Report

X-CAPM: An extrapolative capital asset pricing model Barberis et al. (JFE forthcoming) Roy 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 Investors’ problem • Extrapolator: • Same for rational trader • Clearing condition Equilibrium 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: • 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. Model prediction for ratio based quantities