An Information-Based Theory of Time-Varying Liquidity

Transcription

1 An Information-Based Theory of Time-Varying Liquidity Brett Green UC Berkeley, Haas School of Business joint with Brendan Daley Duke University, Fuqua School of Business Csef-Igier Symposium on Economics and Institutions June 2013

2 Motivation Markets are susceptible to periods of illiqudity. Recent examples include: Real estate (Clayton et al., 2008) Mortgage backed securities (Gorton, 2009; Acharya and Schnabl, 2010; Dwyer and Tkac, 2009) Repo markets (Gorton and Metrick, 2012) Structured credit (Brunnermeier, 2009) Commercial paper (Anderson and Gascon, 2009) Money market funds (Krishnamurthy et al., 2012) We propose an information-based theory to explain such episodes and explore the impact on prices and volatility.

3 Motivation 1000 Issuance of Private-Label Mortgage Backed Securities 20% % 10% 5% % Private Label Issuance ($billions) Percentage of Total US Bond Market (Source: SIFMA)

4 Key Features of the Model The model takes place in a competitive dynamic economy with fully-rational, risk-neutral agents who share a common-prior. The three key features are: 1 Asymmetric Information: the asset owners are privately informed about future cash flows. 2 News: information about cash flows is gradually and stochastically revealed to the market. 3 Shocks: agents are subject to idiosynchratic shocks. Upon arrival, agent is not forced to sell, but is more eager to do so.

5 Key Features of the Model The model takes place in a competitive dynamic economy with fully-rational, risk-neutral agents who share a common-prior. The three key features are: 1 Asymmetric Information: the asset owners are privately informed about future cash flows. 2 News: information about cash flows is gradually and stochastically revealed to the market. 3 Shocks: agents are subject to idiosynchratic shocks. Upon arrival, agent is not forced to sell, but is more eager to do so.

6 Preview of Main Results 1 Time Varying Liquidity. Equilibrium involves periods of full, partial, and zero liquidity. 2 Illiquidity Discount. Illiquidity leads to an endogenous liquidation cost. Buyers anticipate these costs driving prices below fundamentals. 3 Excess Volatility. Bad news gets compounded. Negative signal about fundamentals. Negative signal about future liquidity. 4 (Efficient) Fire Sales. Due to informational externalities. A trade by one owner can reveal information, which facilitates trade by other owners.

7 Related Literature We build on Daley and Green (2012): Single privately informed seller; competetive buyers. News revealed gradually. Trade occurs only once. By incorporating two features: 1 Idiosyncratic (financial/credit/preference) shocks. 2 Multiple shares and multiple informed owners. Focus on the first, consider the second in an extension.

8 Related Literature We build on Daley and Green (2012): Single privately informed seller; competetive buyers. News revealed gradually. Trade occurs only once. By incorporating two features: 1 Idiosyncratic (financial/credit/preference) shocks. 2 Multiple shares and multiple informed owners. Focus on the first, consider the second in an extension.

9 Some Related Literature Asymmetric Information and Liquidity Lucas and McDonald (1990); Korajczyk et al. (1992); Gârleanu and Pedersen (2003); Eisfeldt (2004); Vayanos and Wang (2012)... Transaction costs based theories of illiquidity Amihud and Mendelson (1986); Constantinides (1986); Vayanos (1998, 2004); Lo et al. (2004); Acharya and Pedersen (2005)... Search based theories of illiquidity Duffie et al. (2005, 2007); Vayanos (1998); Vayanos and Wang (2007); Vayanos and Weill (2008)...

10 The Model Agents: Initial owner, A 0 Owner at time t, A t Many potential buyers (the market ) Buyers not modeled directly, though it is possible to do so. Preferences: All agents are risk-neutral and Agents discount future cash flows at rate r

11 The Model The Asset: Single (indivisible) asset of type θ {L, H} Nature chooses θ with P 0 = P(θ = H) The current owner knows θ and accrues (stochastic) cash flow with mean v θ High-value asset pays more: v H > v L Let V θ 0 e rt v θ dt

12 Idiosyncratic Shocks All agents in the economy face idiosyncratic shocks: Publicly observable shocks arrives according to Poisson process with arrival rate λ. Arrival of shock introduces a holding cost c θ. v θ if she has not been hit by a shock (holder) k θ v θ c θ if she has been hit by a shock (seller) Generates gains from repeated trade, but does not force the owner to sell. k H > v L so that shocks are not overly punitive. Preserves strategic considerations.

