Week 1

  • Behavioral finance studies the intersection between finance and psychology
    • How do biases, emotions, social factors affect financial decision making?
  • Uses the rational actor model as a baseline + studies deviations from this actor

Efficient Market Hypothesis (EMH)

  • Efficient Market Hypothesis: Markets are considered efficient if the price of assets reflect all available information; old information is “stale” and useless
    • Weak Form Efficiency: All past market prices and data are fully reflected in current market prices, implies that technical analysis (studying past data) cannot outperform the market
    • Semi-strong Form Efficiency: All public information (not just past asset prices) is completely reflected in asset prices, implies that fundamental analysis (earnings calls, news) cannot outperform the market
    • Strong Form Efficiency: All information (even insider info) is accounted for in stock prices, implies that insiders cannot outperform the market
  • Because you can’t outperform the market via fundamental or technical analysis, active management strategies are not better than passive index funds
  • Diversification is crucial in minimizing risk
  • Minimizing trading costs is crucial to limit costs and maximize returns

Deviations from EMH

  • Behavioral finance argues that asset prices deviate from their EMH values because of psychology and irrational traders (AKA noise traders)
    • An argument against this is that arbitrageurs (AKA rational traders) will quickly correct this noise via calls or puts
    • A counterargument states that arbitrage responses can be costly and risky, and noise traders do not deviate randomly but instead deviate together
  • Delong et. al (1990) showed that noise traders and rational traders can coexist

Testing the EMH

  • Random Walk Model: Fama (1965) tests whether stocks move randomly, proving weak form efficiency since you cannot predict prices based on past prices
  • Event Study Methodology: Fama (1969) analyzed the stock price reaction to fundamental events (earnings calls, mergers) to see how quickly and accurately information is priced into the price of an asset
  • Capital Asset Pricing Model (CAPM): Assesses expected returns based on an asset’s risk, examining whether markets reward higher risk with higher returns
  • Arbitrage Pricing Theory (APT): Uses different risk factors to explain stock returns
  • Fama-French Three (or Four) Factor Model: Extends CAPM by including size and value factors, testing if tthese additional factors predict returns better than CAPM
    • The fourth factor comes from Carhart and is known as the “momentum factor”

Pricing Models via EMH

Present Value

  • Under the EMH, the market price of an asset equals its present value ($PV$)
  • $PV = \sum_{t=1}^n \frac{CF_t}{(1+\delta)^t}$
    • $CF_t$ represents the cash flow in a period t, such as dividends or interest
    • $\delta$ is the discount rate (usually the risk-free interest rate)
    • $n$ is the number of periods, and $n=\infty$ for an indefinite horizon
  • If $PV > P_t$, then the asset is undervalued; if $PV < P_t$, then the asset is overvalued

CAPM

  • Riskier stocks should be worth more since they aren’t as safe
  • The risk on a risky market portfolio is given by ${E[R_m] = R_f + \text{ Equity risk premium}}$
    • $R_m$ is the average market return and $R_f$ is the risk free rate (usually the yield on a 1 year U.S. treasury bill)
  • The equity risk premium can be calculated by $R_m - R_f$
  • Expected stock $i$ returns: $E[R_i] = R_f + \beta_i(E[R_m] - R_f)$
    • $\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)}$
  • $\beta < 0$ means that the stock moves opposite against the market (gold)
  • $\beta < 1$ means that the stock is less volatile than the market (Walmart)
  • $\beta = 1$ means that the stock mirrors market performance
  • $\beta > 1$ means that the stock is more volatile than the market
  • Model falters for high $\beta$ values

Efficiency Measures

  • Risk Premium: $E[R_i] - R_f$
  • Lintner’s Ratio: $\frac{E[R_i] - R_f}{\beta_i}$, should equal 1 if CAPM is right
  • Sharpe Ratio: $\frac{E[R_i] - R_f}{\sigma_i}$
    • The Sharpe ratio should be constant, as CAPM assumes $\sigma_i$ is proportionate to $\beta_i$
  • Jensen’s Alpha: Should equal 0 if CAPM is right
  • Studies have shown that Jensen’s Alpha is positive in portfolios with higher $\beta$

