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 OLG Models Asymmetric Information Expected/Strong Markus K. Brunnermeier Bubble Necessary Conditions Princeton University Limits to Arbitrage Noise Trader Risk December 24, 2014 Synchronization Risk Policy Response Asset Pricing under Asym. Information Overview Limits to Arbitrage

Historical Bubbles

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

Historical Bubbles

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

Limits to Event Tulipmania Crisis of 1763 Crisis of 1772 Arbitrage Time 1634-1637 1763 1772-1773 Historical Place Netherlands Amsterdam, England, Bubbles Hamburg, Scotland Symmetric Information Berlin Pricing Equation Ruling out Bubble asset Tulips Grains, sugar Stocks and OLG Models futures of Asymmetric Information East India Expected/Strong Bubble Necessary Company, Conditions turnpikes, Limits to Arbitrage canals, en- Noise Trader Risk closures, Synchronization Risk building Policy Response construction Asset Pricing Type of bub- Commodity Commodity Securities, under Asym. ble asset real estate Information

Limits to Holder of as- Small-town Merchant London spec- Arbitrage set dealers, bankers ulators, busi-

Historical tavern- ness men Bubbles keepers, Symmetric Information horticultural- Pricing Equation Ruling out ists OLG Models Financier of Sellers of Amsterdam Ayr Bank, Asymmetric Information asset bulbs investors country Expected/Strong Bubble banks Necessary Conditions Expansive No No Yes Limits to Arbitrage monetary Noise Trader Risk policy Synchronization Risk Policy Response Asset Pricing Lending No Yes Yes under Asym. boom Information

Limits to Foreign capi- No No No Arbitrage tal inflows

Historical Financial No No Yes Bubbles deregulation Symmetric Information Severe reces- No No Yes Pricing Equation Ruling out sion OLG Models Banking crisis No Yes Yes Asymmetric Information Spillover No Yes Yes Expected/Strong Bubble to other Necessary Conditions countries Limits to Arbitrage Leaning No No Yes Noise Trader Risk Pricking No No No Synchronization Risk Use of No No No Policy Response quantity instruments Asset Pricing under Asym. Information

Limits to Arbitrage

Historical Only cleaning No Yes No Bubbles

Symmetric Sources Garber Kindleberger Hamilton Information (1989, 1990), (2005), Schn- (1956), Hop- Pricing Equation Ruling out Kindleberger abel & Shin pit (1986), OLG Models Asymmetric (2005) (2004) Kindleberger Information Expected/Strong (2005), Bubble Necessary Sheridan Conditions (1960) Limits to Arbitrage Noise Trader Risk Synchronization Risk Policy Response Asset Pricing under Asym. Information

Limits to Event Latin Amer- Railway ma- Panic of 1857 Arbitrage ica Mania nia

Historical Time 1824-1825 1840s 1856-1857 Bubbles Place England England United States Symmetric Information Bubble asset Securities Railway re- Railroad Pricing Equation Ruling out of Real lated stocks stocks and OLG Models and imagi- and property bonds, land Asymmetric Information nary South classes, corn Expected/Strong Bubble Necessary American Conditions governments Limits to Arbitrage and mines, Noise Trader Risk joint stock Synchronization Risk companies Policy Response Type of bub- Securities, Securities, Securities, ble asset commodity commodity real estate Asset Pricing Holder of as- Widely held Widely held under Asym. set Information

Limits to Financier of Country Bank of Eng- Banks, for- Arbitrage asset banks, Bank land, govern- eign investors

Historical of England ment Bubbles Expansive Yes Yes Yes Symmetric Information monetary Pricing Equation Ruling out policy OLG Models Lending Yes Yes Yes Asymmetric Information boom Expected/Strong Bubble Foreign capi- No Yes Yes Necessary Conditions tal inflows Limits to Arbitrage Financial No No No Noise Trader Risk deregulation Synchronization Risk Severe reces- Yes Yes Yes Policy Response sion Asset Pricing under Asym. Banking crisis Yes Yes Yes Information Spillover Yes No Yes Limits to Arbitrage to other countries Historical Bubbles Leaning Yes No No Symmetric Pricking Possibly No No Information Pricing Equation Use of No No No Ruling out OLG Models quantity Asymmetric Information instruments Expected/Strong Bubble Only cleaning Yes Yes Yes Necessary Conditions Sources Bordo Evans Calomiris & Limits to Arbitrage (1998), Co- (1849), Schweikhart Noise Trader Risk nant (1915), Kindleberger (1991), Synchronization Risk Kindleberger (2005), Kindleberger Policy (2005), Neal Ward-Perkins (2005) Response (1995) (1950), WEO (2003) Asset Pricing under Asym. Information