13 Idiosyncratic Shocks All agents in the economy face idiosyncratic shocks: Publicly observable shocks arrives according to Poisson process with arrival rate λ. Arrival of shock introduces a holding cost c θ. v θ if she has not been hit by a shock (holder) k θ v θ c θ if she has been hit by a shock (seller) Generates gains from repeated trade, but does not force the owner to sell. k H > v L so that shocks are not overly punitive. Preserves strategic considerations.

14 News Arrival Brownian motion drives the arrival of news. A publicly observable score process (X t ) evolves according to: where µ H µ L dx t = µ θ dt + σdb t The quality (or speed) of the news is measured by the signal-to-noise ratio: φ µ H µ L σ One possible interpretation: News=cashflows: µ θ = v θ

15 News Arrival Brownian motion drives the arrival of news. A publicly observable score process (X t ) evolves according to: where µ H µ L dx t = µ θ dt + σdb t The quality (or speed) of the news is measured by the signal-to-noise ratio: φ µ H µ L σ One possible interpretation: News=cashflows: µ θ = v θ

16 News Arrival Brownian motion drives the arrival of news. A publicly observable score process (X t ) evolves according to: where µ H µ L dx t = µ θ dt + σdb t The quality (or speed) of the news is measured by the signal-to-noise ratio: φ µ H µ L σ One possible interpretation: News=cashflows: µ θ = v θ

17 Timing Infinite-horizon, continuous-time setting Trading mechanism: at every t Buyers make offers. Owner decides which offer to accept (if any). Alternative: Seller post price. Owners that trade exit the economy. News and shocks are realized and repeat. Buyers make (public) offers Owner accepts or rejects Shock arrives with probability λdt News is revealed about the asset dx t Buyers make offers dt First best benchmark: Shocked owners (sellers) trade immediately. Informational friction inhibits efficiency.

18 Timing Infinite-horizon, continuous-time setting Trading mechanism: at every t Buyers make offers. Owner decides which offer to accept (if any). Alternative: Seller post price. Owners that trade exit the economy. News and shocks are realized and repeat. Buyers make (public) offers Owner accepts or rejects Shock arrives with probability λdt News is revealed about the asset dx t Buyers make offers dt First best benchmark: Shocked owners (sellers) trade immediately. Informational friction inhibits efficiency.

19 Market Beliefs and Buyers Strategy Buyers begin with common prior: P 0 = P t=0 (θ = H) At time t, buyers know: (i) The path of news arrival, shocks and offers up to time t (ii) All times prior to t (if any) at which the asset has traded Buyers Strategy The buyer s strategy is a bid process W Equilibrium Beliefs W t (ω) is the (maximal) bid made in the history (t, ω) Let P denote the equilibrium belief process held by buyers: P t (ω) = P(θ = H F B t ) ( ) Define Z = ln P 1 P : beliefs in z-space Skip Ahead

20 Market Beliefs and Buyers Strategy Buyers begin with common prior: P 0 = P t=0 (θ = H) At time t, buyers know: (i) The path of news arrival, shocks and offers up to time t (ii) All times prior to t (if any) at which the asset has traded Buyers Strategy The buyer s strategy is a bid process W Equilibrium Beliefs W t (ω) is the (maximal) bid made in the history (t, ω) Let P denote the equilibrium belief process held by buyers: P t (ω) = P(θ = H F B t ) ( ) Define Z = ln P 1 P : beliefs in z-space Skip Ahead

21 Market Beliefs and Buyers Strategy Buyers begin with common prior: P 0 = P t=0 (θ = H) At time t, buyers know: (i) The path of news arrival, shocks and offers up to time t (ii) All times prior to t (if any) at which the asset has traded Buyers Strategy The buyer s strategy is a bid process W Equilibrium Beliefs W t (ω) is the (maximal) bid made in the history (t, ω) Let P denote the equilibrium belief process held by buyers: P t (ω) = P(θ = H F B t ) ( ) Define Z = ln P 1 P : beliefs in z-space Skip Ahead

