X-CAPM: An extrapolative capital
asset pricing model
Barberis et al. (JFE forthcoming)
• Greenwood and Shleifer (2014): investors hold
extrapolative expectations
• Hard to be justified by traditional models
• This paper: address the survey evidence,
hopefully still able to explain some AP
What is in the paper?
• Analytically solve a heterogeneous agents
consumption based model
• Simulate the model
• Match some moments
How does the model work?
• Rational investors and price extrapolators differ only in
expectations about future stock return
• Extrapolators cause the jump to be amplified,
mispricing created by wrong expectation
• Can we explain by traditional model? No, because
stock price go up implies either risk aversion or
perceived risk go down.
• Rational investors know the decisions of extrapolators,
hence do not aggressively counteract the
• But ultimately low dividends bring the overvalued
stock back, and then extrapolators start selling
Special assumptions
• Dividend level follows an arithmetic Brownian
• Investor preferences (CARA, not CRRA):
exponential utility
– more natural to work with quantities defined in
terms of differences rather than ratios, e.g. price
changes rather than returns, “price-dividend
difference” rather than price-dividend ratio.
• Risk free rate, an exogenous constant
• Two type of assets:
– A risk free asset with perfectly elastic supply and
constant interest rate r
– Risky asset with fixed supply Q
• Dividend: arithmetic Brownian motion
• Two type of agents: a continuum of rational
investors and extrapolators
• Extrapolator form beliefs about future price changes
on stock market
• Sentiment (momentum):
• Assume extrapolator’s expectation of the speed of
change in stock prices:
• Price process: no dividends in it
• Assume extrapolators know sigma_p
• Rational investor has correct belief about
dividend process and price process.
• Know how the extrapolators form their beliefs
and trade accordingly
• Both are price takers
Investors’ problem
• Extrapolator:
• Same for rational trader
• Clearing condition
Eqm vs R – stock price
• Rational:
• Equilibrium in presence of extrapolators
Eqm vs E – Stock price process
• Extrapolators:
• Rational investors:
• Eqm
• Extrapolators: expected instantaneous price change
depends positively on the S_t.
• Rational: depends on dividends.
• In equilibrium: depends negatively on S_t.
Eqm vs E – sentiment process
Extrapolators: sentiment follows a random walk if lamda_1 = 1, lamda_0 = 0
In equilibrium: mean-reverting. The higher the beta, the more rapid reverts back
to mean
E vs R – stock price
• Rational:
• Equilibrium:
• When extrapolators are present, consumption policy depends on
• b^e>b^r: extrapolators increase their consumption more due to
income effect
• a^e and a^r are both negative: when sentiment deviates
substantially from its long- run mean, both types increase their
Empirical implication
• Predictive power of D/r –P for future price
• Autocorrelation of P-D/r
• Volatility of price changes and of P-D/r
• Autocorrelation of price changes
• Correlation of consumption changes and prices
• Predictive power of surplus consumption
• Equity premium and Sharpe ratio
Predictive power of D/r –P
• Analogous to Cochrane(2011) regressions, dividend price
change can be expressed as.
• As a matter of accounting, the three regression coefficients
must sum to approximately one at long horizons.
– Price change on the current dividend-price change;
– Dividend change on the current dividend-price change;
– future dividend-price change on the current dividend-price
• For a fixed horizon,
the predictive
power of D/r - P is
stronger for low μ
(few rational
• The predictive
power of D/r - P is
weaker for low β
(more persistent)
Predictive power of D/r –P
• Good cash-flow news stock prices , extrapolators’
expectations , push further current stock price , D/r- P .
• But the stock market is now overvalued, subsequent price
change .
• Predictability stems on extrapolators, so predictive power is
stronger for low μ
• Low β implies high persistent, takes longer to correct
overvaluation, lower predictive power
Volatility of price changes and of P-D/r
lower μ , higher volatility
β does not matter too much in volatility of price change
Volatility of price changes and of P-D/r
• A good cash-flow shock, price , extrapolators push stock
prices up further. Rational investors counter act this
overvaluation, but only mildly: they know that
extrapolators will continue to bid.
• The larger the fraction of extrapolators (low μ) in the
economy, the more excess volatility there is in price
• Excess volatility is insensitive to β. Surprising?
– extrapolators’ beliefs are more varying when β is high, higher β
higher volatility
– However, precisely because extrapolators change their beliefs
more quickly when β is high, any mispricing will correct more
quickly in this case, so rational traders trade more aggressively
against the extrapolators, dampening volatility
Predictive power of surplus
• Surplus consumption
difference predicts
subsequent price
changes with
negative sign
• this predictive power
is strong for low μ
and high β
Predictive power of surplus
• Good cash-flow news -> Extrapolators' expectation ->
consume more -> aggregate consumption , the surplus
consumption .
• Since the stock market is overvalued at this point, the
subsequent price change
• The surplus consumption difference predicts future price
changes with a negative sign.
Model prediction for ratio based

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