Asset Pricing under Asym. Information

Limits to

Historical Bubbles

Symmetric Asset Pricing under Asymmetric Information Information Pricing Equation Bubbles & Ruling out Asymmetric Information Expected/Strong Bubble Markus K. Brunnermeier Necessary Conditions Limits to Princeton University Arbitrage Risk Synchronization August 17, 2007 Risk Asset Pricing under Asym. Information

Limits to Overview Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation • All agents are rational Ruling out Asymmetric • Bubbles under symmetric information Information • Bubbles under asymmetric information Expected/Strong Bubble Necessary Conditions • Interaction between rational arbitrageurs and behavioral Limits to traders - Limits to Arbitrage Arbitrage Noise Trader • Fundamental risk Risk Synchronization • Risk Noise trader risk + Endogenous horizons of arbs • Synchronization risk Asset Pricing under Asym. Information

Limits to Historical Bubbles Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out • 1634-1637 Dutch Tulip Mania (Netherlands) Asymmetric Information • Expected/Strong 1719-1720 Mississippi Bubble (France) Bubble Necessary • Conditions 1720 South Sea Bubble (England) Limits to • 1990 Japan Bubble Arbitrage Noise Trader Risk • 1999 Internet/Technology Bubble Synchronization Risk Asset Pricing under Asym. Information

Limits to A Technology Company Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation • Company X introduced a revolutionary wireless Ruling out communication technology. Asymmetric Information • It not only provided support for such a technology but also Expected/Strong Bubble Necessary provided the informational content itself. Conditions Limits to • It’s IPO price was $1.50 per share. Six years later it was Arbitrage Noise Trader traded at $ 85.50 and in the seventh year it hit $ 114.00. Risk Synchronization • Risk The P/E ratio got as high as 73. • The company never paid . Asset Pricing under Asym. Information

Limits to The Story of RCA in 1920’s Arbitrage Company: Radio Corporate of America (RCA) Historical Technology: Radio Bubbles Years: 1920s Symmetric Information Pricing Equation Ruling out Asymmetric Information Expected/Strong Bubble Necessary Conditions Limits to Arbitrage Noise Trader Risk Synchronization Risk

Figure: RCA’s Price from Dec 25 to Dec 50.

• RCA peaked at $ 397 in Feb. 1929, down to $ 2.62 in May 1932

• RCA’s stock price was below $ 25 at least until 1950 Asset Pricing under Asym. Information

Limits to NASDAQ and “Neuer Markt” Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out Asymmetric Information Expected/Strong Bubble Necessary Conditions Limits to Arbitrage Noise Trader Risk Synchronization Risk Figure: NASDAQ and Neuer Markt during “Technology Bubble”. Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage

Historical Bubbles Symmetric • Keynes’ distinction between and -run Information Pricing Equation investment: Ruling out Asymmetric • Speculation: Buy “overvalued” in the hope to sell it to Information someone else at an even higher price Expected/Strong Bubble • Investing: strategy Necessary Conditions Limits to • Fundamental value: Was ist das? Arbitrage Noise Trader “highest WTP” if one forces agents to buy & hold the Risk Synchronization asset Risk no uncertainty: discounted value of dividends uncertainty w/ risk-neutral agent: expected discounted value uncertainty w/ risk-averse agents: take expectations w.r.t. EMM Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out • Problem of Keynes’ buy and hold definition of Asymmetric Information fundamental value: Expected/Strong Bubble • Retrade does also occur to dynamically complete the Necessary Conditions market (not only for speculation). Limits to • With retrade a different allocation can be achieved and Arbitrage Noise Trader hence the EMM is different. Risk Synchronization • Allow for retrade and take EMM which leads to highest Risk fundamental value. Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage

