Expected Return Models

Event studies are a widely used method in finance and economics to measure the impact of events on stock returns. These studies involve estimating the abnormal returns associated with a particular event and analyzing the statistical significance of these returns to determine the event’s effect. Various models have been developed to calculate the expected returns and abnormal returns, each with its own set of assumptions and considerations. In this summary, we discuss seven popular models used for event studies, including

• Market Model,
• Market Adjusted Model,
• Comparison Period Mean Adjusted Model,
• Market Model with Scholes-Williams beta estimation,
• Market Model with GARCH(1, 1) and EGARCH(1, 1) error estimation,
• Fama-French 3 Factor Model, and
• Fama-French-Momentum 4 Factor Model.

We provide an extensive description of each model, along with their respective formulas, advantages, and disadvantages. By understanding the different models and their characteristics, researchers can choose the most appropriate model for their specific event study and better interpret the results.

Which Model Should You Select?

Selecting the correct model for an event study depends on several factors, including data availability, the characteristics of the stocks or assets being analyzed, the nature of the event, and the specific research question being addressed. Here is a decision support summary to help you choose the most appropriate model:

1. Data availability:
   • If you have limited data or missing values, consider using the Market Adjusted Model or the Comparison Period Mean Adjusted Model, as they require fewer data points than other models.
   • If you have access to high-quality data, including factor data, consider using more sophisticated models like the Fama-French 3 Factor Model Model or the Fama-French-Momentum 4 Factor Model.
2. Stock characteristics:
   • For stocks with infrequent trading or non-synchronous trading, consider using the Market Model with Scholes-Williams beta estimation.
   • For stocks exhibiting time-varying volatility, consider using the Market Model with GARCH(1, 1) or EGARCH(1, 1) error estimation.
3. Nature of the event:
   • If the event is expected to have a significant impact on the market or systematic risk factors, consider using multi-factor models like the Fama-French 3 Factor Model or the Fama-French-Momentum 4 Factor Model.
   • If the event is primarily firm-specific and is not expected to have a substantial impact on the market or systematic risk factors, simpler models like the Market Model, Market Adjusted Model, or Comparison Period Mean Adjusted Model might suffice.
4. Research question:
   • If your research question focuses on the impact of systematic risk factors or the sources of abnormal returns, consider using multi-factor models like the Fama-French 3 Factor Model or the Fama-French-Momentum 4 Factor Model.
   • If your research question is mainly concerned with the existence or magnitude of abnormal returns, simpler models like the Market Model, Market Adjusted Model, or Comparison Period Mean Adjusted Model might be appropriate.

In summary, the choice of the appropriate model depends on the specific context of your event study and the available data. It is essential to consider the advantages and disadvantages of each model and select the one that best addresses your research question while accounting for the characteristics of the stocks and the event under investigation. Additionally, it is always a good idea to conduct robustness checks using alternative models to ensure the validity of your findings.

Statistical Models

Market Model

The Market Model is a widely used method in event studies to estimate the expected returns of a stock and calculate its abnormal returns during an event window. The model is based on a simple linear regression framework and captures the relationship between a stock’s return and the return of a market index, such as the S&P 500 or the Dow Jones Industrial Average. The underlying assumption of the Market Model is that a stock’s return is primarily influenced by market movements, along with a stock-specific idiosyncratic component.

The Market Model can be represented by the following equation:

\[ R_ = \alpha_i + \beta_i \cdot R_ + \varepsilon_ \]

• \(R_\) is the return of stock i at time t
• \(\alpha_i\) is the intercept term, representing the average stock return that is not explained by market movements
• \(\beta_i\) is the slope term (beta coefficient), representing the sensitivity of the stock return to the market return
• \(R_\) is the market return at time t
• \(\varepsilon_\) is the idiosyncratic component of the stock return, representing firm-specific factors not captured by the market return.

