By Force of Habit: A Consumption-Based Explanation of Aggregate - PowerPoint PPT Presentation

By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior by John Y. Campbell and John H. Cochrane (JPE, 1999) Pau Roldan NYU February 25, 2014 1 / 45

Motivation Explaining Aggregate Stock Market Behavior ◮ Classical consumption-based models of asset pricing failed to explain aggregate stock market phenomena. ◮ As of 1999, models missed the fundamental sources of risk driving expected returns. ◮ In the data : ◮ Risk premia is countercyclical (higher at business cycle troughs). ◮ Excess returns are forecastable, predicted by variables that are correlated with or predict business cycle. ◮ Countercyclical variation in stock market volatility. 2 / 45

Motivation Explaining Aggregate Stock Market Behavior ( ctd. ) ◮ Campbell and Cochrane presented an extension of the consumption-based model that empirically matches: ◮ Procyclical variation in stock prices. ◮ Level and volatility of price/dividend ratios and long-horizon forecastability of stock returns. ◮ Both short- and long-run equity premia with slow countercyclical variation in spite of constant risk-free rate. ◮ Key ingredient : slow-moving external habit in preferences. ◮ External habit adjusts the curvature of utility. ◮ In bad times, consumption declines toward habit level, curvature rises and thus risky asset prices fall and expected returns rise. 3 / 45

Motivation Habit ◮ Habit formation introduces preference for high consumption surplus (i.e, over and above habit level) rather than high absolute levels . ◮ This allows for new interpretation of risk premia: ◮ Investors fear stocks because they do badly when surplus consumption ratios are low (recessions) and not because stock returns are correlated with declines in absolute levels of consumption. ◮ This different channel helps close Mehra and Prescott’s equity premium gap. 4 / 45

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 5 / 45

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 6 / 45

Model Preferences ◮ The representative agent problem is to maximize: � + ∞ � δ t ( C t − X t ) 1 − γ − 1 � U ( C , X ) := E 0 1 − γ t =0 where X t is level of habit, δ ∈ (0 , 1) and γ > 0. ◮ Define surplus consumption ratio by: S t := C t − X t C t ◮ S t increases with consumption. ◮ In recessions , S t approaches zero as consumption approaches habit level. ◮ In booms , S t approaches one as consumption rises relative to habit. 7 / 45

Model Preferences ( ctd. ) ◮ Note that risk aversion (local curvature) is countercyclical : η t := − C t u cc ( C t , X t ) = γ u c ( C t , X t ) S t ◮ Habit is external (Abel (1990)). ◮ An individual’s habit level depends on the history of aggregate consumption rather than on the individual’s own past consumption. 8 / 45

Model Consumption and Surplus consumption processes ◮ To generate slow mean reversion in price/dividend ratios, persistence in volatility and long-horizon return forecast ability, we need habit to move slowly in response to consumption. ◮ Define: t := C a t − X t S a C a t where C a t is average consumption. ◮ Assume s a t := log S a t is a heteroskedastic AR(1) process: s a s + φ s a t + λ ( s a t )[ c a t +1 − c a t +1 = (1 − φ )¯ t − g ] where λ ( s a t ) is called sensitivity function . ◮ Consumption is log-normal: v t +1 ∼ iid N (0 , σ 2 ) ∆ c t +1 = g + v t +1 , 9 / 45

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 10 / 45

Model SDF and Sharpe ratio ◮ Implied stochastic discount factor (SDF) is: � S t +1 � − γ δ u c ( C t +1 , X t +1 ) C t +1 M t +1 := = δ u c ( C t , X t ) S t C t δ G − γ exp {− γ ( s t +1 − s t + v t +1 ) } = ◮ Recall that Sharpe ratio is bounded above by market price of risk (Hansen and Jagannathan (1991)): E t [ R e t +1 ] t +1 ) σ t [ M t +1 ] E t [ M t +1 ] ≤ σ t [ M t +1 ] t +1 ] = − ρ t ( M t +1 , R e σ t [ R e E t [ M t +1 ] ◮ In Campbell and Cochrane, largest Sharpe ratio is E t [ R e t +1 ] � e [ γσ (1+ λ ( s t ))] 2 − 1 ≈ γσ (1 + λ ( s t )) max t +1 ] = σ t [ R e j ∈ [ N ] ◮ Choice of λ ( s t ) must exhibit λ ′ ( s t ) < 0 so that risk prices are higher in bad times (when s t is low). 11 / 45