22 Owner s Strategy and Sequential Rationality Owner s Strategy The strategy of an owner is a stopping rule τ. Definition (Sequential Rationality) Given W, an owner s strategy is sequentially rational if for all histories, it solves: [ τ ] sup Et θ e rs (v θ I s c θ )ds + e r(τ t) W τ Ft S τ t (SP θ )

23 Owner s Strategy and Sequential Rationality Owner s Strategy The strategy of an owner is a stopping rule τ. Definition (Sequential Rationality) Given W, an owner s strategy is sequentially rational if for all histories, it solves: [ τ ] sup Et θ e rs (v θ I s c θ )ds + e r(τ t) W τ Ft S τ t (SP θ )

24 Equilibrium Concept Definition An equilibrium is a triple (τ, W, Z): Given W, the owner s strategy is sequentially rational. Given τ and Z, W is such that Buyers earn zero profit. No (profitable) deals exist. Market beliefs, Z, are consistent with Bayes rule whenever possible.

25 Equilibrium Beliefs In equilibrium, the market beliefs evolves based on news as well as: That is, The owner s equilibrium strategy and Previous trades (or lack thereof) dz t = dẑ }{{} t + dq }{{} t updating based only on news updating based on trades Where dq t is the information in whether trade occurred at time t. For example, suppose trade does not occur at time t: If strategies call for trade with probability zero: dq t = 0 If strategies call for a low type to trade with positive probability and a high type to trade with probability zero: dq t > 0

26 Equilibrium Beliefs In equilibrium, the market beliefs evolves based on news as well as: That is, The owner s equilibrium strategy and Previous trades (or lack thereof) dz t = dẑ }{{} t + dq }{{} t updating based only on news updating based on trades Where dq t is the information in whether trade occurred at time t. For example, suppose trade does not occur at time t: If strategies call for trade with probability zero: dq t = 0 If strategies call for a low type to trade with positive probability and a high type to trade with probability zero: dq t > 0

27 Equilibrium Beliefs In equilibrium, the market beliefs evolves based on news as well as: That is, The owner s equilibrium strategy and Previous trades (or lack thereof) dz t = dẑ }{{} t + dq }{{} t updating based only on news updating based on trades Where dq t is the information in whether trade occurred at time t. For example, suppose trade does not occur at time t: If strategies call for trade with probability zero: dq t = 0 If strategies call for a low type to trade with positive probability and a high type to trade with probability zero: dq t > 0

28 Equilibrium Beliefs In equilibrium, the market beliefs evolves based on news as well as: That is, The owner s equilibrium strategy and Previous trades (or lack thereof) dz t = dẑ }{{} t + dq }{{} t updating based only on news updating based on trades Where dq t is the information in whether trade occurred at time t. For example, suppose trade does not occur at time t: If strategies call for trade with probability zero: dq t = 0 If strategies call for a low type to trade with positive probability and a high type to trade with probability zero: dq t > 0

29 Equilibrium Description Equilibrium is stationary w.r.t. (z, i); any history such that: Market beliefs are z The owner s status is i i = 1 indicates seller (positive holding cost) i = 0 indicates holder (zero holding cost)

30 Equilibrium Characterization Theorem There exists an equilbrium. It is characterized by (α, β) R 2 and B(z) : R R and the following three regions when owner is a seller i = 1, 1 If z β: the market is fully liquid. Bid is B(z) and both types accept w.p.1. 2 If z α: the market is partially liquid. Bid is V L. High type rejects. Low type mixes. 3 If z (α, β): the market is fully illiquid. Bid is rejected w.p.1. When the owner is a holder (i = 0), it is common knowledge there are no gains from trade and trade does not occur.

