Asset Pricing under Asym. Information Limits to Arbitrage Historical Bubbles Symmetric Asset Pricing under Asymmetric Information Information Pricing Equation Bubbles & Limits to Arbitrage Ruling out Asymmetric Information Expected/Strong Bubble Markus K. Brunnermeier Necessary Conditions Limits to Princeton University Arbitrage Noise Trader 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 short 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 dividends. 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 Stock 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 speculation and long-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: Buy and hold 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 investors) • 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 dividend 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 position 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)