> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-iv-the-investor-ecology/chapter_15_behavioral.md). # Chapter 15: Behavioral Finance *Part IV: The Investor Ecology — Financial Economics: Claims, Prices, and Holders* *** ## Opening Episode: Ten Thousand Accounts Sometime in the early 1990s a large American discount brokerage handed a graduate student the trading records of ten thousand of its accounts. The records ran from 1987 to 1993 and contained every position and every transaction: what each household owned, what it bought, what it sold, and on what date. No survey instrument, no laboratory, no hypothetical gamble. Just what people did with their own money when nobody was watching. Terrance Odean asked one question of the data, and it was a question that only this kind of data can answer. On any day a household holds several positions, some trading above what it paid and some below. When it decides to sell something, which does it sell? The arithmetic has to be done carefully, because the answer is not simply "winners" — a household in a rising market holds mostly winners, so it will sell mostly winners for purely mechanical reasons. Odean's fix was to compute, for every day on which an account sold anything, the fraction of the gains available that were realized and the fraction of the losses available that were realized. A household indifferent between the two would realize gains and losses at the same rate. These households realized gains at roughly half again the rate at which they realized losses, and the gap was there in every year, in accounts of every size. Two features of the result are what made the paper matter. The first is the exception. The asymmetry shrank sharply in December, and December is when the US tax code rewards realizing losses. So the households knew the rule; they applied it once a year, under deadline, and ignored it for the other eleven months. Whatever was going on, it was not ignorance of the tax treatment. The second is the cost. Odean followed the two sets of stocks forward. The winners that were sold went on to outperform the losers that were kept, by something on the order of three percentage points over the following year. The behavior was not a wash. It was a transfer, made by households, to whoever was on the other side. ![Figure 15.3: The disposition effect](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-bdb565abf0e489414cbdb3840857530fdcc6871a%2Ffig_15_03_the_disposition_effect.png?alt=media) **Figure 15.3: The disposition effect.** The two magnitudes this section states, and only those. Panel (a) is the asymmetry: a household indifferent between gains and losses would realize each at the same rate, and Odean's ten thousand accounts realized gains at about half again the rate of losses, in every year of the sample and in accounts of every size. Panel (b) is what it cost — the winners sold beat the losers kept by something near three percentage points over the following year, so the behavior was a transfer rather than a wash. What is deliberately *not* drawn is the monthly series the exhibit plan originally asked for. The brokerage records are not redistributable and this chapter prints no monthly figures, so a twelve-point curve would be twelve numbers with no source behind them. The December exception is the sharpest thing in the paper and it is stated here in words for the same reason: the asymmetry shrinks in the one month the tax code rewards realizing losses, which is how we know these households understood the rule and applied it once a year — but how far it shrinks is not a number this book can supply. *Source: Odean (1998), as reported in §15.1; the underlying brokerage records are not redistributable.* Chapter 14 recorded this pattern in §14.2 among the things households do and pointedly declined to explain it. That was the division of labor: Chapter 14 documents behavior, this chapter accounts for it. And the accounting has a specific shape. The households in Odean's file were not making random errors, which would average out across ten thousand accounts and leave prices undisturbed. They were making the *same* error, in the same direction, at the same time — because the error follows from a feature of how people evaluate gains and losses that is stable across people. Correlated error is the only kind that can reach prices. Which sets up the chapter's second half, and its harder question. If ten thousand households are systematically wrong in a documented, published, and easily measured way, why is there any money left to be made from it? Chapter 3's opening episode already showed the answer's outline: in March 2000 the market priced 3Com's non-Palm business at negative twenty billion dollars, the arithmetic was printed in the newspapers, and the mispricing persisted anyway, because you cannot sell what you cannot borrow. Behavioral finance is two claims, not one. Investors err in patterned ways, and the machinery for trading against those errors is itself owned, financed, and constrained. The first claim is about psychology. The second is about balance sheets, and it is what makes this a chapter in Part IV rather than a footnote to Part II. *** ## 15.1 Preferences: Prospect Theory Expected utility theory has an agent who evaluates gambles by their consequences for terminal wealth, weighting outcomes by their probabilities. Kahneman and Tversky's 1979 paper, built on choices between simple laboratory gambles, replaced each of those clauses with something else. The replacement has four parts, and each has a fingerprint in financial data. **Reference dependence.** Utility is defined over gains and losses relative to a reference point, not over levels of wealth. The purchase price of a stock, the previous peak of a portfolio, last year's bonus, the price paid for a house — these are not arguments of a well-specified utility function, but they are what people actually compare outcomes to. This is the sharpest break with the standard model, because it makes the *same* terminal wealth pleasant or unpleasant depending on where the agent started. **Loss aversion.** The function is steeper below the reference point than above it. Losing hurts more than the equivalent gain pleases, by a factor that laboratory work puts at roughly two. **Diminishing sensitivity.** The function is concave in gains and convex in losses: the second hundred dollars of gain adds less than the first, and the second hundred dollars of loss subtracts less than the first. Convexity below the reference point means risk *seeking* over losses — a person already behind will accept a fair gamble that offers a chance of getting back to even. **Probability weighting.** Probabilities enter through a transformation that overweights small probabilities and underweights moderate to large ones. A one-in-a-thousand chance is treated as though it were rather more than one in a thousand; the difference between a 60 percent and a 65 percent chance registers as less than five points. Putting the first three together gives the value function, which for numerical work is usually written with a power curvature and a loss-aversion coefficient $$\ell$$: $$ v(x) = \begin{cases} x^{0.88}, & x \ge 0\cr -\ell(-x)^{0.88}, & x < 0 \end{cases} \qquad \ell \approx 2.25 $$ ![Figure 15.1: The prospect-theory value function](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-7115939046d4128d540bc2b64fc74ff717a0e58a%2Ffig_15_01_prospect_value_function.png?alt=media) **Figure 15.1: The prospect-theory value function.** v(x) = x^0.88 for gains and −2.25(−x)^0.88 for losses, with the kink at the reference point marked and the two worked gambles of §15.1 plotted on it. The lower arm is steeper than the upper one and both arms flatten as they run away from the kink; the kink is what makes the location of the reference point the most consequential thing about a decision. *Source: Author's construction; functional form and parameters as in Tversky and Kahneman (1992).* Here $$x$$ is the outcome measured *as a deviation from the reference point*, and the exponent and coefficient are the values Tversky and Kahneman fitted to experimental choices in their 1992 restatement. Figure 15.1 plots it, with the two gambles worked below marked on the curve. Described in words: an S laid on its side, bent through the origin, with the lower arm steeper than the upper one and both arms flattening as they run away from the kink. The kink at zero is the whole theory in one feature. It is what makes the location of the reference point the most consequential thing about a decision. A two-outcome example fixes the magnitudes. Offer a coin flip: heads you win 200 dollars, tails you lose 100. Expected monetary value is 50 dollars, and any risk-neutral agent takes it; a plausibly risk-averse expected-utility agent with any normal level of wealth takes it too, because a gamble this small is a rounding error against lifetime wealth. Evaluate it with the function above, holding probabilities at their true values so that loss aversion is isolated. The gain is worth $$200^{0.88} \approx 106$$; the loss is worth $$-2.25 \times 100^{0.88} \approx -129$$. The prospect's value is $$0.5(106) + 0.5(-129) \approx -12$$. It is rejected. To make an even-money gamble against a $100 loss acceptable, the winning outcome has to be raised to roughly $250. Rabin's calibration argument is what turns this example into a reason to change the theory rather than the parameter. An expected-utility agent who declines a modestly favorable small gamble at every level of wealth must, by the curvature that declining it requires, decline large gambles with enormous upside — refusals that nobody defends and nobody observes. Small-stakes risk aversion and large-stakes risk aversion cannot both be delivered by the concavity of a utility function over wealth. A kink at a reference point delivers the first and leaves the second alone. The same function delivers the risk seeking that drives the chapter's opening episode. Take a household already down 100 dollars on a position. Sitting still is worth $$-129$$. Now offer it a coin flip between recovering to even and losing another 100: the outcomes are $$0$$ and $$-200$$, worth $$0$$ and $$-2.25 \times 200^{0.88} \approx -238$$, so the gamble is worth about $$-119$$. The gamble is *preferred* to the certain loss, even though it is fair and the household is nominally risk-averse. Diminishing sensitivity has made the second hundred dollars of loss cheaper than the first, and the prospect of erasing the loss entirely is worth the risk. That is the disposition effect. A position trading below its purchase price puts the household on the convex arm, where it will gamble; a position trading above it puts the household on the concave arm, where it will lock in. Shefrin and Statman named the pattern in 1985 and derived it from exactly this shape; Odean measured it in trades thirteen years later. The theory came first, which is worth saying, because behavioral finance is often caricatured as a search for stories to fit anomalies. The fourth element has its own picture, and its own price. Figure 15.2 draws the weighting function and then does one thing with it. ![Figure 15.2: Probability weighting](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-25a36814a96586f2a62d0ac73517e642c179be89%2Ffig_15_02_probability_weighting.png?alt=media) **Figure 15.2: Probability weighting.