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Ch.9 The Consumption Capital Asset Pricing Model. 9.1 Introduction 9.2 The Representative Agent Hypothesis and Its Notion of Equilibrium 9.3 An Exchange (Endowment) Economy 9.4 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM
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Ch.9 The Consumption Capital Asset Pricing Model • 9.1 Introduction • 9.2 The Representative Agent Hypothesis and Its Notion of Equilibrium • 9.3 An Exchange (Endowment) Economy • 9.4 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM • 9.5 Testing the Consumption CAPM: The Equity Premium Puzzle • 9.6 Testing the Consumption CAPM: Hansen-Jagannathan Bounds • 9.7 Some Extensions • 9.8 Conclusions
9.2 The Representative Agent Hypothesis and Its Notion of Equilibrium • 9.2.1 An Infinitely Lived Representative Agent • Avoid terminal period problem • Equivalence with finite lives if operative bequest motive • 9.2.2 On the Concept of a « No-Trade » Equilibrium • Positive net supply: the representative agent willingly hold total supply • Zero net supply: at the prevailing price, supply = demand = 0
9.3 An Exchange (Endowment) Economy • Recursive trading – many periods; investment decisions are • made one period at a time, taking due account of their impact • on the future state of the world • One perfectly divisible share • Dividend = economy’s total output • Output arises exogenously and stochastically (fruit tree) • Stationary stochastic process
Probability Transition Matrix • Lucas fruit tree • Grafting an aggregate ouput process • Rational expectations economy: knowledge of the • economic structure and the stochastic process
(9.1) F.O.C:
(9.2) Definition of an equilibrium (i) , i.e., the representative agent owns the entire security; (ii) , i.e., ownership of the entire security entitles the agent to all the economy's output and, (iii) , i.e., the agents’ holdings of the security are optimal given the prevailing prices. Substituting (ii) into (iii) informs us that the equilibrium price must satisfy: Euler Equation
(9.3) (9.4) (9.5) • Discounting at the IMRS of the representative agent! • Assume risk neutrality:
(9.6) (9.7) 9.3.2 Interpreting the Exchange Equilibrium =>Link between discount factor and risk-free rate in a risk neutral world
(9.8) , or, rearranging, , or (9.9)
Interpreting the CCAPM • Risk premium is large for those securities paying high returns when consumption is high (MU is low) and low returns when consumption is low. • Intuition not far from CAPM, but CCAPM adopts consumption smoothing perspective • The key to an asset’s value is its covariation with the MU of consumption rather than the MU of wealth
Let (9.10) One step further: towards a CAPM equation Marginal utility is inversely proportional to ct
(9.11) , or , or (9.12) (9.13) 9.3.3 The Formal Consumption CAPM c denotes portfolio most correlated with consumption if bc,ct =1
1. 2. 9.4 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Finite states Continuum of states
N-period claims N period risk free zero discount bond: (9.14) Valuing future cash flows revisited: Discounting at the IMRS = valuing at A-D prices!
(9.15) (9.16)
(9.18) (9.20) 2(.00123) = .002 = (ln(ER)-ln(ERf) ER – ERf (9.19)
(9.21) (9.22) 9.6 Testing the Consumption CAPM: Hansen-Jagannathan Bounds
, or , or , or , or
(9.24) = . 99 (.967945) = .96 for g = 2.