Limits to Event Gr¨underkrise Chicago real Crisis of 1882 Arbitrage estate boom

Historical Time 1872-1873 1881-1883 1881-1882 Bubbles Place Germany, Chicago France Symmetric Information Austria Pricing Equation Ruling out Bubble asset Stocks, New-built Stocks of new OLG Models railroads, apartments, banks Asymmetric Information houses, land houses from Expected/Strong Bubble Necessary property foreclosure Conditions proceedings, Limits to Arbitrage land Noise Trader Risk Synchronization Risk Policy Response Asset Pricing Type of bub- Securities, Real estate Securities under Asym. ble asset real estate Information

Limits to Holder of as- Widely held Widely held Arbitrage set

Historical Financier of Banks Households Banks, Bubbles asset caisses de Symmetric Information reports, Pricing Equation Ruling out individuals OLG Models Expansive Yes Yes No Asymmetric Information monetary Expected/Strong Bubble policy Necessary Conditions Lending Yes No Yes Limits to Arbitrage boom Noise Trader Risk Foreign capi- Yes No Yes Synchronization Risk tal inflows Policy Response Asset Pricing Financial Yes No No under Asym. Information deregulation Limits to Severe reces- Yes No Yes Arbitrage sion Historical Banking crisis Yes No Yes Bubbles

Symmetric Spillover Yes No No Information to other Pricing Equation Ruling out countries OLG Models Asymmetric Leaning Yes No No Information Expected/Strong Pricking No No No Bubble Necessary Use of No No No Conditions Limits to quantity Arbitrage instruments Noise Trader Risk Synchronization Only cleaning No No Yes Risk Sources Burhop Hoyt (1933) Conant Policy Response (2009), Co- (1915), nant (1915), Kindleberger McCartney (2005), (1935) White (2007) Asset Pricing under Asym. Information Event Panic of 1893 Norwegian Real estate Limits to Arbitrage crisis bubble in the US Historical Bubbles Time 1890-1893 1895-1905 1920-1926 Symmetric Place Australia Norway United States Information Pricing Equation Bubble asset Mining Land prop- Residential Ruling out OLG Models shares, land erty, new housing, Asymmetric Information property homes, real securitization Expected/Strong Bubble estate shares Necessary Conditions Type of bub- Securities, Real estate Real estate Limits to ble asset real estate Arbitrage Noise Trader Holder of as- Borrowers, Construction Households, Risk Synchronization Risk set banks, for- sector, man- banks Policy eign investors ufacturers, Response brokers, stock market investors Financier of Pastoral Commercial Non- asset companies, banks institutional building so- lending, sav- cieties, land ings & loans, mortgage mutual sav- companies, ings banks, trading banks commer- cial banks, insurance banks Asset Pricing Expansive Yes Yes Yes under Asym. monetary Information

Limits to policy Arbitrage Lending Yes Yes Yes

Historical boom Bubbles Foreign capi- Yes Yes No Symmetric Information tal inflows Pricing Equation Ruling out Financial No No Yes OLG Models deregulation Asymmetric Information Severe reces- Yes No No Expected/Strong Bubble sion Necessary Conditions Banking crisis Yes Yes Yes Limits to Arbitrage Spillover Yes No No Noise Trader Risk to other Synchronization Risk countries Policy Response Asset Pricing under Asym. Information

Limits to Arbitrage Leaning No No Yes Pricking No No No Historical Bubbles Use of No No Yes Symmetric quantity Information Pricing Equation instruments Ruling out OLG Models Only cleaning Yes Yes No Asymmetric Information Sources Conant Gerdrup White (2009) Expected/Strong Bubble (1915), (2003) Necessary Conditions Lauck Limits to (1907), Arbitrage Noise Trader Merrett Risk Synchronization Risk (1997) Policy Response Asset Pricing under Asym. Information Event German stock US stock Lost decade Limits to Arbitrage price bubble price bubble Time 1927 1928-1929 1985-2003 Historical Bubbles Place Germany United States Japan Symmetric Bubble asset Stocks Stocks, real Stocks, real Information Pricing Equation estate estate Ruling out OLG Models Type of bub- Securities Securities, Securities, Asymmetric Information ble asset Real estate Real estate Expected/Strong Bubble Holder of as- Wealthy Widely held Widely held Necessary Conditions set individuals, Limits to institutional Arbitrage Noise Trader investors, Risk Synchronization Risk banks Policy Financier of Banks, for- Domestic Trusts, banks Response asset eign investors banks, later private in- vestors, corporation and foreign banks Asset Pricing Expansive No No Yes under Asym. monetary Information

