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