∗ Historical • with stochastic discount factor mt (or pricing kernel m ) Bubbles t the price of an asset is given by Symmetric Information Pricing Equation Ruling out mt pt = Et [mt+1 (pt+1 + dt+1)] Asymmetric Information where m is related to MRS (divided by prob. of state) Expected/Strong t+1 Bubble Necessary • Alternatively one can also write pricing equation in terms Conditions Limits to of the equivalent martingale measure Arbitrage Noise Trader   Risk Qˆ 1 Synchronization p = E (p + d ) Risk t t f t+1 t+1 1 + rt • Securities with Finite Maturity • Reiterate pricing equation • Backwards induction rules out bubbles Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage • Historical Securities with Infinite Maturity Bubbles • Backwards induction argument fails since there is no well Symmetric defined final period Information Pricing Equation • “Lack of market clearing at t = ∞” Ruling out f • Split the price in a fundamental component pt and a Asymmetric Information bubble component bt . Expected/Strong Bubble • By pricing equation, we get the following expectational Necessary Conditions difference equation Limits to   ˆ 1 Arbitrage Q bt = Et bt+1 Noise Trader f Risk 1 + rt Synchronization Risk • Example 1: deterministic bubble ⇒ has to grow at the risk-free rate • Example 2 (Blanchard & Watson 1982): (risk-neutral ) • bubble bursts in each period with prob. (1 − π), persists with prob. π f 1+rt • ⇒ bubble has to grow by a factor π (if it doesn’t burst) Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage • Historical How can we rule out bubbles? Bubbles • Negative bubbles (Blanchard & Watson 1982, Diba & Symmetric Grossman 1988) Information • For b < 0 difference equation implies that p will become Pricing Equation t t Ruling out negative. Asymmetric • Free disposal rules out negative prices. Information Expected/Strong • Positive bubbles on assets with positive net supply if g < r Bubble Necessary (Brock, Scheinkman, Tirole 85, Santos & Woodford 97) Conditions • Argument: (bubbles would outgrow the economy if r > g) Limits to Arbitrage • At any point in time t + τ, the aggregate wealth of the Noise Trader economy contains bubble component b . Risk τ Synchronization • NPVt of aggregate wealth Wt+τ does not converge to Risk zero as τ → ∞ • If aggregate consumptiont+τ is bounded or grows at a rate g < r, NPVt+τ (Ct+τ ) → 0 as τ → ∞. • Household wealth exceeds PV of C for all t + τ sufficiently far in the future. • This is inconsistent with optimization since household would consume part of wealth. Asset Pricing under Asym. Information

Limits to Bubbles under Symmetric Information Arbitrage

Historical Bubbles

Symmetric Information • 5 Counter Examples (Santos and Woodford (1997)): Pricing Equation Ruling out • Example 1: fiat money (=bubble) in OLG models Asymmetric • allows (better) intergeneral transfers Information Expected/Strong • without bubble households want to save more and hence Bubble Necessary MRS “implicit r”< g Conditions (can lead to overaccumulation of private capital and Limits to Arbitrage hence, dynamic inefficiency (see also Abel et al. (1989))) Noise Trader Risk • Example 2: ... Synchronization Risk • Common theme: Pure existence of a bubble enlarges the trading space. leads to different allocation and EMM Asset Pricing under Asym. Information

Limits to Bubbles under Asymmetric Information Arbitrage

Historical Bubbles

Symmetric Information • “Dynamic Knowledge Operator” Pricing Equation Ruling out Asymmetric i n dynamic t o Information Kt (E) = ω ∈ Ω : Pi (ω) ⊆ E Expected/Strong Bubble Necessary Conditions • Expected Bubbles versus Strong Bubbles Limits to Arbitrage • expected bubble: Noise Trader Risk pt > every agents marginal valuation at a date state (t, ω) Synchronization Risk • strong bubble: (arbitrage?) pt > all agents know that no possible realization can justify this price. Asset Pricing under Asym. Information

Limits to Necessary Conditions for Bubbles Arbitrage

Historical Bubbles Symmetric • Model setup in Allen, Morris & Postlewaite 1993: Information Pricing Equation risky asset pays dividend dT (ω):Ω 7−→ + at t = T Ruling out R Asymmetric • Necessary Conditions for Expected Bubbles Information Expected/Strong 1 Initial allocation (interim) Pareto inefficient (Tirole 1982) Bubble Necessary Conditions • rational traders is not willing to buy “bubble asset” since Limits to some traders have realized their gains leaving a negative Arbitrage sum game for the buyers Noise Trader Risk Synchronization 2 Short-sale constraint strictly binds at some future time in Risk some contingency for all i • only don’t sell to the limit now, since shorting might be more profitable in the future Asset Pricing under Asym. Information

