Kyle Model Governance Atoms Bonds. Presence of an Informed Trader and Other Atoms in Kyle Models. Kyle Model Governance Atoms Bonds

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1 Outline 1 Continuous time Kyle (1985) model 2 Kerry Back, Tao Li, and Alexander Ljungqvist, Liquidity and Governance 3 Kerry Back, Tao Li, and Kevin Crotty, Detecting the Presence of an Informed Trader and Other Atoms in Kyle Models 4 Kerry Back and Kevin Crotty, The Informational Role of Stock and Bond Volume Continuous Time Kyle Model Risk-free rate = Single risky asset Value w is announced at date 1 Single risk-neutral informed trader Observes unbiased signal ṽ at date X t = # shares held by informed trader at date t Noise (liquidity) trades a (, σ z ) Brownian motion Z Risk neutral competitive (Bertrand) market makers force price to expected value, given information Y def = X + Z

2 Equilibrium Conditions Competitive pricing: P t = E [ ṽ F Y t ] Look for equilibrium with P t = p(t, Y t ) Informed trader chooses θ t = dx t /dt to maximize [ ] 1 E [ṽ p(t, Y t )]θ t dt ṽ def Constructing Equilibrium We have an equilibrium if Proof: p(t, Z t ) = F Z martingale (p t + σ 2 p yy /2 = ), p(1, Y 1 ) = ṽ (no money left on table), Y = F Y martingale (inconspicuous insider trades) If p(t, Z t ) = F Z martingale, then any strategy is optimal for the insider provided p(1, Y 1 ) = ṽ Y = F Y Y = d Z (Levy s characterization) p(t, Z t ) = F Z martingale and Y = d Z p(t, Y t ) = F Y martingale p(t, Y t ) = F Y martingale and p(1, Y 1 ) = ṽ p(t, Y t ) = E [ ṽ Ft Y ]

3 Liquidity and Governance The presence of large shareholders is generally believed to mitigate agency problems But will a large shareholder take the trouble to intervene in corporate governance or will she surreptitiously sell her shares (take the Wall Street Walk) instead? Liquidity can be good for governance (makes it easier to accumulate a large block) or bad for governance (makes it easier to exit) Closest reference: Maug (JF, 1998) solves a single period Kyle model Cannot analyze how blockholder responds to market Model Blockholder owns X shares at date Governance event after the close of trading at date 1 Blockholder must own B to affect governance Costs c to intervene Intervention changes value of firm from L to H Intervention is both possible and profitable if X 1 B, X 1 ξ def = c/(h L) Assume c is normally distributed and private information Liquidity traders and market makers as before

4 Equilibrium Conditions The investor s date 1 shares X 1 are worth V (X 1, ξ) to her, where { Lx if x < max(b, ξ), V (x, ξ) = Lx + (H L)(x ξ) otherwise She trades to maximize [ E V (X 1, ξ) 1 P t θ t dt P 1 X 1 ξ ], where X 1 is a possible discrete order at the close of trading Equilibrium prices must satisfy p(t, Y t ) = L + (H L) prob ( X 1 max(b, ξ) F Y t ) Equilibrium Informed Trading Define δ = σ z σ 2 ξ + σ2 z, A = δ(x µ ξ ) 1 + δ The equilibrium trading strategy is θ t = δ(µ ξ ξ Z t ) Y t, (1 t)(1 δ) { (B X 1 ) + if Z 1 µ ξ ξ + A/δ, X 1 = otherwise In equilibrium, the large trader becomes active iff Z 1 µ ξ ξ + A/δ Y 1 A Y 1 A

5 Equilibrium Prices Equilibrium prices are p(1, y) = L + (H L)1 {y A}, p(t, y) = E[p(1, Z 1 ) Z t = y] Also, X 1 = X δ( ξ µ ξ ) (1 + δ)z 1 Example X = µ ξ (so the large trader would become active with 5% probability if there were no trade) This implies A =, so the large trader becomes active iff Y 1 1 share outstanding, µ ξ = 1, σ ξ = 2, σ z = 5 Consider ξ = µ ξ, so the informed trader becomes active iff Z 1 These imply δ = 93 and X 1 = X 193 Z 1

6 Y X - A Z Cumulative Orders Market Large Investor Probability of Large Investor Becoming Active Probability of Intervention The unconditional probability of the blockholder becoming active is N(A/σ z ), where N is the standard normal cdf It is increasing in σ z if A < X < µ ξ and decreasing in σ z otherwise Thus, liquidity improves governance when X is small and harms governance when X is large Same conclusion as Maug (JF, 1998)

7 IPO Mechanisms Suppose X is acquired when the firm goes public Different IPO mechanisms (Stoughton-Zechner, JFE, 1998): 1 Take-it-or-leave-it offers 2 Walrasian 3 Discriminatory pricing 4 Discriminatory pricing with a take-it-or-leave-it offer to the large investor 5 Non-discriminatory pricing with a take-it-or-leave-it offer to the large investor and rationing of small investors No equilibrium in (2) Mechanisms (1), (4) and (5) produce higher revenue than (3) and produce a large X We conclude that post-ipo liquidity is harmful for governance Kyle Model with Atoms Examples with discontinuous distribution function for ṽ: w = true value Large trader gets unbiased signal s with probability φ, so ṽ = { s with probability φ, µ with probability 1 φ, where µ = E[ s] Bankruptcy risk positive probability that ṽ = Bernoulli distribution: ṽ = L or ṽ = H

