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

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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.

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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.

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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

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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

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Model

Preferences

◮ The representative agent problem is to maximize:

U(C, X) := E0 +∞

• t=0

δt (Ct − Xt)1−γ − 1 1 − γ

• where Xt is level of habit, δ ∈ (0, 1) and γ > 0.

◮ Define surplus consumption ratio by:

St := Ct − Xt Ct

◮ St increases with consumption.

◮ In recessions, St approaches zero as consumption approaches

habit level.

◮ In booms, St approaches one as consumption rises relative to

habit.

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Model

Preferences (ctd.)

◮ Note that risk aversion (local curvature) is countercyclical:

ηt := −Ctucc(Ct, Xt) uc(Ct, Xt) = γ St

◮ 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.

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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:

Sa

t := C a t − Xt

C a

t

where C a

t is average consumption. ◮ Assume sa t := log Sa t is a heteroskedastic AR(1) process:

sa

t+1 = (1 − φ)¯

s + φsa

t + λ(sa t )[ca t+1 − ca t − g]

where λ(sa

t ) is called sensitivity function. ◮ Consumption is log-normal:

∆ct+1 = g + vt+1, vt+1 ∼ iid N(0, σ2)

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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

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Model

SDF and Sharpe ratio

◮ Implied stochastic discount factor (SDF) is:

Mt+1 := δ uc(Ct+1, Xt+1) uc(Ct, Xt) = δ St+1 St Ct+1 Ct −γ = δG −γ exp{−γ(st+1 − st + vt+1)}

◮ Recall that Sharpe ratio is bounded above by market price of risk

(Hansen and Jagannathan (1991)): Et[Re

t+1]

σt[Re

t+1] = −ρt(Mt+1, Re t+1) σt[Mt+1]

Et[Mt+1] ≤ σt[Mt+1] Et[Mt+1]

◮ In Campbell and Cochrane, largest Sharpe ratio is

max

j∈[N]

Et[Re

t+1]

σt[Re

t+1] =

• e[γσ(1+λ(st))]2 − 1 ≈ γσ(1 + λ(st))

◮ Choice of λ(st) must exhibit λ′(st) < 0 so that risk prices are higher

in bad times (when st is low).

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Model

Risk-free rate

◮ Since rf t = 1/Et[Mt+1], log risk-free rate is

rf

t = − log δ + γg − γ(1 − φ)(st − ¯

s) − γ2σ2 2 (1 + λ(st))2

◮ Two forces drive rf t :

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Model

Risk-free rate

◮ Since rf t = 1/Et[Mt+1], log risk-free rate is

rf

t = − log δ + γg − γ(1 − φ)(st − ¯

s) − γ2σ2 2 (1 + λ(st))2

◮ Two forces drive rf t :

◮ Intertemporal substitution: When surplus consumption ratio

is low, there is incentive to borrow and risk-free rate goes up.

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Model

Risk-free rate

◮ Since rf t = 1/Et[Mt+1], log risk-free rate is

rf

t = − log δ + γg − γ(1 − φ)(st − ¯

s) − γ2σ2 2 (1 + λ(st))2

◮ Two forces drive rf 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 (σ2) rises

willingness to save and drives down risk-free interest rate.

◮ In the data, rf t is fairly constant. So either φ ≈ 1 or, again,

λ′(st) < 0.

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Model

Choosing sensitivity function

◮ Function λ(st) chosen to satisfy there conditions:

• 1. Constant r f

t (i.e, λ′(st) < 0).

• 2. Predetermined habit at steady state (i.e, st = ¯

s at SS).

• 3. Stability near steady state (habit moves negatively with

consumption everywhere).

◮ Use:

λ(st) = √

1−2(st−¯ s)−1 ¯ S

− 1 if st ≤ smax

• therwise

where ¯ S is SS surplus consumption ratio and smax ensures that st ≤ smax, 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.