31 Equilibrium Characterization Theorem There exists an equilbrium. It is characterized by (α, β) R 2 and B(z) : R R and the following three regions when owner is a seller i = 1, 1 If z β: the market is fully liquid. Bid is B(z) and both types accept w.p.1. 2 If z α: the market is partially liquid. Bid is V L. High type rejects. Low type mixes. 3 If z (α, β): the market is fully illiquid. Bid is rejected w.p.1. When the owner is a holder (i = 0), it is common knowledge there are no gains from trade and trade does not occur.

32 Equilibrium Characterization Theorem There exists an equilbrium. It is characterized by (α, β) R 2 and B(z) : R R and the following three regions when owner is a seller i = 1, 1 If z β: the market is fully liquid. Bid is B(z) and both types accept w.p.1. 2 If z α: the market is partially liquid. Bid is V L. High type rejects. Low type mixes. 3 If z (α, β): the market is fully illiquid. Bid is rejected w.p.1. When the owner is a holder (i = 0), it is common knowledge there are no gains from trade and trade does not occur.

33 Equilibrium Characterization Theorem There exists an equilbrium. It is characterized by (α, β) R 2 and B(z) : R R and the following three regions when owner is a seller i = 1, 1 If z β: the market is fully liquid. Bid is B(z) and both types accept w.p.1. 2 If z α: the market is partially liquid. Bid is V L. High type rejects. Low type mixes. 3 If z (α, β): the market is fully illiquid. Bid is rejected w.p.1. When the owner is a holder (i = 0), it is common knowledge there are no gains from trade and trade does not occur.

34 Sample Path of Play Initial Belief β Market Belief (z) α t

35 Sample Path of Play β Both type sellers trade right away Beliefs remain unchanged Market Belief (z) α t

36 Sample Path of Play New holder consumes flow and waits for shock β Market beliefs evolve according to news Market Belief (z) α t

37 Sample Path of Play New holder consumes flow and waits for shock β Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Market Belief (z) α t

38 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Suppose a liquidity shock arrives here α t

39 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers waits in hopes of good news α t

40 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers wait in hopes of good news α Sufficient bad news leads to a partial sell off Low type mixes at α High type always rejects t

41 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers wait in hopes of good news α Sufficient bad news leads to a partial sell off Low type mixes at α High type always rejects If trade occurs at α, beliefs place probability one on a low type forever after t

42 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers wait in hopes of good news α Beliefs reflect at α if trade does not occur (and i=1) t

43 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers wait in hopes of good news High-type seller always eventually reaches β α Beliefs reflect at α if trade does not occur (and i=1) t

44 Sample Path of Play New holder consumes flow and waits for shock β Market Belief (z) Market beliefs evolve according to news Bad news drives beliefs into the no-trade region Both type sellers wait in hopes of good news At which point, trade occurs immediately High-type seller always eventually reaches β α Beliefs reflect at α if trade does not occur (and i=1) t

45 Proof by Construction Step 1: Take B and (α, β) as given. Construct seller value functions F L, F H through ODEs and two sets of boundary conditions: Physical conditions (e.g., value matching). Necessary local Optimality conditions (e.g., smooth pasting). Step 2: Taking F L, F H as given. Construct holder value functions G L, G H through ODEs and boundary conditions. Step 3: Taking G L, G H as given, a buyers value is the expected value to a holder given both types sell: B(z) = E[G θ (z) z] Step 4: Show there exists a fixed point of the system in Steps 1-3. Step 5: Verify necessary optimality conditions are sufficient.

46 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. V H 2 For high enough z, H has little to gain by waiting for good news so she exercises. Asset Value 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L α Market Belief (z) β B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

47 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. V H 2 For high enough z, H has little to gain by waiting for good news so she exercises. Asset Value 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L α Market Belief (z) β B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

48 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. 2 For high enough z, H has little to gain by waiting for good news so she exercises. V H Asset Value refl: F H (α) = 0 vm: FH(β) = B(β) sp: F H (β) = B (β) 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L α Market Belief (z) β FH B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

49 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. V H vm: FL(β) = B(β) 2 For high enough z, H has little to gain by waiting for good news so she exercises. Asset Value 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L vm: FL(α) = VL sp: F L (α) = 0 α Market Belief (z) β FL B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