8 Notation F = cdf of Z 1 H = right-continuous cdf of ṽ R = right-continuous inverse of ṽ R(a) = inf{v H(v) > a} for a 1 Smirnov transform: Take x uniform on [, 1] Then R( x) d = ṽ V = {R(a) R a 1} Then, prob(ṽ V ) = 1 Theorem There is an equilibrium in which P t = p(t, Y t ) with p(t, y) = E[R F(Z 1 ) Z t = y] The equilibrium informed trading strategy is θ t = q(t, Y t, ṽ), where, for v V, q(t, y, v) = E[Z 1 Z t = y, R F(Z 1 ) = v] y 1 t

9 Informed Trading Strategy If v V is not an atom, then there is a unique y such that R F(y) = v, and q(t, y, v) = F 1 H(v) y 1 t This is the drift of a Brownian bridge terminating at F 1 H(v) If v is an atom, then R F(y) = v for each y [H (v), H(v)), and q(t, y, v) = E[Z T Z t = y, F 1 (H (v)) Z T F 1 (H(v))] T t As with a Brownian bridge, R F(Y 1 ) = ṽ and Y is an F Y martingale Detecting Informed Trader Assume the signal s has a continuous strictly increasing cdf G Set y = F 1 (φg(µ)) and y 1 = F 1 (1 φ + φg(µ)) Then, ( ) G 1 F (y) φ if y < y, R F(y) = µ ( ) if y z y 1, G 1 F (y) 1+φ if y > y 1 The equilibrium trading strategy is q(t, y, v) = φ F 1 (φg(v)) y 1 t if v < µ, E[Z 1 Z t =y,z Z 1 z 1 ] y 1 t if v = µ, F 1 (1 φ+φg(v)) y 1 t if v > µ

10 Probability of Information The conditional probability that the strategic trader is informed, given market makers information at any date t < T, is δ(t, Y t ), where δ(t, y) = prob(z T < z Z t = y) + prob(z T > z 1 Z t = y) ( ) ( ) z y = N σ z1 y + 1 N T t σ T t Assume s is lognormally distributed At date t = 1/2, 1 φ = 1 8 δ(t, Yt) 6 φ = 75 4 φ = 5 2 φ = Y t /σ t

11 The proportional spread, scaled by the adverse selection parameter β def = stdev(log s)/σ z is 12 λ(t, Yt)/βPt φ = 1 φ = 75 4 φ = 5 2 φ = Y t /σ t Debt and Equity with Taxes and Bankruptcy Zero-coupon bond with face value D maturing at date 1 Firm s gross value ṽ at date 1 shared among bondholders, shareholders, taxes, and the deadweight costs of bankruptcy Total payout to bondholders and shareholders is x def = (1 α)ṽ if ṽ < D, ṽ if D ṽ < D + E, τ(d + E) + (1 τ)ṽ if ṽ D + E Bondholders get min( x, D), and shareholders get ( x D) +

12 Trading Model No private information when the time the bond is issued Single risk-neutral trader observes ṽ immediately after the issue Vector of stock and bond liquidity trades a (, Σ) Brownian motion Default and Solvency Regions Equilibrium is defined in terms of a strictly increasing function G of date 1 bond and stock orders (y b, y s ) Define D = {y G(y) < }, S = {y G(y) } In equilibrium, Y T D if and only if x < D D is the default region, and S is the solvency region

13 Equilibrium Prices Define functions π b and π s by prob(z T D and Z b T a) = prob( x π b (a)), prob(z T S and Z s T a) = prob(d < x D + π s (a)), Set p b (1, y) = p s (1, y) = { π b (y b ) if y D, D if y S, { if y D, π s (y s ) if y S, Set p = (p b, p s ) and, for t < 1, p(t, y) = E[p(1, Z 1 ) Z t = y] Equilibrium Informed Trades For each x < D, set A(x) = {y D π b (y b ) = x} For x > D, set A(x) = {y S π s (y s ) = x D} The equilibrium informed trading strategy is q(t, y, x) = 1 1 t ( E [ Z 1 Z t = y, Z 1 A(x) ] y )

14 Kyle s Lambdas The vector P t of prices evolves as dp t = Λ t dy t, where Λ = ( p b / y b p b / y s ) p b / y b p b / y s The matrix Λ t is symmetric and positive definite Positive definiteness each security is relatively more sensitive to its own orders Boundary between D and S Impose the following condition on G: G(y b, y s ) = G(y b, y s )/ y s G(y b, y s )/ y b = πs (y s ) D π b (y b ) In a single-security Kyle model, the informed trader neither makes nor loses money by trading in one direction and then immediately reversing the trades With multiple securities, there are many possible paths one can take to move from one inventory pair (y b, y s ) to another (w b, w s ) Trading from one to the other and then back along any paths generates neither gains nor losses when the condition on G holds

15 -2-1 Stock Trades Y s t Bond Trades Y b t This shows the credit spread in percentage points at date t = 1/2 as a function of the cumulative stock trades Y s t standard deviations (σ s t and σb t, respectively) and cumulative bond trades Y b t, measured in units of their -2-1 Stock Trades Y s t Bond Trades Y b t This shows the proportional bond spread in basis points (1, λ bb t /Pt b ) at date t = 1/2 as a function of the cumulative stock trades Yt s and cumulative bond trades Yt b, measured in units of their standard deviations (σ s t and σb t, respectively)

16 -2-1 Stock Trades Y s t Bond Trades Y b t This shows the proportional stock spread in basis points (1, λ ss t /Pt s ) at date t = T /2 as a function of the cumulative stock trades Yt s and cumulative bond trades Yt b, measured in units of their standard deviations (σ s t and σb t, respectively) Empirics 3 years of daily stock and bond transactions data for 334 firms Elements of Λ are positive, and Λ is positive definite Λ ss /P s (proportional stock spread) declines with stock purchases and with bond purchases