9.1 The Representative Agent Hypothesis and its Notion of Equilibrium 9.2 An Exchange (Endowment) Economy 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds Asset Pricing Chapter IX. The Consumption Capital Asset Pricing Model June 20, 2006 Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium 9.2 An Exchange (Endowment) Economy 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds 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 Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds 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 Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds Probability Transition Matrix Table 9.1: Three-State Probability Transition Matrix Output in Period t + 1 Y 1 Y 2 Y 3 Y 1 π 11 π 12 π 13 Y 2 Output in Period t π 21 π 22 π 23 = T Y 3 π 31 π 32 π 33 Y t + 1 ≤ Y j , | Y t = Y i � � G ( Y t + 1 | Y t ) = Prob . Lucas fruit tree Grafting an aggregate output process Rational expectations economy: knowledge of the economic structure and the stochastic process Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds � ∞ � δ t U (˜ { z t + 1 } E max � c t ) t = 0 s.t. c t + p t z t + 1 ≤ z t Y t + p t z t z t ≤ 1 , ∀ t F .O.C: � � �� p t + 1 + ˜ U 1 ( c t ) p t = δ E t U 1 (˜ c t + 1 ) ˜ Y t + 1 (1) � p t + 1 ( Y j )+ Y j � � U 1 ( c t ( Y i )) p t ( Y i ) = δ U 1 ( c t + 1 ( Y j )) π ij j Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds Definition of an equilibrium For the entire economy to be in equilibrium, it must, therefore, be true that: (i) z t = z t + 1 = z t + 2 = ... ≡ 1, in other words, the representative agent owns the entire security; (ii) c t = Y t , that is, ownership of the entire security entitles the agent to all the economy’s output and, � � �� p t + 1 + ˜ U 1 (˜ ˜ (iii) U 1 ( c t ) p t = δ E t c t + 1 ) Y t + 1 , or, the agents’ holdings of the security are optimal given the prevailing prices. Substituting (ii) into (iii) informs us that the equilibrium price must satisfy: � � U 1 ( ˜ p t + 1 + ˜ Y t + 1 )(˜ U 1 ( Y t ) p t = δ E t Y t + 1 ) (2) Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds �� �� p h , t + 1 + ˜ U 1 (˜ c t + 1 ) (˜ p h , t U 1 ( c t ) = δ E t Y h , t + 1 (3) ∞ � � U 1 ( ˜ Y t + τ ) � ˜ p t = E t δ τ Y t + τ (4) , U 1 ( Y t ) τ = 1 Discounting at the IMRS of the representative agent! Assume risk neutrality ∞ ∞ � � ˜ Y t + τ δ τ � � ˜ � � p t = E t Y t + τ = E t (5) , ( 1 + r f ) τ τ = 1 τ = 1 Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds Interpreting the Exchange Equilibrium 1 + r j , t + 1 = p j , t + 1 + Y j , t + 1 p j , t � U 1 (˜ c t + 1 ) � U 1 ( c t ) ( 1 +˜ 1 = δ E t r j , t + 1 ) (6) q b t U 1 ( c t ) = δ E t { U 1 (˜ c t + 1 ) 1 } � U 1 (˜ 1 c t + 1 ) � = q b t = δ E t , (7) 1 + r f , t + 1 U 1 ( c t ) Link between discount factor and risk-free rate in a risk neutral world. Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds � U 1 (˜ � U 1 (˜ c t + 1 ) � c t + 1 ) � � 1 +˜ � U 1 ( c t ) , ˜ 1 = δ E t E t r j , t + 1 + δ cov t r j , t + 1 U 1 ( c t ) (8) � U 1 (˜ 1 + r j , t + 1 c t + 1 ) � U 1 ( c t ) , ˜ 1 = + δ cov t r j , t + 1 , or, rearranging, 1 + r f , t + 1 1 + r j , t + 1 � U 1 (˜ c t + 1 ) � U 1 ( c t ) , ˜ = 1 − δ cov t r j , t + 1 , or 1 + r f , t + 1 � U 1 (˜ c t + 1 ) � � � r j , t + 1 − r f , t + 1 = − δ 1 + r f , t + 1 U 1 ( c t ) , ˜ cov t r j , t + 1 . (9) Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds 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. Asset Pricing
9.1 The Representative Agent Hypothesis and its Notion of Equilibrium Probability Transition Matrix 9.2 An Exchange (Endowment) Economy Interpreting the Exchange Equilibrium 9.3 Pricing Arrow-Debreu State-Contingent Claims with the CCAPM Interpreting the CCAPM 9.4 Testing the Consumption CAPM: The Equity Premium Puzzle The Formal Consumption CAPM 9.5 Testing the Consumption CAPM: Hansen-Jagannathan Bounds One step further: towards a CAPM equation Let U ( c t ) = ac t − b 2 c 2 t U 1 ( c t ) = a − bc t Marginal utility is inversely proportional to c t r j , t + 1 , a − b ˜ � � c t + 1 � � ˜ r j , t + 1 − r f , t + 1 = − δ 1 + r f , t + 1 cov t a − bc t 1 � � � ˜ r j , t + 1 , ˜ � = − δ 1 + r f , t + 1 cov t c t + 1 ( − b ) , or a − bc t � � δ b 1 + r f , t + 1 � ˜ r j , t + 1 , ˜ � r j , t + 1 − r f , t + 1 = cov t c t + 1 . (10) a − bc t Asset Pricing
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