Asset Pricing. Chapter IX. The Consumption Capital Asset Pricing Model. June 20, 2006

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1 Chapter IX. The Consumption Capital Model June 20, 2006

2 The Representative Agent Hypothesis and its Notion of Equilibrium An infinitely lived Representative Agent Avoid terminal period problem Equivalence with finite lives if operative bequest motive 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

3 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM 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

4 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM Probability Transition Matrix Table 9.1: Three-State Probability Transition Matrix Output in Period t +1 Output in Period t Y 1 Y 2 Y 3 Y 1 Y 2 Y 3 π 11 π 12 π 13 π 21 π 22 π 23 π 31 π 32 π 33 = T G(Y t+1 Y t ) = Prob ( Y t+1 Y j, Y t = Y i). Lucas fruit tree Grafting an aggregate output process Rational expectations economy: knowledge of the economic structure and the stochastic process

5 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM F.O.C: max E {z t+1 } ( δ t U( 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 ( )} U 1 (c t )p t = δe t {U 1 ( c t+1 ) p t+1 +Ỹ t+1 (1) U 1 (c t (Y i ))p t (Y i ) = δ j ( U 1 (c t+1 (Y j )) p t+1 (Y j )+Y j) π ij

6 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM 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, )} (iii) U 1 (c t )p t = δe t {U 1 ( c t+1 ) p t+1 + Ỹt+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 (Y t )p t = δe t {U 1 (Ỹt+1)( p t+1 + Ỹt+1) (2)

7 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM p h,t U 1 (c t ) = δe t {[U 1 ( c t+1 ) ( p h,t+1 + Ỹh,t+1 ]} (3) p t = E t τ=1 δ τ [ U 1 (Ỹt+τ ) Ỹ t+τ ], (4) U 1 (Y t ) Discounting at the IMRS of the representative agent! Assume risk neutrality p t = E t τ=1 [ ] [ ] δ τ Ỹt+τ Ỹ t+τ = E t (1 + r f ) τ, (5) τ=1

8 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM Interpreting the Exchange Equilibrium 1 + r j,t+1 = p j,t+1 + Y j,t+1 p j,t 1 = δe t { U1 ( c t+1 ) U 1 (c t ) (1+ r j,t+1) } (6) q b t U 1(c t ) = δe t {U 1 ( c t+1 )1} r f,t+1 = q b t = δe t { } U1 ( c t+1 ), (7) U 1 (c t ) Link between discount factor and risk-free rate in a risk neutral world.

9 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM { } { } U1 ( c t+1 ) { } U1 ( c t+1 ) 1 = δe t E t 1+ r j,t+1 + δcovt U 1 (c t ) U 1 (c t ), r j,t+1 (8) 1 = 1 + r ( ) j,t+1 U1 ( c t+1 ) + δcov t 1 + r f,t+1 U 1 (c t, r j,t+1, or, rearranging, ( ) 1 + r j,t+1 U1 ( c t+1 ) = 1 δcov t 1 + r f,t+1 U 1 (c t ), r j,t+1, or r j,t+1 r f,t+1 = δ ( ( ) ) U1 ( c t+1 ) 1 + r f,t+1 covt U 1 (c t ), r j,t+1. (9)

10 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM 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.

11 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM One step further: towards a CAPM equation Let U(c t ) = ac t b 2 c2 t U 1 (c t ) = a bc t Marginal utility is inversely proportional to c t r j,t+1 r f,t+1 = δ ( ( ) 1 + r f,t+1 covt r j,t+1, a b c ) t+1 a bc t = δ ( ) 1 ) 1 + r f,t+1 cov t ( r j,t+1, c t+1 ( b), or a bc t r j,t+1 r f,t+1 = δb ( ) 1 + r f,t+1 ) cov t ( r j,t+1, c t+1. (10) a bc t

12 Probability Transition Matrix Interpreting the Exchange Equilibrium Interpreting the CCAPM The Formal Consumption CAPM The Formal Consumption CAPM r c,t+1 r f,t+1 = [ ( )] δb 1 + rf,t+1 a bc t "c" denotes portfolio most correlated with consumption r j,t+1 r f,t+1 r c,t+1 r f,t+1 = cov t ( r j,t+1, c t+1 ) cov t ( r c,t+1, c t+1 ), or cov t ( r c,t+1, c t+1 ). (11) r j,t+1 r f,t+1 r c,t+1 r f,t+1 = cov t( r j,t+1, c t+1) var( c t+1 ), or cov t ( r c,t+1, c t+1 ) var( c t+1 ) r j,t+1 r f,t+1 = β j,c t β c,ct [ r c,t+1 r f,t+1 ] (12) r j,t+1 r f,t+1 = β j,ct ( rc,t+1 r f,t+1 ). (13) if β c,ct = 1

