Zhi Da, University of Notre Dame

EXTRAPOLATIVE BELIEFS IN THE CROSS-SECTION: WHAT CAN WE LEARN FROM THE CROWDS?
Zhi Da, University of Notre Dame Xing Huang, Washington University in St. Louis Lawrence Jin, California Institute of Technology SWUFE Seminar July 18, 2018

Central questions in finance
Motivation Central questions in finance How do investors form expectations? Survey evidence shows that investors display extrapolative expectations. Greenwood and Shleifer (2014) | Koijen, Schmeling, and Vrugt (2015) | Kuchler and Zafar (2016) | Armona, Fuster and Zafar (2016) | Amromin and Sharpe (2013) | Frankel and Froot (1987) | Piazzesi and Schneider (2009) How do such expectations affect asset prices? Extrapolative expectations help explain empirical facts about asset prices Greenwood and Shleifer (2014) | Koijen, Schmeling, and Vrugt (2015) | Barberis, Greenwood, Jin, Shleifer (2015)

Central questions in finance
Motivation Central questions in finance Extrapolation models have been empirically tested primarily with aggregate market-level data There has not been a sufficient amount of empirical support for extrapolative expectations in the cross- section, mostly due to data limitation. The micro-foundation of extrapolation, especially how investors form beliefs when facing multiple assets, remains unclear. Possible sources of extrapolative expectations include “belief in the law of small numbers” (Barberis, Shleifer, and Vishny, 1998; Rabin, 2002), availability heuristic (Jin 2015; Gennaioli, Shleifer, and Vishny 2012), and experience effect (Malmendier and Nagel 2011, 2016), among others. However, these studies of belief formation have primarily focused on a single risky asset.

Belief formation in the cross-section
In our paper Belief formation in the cross-section We study the following questions: How do individuals form their beliefs about future returns in the cross-section? How do such beliefs affect asset prices cross-sectionally? We use a novel dataset from a crowdsourcing platform (Forcerank App) Forcerank collects expectations on future stock performances over specified horizons (e.g. one week). Users who contribute on Forcerank are highly diverse and geographically distributed.

Forcerank t t+1 t+2 Shoutout on media for the winner accumulating the most amount of points within the past 13 weeks Rank your stocks by expected performance over the next week (t+1) Track your rankings against your peers' and actual performance

Good features of the data
Forcerank Social media Analyst target price Precise quantitative information Clearly-specified horizon Heterogeneous users Fixed set of stocks Blind setting

Preview results Forcerank beliefs negatively predict future stock returns Robust for Fama-French 5 factors, momentum factor, short-term reversal

Extrapolative beliefs in the cross-section
Preview results Extrapolative beliefs in the cross-section Individuals extrapolate from past returns More weight on more recent returns Return predictability in the cross-section Stronger for stocks with more extrapolators and higher degree of extrapolation bias Out-of-sample: Return predictability extends to stocks that are not covered by the platform. Decomposition: Both the extrapolative component and the residual component have predictive power

Roadmap 3 1 2 4 Return predictability of Forcerank score
Heterogeneity across extrapolators Heterogeneity across extrapolation bias Out-of-sample test 3 Linear model Exponential decay model Prof vs. non-prof 1 An extrapolative model DATA & STATISTICS Game example Sample statistics Model setup Cross-sectional prediction 2 4 Extrapolative Expectation Return predictability

Sample statistics Sample period :
Main analysis: 2016/02 – 2017/12 The same game is repeatedly conducted every week on the platform. multiple weekly contests for the same game. Contest (ticker-week) panel: 1396 contests 293 tickers An contest (a sequence of entries ranking 10 tickers based on expected weekly returns) contains: 20 entries on average

Sample statistics Main analysis: 2016/02 – 2017/12

Heterogeneity across extrapolators Heterogeneity across extrapolation bias Out-of-sample test 3 Linear model Exponential decay model Prof vs. non-prof 1 An extrapolative model DATA & STATISTICS Contest example Sample statistics Model setup Cross-sectional prediction 2 4 Extrapolative Expectation Return predictability

Extrapolative expectations
Linear model Individuals extrapolate from past returns Higher weights on more recent returns

