The Cumulative Abnormal Return (CAR) is a measure used in finance and accounting to aggregate the abnormal returns of a security over a specified time window. The abnormal return for a given day is defined as the actual return minus the expected return. The Cumulative Abnormal Return formula is a crucial tool in financial analytics, especially in event studies. It provides a quantitative measure of how a specific event—like a major corporate announcement, regulatory change, or significant market movement—affects the price of a security or an entire portfolio over a specified time frame.
The Cumulative Abnormal Return is the sum of the abnormal returns over the event window.
What is Cumulative Abnormal Return Formula?
- The formula for Cumulative Abnormal Return over a specific time window, T1 T2, is as follows:
In this formula:
- CARi,T1,T2 = Cumulative Abnormal Return for security i from time T1 to T2
- ARit = Abnormal Return for security i at time t.
2. The Formula for Abnormal Return ARit is:
In this formula:
- Rit = Actual return of security i at time t.
- E(Rit) = Expected return of security i at time t, which could be determined using various models, such as the market model.
The abnormal returns from the formula represent the difference between the actual returns and the expected returns, indicating how the stock performed in comparison to market expectations on a daily basis during the specified event window.
3. The Formula for Expected Return E(Rit) is:
In this formula:
- αi is the intercept
- βi is the slope or market sensitivity
- Rmt is the return of the market index on day
The expected returns result from the formula give an estimate of how the stock should perform given the performance of the market, helping to later calculate the abnormal returns which show the deviation of actual returns from the expected.
How to calculate Cumulative Abnormal Return with formulas?
Real example of Cumulative Abnormal Return formula. To make it easier to understand the Cumulative Abnormal Return formula, I’ve prepared a realistic scenario featuring a fictional company named PharmaHeal Inc. and the S&P 500 returns for a 5-day event window (March 1st to March 5th). The α (Intercept) and β (Slope) are 0.3% and 1.1 respectively. PharmaHeal Inc. has been in the race to develop a vaccine for a recent viral outbreak. On March 3rd, they announce successful Phase 3 clinical trial results for their vaccine candidate.
| Date | FinCorp Inc. Return (R_it) | S&P 500 Return (R_mt) |
|---|---|---|
| March 1st | 2.00% | 1.00% |
| March 2nd | 1.50% | 0.80% |
| March 3rd (Announcement Day) | 3.00% | 1.50% |
| March 4th | -1.00% | -0.50% |
| March 5th | 2.50% | 1.20% |
Now I have to calculate for each day separately Expected Returns, Daily Abnormal Returns and finally Cumulative Abnormal Return.
1. Event Window: In this example Let’s pick a 5-day event window: 2 days before the announcement to capture any potential information leaks, the announcement day, and 2 days after. So, our event period is from March 1st to March 5th.
2. Expected Returns Calculation for each day
The expected returns for PharmaHeal Inc. on each day of the event window were calculated using the market model formula E(Rit)=αi+βi⋅Rmt, where: αi is the intercept (0.3%), βi is the slope or market sensitivity (1.1), mt is the return of the market index (S&P 500) on day t.
- March 1st: E(Rit)=0.3%+(1.1⋅1.0%)=1.4%
- March 2nd: E(Rit)=0.3%+(1.1⋅0.8%)=1.2%
- March 3rd: E(Rit)=0.3%+(1.1⋅1.5%)=1.95%
- March 4th: E(Rit)=0.3%+(1.1⋅−0.5%)=0.15%
- March 5th: E(Rit)=0.3%+(1.1⋅1.2%)=1.62
3. Daily Abnormal Returns Calculation for each day
The abnormal returns for PharmaHeal Inc. were calculated for each day in the event window using the formula ARit=Rit−E(Rit), where: Rit is the actual return of PharmaHeal Inc. on day t, E(Rit) is the expected return on day t>.