** Panel (a) is the weighting function in the rank-dependent form Problem 3 uses, at Tversky and Kahneman's fitted parameter, against the forty-five-degree line a probability would follow. Small probabilities lie above it and everything else lies below, with the crossing at about 0.34. A one-percent chance enters a valuation weighted as 0.0553 — as though it were five and a half percent — and a ninety-nine percent chance enters at 0.9116, so the two weights sum to 0.967 rather than to one. That failure to sum is not a mistake in the arithmetic: the weights are applied to ranked outcomes through a transformation of the cumulative distribution, so no coherent belief corresponds to them, and calling them subjective probabilities is the one reading the theory forbids. Panel (b) prices two securities with that function. Both have an expected payoff of exactly one dollar; one pays a hundred with probability one percent and the other pays two with probability one half. A weighting investor pays 3.72 for the first and 0.75 for the second — a factor of five apart, on identical expectations — which is an expected return of minus 73 percent on the lottery against plus 34 percent on the coin flip. That is the whole of the lottery-stock result. It says that positively skewed securities should be overpriced, that the overpricing is a preference rather than an error, and that the reason a diversified household still holds a handful of volatile names may be that diversification destroys the feature it is paying for. **Table 15.1: The four elements and where each shows up in financial data** | Element of prospect theory | Financial fingerprint | | -------------------------- | ------------------------------------------------------------------------------------------------------------------------------------ | | Reference dependence | Disposition effect (Ch 14 §14.2); nominal loss aversion by house sellers, who set higher asking prices when a sale would book a loss | | Loss aversion | Non-participation in equity markets (Ch 14 §14.2); the equity premium under short evaluation horizons (§15.2) | | Diminishing sensitivity | Risk seeking over losses: holding and doubling down on positions already under water | | Probability weighting | Demand for positively skewed "lottery" stocks and IPOs; deliberate under-diversification | *Source: Author's construction; elements as in Kahneman and Tversky (1979) and Tversky and Kahneman (1992).* The last row deserves its own line, because it inverts a standard result. Barberis and Huang showed that an investor who overweights small probabilities will pay up for a security with a small chance of a very large payoff, and will do so *even holding a diversified portfolio elsewhere* — which means positively skewed stocks should be overpriced and should earn low average returns. The empirical counterpart has held up across markets and sample periods: stocks with lottery-like characteristics, however measured, underperform. And the same preference rationalizes the under-diversification Chapter 14 §14.2 documented. A diversified portfolio has no skewness; a concentrated position in one volatile stock does. A household that wants a lottery ticket cannot get one by holding the market, so the handful of names in the median direct stockholder's account may not be a failure to understand diversification. It may be a preference for the thing diversification destroys. > **Box 15.1 — How a discount brokerage keeps records** > > The chapter's best evidence — Odean's, and the literature that followed it — comes from account-level records at a discount brokerage, and the properties of that data explain both why the results are credible and why the chapter's data exercise cannot reproduce them. > > **What the file contains.** For a large sample of accounts over several years: every trade, with its date, security, direction, quantity and price; end-of-period position statements, so holdings are known and not inferred; and, for a subsample, demographic information supplied at account opening. The critical property is that the same household is observed deciding repeatedly. A researcher can compute what a household *could* have sold on a day it sold something, which is what makes §15.1's proportion-of-gains-realised measure possible at all. A market-wide dataset cannot do that, because it does not know which shares belong to whom or what they cost. > > **What it omits.** Accounts at other institutions, so a household's portfolio is observed only in part; retirement accounts, which by Chapter 14's Table 14.1 hold most household equity exposure; the household's non-financial position; and the reason for any trade. A liquidity-driven sale and a disposition-driven sale look identical. > > **Why it is not redistributable.** The brokerage supplied the data under an agreement, and the accounts belong to identifiable people. Even stripped of names, a transaction history is close to a fingerprint. The data went to a small number of researchers and cannot be posted. > > That is a fact about this chapter's evidence base rather than a complaint about it. The most influential results in behavioural finance rest on proprietary files that a reader cannot open, and the replication that is possible — on later brokerage samples, on other countries' tax records, on exchange-level open-close data — is replication of the *pattern* rather than of the study. Chapter 6 §6.4's post-publication decay has no counterpart here, because nobody can trade on what nobody can see. *** ## 15.2 Mental Accounting, Narrow Framing, and the Equity Premium Prospect theory says outcomes are evaluated relative to a reference point. It does not say which outcomes get grouped together before the evaluation happens, and that turns out to matter as much as the curvature. Tversky and Kahneman's 1981 demonstration is the cleanest exhibit. A public-health program with fixed outcomes and fixed probabilities is described once in lives saved and once in lives lost; the majority choice flips from the certain option to the gamble, with nothing changed but the wording. Framing is not a distortion applied to a decision that exists independently of it. It is part of what the decision is. **Mental accounting** is the practice of sorting money into separate notional accounts and applying different rules to each — the retirement account, the college fund, the play money, the mortgage that is paid down while a credit card balance revolves at a much higher rate. Chapter 14 §14.5 recorded one expensive consequence without naming it: households commonly put bonds in the taxable account and equities in the tax-deferred one, exactly inverting the tax-efficient location, because the retirement account is labeled long-term and therefore receives the long-term asset. The labels are doing work that the tax code is not. **Narrow framing** is the general case: evaluating each gamble in isolation rather than as one more component of a portfolio and of lifetime wealth. Standard theory insists on the broad frame, and the insistence has teeth — the reason a small fair-ish gamble should be accepted is precisely that it is small *relative to everything else*, which is a fact about the frame, not about the gamble. The most consequential application is Benartzi and Thaler's account of the equity premium. Combine loss aversion with a short evaluation period and something striking happens. Over a day, equities lose money roughly half the time; over a decade, almost never. An investor who checks annually therefore experiences equities as a sequence of frequent losses, each weighted more than twice as heavily as an equivalent gain; an investor who checks once a decade barely experiences losses at all. Benartzi and Thaler asked what evaluation period would make a loss-averse investor indifferent between stocks and bonds at the historical return distributions, and the answer came out at about one year — which is, suggestively, the frequency at which portfolios are reported, taxed, and reviewed. Call this **myopic loss aversion**: not loss aversion alone, and not short horizons alone, but their combination. Chapter 5 poses the equity premium puzzle as a challenge to the consumption-based model, where the required risk aversion is implausibly large. Myopic loss aversion answers it by changing the preference and the frame rather than the coefficient. Two things should be said about the answer. It has direct experimental support: subjects shown portfolio outcomes less frequently allocate more to the risky asset, which is a strange prediction for any rational model to make and it holds up. And it is not free — it purchases the equity premium at the cost of making an aggregate price depend on how often investors happen to look at their statements, which is not a deep parameter and is not stable across institutions or decades. The frame also reaches back to Chapter 14's inertia. Status quo bias, procrastination, and choice overload — the mechanisms behind the Madrian-Shea default effect and the near-zero median reallocation rate of Chapter 14 §14.2 — are not prospect theory, but they share its logic: the decision that gets made depends on how the problem is presented, and a decision not made is itself an allocation. There is an irony in the combination. A household that never looks at its retirement account is protected from myopic loss aversion by the same inattention that leaves it at a three percent contribution rate. *** ## 15.3 Beliefs: Overconfidence, Extrapolation, Attention Prospect theory changes preferences. A separate and equally productive tradition leaves preferences alone and changes beliefs. ### Overconfidence and trading volume The single largest quantitative puzzle in individual investor behavior is not any particular trade. It is the sheer volume of trading. Rational models with common priors struggle to generate much trade at all: if I want to buy from you at a price you want to sell at, one of us should wonder why the other is willing. Overconfidence supplies the missing disagreement. People overestimate the precision of their own information, and the prediction is sharp — overconfident investors trade more, and trade to their cost. Odean tested it on the brokerage records and found that the stocks individuals bought subsequently underperformed the stocks they sold, before costs, so the trades were not merely expensive but wrong. Barber and Odean then sorted households by turnover: the most active accounts underperformed the least active by something on the order of six or seven percentage points a year over the sample, with gross returns similar across groups and the gap almost entirely explained by trading costs and bad selection. The cleanest identification came from an unlikely direction. Psychology finds overconfidence more pronounced in men than in women, particularly in domains coded as