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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
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 overvaluation.
• But ultimately low dividends bring the overvalued stock back, and then extrapolators start selling
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
Setting
• 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
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
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
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:
• 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
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
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
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
• 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)
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 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
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
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.