Example Asset Pricing Questions

Behavioral biases, like herd behavior or overconfidence, can limit arbitrage opportunities.

For example, if a significant number of investors irrationally overvalue a stock based on popular trends (herd behavior), its price may remain artificially high despite its fundamentals indicating a lower value. An arbitrageur recognizing this mispricing might avoid shorting the stock, anticipating that these behavioral biases could sustain the overvaluation longer than the arbitrageur can remain solvent, thereby limiting the opportunity for profitable arbitrage. A nice example are the events around GameStop.

GameStop Case Study
In early 2021, GameStop (GME), a struggling video game retailer, became the focal point of a massive stock market event driven by retail investors. Despite the company's weak fundamentals, the stock price soared due to a combination of factors, primarily driven by discussions on social media platforms like Reddit, particularly the subreddit r/WallStreetBets.

Behavioral Biases at Play:

• Herd Behavior
  Definition: Herd behavior occurs when investors follow the actions of a larger group, often ignoring their own analysis or the underlying fundamentals of an investment.
  GameStop Example: A significant number of retail investors bought GameStop shares simply because others were doing so, amplifying the stock's upward momentum. This created a feedback loop where rising prices attracted more buyers, further driving up the price.
• Overconfidence
  Definition: Overconfidence bias leads investors to overestimate their knowledge or predictive abilities regarding stock performance.
  GameStop Example: Many retail investors believed they could outsmart institutional investors and hedge funds, which had heavily shorted GameStop. This overconfidence led them to continue buying and holding the stock, despite its skyrocketing price and the apparent misalignment with the company's financial health.

Impact on Arbitrage Opportunities:

• Sustained Overvaluation
  Challenge for Arbitrageurs: Traditional arbitrage strategies, such as short selling overvalued stocks, were severely challenged. Even though fundamental analysis indicated that GameStop was overpriced, the stock remained at high levels due to persistent buying driven by behavioral biases.
• Risk of Insolvency
  For arbitrageurs, the risk of shorting GameStop was heightened by the possibility that the stock's price could continue to rise due to herd behavior and overconfidence. As the adage goes, "the market can stay irrational longer than you can stay solvent." This risk of potential insolvency from continuous margin calls deterred many from taking arbitrage positions.

The extreme volatility led to trading halts and increased margin requirements by brokers, which further complicated arbitrage efforts. For instance, Robinhood and other brokerage firms restricted trading of GameStop and other highly volatile stocks, citing the need to meet regulatory deposit requirements.

The unprecedented influence of social media on trading decisions created a new layer of complexity. The collective action of retail investors, motivated by a mix of financial gain and a desire to challenge institutional power, led to price movements that defied traditional financial logic.

The GameStop saga illustrates how behavioral biases can create an environment where arbitrage opportunities are not only limited but also highly risky. The interplay of herd behavior, overconfidence, and social media dynamics sustained the overvaluation of GameStop stock far beyond what traditional market fundamentals would suggest. This case study underscores the importance of considering psychological factors and market sentiment in financial markets, especially in an era where information and collective action can rapidly influence stock prices.

The Capital Asset Pricing Model (CAPM) is a foundational model in finance that explains the relationship between systematic risk and expected return for assets, particularly stocks. However, it has limitations in explaining the variations in stock returns. The Fama-French three-factor and five-factor models expand upon CAPM by introducing additional factors to better capture the nuances of asset returns. Here's a breakdown of how these models expand upon CAPM.

The CAPM is based on the following formula: \begin{equation}E[R_i] = R_f + \beta_i (E[R_m] - R_f) \end{equation} where

• $E[R_i]$ is the expected return of asset $i$.
• $R_f$ is the risk-free rate.
• $\beta_i$ is the sensitivity of the asset's returns to the market returns.
• $E[R_m]$ is the expected return of the market portfolio.
• $E[R_m] - R_f$ is the market risk premium.

The Fama-French models expand upon CAPM by introducing additional factors to explain asset returns. The Fama-French three-factor model expands on CAPM by adding two more factors: \begin{equation}E[R_i] = R_f + \beta_i (E[R_m] - R_f) + SMB \cdot s_i + HML \cdot h_i \end{equation} where

• SMB (Small Minus Big): Accounts for the size effect, capturing the excess returns of small-cap stocks over large-cap stocks.
• HML (High Minus Low): Accounts for the value effect, capturing the excess returns of value stocks (high book-to-market ratio) over growth stocks (low book-to-market ratio).

These additional factors address the observation that small-cap stocks and value stocks tend to outperform large-cap and growth stocks, respectively, which CAPM does not fully explain. The Fama-French five-factor model further extends the three-factor model by adding two more factors: \begin{equation}E[R_i] = R_f + \beta_i (E[R_m] - R_f) + SMB \cdot s_i + HML \cdot h_i + RMW \cdot r_i + CMA \cdot c_i\end{equation} where

• RMW (Robust Minus Weak): Accounts for profitability, capturing the excess returns of stocks with robust (high) profitability over those with weak (low) profitability.
• CMA (Conservative Minus Aggressive): Accounts for investment patterns, capturing the excess returns of firms that invest conservatively over those that invest aggressively.

These additional factors are based on empirical evidence showing that more profitable companies and those that invest conservatively tend to yield higher returns.

