Asset Pricing Chapter IX. The Consumption Capital Asset Pricing - - PowerPoint PPT Presentation

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Asset Pricing Chapter IX. The Consumption Capital Asset Pricing - - PowerPoint PPT Presentation

Jan 03, 2023 228 likes •459 views

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

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SLIDE 1

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

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SLIDE 2

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

  • f 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 =

Asset Pricing

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SLIDE 3

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 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

Asset Pricing

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SLIDE 4

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 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 Y 1 Y 2 Y 3 Output in Period t Y 1 Y 2 Y 3   π11 π12 π13 π21 π22 π23 π31 π32 π33   = T G(Yt+1 | Yt) = Prob

  • Yt+1 ≤ Y j, |Yt = Y i

.

Lucas fruit tree Grafting an aggregate output process Rational expectations economy: knowledge of the economic structure and the stochastic process Asset Pricing

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SLIDE 5

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

max

{zt+1} E

∞

  • t=0

δtU(˜ ct)

  • s.t. ct + ptzt+1 ≤ ztYt + ptzt

zt ≤ 1, ∀t F .O.C: U1(ct)pt = δEt

  • U1(˜

ct+1)

  • ˜

pt+1+ ˜ Y t+1

  • (1)

U1(ct(Y i))pt(Y i) = δ

  • j

U1(ct+1(Y j))

  • pt+1(Y j)+Y j

πij

Asset Pricing

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SLIDE 6

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 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) zt = zt+1 = zt+2 = ... ≡ 1, in other words, the representative agent owns the entire security; (ii) ct = Yt, that is, ownership of the entire security entitles the agent to all the economy’s output and, (iii) U1(ct)pt = δEt

  • U1(˜

ct+1)

  • ˜

pt+1 + ˜ Yt+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: U1(Yt)pt = δEt

  • U1( ˜

Yt+1)(˜ pt+1 + ˜ Yt+1)

  • (2)

Asset Pricing

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SLIDE 7

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

ph,tU1 (ct) = δEt

  • U1 (˜

ct+1) (˜ ph,t+1 + ˜ Yh,t+1

  • (3)

pt = Et

∞

  • τ=1

δτ

  • U1( ˜

Yt+τ) U1(Yt) ˜ Yt+τ

  • ,

(4) Discounting at the IMRS of the representative agent! Assume risk neutrality pt = Et

∞

  • τ=1

δτ ˜ Yt+τ

  • = Et

∞

  • τ=1
  • ˜

Yt+τ (1 + rf)τ

  • ,

(5)

Asset Pricing

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SLIDE 8

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

Interpreting the Exchange Equilibrium

1 + rj,t+1 = pj,t+1 + Yj,t+1 pj,t 1 = δEt U1(˜ ct+1) U1(ct) (1+˜ r j,t+1)

  • (6)

qb

t U1(ct) = δEt {U1(˜

ct+1)1} 1 1 + rf,t+1 = qb

t = δEt

U1(˜ ct+1) U1(ct)

  • ,

(7) Link between discount factor and risk-free rate in a risk neutral world.

Asset Pricing

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SLIDE 9

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

1 = δEt U1(˜ ct+1) U1(ct)

  • Et
  • 1+˜

r j,t+1

  • + δcovt

U1(˜ ct+1) U1(ct) ,˜ r j,t+1

  • (8)

1 = 1 + r j,t+1 1 + rf,t+1 + δcovt U1(˜ ct+1) U1(ct) ,˜ r j,t+1

  • , or, rearranging,

1 + r j,t+1 1 + rf,t+1 = 1 − δcovt U1 (˜ ct+1) U1 (ct) ,˜ rj,t+1

  • , or

r j,t+1 − rf,t+1 = −δ

  • 1 + rf,t+1
  • covt

U1(˜ ct+1) U1(ct) ,˜ rj,t+1

  • .

(9)

Asset Pricing

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SLIDE 10

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 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.

Asset Pricing

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SLIDE 11

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

One step further: towards a CAPM equation Let U(ct) = act − b

2c2 t

U1(ct) = a − bct Marginal utility is inversely proportional to ct r j,t+1 − rf,t+1 = −δ

  • 1 + rf,t+1
  • covt
  • ˜

r j,t+1, a − b˜ ct+1 a − bct

  • =

−δ

  • 1 + rf,t+1
  • 1

a − bct covt ˜ r j,t+1,˜ ct+1

  • (−b), or

r j,t+1 − rf,t+1 = δb

  • 1 + rf,t+1
  • a − bct

covt ˜ r j,t+1,˜ ct+1

  • .

(10)

Asset Pricing

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SLIDE 12

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

The Formal Consumption CAPM

r c,t+1 − rf,t+1 =

  • δb
  • 1 + rf,t+1
  • a − bct
  • covt (˜

r c,t+1,˜ ct+1) . (11) "c" denotes portfolio most correlated with consumption r j,t+1 − rf,t+1 r c,t+1 − rf,t+1 = covt (˜ r j,t+1,˜ ct+1) covt (˜ r c,t+1, ˜ ct+1), or r j,t+1 − rf,t+1 r c,t+1 − rf,t+1 =

covt(˜ r j,t+1,˜ ct+1) var(˜ ct+1) covt(˜ r c,t+1,˜ ct+1) var(˜ ct+1)

, or r j,t+1 − rf,t+1 = βj,ct βc,ct

  • r c,t+1 − rf,t+1
  • (12)

¯ rj,t+1 − rf,t+1 = βj,ct ¯ rc,t+1 − rf,t+1

  • .

