## I. INTRODUCTION — Definitions, causes, and paper roadmap

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### Definitions and scope
- Bubble defined as a deviation of the market price from the asset’s fundamental value.
- Positive bubble condition used:
  - Pt > Et[∑_{τ=t+1}^∞ CF_τ (1+r)^{τ−t}], where r is the appropriate discount rate.
  - Alternative upper-bound: Pt > Et[∑_{τ=t+1}^∞ CF_τ (1+r_f)^{τ−t}].
- When cash flows are positively correlated with market risk, required return > risk-free rate; discounted-cash-flow formula gives an upper bound for fair values.
- Upper-bound forecasting strategy: use an upper bound of forecasted cash flows from comparable firms when forecasts are difficult.

### Causes, models, and stylized features
- Early literature: rational-agent models where bubbles can exist.
- Newer approaches relax perfect rationality and introduce:
  - Behavioral models explaining initiation, bursting, and arbitrage failure.
  - Rational models with incentive problems, market frictions, and non-traditional preferences.
- New-generation rational-model channels:
  - Herding incentives among professional managers.
  - Limited-liability compensation structures encouraging riding bubbles.
  - Information intermediaries with misaligned incentives.

### Trading frictions and short selling
- Short selling costs/risks that prolong positive overvaluation:
  - Potentially unlimited loss.
  - Borrowed asset may be called back prematurely.
  - Fees appearing as low interest on margin accounts.
- Historical/regulatory hostility to short sellers documented.
- Empirical finding: firms initiating actions against short sellers in Lamont (2004) lost 42 percent of market capitalization over the next three years.

### Paper roadmap
- Reviews famous bubble episodes, the classic model, new-generation rational models, behavioral models, and implications for the subprime mortgage bubble.

---

### II. BUBBLES THROUGH HISTORY — Famous episodes and stylized facts

### Chronological examples (selected)
- Tulip bubble (Holland, 1634–February 1637)
  - Began in 1634; burst in February 1637.
  - At peak, a single tulip bulb sold for an equivalent of $60,000 today.
- South Sea bubble (1720)
  - Stock rose over 700 percent in first half of 1720; fell to about 50 percent above its start-of-year level by year-end.
- Mississippi bubble (1717–1721)
  - Share price: c. 500 livres tournois in January 1719 → 10,000 livres in December 1719 → back to 500 livres by September 1721.
  - Company exchange for paper money induced runaway inflation with a monthly rate of 23 percent.
- U.S. Crash (October 1929 and aftermath)
  - By end-1932 Manhattan real estate prices had fallen by 67 percent from Q3 1929 and remained depressed for the decade.
- Japan (1980–1993)
  - Stock market rose 373 percent in real terms to 1989 peak; fell by 50 percent over next three years.
  - Land prices almost tripled in late 1980s; market value of all land in Japan at 1990 peak was four times U.S. land value.
  - By end-1993 land prices had dropped by almost 50 percent.
- Dot-com bubble (1995–March 2000 peak)
  - NASDAQ Composite: 775.20 in January 1995; 2,505.89 in January 1999; 5,048.62 peak on March 10, 2000; declined to low of 1,314.85 in August (post-peak).

### Empirical stylized facts and market episodes
- Internet IPOs: many new internet IPOs never generated profits; many firms offered services for free to capture market share.
- Chinese warrant bubbles (2005–2008)
  - Put warrants with maturities between six months and two years on 18 Chinese companies traded at inflated prices despite near-certain expiration worthless.
  - Contributing factors: differences of opinion about underlying assets and a legal ban on short selling any financial securities in China.
- Cross-market observations:
  - Many emerging-market bubbles tied to cheap credit (Reinhart and Rogoff (2009)).
  - Real estate prices take longer to recover from crashes than equity prices.
- Bubble dynamics:
  - Bubbles can burst suddenly (October 1929) or deflate gradually (dot-com).
  - Deflation period generally much shorter than build-up period.
  - Speculative attacks or regulatory changes produce quicker deflation than gradual sentiment reversal.

---

### Core theoretical decomposition and prominent models

### Price decomposition and rational-bubble condition
- General decomposition: Pt = Pfair_t + Bt.
- For infinitely-lived asset:
  - Pt = Et[ ∑_{τ=t+1}^{∞} CF_τ/(1+r)^{τ−t} ] + lim_{T→∞} Et[ B_T/(1+r)^{T−t} ].
- Growth-rate condition for rational bubbles:
  - If bubble grows at rate r_B:
    - r_B < r ⇒ present value is zero (cannot exist).
    - r_B > r ⇒ present value is infinite (cannot exist).
    - Only r_B = r allows a persistent bubble component.
- Implications:
  - Bubbles cannot exist for assets with an upper bound on price or when required return exceeds economy growth.
- Finitely-lived assets:
  - Common-knowledge terminal bursting prevents bubbles, except when absence of common knowledge and short-sale constraints exist (Allen, Morris, and Postlewaite (1993)).

