Dynamic Trading and Asset Prices: Keynes vs. Hayek

Transcription

1 Dynamic Trading and Asset Prices: Keynes vs. Hayek Giovanni Cespa 1 and Xavier Vives 2 1 CSEF, Università di Salerno, and CEPR 2 IESE Business School C6, Capri June 27, 2007

2 Introduction Motivation (I) Do asset prices rely excessively on public information?

3 Introduction Motivation (I) Do asset prices rely excessively on public information? Keynes (GT) pioneered the vision of stock markets as beauty contests where investors try to guess not the fundamental value of an asset but the average opinion of other investors, and end up chasing the crowd. This view portrays a stock market dominated by herding, behavioral biases, fads, booms and crashes (see e.g. Shiller (2000)), and goes against the tradition of considering market prices as aggregators of the dispersed information in the economy advocated by Hayek (1945). According to the latter view prices reflect, perhaps noisily, the collective information that each trader has about the fundamental value of the asset (see e.g. Grossman (1989)). We address the tension between the Keynesian and the Hayekian visions in a dynamic finite horizon market where investors, except noise traders, have no behavioral bias and hold a common prior on the fundamentals.

4 Introduction Motivation (II) When does the price rely excessively on public information?

5 Introduction Motivation (II) When does the price rely excessively on public information? Allen, Morris, and Shin (2006): excessive reliance arises if the asset price is on average farther away from the fundamentals than investors consensus expectation (optimal statistical weight). By the same token, whenever the price is instead closer to the fundamental (than investors average opinion), we could say that we have insufficient reliance on public information. AMS (2006) argue that prices always rely excessively on public information in a dynamic model with risk averse short term traders, differential information, and independent noisy supply across periods the Keynesian paradigm. Does Keynes always prevail over Hayek?

6 Introduction Motivation (II) When does the price rely excessively on public information? Allen, Morris, and Shin (2006): excessive reliance arises if the asset price is on average farther away from the fundamentals than investors consensus expectation (optimal statistical weight). By the same token, whenever the price is instead closer to the fundamental (than investors average opinion), we could say that we have insufficient reliance on public information. AMS (2006) argue that prices always rely excessively on public information in a dynamic model with risk averse short term traders, differential information, and independent noisy supply across periods the Keynesian paradigm. Does Keynes always prevail over Hayek? There are reasons to believe that the result may depend on traders speculative horizon and on the properties of the noise trading process.

7 Introduction Discussion However... (1) In a market with differential information the asset price relation to the fundamentals depends on traders speculative behavior: (a) In a static market agents speculate on the difference between price and liquidation value. As they hold private information on the latter, the price dependence from the fundamentals is positively related to the relative quality of such information. (b) In a dynamic market traders also speculate on short-run price differences. This makes the relation between price and fundamentals also dependent on short-term speculative behavior. (2) Furthermore, if noise traders demand (stock) is independent across periods, patterns of liquidity supply become more easily predictable. This leads traders to accommodate more actively order imbalances, dampening the informational impact of speculators positions. We relax the independence assumption for noise trading and analyze a model where traders have either a long or a short-term speculative horizon, and residual uncertainty can affect the liquidation value.

8 Introduction Main Results When the market is populated by long-term traders If no residual uncertainty affects the liquidation value, and noise trading follows a random walk, prices assign the optimal statistical weight to public information. When traders have short horizons,

9 Introduction Main Results When the market is populated by long-term traders If no residual uncertainty affects the liquidation value, and noise trading follows a random walk, prices assign the optimal statistical weight to public information. When additional pay-off uncertainty and noise trade predictability are added, traders also speculate on short-run price differences, and both excessive and insufficient reliance on public information can arise, yielding a Keynesian and a Hayekian equilibrium. When traders have short horizons,

10 Introduction Main Results When the market is populated by long-term traders If no residual uncertainty affects the liquidation value, and noise trading follows a random walk, prices assign the optimal statistical weight to public information. When additional pay-off uncertainty and noise trade predictability are added, traders also speculate on short-run price differences, and both excessive and insufficient reliance on public information can arise, yielding a Keynesian and a Hayekian equilibrium. When traders have short horizons, There typically are two equilibria ranked by traders responsiveness to private information. In the equilibrium with low (high) signal responsiveness there is excessive (insufficient) reliance on public information.

11 Introduction Main Results When the market is populated by long-term traders If no residual uncertainty affects the liquidation value, and noise trading follows a random walk, prices assign the optimal statistical weight to public information. When additional pay-off uncertainty and noise trade predictability are added, traders also speculate on short-run price differences, and both excessive and insufficient reliance on public information can arise, yielding a Keynesian and a Hayekian equilibrium. When traders have short horizons, There typically are two equilibria ranked by traders responsiveness to private information. In the equilibrium with low (high) signal responsiveness there is excessive (insufficient) reliance on public information. Finally, if noise trading follows a random walk and there is sufficiently large residual uncertainty on the liquidation value, in equilibrium there always is excessive reliance on public information.