Limits to policy Arbitrage Lending No Yes Yes

Historical boom Bubbles Foreign capi- Yes Yes No Symmetric Information tal inflows Pricing Equation Ruling out Financial No No No OLG Models deregulation Asymmetric Information Severe reces- Yes Yes Yes Expected/Strong Bubble sion Necessary Conditions Banking crisis Yes Yes Yes Limits to Arbitrage Spillover No Yes No Noise Trader Risk to other Synchronization Risk countries Policy Response Asset Pricing under Asym. Information Limits to Leaning Yes Yes Yes Arbitrage Pricking Yes Yes Yes Historical Bubbles Use of Yes No Yes

Symmetric quantity Information instruments Pricing Equation Ruling out Only cleaning No No No OLG Models Asymmetric Sources Voth (2003) Kindleberger Kaufman et Information Expected/Strong (2005), al. (2003), Bubble Necessary White (1990) Kindleberger Conditions Limits to (2005), Arbitrage Patrick Noise Trader Risk Synchronization (1998), Risk Posen (2003) Policy Response Asset Pricing under Asym. Information

Limits to Event Scandinavian Scandinavian Scandinavian Arbitrage crisis: Nor- crisis: Swe- crisis: Fin-

Historical way den land Bubbles Time 1988-1992 1990-1992 1991-1992 Symmetric Information Place Norway Sweden Finland Pricing Equation Ruling out Bubble asset Commercial Commercial Land, OLG Models real estate, real estate dwellings Asymmetric Information housing Expected/Strong Bubble Necessary Type of bub- Real estate Real estate Securities, Conditions ble asset Real estate Limits to Arbitrage Noise Trader Risk Synchronization Risk Policy Response Asset Pricing Holder of as- Firms, house- Households, under Asym. set holds business Information

Limits to Financier of Banks Banks, Banks Arbitrage asset finance com-

Historical panies Bubbles Expansive No No No Symmetric Information monetary Pricing Equation Ruling out policy OLG Models Lending Yes Yes Yes Asymmetric Information boom Expected/Strong Bubble Foreign capi- Yes No Yes Necessary Conditions tal inflows Limits to Arbitrage Financial Yes Yes Yes Noise Trader Risk deregulation Synchronization Risk Severe reces- Yes Yes Yes Policy Response sion Asset Pricing under Asym. Information Banking crisis Yes Yes Yes Limits to Arbitrage Spillover No No No

Historical to other Bubbles countries Symmetric Information Leaning No Yes Yes Pricing Equation Ruling out Pricking No No No OLG Models Use of No Yes Yes Asymmetric Information quantity Expected/Strong Bubble instruments Necessary Conditions Only cleaning Yes No No Limits to Arbitrage Sources Gerdrup Englund Nyberg Noise Trader Risk (2003), Vale (1999), Her- (1994) Synchronization Risk (2004) ring, Wachter Policy Response (1998) Asset Pricing under Asym. Information

Limits to Event Asian crisis Dotcom bub- Real estate Arbitrage ble bubble in

Historical Australia Bubbles Time 1997-1998 1995-2000 2002-2004 Symmetric Information Place Thailand United States Australia Pricing Equation Ruling out Bubble asset Stocks, hous- New technol- Dwelling OLG Models ing, commer- ogy company Asymmetric Information cial real es- stocks Expected/Strong Bubble Necessary tate Conditions Type of bub- Securities, Securities Real estate Limits to Arbitrage ble asset real estate Noise Trader Risk Holder of as- Households, Households Synchronization Risk set retail in- Policy Response vestors Asset Pricing Financier of Finance and Venture capi- Banks, mort- under Asym. asset securities talists gage origina- Information

Limits to companies, tors Arbitrage banks

Historical Expansive Yes Yes No Bubbles monetary Symmetric Information policy Pricing Equation Ruling out Lending Yes No Yes OLG Models boom Asymmetric Information Foreign capi- Yes Yes No Expected/Strong Bubble tal inflows Necessary Conditions Financial Yes Yes Yes Limits to Arbitrage deregulation Noise Trader Risk Severe reces- Yes No No Synchronization Risk sion Policy Response Asset Pricing under Asym. Information Banking crisis Yes No No

Limits to Spillover Yes No No Arbitrage to other Historical countries Bubbles Leaning No Yes Yes Symmetric Information Pricking No Yes No Pricing Equation Ruling out Use of No No Yes OLG Models Asymmetric quantity Information instruments Expected/Strong Bubble Necessary Only cleaning Yes No No Conditions Sources Collyns, Sen- BIS BIS 76th An- Limits to Arbitrage hadji (2002), 70th/71st nual Report, Noise Trader Risk Corsetti et al. Annual Bloxham, Synchronization Risk (1999) Report, Kent (2010) Policy Response Greenspan (2002) Asset Pricing under Asym. Event Subprime Spanish hous- Information

