Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion Learning About Rare Disasters Implications for Consumption and Asset Prices Max Gillman & Michal Kejak & Michal Pakoš CERGE-EI June 4, 2014 Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion What We Do Minimal extension of the Mehra-Prescott-Rietz asset-pricing framework to an incomplete information setting Two key ingredients variable growth persistence Learning about growth persistence magnifies economic uncertainty relaxed independence axiom the recursive Epstein-Zin preferences configured so that early resolution of uncertainty is preferred. Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Endogenous Uncertainty Shocks countercyclical variation in the forecast-error variance Var t { g t , T } of the T-period consumption growth rate � C t + T � g t , T = log C t Fact Our model endogenously generates Var t { g t , T } Larger in Recessions compared to Expansions Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Endogenous Uncertainty Shocks (Cont’d) Matching monotonic patterns across the phases of the business cycle Fact Our model endogenously generates Var t { g t , T } ց in Expansions Var t { g t , T } ր in Recessions Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Equity Prices Procyclical variation in price-dividend ratios Countercyclical variation in risk premiums return volatility Sharpe ratios These effects naturally induce leverage effect mean reversion of excess returns predictability excess return consumption volatility from price-dividend ratio Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Equity Prices across Phases of Business Cycle Fact Our model endogenously generates Risk Premium Return Volatility ր in Recessions Sharpe Ratio Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Real Bond Prices Matching the real yield curve level variability persistence of yields Real(!) Yield Curve slopes ց in Expansions Real(!) Yield Curve slopes ր in Recessions Matching bond risk premiums level variability persistence Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Matching the Consumption-Growth-Rate Moments Modeling Approach Matching the Asset-Pricing Moments Implications for Consumption and Asset Prices Conclusion Equity Option Prices Our preliminary results indicate the implied volatility curves of S&P 500 index options in our model mildly downward sloping display negligible curvature Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion Two-State Markov Chain Two−State Markov Chain Expansion Sojourn Time Short Recession Sojourn Time Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion Fast Tail Slimming λ −1 = 1 Year 1 0.9 0.8 0.7 Exponential Density 0.6 0.5 0.4 Prob { No Lost Decade } ≈ 1.0 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Recession Sojourn Time (years) Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion Slow Tail Slimming λ −1 = 10 Years 0.1 0.09 0.08 0.07 Exponential Density 0.06 0.05 0.04 0.03 0.02 Prob { Lost Decade } ≈ 37% 0.01 0 0 10 20 30 40 50 60 70 80 90 100 Recession Sojourn Time (years) Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Modeling Approach Implications for Consumption and Asset Prices Conclusion Learning About Tail Slimming Two−State Semi−Markov Chain Expansion Sojourn Time Short Recession Sojourn Time Long Recession Sojourn Time Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Standard Asset-Pricing Framework representative agent endowment economy á la Lucas (1978) and Mehra-Prescott (1985) with recursive Epstein-Zin preferences 1 �� 1 − 1 � � 1 − 1 1 − 1 � 1 − e − δ � � θ θ e − δ C V 1 − γ � 1 − γ J t = θ + E t t t + 1 where θ = elasticity of intertemporal substitution, γ = relative risk aversion and δ = subjective discount rate with hidden states of the macroeconomy = ⇒ Bayesian learning Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Consumption Dynamics Subject the endowment growth rate � C t + 1 � g t + 1 = log C t to hidden regime shifts σ u ε t + 1 g t + 1 = µ ( S t ) + � �� � � �� � Predictable Component Consumption Growth Surprise Consumption = Endowment in equilibrium Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Predictable Component µ ( S t ) Specification Consider a three-state Markov chain S t ∈ { 0 = Expansion , 1 = Recession , 2 = Lost Decade } with the transition probability matrix p 1 q × p 1 ( 1 − q ) × p 1 P = 1 − p 2 p 2 0 1 − p 3 0 p 3 and consumption growth rates µ ( 1 ) , µ ( 2 ) and µ ( 3 ) Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Semi-Markov Property Consumption Growth Rates Fact The growth rates in the recession and the lost decade are exactly equal, µ ( 2 ) = µ ( 3 ) Our maximum-likelihood estimates µ ( 2 ) = µ ( 3 ) ≈ − 0 . 79 % per year The model is best thought of as two-state semi-Markov model the recession sojourn time is not exponentially distributed the hazard rate of ending the recession is not constant Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Difference from Rietz (1988, JME) Fact The growth rates in the recession and the lost decade are significantly different, µ ( 2 ) ≈ − 0 . 79 % per year µ ( 3 ) ≈ − 79 % per year Moreover, recessions S t ∈ { 1 , 2 } are not persistent Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
Introduction and Related Literature Preview of Results Learning about Tail Slimming Preferences Modeling Approach Implications for Consumption and Asset Prices Conclusion Markov vs. Semi-Markov Chain Consider standard two-state Markov model recessions last about one year, hence the hazard rate λ = 1 probability of observing a lost decade is P { Low-Growth Sojourn Time > 10 years | S � = Expansion } ˆ ∞ λ e − λτ d τ = exp ( − 10 × λ ) = exp ( − 10 ) = 0 . 00005 = 10 Max Gillman & Michal Kejak & Michal Pakoš Learning About Rare Disasters
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