Signal or noise? Uncertainty and learning whether other traders are informed

Transcription

1 Signal or noise? Uncertainty and learning whether other traders are informed Snehal Banerjee (Northwestern) Brett Green (UC-Berkeley) AFA 2014 Meetings July 2013

2 Learning about other traders Trade motives arise for a variety of reasons: Hedging and risk sharing Investment opportunities Liquidity needs (or time preferences) Different beliefs or information Asset pricing models typically assume participants know why others trade. Characteristics of others are common knowledge. Even though their information about fundamentals is not. Unreasonable degree of sophistication? Especially among uninformed or naive traders. This paper: explore asset pricing implications when investors are uncertain and subsequently learn about other traders in the market.

3 What we do in this paper More specifically, we consider a setting in which Rational traders are uncertain whether others are trading on an informative signal or noise/sentiment. Over time, they learn by observing prices and dividends. Generates a rich set of return dynamics: Stochastic, persistent expected returns and volatility, with i.i.d. shocks Price reacts asymmetrically to good news vs. bad news Volatility clustering Relation between information quality and returns varies over time Underlying Mechanism: Multi-dimensional uncertainty leads to non-linearity in prices Learning generates persistence

4 Related Literature Uncertainty about others Participation / proportion of informed: Cao, Coval and Hirshleifer (2002), Romer (1993), Lee (1998), Avery and Zemsky (1998), Gao, Song and Wang (2012) Risk Aversion: Easley, O Hara, and Yang (2013) Existence/Precision of informed signal: Gervais (1997), Li (2012) Non-linear prices (outside CARA/Normal) Ausubel (1990), Foster and Viswanathan (1993), Rochet and Vila (1994), DeMarzo and Skiadas (1998), Barlevy and Veronesi (2000), Spiegel and Subrahmanyam (2000), Breon-Drish (2010), Albagli, Hellwig, and Tsyvinski (2011) Learning about fundamentals Regime switching models: David (1997), Veronesi (1999), David and Veronesi (2008, 2009). Lack of common prior Heterogeneous Beliefs: Harrison and Kreps (1978) etc. With learning: Banerjee et al. (2009)

5 Setup: Payoffs and Preferences Infinite horizon, two securities. Risk-free asset with return normalized to R 1 + r Risky asset in fixed supply (Z) pays dividends where d t+1 N ( 0, σ 2), ρ < 1. D t+1 = ρd t + (1 ρ)µ + d t+1 Competitive OLG investors, live for two periods, mean-variance preferences over final period wealth. Each investor i submits limit order x i,t, taking prices as given. x i,t = E i,t [P t+1 + d t+1 ] RP t αvar i,t [P t+1 + d t+1 ] Market clearing price, P t, must be such that x i,t = Z i

6 Model Setup: Types of Investors Three types of investors: 1) Informed (I ) (e.g., institutions). Receive informative signals S I,t = d t + ɛ t 2) Noise/Sentiment (N) (e.g., retail). Subject to sentiment shocks, S N,t = u t + ɛ t, which they incorrectly believe are informative about dividends 3) Uninformed (U) (e.g., arbs). No private signal. Infer information from prices.

7 Evolution of Investor Composition Key Point of Departure: At any t, either I or N traders are present. U is uncertain about this. Let θ t {I, N} denote which type of other trader is present at t. We consider two different specifications for θ t : 1) i.i.d. with π 0 = Pr(θ t = I ) for all t. 2) Follows a Markov switching process with q = Pr(θ t = i θ t 1 = i) > 1 2 With serial correlation in θ, learning generates novel pricing dynamics.

8 Characterizing Equilibria Ala Kreps (1977), we assume investors can observe residual supply. No noise traders, fixed supply = price can reveal signal to U. Definition An equilibrium is signal revealing if U investors can infer the signal, S θ,t, from the price and residual supply. Important: Signal-revealing equilibria Fully-revealing equilibria U investors are still uncertain about fundamentals because they do not know whether θ t is informed.

9 Learning about dividends Investor θ s beliefs about d t+1 from standard Kalman filtering: where λ = E θ,t [d t+1 ] = λs θ,t and var θ,t [d t+1 ] = σ 2 (1 λ) σ2 σ 2 +σ 2 ε measures information quality Conditional on π t = Pr(θ = I I U,t ), investor U s beliefs are: E U,t [d t+1 ] = π t λs θ,t var U,t [d t+1 ] = π t σ 2 (1 λ) + (1 π t )σ 2 + }{{} π t (1 π t )(λs θ,t ) 2 }{{} expectation of cond. variance variance of cond. expectation Note: When U is uncertain about θ t, variance is with S 2 θ,t.