Limits to housing ing bubble Arbitrage bubble

Historical Time 2007− 2008− Bubbles Place United States Spain Symmetric Information Bubble asset Subprime Pricing Equation Ruling out mortgages OLG Models Asymmetric Type of bub- Real estate Real estate Information ble asset Expected/Strong Bubble Necessary Holder of as- Widely held Widely held Conditions set Limits to Arbitrage Noise Trader Risk Synchronization Risk Policy Response Asset Pricing Financier of Banks Banks, saving under Asym. asset banks Information

Limits to Expansive Yes Yes Arbitrage monetary

Historical policy Bubbles Lending Yes Yes Symmetric Information boom Pricing Equation Ruling out Foreign capi- Yes Yes OLG Models tal inflows Asymmetric Information Financial Yes No Expected/Strong Bubble deregulation Necessary Conditions Severe reces- Yes Yes Limits to Arbitrage sion Noise Trader Risk Banking crisis Yes Yes Synchronization Risk Spillover Yes No Policy Response to other countries Asset Pricing under Asym. Information

Limits to Arbitrage Leaning No Yes

Historical Pricking No No Bubbles Use of No Yes Symmetric Information quantity Pricing Equation Ruling out instruments OLG Models Only cleaning Yes No Asymmetric Information Sources Brunnermeier BoS (2012), Expected/Strong Bubble Necessary (2009), Gor- Carballo-Cruz Conditions ton and (2011) Limits to Arbitrage Metrick Noise Trader Risk (2012) Synchronization Risk Policy Response Asset Pricing under Asym. Information A Technology Company Limits to Arbitrage

Historical Bubbles

Symmetric Information • Company X introduced a revolutionary wireless Pricing Equation Ruling out communication technology. OLG Models Asymmetric • It not only provided support for such a technology but also Information Expected/Strong provided the informational content itself. Bubble Necessary • Conditions It’s IPO price was $1.50 per share. Six years later it was Limits to traded at $ 85.50 and in the seventh year it hit $ 114.00. Arbitrage Noise Trader • The P/E ratio got as high as 73. Risk Synchronization Risk • The company never paid dividends. Policy Response Asset Pricing under Asym. Information The Story of RCA in 1920’s Limits to Arbitrage

Historical Bubbles Symmetric Company: Radio Corporate of America (RCA) Information Pricing Equation Technology: Radio Ruling out OLG Models Years: 1920s Asymmetric Information Expected/Strong Bubble Figure : RCA’s Stock Price from Dec 25 to Dec 50. Necessary Conditions Limits to Arbitrage • RCA peaked at $ 397 in Feb. 1929, down to $ 2.62 in May 1932 Noise Trader Risk Synchronization • RCA’s stock price was below $ 25 at least until 1950 Risk Policy Response Asset Pricing under Asym. Information NASDAQ and “Neuer Markt” Limits to Arbitrage

Historical Bubbles

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

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

Historical Bubbles

Symmetric Information Pricing Equation • Problem of Keynes’ buy and hold definition of Ruling out OLG Models fundamental value: Asymmetric Information • Retrade does also occur to dynamically complete the Expected/Strong Bubble market (not only for speculation). Necessary Conditions • With retrade a different allocation can be achieved and Limits to hence the EMM is different. Arbitrage • Allow for retrade and take EMM which leads to highest Noise Trader Risk fundamental value. Synchronization Risk Policy Response Asset Pricing under Asym. Information Bubbles under Symmetric Information Limits to Arbitrage ∗ • with stochastic discount factor mt (or pricing kernel mt ) Historical Bubbles the price of an asset is given by Symmetric Information Pricing Equation mt pt = Et [mt+1 (pt+1 + dt+1)] Ruling out OLG Models Asymmetric where mt+1 is related to MRS (divided by prob. of state) Information Expected/Strong • Alternatively one can also write pricing equation in terms Bubble Necessary of the equivalent martingale measure Conditions Limits to   Arbitrage Qˆ 1 Noise Trader pt = Et f (pt+1 + dt+1) Risk 1 + r Synchronization t Risk Policy • Securities with Finite Maturity Response • Reiterate pricing equation • Backwards induction rules out bubbles Asset Pricing under Asym. Information Bubbles under Symmetric Information Limits to Arbitrage • Securities with Infinite Maturity Historical • Backwards induction argument fails since there is no well Bubbles defined final period Symmetric Information • “Lack of market clearing at t = ∞” Pricing Equation • Split the price in a fundamental component pf and a Ruling out t OLG Models bubble component bt . Asymmetric • By pricing equation, we get the following expectational Information Expected/Strong difference equation Bubble Necessary   Conditions Qˆ 1 Limits to bt = Et f bt+1 Arbitrage 1 + rt Noise Trader Risk • Example 1: deterministic bubble Synchronization Risk ⇒ has to grow at the risk-free rate Policy • Example 2 (Blanchard & Watson 1982): (risk-neutral Response 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 Bubbles under Symmetric Information Limits to Arbitrage • How can we rule out bubbles? Historical • Negative bubbles (Blanchard & Watson 1982, Diba & Bubbles Grossman 1988) Symmetric Information • For bt < 0 difference equation implies that pt will become Pricing Equation negative. Ruling out OLG Models • Free disposal rules out negative prices. Asymmetric • Positive bubbles on assets with positive net supply if g < r Information Expected/Strong (Brock, Scheinkman, Tirole 85, Santos & Woodford 97) Bubble Necessary • Argument: (bubbles would outgrow the economy if r > g) Conditions • At any point in time t + τ, the aggregate wealth of the Limits to Arbitrage economy contains bubble component bτ . Noise Trader Risk • NPVt of aggregate wealth Wt+τ does not converge to Synchronization Risk zero as τ → ∞ • Policy If aggregate consumptiont+τ is bounded or grows at a Response 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 Bubbles under Symmetric Information Limits to Arbitrage