(13) if βc,ct = 1

Asset Pricing

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SLIDE 13

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

Finite states

1

U1 (c(s)) q (st+1 = s′; st = s) = δU1 (c(s′)) prob (st+1 = s′; st = s) ,

  • r

q (st+1 = s′; st = s) = δ U1 (c(s′)) U1 (c(s)) prob (st+1 = s′; st = s) . Continuum of states

2

q (s′; s) = δ U1 (c(s′)) U1 (c(s)) f (s′; s) q (s′; s) = δf (s′; s) = δπss.

Asset Pricing

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SLIDE 14

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

N-period claims N period risk free zero discount bond: qbN

t

(s) = δN

s′

U1 (c(s′)) U1 (c(s)) prob

  • st+N = s′; st = s
  • (14)

Value future cash flows revisited: pt = Et

∞

  • τ=1

δτ U1(ct+τ) U1(ct) Yt+τ

  • =

∞

  • τ=1
  • s′

δτ U1(ct+τ(s′)) U1(ct) Yt+τ(s′)

  • prob
  • st+τ = s′; st = s
  • =
  • τ
  • s′

qτ(s′, s)Yt+τ(s′), (15) Discounting at the IMRS = valuing at A-D prices!

Asset Pricing

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SLIDE 15

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

pt =

∞

  • τ=1
  • Et [Yt+τ]
  • 1 + cov(U1(˜

ct+τ),˜ Yt+τ) Et[U1(˜ ct+τ)]Et[˜ Yt+τ]

  • (1 + rf,t+τ)τ

. (16) pt =

∞

  • τ=1

Yt+τ (1 + rf,t+τ)τ . pt =

∞

  • τ=1

E[ ˜ Yt+τ] (1 + rf,t+τ)τ (17)

Asset Pricing

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SLIDE 16

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

Table 9.3 : Properties of U.S. Asset Returns U.S. Economy (a) (b) r 6.98 16.54 rf .80 5.67 r − rf 6.18 16.67 (a) Annualized mean values in percent; (b) Annualized standard deviation in percent. Source: Data from Mehra and Prescott (1985).

Asset Pricing

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SLIDE 17

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

U (c) = c1−γ 1 − γ . U1(ct+1) U1(ct) = ct+1 ct −γ . (18) 1 = δEt ˜ ct+1 c t −γ Rt+1

  • = δ(x)−γ ¯

R, (19)

Asset Pricing

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SLIDE 18

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

pt = vYt vYt = δEt

  • v ˜

Y t+1+ ˜ Y t+1 U1(˜ ct+1) U1(ct)

  • .

v = δE

  • (v+1)

˜ Yt+1 Yt ˜ x−γ

t+1

  • .

v = δE

  • (v+1)˜

x1−γ = δE ˜ x1−γ 1 − δE {˜ x1−γ}. Rt+1 ≡ 1 + rt+1 = pt+1 + Yt+1 pt = v + 1 v Yt+1 Yt = v + 1 v xt+1. Et

  • ˜

Rt+1

  • = E
  • ˜

Rt+1

  • = v + 1

v E (˜ xt+1) = E (˜ x) δE {˜ x1−γ}.

Asset Pricing

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SLIDE 19

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

Rf,t+1 ≡ 1 qb

t

=

  • δEt

U1(˜ ct+1) U1(ct) −1 = 1 δ 1 E {˜ x−γ}, (20) E

  • ˜

Rt+1

  • Rf

= E {˜ x} E {˜ x−γ} E ˜ x1−γ = exp

  • γσ2

x

  • ,

(21) ln (ER) − ln (Rf) = γσ2

x.

(22) ln (ER) − ln (Erf ) σ2

x

= 1.0698 − 1.008 (.0357)2 = 50.24 = γ. 2(.00123) = .002 = (ln(ER) − ln(ERf) ∼ = ER − ERf (23)

Asset Pricing

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SLIDE 20

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

p(st) = Et[mt+1(˜ st+1)Xt+1(˜ st+1); st], (24) mt+1(˜ st+1) = δU1(ct+1(˜ st+1)) U1(ct) . pt = Et[ ˜ mt+1 ˜ Xt+1]. (25) 1 = Et[ ˜ mt+1 ˜ Rt+1], 1 = E[ ˜ m ˜ R]

Asset Pricing

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SLIDE 21

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

E[ ˜ m(˜ Ri − ˜ Rj)] = 0,

  • r

E[ ˜ m ˜ Ri−j] = 0, E ˜ mE ˜ Ri−j + cov( ˜ m, ˜ Ri−j) = 0,

  • r

E ˜ mE ˜ Ri−j + ρ( ˜ m, ˜ Ri−j)σmσRi−j = 0,

  • r

E ˜ Ri−j σRi−j + ρ( ˜ m, ˜ Ri−j) σm E ˜ m = 0,

  • r

E ˜ Ri−j σRi−j = −ρ( ˜ m, ˜ Ri−j) σm E ˜ m . (26)

Asset Pricing

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SLIDE 22

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

σm E ˜ m >

  • E ˜

Ri−j

  • σRi−j

. (27) σm E ˜ m > |E(˜ rM − rf)| σrM−rf = .062 .167 = .37. E ˜ m = δ exp(−γµx + 1 2γ2σ2

x) = .99(.967945) = .96 for γ = 2.

Asset Pricing