Interpretation

Interpretation of \(\alpha\) and \(\beta\)

• \(\alpha\) represents the expected return of security i when the market return is zero. It is the portion of the security’s return that cannot be explained by the market’s return, reflecting the security’s unique characteristics or the stock-specific risk. A positive alpha indicates that the security is expected to outperform the market when the market return is zero, while a negative alpha suggests underperformance relative to the market under the same condition.
• \(\beta\) measures the sensitivity of the security’s return to changes in the market return. It indicates the degree to which the security’s return moves in response to a change in the market return. A beta of 1 implies that the security is expected to move in tandem with the market, meaning that a 1% increase (or decrease) in the market return would typically result in a 1% increase (or decrease) in the security’s return. A beta greater than 1 suggests that the security is more volatile than the market, while a beta less than 1 indicates that the security is less volatile than the market. A negative beta means that the security’s return moves in the opposite direction of the market return.

Interpretation of \(\varepsilon\)

In the Market Model represents the error term or residual, capturing the portion of the security’s return that cannot be explained by its relationship with the market return. It reflects the idiosyncratic or firm-specific risk that is not accounted for by the market return.

\(\varepsilon_\) is the difference between the actual return of security i at time t and the return predicted by the model.

• Randomness: \(\varepsilon_\) represents the random component in the return of security i that is not captured by the model. It indicates that there are factors beyond the market return that influence the security’s return.
• Idiosyncratic Risk: Epsilon captures the firm-specific risk or unsystematic risk associated with security i. This risk can be attributed to factors such as company management, industry conditions, or other firm-specific events that affect the security’s return but are not related to the market return. Idiosyncratic risk can be diversified away through portfolio diversification.
• Model Limitations: A large \(\varepsilon_\) may suggest that the Market Model is not fully capturing the relationship between the security’s return and the market return. It might indicate the need for additional variables or a more complex model to better explain the security’s return behavior.

In summary, \(\varepsilon_\) represents the unexplained portion of the security’s return that arises from firm-specific factors or model limitations, and it reflects the idiosyncratic risk associated with the security.

Abnormal Return Calculation

The abnormal return ( \(AR_\) ), which represents the deviation of the actual stock return from its expected return, is calculated as:

\[ AR_ = R_ - (\alpha_i + \beta_i \cdot R_) \]

The abnormal returns can be aggregated across stocks and/or time to assess the overall impact of the event on the stock returns.

Advantages

• Simple and easy to understand, making it a popular choice for many researchers.
• Requires relatively fewer data inputs compared to more complex models.
• Suitable for a wide range of events and research questions.

Disadvantages

• Assumes that the relationship between stock returns and market returns is constant over time.
• Does not account for other risk factors that might influence stock returns, such as firm size, book-to-market ratios, or momentum.
• May not be appropriate for situations where non-synchronous trading or time-varying volatility are significant concerns.

Despite its limitations, the Market Model has been extensively used in event studies due to its simplicity and ease of implementation. However, researchers should be aware of the model’s assumptions and consider using alternative models when appropriate to account for specific factors or characteristics of the event under investigation.

Market Adjusted Model

The Market Adjusted Model is another simple approach used in event studies to estimate the expected returns of a stock and calculate its abnormal returns during an event window. This model is less complex than the Market Model, as it assumes that a stock’s expected return is equal to the market return, without considering any stock-specific factors. The Market Adjusted Model is particularly useful in situations where the estimation of individual stock parameters (such as alpha and beta) is not feasible or desired, and a basic benchmark for comparison is needed.

The Market Adjusted Model can be represented by the following equation:

• \(R_\) is the return of stock i at time t
• \(R_\) is the market return at time t
• \(\varepsilon_\) is the idiosyncratic component of the stock return, representing firm-specific factors not captured by the market return

In this model, the expected return for the stock during the event window is equal to the market return:

The abnormal return (AR_), which represents the deviation of the actual stock return from its expected return, is calculated as:

The abnormal returns can be aggregated across stocks and/or time to assess the overall impact of the event on the stock returns.

Advantages:

• Very simple and easy to implement, requiring only market return data.
• Does not require the estimation of stock-specific parameters, such as alpha and beta.
• Suitable for a quick assessment of the event impact or when data availability is limited.