12 Introduction Intuition (I) A trader observes a positive signal and faces a positive order flow

13 Introduction Intuition (I) A trader observes a positive signal and faces a positive order flow Good news make him increase his long position in the asset. His reaction to the order imbalance is either to accommodate it counting on a future noise trade reversal or to follow the market attributing the imbalance to informed speculators activity and further increase his long position. In the former case, the trader s speculative position is partially offset by his market making position. Thus, the impact of private information on the price is partially sterilized by traders market-making activity. This loosens the price from the fundamentals, yielding excessive reliance on public information.

14 Introduction Intuition (I) A trader observes a positive signal and faces a positive order flow Good news make him increase his long position in the asset. His reaction to the order imbalance is either to accommodate it counting on a future noise trade reversal or to follow the market attributing the imbalance to informed speculators activity and further increase his long position. In the latter case, the trader s speculative position is partially enhanced by his short-run speculative position. Thus, the impact of private information on the price is partially reinforced by traders short-run speculative activity. This tightens the price to the fundamentals, yielding insufficient reliance on public information.

15 Introduction Intuition (I) A trader observes a positive signal and faces a positive order flow Good news make him increase his long position in the asset. His reaction to the order imbalance is either to accommodate it counting on a future noise trade reversal or to follow the market attributing the imbalance to informed speculators activity and further increase his long position. In the latter case, the trader s speculative position is partially enhanced by his short-run speculative position. Thus, the impact of private information on the price is partially reinforced by traders short-run speculative activity. This tightens the price to the fundamentals, yielding insufficient reliance on public information.

16 Introduction Intuition (II) With long-term traders, correlation across noise trade increments helps predicting future noise trade shocks, and tilts traders towards accommodating order imbalances. Residual uncertainty over the liquidation value enhances the hedging properties of future positions, boosting traders signal responsiveness and yielding more aggressive short-term speculation. Depending on the intensity of the correlation across noise trade increments, excessive or insufficient reliance on public information arises. Conversely, absent correlation across noise trade increments and residual uncertainty, the price assigns the optimal statistical weight to the fundamentals. With short-term traders, the lack of future trading opportunities eliminates hedging possibilities. This may give rise to self-fulfilling equilibria. If traders anticipate an order imbalance due to noise traders, they scale down their aggressiveness and accommodate it.

17 Introduction Intuition (II) With long-term traders, correlation across noise trade increments helps predicting future noise trade shocks, and tilts traders towards accommodating order imbalances. Residual uncertainty over the liquidation value enhances the hedging properties of future positions, boosting traders signal responsiveness and yielding more aggressive short-term speculation. Depending on the intensity of the correlation across noise trade increments, excessive or insufficient reliance on public information arises. Conversely, absent correlation across noise trade increments and residual uncertainty, the price assigns the optimal statistical weight to the fundamentals. With short-term traders, the lack of future trading opportunities eliminates hedging possibilities. This may give rise to self-fulfilling equilibria. If traders anticipate an order imbalance due to informed traders, they scale up their aggressiveness and follow the market.

18 Introduction Intuition (II) With long-term traders, correlation across noise trade increments helps predicting future noise trade shocks, and tilts traders towards accommodating order imbalances. Residual uncertainty over the liquidation value enhances the hedging properties of future positions, boosting traders signal responsiveness and yielding more aggressive short-term speculation. Depending on the intensity of the correlation across noise trade increments, excessive or insufficient reliance on public information arises. Conversely, absent correlation across noise trade increments and residual uncertainty, the price assigns the optimal statistical weight to the fundamentals. With short-term traders, the lack of future trading opportunities eliminates hedging possibilities. This may give rise to self-fulfilling equilibria. If traders anticipate an order imbalance due to informed traders, they scale up their aggressiveness and follow the market.

19 Introduction Related Literature Noisy, dynamic REE: Brown and Jennings (1989), Grundy and McNichols (1989), He and Wang (1995), Vives (1995), and Cespa (2002). Higher order expectations in asset pricing: Allen, Morris and Shin (2006), Bacchetta and van Wincoop (2006a, 2006b), Vitale (2006), Banerjee, Kaniel and Kremer (2007). Bubbles: 1 No common prior: Cao and Ou-Yang (2005), Biais and Bossaerts (1998), Kandel and Pearson (1995). 2 Behavioral bias: Abreu and Brunnermeier (2003), Brunnermeier and Nagel (2004).

20 The Static Benchmark One-period asset market where a single risky asset with liquidation value v N (v, τ 1 v ) and a riskless asset are traded. Continuum of CARA (γ 1 ) informed speculators in the unit interval. Each one observes s i1 = v + ɛ i1, where ɛ i1 N (0, τ 1 ɛ 1 ) and submits a limit order X i1 (s i1, p 1 ) to maximize expected utility of W i1 = (v p 1 )x i1. Noise traders submit random demand u 1 N (0, τ 1 u ). Assume v, u 1 and ɛ i1 independent for all i, and that the SLLN holds: R 1 0 s i1di = v a.s.

