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AssetTheory_SL

AssetTheory_SL - Asset pricing Prof Lutz Hendricks L...

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Asset pricing Prof. Lutz Hendricks August 10, 2009 L. Hendricks () Asset pricing August 10, 2009 1 / 44
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Issues 1 What determines the rates of return / prices of various assets? 2 How can risk be measured and priced? We use the Lucas fruit tree model. The implications are far more general than the simple model. L. Hendricks () Asset pricing August 10, 2009 2 / 44
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The Lucas Fruit Tree Model We study the model introduced by Lucas (1978). Agents : A single representative household. Preferences : max E 0 t = 0 β t u ( c t ) (1) E 0 is the expectation as of time t = 0 . L. Hendricks () Asset pricing August 10, 2009 3 / 44
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The Model Technology This is an endowment economy. There are K identical fruit trees. Each tree yields d t units of consumption goods in period t . d t is random and the same for all trees. Trees cannot be produced. Fruits cannot be stored. L. Hendricks () Asset pricing August 10, 2009 4 / 44
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The Model Technology The aggregate resource constraint: c t = Kd t (2) Assume that d is a °nite Markov chain with transition matrix π ( d 0 , d ) . An important feature: All uncertainty is aggregate . There are no opportunities for households to insure each other. This is why we can work with a representative household. L. Hendricks () Asset pricing August 10, 2009 5 / 44
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The Model Markets There are markets for fruits and for trees. There is also a one period bond, issued by households (in zero net supply). Its purpose is to determine a risk-free interest rate. L. Hendricks () Asset pricing August 10, 2009 6 / 44
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Household problem The household starts out with bonds ( b 0 ) and shares ( k 0 ). At each date, he chooses c t , b t + 1 , k t + 1 . The budget constraint is p t k t + 1 + b t + 1 = R t b t + ( p t + d t ) k t ° c t (3) Notation: p : the price of trees. Suppressing dependence on the state. R : the real interest rate on bonds. the price of bonds is normalized to 1 (how?). L. Hendricks () Asset pricing August 10, 2009 7 / 44
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Household problem V ( k , b , d ) = max u ( c ) + β EV ° k 0 , b 0 , d 0 ± (4) subject to Rb + ( p + d ) k ° c + pk 0 ° b 0 = 0 (5) L. Hendricks () Asset pricing August 10, 2009 8 / 44
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Household problem First-order conditions: c : u 0 ( c ) = λ k 0 : λ p = EV k ( k 0 , b 0 , d 0 ) b 0 : λ = EV b ° k 0 , b 0 , d 0 ± Envelope: V k = λ ( p + d ) V b = λ R L. Hendricks () Asset pricing August 10, 2009 9 / 44
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Household problem Euler equations u 0 ( c t ) = β E t ² u 0 ( c t + 1 ) R t + 1 ³ = β E t ´ u 0 ( c t + 1 ) p t + 1 + d t + 1 p t µ L. Hendricks () Asset pricing August 10, 2009 10 / 44
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Household problem Solution A solution consists of state contingent plans f c ( d t ) , k ( d t ) , b ( d t ) g for all histories d t . These satisfy: 2 Euler equations 1 budget constraint. b 0 and k 0 given. Transversality: lim t ! E 0 β t u 0 ( c t ) [ b t + p t k t ] = 0 . L. Hendricks () Asset pricing August 10, 2009 11 / 44
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Market clearing For every history we need: Bonds: b t = 0 Trees: k t = K t Goods: c t = K t d t There is no trade in equilibrium! L. Hendricks () Asset pricing August 10, 2009 12 / 44
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Competitive Equilibrium A CE consists of: 1 an allocation: f c ( d t ) , b ( d t ) , k ( d t ) g .
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