### Intrinsic bubbles (Froot and Obstfeld (1991))
- Bubble function of fundamentals: B(CF_t) = c CF_t^{λ}.
- Rational-bubble requirement: 1 + r = (1 + g)^{λ} when expected cash-flow growth = g.
- Market price example:
  - Pfair_t = CF_t (1+g)/(r−g) (Gordon growth form).
  - Pt = CF_t (1+g)/(r−g) + c CF_t^{λ}.
- Empirical estimates:
  - Using S&P 500 dividends 1900–1988, intrinsic bubble component estimated significantly positive; non-bubble component just under 50 percent of index value as of 1988.
  - Extending to December 31, 2010 with aggregate dividends, bubble estimated at 67.84 percent of S&P 500 index price.

---

### The new generation of rational models — mechanisms creating and sustaining bubbles

### Channels identified
- (a) Herding
  - Relative-wealth preferences (DeMarzo, Kaniel, and Kremer (2008))—agents “keep up with the Joneses.”
  - Reputation-based herding (Scharfstein and Stein (1990))—managers mimic to protect labor-market reputation.
  - Investor flows and mutual funds (Lamont and Frazzini (2008))—“dumb money” flows toward high-sentiment funds that later underperform.
  - Media attention amplifies herding.
- (b) Limited liability
  - Limited downside encourages risky strategies and bubble participation (Allen and Gorton (1993); Allen and Gale (2000)).
  - Convex payoff from limited liability increases preference for risk.
- (c) Perverse incentives of intermediaries
  - Equity analysts may issue “strong buy” recommendations despite private pessimism due to fear of losing management access, cross-subsidization, and commission-linked pay.
  - Rating agencies and auditors paid by firms have incentives to avoid negative signals.
  - Executive compensation and implicit guarantees encourage short-term profit-seeking.

---

### Behavioral models — four classes that generate bubbles

### Model 1: Differences of opinion + short-sale constraints
- Optimistic, dogmatic investors push prices above fair value because pessimists are constrained from shorting.
- Price declines when uncertainty resolves or constraints relax (Miller (1977); Scheinkman and Xiong (2003)).
- Empirical support: analyst forecast dispersion correlates with future underperformance (Diether, Malloy, and Scherbina (2002)); stocks owned by few mutual funds more likely overvalued (Chen, Hong, and Stein (2001)).

### Model 2: Feedback trading (momentum)
- Traders condition on past price movements; positive returns attract more buyers in self-reinforcing loops.
- Media amplify feedback trading (Shiller (2002); Bhattacharya and others (2009)).
- Feedback models resemble Ponzi dynamics: growth sustained by inflows; collapse when inflows end (Hong and Stein (1999); DeLong and others (1990)).

### Model 3: Biased self-attribution
- Investors overweight confirming signals and discount contradictory ones (Daniel, Hirshleifer, and Subrahmanyam (1998)).
- Public signals that confirm priors lead to overconfident upward revisions and bubble-like moves; deflation as contradictory signals accumulate.

### Model 4: Representativeness heuristic + conservatism bias
- Overreaction to salient “strong” news and underreaction to routine evidence (Barberis, Shleifer, and Vishny (1998)).
- Generates momentum and long-run reversals; can produce both positive and negative bubbles.

### Model comparisons and predictions
- Model 1 generates only positive bubbles; Models 2–4 can generate positive and negative bubbles.
- Models 1, 3, and 4 predict bubbles more likely in hard-to-value assets.
- Common initiation mechanisms: overreaction to potentially informative signals and plausible narratives (technology, land scarcity, securitization).

---

### Initiation, deflation, trading patterns, and limits to arbitrage

### How bubbles are initiated
- Financial accelerator (Bernanke, Gertler, Gilchrist (1999)): value-increasing shock lowers borrowing costs, raises investment and cash flows, further raising prices.
- Money illusion in housing (Brunnermeier and Julliard (2008)):
  - V/Rent should equal 1/r_real; if agents use r_nom, V/Rent = 1/r_nom.
  - Since r_nom = r_real + inflation, price-to-rent ratio is decreasing in expected inflation; unexpected fall in inflation expectations raises house prices.
  - Empirical support: price-to-rent changes negatively related to expected inflation changes but not to real rate changes.
- Advisor incentives and disagreement (Hong, Scheinkman, and Xiong (2008)): optimistic advisors exaggerate; uninformed investors ignore incentive bias, creating bubbles.
- Emerging markets: credit expansion after financial liberalization often triggers bubbles (Reinhart and Rogoff (2009)).