Historical Bubbles • 5 Counter Examples (Santos and Woodford (1997)): Symmetric • Example 1: fiat money (=bubble) in OLG models Information Pricing Equation • allows (better) intergeneral transfers Ruling out OLG Models • without bubble households want to save more and hence Asymmetric MRS “implicit r”< g Information (can lead to overaccumulation of private capital and Expected/Strong Bubble hence, dynamic inefficiency (see also Abel et al. (1989))) Necessary Conditions • Geerolf (2014) overturns result with OECD data: Limits to sufficient conditions for dynamic efficiency are not Arbitrage satisfied (e.g. Japan is unambiguously inefficient) Noise Trader Risk • Synchronization Example 2: ... Risk • Common theme: Policy Response Pure existence of a bubble enlarges the trading space. leads to different allocation and EMM. Asset Pricing under Asym. Information Overlapping Generations Limits to Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation • Samuelson (1958) considers an infinite-horizon economy Ruling out OLG Models with two-period lived overlapping agents Asymmetric • Population growth rate = n Information Expected/Strong t t Bubble • Preferences given by u(ct , ct+1) Necessary u1 Conditions • Pareto optimal allocation satisfies = 1 + n u2 Limits to Arbitrage • OLG economies have multiple equilibria that can be Noise Trader Risk Pareto ranked Synchronization Risk Policy Response Asset Pricing under Asym. Information OLG: Multiple Equilibria Limits to Arbitrage

Historical • Assume: Bubbles

Symmetric t t t t Information u(ct , ct+1) = log ct + β log ct+1 Pricing Equation Ruling out t t OLG Models Endowment: yt = e, yt+1 = 1 − e Asymmetric Information • Assume complete markets and interest rate r Expected/Strong Bubble • Agents FOC implies: Necessary Conditions t Limits to ct+1 Arbitrage = 1 + r Noise Trader t Risk βct Synchronization Risk Policy • Response For r = n, this corresponds to the Pareto Solution • 1−e For r = βe − 1, agents will consume their endowment • Autarky solution is clearly Pareto inferior Asset Pricing under Asym. Information OLG: Completion with Durable Asset Limits to Arbitrage

Historical Bubbles

Symmetric • Autarky solution is the unique equilibrium implemented in Information a sequential exchange economy Pricing Equation Ruling out • Young agents cannot transfer wealth to the next period OLG Models Asymmetric • . . . relates to Chris Sims’s lecture Information Expected/Strong • A durable asset provides a store of value Bubble Necessary • Effective store of value reflects market liquidity Conditions • Pareto solution can be attained as a competitive Limits to Arbitrage equilibrium in which the price level grows at same rate as Noise Trader Risk the population, i.e. bt+1 = (1 + n)bt Synchronization Risk • Old agents trade durable asset for young agents’ Policy consumption goods Response Asset Pricing under Asym. Information OLG: Production Limits to Arbitrage

Historical • Diamond (1965) introduces a capital good and production Bubbles • Constant-returns-to-scale (CRS) production: Symmetric Information Pricing Equation Yt = F (Kt , Lt ) Ruling out OLG Models Asymmetric • Optimal level of capital is given by the golden rule, i.e. Information Expected/Strong 0 ? Bubble f (k ) = n Necessary Conditions Limits to Arbitrage Here, lowercase letters signify per capita values Noise Trader Risk • Individual (and firm) optimization implies that: Synchronization Risk Policy u1 0 Response = 1 + r = 1 + f (k) u2 It is possible that r < n ⇒ k > k? ⇒ Pareto inefficient Asset Pricing under Asym. Information OLG: Production & Efficiency Limits to Arbitrage

