Higher Order Expectations, Illiquidity, and Short-term Trading

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1 Higher Order Expectations, Illiquidity, and Short-term Trading Giovanni Cespa 1 and Xavier Vives 2 1 Cass Business School, and CEPR 2 IESE Business School

2 Introduction Liquidity and asset pricing: role of private information. Persistent liquidity trading with heterogeneous information leads to multiple equilibria. HOEs on liquidity trading: joint theory of illiquidity and reliance on public and private information. Implications for asset pricing.

3 Liquidity trading persistence Persistence in liquidity: Well documented empirically (Hendershott and Seasholes (2009), Chordia and Subrahmanyam (2004)). Makes the aggregate demand for the asset depend on two fundamentals : a short- and a long-lived one. With limited risk-bearing capacity, it can create temporary price pressure, implying predictability (Hendershott and Seasholes (2009)). We uncover a novel implication of persistence.

4 Preview of results With short term, heterogeneously informed investors and persistent liquidity trading multiple equilibria with different self-fulfilling levels of liquidity: With limited risk-bearing capacity, prices respond to liquidity trading independently from the arrival of news. Investors with short term horizon potentially face higher liquidation risk. However, the more aggressively investors respond to private information, the more prices reflect fundamentals, allowing to better disentangle the liquidity component of the aggregate demand, which facilitates forecasting the next period aggregate demand and the liquidation price, lowering the risk investors bear....

5 Preview of results (ctd.) β = 0 High Illiq. Eq. β (0, 1] Low Illiq. Eq. Reliance on public information High High Low Illiquidity High High Low Expected returns High High Low Expected volume of informational trading Low Low High Price informativeness Low Low High Return correlation at long horizons Return correlation at short horizons ± +

6 Related literature Higher Order Expectations (HOEs) and asset prices: Allen, Morris, and Shin (2006), Bacchetta and van Wincoop (2008), Kondor (2009), Nimark (2007), Biais and Bossaerts (1998), Cao and Ou-Yang (2005), Banerjee, Kaniel, and Kremer (2009), Ottaviani and Sørensen (2009). Short term trading: Brown and Jennings (1989), Froot, Scharfstein, and Stein (1992), Dow and Gorton (1994), Vives (1995), Cespa (2002), Albagli (2011). Ability of non-informational order imbalances to predict returns: Hendershott and Seasholes (2009), Coval and Stafford (2007). Anomalies in financial markets: Daniel, Hirshleifer, and Subrahmanyam (1998), Vayanos and Woolley (2010).

7 Plan of the talk Model and notation. The 2-period market with heterogeneous information. Implications for reliance on public information. Implications for asset pricing.

8 Assets Two assets are traded during 2 periods (n = 1, 2) by CARA informed investors (rational investors) and liquidity traders: Riskless asset with zero (net) return. Risky asset with final liquidation value v N ( v, τ 1 v ).

9 Rational investors Continuum of investors in [0, 1] have short term horizons: at n they maximize E [ exp { 1 ((p n+1 p n)x in) } s in, p n]. Short horizons can be justified on grounds of incentive reasons related to performance evaluation. At n, i [0, 1] receives a signal s in = v + ɛ in, ɛ in N (0, τ 1 ɛ ), with ɛ in ɛ jn, ɛ in ɛ in+1, and also orthogonal to all the other random variables in the model.

10 Rational investors (ctd.) Each investor i observes the sequence of prices up to period n: p n {p t} n t=0, n = 1, 2. Submits a demand schedule X n(s in, p n 1, p n) = a ns in ϕ n (p n ), denoting the desired position in the risky asset. Convention: 1 s indi = v, a. s. 0

11 Liquidity traders Liquidity traders demand: Persistence in liquidity trading: Chordia and Subrahmanyam (2004), Hendershott and Seasholes (2009). Liquidity trading follows an AR(1) process: θ n = βθ n 1 + u n, with β [0, 1], and u n N (0, τ 1 u ) orthogonal to all other random variables. Note: 1 β = 0 θ n follows a random walk u n = θ n θ n 1 Kyle (1985), Vives (1995). Noise trading is i.i.d. periods. across AMS (2006).

12 Liquidity traders (ctd.) Suppose u n N (0, 1) and β =

13 Liquidity traders (ctd.) Suppose u n N (0, 1) and β = 1/

14 Liquidity traders (ctd.) Suppose u n N (0, 1) and β =

15 Timeline Let x n 1 0 x indi 1 Liquidity traders submit θ 1 = u 1. 1st period informed investors submit x 1. Market clears: x 1 + θ 1 = 0. 2 Liquidity traders submit θ 2 = u 2 + βθ 1. 2nd period informed investors submit x 2. 1st period investors revert. Market clears: x 2 + θ 2 = 0. 3 The asset is liquidated.