### Why bubbles burst or deflate
- Model 1: burst when uncertainty resolved or short-sale constraints relaxed.
  - Example: stocks with high analyst disagreement show large declines around earnings announcements (Scherbina (2008)).
  - Internet IPO lock-up expirations: 80 percent–85 percent of shares often held by insiders; many lock-ups expired Oct 1999–Apr 2000, unlocking almost $300 billion; Morgan Stanley internet index fell from 1,030 on March 1, 2000 to 430 on April 30, 2000; typical lock-up: 180 days.
- Model 2: deflation when new capital supply exhausted; end often with widespread retail participation or credit tightening (e.g., China tripled security transactions tax on May 30, 2007).
- Models 3 and 4: deflate when sentiment reverses or when a strong negative signal appears.
- Finitely-lived assets: known expiration ensures deflation (Chinese warrants showed accelerating price decreases six days before expiration and heavy trading).
- Arbitrage attacks: bubbles burst when arbitrageurs short sufficiently large amounts.

### Trading volume patterns and life cycle
- Bubbles often have abnormally high trading volume.
  - Dot-com example: monthly turnover of internet stocks exceeded 50 percent in 12 of 24 months before the internet index peak in February 2000; average turnover for non-internet stocks was 10–15 percent; after decline internet-stock turnover fell to market average.
- Model 2 lifecycle: low volume early, drastic increase during middle speculative stage, volume fall as inflows slow.
- Supply responses: IPOs, SEOs, start-ups, new housing construction.
- Later stages: increased fraud and illusion-preserving trades as participants attempt to offload overvalued assets.

### Why mispricings persist — frictions and risks
- Key frictions and risks that limit arbitrage:
  - Risk that bubble continues to grow, inducing margin calls and forced scaling back (Shleifer and Vishny (1997); Xiong (2001); Gromb and Vayanos (2002)).
  - Fundamental-change risk: shifts in fundamentals can make arbitrage unprofitable.
  - Informational asymmetries raise trading costs for large trades (Sadka and Scherbina (2010)).
  - Coordination failure: sequential awareness and short-sale limits impede coordinated attacks (Abreu and Brunnermeier (2003)).
- Abreu and Brunnermeier (2003) mechanism:
  - Overvaluation arises when market grows at rate g after t0 while fair value grows at r.
  - Bubble component B_t = P0[(1+g)^{t−t0} − (1+r)^{t−t0}] grows with time and with (g − r).
  - A critical fraction k of arbitrageurs must short to crash the bubble; uncertainty and coordination needs induce arbitrageurs to ride the bubble rather than attack prematurely.

---

### Experimental evidence and theoretical implications

### Experimental findings
- Bubbles observed in 14 out of 22 experiments replicating Smith, Suchanek, and Williams (1988).
- Typical experimental setup: 15 (or 30) trading periods, each lasting up to 240 seconds.
- Dividend distributions common knowledge; short selling typically not allowed.
- Mean price in first trading period always below expected value of future dividends (consistent with risk aversion).
- Collapse of bubbles preceded by decline in “buy” vs. “sell” offers and accompanied by lower trading volume than during bubble rise.
- Mixing experienced and inexperienced traders:
  - Inclusion of experienced traders who had previously participated in three rounds substantially reduced or eliminated bubbles even when experienced fraction was only one-third.
- Short sale permission mixed results:
  - Ackert, Church, and Deaves (2002): allowing short selling made markets more efficient.
  - Naruvy and Noussair (2006): permitting short selling did not eliminate bubble-like properties.

### Theoretical contributions about rational traders
- Rational investors do not necessarily eliminate bubbles when:
  - There is no common knowledge of a bubble’s existence.
  - Feedback traders are present.
  - Key agents face limited liability.
- Arbitrageurs may optimally ride bubbles and delay attacks; waiting time before attack increases with:
  - Disagreement about bubble start date.
  - Fraction k required to succeed in attack.
  - Excess growth rate g−r.
- Empirical consistency:
  - Brunnermeier and Nagel (2004): hedge funds rode the dot-com bubble and sold out before deflation.
  - Some investment banks and hedge funds rode MBS/CDO exposures and later switched to CDS shorts.

---

### Subprime mortgage bubble — selected model-consistent findings

### Financial innovations and incentives
- Securitization (MBS → CDOs) diversified regional idiosyncratic risk and lowered mortgage rates, amplifying house price rises.
- Moral hazard from securitization:
  - Originators did not hold loans, incentivizing volume over quality.
  - Example: average down payment for Alt A borrowers fell from 14 percent in 2000 to 2.7 percent in 2006.
- Rating agency failures:
  - Models used recent house-price data and assumed repossession recoveries; these assumptions failed when prices declined.
  - Assumed cross-regional correlation as low as 30 percent proved close to 100 percent during declines.
  - Loan pools sometimes rated on average FICO while pooling high- and low-FICO borrowers, understating risk.

### Macro developments and market impacts
- Federal Reserve policy 2004–2006: Fed raised rates 17 times, from 1 percent to 5.25 percent.
- Mortgage resets and house-price declines:
  - Many teaser-rate mortgages reset to market rates, contributing to defaults.
  - Case-Shiller 20-city index: fell by 9.03 percent in 2007 and by 18.61 percent in 2008.