50 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. V H vm: FL(β) = B(β) 2 For high enough z, H has little to gain by waiting for good news so she exercises. Asset Value 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L vm: FL(α) = VL sp: F L (α) = 0 α Market Belief (z) β FL B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

51 Intuition Take B as given: 1 H can always get B(z) if she wants it. For z < β, she does better by not exercising the option. V H 2 For high enough z, H has little to gain by waiting for good news so she exercises. Asset Value 3 L can always get V L if he wants it. But for z (α, β), he does better to mimic H. V L α Market Belief (z) β FH FL B 4 L s prospects of reaching β decrease with z. At z = α, he is just indifferent = willing to mix. Figure : Constructing F θ from B

52 Buyer and Holder Values Of course, B, depends on a holder s value, G θ, which in turn depends on a seller s value, F θ. Fixed point characterizes this interdependence. Useful to compare to two benchmark cases. 1 Benchmark 1: No private information. All agents symmetrically uninformed about θ. 2 Benchmark 2: No shocks. Set λ = 0.

53 Benchmark 1: No Private Information Suppose owners and buyers are commonly uninformed about θ. Then, upon arrival of a shock: (i) A seller has no reason to delay trade. (ii) Given any market belief z, buyers are willing to pay the expected fundamental value of the asset. B(z) = Ψ(z) E [V θ z] (iii) Therefore, the market is fully liquid for all z and trade occurs immediately at Ψ(z).

54 Benchmark 1: No Private Information Suppose owners and buyers are commonly uninformed about θ. Then, upon arrival of a shock: (i) A seller has no reason to delay trade. (ii) Given any market belief z, buyers are willing to pay the expected fundamental value of the asset. B(z) = Ψ(z) E [V θ z] (iii) Therefore, the market is fully liquid for all z and trade occurs immediately at Ψ(z).

55 Benchmark 2: No Financial Shocks When λ = 0, there is a unique three-region equilibrium. (i) Because holders never face the need to resell, G θ (z) = v θ. (ii) Buyers still face potential adverse selection, but need not worry about future liquidation costs. Their (unconditional) value is: (iii) The asset trades only once and. B(z) = Ψ(z) Price = fundamental value

56 Benchmark 2: No Financial Shocks When λ = 0, there is a unique three-region equilibrium. (i) Because holders never face the need to resell, G θ (z) = v θ. (ii) Buyers still face potential adverse selection, but need not worry about future liquidation costs. Their (unconditional) value is: (iii) The asset trades only once and. B(z) = Ψ(z) Price = fundamental value

57 Benchmark 2: Equilibrium Value Functions Equilibrium Asset Values without Shocks V H Asset Value G H F H G L F L Ψ B V L α β Market Belief (z)

58 The Illiquidity Discount Two properties from benchmarks: 1 B(z) = Ψ(z), and 2 Asset always trades at fundamental value. With both private information and shocks, these no longer hold. Holder face the potential of costly future liquidation. G θ < v θ r As a result, buyers shade bids below fundamentals. Proposition (The Illiquidity Discount) When the market is fully liquid, trade takes place at a price strictly below the fundamental value. B(z) = Ψ(z) δ(z) }{{} illiquidity discount

59 The Illiquidity Discount Two properties from benchmarks: 1 B(z) = Ψ(z), and 2 Asset always trades at fundamental value. With both private information and shocks, these no longer hold. Holder face the potential of costly future liquidation. G θ < v θ r As a result, buyers shade bids below fundamentals. Proposition (The Illiquidity Discount) When the market is fully liquid, trade takes place at a price strictly below the fundamental value. B(z) = Ψ(z) δ(z) }{{} illiquidity discount

60 The Illiquidity Discount Two properties from benchmarks: 1 B(z) = Ψ(z), and 2 Asset always trades at fundamental value. With both private information and shocks, these no longer hold. Holder face the potential of costly future liquidation. G θ < v θ r As a result, buyers shade bids below fundamentals. Proposition (The Illiquidity Discount) When the market is fully liquid, trade takes place at a price strictly below the fundamental value. B(z) = Ψ(z) δ(z) }{{} illiquidity discount

