Information Percolation, Momentum, and Reversal

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1 Reversal Daniel Andrei Julien Cujean a b c Banque de France, Mars 2014 Reversal 0 / 14

2 Time-Series Momentum and Reversal: Evidence T-Statistic T-Statistic by Month, All Asset Classes Month Lag Source: Moskowitz, Ooi, and Pedersen (2012) Reversal 1 / 14

3 Behavioral Theories of Momentum and Reversal Barberis, Shleifer, and Vishny (1998) Daniel, Hirshleifer, and Subrahmanyam (1998) Hong and Stein (1999) Momentum Conservatism (underreaction) Biased self-attribution (continuing overreaction) Newswatchers (underreaction) Reversal Representativeness heuristic (overreaction) Overconfidence (overreaction) Momentum Traders aaaa (overreaction) Challenges (Moskowitz, Ooi, and Pedersen, 2012): Markets vary widely in terms of type of investors, yet the pattern of returns is the same No apparent link between measures of investor sentiment and time series momentum Reversal 2 / 14

4 Behavioral Theories of Momentum and Reversal Barberis, Shleifer, and Vishny (1998) Daniel, Hirshleifer, and Subrahmanyam (1998) Hong and Stein (1999) Momentum Conservatism (underreaction) Biased self-attribution (continuing overreaction) Newswatchers (underreaction) Reversal Representativeness heuristic (overreaction) Overconfidence (overreaction) Momentum Traders aaaa (overreaction) Challenges (Moskowitz, Ooi, and Pedersen, 2012): Markets vary widely in terms of type of investors, yet the pattern of returns is the same No apparent link between measures of investor sentiment and time series momentum Reversal 2 / 14

5 A Rational Theory of Momentum and Reversal An equilibrium model with multiple agents Momentum traders Contrarians Partial information aggregation (underreaction) The information diffuses through word-of-mouth communication (Duffie and Manso, 2007) Prices play an informational role (Grossman and Stiglitz, 1980) Momentum trading is profitable and yet momentum persists Rumor spreads can generate reversals (overreaction) Reversal 3 / 14

6 A Rational Theory of Momentum and Reversal An equilibrium model with multiple agents Momentum traders Contrarians Partial information aggregation (underreaction) The information diffuses through word-of-mouth communication (Duffie and Manso, 2007) Prices play an informational role (Grossman and Stiglitz, 1980) Momentum trading is profitable and yet momentum persists Rumor spreads can generate reversals (overreaction) Reversal 3 / 14

7 A Rational Theory of Momentum and Reversal An equilibrium model with multiple agents Momentum traders Contrarians Partial information aggregation (underreaction) The information diffuses through word-of-mouth communication (Duffie and Manso, 2007) Prices play an informational role (Grossman and Stiglitz, 1980) Momentum trading is profitable and yet momentum persists Rumor spreads can generate reversals (overreaction) Reversal 3 / 14

8 A Rational Theory of Momentum and Reversal An equilibrium model with multiple agents Momentum traders Contrarians Partial information aggregation (underreaction) The information diffuses through word-of-mouth communication (Duffie and Manso, 2007) Prices play an informational role (Grossman and Stiglitz, 1980) Momentum trading is profitable and yet momentum persists Rumor spreads can generate reversals (overreaction) Reversal 3 / 14

9 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

10 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

11 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

12 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

13 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

14 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

15 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

16 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

17 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Each a private signal Z 0 i = Ũ + ɛ i 0 ɛ i 0 N ( ) 0, S 1 Z i 1 = Ũ + ɛ i 1 Z i 2 = Ũ + ɛ i 2 Z i 3 = Ũ + ɛ i 3 Stock P pays off Ũ N ( ) 0, H 1 Noisy supply X t N (0, 1 Φ ) Reversal 4 / 14

18 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Reversal 5 / 14

19 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Agents meet and share their initial signal with intensity λ: µ t (n) = probability density function over the number of signals Reversal 5 / 14

20 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Agents meet and share their initial signal with intensity λ: µ t (n) = probability density function over the number of signals density µt(n) 100% 80% 60% 40% 20% 0% number of signals n Reversal 5 / 14

21 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Agents meet and share their initial signal with intensity λ: µ t (n) = probability density function over the number of signals density µt(n) 100% 80% 60% 40% 20% 0% number of signals n Reversal 5 / 14

