1. X-CAPM: An extrapolative capital asset pricing model Barberis et al
X-CAPM: An extrapolative capital asset pricing model Barberis et al. (JFE forthcoming)
Roy

2. Motivation
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 moments

3. What is in the paper?
Analytically solve a heterogeneous agents consumption based model
Simulate the model
Match some moments

4. 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 overvaluation.
But ultimately low dividends bring the overvalued stock back, and then extrapolators start selling

5. Special assumptions
Dividend level follows an arithmetic Brownian motion
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

6. Setting Two type of assets: Dividend: arithmetic Brownian motion
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

7. Setting
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

8. Setting
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

9. Investors’ problem Extrapolator: Same for rational trader
Clearing condition

10. Equilibrium

11. Eqm vs R – stock price Rational:
Equilibrium in presence of extrapolators

12. 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.

13. Eqm vs E – sentiment process
Extrapolators:
Equilibrium:
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

14. E vs R – stock price Rational: Equilibrium:
When extrapolators are present, consumption policy depends on S_t.
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 consumption

15. Empirical implication
Predictive power of D/r –P for future price changes
Autocorrelation of P-D/r
Volatility of price changes and of P-D/r
Autocorrelation of price changes
Correlation of consumption changes and prices changes
Predictive power of surplus consumption
Equity premium and Sharpe ratio

16. 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 change

17. For a fixed horizon, the predictive power of D/r - P is stronger for low μ (few rational investors)
The predictive power of D/r - P is weaker for low β (more persistent)

18. 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

19. Volatility of price changes and of P-D/r
lower μ , higher volatility
β does not matter too much in volatility of price change

20. 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 changes.
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

21. Predictive power of surplus consumption
Surplus consumption difference predicts subsequent price changes with negative sign
this predictive power is strong for low μ and high β

22. Predictive power of surplus consumption
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.

23. Model prediction for ratio based quantities