16 Notation Investor i s expectation about the liquidation value at time n is E in[v] E[v s in, p n ]. Investor i s forecast precision at time n is τ in 1 Var[v s in, p n ]. The expectation based on public information only is Public precision is E n[v] E[v p n ]. τ n 1 Var[v p n ]. The average expectation (consensus opinion) is Ē n[v] 1 0 E in[v]di = α En v + (1 α En )E n[v], α En : optimal statistical weight to private info.

17 A static market Suppose we removed the second period Liquidity traders submit θ 1 = u 1. 1st period informed investors submit x 1. Market clears: x 1 + θ 1 = 0. Liquidity traders submit θ 2 = u 2 + βθ 1. 2nd period informed investors submit x 2. 1st period investors revert. Market clears: x 2 + θ 2 = 0. The asset is liquidated.

18 A static market Suppose we removed the second period 1. Static strategies, buy and hold : X 1(s i1, p 1) = 2. Static price akin to consensus : p 1 = Vari1[v] Ē1[v] + θ 1 = E 1[v] + Vari1[v] } {{ } Λ 1 Ei1[v] p1. Var i1[v] E 1[θ 1], where λ 1 = α E 1 a 1 }{{} Λ 1 +(1 α E1 ) a1τ u τ 1, but also, E 1[θ 1] moves p 1 away from the efficient price (Hendershott and Seasholes (2004)). 3. The two components of the price comove positively: Cov[E 1[v], E 1[θ 1]] > 0 Var[p 1] > Var[E 1[v]] + Λ 2 1Var[E 1[θ 1]].

19 A static market Suppose we removed the second period 1. Static strategies, buy and hold : X 1(s i1, p 1) = 2. Static price akin to consensus : p 1 = α E1 ( v + θ1 a 1 ) + (1 α E1 )E 1[v] = E 1[v] + Vari1[v] } {{ } Λ 1 Ei1[v] p1. Var i1[v] E 1[θ 1], where λ 1 = α E 1 a 1 }{{} Λ 1 +(1 α E1 ) a1τ u τ 1, but also, E 1[θ 1] moves p 1 away from the efficient price (Hendershott and Seasholes (2004)). 3. The two components of the price comove positively: Cov[E 1[v], E 1[θ 1]] > 0 Var[p 1] > Var[E 1[v]] + Λ 2 1Var[E 1[θ 1]].

20 A 2-period market Equilibrium Proposition At a linear equilibrium of the market the price is given by ( ) p n = α Pn v + θn + (1 α Pn )E n[v] a n = E n[v] + Λ ne n[θ n] where, while Λ 2 = 1/(τ i2). Λ 1 = Vari1[p2] + βλ 2, Short-term investment horizons imply: A change in the inventory component of illiquidity Λ 1. A change in the weight assigned by the price to noisy private information α P1.

21 The change in Λ 1 The impact of liquidity trading persistence In the static market: 1 Liquidity traders submit θ1 = u1. 1st period informed investors submit x1. Market clears: x1 + θ1 = 0. 2 Liquidity traders submit θ2 = u2 + βθ1. 2nd period informed investors submit x2. 1st period investors revert. Market clears: x2 + θ2 = 0. 3 The asset is liquidated. Then: (a) Investors hold the asset until date 3. (b) The price reflects the (risk-tolerance weighted) uncertainty about v: p 1 = E 1[v] + Vari1[v] E 1[θ 1]. } {{ } Λ 1

22 The change in Λ 1 (ctd.) The impact of liquidity trading persistence In the dynamic market, due to short horizons there is re-trading at date 2: X 1(s i1, p 1) = Ei1[p2] p1. Var i1[p 2] (a) First period investors face uncertainty over the liquidation price which bites more, the higher is their risk aversion: Var i1[p 2]. (b) Because of persistence in liquidity, θ 1 also affects the second period investors (βθ 1), and this adds to first period investors uncertainty over the liquidation price: βλ 2. Overall, the first period price reaction to the (expected) liquidity shock is Λ 1 = Vari1[p2] + βλ 2.

23 The change in α P1 Prices and consensus An alternative expression for the price p 1 = Ē1[v] + α P 1 α E1 a 1 E 1[θ 1] + α E 1 a 1 θ 1. Short-term trading yields a systematic departure of the price from consensus (the static price). What does this imply? Corollary At a linear equilibrium Cov[p 1, v] > Cov[Ē1[v], v] α P 1 > α E1. With short term speculation the price is either systematically drawn closer or farther away from fundamentals compared to consensus.