22 Information Percolation in Centralized Markets There is a continuum i [0,1] of investors with CARA= 1 γ utility Agents meet and share their initial signal with intensity λ: µ t (n) = probability density function over the number of signals density µt(n) 100% 80% 60% 40% 20% 0% number of signals n Reversal 5 / 14

23 Equilibrium Prices and Optimal Trading Strategies There is a continuum i [0,1] of investors with CARA= 1 γ utility P 0 P 1 P 2 P 3 4 Reversal 6 / 14

24 Equilibrium Prices and Optimal Trading Strategies There is a continuum i [0,1] of investors with CARA= 1 γ utility P 0 P 1 P 2 P 3 4 P t = Kt H K Ũ t 1+γ 2 SΩ j Φ X t j=0 γk t j Reversal 6 / 14

25 Equilibrium Prices and Optimal Trading Strategies There is a continuum i [0,1] of investors with CARA= 1 γ utility P 0 P 1 P 2 P 3 4 D t i D [( t 1 i = γ SΩ t + at i P t = Kt H K Ũ t 1+γ 2 SΩ j Φ X t j=0 γk t j marginal info advantage )( Z i t P t 1 ) ( cummulative info advantage K t 1+γ 2 SΦΩ t + A i t )( P t P t 1 )] Reversal 6 / 14

26 Equilibrium Prices and Optimal Trading Strategies There is a continuum i [0,1] of investors with CARA= 1 γ utility P 0 P 1 P 2 P 3 4 D t i D [( t 1 i = γ SΩ t + at i P t = Kt H K Ũ t 1+γ 2 SΩ j Φ X t j=0 γk t j marginal info advantage )( Z i t P t 1 ) ( cummulative info advantage K t 1+γ 2 SΦΩ t + A i t E[ D t i D t 1 i P t P ] t 1 = mt ( P i t P ) t 1 )( P t P t 1 )] Reversal 6 / 14

27 Momentum and Momentum Trading 0 m i Meeting intensity λ Reversal 7 / 14

28 Momentum and Momentum Trading m i Momentum trading Contrarian trading Meeting intensity λ Reversal 7 / 14

29 Momentum and Momentum Trading 0 Momentum trading m i Average Contrarian trading Meeting intensity λ Reversal 7 / 14

30 Momentum and Momentum Trading m i % percentile Average 5% percentile Momentum trading Contrarian trading Meeting intensity λ Reversal 7 / 14

31 Momentum and Momentum Trading m i % percentile Average 5% percentile Momentum trading Contrarian trading Meeting intensity λ Reversal 7 / 14

32 Momentum and Momentum Trading Returns Autocorrelation m i % percentile Average 5% percentile Momentum trading Contrarian trading Meeting intensity λ Reversal 7 / 14

33 Relation Between Size and Momentum Momentum NYSE/AMEX-Based Size Quintiles Source: Hong, Lim, and Stein (2000) Reversal 8 / 14

34 Market Learning and Momentum P t = K t H K t Ũ t j=0 1 + γ 2 SΩ j Φ γk t X j Trend T λ = 0 λ = 1 λ = Time t Reversal 9 / 14

35 Market Learning and Momentum P t = K t H K t Ũ t j=0 1 + γ 2 SΩ j Φ γk t X j Trend T λ = 0 λ = 1 λ = Time t Reversal 9 / 14

36 Momentum Profits Momentum Profits = E [ 3 t=1 ] ( ) 1 1 Pt>0 P Pt<0 t+1 Momentum Profits Meeting Intensity λ Reversal 10 / 14

37 Momentum Profits Momentum Profits = E [ 3 t=1 ] ( ) 1 1 Pt>0 P Pt<0 t+1 Momentum Profits Meeting Intensity λ Reversal 10 / 14

38 Rumors, Price Overshooting, and Reversals Z t i = Ũ + Ṽ + ɛi t ( ) Ṽ N 0, 1 ν Z i 3 = Ũ + ɛi 3 Ũ N ( ) 0, 1 H Ũ + Ṽ = Average Price Ũ = ; ; Time Reversal 11 / 14

39 Rumors, Price Overshooting, and Reversals Z t i = Ũ + Ṽ + ɛi t ( ) Ṽ N 0, 1 ν Z i 3 = Ũ + ɛi 3 Ũ N ( ) 0, 1 H Ũ + Ṽ = Average Price Ũ = Time Reversal 11 / 14

40 Rumors, Price Overshooting, and Reversals Z t i = Ũ + Ṽ + ɛi t ( ) Ṽ N 0, 1 ν Z i 3 = Ũ + ɛi 3 Ũ N ( ) 0, 1 H Ũ + Ṽ = Average Price Ũ = λ = Time Reversal 11 / 14

