Ambiguity and Information Processing in a Model of Intermediary Asset Pricing

Transcription

1 Ambiguity and Information Processing in a Model of Intermediary Asset Pricing Leyla Jianyu Han 1 Kenneth Kasa 2 Yulei Luo 1 1 The University of Hong Kong 2 Simon Fraser University December 15, / 21

2 Introduction Heterogeneity in information processing capacity Financial intermediaries (specialists) are assumed to possess greater channel capacity (Rational Inattention (Sims, 23)). Households purchase this capacity by delegating investments to intermediaries. Although households could manage their portfolios themselves, most choose not to do so. Two frictions in financial contract: Incentive constraint arises from a moral hazard problem, requires a minimum capital for risk-sharing (He-Krishnamurthy, 212). Participation constraint depends on the heterogeneity in channel capacity. 2 / 21

3 Introduction Heterogeneity in information processing capacity Financial intermediaries (specialists) are assumed to possess greater channel capacity (Rational Inattention (Sims, 23)). Households purchase this capacity by delegating investments to intermediaries. Although households could manage their portfolios themselves, most choose not to do so. Two frictions in financial contract: Incentive constraint arises from a moral hazard problem, requires a minimum capital for risk-sharing (He-Krishnamurthy, 212). Participation constraint depends on the heterogeneity in channel capacity. Heterogeneity in beliefs Knightian uncertainty/ambiguity/robustness (Hansen-Sargent, 28)) When volatility increases, so does ambiguity, the drift distortions produce endogenous heterogeneous beliefs. When volatility is high specialists become relatively pessimistic, and this tightens the capital constraint and accelerates the onset of a financial crisis. 3 / 21

4 Market Structure Expert Wealth K t! " #! " % " % " riskless Intermediary Capital % ". 5 " / ". - "/ ". (1 " )/ ". 1 5 " 1 / ". riskless / " / ", Household Wealth! ", - % ",! ", % ", riskless Effective risk sharing constraint: ε h t mε t. m reflects the financial constraint due to agency friction and ambiguity. Participation constraint: k t a 3 (Σ Σ h ). a 3 <, κ > κ h Σ < Σ h 4 / 21

5 Risk Premium Sharpe Ratio Two Agents Pessimism.25 Two Agents Ambiguity Difference Volatility 2 Expert Scaled Wealth / 21

6 Risk Premium Sharpe Ratio Two Agents Pessimism 1-4 Two Agents Ambiguity Difference Volatility Expert Scaled Wealth / 21

7 Model Structure Risky asset dividend is governed by stochastic growth rate g t and volatility σ t, dd t D t = g t dt + σ t dz t, (1) Assume the volatility σ t is a two-state Markov chain with state space Σ d = {σ H, σ L }, where σ H > σ L. The intensity matrix is [ ] λh λ H. (2) λ L λ L Unobservable growth rate follows a (known) mean-reverting process dg t = ρ g (ḡ g t ) dt + σ g dz u t (3) Agents observe only a noisy signal containing imperfect information ds t = g t dt + σ s dz s t (4) 7 / 21

8 Capacity-Constrained Kalman Filter The Kalman filter of learning is dĝ t = ρ g (ḡ g t ) dt + Σ t dẑt + Σ t dẑ t s (5) σ t σ s [ ( 1 dσ t = σg 2 2ρ g Σ t Σ 2 t σt )] σs 2 dt (6) Σ t: signal/noise ratio (estimation variance of the unobserved state). Investor has a finite information-processing capacity (Sims, 23) H (g t+ t I t ) H (g t+ t I t+ t ) κ t, (7) The Kalman gain is constrained by the agent s channel capacity Risky asset return 1 Σ t 2 σs 2 κ. (8) dr t = D tdt + dp t P t = µ R,t dt + σ R,t dz t. (9) 8 / 21

9 Household Robust Consumption/Portfolio Rules Objective ( ) V ĝt h, Σ h t, Wt h ; Yt h = sup infe {Ct h,εh t } νt h [ s.t. dwt h = ε h t (π R,t k t) + r twt h Ct h [ e ρh t ln Ct h + 1 ( ) ] 2 ν h 2θ h t dt (1) ] ) dt + σw h,t (νt h dt + dẑt, (11) Optimal rules dĝt h = ρ g (ḡ g t) dt + Σh t dẑ t + Σh t dẑt s (12) σ t σ [ s ( ) Σ h 2 ( ] ) dσ h t = σg 2 2ρ g Σ h t 2 t 2κ h Σ h t dt (13) ν h t ε h t σ 2 t = θh ρ h ε h t σ R,t W h t C h t (14) = ρ h W h t (15) = π R,t k t W h γ h σr,t 2 t (16) Effective HH risk aversion γ h = 1 + θh ρ h ; θ h : HH ambiguity aversion degree. 9 / 21