### Market instruments and dynamics
- CDS contracts and the ABX index implicitly relaxed short-sale constraints in MBS markets, contributing to repricing toward fundamentals.
- Sophisticated investors rode and later bet against the housing bubble; timing affected losses and gains.

### Institutional and policy responses
- Financial Crisis Inquiry Commission identified warning signs: explosion in risky subprime lending, unsustainable house price rises, unscrupulous lending practices, steep increases in homeowner mortgage debt, and spikes in trading in mortgage-backed products.
- Post-crisis legislative response: enactment of the Dodd-Frank Wall Street Reform and Consumer Protection Act to improve monitoring, incentives, and investor protection.

---

### Policy implications and recommendations

### Prevention-focused suggestions from the literature
- Remove short sale restrictions whenever possible to reduce bubble likelihood.
- Improve financial education to mitigate investor irrationality effects.
- Mitigate limited liability incentive structures and expose agents to downside risk to reduce moral hazard and excessive risk-taking.

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*Source: _wp1345 (content as supplied).*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### I. INTRODUCTION

### Definitions and scope
- A bubble is defined as a deviation of the market price from the asset’s fundamental value.
- Positive bubble definition used in the paper:
  - Pt > Et[∑_{τ=t+1}^∞ CF_τ (1+r)^{τ−t}], where r is the appropriate discount rate.
  - Alternative upper-bound definition using the risk-free rate:
    - Pt > Et[∑_{τ=t+1}^∞ CF_τ (1+r_f)^{τ−t}].
- When an asset’s cash flows are positively correlated with market risk, the required rate of return is strictly greater than the risk-free rate; the discounted-cash-flow formula thus represents an upper limit of the justifiable range of fair values.
- When forecast of future cash flows is difficult, an upper bound of forecasted cash flows for other firms in the same industry or asset class may be used.

### Causes, models, and stylized features
- Early literature: models with fully rational agents in which bubbles could exist.
- Recent literature broadens focus by relaxing perfect rationality and by introducing:
  - Behavioral models that explain bubble initiation, bursting conditions, and failures of arbitrage.
  - Rational models with incentive problems, market frictions, and non-traditional preferences.
- New-generation rational models emphasize:
  - Incentives to herd among professional money managers.
  - Limited liability compensation structures encouraging investment in bubbles.
  - Information intermediaries whose incentives are not always compatible with reporting negative information.
- Behavioral models provide explanations for:
  - How bubbles may be initiated.
  - Conditions under which bubbles burst.
  - Why arbitrage forces may fail to restore prices to fundamentals.
  - Why many bubble episodes are accompanied by high trading volume; experimental studies support behavioral findings.

### Trading frictions and short selling
- Short selling is associated with costs and risks that can make positive overvaluation more persistent than undervaluation:
  - Potentially unlimited loss.
  - Risk that the borrowed asset will be called back prematurely.
  - Fees that manifest as low interest rates paid on the margin account.
- Historical and regulatory hostility toward short sellers is noted (examples mentioned include bans in 18th and 19th century England and restrictions in modern countries).
- Empirical evidence cited: firms that started actions against short sellers in Lamont (2004) ended up losing 42 percent of their market capitalization over the next three years, consistent with short sellers identifying overvaluation.

### Paper roadmap
- After a brief summary of famous bubble episodes and a description of the classic model, the paper reviews:
  - New-generation rational models.
  - Behavioral models of bubbles.
- Concludes with how insights from new models help understand the evolution of the recent subprime mortgage bubble.

*Italic line: Source: _wp1345 - References . . . (content as supplied)*

### II. BUBBLES THROUGH HISTORY

### Overview
- Section provides, in chronological order, a brief overview of famous examples of bubbles observed throughout history.
- The list is acknowledged as incomplete.

### Tulip bubble (Holland, 1634–February 1637)
- Began in 1634 and burst in February 1637.
- At the bubble’s peak, a single tulip bulb sold for an equivalent of $60,000 today.

### South Sea bubble (1720)
- Involved the market price of the South Sea Company (an English firm with no assets but claims of large profits).
- During the first half of 1720 the company’s stock price rose by over 700 percent, then fell during the second half of 1720 to about 50 percent above what it had been at the start of the year.

### Mississippi bubble (Compagnie d’Occident / Compagnie des Indes; 1717–1721)
- Company founded in August 1717; by May 1719 obtained broader trade control.
- Share price rose from around 500 livres tournois in January 1719 to 10,000 livres in December 1719.
- In January 1720 prices began to fall as investors sold to turn gains into gold; company attempted to exchange shares for paper money, causing runaway inflation with a monthly rate of 23 percent.
- By September 1721 the price had dropped to its pre-bubble price of 500 livres.