41 Rumors, Price Overshooting, and Reversals Z t i = Ũ + Ṽ + ɛi t ( ) Ṽ N 0, 1 ν Z i 3 = Ũ + ɛi 3 Ũ N ( ) 0, 1 H Ũ + Ṽ = Average Price Ũ = λ = 0 λ = Time Reversal 11 / 14

42 Rumors, Price Overshooting, and Reversals Z t i = Ũ + Ṽ + ɛi t ( ) Ṽ N 0, 1 ν Z i 3 = Ũ + ɛi 3 Ũ N ( ) 0, 1 H Ũ + Ṽ = Average Price Ũ = Time λ = 0 λ = 1 λ = 2.5 Reversal 11 / 14

43 Rumors, Price Overshooting, and Reversals Corr(P 2 P 1,P 1 P 0 ) Corr(P 3 P 2,P 2 P 1 ) Meeting intensity λ Reversal 12 / 14

44 The Intuition is Robust to a Steady State World D t D t+1 D t+2 D t+3 X t 3 X t 2 X t 1 X t Reversal 13 / 14

45 The Intuition is Robust to a Steady State World D t D t+1 D t+2 D t+3 X t 3 X t 2 X t 1 X t P t = b 3 X t 3 b 2 X t 2 b 1 X t 1 b 0 X t + a 0 D t + a 1 D t+1 + a 2 D t+2 + a 3 D t+3 Reversal 13 / 14

46 The Intuition is Robust to a Steady State World D t D t+1 D t+2 D t+3 X t 3 X t 2 X t 1 X t P t = b 3 X t 3 b 2 X t 2 b 1 X t 1 b 0 X t + a 0 D t + a 1 D t+1 + a 2 D t+2 + a 3 D t+3 Reversal 13 / 14

47 The Intuition is Robust to a Steady State World D t D t+1 D t+2 D t+3 X t 3 X t 2 X t 1 X t P t = b 3 X t 3 b 2 X t 2 b 1 X t 1 b 0 X t + a 0 D t + a 1 D t+1 + a 2 D t+2 + a 3 D t+3 Reversal 13 / 14

48 The Intuition is Robust to a Steady State World D t D t+1 D t+2 D t+3 X t 3 X t 2 X t 1 X t P t = b 3 X t 3 b 2 X t 2 b 1 X t 1 b 0 X t + a 0 D t + a 1 D t+1 + a 2 D t+2 + a 3 D t+3 Cov(P t+2 P t+1,p t+1 P t ) is generally amplified by information percolation For some calibrations, Cov(P t+2 P t+1,p t+1 P t ) can turn from negative (reversals) to positive (momentum) Intuition about momentum/contrarian trading holds in this case as well Reversal 13 / 14

49 Further Questions Multiple asset setting (Moskowitz, Ooi, and Pedersen (2012) show that correlations of time series momentum strategies across asset classes are larger than the correlations of the asset classes themselves) Endogenous meeting intensity (explaining momentum in response to fundamental impulses such as earnings announcements or analysts forecast revisions) Truth telling, influence of an investor on others (talking about winners or losers, thought contagion) Reversal 14 / 14

50 References I Barberis, N., A. Shleifer, and R. Vishny (1998, September). A model of investor sentiment. Journal of Financial Economics 49(3), Daniel, K., D. Hirshleifer, and A. Subrahmanyam (1998, December). Investor psychology and security market under- and overreactions. Journal of Finance 53(6), Duffie, D. and G. Manso (2007, May). Information percolation in large markets. American Economic Review 97(2), Grossman, S. J. and J. E. Stiglitz (1980). On the impossibility of informationally efficient markets. The American Economic Review 70(3), pp Hong, H., T. Lim, and J. C. Stein (2000). Bad news travels slowly: Size, analyst coverage, and the profitability of momentum strategies. The Journal of Finance 55(1), pp Hong, H. and J. C. Stein (1999, December). A unified theory of underreaction, momentum trading, and overreaction in asset markets. Journal of Finance 54(6), Moskowitz, T. J., Y. H. Ooi, and L. H. Pedersen (2012). Time series momentum. Journal of Financial Economics 104(2),

Information Percolation, Momentum, and Reversal

Information Percolation, Momentum, and Reversal Daniel Andrei Julien Cujean January 6, 216 Abstract We propose a rational model to explain time-series momentum. The key ingredient is word-of-mouth communication,