Historical Bubbles Symmetric • Diamond recommends issuing government debt at interest Information Pricing Equation rate r Ruling out OLG Models • Tirole (1985) introduces a rational bubble asset trading at Asymmetric Information price bt : Expected/Strong 1 + r Bubble t+1 Necessary bt+1 = bt Conditions 1 + n Limits to Arbitrage • Both solutions crowd out investment and increase r Noise Trader Risk • If baseline economy is inefficient, then an appropriately Synchronization Risk chosen debt issuance or bubble size can achieve Pareto Policy optimum with r = n Response Asset Pricing under Asym. Information OLG: Crowding-out vs. Crowding-in Limits to Arbitrage

Historical Bubbles

Symmetric Information Pricing Equation • Depending on the framework, government debt and Ruling out OLG Models presence of bubbles can have two opposite effects: Asymmetric Information 1 Crowding-out refers to the decreased investment to Expected/Strong increase in the supply of capital Bubble Necessary 2 Conditions Crowding-in refers to increased investment due to Limits to improved risk transfer Arbitrage Noise Trader • Woodford (1990) explores both of these effects Risk Synchronization Risk Policy Response Asset Pricing under Asym. Information Bubbles and Credit Frictions Limits to Arbitrage

Historical Bubbles • Symmetric Samuelson-Tirole model implications are hard to reconcile Information with the following stylized facts: Pricing Equation Ruling out 1 bubbles seem to pop up & burst (not deterministic) in reality OLG Models 2 bubbles are associated with consumption booms, as well as Asymmetric Information rapid expansions in capital stock and output Expected/Strong Bubble • Martin and Ventura (2012) address these shortcomings in Necessary Conditions an OLG framework by introducing: Limits to Arbitrage – investor sentiment shocks Noise Trader Risk – capital market imperfections Synchronization Risk • Takeaway: bubbles are not only reduce inefficient invest- Policy Response ments, but also increase efficient ones Asset Pricing under Asym. Information Bubbles and Credit Frictions: Model Limits to Arbitrage

Historical Bubbles

Symmetric Information • Risk-neutral individuals; utility: Uit = Et [cit+1] Pricing Equation Ruling out – each generation contains a measure 1 of individuals OLG Models – live for two periods & supply 1 unit of labour when young Asymmetric 1−α α Information • Technology: F (lt , kt ) = lt kt Expected/Strong Bubble – fraction  ∈ [0, 1] of productive individuals produce 1 unit of Necessary Conditions capital with one unit of output; unproductive produce δ < 1 Limits to units of capital with 1 unit of output Arbitrage Noise Trader Risk • Financial Friction: no borrowing allowed ⇒ unproductive Synchronization Risk investors have to make own investments Policy Response Asset Pricing under Asym. Information Bubbles and Credit Frictions: Model Limits to Arbitrage

Historical • Dynamics of capital stock in the presence of bubbles: Bubbles  Symmetric b + bP Information  α P t t  Ask + (1 − δ)b − δbt if < 1 Pricing Equation  t t (1 − )skα Ruling out k = t OLG Models t+1 P  α bt + bt Asymmetric  sk − bt if ≥ 1 Information  t (1 − )skα Expected/Strong t Bubble Necessary Conditions • Crowding-out: when old sell bubble to young, consumption Limits to Arbitrage grows and investment falls; bubble crowds out unproductive Noise Trader Risk investments first, then productive investments. Average in- Synchronization Risk vestment efficiency rises and crowding-out effect minimized. Policy Response • Crowding-in: when productive young sell bubble to unpro- ductive young, productive investments replace unproductive ones. This further raises average investment efficiency. Asset Pricing under Asym. Information Bubbles under Asymmetric Information Limits to Arbitrage

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

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

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

Historical Bubbles • Main Result: Strong bubble can be ruled out at time t if   Symmetric G G G dT =0 Information Kt Kt+1 ···KT −1 ET = {ω ∈ Ω|pt (ω) = 0} Pricing Equation Ruling out • (That is, it is mutual knowledge in t that in period t + 1 it OLG Models Asymmetric will be mutual knowsledge that ... in (T − 1) it will be Information mutual knowledge that dT = 0.) Expected/Strong Bubble • Sketch argument: Necessary Conditions if it is mutual knowledge at T − 1 that dT = 0, then Limits to pT −1 = 0. Arbitrage Noise Trader • if it is mutual knowledge at T − 2 that pT −1 = 0, then Risk Synchronization pT −2 = 0. Risk • ... Policy Response • 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 Arbitrage - Overview Limits to Arbitrage