masculine, of which finance is one. Barber and Odean's "boys will be boys" paper used the account-holder gender field in the brokerage records and found that men traded roughly 45 percent more than women, and that the extra trading cost them: the reduction in net returns from trading was on the order of two and a half percentage points a year for men against something under two for women, with the gap widest between single men and single women. The variable is plausibly unrelated to information and closely related to the psychological trait, which is about as close to an experiment as this literature gets outside a laboratory. ### Extrapolation, and why it matters for asset pricing Ask investors what they expect the stock market to return over the next year. Greenwood and Shleifer assembled six such survey series — individual investors, chief financial officers, newsletter writers, and others — spanning several decades, and established three things. The series agree with each other, which means they are measuring something real rather than survey noise. They are strongly **extrapolative**: expected returns are high after the market has risen and when prices are high relative to fundamentals, low after it has fallen. And they do not forecast returns. If anything the relationship runs the wrong way, since high prices are followed on average by low returns. Then the finding that makes this a chapter in an asset pricing book rather than a curiosity. Model-based expected returns — the ones that come out of the return-predictability regressions in Chapter 7 — move *opposite* to the survey expectations. The rational account of predictability says required returns are countercyclical: investors demand more compensation to hold equities in bad times, when prices are low, which is precisely what generates the predictability that Chapter 7's regressions detect. Chapter 5's habit-formation model is the leading formalization, with risk aversion rising as consumption falls toward habit. The survey respondents are doing the opposite. They expect the most exactly when the model says they should be *demanding* the most, and expect least when prices are cheapest. The two accounts fit the same predictability evidence and disagree completely about the mechanism. In the rational version, high prices forecast low returns because holders are content to accept less. In the extrapolative version, high prices forecast low returns because holders expected too much and will be disappointed. Both are consistent with the regression. Only one is consistent with the surveys, and only one is consistent with the fact that flows into equity funds are strongly positively related to the survey expectations — investors are not merely saying it, they are acting on it. Chapter 7 stages this as the Fama-Shiller disagreement; the survey evidence is the sharpest piece of testimony the behavioral side has, and it is why extrapolative models of expectation formation now sit alongside habit and long-run risk as candidate explanations of the same facts rather than beneath them. ### Limited attention The third belief mechanism is not error but scarcity: attention is a finite resource, and information that is not attended to is not priced immediately. Post-earnings-announcement drift is the standing exhibit. Prices respond to an earnings surprise on the day, and then continue drifting in the same direction for weeks — which is a violation of semi-strong efficiency in its most literal form (Chapter 7), and a violation that has survived four decades of scrutiny. The attention interpretation gets its traction from the timing. The immediate response is weaker, and the subsequent drift stronger, for announcements made on Fridays, and for announcements that land on days crowded with other companies' announcements. Nothing about the information differs; only the number of eyes on it. Attention also shapes what households buy, through an asymmetry Barber and Odean identified and that is obvious once stated. Selling is a choice among the handful of stocks you already own. Buying is a choice among thousands, and no household searches thousands. So the buy decision is made from whatever subset attention has already delivered — stocks in the news, stocks with extreme returns, stocks with abnormal volume. Individual investors are net buyers of attention-grabbing stocks; institutions, whose search process is systematic, are not. This is a mechanism for correlated demand that requires no error at all, only a search cost, and it points directly at §15.4's question. *** ## 15.4 From Individuals to Prices: Behavioral Asset Pricing Nothing so far establishes that any of this matters for prices. The standard objection is Friedman's and it is a good one: errors that are independent across investors cancel, and errors that are correlated invite arbitrage. Either way prices survive. The first half of the objection has been answered already. The biases of §§15.1-15.3 are not independent draws. Reference points cluster because purchase prices cluster; extrapolation is driven by the same public return history for everyone; attention is drawn by the same news. Correlated demand is the default, not the special case. The second half is the hard part, and it has a formal answer. DeLong, Shleifer, Summers and Waldmann's noise-trader model turns the arbitrage argument against itself. Suppose some investors' demand is driven by sentiment that fluctuates unpredictably. An arbitrageur who sees an asset trading above fundamental value and shorts it takes on the fundamental risk of the asset — and *also* the risk that sentiment gets more extreme before it corrects, driving the price further away and generating a loss on a position that was right. Call this **noise-trader risk**. It is created entirely by the presence of the noise traders, it cannot be hedged, and because it is a genuine risk borne by a finite-horizon arbitrageur, it limits the size of the position rationally taken. Mispricing survives in equilibrium. The model has two further implications that are easy to miss and are the reason it is still taught. Noise traders bear risk they themselves create, so they can earn *higher* expected returns than the arbitrageurs trading against them, which means the survival argument — that the irrational are selected out — does not go through. And prices are excessively volatile relative to fundamentals, because sentiment is an extra shock. The closed-end fund puzzle, in which funds trade at discounts that move together across funds and correlate with small-stock returns even though each fund's assets are marked daily, was the canonical early application. **Sentiment**, measured. If sentiment moves prices, it should be possible to construct an index of it and test what it predicts. Baker and Wurgler did so, extracting a common factor from a set of market-based proxies — the closed-end fund discount, share turnover, the number of IPOs and their first-day returns, the equity share of new issues, and the premium at which dividend payers trade relative to non-payers. The index is not a measure of the *level* of mispricing, which is unobservable. It is a measure of a time-varying tilt. The prediction is cross-sectional, and it is the useful part. Sentiment should move the prices of stocks that are both hard to value and hard to arbitrage, and leave the rest roughly alone. So following high-sentiment periods, subsequent returns should be relatively *low* for small, young, unprofitable, non-dividend-paying, high-volatility, extreme-growth and distressed stocks, and relatively high for those same stocks following low-sentiment periods; bond-like, profitable, dividend-paying stocks should show little conditional pattern either way. That is what the data show. The result is not that sentiment predicts the market — the aggregate evidence is much weaker — but that it predicts the *spread* between the arbitrage-resistant and arbitrage-friendly ends of the cross-section. Which is the theory's actual content: sentiment reaches prices where the correcting mechanism is weakest. ![Figure 15.4: Does sentiment forecast anything](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-a4cf4ea0a66bed6cf07b0951a46ca7f3b5c20d81%2Ffig_15_04_does_sentiment_forecast_anything.png?alt=media) **Figure 15.4: Does sentiment forecast anything?** The AAII Sentiment Survey's bull-bear spread — the weekly retail poll, averaged to months — against the market, in both directions, from 1987 to 2025. Panel (a) is the test the aggregate claim implies: sentiment now against the market's return over the following twelve months. The relation has the contrarian sign the theory wants and a correlation of −0.17, which is to say it is there and it is weak. Panel (b) reverses the arrow and finds a relation more than twice as strong, at +0.41: the survey is far better explained by the returns that preceded it than by the returns that follow it. Retail sentiment, measured this way, is substantially a report on the recent past — which is the honest content of §15.4's scorecard row, drawn. Three things the figure does not establish. It is the *aggregate* test, and the chapter's claim is that what survives is the cross-sectional prediction — the spread between the arbitrage-resistant and arbitrage-friendly ends — on which no aggregate scatter can speak. The twelve-month windows overlap, so 456 monthly observations contain nearer forty independent ones, and neither correlation should be read as a precise estimate. And AAII polls a self-selected membership of individual investors, which is a particular slice of Chapter 14's households rather than Chapter 4's marginal investor. *Source: AAII Sentiment Survey; Kenneth French's data library (value-weighted market return including dividends). Author's calculations.