24 The change in α P1 (ctd.) Strategies Return predictability: E 1[p 2 p 1] = Vari1[p2] E 1[θ 1] E 1[θ 1] = E1[p2 p1], Var i1[p 2] so that for any estimate of θ 1, the market anticipates reversion. However, Corollary At a linear equilibrium, a rational investor s strategy in the first period is given by X 1(s i1, p 1) = a1 α E1 (E i1[v] p 1) + α P 1 α E1 α E1 E 1[θ 1]. Short term strategies depend on whether { αp1 > α E1 Trend chasing α P1 < α E1 Contrarian

25 Trend chasers and contrarians Suppose E 1[θ 1] > 0: { Fundamentals information: v > E1[v] E 1[θ 1] = a 1(v E 1[v]) + θ 1 Liquidity trading: θ 1 > 0. If α P1 > α E1, (a) The price reflects better the fundamentals than the static price. (b) But the departure from the static price is due to short-term trading. (c) Hence, short term trading is tying the price closer to fundamentals (compared to consensus) and E 1[θ 1] > 0 is possibly signalling fundamentals information! Short term investors chase the trend.

26 Multiple equilibria Proposition Linear equilibria always exist. If 1. β (0, 1] : (a) There are two equilibria in which a 2 = τ ɛ, and { 0 < a 1 < τ ɛ a 1 α P1 < α E1 λ 2 > 0 τ ɛ < a1 α P1 > α E1 λ 2 < 0 (b) Furthermore, λ 2(a 1 ) < λ 2(a 1), and prices are more informative along the low illiquidity equilibrium. 2. β = 0 : only the high illiquidity equilibrium survives: a 2 = τ ɛ, a 1 < τ ɛ, and investors are contrarians: α P1 < α E1.

27 The role of persistence Intuition An investor at 1 uses his private signal to anticipate p 2: Due to persistence: 1. Can anticipate part of the 2nd period liquidity demand: E i1[p 2] = Λ 2a 2E i1[v] + (1 Λ 2a 2)E i1[e 2[v]] + Λ 2βE i1[θ 1]. 2. Potentially face an increase in the liquidation price risk: Var i1[p 2] = Var i1[e 2[v]]+Λ 2 2Var i1[e 2[θ 2]]+2Λ 2Cov i1[e 2[v], E 2[θ 2]]. However, this risk is endogenous and as a 1 increases 1. E 1[v] and (for a 1 large enough) E 2[v] are closer to the fundamentals. 2. Investors can better separate the impact of v from that of θ 1: Var[E i1[θ 1]] a 1 < For β > 0 this improves the forecast of E 2[θ 2], reducing the uncertainty over p 2 (Cov i1[e 2[v], E 2[θ 2]] becomes negative). These two effects generate a double feedback loop yielding multiple equilibria.

28 HOEs and reliance on public information Consider a 3-period model. In period 3: 1 0 Hence, X 3(s i3, p 3 )di+θ 3 = 0, due to normality and CARA In period 2 p 3 = Ē3[v] + Λ3θ3, where Λ3 = Vari3[v]/. Vari2[p3] p 2 = Ē2[p3] + θ 2. Substituting the above obtained expression for p 3: [ p 2 = Ē2 Ē 3[v] + Vari3[v] ] θ 3 + Vari2[p3] θ 2 = Ē2 [Ē3[v] ] + Vari3[v] βē2 [θ2] + Vari2[p3] θ 2. Ē3[v] p3 +θ 3 = 0. Var i3[v]

29 HOEs and reliance on public information Consider a 3-period model. In period 3: 1 0 Hence, X 3(s i3, p 3 )di+θ 3 = 0, due to normality and CARA In period 2 p 3 = Ē3[v] + Λ3θ3, where Λ3 = Vari3[v]/. Vari2[p3] p 2 = Ē2[p3] + θ 2. Substituting the above obtained expression for p 3: [ p 2 = Ē2 Ē 3[v] + Vari3[v] ] θ 3 + Vari2[p3] θ 2 = Ē2 [Ē3[v] ] + Vari3[v] βē2 [θ2] + Vari2[p3] θ 2. Ē3[v] p3 +θ 3 = 0. Var i3[v]

30 HOEs and reliance on public information (ctd.) In period 1: p 1 = Ē1 [Ē2 [Ē3 [v] ]] + Vari3[v] βē1 [Ē2[θ 2] ] + Vari2[p3] βē1[θ1]+ Vari1[p2] θ 1, The Beauty contest: when β > 0 the equilibrium price reflects HOEs 1. Liquidation value (as in Allen, Morris, and Shin (2006)). 2. Liquidity trades in periods 1, and 2.