Historical • Efficient Market Hypothesis - 3 levels of justifications Bubbles • All traders are rational, since behavioral will not survive in Symmetric Information the long-run (their wealth declines) Pricing Equation • Behavioral trades cancel each other on average Ruling out OLG Models • Rational arbitrageurs correct all mispricing induced by Asymmetric behavioral traders. Information Expected/Strong • Bubble Fama/Friedman contra Keynes Necessary Conditions • “If there are many sophisticated traders in the market, Limits to they may cause these “bubbles” to burst before they really Arbitrage Noise Trader get under way.” (Fama 1965) Risk • Synchronization “It might have been supposed that competition between Risk expert professionals, possessing judgment and knowledge Policy Response beyond that of the average private investor, would correct the vagaries of the ignorant individual left to himself.” (Keynes 1936) Asset Pricing under Asym. Information Limits to Arbitrage - Overview Limits to Arbitrage

Historical Bubbles

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

Historical Bubbles

Symmetric Information Pricing Equation Ruling out • Idea: Arbitrageurs do not fully correct the mispricing OLG Models caused by noise traders due to Asymmetric Information • arbitrageurs short horizons Expected/Strong Bubble • Necessary arbitrageurs risk aversion (face noise trader risk) Conditions Limits to • Noise traders survive in the long-run Arbitrage Noise Trader (they are not driven out of the market.) Risk Synchronization Risk Policy Response Asset Pricing under Asym. Information Noise Trader Risk - DSSW 1990a Limits to Arbitrage • Model Setup of DSSW 1990a

Historical • OLG model Bubbles • agents live for 2 periods Symmetric • make portfolio decisions when they are young Information • Pricing Equation 2 assets Ruling out OLG Models safe asset s pays fixed real dividend r Asymmetric perfect elastic supply Information Expected/Strong numeraire, i.e. ps = 1 Bubble • Necessary unsafe asset u pays fixed real dividend r Conditions no elastic supply of X sup = 1 Limits to Arbitrage price at t = pt Noise Trader • Fundamental value of s = Fundamental value of u Risk Synchronization (perfect substitutes) Risk • agents Policy Response • 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 Noise Trader Risk - DSSW 1990a Limits to Arbitrage • Individual demand Historical Bubbles • arbitrageur’s Symmetric E [W ] − γVar [W ] = Information a a 2 Pricing Equation c0 + xt [r + Et [pt+1] − pt (1 + r)] − γ (xt ) Vart [pt+1] Ruling out OLG Models • noise traders Asymmetric E [W ] − γVar [W ] = Information n a 2 Expected/Strong c0 + xt [r + Et [pt+1] + ρt − pt (1 + r)] − γ (xt ) Vart [pt+1] Bubble Necessary • Taking FOC Conditions arbitrageurs: x a = r+Et [pt+1]−(1+r)pt Limits to t 2γVart [pt+1] Arbitrage noise traders: x n = r+Et [pt+1]−(1+r)pt + ρt Noise Trader t 2γVart [pt+1] 2γVart [pt+1] Risk Synchronization Risk • Market Clearing a n Policy (1 − µ) xt + µxt = 1 Response 1 p = [r + E [p ] − 2γVar [p ] + µρ ] t 1 + r t t+1 t t+1 t Asset Pricing under Asym. Information Noise Trader Risk - DSSW 1990a Limits to Arbitrage Solve recursively,

Historical 1 Bubbles pt+1 = [r + Et+1 [pt+2] − 2γVart+1 [pt+2] + µρt+1] Symmetric 1 + r Information 1 ∗ Pricing Equation E [p ] = [r + E [p ] − 2γVar [p ] + µρ ] Ruling out t t+1 t t+2 t t+2 OLG Models 1 + r Asymmetric Information we will see later that Vart [pt+τ ] is a constant for all τ. Expected/Strong Bubble Solve first order difference equation Necessary Conditions ∗ ∗ Limits to µ (ρt − ρ ) µρ 2γ Arbitrage pt = 1 + + − Vart [pt+1] Noise Trader 1 + r r r Risk Synchronization Risk Note that ρt is the only random variable. Hence, Policy 2 2 µ σρ Response Var [p ] = Var [p ] = 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 Noise Trader Risk - DSSW 1990a Limits to Arbitrage ∗ ∗ 2 2 µ (ρt − ρ ) µρ (2γ) µ σρ Historical pt = 1 + + − Bubbles 1 + r r r (1 + r) Symmetric Information where Pricing Equation Ruling out • 1 = fundamental value ∗ OLG Models • µ(ρt −ρ ) = deviation due to current misperception of noise Asymmetric 1+r Information traders Expected/Strong ∗ Bubble • µρ = average misperception of noise traders Necessary r Conditions 2 2 (2γ)µ σρ Limits to • − = arbitrageurs’ risk-premium Arbitrage r(1+r) Noise Trader Risk • Homework: Synchronization Risk 1 Check limiting cases Policy 1 γ → 0 Response 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 Do Noise Traders Die Out over Limits to Arbitrage Time (Evolutionary Argument) Historical • Relative Expected Returns Bubbles