Limits to Necessary Conditions for Bubbles Arbitrage

Historical Bubbles • Additional Necessary Conditions for Strong Bubbles Symmetric Information 1 asymmetric information is necessary since Pricing Equation Ruling out traders must believe that the other traders do not know Asymmetric this fact. Information Expected/Strong 2 Net trades of all traders cannot be CK (since CK of Bubble Necessary actions negates asymmetric info about events) Conditions ⇒ no bubbles in economies with only two types of traders. Limits to Arbitrage • Noise Trader Morris, Postelwaite & Shin (1995)-Model setup Risk Synchronization • Risk now, all agents are risk-neutral i  i  • pT = dT and pt = maxi Et pt+1|Pt for all ω ∈ Ω and t = 1, ..., T . • Let’s focus on ω, where dT = 0, dT =0 ET : {ω ∈ Ω|dT (ω) = 0} Asset Pricing under Asym. Information

Limits to Necessary Conditions for Bubbles Arbitrage

Historical Bubbles • Main Result: Strong bubble can be ruled out at time t if Symmetric   G G G dT =0 Information K K ···K E = {ω ∈ Ω|pt (ω) = 0} Pricing Equation t t+1 T −1 T Ruling out Asymmetric • (That is, it is mutual knowledge in t that in period t + 1 it Information will be mutual knowsledge that ... in (T − 1) it will be Expected/Strong Bubble mutual knowledge that dT = 0.) Necessary Conditions • Sketch argument: Limits to if it is mutual knowledge at T − 1 that d = 0, then Arbitrage T Noise Trader pT −1 = 0. Risk Synchronization • if it is mutual knowledge at T − 2 that pT −1 = 0, then Risk pT −2 = 0. • ... • Since knowledge can only improve over time. If it is at t already (T − t)-mutual knowledge that dT = 0, pt = 0. Asset Pricing under Asym. Information

Limits to Limits to Arbitrage - Overview Arbitrage

Historical Bubbles • Efficient Market Hypothesis - 3 levels of justifications

Symmetric • All traders are rational, since behavioral will not survive in Information Pricing Equation the long-run (their wealth declines) Ruling out • Behavioral trades cancel each other on average Asymmetric • Rational arbitrageurs correct all mispricing induced by Information Expected/Strong behavioral traders. Bubble Necessary Conditions • Fama/Friedman contra Keynes Limits to • “If there are many sophisticated traders in the market, Arbitrage Noise Trader they may cause these “bubbles” to burst before they really Risk Synchronization get under way.” (Fama 1965) Risk • “It might have been supposed that competition between expert professionals, possessing judgment and knowledge beyond that of the average private , would correct the vagaries of the ignorant individual left to himself.” (Keynes 1936) Asset Pricing under Asym. Information

Limits to Limits to Arbitrage - Overview Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out Asymmetric • Reasons for limits to arbitrage Information Expected/Strong • Fundamental risk Bubble Necessary • Noise trader risk (DSSW 1990a, Shleifer & Vishny 1997) Conditions • Synchronization risk (Abreu & Brunnermeier 2002, 2003) Limits to Arbitrage Noise Trader • Special case of market frictions (incl. liquidity) Risk Synchronization Risk Asset Pricing under Asym. Information

Limits to Noise Trader Risk Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out • Idea: Arbitrageurs do not fully correct the mispricing Asymmetric Information caused by noise traders due to Expected/Strong Bubble • arbitrageurs short horizons Necessary Conditions • arbitrageurs risk aversion (face noise trader risk) Limits to Arbitrage • Noise traders survive in the long-run Noise Trader Risk Synchronization (they are not driven out of the market.) Risk Asset Pricing under Asym. Information

Limits to Noise Trader Risk - DSSW 1990a Arbitrage • Model Setup of DSSW 1990a Historical • Bubbles OLG model • agents live for 2 periods Symmetric Information • make portfolio decisions when they are young Pricing Equation • 2 assets Ruling out Asymmetric Information safe asset s pays fixed real dividend r Expected/Strong perfect elastic supply Bubble numeraire, i.e. p = 1 Necessary • s Conditions unsafe asset u pays fixed real dividend r Limits to sup Arbitrage no elastic supply of X = 1 Noise Trader price at t = pt Risk Synchronization • Fundamental value of s = Fundamental value of u Risk (perfect substitutes) • agents • mass of (1 − µ) of arbitrageurs • mass of µ of noise traders, who misperceive next period’s ` ∗ 2 ´ ∗ price by ρt ∼ N ρt , σρ ,(ρ measures bullishness) • CARA utility function U (W ) = − exp {−2γW } with certainty equivalent E [W ] − γVar [W ]. Asset Pricing under Asym. Information