Graphs 15 / 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

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Model

Pricing claims to consumption

◮ Log surplus consumption ratio st is the only state. ◮ Stocks are modeled as claims to the consumption stream. ◮ The Euler conditional pricing equation holds:

Et[Mt+1Rt+1] = 1 with Rt+1 := Pt+1 + Dt+1 Pt

◮ Thus, the price/consumption and price/dividend ratios solve

Pt Ct (st) = Et

• Mt+1

Ct+1 Ct

• 1 + Pt+1

Ct+1 (st+1)

• Pt

Dt (st) = Et

• Mt+1

Dt+1 Dt

• 1 + Pt+1

Dt+1 (st+1)

• ◮ Dividends law of motion is exogenous:

∆dt+1 = g + wt+1, wt+1 ∼ iid N(0, σ2

w)

with CORR[wt, vt] = ρ (weak for US data).

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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

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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).

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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.

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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

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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

Evaluation

Numerical Solution (ctd.)

◮ The price/dividend ratios of the consumption and dividend

claims.

◮ Increase linearly with surplus ratio (procyclicality). ◮ When consumption is close to habit, there is a high utility

curvature which depresses price relative to dividends.

◮ Distribution is negatively skewed as well. 23 / 45

Evaluation

Numerical Solution (ctd.)

◮ Conditional expected claim returns and risk-free rate.

◮ Risk-free rate is constant by construction. ◮ Expected returns rise as consumption declines toward habit. ◮ Therefore, risk premium is countercyclical. 24 / 45

Evaluation

Numerical Solution (ctd.)

◮ Conditional standard deviations of claim returns.

◮ Price declines increase volatility (leverage effect). ◮ Conditional variance in return is highly autocorrelated. ◮ Conditional variation in volatility of returns is countercyclical. 25 / 45

Evaluation

Numerical Solution (ctd.)

◮ Maximal (HJ bound) and actual conditional Sharpe ratios.

◮ Consumption claim is nearly conditionally mean-variance

efficient (it almost attains HJ bound).

◮ Dividend claim is subject to nonlinear shocks and has higher

variance, so lower Sharpe ratios and less efficient.

◮ Countercyclical Sharpe ratios: in recessions, prices are low and

Sharpe ratios are high.

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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

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Evaluation

Simulated Data

◮ Simulation of 500,000 dates of monthly data. ◮ Comparison with historical data:

◮ Model matches mean and st.dev. of excess stock returns. ◮ Model accounts for volatility of stock prices. ◮ Discount factor δ = 0.89 and constant risk-free rate so both

equity premium and risk-free rate puzzles are “solved”.

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Evaluation

Simulated Data (ctd.)

◮ Long-horizon regressions of log excess returns on log

price/dividend ratio:

◮ Coefficients are negative: high prices imply low expected

returns.

◮ Coefficients increase linearly with horizon, and R2 gets higher. ◮ Model’s predictions match well postwar data. 29 / 45

Evaluation

Simulated Data (ctd.)

◮ Volatility tests and variance decompositions:

◮ Variation in price/dividend ratios is explained mostly by

expected return variation.

◮ Correlations between consumption growth and stock returns:

◮ Unlike the standard C-CAPM model, no perfect correlation. ◮ In the data, there is very little contemporaneous correlation. 30 / 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

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Evaluation

Historical Data

◮ Feeding model with actual data produces:

◮ Cyclical behavior and long-term fluctuations are accounted for. ◮ Yet, bad performance in latest years. 32 / 45

Equity Premium and Risk-Free Rate puzzles

◮ In the standard cansumption-based model with Mt+1 = δ

• Ct+1

Ct

−η , E[Re] σ(Re) ≈ ησ r f

t

= − log δ + ηg − η2 σ2 2

◮ EP puzzle:

To get Sharpe ratio of 0.5 with σ = 1.22, need η ≥ 41.

◮ RFR puzzle:

With η = 41 and g = 1.89, need δ = 1.9 > 1 to get r f

t = 0.01.