* **Bubbles**, in one paragraph, because the episodes belong elsewhere. Two ingredients generate a price above any investor's own valuation. Miller's observation is that with binding short-sale constraints, price reflects the beliefs of the optimists alone, since the pessimists are locked out rather than merely quiet — which is exactly the Palm mechanism of Chapter 3, where ninety-five percent of the shares could not be lent. Add heterogeneous beliefs that turn over, and Scheinkman and Xiong's resale-option logic pushes price above even the optimist's valuation, because each holder expects to sell to someone more optimistic still. The empirical work on the late-1990s episode fits: short interest in internet stocks ran against the lendable float, and sophisticated investors were on balance riding the rise rather than leaning against it. Whether such episodes are identifiable in real time is Chapter 7's predictability debate, and the crisis-scale version is the 2008 volume's. > **Box 15.2 — GameStop, January 2021** > > The episode is worth one box because three of this chapter's mechanisms operated at once, and because it is usually told as a story about a crowd when it is better read as a story about a hedging chain. > > The setup was a stock with unusually high short interest, held short by a small number of funds, against a float that was small relative to that short position. Through January 2021 retail order flow into the stock and into short-dated call options on it rose sharply, coordinated loosely through social media and executed through brokerages that had recently removed commissions. > > Section 15.3's attention channel supplies the first mechanism. Attention is scarce, purchases are constrained by it in a way sales are not, and a stock that becomes salient attracts buying from people who were not previously choosing between it and anything else. The second mechanism is Chapter 8 §8.6's. A dealer who sells a call is short gamma; to stay hedged as the stock rises, he must buy more stock, and the more of it he has sold the more he must buy. Concentrated buying of short-dated out-of-the-money calls therefore converts directly into demand for the underlying — mechanically, at whatever price. The third is the short side: a short position that moves against its holder produces margin calls, and closing a short means buying, into the same market. > > What the episode is not is a demonstration that sentiment moves prices in the ordinary case. The float was small, the short interest extreme, and the option activity concentrated in a way that is rare. Section 15.4's question — whether correlated household error reaches prices generally — is not answered by an event chosen for being extraordinary. > > Read it instead as the clearest available illustration of §15.5's asymmetry. The arbitrageur's position was correct on fundamentals and unfundable on the path, and the funds that closed did so because their financing said to, not because their view changed. *** ## 15.5 Limits to Arbitrage: Shleifer and Vishny Textbook arbitrage is a person with a mispricing and a balance sheet. Shleifer and Vishny's contribution was to notice that in the actual world these are two different people. Arbitrage is delegated. It is performed by a small number of specialists — hedge funds, proprietary desks, relative-value books — who know their markets deeply and are therefore, by construction, undiversified. The capital they deploy belongs to investors who do not know those markets, cannot evaluate a position on its merits, and can therefore judge the manager only by realized returns. That is the whole model, and everything follows from it. Consider a manager who has correctly identified an asset trading twenty percent below fundamental value and has put the fund into the position. Sentiment worsens; the gap widens to thirty percent. Two things now happen at once. The opportunity has become *better*: the expected return on the next dollar committed has risen. And the fund has just posted a large loss. Its investors — reasonably, given that they cannot distinguish a manager who is early from a manager who is wrong — redeem. The manager, who should be buying, is selling. This is **performance-based arbitrage**, and its signature is that the supply of arbitrage capital contracts exactly when the demand for it is highest. The consequences run in several directions and are worth separating. Arbitrage is weakest precisely where it is needed most. In normal times, when mispricings are small, capital is plentiful and the mechanism works about as the textbook says. In extreme states, when mispricings are large, capital withdraws. So the standard defense of market efficiency — that large mispricings would attract capital — is exactly backwards about the states in which the defense is being invoked. Arbitrageurs will decline trades they believe in. Knowing that the redemption mechanism exists, a rational manager sizes positions to survive the drawdown rather than to maximize expected value, holds cash against the possibility of the gap widening, and prefers trades with a *catalyst* — a merger closing, an index reconstitution, a maturity date — over trades that are merely cheap and might stay cheap. This is why convergence trades cluster around events and why "value" positions with no convergence date are the last to be arbitraged. Arbitrage can amplify rather than correct. A manager who anticipates forced selling by others has reason to sell first, and the liquidation of a crowded position is a price move that has nothing to do with anyone's view of value. Chapter 6 §6.8's discussion of factor crowding is the same phenomenon observed in the cross-section. Note what this section is and is not doing. **Chapter 16 §16.5 is this book's canonical statement that constrained capital moves prices**, and it should be read as the general case: leverage ratios, capital charges, haircuts, redemption queues, and mandates, each capable of forcing a sale that fundamentals do not warrant, with the margin-spiral dynamics worked through on the balance sheets there. This section supplies the complement that §16.5 explicitly points back to. Chapter 16 explains why the arbitrage capital is *not there* when it is needed. Shleifer and Vishny explain why an arbitrageur who *has* capital may still decline the trade — because the risk being priced is not the risk of being wrong, but the risk of being right too early in front of an investor who cannot tell the difference. The two mechanisms are separable in principle and inseparable in practice, and the September 1998 unwind of a large relative-value fund is the standing illustration of both operating at once. Three further limits belong in the same account and are developed elsewhere in this book rather than here. **Short-sale constraints** are the mechanical limit, and Chapter 3's opening episode is the worked exhibit: a mispricing of twenty billion dollars, published in the financial press, surviving because Palm's lendable float was tiny, borrow costs ran to tens of percent a year, lenders could recall at will, and the convergence date was contingent on a tax ruling. Chapter 3 §3.2 makes the general point that the no-arbitrage assumption is an assumption about a trade, and trades are executed by holders with balance sheets. This chapter's addition is that the holder doing the executing is usually spending somebody else's money. **Career risk** is the delegation problem one level down, inside the institution. Chapter 16 §16.3 sets it out with the Keynes epigraph it deserves — better for reputation to fail conventionally than to succeed unconventionally — and traces it through benchmark-hugging, window dressing, and the asymmetric flow-performance relationship. A contrarian position that is wrong for two quarters ends a career in a way that a consensus position that is wrong for two quarters does not. The relevant point here is that career risk and performance-based arbitrage are the same mechanism with different principals: in one the money leaves, in the other the manager does. **Fundamental risk and horizon**, finally, are what remain even with permanent capital and no career. Very few mispricings are true arbitrages in Chapter 3's sense. Most are bets that require a model to identify, and a manager who is confident in the model is still exposed to being wrong about it — which is a reason for the mispricing to persist that no amount of patient capital removes. *** ## 15.6 What Behavioral Finance Is Not Three boundaries, because the field's reputation suffers from being asked to be more than it is. **It is not a theory of corporate decisions in this chapter.** Managers are people, and the psychology of §§15.1-15.3 applies to them as it applies to households. But their decisions are made inside a capital structure, a governance system, and a market for corporate control, and those are prerequisites this chapter does not have. So the behavioral-corporate material sits with its prerequisites: equity market timing as a third theory of capital structure alongside trade-off and pecking order is Chapter 23 §23.4, and managerial hubris in acquisitions is Chapter 22 §22.5, next to the merger evidence it is trying to explain. **It is not the claim that institutions are rational and households are not.** Institutional traders exhibit the same biases in the laboratory, and some in the field: disposition effects appear in professional books, and extrapolation appears in the return expectations of chief financial officers, who are as sophisticated a survey population as one can assemble. What differs is not the psychology but the machinery around it — risk limits, mandates, mark-to-market discipline, and a boss. That machinery removes some biases and creates others. Herding and benchmark-hugging are institutional pathologies with no household counterpart, and they are generated by the *solution* to the agency problem, not by its absence. The right statement is that the biases which reach prices are the ones the institutional structure fails to arbitrage away, which is a joint proposition about psychology and about industrial organization. **It is not immune to the replication problem.** The 2010s brought a reckoning to the anomalies literature and behavioral finance was implicated, because a great many published anomalies had been offered as behavioral evidence. Large-scale replication efforts found that a substantial share of published cross-sectional predictors fail to survive standard corrections — equal-weighting that overstates the contribution of microcaps, appropriate multiple-testing thresholds, and out-of-sample periods. Chapter 6 §6.4 works through the statistics and the factor zoo. Its sharpest single result belongs here too: McLean and Pontiff document that predictor returns are meaningfully lower out of sample even before publication, and lower again after publication, with the post-publication decay concentrated in exactly the predictors that are cheapest to trade. Read one way this is damaging to behavioral finance, and some of the anomalies it built on have indeed faded. Read another way it is the theory working. If a documented mispricing attracts capital and shrinks, that is limits to arbitrage being relaxed by publication, and the pattern of which anomalies decay — the liquid ones, the ones without short-sale constraints — is the pattern §15.5 predicts. What did *not* fade is instructive: the household behaviors of §§15.1-15.3 replicate across countries, brokerages, and decades. It is the trading strategies built on them that decayed, not the psychology. *** ## 15.7 Who Profits from Whose Biases? Part IV asks of every claim who holds it and what their constraints do to its price. This chapter's version of the question is about a transfer. The transfer is real and it is measurable. Odean's households sold the wrong stocks. Barber and Odean's most active accounts gave up several points a year. The most careful accounting comes from Taiwan, where a complete record of exchange transactions by investor type allowed Barber, Lee, Liu and Odean to compute the aggregate: individual investors' trading losses ran to something on the order of two percent of the country's GDP annually over the period studied, with institutions and dealers on the other side. Whatever else household bias is, it is a payment. The obvious next sentence is that arbitrageurs collect it and correct the prices in the process, and the sentence is wrong in three specific ways this book has now assembled the parts to state. **The correcting capital is other people's, and it leaves at the wrong time.