Cumulative abnormal returns

What Is Cumulative Abnormal Returns?

Cumulative abnormal returns (CAR) represent the sum of a security's or portfolio's daily or periodic abnormal return over a specified period, typically surrounding a particular event. This metric falls under the broader financial category of performance measurement and is a cornerstone of event study methodology. Unlike a single abnormal return, which captures a deviation from expected performance on a specific day, cumulative abnormal returns aggregate these deviations, providing a comprehensive view of the total abnormal performance around an event. This aggregation is crucial for assessing the impact of new information or corporate actions on shareholder wealth in the financial markets.

History and Origin

The concept of measuring abnormal returns and their aggregation to understand market reactions gained prominence with the development of the Efficient Market Hypothesis (EMH). Early work in the 1960s, notably by academics such as Eugene Fama, Lawrence Fisher, Michael Jensen, and Richard Roll, laid the groundwork for modern event studies. These studies aimed to determine if and how quickly stock prices adjust to new information. The methodology for calculating abnormal returns and cumulative abnormal returns was further refined by researchers like Stephen Brown and Jerold Warner in the 1980s, establishing standard procedures for empirical analysis in financial economics. Today, models like the Fama-French multi-factor models, which are documented on resources such as the Kenneth R. French Data Library, are commonly employed to estimate expected returns.

Key Takeaways

Cumulative abnormal returns (CAR) sum up a security's abnormal return over a period surrounding an event.
CAR is a key metric in event study methodology, used to assess the impact of specific events on security prices.
A positive CAR indicates that the security outperformed its expected return, while a negative CAR suggests underperformance.
The calculation of CAR requires defining a normal return, often using financial models like the Capital Asset Pricing Model (CAPM) or multi-factor models.
CAR helps researchers and analysts understand market efficiency and the market's reaction to corporate news or macroeconomic announcements.

Formula and Calculation

The calculation of cumulative abnormal returns involves two primary steps: first, determining the abnormal return for each period within the event window, and second, summing these individual abnormal returns.

The abnormal return (AR) for a given security (i) at time (t) is typically defined as:

AR_{i,t} = R_{i,t} - E(R_{i,t})

Where:

(R_{i,t}) = The actual return of security (i) at time (t).
(E(R_{i,t})) = The expected or "normal" return of security (i) at time (t) in the absence of the event. The expected return is often derived from an asset pricing model like the Capital Asset Pricing Model (CAPM) or a multi-factor model, based on historical market returns and the security's risk characteristics.

The Cumulative Abnormal Return (CAR) over an event window from time (T_1) to (T_2) is then calculated as the sum of the abnormal returns during that period:

CAR_{i, (T_1, T_2)} = \sum_{t=T_1}^{T_2} AR_{i,t}

For a sample of (N) firms experiencing similar events, researchers might also calculate the Cumulative Average Abnormal Return (CAAR), which is the average of the individual CARs across all firms in the sample.

Interpreting the Cumulative Abnormal Returns

Interpreting cumulative abnormal returns involves assessing both the magnitude and statistical significance of the accumulated abnormal performance. A positive CAR suggests that, on average, the security or group of securities experienced returns higher than what would be expected given its systematic risk and the prevailing market returns during the event window. Conversely, a negative CAR indicates underperformance relative to expectations.

For example, a significant positive CAR around a new product announcement could imply that the market viewed the announcement favorably, leading to a temporary increase in the stock's value beyond its normal fluctuations. The longer the event window, the more potential for other factors to influence stock performance, making short-term windows generally preferred for isolating event-specific impacts. The interpretation also heavily relies on the chosen model for expected returns, as different models may yield different baseline expectations for a security's risk-adjusted return.

Hypothetical Example

Consider a hypothetical technology company, "InnovateTech Inc.," that announces a groundbreaking new product on Day 0. An analyst wants to study the impact of this announcement on its stock prices over a 5-day event window, from two days before the announcement (Day -2) to two days after (Day +2).

The analyst uses a market model to calculate the expected daily return for InnovateTech, based on its historical beta and the market's performance.

Day	Actual Return ((R_{i,t}))	Expected Return ((E(R_{i,t})))	Abnormal Return ((AR_{i,t}))
Day -2	0.5%	0.1%	0.4%
Day -1	0.2%	0.1%	0.1%
Day 0	3.0%	0.1%	2.9%
Day +1	1.5%	0.1%	1.4%
Day +2	0.8%	0.1%	0.7%

To calculate the cumulative abnormal returns for this 5-day window, the analyst sums the daily abnormal returns:

CAR_{InnovateTech, (-2, +2)} = 0.4\% + 0.1\% + 2.9\% + 1.4\% + 0.7\% = 5.5\%

This CAR of 5.5% suggests that InnovateTech's stock price experienced a positive cumulative increase of 5.5% beyond what would normally be expected during the 5-day period surrounding the product announcement. This indicates a favorable market reaction and could inform future investment decisions.