### U.S. Crash (October 1929 and aftermath)
- The Roaring Twenties expansion halted in October 1929 with a crash involving both stock and real estate prices.
- Using a hand-collected dataset, by the end of 1932 real estate prices in Manhattan had fallen by 67 percent from the third quarter of 1929 and stayed down for the remainder of the decade.
- Collapse of the housing sector greatly depleted household wealth, contributing to the severity of the Great Depression.

### Japan (1980–1993)
- From 1980 to the peak in 1989, the Japanese stock market rose 373 percent in real terms, then fell by 50 percent in the next three years.
- Land prices almost tripled in the second half of the 1980s; at its peak in 1990, the market value of all the land in Japan was four times the land value in the United States.
- By the end of 1993, Japanese land prices had dropped by almost 50 percent.
- Some argue the collapse slowed economic growth thereafter.

### Dot-com (internet) bubble (1995–March 2000 peak)
- Started around 1995; rapid growth in internet sector and supply of internet IPOs fueled the bubble.
- NASDAQ Composite indices:
  - 775.20 in January 1995.
  - 2,505.89 in January 1999.
  - 5,048.62 peak on March 10, 2000.
  - Declined to a low of 1,314.85 in August (year implied by context following peak).

*Italic line: Source: _wp1345 - References . . . (content as supplied)*

### 2002. During the bubble period, investment banks responded to the high demand for inter-

### _wp1345 - 2002. During the bubble period, investment banks responded to the high demand for inter-

### Empirical examples and stylized facts
- Internet IPOs during the bubble period: a large fraction of new internet IPOs never generated any profit; many firms offered services for free to capture market share and later monetize — many had identical business models and competed in the same market, ensuring a large fraction would eventually fail.
- Chinese warrant bubbles (Yu and Xiong (2011)):
  - Observed in the 2005-2008 period.
  - Occurred in put warrants with long maturities, ranging between six months and two years, issued on 18 Chinese companies.
  - Warrants were all but sure to expire worthless yet traded at inflated prices and high volumes throughout contract lives.
  - Contributing factors: (a) differences of opinion about the underlying asset, and (b) legal ban on short selling any financial securities in China, including warrants.
- Historical and cross-market observations:
  - Reinhart and Rogoff (2009): many bubbles in emerging markets; many bubbles instigated by cheap credit.
  - Real estate prices take longer to recover from crashes than equity prices; consistent with U.S. Depression evidence.
- Bubble dynamics:
  - Bubbles can burst suddenly (e.g., stock market crash in October 1929) or deflate gradually (e.g., dot-com bubble).
  - Evidence indicates the deflation period is generally much shorter than the build-up period.
  - Speculative attacks or regulatory changes tend to produce quicker deflation than sentiment reversals.

### Core theoretical decomposition and classical rational-bubble condition
- Price decomposition for an asset:
  - Pt = Pfair_t + Bt (price = fair value + bubble component).
  - For an infinitely-lived asset, Pt = Et[ sum_{τ=t+1}^{∞} CF_τ/(1+r)^{τ−t} ] + lim_{T→∞} Et[ B_T/(1+r)^{T−t} ].
- Growth-rate condition for a rational bubble on an infinitely-lived asset:
  - If bubble grows at rate r_B, then:
    - r_B < r ⇒ present value of bubble is zero (cannot exist).
    - r_B > r ⇒ present value is infinite (cannot exist).
    - Only r_B = r allows a bubble component to persist.
- Implications:
  - Bubbles cannot exist for assets with an upper bound on price (e.g., assets with close substitutes).
  - If asset’s required rate of return exceeds economy growth, a bubble cannot exist.
- Finitely-lived assets:
  - Backward-induction implies a bubble cannot exist if it is common knowledge that it will burst at known terminal time T.
  - Exception: absence of common knowledge and binding short-sale constraints can allow bubbles in finitely-lived assets (Allen, Morris, and Postlewaite (1993)).

### Intrinsic bubbles (Froot and Obstfeld (1991))
- Bubble as a function of fundamentals:
  - B(CF_t) = c CF_t^{λ}, where c and λ are constants.
  - Rational-bubble requirement leads to 1 + r = (1 + g)^{λ} when expected cash-flow growth rate = g and bubble must grow at rate r.
- Market price specification:
  - Pfair_t = CF_t (1+g)/(r−g) (Gordon growth form).
  - Pt = CF_t (1+g)/(r−g) + c CF_t^{λ}.
- Empirical findings:
  - Using S&P 500 data 1900-1988 and dividends as proxy for cash flows, intrinsic bubble component estimated significantly positive; the non-bubble component was just under 50 percent of index value as of 1988.
  - Extending dataset to December 31, 2010 and using aggregate dividends, the bubble is estimated at 67.84 percent of the S&P 500 index price.