** That is §15.5, and Chapter 16 §16.5's canonical statement is the general case. Prices are corrected up to the point at which constrained capital finds it worthwhile and safe to correct them, and that point moves — it recedes in exactly the states where mispricing is widest. **Most of the household sector is not on the other side of any trade.** Chapter 14 §14.6 is the essential qualification, and it cuts against the caricature in both directions. The household demand that has been growing for four decades is the default-driven, payroll-dated, glide-path-allocated flow into retirement vehicles, and that flow does not trade. It cannot be exploited by an arbitrageur because it does not respond to anything, and it does not exhibit the disposition effect because it does not sell. The behaviors of this chapter belong overwhelmingly to the *other* household demand curve — the wealthy direct holder of §14.6, the taxable brokerage account, the person choosing among thousands of stocks with limited attention. Behavioral finance describes a shrinking share of household activity and, because that share is where all the discretionary trading happens, an undiminished share of household *trading*. **Much of the transfer is collected as fees rather than as corrected prices.** Chapter 17's Berk-Green equilibrium is the mechanism: skill accrues to the manager, not to the investor, because capital flows to skill until the expected alpha net of fees is competed to zero. An intermediary that identifies a behavioral mispricing captures the value of doing so, and the household on the other side of the trade is frequently the same household paying the fee to a different intermediary. Chapter 18 asks what fee and lockup structures do to arbitrage capital's patience, which is the same question §15.5 asks from the other end. Which is where a chapter on a live literature owes its reader a grading rather than a warning. Section 15.6's three boundaries, and the findings of §§15.1-15.5 they bound, sort as follows — on the schema Chapter 6 §6.4 uses for the cross-section and Chapter 20 §20.5 for the demand system. **Table 15.2: The state of the evidence** | Claim | Status | | -------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | The household biases of §§15.1-15.3 are real and replicate | **Established.** The disposition effect, the turnover-performance gap, and attention-driven buying are measured in trades rather than in surveys, and reproduce across countries, brokerages, and decades | | Prospect theory describes choice under risk better than expected utility does | **Established** as description. The reference point is supplied by the application rather than by the theory, which is where the discipline has to come from | | Survey expectations are extrapolative and run opposite to model-based expected returns | **Widely accepted.** Six independent series agree with each other, fail to forecast returns, and are followed by fund flows | | Correlated error reaches prices only where arbitrage is limited | **Widely accepted** as the field's logical structure; the *level* of mispricing it permits is unobservable, which is why the sentiment index measures a tilt rather than a level | | Sentiment moves the aggregate market | **Contested.** What holds is the cross-sectional prediction — low subsequent returns at the hard-to-value, hard-to-arbitrage end after high sentiment; the aggregate evidence is much weaker | | The published anomalies built on behavioral evidence survive replication | **Contested.** A substantial share fail standard corrections, and predictor returns fall out of sample and again after publication, most where arbitrage is cheapest | *Source: Author's assessment of the literature discussed in §§15.1-15.6.* Which leaves the picture Part IV has been building toward. The household of Chapter 14 arrives at the market with the psychology of this chapter, meets the institutions of Chapters 16 and 17, and hands over some money. Those institutions are not a frictionless correction mechanism; they are holders with liabilities, mandates, redemption terms, and careers. The equilibrium is not the one in which arbitrageurs eliminate everything and it is not the one in which sentiment sets prices unopposed. It is a mixture whose weights depend on who holds the claim and what binds them — which is a demand-system question, and Chapter 20 estimates it. *** ## Elsewhere in the Series * **Market psychology in crisis form** — the 2008 crisis volume. Panic, the collapse of confidence in models, and the behavior of professional risk-takers under extreme stress are narrated there; §15.4's one paragraph on bubbles is deliberately not a substitute. * **Performativity: models that shape the behavior they describe** — this book, Chapter 7 §7.6. Prospect theory says preferences respond to framing; the performativity literature says market participants' framing responds to the models they are taught. The two are closer than either literature admits, and Chapter 7 §7.6 owns the argument. * **Behavioral corporate finance** — this book, Chapters 22 and 23. Managerial hubris and acquirer returns in Chapter 22; equity market timing as a theory of capital structure in Chapter 23. Both sit with their prerequisites, not here. * **Within this book:** the household facts this chapter explains — Chapter 14, especially §§14.2, 14.5, 14.6. The canonical constrained-capital statement this chapter's §15.5 feeds — Chapter 16 §16.5, with career risk in §16.3. The Palm episode as the worked short-sale-constraint exhibit — Chapter 3. The equity premium puzzle myopic loss aversion addresses, and habit as the rational mirror of extrapolation — Chapter 5. The replication statistics and the factor zoo — Chapter 6. Predictability and the Fama-Shiller disagreement — Chapter 7 §7.4, which cites §15.3's survey evidence as the behavioral side's sharpest testimony; the compact limits-to-arbitrage statement for course use, which points back to §§15.4-15.5 — Chapter 7 §7.5. Berk-Green and the capture of alpha by managers — Chapter 17. Arbitrage capital and fund structure — Chapter 18. Demand-system estimation — Chapter 20. *** ## Summary 1. **The disposition effect is measured in trades, not surveys.** Odean's ten thousand discount-brokerage accounts, 1987-1993, realized gains at roughly half again the rate at which they realized losses, with the gap narrowing in December when the tax code rewards the opposite. The winners sold went on to outperform the losers kept. The behavior was costly and it was not ignorance of the tax rule. 2. **Prospect theory replaces four features of expected utility at once.** Outcomes are evaluated as gains and losses relative to a reference point; losses count roughly twice as heavily as equivalent gains; sensitivity diminishes in both directions, making the value function convex over losses and hence risk-seeking there; and small probabilities are overweighted. The calibrated value function rejects a 50/50 gamble of $$+200$$ against $$-100$$ and requires a gain of roughly $250 to accept it. 3. **Each element leaves a fingerprint.** Reference dependence gives the disposition effect and nominal loss aversion in housing; convexity over losses gives the holding and doubling-down of underwater positions; probability weighting gives demand for lottery-like stocks, which are correspondingly overpriced, and rationalizes the under-diversification of Chapter 14 §14.2 as a preference for skewness rather than a failure to diversify. 4. **Framing determines what gets evaluated.** Mental accounting explains the inverted asset location of Chapter 14 §14.5. Narrow framing plus loss aversion gives myopic loss aversion, and Benartzi and Thaler's calculation that an evaluation period of about one year makes stocks and bonds equally attractive at historical distributions — an answer to Chapter 5's equity premium puzzle that changes the frame rather than the risk aversion coefficient, at the cost of making an aggregate price depend on how often people look. 5. **Overconfidence generates the trading volume that rational models cannot.** Individuals' purchases underperform their sales; the highest-turnover accounts underperform the lowest by several percentage points a year; and men, whom psychology finds more overconfident, trade substantially more than women and lose correspondingly more to it. 6. **Survey expectations are extrapolative, and they run opposite to model-based expected returns.** Greenwood and Shleifer's six series agree with each other, rise after the market rises, fail to forecast returns, and are negatively correlated with the required returns that Chapter 7's predictability regressions imply and Chapter 5's habit model rationalizes. Both accounts fit the regression evidence; only one fits the surveys, and fund flows follow the surveys. 7. **Limited attention is scarcity, not error.** Post-earnings-announcement drift is stronger for Friday and crowded announcements. Buying is a search over thousands of names and selling a choice among the few already held, so individuals are net buyers of attention-grabbing stocks — a mechanism for correlated demand requiring no bias at all. 8. **Individual error reaches prices only if it is correlated and arbitrage is limited.** Correlation is the default, since reference points, return histories, and news are shared. DeLong, Shleifer, Summers and Waldmann show that arbitrage against sentiment carries noise-trader risk — the risk that mispricing widens before it converges — which bounds the position, allows noise traders to survive and even to out-earn arbitrageurs, and generates excess volatility. Baker and Wurgler's sentiment index predicts the cross-sectional *spread*: high sentiment is followed by low relative returns on small, young, unprofitable, volatile and distressed stocks, precisely the hard-to-value and hard-to-arbitrage end. 9. **Shleifer and Vishny make behavioral finance a chapter about institutions.** Arbitrage is delegated, specialized, and funded by investors who can judge only realized returns, so a widening mispricing triggers redemptions exactly when the opportunity is best. Arbitrageurs therefore size to survive drawdowns, hold cash, prefer trades with catalysts, and sometimes amplify rather than correct. This is the complement to Chapter 16 §16.5's canonical constrained-capital statement: §16.5 explains why the capital is absent, this explains why present capital declines the trade. Short-sale constraints (Chapter 3's Palm episode) and career risk (Chapter 16 §16.3) complete the account. 10. **The transfer is real; the correction is partial.