31 HOEs and reliance on public information (ctd.) In period 1: p 1 = Ē1 [Ē2 [Ē3 [v] ]] + Vari3[v] βē1 [Ē2[θ 2] ] + Vari2[p3] βē1[θ1]+ Vari1[p2] θ 1, The Beauty contest: when β > 0 the equilibrium price reflects HOEs 1. Liquidation value (as in Allen, Morris, and Shin (2006)). 2. Liquidity trades in periods 1, and 2.

32 HOEs and reliance on public information (ctd.) When β = 0, only HOEs about fundamentals matter and: Cov[p 1, v] < Cov[Ē1[v], v], and p 1 relies heavily on public information (compared to consensus), as in Allen et al. (2006). However, Corollary When β > 0, and { a 1 Cov[p 1, v] < Cov[Ē1[v], v] a 1 = a1 Cov[p 1, v] > Cov[Ē1[v], v]. When β > 0, HOEs about liquidity trading affect prices, and short term speculation can make prices better predictors of fundamentals (compared to consensus).

33 Asset pricing implications Momentum and reversal Consistent with empirical evidence on return regularities: Corollary When N = 2: 1. For all β [0, 1], Cov[p 2 p 1, p 1 v] < For β (0, 1], Cov[v p 2, p 1 v] < 0. For β = 0, Cov[v p 2, p 1 v] = For β (0, 1], along the equilibrium with low illiquidity Cov[v p 2, p 2 p 1] > 0. Along the equilibrium with high illiquidity, for τ v < ˆτ v, there exists a value ˆβ such that for all β > ˆβ, Cov[v p 2, p 2 p 1] > 0. If β = 0, Cov[v p 2, p 2 p 1] < 0.

34 Asset pricing implications Momentum and reversal: mean price path Differently from behavioral finance literature: E[p n v] n 2 3

35 Asset pricing implications Volume and return predictability Volume of informational trading: 1 1 V 2 E [ X2(s i2, z 2 ) X 1(s i1, z 1) ] di E [ X 2(θ 2) X 1(θ 1) ] di 0 0 Corollary When N = 2, for all β (0, 1] 1. The expected volume of informational trading is higher along the low illiquidity equilibrium. When β = 0 only the equilibrium with a low volume of informational trading survives. 2. Expected returns are higher along the high illiquidity equilibrium.

36 Asset pricing implications Volume and return predictability (ctd) A high volume of informational trading predicts momentum, consistently with Llorente et al. (2002). However, With high persistence (β high very different trading frequencies), momentum is compatible with both trend chasing and contrarian behavior Liquidity-motivated trades also may generate positively autocorrelated returns. With low persistence momentum is only compatible with trend chasing behavior. If investors chase the trend, they are on the equilibrium with momentum at short horizons.

37 Asset pricing implications Illiquidity and expected returns Due to the law of iterated expectations: This implies Corollary When u n N (ū, τ 1 u ), E[p n p n 1] = Λ ne[θ n] Λ n 1E[θ n 1]. E[p 1 v] = Λ 1ū E[p 2 p 1] = (Λ 2(1 + β) Λ 1) ū E[v p 2] = Λ 2(1 + β)ū.

38 Asset pricing implications Illiquidity and expected returns (ctd) Denoting by Λ H n and Λ L n, the inventory component of market illiquidity associated with the high and low illiquidity equilibrium at time n: Corollary Λ L n < Λ H n, for n = 1, 2. As investors anticipate worse market conditions along the high illiquidity equilibrium, The expected compensation required to absorb the demand shock from liquidity traders in the first and second period is higher, implying a higher risk premium for first and third period returns. Numerical simulations show that a similar result also holds in the second period.

39 Asset pricing implications Illiquidity and expected returns (ctd) Λ1ū β (a) (Λ2(1 + β) Λ1)ū β (b) Λ2(1 + β)ū β (c)

40 Asset pricing implications Illiquidity and expected returns (ctd) Further implication of the figure: As β increases from 0 to 1, the risk premia at all trading dates increase, along each equilibrium. As β becomes larger, a higher fraction of liquidity traders positions at any date remains in the market, augmenting the inventory risk born by rational investors. As a consequence, the expected premium required to absorb liquidity trades increases.

41 Summary β = 0 High Illiq. Eq. β (0, 1] Low Illiq. Eq. Reliance on public information α P1 < α E1 α P1 < α E1 α P1 > α E1 Illiquidity High High Low Expected returns High High Low Expected volume of informational trading Low Low High Price informativeness Low Low High Return correlation at long horizons Return correlation at short horizons ± +

42 Conclusions When a market is populated by short term investors, With persistence in liquidity trading and heterogeneous information, short term horizons yield multiple equilibria which can be ranked in terms of illiquidity. Implications for reliance on public and private information, and price informativeness. Implications for asset pricing: momentum, volume and return predictability, and expected returns.

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