### The new generation of rational models — channels creating and sustaining bubbles
- Broad mechanisms: non-standard preferences and incentives that cause herding; limited liability leading to preference for riding bubbles; perverse incentives of information intermediaries.
- (a) Herding:
  - Relative-wealth preferences (DeMarzo, Kaniel, and Kremer (2008)): “keeping up with the Joneses” causes agents to join bubbles to avoid falling behind peers.
  - Reputation-based herding (Scharfstein and Stein (1990)): managers mimic others to protect labor-market reputation; equilibrium can have second mover mimic first mover regardless of private signal.
  - Investor flows and mutual funds (Lamont and Frazzini (2008)): investors allocate money dynamically to funds investing in high-sentiment stocks, producing the “dumb money” phenomenon—sentiment-driven funds favored during bubbles and earning sub-par returns later.
  - Media attention amplifies herding by directing investor attention to performing assets.
- (b) Limited liability:
  - Limited downside induces agents (unskilled managers, borrowers) to prefer risky strategies and ride bubbles (Allen and Gorton (1993); Allen and Gale (2000)).
  - Convex payoff from limited liability increases preference for risk and bubble participation.
- (c) Perverse incentives of intermediaries:
  - Equity analysts may issue “strong buy” recommendations while privately pessimistic: reasons include fear of losing management access, cross-subsidization by investment banking business, and compensation linked to trading commissions.
  - Rating agencies and auditors paid by firms have incentives to avoid negative signals and may assign overly positive ratings or overlook irregularities.
  - Executive compensation and implicit guarantees can encourage short-term profit-seeking over long-term honesty.

### Behavioral models — four classes that generate bubbles
- General feature: at least one group of agents departs from perfect rationality.
- Model 1: Differences of opinion + short-sale constraints
  - Optimistic, dogmatic investors push prices above fair value because pessimists cannot sell short.
  - Price falls when uncertainty resolves or constraints relax (Miller (1977); Scheinkman and Xiong (2003)).
  - Empirical support: dispersion in analyst forecasts correlates with future underperformance (Diether, Malloy, and Scherbina (2002)); stocks owned by few mutual funds more likely overvalued (Chen, Hong, and Stein (2001)).
- Model 2: Feedback trading (momentum)
  - Traders condition on past price movements; positive returns attract buyers who push prices higher, attracting more buyers in a self-reinforcing loop.
  - News media amplify feedback trading (Shiller (2002); Bhattacharya and others (2009)).
  - Feedback models resemble Ponzi dynamics: growth sustained by inflow of new money; collapse when inflows dry up (Hong and Stein (1999); DeLong and others (1990)).
- Model 3: Biased self-attribution
  - Investors overweight signals confirming their private beliefs and discount contradictory signals (Daniel, Hirshleifer, and Subrahmanyam (1998)).
  - Public signals that confirm private signals lead to overconfident upward revisions and bubble-like price moves; deflation occurs as contradictory signals accumulate.
- Model 4: Representativeness heuristic + conservatism bias
  - Investors overreact to salient “strong” news (representativeness) and underreact to routine evidence (conservatism), leading to over-extrapolation or underreaction followed by overreaction (Barberis, Shleifer, and Vishny (1998)).
  - Generates momentum and long-run reversals; can produce both positive and negative bubbles.
- Model comparisons:
  - Model 1 generates only positive bubbles; models 2–4 can generate positive and negative bubbles.
  - Models 1, 3, and 4 predict bubbles more likely in hard-to-value assets.

### How bubbles are initiated
- Behavioral initiation: prices overreact to potentially informative signals about fundamentals.
- Plausible narratives often exist (technology promise, land scarcity, securitization of mortgages).
- Financial accelerator (Bernanke, Gertler, Gilchrist (1999)): a value-increasing shock lowers borrowing costs, stimulates investment, raises cash flows and prices, producing feedback on investment.
- Money illusion in housing (Brunnermeier and Julliard (2008)):
  - House price-to-rent ratio V/Rent should equal 1/r_real; if agents discount rents with nominal rate (money illusion), V/Rent = 1/r_nom.
  - Since r_nom = r_real + inflation, price-to-rent ratio is decreasing in expected inflation; an unexpected fall in inflation expectations raises house prices.
  - Empirical support: changes in price-to-rent ratios negatively related to changes in expected inflation but not to changes in real interest rates.
- Advisor incentives and disagreement (Hong, Scheinkman, and Xiong (2008)): sets of advisors disagree about new technology; optimistic advisors exaggerate; uninformed investors who ignore incentive bias create bubbles.
- Emerging markets: credit expansion after financial liberalization often triggers bubbles (Reinhart and Rogoff (2009)).