21 The Static Benchmark One-period asset market where a single risky asset with liquidation value v N (v, τ 1 v ) and a riskless asset are traded. Continuum of CARA (γ 1 ) informed speculators in the unit interval. Each one observes s i1 = v + ɛ i1, where ɛ i1 N (0, τ 1 ɛ 1 ) and submits a limit order X i1 (s i1, p 1 ) to maximize expected utility of W i1 = (v p 1 )x i1. Noise traders submit random demand u 1 N (0, τ 1 u ). Assume v, u 1 and ɛ i1 independent for all i, and that the SLLN holds: R 1 0 s i1di = v a.s. In this set-up (Hellwig 1980, Admati, 1985) there exists a unique equilibrium in linear strategies where X i1 (s i1, p 1 ) = (γ/var[v s i1, z 1 ])(E[v s i1, z 1 ] p 1 ), z 1 = a 1 v + u 1, and a 1 = γτ ɛ1.

22 The Static Benchmark We use the above CARA-normal framework to investigate conditions under which the equilibrium price deviates systematically from the liquidation value with respect to traders average expectations. When»Z 1 E [p 1 v v] > E E[v s i1, p 1 ]di v v, we will say that the price displays excessive reliance on public information (similarly as in AMS (2006)). 0

23 The Static Benchmark We use the above CARA-normal framework to investigate conditions under which the equilibrium price deviates systematically from the liquidation value with respect to traders average expectations. When»Z 1 E [p 1 v v] < E E[v s i1, p 1 ]di v v, we will say that the price displays excessive reliance on private information. 0

24 The Static Benchmark We use the above CARA-normal framework to investigate conditions under which the equilibrium price deviates systematically from the liquidation value with respect to traders average expectations. When»Z 1 E [p 1 v v] < E E[v s i1, p 1 ]di v v, we will say that the price displays excessive reliance on private information. 0 Result: in the static benchmark the price assigns the optimal statistical weight to public information.

25 The Static Benchmark 6-Steps proof: Step 1 Owing to normality: E[v s i1, z 1 ] = α E1 s i1 + (1 α E1 )E[v z 1 ], where α E1 = τ ɛ1 /(τ 1 + τ ɛ1 ).

26 The Static Benchmark 6-Steps proof: Step 1 Owing to normality: Step 2 Hence, where α E1 = τ ɛ1 /(τ 1 + τ ɛ1 ). where E 1 [v] = R 1 0 E[v s i1, p 1 ]di. E[v s i1, z 1 ] = α E1 s i1 + (1 α E1 )E[v z 1 ], E[v s i1, p 1 ] = α E1 v + (1 α E1 )E[v z 1 ],

27 The Static Benchmark 6-Steps proof: Step 1 Owing to normality: Step 2 Hence, where α E1 = τ ɛ1 /(τ 1 + τ ɛ1 ). E[v s i1, z 1 ] = α E1 s i1 + (1 α E1 )E[v z 1 ], E[v s i1, p 1 ] = α E1 v + (1 α E1 )E[v z 1 ], where E 1 [v] = R 1 E[v s 0 i1, p 1 ]di. Step 3 At a linear equilibrium with private signal responsiveness a 1, p 1 = α P1 v + u «1 + (1 α P1 )E[v z 1 ], a 1 where α P1 = a 1 /(γ(τ 1 + τ ɛ1 )).

28 The Static Benchmark Step 4 Since and E ˆE 1 [v] v v = (1 α E1 )(E[v z 1 ] v), E[p 1 v v] = (1 α P1 )(E[v z 1 ] v),

29 The Static Benchmark Step 4 Since and E ˆE 1 [v] v v = (1 α E1 )(E[v z 1 ] v), E[p 1 v v] = (1 α P1 )(E[v z 1 ] v), Step 5 There is excessive reliance on public information whenever that is whenever a 1 < γτ ɛ1. α P1 < α E1,

30 The Static Benchmark Step 4 Since and E ˆE 1 [v] v v = (1 α E1 )(E[v z 1 ] v), E[p 1 v v] = (1 α P1 )(E[v z 1 ] v), Step 5 There is excessive reliance on public information whenever that is whenever a 1 < γτ ɛ1. α P1 < α E1, Step 6 In equilibrium a 1 = γτ ɛ1, (i.e. α P1 = α E1 = τ ɛ1 /(τ 1 + τ ɛ1 )), and the price is as farther away from the fundamentals than traders average expectations.