Arbitrage Pricing Theory (APT)

  • Model: $E[R_i] = R_f + \gamma_{1,i}F_1 + \cdots + \gamma_{n,i}F_n$
    • $F_i$ are firm specific factors
  • APT assumes less about investor behavior and market conditions
  • Generalizes CAPM; adds more factors aside from just $\beta$

Fama-French Factor Model

  • Uses specific factors as opposed to generalized ones
  • Four factor model
    • $R_M - R_f$: Market risk premium
    • $SMB$ (Small Minus Big): Size premium captures the excess returns of small-cap stocks over large-cap stocks
    • $HML$ (High Minus Low): Value premium captures the excess returns of high book-to-market stocks over low book-to-market stocks
    • $RMB$ (Robust Minus Weak): Profitability premium, captures the excess returns of stocks with robust operating profitability over stocks with weak profitability
    • Fifth factor, $CMA$ (Conservative Minus Aggressive): Investment premium, captures the excess of returns of firms with conservative vs. aggressive investment strategies

Markovitz Mean-Variance Approach

  • The mean-variance rule allows investors to minimize risk and become “rational”
    • Investment decisions should be determined by the mean and variance
    • If same mean, choose stock with lower variance
    • If same variance, choose stock with higher mean
  • Falters when the choice is between two stocks where one has both a higher mean and a higher variance
  • Rule is only appropriate when returns are normally distributed and agents have quadratic utility functions

Expected Utility Theory (EUT)

  • Von Neumann and Morgenstern’s axioms of rationality are used to define the EUT, the leading theory of decision-making under risk
  • Axioms
    • Completeness: Every pair of alternatives can be compared
    • Transitivity: Preferences are consistent across choices
    • Continuity: Preference between lotteries should not change abruptly with small changes in probability
      • If $A\succeq B \succeq C$, there exists $p,q\in [0,1]$ such that $pA+ (1-pC)\succeq B \succeq qA + (1-q)C$
    • Independence: Consistency of preferences between lotteries regardless of a common third prospect
      • If $A\succeq B$, then $pA + (1-p)C \succeq pB + (1-p)C$ for some $p\in [0,1]$
  • If the axioms are satisfied, then a utility function $u(x)$ exists
  • In uncertain settings, decision makers will aim to maximize their expected utility
  • Utility is ordinal, not cardinal
  • Can be defined in terms of wealth $w$
  • Risk averse people have a concave utility function, risk neutral have a linear utility function, and risk loving have a convex utility function
  • The Allais Paradox and Ellsberg’s Paradox show that the axiom of independence is often violated
    • Linked to uncertainty and regret
  • Research has shown that the way an action is framed can bias irrational agents into becoming rational
    • Default option bias: People often choose the default option

Week 2

Challenges to EMH

  • The Grossman-Stiglitz Paradox stipulates that a perfectly efficient market is impossible to achieve
    • Main idea: If markets are already efficient and info is priced in, then traders have no incentive to spend resources to gain more information, so perfect market efficiency is theoretically impossible
    • Model: Traders can be informaed or uninformed, but information about fundamentals incurs a cost
    • Assumptions: EMH, rational expectations, information costs
    • Equilibrium: There is an equilibrium fraction of informed traders based on information cost and trading gains based on info
    • Main result: The fraction of informed traders is greater than 0 but less than 1; must have inefficiency to motivate traders, but prices should reflect most publicly available information
  • Shiller (1981) argued that stocks are more volatile than can be explained by the changes in dividends
    • Main idea: The predicted value of a stock should have more volatility (variance) than the actual value of a stock; empirically proved false, suggesting excess volatility
    • Finds that fundamental information is not the only factor in pricing stocks
  • DeBondt and Thaler (1985) gave evidence that the stock market will overreact to news; extreme movements are often followed by large adjustments
    • Main idea: Look at winner and loser stocks, see how their returns evolve over time
    • Finds that winner stocks in a three year period will go down in the next three year period and vice versa for loser stocks
    • Goes against EMH; returns are predictable
  • EMH states that non-fundamental information should not affect prices, but many papers disagree
    • January effect, documented by Klein (1983), shows that stock prices increase in January
    • Index fund inclusion, shown in Harris and Gurel (1986) and Shleifer (1986), states that stocks recently added to the S&P 500 increase in price on average

