A Theory of Asset Prices based on Heterogeneous Information and Limits to Arbitrage Elias Albagli USC Marhsall Christian Hellwig Toulouse School of Economics Aleh Tsyvinski Yale University September 20, 2011
Motivation This paper: asset pricing theory based on heterogeneous info and limited arbitrage Parsimonious: all results derive from these 2 elements General: tractability allows analysis of wide class of securities Central message: systematic departure of prices from fundamentals Beliefs are heterogeneous: private signal + price Price = expectation of marginal trader Noisy info aggregation prices expected dividends (cond. on public info) Price over/undervaluation (depending on payoff structure) Price volatility can exceed realized dividend volatility Variety of applications: M&M capital structure irrelevance Excess volatility of stock returns Price reaction to public announcements
Relation with Literature 1. Information aggregation Grossman and Stiglitz (AER 80); Hellwig (JET 80); Diamond and Verrecchia (JFE 81); Wang (REStud 93). 2. Heterogeneous Beliefs and Bubbles Harrison and Kreps (QJE 78); Scheinkman and Xiong (JPE 03); Abreu and Brunnermeier (ECT 03). 3. Finance puzzles M&M Capital structure irrelevance: Myers (JF 84); Myers and Majluf (JFE 84). Excess return volatility: Shiller (AER 81), Campbell and Shiller (RFS 88), Cochrane (RFS 92). Stock price under/overreaction: Barberis et al. (JFE 98); Daniel et al. (JF 98); Hong and Stein (JF 99).
Outline of Talk 1. Setup 2. Information Aggregation Wedge 3. Applications 4. Robustness
Setup
Environment Single risky asset in unit supply Pays π(θ); Fundamental: θ N(0, σ 2 θ ) Dividend function: π ( ) > 0, otherwise unrestricted Two dates: Trading in financial market (t = 0) Payoffs realized (t = 1)
Financial Market: t = 0 Informed traders: i [0, 1] Risk-neutral Limits to arbitrage: Can buy at most 1 share, and cannot short-sell Observe private signal x i N(θ, β 1 ), share price P Buy (d i = 1)/don t buy (d i = 0): { 1 if E[π(θ) xi, P] P d(x, P) = 0 otherwise Aggregate informed demand: D(θ, P) = d(x, P)dΦ( β(x θ)) Noisy demand: Φ(u); u N(0, σ 2 u)
Equilibrium Definition A Perfect Bayesian Equilibrium (PBE) consists of 1. Price function P (θ, u) 2. Informed traders demands d(x, P) 3. Posterior beliefs H(θ x, P) for informed traders s.t., (i) Informed traders demands are optimal (given beliefs) (ii) The asset market clears (iii) Posterior beliefs satisfy Bayes rule
Trader Optimality: Threshold Strategy Expected dividend, and demand d(x, P): monotone in x Trading strategy: signal threshold x(p) 1 if x i > ˆx(P) d(x, P) = 0 (0, 1) if x i < ˆx(P) if x i = ˆx(P) Price = dividend expectation of marginal trader (x i = ˆx(P)) P = E[π(θ) x i = ˆx(P), P] = π(θ)dh(θ ˆx(P), P)
Market Clearing D(θ, P) + Φ(u) = 1; Φ( β( x(p) θ)) = Φ(u) ˆx(P) = θ + 1/ β u z P: aggregates private info P informationally equivalent to ˆx(P) = z z: endogenous public signal Increasing in fundamental θ, noisy demand u θ z N(z, σ 2 u/β); precision of z: β/σ 2 u β: private info precision; σ 2 u: noisy demand variance
Proposition: Asset Market Equilibrium Unique equilibrium: price P π (z) and traders threshold ˆx(p) = z = Pπ 1 (p), P π(z) = = π(θ)dφ σ 2 θ + β + βσ 2 u π(γ P z + σ θ 1 γp u)φ(u)du θ σ 2 θ 2 β + βσu + β + βσ 2 u } {{ } γ P z Marginal trader pricing share conditions on private signal x i = z; public signal z Bayesian weight γ P on signal z; residual uncertainty = 1 γ P Expected dividend, conditional on public signal z only V π(z) = = π(θ)dφ σ 2 θ + βσ 2 u π(γ V z + σ θ 1 γv u)φ(u)du θ σ 2 θ 2 βσu + βσ 2 u } {{ } γ V z Bayesian weight γ V (< γ P ) on signal z; residual uncertainty = 1 γ V
Information Aggregation Wedge
Information Aggregation Wedge Information aggregation wedge: W π(z) P π(z) V π(z) Marginal trader puts higher weight on market signal z than outsider who only observes the price (γ P > γ V ) P π(z) = V π(z) = π(θ)dφ π(θ)dφ σ 2 θ σ 2 θ + β + βσ 2 u + βσ 2 u θ θ β + βσu 2 z σ 2 θ + β + βσu 2 }{{} σ 2 θ βσ 2 u + βσ 2 u } {{ } γ V γ P z But V π(z) is the correct metric for valuing the unconditional dividend: = E[π(θ)] = E[V π(z)]