### Why bubbles burst or deflate
- Model 1: burst when uncertainty resolved or short-sale constraints relaxed.
  - Empirical example: large price declines around earnings announcements for stocks with high analyst disagreement (Scherbina (2008)).
  - Internet IPO lock-up expirations (Ofek and Richardson (2003)):
    - Between 80 percent and 85 percent of shares of many new internet IPOs were held by insiders, venture capitalists, and angel investors restricted by lock-up provisions.
    - Many lock-ups expired between October 1999 and April 2000, unlocking almost $300 billion in shares in a short period.
    - Morgan Stanley internet index fell from 1,030 on March 1, 2000 to 430 on April 30, 2000.
    - Typical lock-up length: 180 days.
- Model 2: deflation when supply of new capital is exhausted; end often coincides with widespread participation by non-sophisticated investors or credit tightening.
  - Governments can tighten credit or increase taxes/transactions taxes to deflate bubbles (e.g., China tripled security transactions tax on May 30, 2007).
- Models 3 and 4: deflate when sentiment reverses or when a “strong” negative signal appears (e.g., stock market crash as a negative signal for other asset classes).
- Finitely-lived asset bubbles (e.g., Chinese warrants): known expiration guarantees eventual deflation; observed gradual deflation with accelerating price decreases six days before expiration and heavy trading (Yu and Xiong (2011)).
- Arbitrage attacks: bubbles burst when arbitrageurs sell short sufficient amounts of the overvalued asset.

### Trading volume patterns and life-cycle
- Bubbles are frequently accompanied by abnormally high trading volume.
  - Example: during dot-com bubble, monthly turnover of internet stocks exceeded 50 percent in 12 out of 24 months preceding the internet index peak in February 2000; average turnover for non-internet stocks was 10-15 percent; after decline turnover of internet stocks dropped to market average.
- Model 2 (feedback trading): trading volume low in early stage, drastically increases during middle speculative stage as more investors notice past price increases and enter; volume may fall as inflows slow and bubble growth decelerates.
- Supply response during high demand often via IPOs, SEOs, start-ups, and new housing construction.
- Later stages see more fraud and illusion-preserving trades as participants attempt to offload overvalued assets.
- Model 1 extensions (Baker and Stein (2004); Scheinkman and Xiong (2003, 2004)): bubbles coincide with active trading when irrational investors are optimistic and underreact to order-flow information; bubble size increases with overconfidence and rapid opinion evolution, decreases with trading costs.

### Why mispricings are not fully arbitraged away
- Key frictions and risks:
  - Risk that bubble continues to grow, forcing short sellers to face margin calls and scale back (Shleifer and Vishny (1997); Xiong (2001); Gromb and Vayanos (2002)).
  - Fundamental-change risk: the asset’s fundamentals may shift, eliminating apparent mispricing and making arbitrage unprofitable.
  - Informational asymmetries: large trades face high trading costs because traders may be informed; increased costs reduce arbitrage profitability (Sadka and Scherbina (2010)).
  - Coordination failure: when many small arbitrageurs needed to coordinate, sequential awareness and limited short-sale capacity impede collective attack (Abreu and Brunnermeier (2003)).
- Abreu and Brunnermeier (2003) mechanism:
  - Overvaluation arises when market continues to grow at high rate g after t0 while fair value grows at r.
  - Bubble component B_t = P0[(1+g)^{t−t0} − (1+r)^{t−t0}] grows with time and with (g − r).
  - Short-sale limits and sequential awareness imply a critical fraction k of arbitrageurs must short to crash bubble; lack of coordination and uncertainty about burst timing deter premature arbitrage.
  - Arbitrageurs may trade with the mispricing (amplify) rather than against it if doing so lets them resell at higher prices later.

*Italicized source attribution: content derived from the supplied IMF PDF chapter/section.*

### conclusion of this model is that—much as in DeLong and others (1990)—upon becoming

### VII. CONCLUSION

### Historical perspective and overarching insight
- Quotation capturing recurring nature of market booms and panics: “Nowhere does history indulge in repetitions so often or so uniformly as in Wall Street. When you read contemporary accounts of booms or panics, the one thing that strikes you most forcibly is how little either stock speculation or stock speculators today differ from yesterday. The game does not change and neither does human nature.” (Reminiscences of a Stock Operator, 1923, p. 180).
- Over almost 90 years, many instances of booms and crashes across asset markets have been observed, motivating extensive academic work to understand persistent bubble phenomena.
- Recent literature shifted from showing bubbles can exist in perfectly rational markets to modeling observed bubble dynamics often by relaxing perfect rationality.

### Taxonomy of behavioral models of bubbles
- Four types of behavioral models:
  - Models emphasizing short sale constraints with diverging investor beliefs about fair value.
  - Feedback-trader models: traders form demands assuming recent price trends will continue, thereby making them continue.
  - Self-attribution bias models: investors overweight public signals that confirm priors and underweight contradictory signals, generating overreaction.
  - Representativeness heuristic models combined with conservatism bias: investors overreact to attention-catching news and are slow to update faulty models.