- March 1st: ARit=2.0%−1.4%=0.6%
- March 2nd: ARit=1.5%−1.2%=0.3%
- March 3rd: ARit=3.0%−1.95%=1.05%
- March 4th: ARit=−1.0%−0.15%=−1.15%
- March 5th: ARit=2.5%−1.62%=0.88%
4. Cumulative Abnormal Return (CAR) Calculation
Investors and analysts want to understand the market’s reaction to this announcement. They decide to analyze the Cumulative Average Abnormal Return around the announcement date to determine the aggregated market-adjusted effect. Logically I use Cumulative Abnormal Return formula:
CARi,March1st,March5th=ARi,March1st+ARi,March2nd+March3rd+March4th+March5th
CAR=0.6%+0.3%+1.05%−1.15%+0.88%=1.68%
CAR = 1.68%
The analysis of the data for PharmaHeal Inc. reveals varying daily abnormal returns during the 5-day event window, calculated using the formula for abnormal returns. The positive abnormal returns on March 1st, 2nd, 3rd, and 5th suggest that the stock performed better than expected on these days. Conversely, the negative abnormal return on March 4th indicates underperformance. The Cumulative Abnormal Return (CAR) of 1.68% over this period suggests a favorable overall abnormal performance, calculated with the Cumulative Abnormal Return formula. This analysis provides insight into how specific events or market conditions during this period might have impacted PharmaHeal Inc.’s stock performance relative to the market.
Easy Еxample of Cumulative Abnormal Return formula
Let’s use another a real-life example involving a hypothetical tech company, let’s call it “TechFin Corp.”
Imagine TechFin Corp. has been a moderately performing tech company for the past several years. However, on February 15th, they announce the development of a groundbreaking technology that has the potential to revolutionize the industry.
Financial analysts and investors are keen to determine the impact of this announcement on the company’s stock price. Specifically, they want to see if the stock provided any abnormal returns in the days following the announcement, suggesting that the market views this new technology very favorably (or unfavorably).
Event Study Steps:
- Event Window: Let’s analyze a 5-day event period: 2 days before the announcement (to catch any information leakage or early reactions) and 3 days after the announcement. So, our event time is from February 13th to February 17th.
- Calculate Expected Returns: Using historical stock data and a relevant market index (like the S&P 500) for, say, the past 250 trading days before our event window, we can determine the expected return of TechFin Corp. stock for each day in our event time. This is typically done using regression analysis to find a relationship between TechFin Corp.’s returns and the broader market’s returns.
- Calculate Daily Abnormal Returns: For each day in the event window, we subtract the expected return (calculated in step 2) from the actual return of TechFin Corp.’s stock.
For simplicity, let’s say these are our abnormal returns:
- February 13th: -1% (maybe due to some negative industry news unrelated to TechFin)
- February 14th: 2% (perhaps some leaks about the announcement)
- February 15th (Announcement Day): 7% (big positive reaction!)
- February 16th: 3% (continued positive momentum)
- February 17th: -2% (a slight pullback as some investors take profits)
Calculate Cumulative Abnormal Return wit formula
Sum up the daily abnormal returns over the event window:
CAR = (-1% + 2% + 7% + 3% – 2%) = 9%
The 9% CAR, as determined with the Cumulative Abnormal Return formula, indicates that, over the 5-day window surrounding the announcement, TechFin Corp.’s stock achieved a cumulative return of 9% more than what was expected based on its historical relationship with the market. This Cumulative Abnormal Return formula suggests a very positive market reaction to the groundbreaking technology announcement.
This kind of analysis helps investors and analysts gauge the significance of specific events on stock performance and informs future investment decisions. Cumulative Abnormal Return formula is very important in analytics.
I am a crypto journalist and blockchain expert. I like technology and started reading about bitcoin in 2013. Crypto is my passion and I like to write about cryptocurrencies.
P.S.
When I wrote the article “Cumulative Abnormal Return Formula” I analyzed statistics from various reliable sources. Always verified information from the Genesis code.