Information Wedge in Linear Case Symmetric Risk P(z) V(z) 0 E[P(z)]=E[V(z)] z Price more responsive to z than expected dividend
Intuition: Shift in Marginal Trader s Identity Key intuition: for each realization of z, marginal trader is a different agent Higher z (due to θ, and/or u) has two effects on beliefs Higher θ shifts up distribution of signals x s : higher demand higher ˆx(P) Higher u lowers net supply higher ˆx(P) to deter buying by informed Expectations of new marginal trader pricing shares raised through both effects Higher expectations due to market signal (just like anyone else) Higher expectations due to shift in identity (this is the extra kick) Double weighting of market info z is rational (Bayesian updating)
Unconditional Wedge Lemma (unconditional wedge): The unconditional wedge is given by ( W π (σ P ) E[W (z)] = π (θ) π ( θ) ) ( ( ) ( )) θ θ Φ Φ dθ, 0 }{{} σ θ σ P }{{} sign magnitude Sign: related to curvature of π( ) Magnitude given by informational frictions σ 2 P Marginal trader s posterior θ z: N(γ P z, (1 γ P )σ 2 θ ) Prior: z N(0, σ 2 θ /γ V ) Compounded distribution: θ N(0, (1 γ P )σθ 2 + γ2 P σ2 θ /γ V ) }{{} σ 2 P >σ2 θ Marginal trader overweights tails of θ distribution (overreacts to z) Pricing of shares as if θ N(0, σp 2 ), rather than N(0, σ2 θ ) (fatter tails)
Results: Over/Under-Valuation Definition (risk type): a dividend function π(θ), θ > 0, (i) Has symmetric risks if π (θ) = π ( θ), (ii) Has upside risks if π (θ) π ( θ), (iii) Has downside risks if π (θ) π ( θ), (iv) If π 1 (θ) π 1 ( θ) π 2 (θ) π 2 ( θ), π 1( ) has more downside (less upside) risk than π 2( ) Theorem (value bias): (i) If π( ) has symmetric risk, W π (σ P ) = 0 (ii) If π( ) has upside risk, W π (σ P ) > 0 (iii) If π( ) has downside risk, W π (σ P ) < 0 (iv) W π (σ P ) increasing in info frictions σ P (v) If π 1 ( ) has more downside (less upside) risk than π 2 ( ), W π2 (σ P ) W π1 (σ P ) increasing in σ P
Risk Types and Information Wedges Symmetric Risk P(z) V(z) 0 E[P(z)]=E[V(z)] z Expected wedge = 0 (as in CARA-normal)
Risk Types and Information Wedges Upside Risk P(z) V(z) E[P(z)] 0 E[V(z)] W(z) z Expected wedge > 0: Overpriced security (on expectation)
Risk Types and Information Wedges Downside Risk P(z) V(z) E[V(z)] 0 E[P(z)] W(z) z Expected wedge < 0: Underpriced security (on expectation)
Formal Results: Volatility Theorem (excess variability): For any payoff function π ( ) with symmetric, upside or downside risks, (i) E((π(θ) π(0)) 2 ) > E ( (V π (z) V π (0)) 2) (ii) E ( (P π (z) P π (0)) 2) > E ( (V π (z) V π (0)) 2) Prices more volatile than expected dividends (ii) E ( (P π (z) P π (0)) 2) > E ( (π (θ) π (0)) 2), if σ 2 u and/or β high enough Prices more volatile than realized dividends, in the absence of a SDF Compare with West (Ect, 1988): variability of posterior expectation < variability of realized dividends our model: change in identity delivers the extra volatility
Recap: Key Results thus far Parsimonious model of info aggregation Applies to arbitrary (monotone) payoff functions Tractability arises from risk-neutral setup with limited arbitrage Main result: Information aggregation wedge Prices overreact to market info due to identity effect Leads to average over/undervaluation (depending on curvature of π( )) Leads to excess volatility of prices
Applications
Application 1: M&M Dividend Split Irrelevance Suppose dividend is split in 2 and sold in separate markets π( ) = π 1 ( ) + π 2 ( ) π 1 ( ) has downside risk, π 2 ( ) has upside risk Market characteristics ( ) Informed traders active in one market only, observe x i,j N θ, β 1 j Noisy demands: ( ) (( u1 0 = N u 2 0 ) (, σu,1 2 ρσ u,1 σ u,2 ρσ u,1 σ u,2 σu,2 2 )) Consider informationally segmented markets Traders in mkt j don t observe price P j Results also hold in info connected markets (see paper) Market characterized fully by info frictions σ P,j