** Household trading losses have been measured at the scale of a couple of percent of GDP in the one market with complete transaction records. But the correcting capital is other people's and leaves at the wrong time; the fastest-growing part of household demand is default-driven and does not trade at all (Chapter 14 §14.6); and much of the surplus is captured as fees rather than as corrected prices (Chapter 17). The behavioral anomalies that faded after publication faded where arbitrage was cheapest, which is the theory working, not failing. *** ## Key Terms * **Prospect theory**: Kahneman and Tversky's descriptive theory of choice under risk, in which outcomes are evaluated as gains and losses relative to a reference point, through a value function that is concave over gains, convex and steeper over losses, and in which probabilities enter through a weighting function that overweights small probabilities * **Loss aversion**: The property that the value function is steeper below the reference point than above it, with a coefficient estimated at roughly two; the source of non-participation, of myopic loss aversion, and of the equity premium in behavioral accounts * **Disposition effect**: The tendency to realize gains at a higher rate than losses; predicted by reference dependence with diminishing sensitivity, measured in brokerage records by Odean, and costly because the winners sold outperform the losers kept * **Mental accounting**: The practice of assigning money to separate notional accounts governed by different rules, so that fungible dollars are treated as non-fungible; the source of inverted asset location and of layered "safety-then-aspiration" portfolios * **Noise-trader risk**: The risk, borne by an arbitrageur trading against sentiment, that the mispricing widens before it converges; created by the noise traders themselves, unhedgeable, and sufficient to bound arbitrage positions and allow mispricing in equilibrium * **Sentiment**: A common, time-varying component of investor demand not justified by fundamentals; measured by Baker and Wurgler as a factor extracted from market-based proxies, and predictive of the cross-sectional spread between hard-to-arbitrage and easy-to-arbitrage stocks * **Limits to arbitrage**: The set of reasons — delegated and performance-sensitive capital, short-sale constraints, career risk, fundamental risk, and horizon — why a known mispricing is not traded away; the reason behavioral finance is a claim about institutions and not only about psychology * **Extrapolative expectations**: Beliefs about future returns formed by projecting recent returns forward; documented across six independent survey series, procyclical, and negatively correlated with the countercyclical required returns implied by rational models of predictability *** ## Readings ### Required * Kahneman, D. and A. Tversky (1979). "Prospect Theory: An Analysis of Decision under Risk." *Econometrica* 47(2): 263-291. *The founding paper, and short. Read the numbered problem pairs first and answer them yourself before reading the analysis; the point is that you will make the same choices the subjects made. The value function of §15.1 is Figure 3.* * Shleifer, A. and R. Vishny (1997). "The Limits of Arbitrage." *Journal of Finance* 52(1): 35-55. *The structural argument that makes this a Part IV chapter. Read section II's model for the mechanism and section IV for the implications, particularly the claim that arbitrage is least effective in extreme states — the exact opposite of the standard efficiency defense.* ### Recommended * Barberis, N. and R. Thaler (2003). "A Survey of Behavioral Finance." In G. Constantinides, M. Harris and R. Stulz (eds.), *Handbook of the Economics of Finance*, Volume 1B, Chapter 18. Elsevier. *The best single map of the field, organized exactly as this chapter is: limits to arbitrage first, then psychology, then applications. Use it as the syllabus for anything here treated in a paragraph.* * Tversky, A. and D. Kahneman (1992). "Advances in Prospect Theory: Cumulative Representation of Uncertainty." *Journal of Risk and Uncertainty* 5(4): 297-323. *Where the value function of §15.1 gets the parameters this chapter quotes and where probability weighting is given a usable rank-dependent form. The 1979 paper states the theory; this one makes it something you can compute with, which is what the lottery-stock result requires.* * Rabin, M. (2000). "Risk Aversion and Expected-Utility Theory: A Calibration Theorem." *Econometrica* 68(5): 1281-1292. *The argument added to §15.1: small-stakes risk aversion, taken seriously inside expected utility, implies absurd large-stakes behavior. The Rabin and Thaler "Anomalies" column of 2001 is the readable restatement, and the one to assign.* * Tversky, A. and D. Kahneman (1981). "The Framing of Decisions and the Psychology of Choice." *Science* 211(4481): 453-458. *Four pages, and the source for §15.2's opening. The public-health problem is the demonstration; the accompanying discussion of why a normatively irrelevant description changes the choice is what a finance reader should take away.* * Odean, T. (1998). "Are Investors Reluctant to Realize Their Losses?" *Journal of Finance* 53(5): 1775-1798. *The opening episode. The methodological contribution is the construction of the proportion of gains and losses realized relative to what was available to realize; understand that denominator before reading the results.* * Barber, B. and T. Odean (2001). "Boys Will Be Boys: Gender, Overconfidence, and Common Stock Investment." *Quarterly Journal of Economics* 116(1): 261-292. *Overconfidence identified off a variable that is plausibly unrelated to information.* * DeLong, J. B., A. Shleifer, L. Summers and R. Waldmann (1990). "Noise Trader Risk in Financial Markets." *Journal of Political Economy* 98(4): 703-738. *Why arbitrage does not eliminate sentiment, and why noise traders can survive and out-earn the investors trading against them.* * Greenwood, R. and A. Shleifer (2014). "Expectations of Returns and Expected Returns." *Review of Financial Studies* 27(3): 714-746. *Six survey series, all extrapolative, all pointing the wrong way relative to model-based expected returns. The single most useful piece of evidence in the rational-versus-behavioral debate over predictability.* * Benartzi, S. and R. Thaler (1995). "Myopic Loss Aversion and the Equity Premium Puzzle." *Quarterly Journal of Economics* 110(1): 73-92. *Read against Chapter 5.* * Baker, M. and J. Wurgler (2006). "Investor Sentiment and the Cross-Section of Stock Returns." *Journal of Finance* 61(4): 1645-1680. *The index and, more importantly, the conditional cross-sectional predictions. The index itself is free to download from Wurgler's page and is the basis of the data exercise.* * Barberis, N. and M. Huang (2008). "Stocks as Lotteries: The Implications of Probability Weighting for Security Prices." *American Economic Review* 98(5): 2066-2100. *Probability weighting taken to its asset-pricing conclusion, including the result that a skewed security can be overpriced even for an investor holding a diversified portfolio.* * Barber, B. and T. Odean (2008). "All That Glitters: The Effect of Attention and News on the Buying Behavior of Individual and Institutional Investors." *Review of Financial Studies* 21(2): 785-818. *The buy-versus-sell search asymmetry of §15.3.* * Chen, G., K. A. Kim, J. R. Nofsinger and O. M. Rui (2007). "Trading Performance, Disposition Effect, Overconfidence, Representativeness Bias, and Experience of Emerging Market Investors." *Journal of Behavioral Decision Making* 20(4): 425-451. *An external-validity check on §§15.1-15.3 and on §15.7's transfer statistic, which rests on one market. The biases replicate outside the United States, and experience attenuates some of them but not all.* * Lee, C., A. Shleifer and R. Thaler (1991). "Investor Sentiment and the Closed-End Fund Puzzle." *Journal of Finance* 46(1): 75-109. *Noise-trader theory's first serious empirical outing.* * Gürkaynak, R. (2008). "Econometric Tests of Asset Price Bubbles: Taking Stock." *Journal of Economic Surveys* 22(1): 166-186. *Where §15.4's single paragraph on bubbles points. The survey's conclusion is the useful one for this book: the econometric tests cannot separate a bubble from a misspecified fundamental, which is why the chapter treats short-sale constraints and resale options as the identifiable content rather than the detection problem.* * McLean, R. D. and J. Pontiff (2016). "Does Academic Research Destroy Stock Return Predictability?" *Journal of Finance* 71(1): 5-32. *The decay statistics of §15.6; the mechanics of the replication debate are Chapter 6's.* * Shiller, R. J. (2003). "From Efficient Markets Theory to Behavioral Finance." *Journal of Economic Perspectives* 17(1): 83-104. *The other side of the exchange from Fama (1998). Reading the pair makes §15.6 self-contained: the same evidence on long-horizon returns and excess volatility, and two incompatible readings of it.* * Simon, H. A. (1955). "A Behavioral Model of Rational Choice." *Quarterly Journal of Economics* 69(1): 99-118. *The pre-history §15.6 needs. The claim that agents satisfice within cognitive limits was made respectable long before the anomalies literature, which matters for the boundary the section is drawing: behavioral finance is a continuation of an old argument, not a reaction to recent data.* * Camerer, C., G. Loewenstein and D. Prelec (2005). "Neuroeconomics: How Neuroscience Can Inform Economics." *Journal of Economic Literature* 43(1): 9-64. *One citation to the discipline immediately beyond §15.6's boundary. Useful for seeing what behavioral finance declines to claim: nothing in §§15.1-15.5 requires a mechanism below the level of choice.* *** ## Discussion Questions 1. **Could the disposition effect be rational?** Construct the strongest possible defense of a household that sells winners and holds losers, without invoking any psychology. Candidates: a belief in mean reversion at the individual-stock level; rebalancing toward target weights; the option value of waiting for a position to recover; liquidity needs met by selling whatever has appreciated; transaction-cost thresholds; and the possibility that the household holds private information. For each, state what it predicts *beyond* the raw asymmetry — about December, about the subsequent returns of the two groups, about accounts held in tax-deferred versus taxable form, about repurchases of stocks previously sold. Which predictions does Odean's evidence contradict, and which survive? Then take the harder step: is there any version of the rational account that survives *both* the December pattern and the subsequent-return pattern simultaneously? 2. **Why limits to arbitrage make behavioral finance a theory of institutions.** Suppose that every household bias catalogued in §§15.1-15.3 is real, correlated, and stable, and that arbitrage capital is unlimited, patient, unlevered, and not delegated. Show that essentially none of the pricing implications of §15.4 survive. Now reintroduce delegation alone, holding the psychology fixed, and derive as much of §15.5's implication list as you can. What does the exercise establish about the *logical* structure of the field — is psychology necessary, sufficient, or neither, for a behavioral theory of prices? Finally, relate your answer to Chapter 16 §16.5's canonical statement: if constrained capital moves prices even with fully rational holders, in what sense is the behavioral evidence doing independent work? 3. **Extrapolation versus habit.