Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 2 APPLYING THE BASIC MODEL (a) Show that optimal consumption follows ct = rkt + rβ ∞X j=0 ct = ct−1 +(Et − Et−1) rβ β j Etet+j ∞X j=0 β j et+j i.e., consumption equals permanent income, precisely defined, and consumption follows a random walk whose innovations are equal to innovations in permanent income. (b) Assume that the endowment et follows an AR(1) et = ρet−1 + εt (2.46) (2.47) and specialize (2.46) and (2.47). Calculate and interpret the result for ρ =1and ρ =0. (The result looks like a “consumption function” relating consumption to capital and current income, except that the slope of that function depends on the persistence of income shocks. Transitory shocks will have little effect on consumption, and permanent shocks a larger effect.) (c) Calculate the one period interest rate (it should come out to r of course) and the price of a claim to the consumption stream. e and k are the only state variables, so the price should be a function of e and k. Interpret the time-variation in the price of the consumption stream. (This consumer gets more risk averse as consumption rises to c∗ . c∗ is the bliss point, so at the bliss point there is no average return that can compensate the consumer for greater risk.) 3. Consider again CRRA utility, X j 1−γ Et β ct+j . Consumption growth follows a two-state Markov process. The states are ∆ct = ct/ct−1 = h, l,anda2×2matrixπgoverns the set of transition probabilities, i.e. pr(∆ct+1 = h|∆ct = l) =πl→h. (This is the Mehra-Prescott 1986 model, but it will be faster to do it than to look it up. It is a useful and simple endowment economy.) (a) Find the riskfree rate (price of a certain real payoff of one) in this economy. This price is generated by p b t = Et(mt,t+11). You are looking for two values, the price in the l state and the price in the h state. (b) Find the price of the consumption stream (the price at t of {ct+1,ct+2,...}). To do this, guess that the price/consumption ratio must be a function of state (h,l), and find 52
that function. From SECTION 2.5 PROBLEMS p c ¡ t = Et mt,t+1(p c ¢ t+1 + ct+1 ) find a recursive relation for p c t/ct, and hence find the two values of p c t/ct, one for the h state and one for the l state. (c) Pick β =0.99 and try γ =0.5, 5 (Try more if you feel like it). Calibrate the consumption process to have a 1% mean and 1% standard deviation, and consumption growth uncorrelated over time. Calculate prices and returns in each state. (d) Now introduce serial correlation in consumption growth with γ =5. (You can do this by adding weight to the diagonal entries of the transition matrix π.) What effect does this have on the model? 53
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 and 26: 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: SECTION 2.5 PROBLEMS Since the opti
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

Asset Pricing John H. Cochrane June 12, 2000

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