31 The Static Benchmark Step 4 Since and E ˆE 1 [v] v v = (1 α E1 )(E[v z 1 ] v), E[p 1 v v] = (1 α P1 )(E[v z 1 ] v), Step 5 There is excessive reliance on public information whenever that is whenever a 1 < γτ ɛ1. α P1 < α E1, Step 6 In equilibrium a 1 = γτ ɛ1, (i.e. α P1 = α E1 = τ ɛ1 /(τ 1 + τ ɛ1 )), and the price is as farther away from the fundamentals than traders average expectations.

32 The Static Benchmark Step 4 Since and E ˆE 1 [v] v v = (1 α E1 )(E[v z 1 ] v), E[p 1 v v] = (1 α P1 )(E[v z 1 ] v), Step 5 There is excessive reliance on public information whenever that is whenever a 1 < γτ ɛ1. α P1 < α E1, Step 6 In equilibrium a 1 = γτ ɛ1, (i.e. α P1 = α E1 = τ ɛ1 /(τ 1 + τ ɛ1 )), and the price is as farther away from the fundamentals than traders average expectations. In a static setup α P1 only depends on traders (relative) signal quality. In a dynamic setup it is also affected by traders speculative activity on short-run returns.

33 The Static Benchmark Alternative (more intuitive?), 3-Steps proof: Step 1 With CARA utility, traders aggregate demand is proportional to R 1 0 E[v s i1, p 1 ]di p 1.

34 The Static Benchmark Alternative (more intuitive?), 3-Steps proof: Step 1 With CARA utility, traders aggregate demand is proportional to R 1 0 E[v s i1, p 1 ]di p 1. Step 2 At equilibrium Z 1 x i1 di + u 1 = Z γ(τ 1 + τ ɛ1 )(E[v s i1, p 1 ] p 1 )di + u 1 = 0. Hence, p 1 = E 1 [v] + u 1 γ(τ 1 + τ ɛ1 ).

35 The Static Benchmark Alternative (more intuitive?), 3-Steps proof: Step 1 With CARA utility, traders aggregate demand is proportional to R 1 0 E[v s i1, p 1 ]di p 1. Step 2 At equilibrium Z 1 x i1 di + u 1 = Z γ(τ 1 + τ ɛ1 )(E[v s i1, p 1 ] p 1 )di + u 1 = 0. Hence, p 1 = E 1 [v] + u 1 γ(τ 1 + τ ɛ1 ). Step 3 As u 1 and v are by assumption orthogonal, E[p 1 v] = E[E 1 [v] v], and the result follows.

36 A Dynamic Market with Long-Term Traders Trade takes place during N 2 periods. In period N + 1 the asset is liquidated. Investors maximize the expected utility of their final wealth: W in = NX N 1 X π in = (p n+1 p n)x in + (v p N )x in. n=1 n=1 In period n an informed trader i receives a signal s in = v + ɛ in, where ɛ in N (0, τ 1 ɛ n ), v and ɛ in are independent for all i, n and error terms are also independent both across time periods and traders. Given v, we assume that for all n, R 1 sindi = v a.s. 0 Noise trading follows an AR(1) process θ n = βθ n 1 + u n, with β [0, 1], {u n} N n=1 is an i.i.d. normally distributed random process (independent of all other random variables in the model), with u n N (0, τ 1 u ).

37 A Dynamic Market with Long-Term Traders 8 >< 1 β = >: θ n follows a random walk u n = θ n θ n 1

38 A Dynamic Market with Long-Term Traders 8 >< 1 β = >: θ n follows a random walk u n = θ n θ n 1 Kyle (1985), Vives (1995).

39 A Dynamic Market with Long-Term Traders 8 >< 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

40 A Dynamic Market with Long-Term Traders 8 >< 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).

41 A Dynamic Market with Long-Term Traders 8 >< 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). In general β < 1 implies that supply increments are negatively correlated: Cov[ θ n+1, θ n] = (β 1)τ 1 u 1 β(1 «βn+2 ) < β

42 No Bubbles when Noise Trading follows a Random Walk Suppose β = 1. Then, Proposition In the market with long-term, informed speculators there exists a unique equilibrium in linear strategies. The equilibrium is symmetric. Prices are given by p o = v, p N+1 = v, and for n = 1, 2,..., N, p n = λ nz n + (1 λ n a n)p n 1 and strategies are given by: X in( s in, p n ) = γ τ n + where a n = γ( P n t=1 τ ɛ t ), and! nx τ ɛt (E[v s in, z n ] p n) t=1 = a n( s in p n) + γτ n(e[v z n ] p n), λ n = 1 + γτ u an γ(τ n + P n t=1 τ ɛ t ).

43 No Bubbles when Noise Trading follows a Random Walk The Equilibrium (II) The equilibrium has a static nature: in every period n investors trade as if the asset would be liquidated in the following period n + 1, exploiting all their available information.