** Chapter 7's predictability regressions show that high price-dividend ratios forecast low subsequent returns. Chapter 5's habit model explains this with countercyclical required returns; §15.3's survey evidence shows expectations moving the other way. Design the sharpest test you can that distinguishes the two accounts, being explicit about what each predicts for: survey expectations, mutual fund flows, the cross-section of who is buying and selling at market peaks, option-implied risk premia, and realized consumption growth. Which of these data are available and which are not? Then argue the position that the two accounts are not actually rivals — that some holders extrapolate and others require compensation — and say what such a mixture implies for Chapter 20's estimation problem. 4. **Sentiment and the arbitrage boundary.** Baker and Wurgler's index predicts returns on hard-to-value and hard-to-arbitrage stocks, and not much else. Explain why this pattern is *stronger* evidence for the noise-trader account than an index that predicted the aggregate market would have been. Then attack it: the proxies from which the index is built (closed-end fund discounts, IPO volume, turnover, the dividend premium) are themselves prices or quantities determined in equilibrium, so what exactly is the exclusion restriction that makes the index a measure of sentiment rather than of something fundamental — time-varying risk aversion, say, or the arrival of firms with genuinely uncertain prospects? What would settle it? 5. **The transfer and who ends up with it.** Section 15.7 argues that household trading losses are large, that arbitrageurs do not capture all of them, and that much of the surplus is collected as fees. Trace one dollar lost by a household on a disposition-effect trade through the ecology: who is on the other side of the trade, what do they pay their prime broker and their own investors, and what fraction reaches the ultimate provider of arbitrage capital? Now ask the welfare question the chapter deliberately does not answer. If the household's error is a preference — the skewness demand of §15.1 rather than a mistake — is the transfer a loss at all? What would you need to know about the household to decide, and does the answer differ for the under-diversified lottery-stock holder, the overconfident high-turnover trader, and the inattentive default-driven saver of Chapter 14 §14.3? *** ## Problems **Problem 1 — The value function, worked.** Use §15.1's value function: $$v(x) = x^{0.88}$$ for $$x \ge 0$$ and $$v(x) = -2.25(-x)^{0.88}$$ for $$x < 0$$, with $$x$$ measured as a deviation from the reference point, in dollars. Hold probabilities at their true values throughout, so that loss aversion and diminishing sensitivity are isolated from probability weighting. (a) Evaluate the coin flip that pays 200 on heads and costs 100 on tails. Report the value of each arm and of the prospect, and confirm that it is rejected. (b) Find the winning outcome that makes an even-money gamble against a loss of 100 exactly acceptable. Then show that an even-money gamble is acceptable only when its winning outcome is at least $$2.25^{1/0.88}$$ times its losing one, whatever the stake, and compute that multiple. (c) A household is 100 under water on a position. Compute the value of sitting still and the value of a fair coin flip between recovering to even and losing another 100. Which does it prefer, and by how much? (d) Repeat (c) for a household 200 under water offered a flip between recovering to even and losing another 200. Then show that the gamble is always worth $$2^{0.88}/2$$ of the certain loss, at any stake, and compute that fraction. (e) Use (d) to evaluate the claim that households double down harder the deeper under water they are. Which feature of the functional form produces your answer, and what would a theory need in order to predict that the pull strengthens with the size of the loss? **Problem 2 — Which position gets sold.** A taxable household holds two positions of 100 shares each, both now trading at 60. It bought A at 40 and B at 80. It must raise cash by selling exactly one of them, in full. All figures are in dollars. (a) Taking the purchase price as the reference point, use Problem 1's function to value the realization of each position. Which does the household sell? (b) Now add §14.5's tax rule. Long-term gains and losses are taxed at 20 percent and the household has other realized gains against which a loss can be offset. Compute the after-tax cash raised by each sale and the swing in favour of selling B. (c) Odean found that the winners sold outperform the losers kept by about three percentage points over the following year. Compute the first-year opportunity cost on a position of this size and add it to (b). Then state the one qualification §14.5 attaches to the tax half of your total. (d) The reference point is not given by the theory. Recompute the value of selling B if the household's reference point is instead (i) the price of 60 it has watched for a year, or (ii) a previous peak of 100. Report all three numbers and say what their spread does to the theory's testability. (e) Shefrin and Statman derived the disposition effect in 1985; Odean measured it in 1998. Name the two features of Odean's evidence that any rational rebalancing or mean-reversion account must also reproduce, and say which of the two it cannot. **Problem 3 — Lottery stocks and the weighting function.** Add to Problem 1's value function the rank-dependent weighting function Tversky and Kahneman fitted alongside it. A true probability $$\pi$$ enters the valuation as the decision weight $$ \frac{\pi^{0.61}}{(\pi^{0.61} + (1-\pi)^{0.61})^{1/0.61}} $$ where 0.61 is the curvature they estimated over gains. Two securities pay off in one year, against a riskless rate of zero and a reference point of zero. Security L pays 100 with probability 0.01 and nothing otherwise; security S pays 2 with probability 0.5 and nothing otherwise. All figures are in dollars. (a) Compute the expected payoff of each security. They are equal; report the value. (b) Compute the decision weights attached to probabilities of 0.01, 0.5 and 0.99. Show that the first and third do not sum to one, and explain in one sentence why decision weights are not probabilities. (c) Value each security as the decision weight times the value of its payoff, and convert each to a certainty equivalent in dollars. Report both. (d) Suppose each security trades at its certainty equivalent. Compute the expected return on each. Which is overpriced, by what factor, and how does the answer relate to the last row of Table 15.1? (e) Section 15.1 says the same preference rationalizes the under-diversification of Chapter 14 §14.2. Explain why an investor who already holds a diversified portfolio would still pay up for L. Then name the mechanism from §§15.4-15.5 that keeps L from being arbitraged back to its expected payoff, and say what that mechanism predicts about where in the cross-section the overpricing should be found. **Problem 4 — Noise-trader risk, priced.** A security's fundamental value is 100 and is known to everyone. Next period its price will be 100 plus a sentiment term equal to 0 with probability 0.6 and 40 with probability 0.4. Noise traders hold more than the outstanding supply, so arbitrageurs in aggregate carry a short position of one unit, and they will do so only at a price that compensates them for variance: $$ p = E\[p'] + \lambda \cdot \mathrm{Var}(p') $$ where $$\lambda = 0.02$$ is the price of risk they charge per unit of variance borne. All figures are in dollars. (a) Compute $$E\[p']$$ and $$\mathrm{Var}(p')$$. (b) Compute today's price and its excess over fundamental value. Decompose that excess into the part that is expected future sentiment and the part that is the arbitrageurs' risk premium. (c) Now suppose sentiment next period is certain rather than random: first at 0, then at 16, which is its mean in (a). Compute the price in each case. Use the two answers to say precisely what noise-trader risk is and what it is not. Note also where all of $$\mathrm{Var}(p')$$ came from, given that fundamental value never moves. (d) At the price in (b), compute the arbitrageurs' expected return on the short, its standard deviation, and the Sharpe ratio. (e) Write the excess over fundamental value as a function of $$\lambda$$ and give its derivative. Then bring in §15.5: explain in two sentences why delegated, redeemable capital makes $$\lambda$$ itself depend on the realized price path, and why the two mechanisms together predict that the mispricing is widest exactly where it is already largest. **Problem 5 ★ — An anomaly and its rivals.** A long-short characteristic strategy earns a mean excess return of 0.60 percent a month, with a monthly standard deviation of 3.5 percent, over the 372 months from 1970 through 2000. Over the 288 months from 2001 through 2024 it earns 0.18 percent a month at the same standard deviation. It was first published in 2001. Its decay is concentrated in the large, liquid, cheaply shorted half of the cross-section. (a) Compute the annualized mean, the annualized Sharpe ratio, and the $$t$$-statistic on the mean in each subsample. (b) Chapter 6 §6.4 recommends a working multiple-testing hurdle of about 3.0 on the $$t$$-statistic, against a Bonferroni cutoff of 3.78 for the 316 factors published through 2012. Does the in-sample result clear either? What does clearing the first and not the second establish, and what does it not? (c) McLean and Pontiff find predictor returns roughly 26 percent lower out of sample and roughly 58 percent lower after publication. Compute the second-period mean implied by the post-publication figure and compare it with the 0.18 percent observed. Is this predictor decaying faster or slower than average? (d) Three rivals to the behavioral account: the spread was compensation for risk; the spread was a data-mining artifact; the spread was real but never exceeded trading costs. For each, state one prediction about the post-2001 record that distinguishes it from the behavioral account. Then say which of the three the concentration of the decay in the *liquid* half argues against most directly, and why §15.6 reads that pattern as the theory working rather than failing. (e) Section 15.6 says the anomalies faded and the psychology did not. State the evidence that separates the two claims. Then say what a researcher would have to show in order to establish that the behavioral account of *this* strategy was wrong, rather than merely arbitraged away. *** ## Selected Solutions *Solutions to Problems 1 and 3 follow. Solutions to the remainder are in the instructor materials.* **Problem 1.