Cognitive Biases

  • Kahneman and Tversky made strides in finding that actors are not rational
  • Four main irrationalities
    • Representativeness: People make judgements by a general idea of what the data represents as opposed to the data’s true implications
      • Linked to base rate neglect: individuals ignore the base rate of something happening
    • Conservatism: Individuals underweight new info and emphasize prior beliefs
    • Framing: The different ways a question/problem is told will elicit different answers
    • Anchoring: Individuals will be biased given a reference point
  • Thaler (1985) revealed the bias of mental accounting: money is treated based on its origin, its purpose, or the decision maker’s mental account
    • A mental account can be thought of as a set purpose for money; e.g. a vacation fund vs. a checkings account
    • One application is with dividends: people are more likely to spend money from dividends (since it is “found money”, like a gift or tax refund) as opposed to capital gains income
  • Isolation Effect: Decision makers prioritize differentiating features while disregarding commonalities in order to reduce one’s cognitive load; known as “editing the prospect”
  • Small Probabilities Effect: Rare events are weighted higher than their objective probabilities, influencing the choices people make in order to avoid unlikely losses (insurance) or realize unlikely gains (lottery)
    • Implies that subjective probabilities don’t increase proportionally
  • Reflection Principle: Decision makers are typically risk averse in the positive domain (more willing to take a certain win) and risk-loving in the negative domain (more willing to gamble to potentially lose less)
  • Loss Aversion: People feel a loss more strongly than a gain of equivalent value

Prospect Theory

  • Standard Utility Theory (SUT) assumes that the utility function is smooth and concave
  • Prospect Theory accounts for the above biases by having utility functions that are not necessarily globally concave
    • Involves looking at options, editing, evaluating the edited prospects, and making a choice; different than SUT where one goes immediately from looking at options to making a choice
  • The editing stage is used for DMs to simplify and organize prospects, reducing complex choices into simpler ones; has many stages
    • Coding: Set a reference point and consider gains/losses relative to this point
    • Combination: Similar outcomes are combined to simplify the choice
    • Segregation: Risk-free and risky components are seperated
    • Cancellation: Shared aspects of options are ignored and differences are emphasized
  • Evaluation is based on the edited prospects and relative to the reference point
    • Value (utility) function is defined on gains and losses, where gains are concave and losses are convex
Value function:v(x)={xα,x≥0−λ(−xβ),x<0\text{Value function:}\\ v(x) = \begin{cases} x^\alpha, & x\geq 0\\ -\lambda(-x^\beta), & x < 0 \end{cases}
  • Empirically, $\alpha = \beta = 0.88$ and $\lambda = 2.25$
  • Prospect Theory proposes a probability weighting function were small probabilities are overweighted and moderate/large probabilities are underweighted
    • PWF is non-linear, has inverted S shape
  • π(p)=pγ(pγ+(1−p)γ)1−γ\pi(p) = \frac{p^\gamma}{(p^\gamma + (1-p)^\gamma)^{1-\gamma}} \\
  • $\gamma$ is a parameter that controls the degree of curvature; smaller values have greater curve, while $\gamma=1$ is linear weighting

Week 3

  • Two main ways to explain financial market anomalies
    • Preference based approaches: Use Prospect Theory as opposed to EUT
    • Belief based approaches: Beliefs are not updated correctly after consuming new information