### Experimental findings relevant to bubbles
- Experimental markets (building on Smith, Suchanek, and Williams (1988)) replicate bubbles frequently:
  - A bubble is observed in 14 out of 22 experiments.
  - Typical experimental setup: 15 (or 30) trading periods, each lasting a maximum of 240 seconds.
  - Information on dividend distributions is common knowledge; short selling is not allowed.
  - Mean price in the first trading period is always below the expected value of future dividends, consistent with risk aversion.
  - Collapse of bubbles preceded by a decline in “buy” relative to “sell” offers and accompanied by lower trading volume than during the bubble’s rise.
- Key experimental variations and results:
  - Mixing experienced and inexperienced traders: having experienced traders who had previously participated in three rounds substantially reduced or eliminated bubbles even when the experienced fraction was only one-third.
  - Short sale permission: Ackert, Church, and Deaves (2002) found allowing short selling made markets more efficient; Naruvy and Noussair (2006) found permitting short selling did not eliminate bubble-like properties (high transaction volume, large price swings relative to fundamentals, sustained trading away from fundamentals).
- Experimental literature confirms that lack of common knowledge of rationality among traders is a key driver of bubbles.

### Theoretical contributions about rational traders and bubbles
- New theoretical models show rational investors do not necessarily eliminate bubbles:
  - Realistic innovations include lack of common knowledge of a bubble’s existence, presence of feedback traders, and limited liability for key agents.
  - These assumptions can lead rational traders to trade with the bubble (ride it) rather than against it, aggravating mispricing for some time.
  - Models (e.g., Abreu and Brunnermeier (2003) and related work) show arbitrageurs may optimally ride bubbles and delay attacking them; the waiting time before attack increases with:
    - disagreement among arbitrageurs about when the bubble started;
    - the fraction of arbitrageurs, k, required to succeed in the attack;
    - the “excess” growth rate of the bubble, g−r.
  - If these values are sufficiently large, speculative attack may be deferred until the bubble bursts for exogenous reasons, letting early-informed arbitrageurs reap maximum profits.
- Empirical and historical consistency:
  - Brunnermeier and Nagel (2004) documented hedge funds during the dot-com bubble riding the bubble and selling out before deflation.
  - Sophisticated market participants (e.g., some investment banks and hedge funds) initially took long positions in MBS and CDOs and later switched to short bets (via CDS), consistent with models predicting riding then attacking the bubble.

### Subprime mortgage bubble lessons (selected findings tied to model implications)
- Financial innovations and policies increased attractiveness of housing relative to stocks and bonds; securitization (MBS → CDOs) allowed diversification of idiosyncratic regional risk and helped lower mortgage rates, amplifying house price rises.
- Moral hazard from securitization:
  - Originators no longer held loans on their books, incentivizing volume over quality and proliferating loan types attracting subprime borrowers.
  - Example: average down payment for Alt A borrowers fell from 14 percent in 2000 to only 2.7 percent in 2006.
- Rating agency failures:
  - Models used recent house-price data and assumed repossession would recover loan value; these assumptions failed when prices declined.
  - Assumed cross-regional correlation as low as 30 percent proved close to 100 percent during price declines.
  - Loan pools were sometimes delivered at the bottom of desired rating grades and pooled high- and low-FICO borrowers but rated on average FICO, understating risk.
- Macroeconomic and market developments:
  - Between 2004 and 2006, the Fed raised interest rates 17 times, from 1 percent to 5.25 percent.
  - Many teaser-rate mortgages reset to market rates, contributing to defaults; the largest decline in house prices occurred in 2008.
  - Case-Shiller 20-city index fell by 18.61 percent in 2008 and by 9.03 percent in 2007.
- Market instruments and dynamics:
  - Introduction of CDS contracts and the ABX index implicitly relaxed short sale constraints for MBS markets and contributed to repricing toward fundamentals.
  - Some sophisticated investors rode the housing bubble and then switched to betting against it; timing influenced eventual losses or gains.
- Policy and institutional responses:
  - Financial Crisis Inquiry Commission identified warning signs that, if heeded, might have avoided the crisis: explosion in risky subprime lending, unsustainable house price rises, unscrupulous lending practices, steep increases in homeowner mortgage debt, and spikes in trading activities in mortgage-backed products.
  - Post-crisis: enactment of the Dodd-Frank Wall Street Reform and Consumer Protection Act aiming to improve monitoring, incentives, and investor protection.

### Policy implications and recommendations from the literature
- Prevention-focused suggestions:
  - Remove short sale restrictions whenever possible to reduce the likelihood of bubbles.
  - Provide better financial education to reduce adverse influence of investor irrationality.
  - Mitigate limited liability incentive structures and expose agents to downside risk to reduce moral hazard and excessive risk-taking.

*Source: Conclusion section of the content unit provided.*

### REFERENCES

### _wp1345 - REFERENCES

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*Source: _wp1345 - REFERENCES (content unit).*

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