Extrapolative expectations Exponential decay model: specification
Previously estimated by Greenwood and Shleifer (2014), Barberis et al. (2015) and Cassella and Gulen (2017). 𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 0 + 𝜆 1 𝑠=0 𝑛 𝑤 𝑠 𝑅 𝑖,𝑡−𝑠 + 𝜀 𝑡 Level effect: A scaling factor that captures the overall extent to which individuals respond to past returns. 𝑤 𝑠 = 𝜆 2 𝑠 𝑗=0 𝑛 𝜆 2 𝑗 , 0≤ 𝜆 2 <1 Slope effect: It captures how the weights on past returns decay A lower 𝜆 2 means higher weights on more recent past returns (i.e., shorter memory) A higher 𝜆 1 and a lower 𝜆 2 indicate a higher degree of extrapolation bias

Extrapolative expectations Exponential decay model: estimation horizon
𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 0 + 𝜆 1 𝑠=0 𝑛 𝑤 𝑠 𝑅 𝑖,𝑡−𝑠 + 𝜀 𝑡 𝑤 𝑠 = 𝜆 2 𝑠 𝑗=0 𝑛 𝜆 2 𝑗 , 0≤ 𝜆 2 <1 We use n=12 for exponential decay model estimation

Extrapolative expectations Exponential decay model: estimation
𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 0 + 𝜆 1 𝑠=0 𝑛 𝑤 𝑠 𝑅 𝑖,𝑡−𝑠 + 𝜀 𝑡 𝑤 𝑠 = 𝜆 2 𝑠 𝑗=0 𝑛 𝜆 2 𝑗 , 0≤ 𝜆 2 <1 Returns one month earlier (week t-4) are only 9% as important as returns in the past week (week t)

Surveying CalTech students
Anecdotes Surveying CalTech students How do you come up with the rankings? “based on last week and last month's performance” “based on quick look of past month returns“ “based roughly on last weeks ranks” …… “Although MSFT had a bad return last week, the 1 year graph of MSFT indicates that the price is steadily increasing every month. Thus, I thought that the price would increase next week despite its bad result last week.” “Average of returns over the last 4 months” “Average for last 2 months”

Micro-foundation for Extrapolative expectation
Asymmetric effects of positive past returns vs. negative past returns Higher overall weight and longer memory span for negative past returns Heterogeneous effects across professionals vs. non-professionals Professionals rely less on past returns and have a longer memory span Determinants of extrapolation biases Size, B/M, volatility, volume

positive vs. negative past returns
Asymmetric effects of positive vs. negative past returns Extrapolation is asymmetric Longer memory span for negative past returns

Heterogeneous extrapolative expectations
Professional vs. Non-professional 𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 0 + 𝜆 1 𝑠=0 𝑛 𝑤 𝑠 𝑅 𝑖,𝑡−𝑠 + 𝜀 𝑡 𝑤 𝑠 = 𝜆 2 𝑠 𝑗=0 𝑛 𝜆 2 𝑗 , 0≤ 𝜆 2 <1 Professionals rely less on past returns Professionals have a longer memory span

Determinants of extrapolation bias
We estimate the extrapolation parameters for each stock 𝒊 : 𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 𝑖, 0 + 𝜆 𝑖, 1 𝑠=0 𝑛 𝑤 𝑖, 𝑠 𝑅 𝑖, 𝑡−𝑠 + 𝜀 𝑖, 𝑡 𝑤 𝑖,𝑠 = 𝜆 𝑖,2 𝑠 𝑗=0 𝑛 𝜆 𝑖, 𝑗 , 0≤ 𝜆 𝑖, 2 <1 Larger firms are more visible to investors Higher volatility increases difficulties to infer trend Volume-induced salience leads to longer memory span Back

How does extrapolation affect asset prices?
Cross-sectional predictions of an extrapolation model Finite-horizon: T+1 periods, 𝑡=0, 1, …, 𝑇 Assets: N+1 assets; one risk-free asset ( 𝑅 𝑓 =0) and N risky assets Risky asset 𝒊 : a claim to a single dividend payment at 𝑇 𝐷 𝑖,0 : public information at time 0 𝜀 𝑖,𝑡 : public news at time 𝑡, including both market and firm-specific news

Cross-sectional predictions of an extrapolation model Two types of traders Fundamental traders: 𝜇 𝐹 of the economy Extrapolators: 𝜇 𝐸 =1− 𝜇 𝐹 of the economy Share demand Fundamental traders: 𝑁 𝑡 𝐹 = 1 𝛾 Σ 𝐹 −1 ( 𝐷 𝑡 −𝛾 𝑇−𝑡−1 Σ 𝐹 𝑄− 𝑃 𝑡 ) Extrapolators: 𝑁 𝑡 𝐸 = 1 𝛾 Σ 𝐹 −1 ( 𝜆 0 + 𝜆 1 ′ 𝑆 𝑡 ) Market clearing conditions determine the equilibrium prices Key behavioral assumption