10 Specialist Robust Consumption/Portfolio Rules Objective J (ĝ t, Σ t, W t; Y t) = sup infe {C t,ε t } ν t e ρt [ ln C t + 1 2θ (νt)2 ] dt (17) s.t. dw t = [ε tπ R,t + (q t + r t)w t C t] dt + σ W,t ( ν tdt + dẑ t ) (18) Optimal rules: dĝ t = ρ g (ḡ g t) dt + Σt Σt dẑt + dẑ t s (19) σ t σ s ( ) dσ t = σg 2 2ρ g Σ t Σ2 t 2κΣ 2 σt 2 t dt (2) ν t ε t = θ ε tσ R,t (21) ρ W t C t = ρw t (22) = π R,t W t. (23) γσr,t 2 Effective specialist risk aversion γ = 1 + θ ; θ: specialist s ambiguity aversion. ρ 1 / 21

11 Equilibrium Intermediation market clears, ε h t = 1 β t β t ε t. (24) Stock market clears, ε t + ε h t = P t. (25) Goods market clears, C t + C h t = D t. (26) 11 / 21

12 Risk Sharing Constraint In unconstrained region, Slack risk sharing constraint ε h t kt= < mε t π R,t γ h σr,t 2 Wt h < m π R,t γσr,t 2 W t In constrained region, Binding risk sharing constraint T h t = W h t < mw t. ε h t = mε t W h t mw t = T h t. 12 / 21

13 Risk Sharing Constraint In unconstrained region, Slack risk sharing constraint ε h t kt= < mε t π R,t γ h σr,t 2 Wt h < m π R,t γσr,t 2 W t In constrained region, Binding risk sharing constraint T h t = W h t < mw t. ε h t = mε t W h t mw t = T h t. Effective financial constraint: m γh γ m = 1 + θh /ρ h 1 + θ/ρ m (27) ρ h ρ, θ h = θ γ h γ m m (28) Define scaled specialist wealth as the unique state variable x t = W t /D t. When the risk sharing constraint just starts to bind, x c = 1 mρ h +ρ. 13 / 21

14 Steady State Solution In the steady state, ] Σ = σ [ 2 (κ + ρ g ) + (κ + ρ g ) 2 + (σ g / σ) 2 (29) Value function dσ dκ < (3) J (ĝ t, Σ t, W t ; Y t ) = 1 ρ ln W t + a + a 1 ĝ 2 + a 2 ĝ + a 3 Σ + Y (x t ), a 3 <. (31) dj dκ = dj dσ dσ dκ = a dσ 3 >. (32) dκ Agents with higher channel capacity have higher steady state welfare. κ > κ h k J V = a 3 ( Σ Σ h ) >. (33) Participation Constraint: k t a 3 (Σ Σ h ). Households will remain in the contract as long as the channel capacity difference is sufficiently greater than the intermediation fee. 14 / 21

15 Stationary Wealth Distribution (Constant Volatility) Endogenous Wealth Evolution dx t x t = µ x,tdt + σ x,tdz t..45 Stationary Density 2 Stationary Density left: σ H = 5 right: σ L =.9 15 / 21

16 Simulated Wealth Distribution 16 / 21

17 Probability of Constraint Binds 7 Probability of Falling into Constrained Region 6 5 High Ambiguity Low Ambiguity Probability of Sharpe Ratio Exceed Twice of the Mean:.32% 17 / 21

18 Asset Prices 18 / 21

19 Risk Premium Sharpe Ratio Two Agents Pessimism.25 Two Agents Ambiguity Difference Volatility 2 Expert Scaled Wealth / 21

20 Risk Premium Sharpe Ratio Two Agents Pessimism 1-4 Two Agents Ambiguity Difference Volatility Expert Scaled Wealth / 21

21 Conclusion Heterogeneity in information processing capacity Two frictions in financial contract: Participation constraint depends on the heterogeneity in channel capacity. Incentive constraint requires a minimum capital for risk-sharing, subjected to effective financial constraint. Endogenous heterogeneous beliefs due to ambiguity When volatility is high specialists become relatively pessimistic, and this tightens the capital constraint and accelerates the onset of a financial crisis. 21 / 21