10 Static Benchmark: Uncertainty, no learning about others Consider the static benchmark: Market open at t = 0, all uncertainty realized at t = 1. Set r = 0. There is uncertainty about θ, but no subsequent learning. Traders submit limit orders x i = E i[d 1 ] P αvar i [d 1 ] For θ traders, moment calculations are straightforward x θ = λs θ P ασ 2 (1 λ) Demand is monotone in S θ = equilibrium must be signal revealing. Note: By construction, x θ is identically distributed across θ {N, I }. = U does not learn about θ from prices.

11 Static Benchmark Given that the signal is revealed to U investors: x U = E U[d 1 ] P αvar U [d 1 ] = π 0 (1 λ)σ 2 + (1 π 0 )σ 2 }{{} expectation of conditional variance π 0 λs θ P + π 0 (1 π 0 )(λs θ ) 2 }{{} variance of conditional expectation First Effect: Uncertainty about whether θ is informed = U s conditional variance depends on S θ. = Equilibrium price is non-linear in the signal. = Asymmetric price reaction to news.

12 Asymmetric price reaction Prediction 1: The price reacts more strongly to bad news than it does to good news. Intuition: Good news (positive signals) increase expectations about fundamentals, but also increases U s uncertainty about how to interpret the signal (recall the total variance formula). = Offsetting effects. Bad news (negative signals) reduce expectations about fundamentals and increase U s uncertainty about how to interpret the signal. = Reinforcing effects. Empirical evidence: Campbell and Hentschel (1992), Skinner (1994), Skinner and Sloan (2002)

13 Price Decomposition In the static benchmark, the equilibrium price is given by: P 0 = κλs θ + (1 κ)π 0 λs θ }{{} weighted expectation of d 1 κσ 2 (1 λ)αz }{{} risk premium where κ measures the relative precision of θ s beliefs. κ = var U [d 1 S θ ] var U [d 1 S θ ] + var θ [d 1 S θ ] From earlier calculations, κ depends on π 0. = Return moments vary with U s beliefs. e.g., risk premium hump shaped in π 0. = Time variation in moments in dynamic model with persistance.

14 Dynamic Setting: Learning about θ Returning to the dynamic setting: As before, U will not learn about θ by observing only prices. But U does learn about θ following the dividend realization. If dividend is line with signal = π t increases If dividend is far away from signal = π t decreases. When the composition of other traders exhibits serial correlation, the evolution of beliefs creates persistence and a source of risk. Second Effect (learning about θ): Learning leads to time variation in π t and therefore in risk premia. Learning also generates an additional feedback effect. Evolution of future beliefs = additional volatility, lower prices, higher risk premium.

15 Learning about θ λ=0.2 λ=0.5 λ= π t Surprise in dividend (in std dev) Figure : Updated beliefs after observing P t, d t+1 starting from π t = 0.75, q = 1.

16 Time-varying, persistent return moments Prediction 2: Uncertainty about other traders leads to time-varying, serially correlated risk premia and volatility. Intuition Prices and therefore returns moment are driven by uninformed traders beliefs, π t. Clearly persistent = serial correlation. Beliefs evolve (stochastically) over time = time variation. Uncertainty about future beliefs feeds back into prices and generates additional risk premia and volatility.

17 Volatility Clustering Prediction 3: If π t is sufficiently large, a return surprise, predicts higher volatility and higher expected returns in future periods. Intuition Starting from π t 1, dividend surprises (in either direction) make it less likely that the signal was informative. π t+1 decreases = U s uncertainty about θ increase. Higher uncertainty = higher risk premia and volatility going forward.

18 Information Quality and Returns Prediction 4: The relation between information quality (λ) changes over time and depends on the level disagreement across investors. When investors agree on the interpretation of S θ,t (i.e., π t 1): Higher λ Less posterior uncertainty about fundamentals Lower expected returns and volatility When investors disagree on the interpretation of S θ,t (i.e., π t 0): Higher λ x θ,t more sensitive to signals More noise in prices for U investors Higher expected returns and volatility Result may help reconcile the mixed empirical evidence

19 Comparison to Standard Models The model bridges the gap between two standard settings. In each, U knows whether θ is informed: π 0 = 0: Differences of opinions (DO) benchmark π 0 = 1: Rational expectations (RE) benchmark In both benchmarks, unique equilibrium is signal revealing: Demand is linear in the (inferred) signal and price: x i,t = α i S i,t β i P t The equilibrium price is a linear in the signal: P t = a + bs i,t = Unconditional return moments are constant over time. = Conditional (on the signal) returns are i.i.d, volatility is constant. Main Point: Results derive from uncertainty and learning about θ.

20 What else do we do? Price decomposition and comparative statics in dynamic setting Parameterized example Economically significant magnitudes. Alternative Specifications: Robustness 1) Noisy REE setup

21 Summary Develop a framework to study uncertainty and learning about whether others are informed This uncertainty has rich implications for return dynamics Non-linear price that reacts asymmetrically to good news vs. bad news Stochastic, persistent return moments, even with i.i.d. shocks Volatility clustering Time-varying relation between information quality and return moments