ(Ri ). (17) 1=E(mR i )=E(m)E(R i )+ρ m,R iσ(R i )σ(m) E(R i )=R f σ(m) − ρm,Ri E(m) σ(Ri ). (18) Correlation coefficients can’t be greater than one in magnitude, leading to (1.17). This simple calculation has many interesting and classic implications. 1. Means and variances of asset returns must lie in the wedge-shaped region illustrated in Figure 1. The boundary of the mean-variance region in which assets can lie is called the mean-variance frontier. It answers a naturally interesting question, “how much mean return can you get for a given level of variance?” 2. All returns on the frontier are perfectly correlated with the discount factor: the frontier is generated by ¯ ¯ ¯ρ ¯ m,Ri =1. Returns on the upper part of the frontier are perfectly negatively correlated with the discount factor and hence positively correlated with consumption. They are “maximally risky” and thus get the highest expected returns. Returns on the lower part of the frontier are perfectly positively correlated with the discount factor and hence negatively correlated with consumption. They thus provide the best insurance against consumption fluctuations. 3. All frontier returns are also perfectly correlated with each other, since they are all perfectly correlated with the discount factor. This fact implies that we can span or synthesize any frontier return from two such returns. For example if you pick any single frontier return Rm then all frontier returns Rmv must be expressible as R mv = R f + a ¡ R m − R f¢ for some number a. 4. Since each point on the mean-variance frontier is perfectly correlated with the discount 27
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 and 24: equation (1.2) as SECTION 1.4 CLASS
Page 25: 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 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 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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