Equity Premium Puzzle

  • There is no production; a tree yields a stochastic endowment each period
  • Consumption equals endowment and dividends; $c_t=y_t=d_t$

Diagnostic Expectations

  • Belief-based model
  • Uses the representative heuristic: people judge probability by similarity to a stereotype, leading to base rate neglect and overreaction
    • One example is that people will guess that red-haired people are Irish despite a vast majority of Irish people not having red hair
    • Applies to stocks: a stock with long-run growth is overly predicted to continue to go up in price, but they mean-revert once expectations are not met
  • Agents do not use Bayes’ rule but instead a pyschological adjustment Etdiag[xt+1]=μ+γ(xt−μ)E_t^{\text{diag}} [x_{t+1}] = \mu + \gamma (x_t - \mu)
    • $\mu$ represents the prior belief or historical mean
    • $x$ is the variable to be predicted
    • $\gamma > 1$ is the exaggeration factor
    • The larger the surprise $\vert x_t - \mu \vert$ the greater the overreaction
    • Overreaction in short run, reversion to mean/fundamentals in the medium/long run
    • $\mu$ reflects individual beliefs and varies among people depending on one’s lived experience
  • Bayesian updating would use Kalman filtering Et[xt+1]=Et−1[xt+1]+K(xt−Et−1[xt]),K∈(0,1)E_t[x_{t+1}] = E_{t-1}[x_{t+1}] + K(x_t - E_{t-1}[x_t]), K\in(0,1)
    • Surprise $x_t - E_{t-1}[x_t]$ shifts beliefs proportionally; smooth, statistically sound updating
  • Difference in diagnostic updating can lead to excess volatility

Prospect Theory Explanation

  • Barberis showed that prospect theory can explain the equity premium puzzle (people value wealth and overreact to losses)
  • Excess volatilty puzzle can also be explained by prospect theory; allow loss aversion to depend on prior gains and losses
  • Prior gains = more likely to gamble (“house money effect”), prior losses = less likely to gamble

Narrow Bracketing

  • Explains the phenomenon where people evaluate risky decisions as standalone as opposed to being part of a larger context
  • A stock with returns $(-100, 110, 0.5)$ might be rejected due to the losses it can incur even if it is profitable in the long run

Market Sentiment

  • Investors’ general attitude towards the market can heavily influence a stock’s valuation
    • One measurement is the implied volatility index (VIX); is not consistent, changes with speculation
  • Keynes pointed out animal spirits as a separate force in 1936
  • Cass and Shell (1983) argued that random, extrinsic signals (“sunspots”) can affect outcomes if people think they matter
  • Closed-end funds are broker issued; can only be sold once and never again
    • Can only be traded on secondary markets
    • Prices deviate from their net asset value (NAV)
  • Closed-End Fund Puzzle: CEFs are sold at a premium when first issued and then trade at a discount in secondary markets
    • Can be explained by market sentiment (retail investors dominate the market)
      • Sentiment leads to overpricing at IPO, and shifts in mood leads to discounts
    • Noise traders makes arbitrage too risky
    • Limits to arbitrage: hard to short or replicate the NAV portfolio
  • Market sentiment and mood have been proven to affect asset prices
    • Sunshine effect, weekend/Monday effects, sports outcomes, natural disasters, and tragedies all lead to significant differences in returns the next day

Week 4

Limits to Arbitrage

  • Behavioral finance is based on the fact that irrational actors affect prices, but the arbitrage critique argues that arbitrageurs will corret these mispricings; this critique has various limits
    • Noise trader risk, fundamental risk, implementation costs, statistical limits

Noise Trader Risk

  • Model: The market is comprised of informed traders (arbitrageurs), who know the true value of an asset, and noise traders, who are uninformed
  • Most models will have one risky asset and one risk-free asset that give dividends
  • Arbitrageurs will understand there is a mispricing, but the mispricing could worsen before correcitng, leading to a heavier loss; noise trader risk
  • Noise traders can profit if sentiment persist, allowing them to make profit and making the amount of noise traders permanently non-zero