Practical Applications

Cumulative abnormal returns are widely applied across various domains in finance and economics:

Academic Research: CAR is a primary tool for testing aspects of the Efficient Market Hypothesis. Researchers use it to investigate how quickly and completely new information is incorporated into stock prices. This involves examining the impact of various corporate events, such as dividend announcements, earnings surprises, stock splits, or mergers and acquisitions.
Corporate Finance: Companies use event studies to understand the market's reaction to their strategic decisions. For example, a company contemplating a major capital expenditure or a share buyback program might analyze historical CARs for similar events to anticipate potential market responses.
Regulatory Analysis: Regulatory bodies, like the U.S. Securities and Exchange Commission (SEC), may use event studies to assess the impact of new regulations or significant corporate disclosures on market behavior. Understanding how disclosures, such as those detailed on Investor.gov's guide to corporate filings, affect shareholder wealth is critical for investor protection and market integrity.
Legal Cases: Event studies and CARs can serve as evidence in litigation, particularly in cases involving allegations of securities fraud or insider trading. If a significant abnormal return is observed around a confidential event, it might suggest information leakage.
Investment Management: While less common for daily trading, understanding how certain types of news or events typically affect portfolio performance can help fund managers anticipate market movements and adjust their strategies. For instance, the market's reaction to large corporate deals, such as Microsoft's acquisition of Activision, can be analyzed using CARs to gauge investor sentiment towards similar future transactions.

Limitations and Criticisms

While cumulative abnormal returns are a powerful tool, they are not without limitations and criticisms:

Model Dependence: The accuracy of CARs heavily relies on the model used to estimate "normal" or expected returns. If the chosen model (e.g., Capital Asset Pricing Model or a multi-factor model) fails to capture all relevant factors influencing a security's returns, the calculated abnormal returns may be biased. This can lead to misleading conclusions about the event's true impact.
Event Window Definition: Defining the precise "event date" and the length of the "event window" can be challenging. Information may leak prior to a formal announcement, or the market reaction might extend beyond the chosen window, leading to an underestimation or overestimation of the event's impact.
Cross-Sectional Dependence: When analyzing a sample of firms, their abnormal return might not be independent, especially if they are in the same industry or affected by the same macroeconomic factors. This cross-sectional correlation can inflate the statistical significance of results if not properly addressed in the regression analysis.
Long-Term Performance Measurement: Cumulative abnormal returns are generally more reliable for short-term event windows (days to a few months). For long-term studies (e.g., several years), CARs can suffer from rebalancing biases and skewness, potentially leading to inaccurate inferences about long-run portfolio performance. Alternative measures, such as Buy-and-Hold Abnormal Returns (BHAR), are often considered more appropriate for long horizons, though they also have their own set of challenges. Discussions from academic institutions, like BI Norwegian Business School's lecture notes on Event Study Analysis, highlight these complexities in comparing CAR and BHAR for different horizons.
Confounding Events: Other market or firm-specific news occurring within the event window can confound the results, making it difficult to attribute the observed abnormal returns solely to the event under study.

Cumulative Abnormal Returns vs. Abnormal Return

The terms "cumulative abnormal returns" and "abnormal return" are closely related but refer to different aspects of performance analysis.

An abnormal return (AR) measures the unexpected gain or loss of a security on a single day or for a single defined period. It is the difference between the actual return of a security and its expected return, calculated based on an asset pricing model that accounts for market movements and the security's risk. Essentially, it answers the question: "How much did this stock perform above or below what was predicted for this specific day?"

Cumulative abnormal returns (CAR), on the other hand, represent the sum of these daily or periodic abnormal returns over a specified time frame, known as the event window. CAR aggregates the individual abnormal returns to provide a total measure of unexpected performance over a period. It answers the question: "What was the total unexpected performance of this stock over the entire period surrounding a particular event?" While a single abnormal return provides a snapshot, CAR offers a cumulative view of the event's impact on a security's value. The distinction is crucial because a single day's abnormal return might be small or even negative, but the cumulative effect over several days could be significant, or vice-versa.

FAQs

What is the purpose of calculating cumulative abnormal returns?

The primary purpose of calculating cumulative abnormal returns is to measure the aggregate impact of a specific event (e.g., an earnings announcement, a merger, a regulatory change) on a security's or a company's value, beyond what would be expected from normal market movements. It helps assess how the market reacts to new information.²

How do you determine the "normal return" when calculating CAR?

The "normal return" is the expected return of a security in the absence of a specific event. It is typically estimated using financial models like the Capital Asset Pricing Model (CAPM), which considers the security's beta (its sensitivity to market movements) and the overall market returns. More sophisticated models, such as the Fama-French multi-factor models, are also commonly used.

Can cumulative abnormal returns be negative?

Yes, cumulative abnormal returns can be negative. A negative CAR indicates that, over the specified event window, the security or portfolio underperformed its expected return. This suggests that the market reacted unfavorably to the event, or that the event led to a decrease in the company's perceived value.

What is an "event window" in the context of CAR?

An "event window" is the specific period of time surrounding an event for which abnormal returns are calculated and then summed to derive the cumulative abnormal return. This window can range from a few days to several weeks or months, depending on the nature of the event and the research question. It typically includes days before, during, and after the actual event date to capture pre-announcement information leakage or delayed market reactions.¹

How is the Efficient Market Hypothesis related to cumulative abnormal returns?

The Efficient Market Hypothesis posits that asset prices fully reflect all available information. In an efficient market, any unexpected information should be immediately and fully incorporated into prices, leading to zero abnormal return shortly after an announcement. Therefore, sustained, statistically significant positive or negative cumulative abnormal returns following an event could be seen as evidence inconsistent with certain forms of market efficiency.