Disadvantages:

• Assumes that the stock’s expected return is equal to the market return, ignoring any stock-specific factors.
• Does not account for other risk factors that might influence stock returns, such as firm size, book-to-market ratios, or momentum.
• Less accurate in estimating abnormal returns compared to more complex models, especially when stock-specific factors are significant.

While the Market Adjusted Model is less sophisticated than other models, its simplicity and ease of implementation make it a popular choice for preliminary analyses or when data availability is a concern. However, researchers should be aware of the model’s limitations and consider using alternative models when appropriate to account for specific factors or characteristics of the event under investigation.

Comparison Period Mean Adjusted Model

The Comparison Period Mean Adjusted Model is another relatively simple approach used in event studies to estimate the expected returns of a stock and calculate its abnormal returns during an event window. This model is based on the assumption that a stock’s expected return during the event window is equal to its average return during a comparison period (typically a pre-event period). This model is particularly useful when researchers want to control for a stock’s historical performance and do not wish to rely on market return data.

The Comparison Period Mean Adjusted Model can be represented by the following equation:

\[ R_ = μ_i + \varepsilon_ \]

• \(R_\) is the return of stock i at time t
• \(\mu_i\) is the mean return of stock i during the comparison period
• \(\varepsilon_\) is the idiosyncratic component of the stock return, representing firm-specific factors not captured by the mean return

In this model, the expected return for the stock during the event window is equal to the mean return of the stock during the comparison period:

The abnormal return ( \(AR_\) ), which represents the deviation of the actual stock return from its expected return, is calculated as:

\[ AR_ = R_ - E[R_] = R_ - μ_i \] The abnormal returns can be aggregated across stocks and/or time to assess the overall impact of the event on the stock returns.

Advantages:

• Simple and easy to implement, requiring only historical stock return data.
• Does not require the estimation of stock-specific parameters or market return data.
• Suitable for situations where the focus is on a stock’s historical performance.

Disadvantages:

• Assumes that the stock’s expected return is constant during the event window and equal to its historical mean return.
• Does not account for other risk factors that might influence stock returns, such as market movements, firm size, book-to-market ratios, or momentum.
• Less accurate in estimating abnormal returns compared to more complex models, especially when stock-specific factors or market influences are significant.

The Comparison Period Mean Adjusted Model offers a straightforward way to estimate abnormal returns based on a stock’s historical performance. However, researchers should be aware of the model’s limitations and consider using alternative models when appropriate to account for specific factors or characteristics of the event under investigation.

Market Model with Scholes-Williams beta estimation

The Market Model with Scholes-Williams beta estimation is an extension of the standard Market Model designed to address potential biases arising from non-synchronous trading in the stock and market index data. Non-synchronous trading occurs when the trading hours of a particular stock do not perfectly align with those of the market index, or when stock prices are not continuously updated due to illiquidity, trading halts, or other factors. This can result in inaccurate estimation of the beta coefficient, which in turn affects the calculation of abnormal returns in event studies.

Scholes and Williams (1977) proposed a method to adjust for non-synchronous trading effects by estimating the beta coefficient using a combination of contemporaneous and lagged market returns. The Scholes-Williams beta estimation is based on the following formula:

where \(R_\) is the return of stock i at time t, \(R_\) is the market return at time t, \(R_\) is the lagged market return, and \(\rho\) is an autocorrelation coefficient of the market returns. The autocorrelation coefficient measures the relationship between the market returns at different time lags and is used to weight the lagged market return in the estimation process.

The abnormal return is calculated similarly to the Market Model but uses the Scholes-Williams beta.

Advantages:

• Accounts for the effects of non-synchronous trading, which can bias alpha and beta estimates.
• Provides more accurate abnormal return estimates in the presence of non-synchronous trading.

Disadvantages:

• More complex than the standard Market Model.
• Requires additional data to estimate the Scholes-Williams alpha and beta, including lagged market returns.

For additional literature on the topic and applications of the Scholes-Williams estimation technique, you can refer to the following publications Dimson (1979) , Cohen et al. (1983) , or JAFFE and WESTERFIELD (1985) . These papers provide further insights into the challenges associated with non-synchronous trading, as well as the application of the Scholes-Williams technique in various contexts. While some of these studies might not exclusively focus on the Scholes-Williams method, they do address the issue of non-synchronous trading and its impact on beta estimation.