Fundamental Risk

  • A mispriced asset is still risky because it takes time for prices to revert back to their fundamentals
    • This lag can shift what the fundamentals of an asset are, causing prices to veer further away from the original expectation
  • Can be hedged against, but only partially
    • General strategy: If you believe one stock is overpriced, short it and buy an index of the market it belongs to. If market sentiment changes, then the value of the index will go up, offsetting your loss from the short
    • The index is not a perfect substitute, making the arbitrage fundamentally risk

Agency Problems

  • Arbitrageurs don’t manage their own capital, so investors must decide how much capital to give arbitrageurs
  • Performance-based arbitrage limits arbitrage even further since arbitrage is subject to the whims of short-run performance; poor performance leads to capital withdrawals and forced liquidations
  • Mispricing worsens as arbitrageurs scramble out of bad short-run positions

Costs

  • Transaction fees, requirements, etc. cause arbitrage to be hard to implement in reality
  • Jones and Lamont (2002) show that stocks that are expensive to short remain overpriced
  • Overpriced assets are typically not arbitraged against due to the costs of shorting
  • Empirically analyzing mergers highlights these costs; two companies who are merged should trade at the same price, but often one of the companies will be mispriced due to shorting difficulties

Statistical Limits to Arbitrage

  • Arbitrageurs use statistical estimation to learn the fundamental values of assets; known as statistical arbitrage
  • $\alpha$ opportunities are bounded due to the fact that they must be estimated from past returns, and sampling will always have error
    • Opportunities will still be missed
  • Data is sampled from noisy sources, so there is bound to be overfitting, risk, and estimation error

Overconfidence

  • Overprecision: Overconfidence in accuracy of beliefs, also known as miscalibration
    • Evidence: Studies have been shown that people have narrower confidence intervals than is true
    • Explains disposition effect (hold onto losers and sell winners), excess volatility + momentum, overreaction to information, inaccurate estimates of effects of mergers
  • Overplacement: Belief that one is better than others, such as believing one is above average in driving
    • Evidence: Studies have shown that over 50% of people will call themselves above average
    • Illusory superiority: Most people will rate themselves in the top half
  • High market turnover is evidence that people are overconfident
    • Average share of stock changes hands more than once a year; efficient, informed traders should hold onto stocks for longer due to transaction costs and symmetric info
    • Overplacement: Investors believe they are better than others at trading
    • Overprecision: Investors are more confident in their expectations than is true
  • Leads to excess volatility, lower returns than expected
  • Studies have shown that higher trading frequency leads to lower net returns, mostly due to trading costs
    • Shows that investors are overconfident
  • Overconfidence can be attributed to various things
    • Biology: Overconfidence was required to survive because they needed to hunt, take risks, etc.; especially prevalent in men
    • Biased self-attribution: Belief that good outcomes are due to skill and bad outcomes are due to luck
    • Confirmation bias: People look for evidence that supports what they already believe
  • Overconfidence survives because they can earn higher payoffs in bull markets, they reminisce on early gains, selection bias, and feedback delay

Week 5

Other Modeling Approaches

Rational Inattention

  • No agent has full information, and information comes at a cost; rational agents must choose how to obtain information and for what cost
  • In this model, information acquisition is endogenous, and heterogeneity is derived from preferences towards preferences
  • Contrasts the limit to arbitrage approach, as agents don’t have full information
  • Idea: choose what information structure ($P(s\vert x)$) to follow that gives you information on your perceived state of the world $s$ given the true state of the world $x$
    • Actions are chosen probabilistically; mistakes are optimal due to cognitive limits
    • Higher payoff means that an action is more likely to be chosen
    • Small payoff differences are smoothed out
    • Information constraints reationalize errors and delays in decision-making
  • This leads to under-diversified portfolios, bias towards local stocks, and actively managed mutual funds
    • Home Bias Puzzle: Investors disproportionately hold domestic assets despite theoretically gains through global diversification
      • Explained by even a slight home information edge providing a large incentive to specialize
  • Different from Cognitive Uncertainty; rational inattention states that actors make rational choices from limited information, but cognitive uncertainty states that actors have doubts about their own actions and judgements