13 Finite states 1 U 1 (c(s)) q (s t+1 = s ; s t = s) = δu 1 (c(s )) prob (s t+1 = s ; s t = s), or q (s t+1 = s ; s t = s) = δ U 1 (c(s )) U 1 (c(s)) prob (s t+1 = s ; s t = s). Continuum of states 2 q (s ; s) = δ U 1 (c(s )) U 1 (c(s)) f (s ; s) q (s ; s) = δf (s ; s) = δπ ss.

14 N-period claims N period risk free zero discount bond: q bn t (s) = δ N s U 1 (c(s )) U 1 (c(s)) prob ( s t+n = s ; s t = s ) (14) Value future cash flows revisited: [ ] p t = E t δ τ U1 (c t+τ ) U 1 (c t ) Y t+τ τ=1 [ = δ τ U1 (c t+τ (s ] )) Y t+τ (s ) prob ( s t+τ = s ; s t = s ) U τ=1 s 1 (c t ) = q τ (s, s)y t+τ (s ), (15) τ s Discounting at the IMRS = valuing at A-D prices!

15 p t = τ=1 { { }} E t [Y t+τ ] 1 + cov(u 1( c t+τ ),Ỹt+τ ) E t [U 1 ( c t+τ )]E t[ỹt+τ] (1 + r f,t+τ ) τ. (16) p t = p t = τ=1 τ=1 Y t+τ (1 + r f,t+τ ) τ. E[Ỹt+τ ] (1 + r f,t+τ ) τ (17)

16 Table 9.3 : Properties of U.S. Asset Returns U.S. Economy (a) (b) r rf r r f (a) Annualized mean values in percent; (b) Annualized standard deviation in percent. Source: Data from Mehra and Prescott (1985).

17 U (c) = c1 γ 1 γ. U 1 (c t+1 ) U 1 (c t ) = ( ct+1 c t ) γ. (18) { ( ) γ ct+1 1 = δe t R t+1} = δ(x) γ R, (19) c t

18 p t = vy t { ( ) } U1 ( c t+1 ) vy t = δe t vỹ t+1+ỹ t+1. U 1 (c t ) { } v = δe (v+1) Ỹt+1 x γ t+1 Y t v = δe { (v+1) x 1 γ} = { x 1 γ} δe 1 δe { x 1 γ }. R t r t+1 = p t+1 + Y t+1 = v + 1 p t v ( ) ( ) E t Rt+1 = E Rt+1. Y t+1 = v + 1 E ( x t+1 ) = v = v + 1 x t+1. Y t v E ( x) δe { x 1 γ }.

19 R f,t+1 1 qt b = ( ) E Rt+1 [ { U1 ( c t+1 ) δe t U 1 (c t ) }] 1 = 1 δ R f = E { x} E { x γ } E { x 1 γ} = exp [ γσ 2 x 1 E { x γ }, (20) ], (21) ln (ER) ln (R f ) = γσ 2 x. (22) ln (ER) ln (E rf ) σ 2 x = (.0357) 2 = = γ. 2(.00123) =.002 = (ln(er) ln(er f ) = ER ER f (23)

20 p(s t ) = E t [m t+1 ( s t+1 )X t+1 ( s t+1 ); s t ], (24) m t+1 ( s t+1 ) = δu 1(c t+1 ( s t+1 )). U 1 (c t ) p t = E t [ m t+1 Xt+1 ]. (25) 1 = E t [ m t+1 Rt+1 ], 1 = E[ m R]

21 or E[ m( R i R j )] = 0, E[ m R i j ] = 0, or or or E me R i j + cov( m, R i j ) = 0, E me R i j + ρ( m, R i j )σ m σ Ri j = 0, E R i j σ Ri j E R i j σ Ri j + ρ( m, R i j ) σ m E m = 0, = ρ( m, R i j ) σ m E m. (26)

22 σ m E m > E R i j σ Ri j. (27) σ m E m > E( r M r f ) =.062 σ rm r f.167 =.37. E m = δ exp( γµ x γ2 σ 2 x) =.99( ) =.96 for γ = 2.

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