Six anomalies looking for a model: A consumption based explanation of international finance puzzles

Transcription

1 Six anomalies looking for a model: A consumption based explanation of international finance puzzles Riccardo Colacito Shaojun Zhang (NYU Stern) Colacito (2009) 1 / 17

2 Motivation Six Puzzles Backus-Smith anomaly model : e t+1 = mt+1 m t+1 data : corr( c c, e) = Brandt, Cochrane and Stanta-Clara puzzle model : var( e t+1 ) = var(m t+1) + var(m t+1 ) 2cov(m t+1, m t+1 ) data : std( e) = 0.11, std(m UK ) = 0.39, std(m US ) = 0.37 UIP puzzle model : E t [ e t+1 ] = r t r t 1 2 var t(m t+1) var t(m t+1 ) data : e t+1 = α + β(r t r t ) + ξ t+1 : β = 0 Shaojun Zhang (NYU Stern) Colacito (2009) 2 / 17

3 Motivation Six Puzzles equity premium puzzle (average Sharpe ratios 0.38 for G7) low correlation in fundamentals, but high correlation in stock returns, and higher but not perfect correlation in risk-free rates corr( c, c ) = corr( d, d ) = corr(r stk, rstk) = corr(r rf, rrf ) = Shaojun Zhang (NYU Stern) Colacito (2009) 3 / 17

4 Model Model and Calibration Complete markets Representative agents have risk-sensitive preferences: ( ) Ut+1 U t = (1 δ) log C t + δθ log E t exp θ ( U Ut = (1 δ) log Ct ) + δθ log E t exp t+1 θ with θ = 1 1 γ. Shaojun Zhang (NYU Stern) Colacito (2009) 4 / 17

5 Model Model and Calibration Long-run risk in consumption: log C t+1 log C t = c t+1 = µ c + λ 1 z 1,t + λ 2 z 2,t + λ 1/2 t ε t+1 log Ct+1 log Ct = ct+1 = µ c + λ1z 1,t + λ2z 2,t + λt 1/2 ε t+1 z 1,t = ρ 1 z 1,t 1 + λ 1/2 t 1 ε 1,t z 2,t = ρ 2 z 2,t 1 + λt 1 1/2 2,t λ t = σ(1 ρ λ ) + ρ λ λ t 1 + φ λ λ t 1 ε λ,t Dividend dynamics: λ t = σ(1 ρ λ ) + ρ λ λ t 1 + φ λ λ t 1ε λ,t d t+1 = µ d + λ d1 z 1,t + λ d2 z 2,t + λ 1/2 t ε d,t+1 dt+1 = µ d + λd 1 z 1,t + λd 2 z 2,t + λt 1/2 ε d,t+1 Shaojun Zhang (NYU Stern) Colacito (2009) 5 / 17

6 Baseline Calibration Model and Calibration Shaojun Zhang (NYU Stern) Colacito (2009) 6 / 17

7 Model and Calibration Baseline calibration - long-run risk log C t+1 log C t = c t+1 = µ c + 5.5z 1,t + z 2,t + λ 1/2 t ε t+1 log Ct+1 log Ct = ct+1 = µ c + z 1,t + 5.5z 2,t + λt 1/2 ε t+1 z 1,t = 0.987z 1,t 1 + λ 1/2 t 1 ε 1,t z 2,t = 0.987z 2,t 1 + λt 1 1/2 2,t λ t = 0.006(1 0.95) λ t λ t 1 ε λ,t λ t = 0.006(1 0.95) λ t λ t 1ε λ,t Shaojun Zhang (NYU Stern) Colacito (2009) 7 / 17

8 Benchmark Result Model and Calibration Shaojun Zhang (NYU Stern) Colacito (2009) 8 / 17

9 Solution of the model Inside the black box Log discount rate can be solved as follows: Solution method m t+1 = v 0,m v m s t + 1 θ ( ṽ 1 λ 1/2 t + v 2 λt 1/2 ( v1 λ 1/2 t + ṽ2 λt 1/2 mt+1 = v 0,m vms t + 1 θ s t = [z 1,t, z 2,t, λ t, λt, h t ] h t = λ 1/2 t λt 1/2 η t = [ε 1,t, ε 2,t, ε c,t, ε c,t, ε λ,t, ε λ,t] ) η t+1 ) η t+1 Shaojun Zhang (NYU Stern) Colacito (2009) 9 / 17

10 Inside the black box Volatility of real exchange rate changes Complete market e t+1 = m t+1 m t+1 = (v m v m) s t + 1 θ ( ) (v1 ṽ 1 )λ 1/2 t + (ṽ2 v 2 )λt 1/2 η t+1 Key: pick correlation matrix of the shocks carefully. Shaojun Zhang (NYU Stern) Colacito (2009) 10 / 17

11 Inside the black box Correlation of risk free rates Let s shut down uncertainty shocks for now. Short rates are: r t = r t + λ 1 z 1,t + λ 2 z 2,t r t = r t + λ 1z 1,t + λ 2z 2,t Key: two serially correlated shocks with proper correlation. Shaojun Zhang (NYU Stern) Colacito (2009) 11 / 17

12 Inside the black box Correlation of stock market returns Use Campbell-Shiller decomposition: r d,t+1 = k 0 + d t+1 k 1 dp t+1 + dp t dp t = dp + λ 1 λ d1 1 k 1 ρ 1 z 1,t + λ 2 λ d2 1 k 1 ρ 2 z 2,t dp t = dp + λ 2 λ d2 1 k 1 ρ 2 z 1,t + λ 1 λ d1 1 k 1 ρ 1 z 2,t Key: pick corr(z 1, z 2 ) carefully. Shaojun Zhang (NYU Stern) Colacito (2009) 12 / 17

13 Backus-Smith puzzle Inside the black box Keep uncertainty shocks shut down. Innovations are as follows e = β 1 ε 1 β 2 ε 2 + λε λ ε c c = λε λ ε asymmetric loadings β 1 = β 2 Key: correlation: ρ 1,c = ρ 2,c = 0.06, ρ 1,c = ρ 2,c = 0.6 Shaojun Zhang (NYU Stern) Colacito (2009) 13 / 17

14 UIP puzzle Inside the black box Let s turn on uncertainty shocks again. Solve for slope explicitly: UIP regression e t+1 = α + β(r t rt ) + ξ t+1 ( Model counterpart mt+1 m t+1 = α + β p t + 1 ) 2 q t + ξ t+1 where ˆβ = Var(p t) Cov(p t, q t ) Var(p t q t) p t = E t m t+1 E t m t+1 = A 1 z 1,t A 2 z 2,t + B 1 λ t B 2 λ t q t = Var t m t+1 Var t m t+1 = B 1 λ t + B 2 λ t Key: get Cov(p t, q t ) sufficiently negative. Shaojun Zhang (NYU Stern) Colacito (2009) 14 / 17

15 Stochastic Volatility Sensitivity Analysis Shaojun Zhang (NYU Stern) Colacito (2009) 15 / 17

16 Long-run risk Sensitivity Analysis Shaojun Zhang (NYU Stern) Colacito (2009) 16 / 17

17 Conclusion Sensitivity Analysis It reverse-engineers a plausible consumption process, builds a complete market two-country model with risk-sensitive preferences and long run risk. It can explain the six puzzles. Shaojun Zhang (NYU Stern) Colacito (2009) 17 / 17