Economic Models

Fama-French 3 Factor Model

The Fama-French Three-Factor Model is a widely used asset pricing model that extends the traditional Capital Asset Pricing Model (CAPM) by incorporating additional risk factors that can influence stock returns. Developed by Eugene Fama and Kenneth French in the early 1990s ( (Fama and French 1993) ), the model aims to provide a more comprehensive explanation of the cross-section of stock returns.

The CAPM, while a foundational model in finance, has been criticized for its simplistic assumption that the market risk premium is the sole driver of stock returns. In contrast, the Fama-French Three-Factor Model recognizes that there are other systematic risk factors that can affect stock performance, beyond just the overall market risk.The three key factors included in the Fama-French model are:

1. Market Risk Premium: The difference between the expected return of the market and the risk-free rate. This provides investors with excess return as compensation for the additional volatility of returns over the risk-free rate.
2. Size Premium (SMB): The difference in returns between small-cap and large-cap stocks. This factor captures the tendency of small-cap stocks to outperform large-cap stocks over the long-term.
3. Value Premium (HML): The difference in returns between high book-to-market (value) and low book-to-market (growth) stocks. This factor reflects the observation that value stocks tend to outperform growth stocks.

The mathematical representation of the Fama-French Three-Factor Model is:

\[ R_i - R_f = \alpha + \beta_1 \cdot (R_m - R_f) +\beta_2\cdot \text + \beta_3\cdot\text+\varepsilon \]

• \(R_i\) is the return of the stock or portfolio
• \(R_f\) is the risk-free rate
• \(R_m\) ​ is the return of the market portfolio
• \(\alpha\) is the abnormal return
• \(\beta_1, \beta_2, \beta_3\) are the factor loadings for the market, size, and value factors resp
• \(\varepsilon\) is the error term

The Fama-French Three-Factor Model has been widely used in event studies to estimate expected returns and calculate abnormal returns. It is considered more robust than the CAPM, as it captures additional sources of systematic risk that can influence stock returns.

Advantages:

• Captures additional sources of systematic risk beyond the market risk factor
• Provides a more comprehensive explanation of stock returns compared to the CAPM
• Widely used and accepted in the finance and investment community
• Useful for event studies and performance evaluation

Disadvantages:

• Requires additional data and calculations compared to the CAPM
• The factors may not be stable over time, leading to potential model instability
• The interpretation of the factor loadings can be complex and may require additional analysis
• The model may not fully capture all the sources of systematic risk in the market

The Fama-French Three-Factor Model has been widely used in event studies to estimate expected returns and calculate abnormal returns. It is considered more robust than the CAPM, as it captures additional sources of systematic risk that can influence stock returns.

Literature

• The Econometrics of Financial Markets
• Event Studies for Financial Research
• Econometrics

References

Cohen, Kalman J., Gabriel A. Hawawini, Steven F. Maier, Robert A. Schwartz, and David K. Whitcomb. 1983. “Friction in the Trading Process and the Estimation of Systematic Risk.” Journal of Financial Economics 12 (2): 263–78. https://doi.org/10.1016/0304-405x(83)90038-7.

Dimson, Elroy. 1979. “Risk Measurement When Shares Are Subject to Infrequent Trading.” Journal of Financial Economics 7 (2): 197–226. https://doi.org/10.1016/0304-405x(79)90013-8.

Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56. https://doi.org/10.1016/0304-405x(93)90023-5.

JAFFE, JEFFREY, and RANDOLPH WESTERFIELD. 1985. “The Week-End Effect in Common Stock Returns: The International Evidence.” The Journal of Finance 40 (2): 433–54. https://doi.org/10.1111/j.1540-6261.1985.tb04966.x.

Scholes, Myron, and Joseph Williams. 1977. “Estimating Betas from Nonsynchronous Data.” Journal of Financial Economics 5 (3): 309–27. https://doi.org/https://doi.org/10.1016/0304-405X(77)90041-1.

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