Cross-sectional predictions of an extrapolation model Running the following predictability regression The slope coefficient is Prediction: (1) 𝑏 𝑖 <0 (2) more extrapolators in the economy (higher 𝜇 𝐸 ) a higher degree of extrapolation bias (higher 𝜆 𝑖,1 or lower 𝜆 𝑖,2 ) → a stronger degree of return predictability

Return predictability Fama-MacBeth regression
Consistent with the model prediction Forcerank represents the thinking process of behavioral investors in the market Note: Forcerank score: the consensus submitted in week t based on beliefs about week t+1 return. Predicted score: the fitted value from the nonlinear specification. Residual score: the residual value from the nonlinear specification.

Return predictability
Trading strategy Note: Forcerank score: the consensus submitted in week t based on beliefs about week t+1 return. Predicted score: the fitted value from the nonlinear specification. Residual score: the residual value from the nonlinear specification.

Heterogeneity analysis: more vs. less extrapolators We use the level of institutional ownership (IO) as a proxy for 𝜇 𝐹 More extrapolators: lower than median IO Less extrapolators: higher than median IO Consistent with the model prediction The return predictability are only present when more extrapolators trade on the stocks Note: Forcerank score: the consensus submitted in week t based on beliefs about week t+1 return. Predicted score: the fitted value from the nonlinear specification.

Heterogeneity analysis: extrapolation bias across games 𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖, 𝑡 = 𝜆 0 + 𝜆 1 𝑠=0 𝑛 𝑤 𝑠 𝑅 𝑖,𝑡−𝑠 + 𝜀 𝑡 𝑤 𝑠 = 𝜆 2 𝑠 𝑗=0 𝑛 𝜆 2 𝑗 , 0≤ 𝜆 2 <1 We allow the extrapolation parameters to vary across different game g ‘s: We can also estimate these parameters from realized returns over week t+1 (rational expectations): Prediction: 𝐹𝑜𝑟𝑐𝑒𝑟𝑎𝑛 𝑘 𝑖,𝑔, 𝑡 = 𝜆 𝑔, 0 + 𝜆 𝑔, 1 𝑠=0 𝑛 𝑤 𝑔, 𝑠 𝑅 𝑖, 𝑔, 𝑡−𝑠 + 𝜀 𝑖, 𝑔, 𝑡 𝑤 𝑔,𝑠 = 𝜆 𝑔,2 𝑠 𝑗=0 𝑛 𝜆 𝑔, 𝑗 , 0≤ 𝜆 𝑔, 2 <1 𝑅𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑟𝑎𝑛𝑘𝑖𝑛𝑔 𝑖, 𝑔,𝑡 = 𝜆 𝑔,0 𝑟 + 𝜆 𝑔, 1 𝑟 𝑠=0 𝑛 𝑤 𝑔,𝑠 𝑟 𝑅 𝑖, 𝑔,𝑡−𝑠 + 𝜀 𝑖, 𝑔,𝑡 𝑟 𝑤 𝑔,𝑠 𝑟 = 𝜆 𝑔, 2 𝑟 𝑠 𝑗=0 𝑛 𝜆 𝑔,2 𝑟 𝑗 , 0≤ 𝜆 𝑔,2 𝑟 <1

Heterogeneity analysis: extrapolation bias across games Determinants of extrapolation bias

Out-of-sample test Additional evidence that Forcerank represents the thinking process of behavioral investors in the market Not a simple manifestation of liquidity shocks that explains return reversal Extrapolative beliefs also play a role in explaining short-term price reversal

Return predictability Fama-MacBeth regression: residuals
The Forcerank score may reveal additional investor “sentiment” not associated with past returns. Note: Forcerank score: the consensus submitted in week t based on beliefs about week t+1 return. Predicted score: the fitted value from the nonlinear specification. Residual score: the residual value from the nonlinear specification.

Return predictability Trading strategy: residuals
Note: Forcerank score: the consensus submitted in week t based on beliefs about week t+1 return. Predicted score: the fitted value from the nonlinear specification. Residual score: the residual value from the nonlinear specification.

Conclusions Investors extrapolate from recent past returns when forming expectations about future returns Stronger extrapolation bias among non-professionals. The extrapolative component of Forcerank score negatively predict future returns. Consistent with extrapolation model. Not explained by liquidity-based short-term reversal. The predictive power is stronger for stocks with more extrapolators and a higher degree of extrapolation bias. Forcerank score may capture additional “sentiment” not associated with return extrapolation. Exciting opportunity for future research

Zhi Da, University of Notre Dame

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