44 No Bubbles when Noise Trading follows a Random Walk The Equilibrium (II) The equilibrium has a static nature: in every period n investors trade as if the asset would be liquidated in the following period n + 1, exploiting all their available information. Intuition: suppose N = 2, then a trader s strategy in period 1 is given by: x i1 =

45 No Bubbles when Noise Trading follows a Random Walk The Equilibrium (II) The equilibrium has a static nature: in every period n investors trade as if the asset would be liquidated in the following period n + 1, exploiting all their available information. Intuition: suppose N = 2, then a trader s strategy in period 1 is given by: x i1 = γ E[p 2 p 1 s i1, z 1 ] h 22 {z } Price speculation

46 No Bubbles when Noise Trading follows a Random Walk The Equilibrium (II) The equilibrium has a static nature: in every period n investors trade as if the asset would be liquidated in the following period n + 1, exploiting all their available information. Intuition: suppose N = 2, then a trader s strategy in period 1 is given by: x i1 = γ E[p 2 p 1 s i1, z 1 ] h 22 {z } where h 21 < 0 and h 22 > 0. Price speculation h 21 E[x i2 s i1, z 1 ] h 22 γ(τ 2 + P 2 t=1 τ ɛ t ) {z } Hedging

47 No Bubbles when Noise Trading follows a Random Walk A No-Bubbles Result Corollary For 1 n N, E[p n v v] = E[E n[v] v v].

48 No Bubbles when Noise Trading follows a Random Walk A No-Bubbles Result Corollary For 1 n N, E[p n v v] = E[E n[v] v v]. When the market is populated by long-term traders and noise traders supply shocks follow a random walk, informed traders demand has a static nature, and the price assigns the optimal statistical weight to public information. In contrast to Allen, Morris and Shin (2006), Banerjee, Kaniel and Kremer (2007), and Bacchetta and van Wincoop (2006), in a dynamic model it is not always true that higher order beliefs become relevant.

49 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (I) Suppose β [0, 1), and that the liquidation value traders receive at the terminal date is given by v + δ, where δ N (0, τ 1 δ ). Define z 1 = a 1 v + θ 1, and z 2 = a 2 v + θ 2, where a 2 = a 2 βa 1. Proposition In the 2-period market with long-term, informed speculators, residual uncertainty about the liquidation value, and correlated supply shocks there exists an equilibrium in linear strategies where: X i2 ( s i2, p 2 ) = γ(var[v + δ s i2, z 2 ]) 1 (E[v s i2, z 2 ] p 2 ), X i1 (s i1, p 1 ) = a 1(τ 1 + τ ɛ1 ) (E[v s i1, z 1 ] p 1 ) + (γ + h 21)(βρ 1) τ ɛ1 γ(τ 2 + P 2 t=1 τ ɛ t ) E[θ 1 z 1 ], where a 2 = γ(1 + κ) 1 (τ ɛ1 + τ ɛ2 ), a 1, κ (τ 2 + P 2 t=1 τ ɛ t )/τ δ, ρ a 1 (1 + κ)/(γτ ɛ1 ) > 1, and h 21 are constants.

50 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (II) Second period strategies keep the static property In the second period the price assigns the optimal statistical weight to public information. However: a 2 < a2 STATIC γ(τ ɛ1 + τ ɛ2 ). First period strategies loose the static property, a trader s position is no longer proportional to (E[v s i1, z 1 ] p 1 ): X i1 (s i1, p 1 )= a 1(τ 1 + τ ɛ1 ) (E[v s i1, z 1 ] p 1 ) + (γ + h 21)(βρ 1) τ ɛ1 γ(τ {z } 2 + P 2 t=1 τ ɛ t ) E[θ 1 z 1 ] {z } Static Short-run speculation and excessive reliance on public information can in principle occur. The sign of the speculative component depends on the magnitude of β ρ.

51 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (III) β captures the degree of correlation between supply increments: the lower is β the more likely it is that a given supply shock in the first period reverts in the second period: Cov[θ 1, θ 2 ] = (β 1)τ 1 u 0, (i.e. supply increments are negatively correlated) ρ measures the deviation that residual uncertainty and noise trading correlation induce in traders first period signal aggressiveness with respect to the static aggressiveness, ρ > 1:

52 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (III) β captures the degree of correlation between supply increments: the lower is β the more likely it is that a given supply shock in the first period reverts in the second period: Cov[θ 1, θ 2 ] = (β 1)τ 1 u 0, (i.e. supply increments are negatively correlated) ρ measures the deviation that residual uncertainty and noise trading correlation induce in traders first period signal aggressiveness with respect to the static aggressiveness, ρ > 1: a 1 = γτ ɛ κ 0 1 C A.

53 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (III) β captures the degree of correlation between supply increments: the lower is β the more likely it is that a given supply shock in the first period reverts in the second period: Cov[θ 1, θ 2 ] = (β 1)τ 1 u 0, (i.e. supply increments are negatively correlated) ρ measures the deviation that residual uncertainty and noise trading correlation induce in traders first period signal aggressiveness with respect to the static aggressiveness, ρ > 1: a 1 = γτ ɛ κ 0 Var 2 [v + δ] Var 2 [v] a 1 1 C A.