61 Equilibrium Value Functions Equilibrium Asset Values with Shocks V H Asset Value V L α β Market Belief (z) G H F H G L F L Ψ B

62 The Illiquidity Discount Illiquidity Discount (%) The illiquidity discount as measured by Ψ B Ψ α β

63 Excess Volatility Proposition In fully liquid markets, the volatility of the equilibrium price process is strictly greater than the fundamental volatility. That is B (z) > Ψ (z). Intuition: Starting from any z β, bad news has two affects. 1 Reduces traders expectations about fundamentals. 2 Increases likelihood of future illiquidity. The effect of bad news gets amplified, generating additional volatility.

64 Excess Volatility Proposition In fully liquid markets, the volatility of the equilibrium price process is strictly greater than the fundamental volatility. That is B (z) > Ψ (z). Intuition: Starting from any z β, bad news has two affects. 1 Reduces traders expectations about fundamentals. 2 Increases likelihood of future illiquidity. The effect of bad news gets amplified, generating additional volatility.

65 Excess Volatility Proposition In fully liquid markets, the volatility of the equilibrium price process is strictly greater than the fundamental volatility. That is B (z) > Ψ (z). Intuition: Starting from any z β, bad news has two affects. 1 Reduces traders expectations about fundamentals. 2 Increases likelihood of future illiquidity. The effect of bad news gets amplified, generating additional volatility.

66 Market Efficiency Two ways to measure efficiency: 1 Trade Volume frequency with which asset is (efficiently) transferred. 2 Value Loss fraction of total value realized. L F Ψ(z) E[F θ(z) z], or L G Ψ(z) E[G θ(z) z] Ψ(z) Ψ(z) Note: Focus on value loss measure here. Results similar for volume.

67 Efficiency: Value Loss 20 L F L G 15 Loss in Efficiency (%) α β Figure : Efficiency as it depends on z, i.

68 Efficiency and News Quality 15 φ=0.2 φ=1 φ= Figure : Efficiency may increase or decrease with news quality depending on the initial state.

69 Efficiency and Arrival of Shocks 15 λ=1.5 λ=0.25 λ= Figure : Efficiency decreases with the arrival rate of shocks costly liquidation occurs more frequently.

70 Efficiency and Holding Costs 12 δ=0.9 δ=0.75 δ= Figure : Efficiency can increase or decrease with the severity of the shock. Lower δ corresponds to higher holding costs (δ v θ c θ v θ ).

71 Markets with Multiple Shares We extend the model to a setting with N identical shares. Each agent can own at most one share. Possible Interpretations Dispersion of (informed) ownership and Transparency of trades Purpose? Application to a broader range of markets The role information externalities Robustness

72 Markets with Multiple Shares We extend the model to a setting with N identical shares. Each agent can own at most one share. Possible Interpretations Dispersion of (informed) ownership and Transparency of trades Purpose? Application to a broader range of markets The role information externalities Robustness

73 Information Externalities and Fire Sales Two interesting results from the N-share model: 1 Fire Sales One seller s trade at z = α, reveals θ = L. Other sellers have no (further) reason to delay. Holders sell immediately upon arrival of shock. 2 Implications for Efficiency The presence of other informed sellers leads to faster information revelation. This affects equilibrium asset values. Improves overall market efficiency (in contrast to φ).

74 Information Externalities and Fire Sales Two interesting results from the N-share model: 1 Fire Sales One seller s trade at z = α, reveals θ = L. Other sellers have no (further) reason to delay. Holders sell immediately upon arrival of shock. 2 Implications for Efficiency The presence of other informed sellers leads to faster information revelation. This affects equilibrium asset values. Improves overall market efficiency (in contrast to φ).

75 Final Remarks Presented a theory of time-varying liquidity based on: Private information News Revelation Idiosyncratic Shocks Model also generates An illiquidity discount Excess volatility Fire sales Discussed implications for market efficiency.

76 Final Remarks Presented a theory of time-varying liquidity based on: Private information News Revelation Idiosyncratic Shocks Model also generates An illiquidity discount Excess volatility Fire sales Discussed implications for market efficiency.

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