Symmetric • Difference in returns Information n a Pricing Equation • ∆Rn−a = (xt − xt )[r + pt+1 − pt (1 + r)] Ruling out (1+r)2ρ OLG Models n a ρt t • Aside 1: (xt − xt ) = = 2 2 (2γ)Vart [pt+1] (2γ)µ σρ Asymmetric n a Information (Note for µ → 0, (xt − xt ) → ∞) Expected/Strong Bubble • Aside 2: By market clearing E [r + pt+1 − pt (1 + r)] = Necessary 2 2 Conditions (2γ)µ σρ (2γ) Vart [pt+1] − µρt = 2 − µρt Limits to (1+r) Arbitrage Noise Trader 2 2 Risk (1 + r) (ρt ) Synchronization ⇒ E [∆R ] = ρ − Risk t n−a t 2 2 (2γ) µ σρ Policy Response • 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 Do Noise Traders Survive over Limits to Arbitrage Time (Evolutionary Argument)

Historical Bubbles

Symmetric Information Pricing Equation • Taking unconditional expectations Ruling out OLG Models 2 2 Asymmetric (1 + r) ρ∗ + (1 + r) σ2 Information E [∆R ] = ρ∗− ρ > 0 only if ρ∗ > 0 Expected/Strong n−a 2 2 Bubble (2γ) µ σρ Necessary Conditions Limits to • “Overoptimistic/bullish” traders hold riskier positions and Arbitrage Noise Trader have higher expected returns. Risk Synchronization Risk • In evolutionary process, they will have more off-springs Policy and hence they won’t die out. Response Asset Pricing under Asym. Information Myopia due to Liquidation Risk Limits to Arbitrage • Why are professional arbitrageurs myopic? Historical • Model setup of Shleifer & Vishny (1997) [slightly Bubbles modified] Symmetric Information • Two assets Pricing Equation Ruling out • risk free bond and OLG Models • risky stock with final value v Asymmetric Information • Two types of fund managers: Expected/Strong Bubble Necessary • Good fund managers know fundamental value v Conditions • Bad fund managers have no additional information Limits to (just gamble with “other people’s money”). Arbitrage Noise Trader Risk • Two trading rounds t = 1 and 2 (in t = 3 v is paid out) Synchronization Risk • Individual investors Policy • Response 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 Myopia due to Liquidation Risk Limits to Arbitrage • Price setting: Historical Bubbles • P3 = v

Symmetric • P2 is determined by aggregate demand of fund managers Information and liquidity/noise traders Pricing Equation Ruling out OLG Models • Focus on case where Asymmetric Information 1 P1 < v (asset is undervalued) Expected/Strong Bubble 2 P2 < P1asset price goes even further down in t2 due to Necessary Conditions • sell order by noise traders Limits to • sell order by other informed traders Arbitrage Noise Trader Risk • Performanced-based fund flows (see Chevalier & Ellison 1997) Synchronization Risk • If price drops, the probability increases that the fund Policy Response 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 Myopia due to Liquidation Risk Limits to Arbitrage

Historical Bubbles

Symmetric Information • “Good” manager’s problem who has invested in risky asset Pricing Equation Ruling out • He has to liquidate his position at P2 < P1 (exactly when OLG Models mispricing is the largest!) Asymmetric Information That is, he makes losses, even though the asset was Expected/Strong Bubble initially undervalued. Necessary Conditions • Due to this “early liquidation risk”, at t = 1 a rational Limits to fund manager is reluctant to fully exploit arbitrage Arbitrage Noise Trader opportunities at t = 1. Risk • Synchronization Focus on short-run price movements ⇒ myopia of Risk professional arbitrageurs Policy Response Asset Pricing under Asym. Information Synchronization Risk Limits to Arbitrage

Historical Bubbles

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

Historical • Long-standing debate about the role of monetary policy Bubbles with regard to asset price bubbles: Symmetric Information • “Cleaning” view (Greenspan (1999, 2002)): mitigate the Pricing Equation Ruling out consequences of bursting bubbles rather than trying to de- OLG Models tect and prevent asset price bubbles when they emerge Asymmetric Information • “Leaning” view (BIS): try to prevent the build-up of bub- Expected/Strong bles by reacting early on to upward-trending asset prices Bubble Necessary Conditions • Issues related to “leaning-against the wind”: Limits to 1 Bubbles are hard to identify (as fundamental asset values) Arbitrage Noise Trader 2 Monetary policy instruments are too blunt when aimed at Risk Synchronization containing bubbles (may overly suppress output/inflation) Risk 3 Bubbles could instead be tackled with financial regulation Policy Response • Recent crisis experience tilted views towards more interven- tion, closer to the BIS view (Stein (2013, 2014))