Cognitive Uncertainty

  • Actors have second-order doubts about their actions
    • Leads to internal noise, reliance on intuitive heuristics
  • Idea: $a^*(p)$ represents the DM’s true expected utility-maximizing action, but the DM has an imperfect model of it due to limited cognition
    • DM receives a cognitive signal $s_i = a^(p)+\varepsilon_i$ and holds a prior belief ${a^(p)\sim \mathcal{N}(d_i, \sigma^2)}$ where $d_i$ is the default action
  • Explains underreaction and probability weighting

Information Ambiguity

  • Agents do not trust the quality of news sources (signals); leads to agents having a set of plausible priors as opposed to a single prior
  • Each signal follows $s = \theta + \varepsilon, \varepsilon \sim \mathcal{N}(0, \sigma_s^2)$ where $\sigma_s^2$ is random inside of a range
    • In the Bayesian case, we know what $\sigma_s^2$ is, but with ambiguity we have no idea
  • With multiple priors, agents evaluate actions under a worst-case posterior, overweighting bad news compared to good news
    • AKA assymetric response to news, overweighting of signals that convey bad news, assets should be undervalued when signals are ambiguous

Experience Effects

  • Theoretically, DMs have access to all data throughout all of history
  • Empirical data shows that recent and live experience matters more than history that was not personally experienced
    • Experience Effects: Professionals and amateurs are biased towards their lived experience
  • Bulkley and Harris (1997) showed that analysts overweight recent earnings growth in forecasting long-term earnings and that this overoptimism is often wrong

Week 6

  • A bubble has different meanings depending on who you ask; violates EMH, but behaviorists will justify bubbles by citing herd mentality
  • Historical examples
    • First recorded bubble was in the Netherlands where speculators drove up the price of tulips (Garber (1989))
    • Dot-com bubble was a big recent bubble regarding tech companies
  • Bubbles can be illustrated by the cyclically adjusted price/earnings ratio (CAPE) which has a long-run average of 16-17
  • Rational bubbles require an infinite horizon, grow at some rate $r$, and grow even faster if there is a probability of bursting
    • To remove bubbles, economists use the transverality condition where they set $b_t = 0$; justified if agents live forever + care only about dividends and markets are efficient
      • Actors obviously live finitely and buy for resale (Greater Fool fallacy)
  • Tirole (1985) gives an overlapping generations model of bubbles to justify bubbles using intrinsically worthless fiat money
    • Money has value because it can be resold and because actors will die
  • Cryptocurrency also has value because of bubbles
    • Limited supply leads to resale demand (Harrison and Kreps, 1978)
    • Social and signalling value; owners can express ideology and identity (Sockin and Xiong, 2020)
    • Network access: Owning a coin gives you access to the platform (Catalini and Gans, 2016)
    • Dual-role utility: Tokens provide access to services and incentivize trust (Pagnotta and Buraschi, 2018)
  • Harrison and Kreps showed that the market price will be greater than the fundamental price under one’s beliefs because they believe someone else will buy it for more (heterogenous beliefs)
  • Abreu-Brunnermeier Bubble-Riding Model (2003) shows that bubbles can exist with arbitrageurs because arbitrageurs will disagree when the bubble will burst, leading to the bubble growing larger and larger
  • No-Trade Theorem states that no speculative trade should occur if a market has common priors, symmetric information, and rational agents
    • Bubbles will only persist if there are heterogenous beliefs, assymetric or delayed information, and/or behavioral biases or noise trading