** (a) $$v(200) = 200^{0.88} = 105.90$$ and $$v(-100) = -2.25 \times 100^{0.88} = -129.47$$. The prospect is worth $$0.5(105.90) + 0.5(-129.47) = \mathbf{-11.79}$$, so it is **rejected** — a gamble with an expected value of $$+50$$ dollars, declined. The rejection is entirely loss aversion: the losing arm is weighted 2.25 times as heavily as an equal-sized gain. (b) Indifference requires $$x^{0.88} = 2.25 \times 100^{0.88}$$, so $$x = (2.25 \times 100^{0.88})^{1/0.88} = \mathbf{251.31}$$. In general the winning outcome must satisfy $$x^{0.88} \ge 2.25 y^{0.88}$$, so $$x/y \ge 2.25^{1/0.88} = \mathbf{2.5131}$$, independent of the stake. **The required odds are a property of the preferences, not of the size of the bet** — which is precisely what an expected-utility household with concave utility over wealth cannot deliver, since for it the required odds shrink toward one as the stake shrinks. (c) Sitting still is worth $$v(-100) = \mathbf{-129.47}$$. The flip is worth $$0.5 v(0) + 0.5 v(-200) = 0.5(0) + 0.5(-238.28) = \mathbf{-119.14}$$. The household **prefers the gamble**, by 10.33 — it is risk-*seeking* over losses, which is the fourfold pattern's third quadrant and the mechanism behind holding on to a losing position. (d) Sitting still is $$v(-200) = -238.28$$; the flip is $$0.5 v(-400) = -219.26$$. In general the flip is worth $$0.5 \times 2.25 \times (2y)^{0.88}$$ against a certain $$2.25 \times y^{0.88}$$, so their ratio is $$2^{0.88}/2 = \mathbf{0.9202}$$ at **every** stake: the gamble is always worth about 92 percent of the certain loss, and is always preferred by the same proportional margin. (e) The claim is false in this functional form. The pull toward doubling down is *scale-invariant* — a household 10,000 under water is drawn to the gamble in exactly the same proportion as one 100 under water — because the value function is a power function, and a power function has constant relative curvature. To predict that the pull strengthens with the depth of the loss, a theory would need the *curvature over losses to increase in the size of the loss*, which the Tversky-Kahneman form does not deliver; or it would need something outside the value function, such as a reference point that drifts, a probability weight applied to the recovery, or the mental accounting of §15.2 in which the position is a separate account whose closure is the thing being avoided. *Two of those are testable and one is not, which is roughly the state of the literature.* **Problem 3.** (a) Security L: $$0.01 \times 100 = 1$$. Security S: $$0.5 \times 2 = 1$$. Both have an expected payoff of $$\mathbf{1}$$, so under expected value they are the same security. (b) With $$w(\pi) = \pi^{0.61}/(\pi^{0.61} + (1-\pi)^{0.61})^{1/0.61}$$: | $$\pi$$ | 0.01 | 0.50 | 0.99 | | ---------- | ---------- | ---------- | ---------- | | $$w(\pi)$$ | **0.0553** | **0.4206** | **0.9116** | A probability of one percent is carried into the valuation as though it were five and a half percent, and $$w(0.01) + w(0.99) = 0.967 \ne 1$$. Decision weights are not probabilities because they are applied to *ranked* outcomes rather than to states: the transformation is of the cumulative distribution, so the weights on a partition need not sum to one and no coherent belief corresponds to them. (c) $$V(L) = w(0.01) v(100) = 0.0553 \times 57.54 = 3.180$$, and $$V(S) = w(0.5) v(2) = 0.4206 \times 1.840 = 0.774$$. Converting each back through $$v^{-1}(z) = z^{1/0.88}$$ gives certainty equivalents of $$\mathbf{3.72}$$ for L and $$\mathbf{0.75}$$ for S. (d) If each trades at its certainty equivalent, the expected return on L is $$1/3.72 - 1 = \mathbf{-73.1}$$ **percent** and on S is $$1/0.75 - 1 = \mathbf{+33.8}$$ **percent**. L is overpriced relative to S by a factor of $$3.72/0.75 = \mathbf{4.98}$$, on identical expected payoffs. That is Table 15.1's last row — the lottery-preference row — with a number attached: a security whose payoff is a small chance of a large sum is bid up until its expected return is deeply negative, and the securities that look most like L in the data (small, volatile, high-skewness, low-priced) are exactly the ones with the worst average returns. (e) A diversified investor still pays up for L because the preference is defined over the *prospect*, not over the portfolio. Prospect theory evaluates each gamble against its own reference point rather than integrating it into total wealth, and the weighting function's overweighting of small probabilities does not wash out in a portfolio: the skewness is the point of holding it, and diversifying it away would destroy the feature the investor is paying for. What keeps L from being arbitraged back to 1 is §15.5's limits to arbitrage, and specifically the two constraints that bind hardest here: the correction requires a *short* position, so its losses are unbounded and its cost of borrow is high and variable, and the mispricing can widen before it converges, which for an arbitrageur with outside capital (Shleifer and Vishny) is the risk that ends the fund. The prediction that follows is sharp and testable: the overpricing should be **largest where shorting is most expensive and the security is most lottery-like** — small capitalization, high idiosyncratic volatility, low nominal price, high short interest and low lendable supply — which is where the cross-sectional evidence of §15.4 in fact finds it. *** ## Data Exercise: Does Sentiment Forecast Anything? Every part of the baseline exercise runs on free public data. Parts A and B use survey measures of sentiment; Part C uses the market-based index of §15.4. **Part A — The AAII Sentiment Survey (free).** The American Association of Individual Investors has published a weekly poll of its members since 1987, reporting the share bullish, neutral, and bearish on the market over the next six months. The full history is published as a spreadsheet on the association's Sentiment Survey page and was free without registration when this book went to press. If you find it gated, check the response's content type rather than its status code — a gated page returns successfully and will load as an empty or nonsense table — and proceed with whatever span you can obtain, since the exercise works with fifteen years as well as with thirty. For market returns, use Robert Shiller's monthly US stock market file (S\&P composite price, dividends, earnings, and CPI, free from his Yale page), which also supplies the cyclically adjusted price-earnings ratio you will need later. 1. Construct the bull-bear spread (bullish share minus bearish share) and aggregate it to monthly frequency. Plot it against the S\&P composite index over the full sample and mark the five largest peaks and troughs in the spread. Before running anything, write down which way you expect the relationship to run and why. 2. Regress subsequent one-month, three-month, and twelve-month S\&P total returns on the current bull-bear spread. Report coefficients, standard errors corrected for the overlap in the longer-horizon regressions, and $$R^2$$. Is the sign consistent with sentiment as a contrarian indicator? 3. Now regress the *current* spread on *past* returns at the same three horizons. Compare the two sets of $$R^2$$. Section 15.3's claim is that survey expectations are far better explained by past returns than they are at explaining future ones. Does your version of the claim hold, and by how large a margin? 4. Split the sample at 2008 and repeat. If the relationship is unstable, say what that does to the interpretation — and, more importantly, to any trading rule built on it. **Part B — Shiller's confidence indices (free).** The Yale International Center for Finance publishes the Stock Market Confidence Indices monthly, separately for individual and for institutional investors, in four series: one-year confidence, buy-on-dips confidence, crash confidence, and valuation confidence. The historical spreadsheets are free to download from the center's data page. 1. Plot the individual and institutional valuation confidence indices on one panel, monthly, over the full sample. The series measure the fraction of respondents who think the market is *not* too highly priced. Overlay the CAPE ratio from Part A's Shiller file. 2. Section 15.3 predicts extrapolation: confidence should be high when the market is expensive. Is it? Compute the correlation between each confidence series and the contemporaneous CAPE, and between each and the trailing twelve-month return. 3. Compare individuals with institutions. Are the two groups' series correlated with each other? Does either lead? Section 15.6 argues that institutional and individual psychology differ less than is usually assumed but that the institutional machinery differs a great deal. What do these series show, and what can they not show? 4. The crash confidence index asks respondents for the probability of a catastrophic one-day decline in the next six months. Compare the average reported probability with the historical frequency of such declines. Relate the gap to §15.1's probability weighting, and state carefully why a survey answer about a small probability is *not* the same object as the weighting function in a prospect-theory value calculation. **Part C — The Baker-Wurgler index and the cross-section (free).** Jeffrey Wurgler posts the monthly sentiment index of §15.4, both raw and orthogonalized to macroeconomic conditions, on his NYU faculty page. Ken French's data library supplies free monthly returns on portfolios sorted by size, book-to-market, operating profitability, and volatility. 1. Form the return spread between the smallest and largest size deciles, and between the most and least volatile portfolios available in French's library. 2. Sort months into high- and low-sentiment groups by the prior month's index value (above or below the sample median, and then above or below the top and bottom terciles). Compute average subsequent spread returns in each group. 3. Baker and Wurgler predict that following high sentiment, the small, young, volatile, unprofitable end underperforms. Does the conditional pattern appear in your spreads, and is it stronger with the tercile split than with the median split? Report how many independent months you actually have, and be honest about the resulting standard errors. 4. Repeat using the orthogonalized index. If the results weaken, what does that tell you about whether the raw index is measuring sentiment or measuring the business cycle? Connect your answer to Discussion Question 4. **Part D ★ — Individual trades (licensed or restricted data).** The exercise this chapter would most like to set cannot be run on free data: the brokerage records behind §15.1 and §15.3 are proprietary and are not redistributed. Two partial substitutes, if you have access. 1. With CRSP daily data through WRDS, construct the capital-gains-overhang variable used in the literature that links the disposition effect to momentum — a turnover-weighted average of past prices, standing in for the aggregate reference point of a stock's holders — and test whether it forecasts returns in the cross-section. State clearly which step of the construction is the identifying assumption. 2. With TAQ or an exchange-provided investor-type classification, replicate the accounting behind §15.7's transfer statistic for a market where such data exist. Before running it, write down what a positive number would and would not establish about who is exploiting whom.