Limits to Noise Trader Risk - DSSW 1990a Arbitrage

Historical • Individual demand Bubbles • arbitrageur’s Symmetric Information E [W ] − γVar [W ] = Pricing Equation a a 2 Ruling out c0 + xt [r + Et [pt+1] − pt (1 + r)] − γ (xt ) Vart [pt+1] Asymmetric • noise traders Information E [W ] − γVar [W ] = Expected/Strong Bubble n a 2 Necessary c0 + xt [r + Et [pt+1] + ρt − pt (1 + r)] − γ (xt ) Vart [pt+1] Conditions • Taking FOC Limits to a r+Et [pt+1]−(1+r)pt Arbitrage arbitrageurs: x = t 2γVart [pt+1] Noise Trader n r+Et [pt+1]−(1+r)pt ρt Risk noise traders: x = + Synchronization t 2γVart [pt+1] 2γVart [pt+1] Risk • Market Clearing a n (1 − µ) xt + µxt = 1 1 p = [r + E [p ] − 2γVar [p ] + µρ ] t 1 + r t t+1 t t+1 t Asset Pricing under Asym. Information

Limits to Noise Trader Risk - DSSW 1990a Arbitrage Solve recursively, Historical Bubbles 1 p = [r + E [p ] − 2γVar [p ] + µρ ] Symmetric t+1 t+1 t+2 t+1 t+2 t+1 Information 1 + r Pricing Equation 1 ∗ Ruling out Et [pt+1] = [r + Et [pt+2] − 2γVart [pt+2] + µρ ] Asymmetric 1 + r Information Expected/Strong Bubble we will see later that Vart [pt+τ ] is a constant for all τ. Necessary Conditions Solve first order difference equation Limits to ∗ ∗ Arbitrage µ (ρt − ρ ) µρ 2γ Noise Trader p = 1 + + − Var [p ] Risk t t t+1 Synchronization 1 + r r r Risk Note that ρt is the only random variable. Hence, µ2σ2 Var [p ] = Var [p ] = ρ t t+1 t+1 (1+r)2 µ (ρ − ρ∗) µρ∗ (2γ) µ2σ2 p = 1 + t + − ρ t 1 + r r r (1 + r) Asset Pricing under Asym. Information

Limits to Noise Trader Risk - DSSW 1990a Arbitrage ∗ ∗ 2 2 Historical µ (ρt − ρ ) µρ (2γ) µ σρ Bubbles p = 1 + + − t 1 + r r r (1 + r) Symmetric Information where Pricing Equation Ruling out • 1 = fundamental value ∗ Asymmetric µ(ρt −ρ ) Information • 1+r = deviation due to current misperception of noise Expected/Strong Bubble traders Necessary ∗ Conditions µρ • r = average misperception of noise traders Limits to (2γ)µ2σ2 Arbitrage • − ρ = arbitrageurs’ risk-premium Noise Trader r(1+r) Risk Synchronization Risk • Homework: 1 Check limiting cases 1 γ → 0 2 2 σρ → 0 2 Check whether there is also a fundamental equilibrium, where pt = 1 for all t (no risk ⇒ arbitrageurs buy everything) Asset Pricing under Asym. Information

Limits to Do Noise Traders Die Out over Arbitrage Time (Evolutionary Argument) Historical Bubbles • Relative Expected Returns Symmetric Information • Difference in returns Pricing Equation n a Ruling out • ∆Rn−a = (xt − xt )[r + pt+1 − pt (1 + r)] 2 Asymmetric n a ρt (1+r) ρt Information • Aside 1: (xt − xt ) = = 2 2 (2γ)Vart [pt+1] (2γ)µ σρ Expected/Strong n a Bubble (Note for µ → 0, (xt − xt ) → ∞) Necessary Conditions • Aside 2: By market clearing E [r + pt+1 − pt (1 + r)] = 2 2 Limits to (2γ)µ σρ Arbitrage (2γ) Vart [pt+1] − µρt = 2 − µρt Noise Trader (1+r) Risk Synchronization 2 2 Risk (1 + r) (ρt ) ⇒ Et [∆Rn−a] = ρt − 2 2 (2γ) µ σρ • Taking unconditional expectations 2 ∗ 2 2 ∗ (1 + r) ρ + (1 + r) σρ ∗ E [∆Rn−a] = ρ − 2 2 > 0 only if ρ > 0 (2γ) µ σρ Asset Pricing under Asym. Information