The expansions provided by the Fama-French models are important for better understanding and predicting stock returns. They allow for a more nuanced risk assessment and portfolio management strategy. These models are extensively used in academic research and practical finance for asset pricing, risk management, and investment decision-making.

Prospect Theory, developed by Kahneman and Tversky, proposes that people value gains and losses differently, leading to decision-making that deviates from traditional rational models. In terms of stock returns, it suggests investors are more sensitive to losses than to equivalent gains and may make irrational decisions based on this asymmetry. This theory is relevant as it provides insight into investor behavior, especially under uncertainty, and helps in understanding market anomalies and investor decision-making processes.

Elaboration
The main empirical prediction about prospect theory by Barbaris, Mukherjee and Wang is that stocks whose historical distribution have high (low) prospect theory values will have low (high) subsequent returns. This prediction is expected to hold primarily among small cap stocks, in other words, among stocks for which individual investors play a more important role.

Another hypothesis by the article stated that a stock whose future returns are expected to be positively skewed will be 'overpriced' and will earn a lower average return.

• A positively skewed investment return means there were frequent small losses and a few large gains.
• A negatively skewed investment return means there were frequent small gains and a few large losses.

First, we will elaborate on the quantification of the prospect theory (formed by Tversky and Kahneman (TK)). It starts with the prospect theory value function, given by \begin{equation}
v(x) = \left\{\begin{matrix}
x^{\alpha} & x>0 \\
- \lambda (-x)^{\alpha} & x < 0 \end{matrix}\right. \end{equation} Typical values for the parameters are $\alpha = 0.88$ and $\lambda=2.25$. The value function is plotted in Figure 1, in which we see the investor's risk aversion over gains and the risk seeking over losses. We will elaborate shortly on practical examples.

Furthermore, the weight functions need to be introduced. The main consequences of the weighting functions is that it overweights the tails. Think about a lottery or insurance. With a lottery, you have a small probability p for a big prize and with an insurance we pay monthly for a small probability p for a loss. By overweighting the tail probability, the prospect theory can capture both of these choices. The weighting functions (which will be used later on this lesson) are given as follows. \begin{equation} w^+(p) = \frac{p^{\gamma}}{(p^{\gamma}+(1-p)^{\gamma})^{\frac{1}{\gamma}}} \end{equation} \begin{equation} w^-(p) = \frac{p^{\delta}}{(p^{\delta}+(1-p)^{\delta})^{\frac{1}{\delta}}} \end{equation} The parameters in the three equations above have values $\alpha, \gamma, \delta \in (0,1)$ and $\lambda > 1$.

When thinking about a stock, some of the investors mentally represent it as the distribution of its monthly returns in excess of the market over the past five years. Given a specific stock, we record the stock's return in excess of the market in each of the previous sixty months and then sort these sixty excess returns in increasing order. Suppose that $m$ excess returns are negative and $n=60-m$ are positive. Then, the excess returns are ordered as $(r_{-m}, r_{-m+1}, ..., r_{-1}, r_{1}, ..., r_{n-1}, r_{n})$, which becomes the stock's historical return distribution, with each return having an equal 1/60 probability. Finally, the prospect theory value of this distribution is given as

\begin{equation}
TK = \sum_{i=-m}^{-1} v(r_i) [w^- (\frac{i+m+1}{60}) - w^- (\frac{i+m}{60})] + \sum_{i=1}^{n} v(r_i) [w^+ (\frac{n-i+1}{60}) - w^+ (\frac{n-i}{60})]
\end{equation}

This TK value captures the impression that investors form about a stock, after seeing its historical price fluctuations in a chart.

One important property of TK is that it does not depend on the order in which the sixty past returns occur in time. The justification behind this method is that the TK is capturing an investor's quick, passive reaction to a chart. This reaction is likely to be based on the chart as an integral whole, with the early part of the chart affecting the investor just as much as the later part.

Prospect theory investors are investors that construct their portfolio holdings by taking the tangency portfolio and adjusting it, by increasing their holdings of stocks with high prospect theory values and decreasing their holdings of stocks with low prospect theory values.

The main hypothesis is that TK value of stocks will predict the stock's subsequent return in the cross section. Two methods have been used in order to test this hypothesis:

• The Time Series Test
• The Gama-MacBeth Test

Time Series Test

1. At the start of each month, beginning in July 1931 and ending in December 2010, we sort stocks intro deciles based on their TK-value.
2. Compute the average return of each decile over the entire sample.
3. Report the average return of each decile in excess of the risk-free rate, the four-factor alpha for each decile (the return adjusted by the 3-factor model from Fama & French, 1993, and by momentum), the 5-factor alpha (the 3-factor model and momentum and liquidity) and the characteristics adjusted return.

See the article for an overview of the results. The results support the main hypothesis. The average return on low TK values is significantly larger than the average return for higher TK values.

Fama-MacBeth Test

The Fama-MacBeth test has one advantage over the time series test. This method allows us to examine the predictive power of TK, while controlling for known predictors of the return. Each month, from July 1931 till December 2010, the method runs a cross sectional regression of stocks returns in that month on the TK value that is measured at the start of the month and on variables already known to predict returns.

The results support the main hypothesis. Even after we include the major known predictors of the return, the TK value remains significant.