Model Risk-free rate ◮ Since r f t = 1 / E t [ M t +1 ], log risk-free rate is s ) − γ 2 σ 2 r f (1 + λ ( s t )) 2 t = − log δ + γ g − γ (1 − φ )( s t − ¯ 2 ◮ Two forces drive r f t : 12 / 45

Model Risk-free rate ◮ Since r f t = 1 / E t [ M t +1 ], log risk-free rate is s ) − γ 2 σ 2 (1 + λ ( s t )) 2 r f t = − log δ + γ g − γ (1 − φ )( s t − ¯ 2 ◮ Two forces drive r f t : ◮ Intertemporal substitution : When surplus consumption ratio is low, there is incentive to borrow and risk-free rate goes up. 13 / 45

Model Risk-free rate ◮ Since r f t = 1 / E t [ M t +1 ], log risk-free rate is s ) − γ 2 σ 2 r f (1 + λ ( s t )) 2 t = − log δ + γ g − γ (1 − φ )( s t − ¯ 2 ◮ Two forces drive r f t : ◮ Intertemporal substitution : When surplus consumption ratio is low, there is incentive to borrow and risk-free rate goes up. ◮ Precautionary savings : An increase in uncertainty ( σ 2 ) rises willingness to save and drives down risk-free interest rate. ◮ In the data, r f t is fairly constant. So either φ ≈ 1 or, again, λ ′ ( s t ) < 0. 14 / 45

Model Choosing sensitivity function ◮ Function λ ( s t ) chosen to satisfy there conditions: 1. Constant r f t (i.e, λ ′ ( s t ) < 0). 2. Predetermined habit at steady state (i.e, s t = ¯ s at SS). 3. Stability near steady state (habit moves negatively with consumption everywhere). ◮ Use: � √ 1 − 2( s t − ¯ s ) − 1 − 1 if s t ≤ s max ¯ λ ( s t ) = S 0 otherwise where ¯ S is SS surplus consumption ratio and s max ensures that s t ≤ s max , a.s. ◮ The three conditions above can be shown to be satisfied. Proof ◮ Thus, we get: ◮ Higher sensitivity in crises. ◮ Habit responds positively to consumption everywhere and does not move around SS. 15 / 45 Graphs

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 16 / 45

Model Pricing claims to consumption ◮ Log surplus consumption ratio s t is the only state. ◮ Stocks are modeled as claims to the consumption stream. ◮ The Euler conditional pricing equation holds: R t +1 := P t +1 + D t +1 E t [ M t +1 R t +1 ] = 1 with P t ◮ Thus, the price/consumption and price/dividend ratios solve � � �� P t C t +1 1 + P t +1 ( s t ) = ( s t +1 ) E t M t +1 C t C t C t +1 � � �� P t D t +1 1 + P t +1 ( s t ) = ( s t +1 ) E t M t +1 D t D t D t +1 ◮ Dividends law of motion is exogenous: w t +1 ∼ iid N (0 , σ 2 ∆ d t +1 = g + w t +1 , w ) with CORR [ w t , v t ] = ρ (weak for US data). 17 / 45

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 18 / 45

Evaluation Calibration ◮ Model is compared to two data sets: 1. Post-war (1947-1995) New York Stock Exchange stock index returns, 3-month Treasury bill rate and per capita nondurables and services consumption. 2. Annual data set of S&P500 stock and commercial paper returns (1871-1993) and per capita consumption (1889-1992). 19 / 45

Evaluation Evaluation Exercises ◮ The model is evaluated with three exercises: 1. Solve the model numerically and characterize its behavior. 2. Simulate data by drawing shocks randomly and show how simulated data replicates actual data. 3. Feed the model historical consumption shocks to assess empirically implied movements in asset prices. 20 / 45

Outline Motivation Model Preferences and Technology Marginal utility Pricing claims Evaluation Calibration Numerical Solution Simulated Data Historical Data Equity Premium and Risk-Free Rate puzzles Microeconomic Implications and Objections to the Model Heterogeneity Risk Aversion External versus Internal Habit Conclusions 21 / 45

Evaluation Numerical Solution ◮ The stationary unconditional distribution of the surplus consumption ratio. ◮ Negatively skewed, with important fat tail of low surplus. ◮ Occasional deep recessions are not matched by large booms. 22 / 45