Application 1: M&M Dividend Split Irrelevance Proposition: (i) Seller s revenue is independent of split iff σ P,1 = σ P,2 Markets have identical information frictions (ii) Total expected revenue from π( ) maximized by following split: π1 (θ) = min {π (θ), π (0)}, and π 2 (θ) = max {π (θ) π (0), 0} Assign π 1 to investor pool with lower informational friction (σ P,1) Intuition π1 has more downs. risk than any other π 1, π2 has more ups. risk than any other π 2, Any alternative split {π 1, π 2 } transfers ups. risk from σ P,2 to σ P,1 investors...resulting in a net loss of revenue (lower overall wedge)
Splitting Cash Flows for Arbitrary π(θ) Total Dividend π(θ) π(0) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 > θ (st. dev.) Cash flow group 1 (downside) π(0) π (θ) 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 > θ (st. dev.) Cash flow group 2 (upside) π (θ) 2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 > θ (st. dev.) π 1 has max. downside risk; π 2 max. upside risk
Application 2: Dynamic Trading Dynamic extension: Dividend each period: π(θ t), θ t i.i.d. Traders infinitely lived, discount future at fixed rate δ (0, 1) Price and expected dividend satisfy recursive expression: P π (z t) = E (π (θ t) + δp π(z t+1) x = z t, z t) V π (z t) = E (π(θ t) + δv π(z t+1) z t) And so does the wedge: W π (z t) = w π (z t) + δe (W π (z)) = w π (z t) + δ E (wπ (z)), 1 δ where w π (z t) = E (π (θ t) x = z t, z t) E (π (θ t) z t)
Application 2: Dynamic Trading Proposition (Dynamic Wedge): Suppose that π ( ) is bounded below, increasing, and convex: For any σ P > σ θ, ˆδ < 1 s.t. δ > ˆδ, W (z t) > 0, for all z t. Dynamic model implies: Future expected wedges raise current share price (if π ( ) has upside risk) For high enough δ, share might always be overpriced
Application 3: Public Disclosures How does exogenous public info about θ affect wedge? Let y N(θ, α 1 ) be a public disclosure on θ Same eq. characterization as before, but with extra info ( P π (y, z) = π(θ)dφ σ 2 θ + α + β + βσu (θ 2 ( V π (y, z) = π(θ)dφ σ 2 θ + α + βσu (θ 2 σ 2 θ )) αy + (β + βσ 2 u )z + α + β + βσu 2 )) z + α + βσu 2 σ 2 θ αy + βσ 2 u Public info crowds out impact of z on price and expected dividend In the limit α, wedge dissapears But for finite levels of precision α, impacts are more subtle...
Application 3: Public Disclosures Proposition (Public Disclosures): Consider linear dividend π( ) (holds more generally) ( i) Var (V π (y, z)) increasing in α Standard Blackwell ( ii) For σ 2 u 2, Var (P π (y, z)) increasing in α; Otherwise, Var (P π (y, z)) increasing in α iff α α If noisy demand too volatile, α reduces price overreaction to z But for large enough α, price vol increasing (more responsive to θ) ( iii) Var (W (y, z)) is decreasing in α iff α α (and increasing otherwise), For low α, an increase reduces impact of z on V π (y, z) more than on P π (y, z) Larger wedge But for large enough α, both V π (y, z) and P π (y, z) hardly respond to z Wedge vanishes
Robustness
Robustness Alternative distributional assumptions: let θ on arbitrary smooth prior on R, x i iid cdf F ( θ) satisfying MLRP, Noisy demand D according to cdf G ( ) on [0, 1] Can always characterize wedge in this environment Price impact of information Let noisy demand be elastic: D(u, P) = Φ (u + η (E (π (θ) P) P)) Wedge is inversely related to elasticity η Noise traders arbitrage away the wedge Wedge in CARA-normal setup (noisy REE) Can only solve in the linear case: π ( ) = k > 0 Wedge has two components A constant reflecting discount (premium) for average shares held A symmetric information aggregation wedge Unconditional returns driven by the average compensation for risk
Conclusions Tractable noisy REE framework Heterogeneous beliefs, risk neutrality and limited arbitrage Useful to analyze more general payoff structures Key result: information aggregation wedge Prices overreact to market information Generates excess price/return volatility Generates over/undervaluation on average (depending on shape of payoffs) Applications in finance M&M capital structure irrelevance Excess volatility puzzle Impact of public disclosures