54 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (III) β captures the degree of correlation between supply increments: the lower is β the more likely it is that a given supply shock in the first period reverts in the second period: Cov[θ 1, θ 2 ] = (β 1)τ 1 u 0, (i.e. supply increments are negatively correlated) ρ measures the deviation that residual uncertainty and noise trading correlation induce in traders first period signal aggressiveness with respect to the static aggressiveness, ρ > 1: a 1 = γτ ɛ κ 0 Var 2 [v + δ] Var 2 [v] a 1 ˆλ 2 λ 2 a 1 1 C A.

55 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek The Equilibrium (III) β captures the degree of correlation between supply increments: the lower is β the more likely it is that a given supply shock in the first period reverts in the second period: Cov[θ 1, θ 2 ] = (β 1)τ 1 u 0, (i.e. supply increments are negatively correlated) ρ measures the deviation that residual uncertainty and noise trading correlation induce in traders first period signal aggressiveness with respect to the static aggressiveness, ρ > 1: a 1 = γτ ɛ κ 0 Var 2 [v + δ] ˆλ 2 γ 2 (τ 2 + P 2 t=1 τ ɛ t ) Var 2 [v] λ 2 (τ 2 + τ ɛ1 )(γ 2 + Var 2 [v + δ]var 1 [x i2 ](1 ρ 2 1,{x i2,p 2 } )) a 1 a 1 a 1 1 C A.

56 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Intuition (II) Suppose that based on z 1 the market expects θ 1 > 0: E[θ 1 z 1 ] = a 1 (v E[v z 1 ]) {z } + θ 1 {z} > 0 (1) Good fundamentals (2) Positive noise shock (1) is more likely if a 1 and ρ are high traders speculate on price momentum. However, this is risky the more, the stronger is the negative correlation across supply increments. Therefore, as β decreases, traders may choose to accommodate the noise shock, counting on a supply reversal effect that lowers the risk of holding an unbalanced risky position.

57 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Effect on the Price As first period strategies are not proportional to E[v s i1, z 1 ] p 1, where (γ + h 21 )(βρ 1)τ 1 τ ɛ1 p 1 = E 1 [v] + a 1 γ(τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ) E[θ θ 1 1 z 1 ] + α E1 a 1 = α P1 v + θ «1 + (1 α P1 )E[v z 1 ], a 1 α P1

58 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Effect on the Price As first period strategies are not proportional to E[v s i1, z 1 ] p 1, where (γ + h 21 )(βρ 1)τ 1 τ ɛ1 p 1 = E 1 [v] + a 1 γ(τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ) E[θ θ 1 1 z 1 ] + α E1 a 1 = α P1 v + θ «1 + (1 α P1 )E[v z 1 ], a 1 α P1 α E1 {z} + Optimal S-weight +

59 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Effect on the Price As first period strategies are not proportional to E[v s i1, z 1 ] p 1, where (γ + h 21 )(βρ 1)τ 1 τ ɛ1 p 1 = E 1 [v] + a 1 γ(τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ) E[θ θ 1 1 z 1 ] + α E1 a 1 = α P1 v + θ «1 + (1 α P1 )E[v z 1 ], a 1 α P1 α E1 {z} + (γ + h 21 )(βρ 1)τ 1 τ ɛ1 γ(τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ) {z } Optimal S-weight + ST Speculation

60 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Intuition When conditionally on z 1 traders expect the supply increment to be information driven, β ρ > 1, they reinforce the market weight on the fundamentals with their speculative activity (e.g. if E[θ 1 z 1 ] > 0, speculating on a further price increase), moving the price closer to the fundamentals. When, on the other hand, traders feel the risk of momentum trading to be too high, β ρ < 1, they take the other side of the market (e.g. if E[θ 1 z 1 ] > 0, speculating on a supply increment reversal), widening the distance between p 1 and v.

61 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Bubbles and Reverse Bubbles Corollary In any linear equilibrium of the market with long-term, informed speculators when the fundamentals is affected by residual noise the price displays excessive reliance on public information if and only if βρ < 1. E ˆE 1[v] v v E ˆE 1[v] v 0 1/β ρ

62 Residual Uncertainty and Correlated Noise Trade Increments: Keynes vs. Hayek Keynes vs. Hayek When β = 0 (β = 1) excessive (insufficient) reliance on public information occurs. Corollary In any linear equilibrium of the market with long-term, informed speculators when the fundamentals is affected by residual noise, if β = 0 (β = 1) excessive (insufficient) reliance on public information occurs. On the other hand if no residual uncertainty affects the liquidation value: Corollary In any linear equilibrium of the market with long-term, informed speculators when the fundamentals is not affected by residual noise (i.e. 1/τ δ = 0) there always is excessive reliance on public information.