Limits to Do Noise Traders Survive over Arbitrage Time (Evolutionary Argument) Historical Bubbles

Symmetric Information Pricing Equation Ruling out • Taking unconditional expectations Asymmetric Information 2 ∗ 2 2 Expected/Strong (1 + r) ρ + (1 + r) σ Bubble ∗ ρ ∗ Necessary E [∆Rn−a] = ρ − 2 2 > 0 only if ρ > 0 Conditions (2γ) µ σρ Limits to Arbitrage Noise Trader • “Overoptimistic/bullish” traders hold riskier positions and Risk Synchronization have higher expected returns. Risk • In evolutionary process, they will have more off-springs and hence they won’t die out. Asset Pricing under Asym. Information

Limits to Myopia due to Liquidation Risk Arbitrage • Why are professional arbitrageurs myopic? Historical Bubbles • Model setup of Shleifer & Vishny (1997) [slightly Symmetric modified] Information Pricing Equation • Two assets Ruling out • risk free bond and Asymmetric Information • risky stock with final value v Expected/Strong Bubble • Two types of fund managers: Necessary Conditions • Good fund managers know fundamental value v Limits to • Bad fund managers have no additional information Arbitrage Noise Trader (just gamble with “other people’s money”). Risk Synchronization Risk • Two trading rounds t = 1 and 2 (in t = 3 v is paid out) • Individual investors

• entrust their money F1 to a fund manager without knowing the fund managers’ skill level - “separation of brain and money” • can withdraw their funds in t = 2 • Noise traders submit random demand Asset Pricing under Asym. Information

Limits to Myopia due to Liquidation Risk Arbitrage • Historical Price setting: Bubbles • P3 = v Symmetric Information • P2 is determined by aggregate demand of fund managers Pricing Equation and liquidity/noise traders Ruling out Asymmetric • Focus on case where Information Expected/Strong Bubble 1 P1 < v (asset is undervalued) Necessary Conditions 2 P2 < P1asset price goes even further down in t2 due to Limits to • sell order by noise traders Arbitrage Noise Trader • sell order by other informed traders Risk Synchronization Risk • Performanced-based fund flows (see Chevalier & Ellison 1997) • If price drops, the probability increases that the fund manager is “bad”. • Individual investors withdraw their money at t = 2.   P2 • Shleifer & Vishny assume F2 = F1 − aD1 1 − , where P1 D1 is the amount the fund manager invested in the stock. Asset Pricing under Asym. Information

Limits to Myopia due to Liquidation Risk Arbitrage

Historical Bubbles

Symmetric Information • Pricing Equation “Good” manager’s problem who has invested in risky asset Ruling out • He has to liquidate his position at P < P (exactly when Asymmetric 2 1 Information mispricing is the largest!) Expected/Strong Bubble That is, he makes losses, even though the asset was Necessary Conditions initially undervalued. Limits to • Due to this “early liquidation risk”, at t = 1 a rational Arbitrage Noise Trader fund manager is reluctant to fully exploit arbitrage Risk Synchronization opportunities at t = 1. Risk • Focus on short-run price movements ⇒ myopia of professional arbitrageurs Asset Pricing under Asym. Information

Limits to Synchronization Risk Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation Ruling out • “Bubbles and Crashes” (Abreu & Brunnermeier 2003) Asymmetric Information (for bubbles) Expected/Strong Bubble Necessary • “Synchronization Risk and Delayed Arbitrage” (Abreu & Conditions Brunnermeier 2002) Limits to Arbitrage (for any form of mispricing) Noise Trader Risk Synchronization • see power point slides (file “08 slides Eco525.ppt”) Risk