◮ In Campbell and Cochrane, both puzzles solved through high

curvature due to habits. Since coefficient of RRA is η = γ/¯ S, r f

t = − log δ + ηg −

η ¯ S 2 σ2 2

◮ Also able to rationalize long-run equity premium.

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Implications and Objections

◮ Three possible objections to the model:

• 1. It seems not to allow for heterogeneity across consumers.
• 2. It assumes an implausibly high risk aversion.
• 3. It relies on an external-habit rather than internal-habit

specification.

◮ We should examine if either of these are essential.

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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

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Implications and Objections

Heterogeneity

◮ With the heterogeneity in wealth in the data, poor people

could be below habit, which makes no sense in the power utility specification.

◮ Extensions:

• 1. Allow for different groups with different levels of wealth and let

each agent’s habit be determined by the average consumption

• f his reference group rather than the economy as a whole.
• 2. Allow explicitly for consumer heterogeneity:

◮ Assume individual endowment:

C i

t =

ξi ξ 1/γ (C a

t − Xt) + Xt

with weights ξ := 1

0 ξidi. ◮ This leads to the same marginal utility growth for all agents:

all individuals agree on asset prices.

◮ Same results, only algebraically more unpleasant. 36 / 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

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Implications and Objections

Risk Aversion

◮ Model is populated by agents with high risk aversions. ◮ Campbell and Cochrane argue that this feature is inescapable

in the class of identical-agent models that are consistent with equity premium.

◮ Let V (Wt, W a

t , Sa t ) be value of wealth, Wt. Risk aversion is:

rrat := −WtVWW VW = −∂ log VW ∂ log Wt

◮ By the envelope condition, uc = VW , so

rrat

• Risk aversion

= ∂ log uc ∂ log Ct × ∂ log Ct ∂ log Wt = ηt

• Curvature

× ∂ log Ct ∂ log Wt

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Implications and Objections

Risk Aversion (ctd.)

◮ In standard power utility model, ∂ log Ct ∂ log Wt = 1. ◮ However, in C&C, consumption rises more than proportionally

to an increase in idiosyncratic wealth:

◮ Consumer builds up wealth due to a precautionary motive in

• rder to keep it above habit level.

◮ That is, ∂ log Ct ∂ log Wt ≥ 1, so

rrat ≥ ηt

◮ One way to break the compromise is to create low ∂ log Ct ∂ log Wt

(Constantinides (1990), Boldrin, Christiano and Fisher (1996)), but then long-run equity premium is not explained.

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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

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Implications and Objections

External versus Internal Habit

◮ Using internal as opposed to external habit lowers marginal

utilities at all dates.

◮ Since prices are determined by ratios of marginal utilities,

introducing internal habit does not change aggregate implications:

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Conclusions

◮ A consumption-based model of asset pricing augmented by

external habit preference exhibits:

• 1. Long-predictability of excess returns from dividend/price ratio.
• 2. Mean reversion in returns.
• 3. High stock price and return volatility.
• 4. Persistent movements in stock volatility.

◮ Predictions are in spite of constant risk-free rate, and

rationalize both equity premium and risk-free rate puzzles.

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APPENDIX

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Appendix

Given the assumed λ(st),

• 1. The interest rate is constant:

r f

t = − log δ + γg − γ

2 (1 − φ)

• 2. Differentiating the law of motion of sa

t+1,

∂xt+1 ∂ct+1 ≈ 1 − λ(st) e−st − 1 an approximation that holds near the steady state. To get ∂xt+1

∂ct+1 = 0 at

steady state, we require λ(¯ s) = 1 ¯ S − 1 which holds given the assumptions.

• 3. To ensure determinacy near the steady state, we require

∂ ∂s ∂x ∂c

• s=¯

s

= 0 which means λ′(¯ s) = − 1

¯ S , true given the assumptions. Back 44 / 45

Appendix

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