63 Keynes vs. Hayek Numerical simulations show that 1/τ δ > 0, ˆβ below (above) which excessive (insufficient) reliance on public information occurs. This allows to define Ω {(β, 1/τ δ ) [0, 1] R + βρ(β) = 1} as the set of pairs (β, 1/τ δ ) for which the price assigns the optimal statistical weight to fundamentals /τ δ γ = 4 γ = 2 γ = 1/ β

64 The Market with Short-Term Traders Within the same model analyzed in the previous section, suppose that the liquidation value is not affected by residual uncertainty (i.e. 1/τ δ = 0), and that traders have short-term horizons. In particular: Suppose that traders take a position in period n and unwind it in period n + 1. In any period n a trader i maximizes the expected utility of his short-term profits π in = (p n+1 p n)x in. The private information each trader i receives in every period n is transmitted to the corresponding trader in period n + 1.

65 A General Formula Let N = 2. Owing to CARA and normality, in the second period X i2 ( s i2, p 2 ) = γ(var[v s i2, z 2 ]) 1 (E[v s i2, z 2 ] p 2 ). The corresponding market clearing equation is: Z 1 0 x i2 di + θ 2 = 0. Let Var n[y ] = Var[Y s in, z n ], then the second period equilibrium price is given by p 2 = E 2 [v] + θ 2 γ Var 2[v]. Owing to ST horizons, in the first period a trader s position is given by X i1 (s i1, p 1 ) = γ(var[p 2 s i1, z 1 ]) 1 (E[p 2 s i1, z 1 ] p 1 ). At equilibrium: Z 1 0 x i1 di + θ 1 = 0.

66 A General Formula Solving for the first period equilibrium price yields p 1 = E 1 [p 2 ] + θ 1 γ Var 1[p 2 ]» = E 1 E 2 [v] + βθ 1 γ Var 2[v] + θ 1 γ Var 1[p 2 ]. When β = 0 we are in the AMS (2006) case and the price in period 1 depends on the average expectation in period 1 of the average expectation in period 2 of the fundamentals value plus (risk-adjusted) noise. However, this is no longer the case when β 0. In any period the asset price depends on two components: (1) the market average expected liquidation value and (2) the risk associated with holding a position in the asset which affects both the first and second period price.

67 A General Formula The formula can be generalized to the N 2 period case: When β = 0:»»» p n = E n E n+1 E N 1 E N [v] + Var N[v] β N n θ n γ + Var N 1[p N ] β N (n+1) θ n + Var n+1[p n+2 ] βθ n γ γ + Varn[p n+1] θ n. γ p n = E n ˆE n+1 ˆ E N 1 ˆE N [v] {z } + θn γ Varn[p n+1] The Beauty Contest

68 A General Formula AMS (2006): when averaging over the realizations of noise trading, the price at date n will in general not coincide with the period n average expectation of the fundamentals (the price at N + 1). The mean price path p n gives a higher weight to history relies more on public information than the mean consensus path E n[v]. This is because of the bias towards public information when a Bayesian agent has to forecast the average market opinion knowing that it is formed also on the public information observed by other agents. This also implies that the current price will be always farther away from fundamentals than the average of investors expectations and that it will be more sluggish to adjust.

69 A General Formula However, according to the previous section we know that excessive reliance on public information depend on traders short-run speculative activity. If traders speculative activity is fundamentals related the price ends up being closer to the liquidation value than the average consensus expectation. Based on this intuition: 1 We show how the results in AMS (2006) can be overturned when noise trading is not independent across periods, (i.e. if β (0, 1] and there is no residual uncertainty). 2 We also present an example with large residual uncertainty and β = 1 where the price is farther away from the fundamentals.

70 Short-Term Trading and Bubbles The Equilibrium (I) Suppose β (0, 1], N = 2, then Proposition In the 2-period market with short-term traders there exist two symmetric equilibria in linear strategies where X i2 ( s i2, p 2 ) = (γ/var[v s i2, z 2 ])(E[v s i2, z 2 ] p 2 ), X i1 (s i1, p 1 ) = γρ(τ 1 + τ ɛ1 ) (E[v s i1, z 1 ] p 1 ) + (βρ 1)τ 1 τ 2 + P 2 t=1 τ E[θ 1 z 1 ], ɛ t a 2 = γ(τ ɛ1 + τ ɛ2 ), ρ (a 1 /γτ ɛ1 ), and a 1 is given by the (two) real solutions to a quartic equation, and satisfies 0 < a 1 < a 2 /β < a 1.

71 Short-Term Trading and Bubbles The Equilibrium (II) As in the model with LT traders, the second period strategy replicates the static one. Hence, the price assigns the optimal statistical weight to public information. Lack of a second period hedge yields two equilibria in which a 1 is either larger or smaller than the static solution: a 1 < γτ ɛ1 < (γτ ɛ1 /β) < a 1. Figure Let ρ a 1 /(γτ ɛ1 ), p 1 = α P1 v + θ «1 + (1 α P1 )E[v z 1 ], a 1 where α P1 = α E1 + (βρ 1)τ ɛ1 τ 1 (τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ). Hence, excessive reliance on public information arises in the first period if and only if βρ < 1.

72 Short-Term Trading and Bubbles Corollary In the market with short-term traders along the low (high) trading intensity equilibrium the first period equilibrium price is always farther away (closer) to the fundamentals than traders first period average expectation. Corollary Prices do not necessarily display inertia. Along the high (low) trading intensity equilibrium, the first period price rapidly (slowly) adjusts to the fundamentals. Indeed, as traders overreact to their signal, after the first period the price jumps close to v and then fully adjusts in the following two periods. Thus, differently from AMS (2006), the equilibrium price exhibits inertia if and only if βρ < 1. When β = 0 we have the AMS case. In the market with short term traders when β = 0 there exists a unique equilibrium in linear strategies. In this equilibrium the price always displays excessive reliance on public information.

73 Short-Term Trading and Bubbles Stability Claim: The high trading intensity equilibrium is unstable since the slope of the aggregate excess demand function is positive. In this case buying or selling pressures, drive the market away from equilibrium. Given the realization of the first period informational shock z 1, we can define the second period aggregate excess demand function as follows: XD z 2 + λ 1 2 (1 λ 2 ˆ a 2 )ˆp 1 λ 1 2 p 2, where XD = 0 when the market is in equilibrium and XD 0 otherwise. The slope of excess demand depends on the sign of λ 2 and in the high trading intensity equilibrium we have that λ 2 < 0. Restricting attention to stable equilibria we can conclude that the positive bubble equilibrium in AMS (2006) is robust.

74 The Effect of Residual Uncertainty Suppose that β = 1 and the liquidation value traders receive at the terminal date is given by v + δ, with δ N (0, τ 1 δ ) a noise term independent from all the other random variables. As in the market with LT traders, in the second period strategies keep the static property even though, owing to the higher uncertainty over the liquidation value, traders scale down their aggressiveness: a 2 < γ(τ ɛ1 + τ ɛ2 ). Let ρ a 1 (1 + κ)/(γτ ɛ1 ), then p 1 = «αp1 α E1 θ 1 E 1 [v] + E[θ 1 z 1 ] + α E1 a 1 a 1 = (ρ 1) τ ɛ1 τ 1 E 1 [v] + a 1 (τ 1 + τ ɛ1 )(τ 2 + P 2 t=1 τ ɛ t ) E[θ θ 1 1 z 1 ] + α E1, a 1

75 The Effect of Residual Uncertainty Corollary In any linear equilibrium of the market with short-term traders, for sufficiently large residual uncertainty in the liquidation value, the first period price always displays excessive reliance on public information. As in the market with long-term traders, higher uncertainty over the liquidation value exacerbates the reaction of the second period price to the upcoming net informational addition z 2. Hence, second period traders considerably scale down their aggressiveness (a 2 ) and the second period order flow is mainly noise driven. Speculating on price momentum then becomes risky as traders in the first period cannot hope that further information will confirm their prediction. Hence, traders absorb the expected supply yielding a price which is less anchored to the fundamentals (De Long, et al. (1990)).

76 Discussion and Concluding Remarks In this paper we have studied under what conditions asset prices systematically depart from fundamentals compared to the average market consensus in a variety of market contexts. We proved that when long-term traders populate the market and noise trading follows a random walk, equilibrium prices are as close to fundamentals as investors average opinion. Furthermore, an increase in the uncertainty over the liquidation value coupled with noise shocks predictability leads traders to speculate on short-run price movements yielding excessive (Keynesian equilibrium), and insufficient (Hayekian equilibrium) reliance on public information. Finally, when traders are myopic, multiple equilibria arise, delivering a richer set of possibilities. Depending on the equilibrium that obtains the price can either be closer or farther away from the fundamentals compared to the market consensus.

77 Discussion and Concluding Remarks In contrast to the results put forth by AMS (2006), there are situations where equilibrium prices may work as better aggregators of traders private information than the average consensus opinion. In our setup Keynes beauty contest allegory represents only one possible outcome of the information aggregation process that arises as traders focus their short-term activities on the exploitation of noise trading predictability. However, an alternative outcome which is more in line with Hayek (1945) s view of the market is possible. Indeed, to the extent that with residual uncertainty over fundamentals informed trades drive the order flow, speculators short-term trading activity reinforces the weight the price assigns to fundamentals information. This, in turn, may draw the price closer to the fundamentals.

78 Thanks!

79 Bubbles and Reverse Bubbles E ˆE 1 [v] v v E ˆE 1 [v] v 0 1/β ρ

80 Short-Term Trading and Bubbles β = 0 β = β = a 1