Abnormal Return
In trading, each security has the expected and actual returns, i.e., the net gain or loss on an investment over a specified time period. An AR denotes a difference between those returns. It is important when defining the riskadjusted performance of a security or portfolio.
What Is Abnormal Return?
It refers to unusual profits, either positive or negative, generated by specific portfolios or securities over a certain period of time. In other words, it is a summary of how the actual returns differ from the expected yield. The expected yield is typically based on an index such as the S&P 500 or Dow Jones. An AR can pertain both to individual security and a portfolio of investments.
For example, the actual return of 40% in a mutual fund, which was predicted to average 20% per year, would create a positive abnormal return of 30%. On the other hand, earning 5% would generate a negative AR of 15%.
Abnormal returns are sometimes triggered by events such as lawsuits, IPOs, mergers, company earning or dividend announcements, etc. Such events provide the market with new information, which can impact the price of a security.
The AR is useful in these cases:

When it is necessary to compare the riskadjusted performance of a security or portfolio against the overall market performance.

When an investor would like to determine if they have been properly compensated for the level of risk that they have taken.
Abnormal Return Calculation Example
The calculation of expected return for specific security is based on the riskfree RoR (benchmark index), beta, and expected a market return. This is called the capital asset pricing model (CAPM). For stocks, the AR is calculated by subtracting the benchmark return from the stock return.
Let's suppose that you hold a portfolio of securities and would like to calculate the AR for the previous year. Consider that the riskfree RoR is 2% and the expected return of a benchmark index is 15%.
The portfolio returned 25% with a beta of 1.25 in relation to the benchmark index. Thus, given the amount of risk assumed, the portfolio's return should have been 18.25 %, or (2% + 1.25 x (15%  2%)). As a result, the abnormal return during the previous year was 6.75% or 25 to 18.25%.
Let's consider another example with stocks. For instance, the riskfree RoR is 5% and the expected return of a benchmark index is 12%. A stock returned 9% and had a beta of 2. Based on the capital asset pricing model (CAPM), the stock's expected return is 19%. Therefore, the stock had an AR of 10%.
Cumulative Abnormal Return (CAR)
The CAR refers to the total of all abnormal returns during a small period of time, such as several days, because daily calculation can create a bias in the results. The CAR is used to measure the effect of events that trigger AR. The CAR is also useful for defining how accurately the asset is priced in predicting its performance. As a result, the analysts get longerterm information about the effects of a major event on a stock's price.
To calculate the CAR, follow these steps:

Define the market return for one day.

Define the return on an individual stock for one day.

Subtract the market return from the return on the individual stock to get the abnormal return.

Repeat steps 1  3 for each of the days that fall within your chosen time frame. For example, if you wanted to calculate the CAR of a stock over a five day period, then you would need to repeat the first three steps five times.

To get the CAR, add the abnormal returns from each of the days. For example, if you were calculating the CAR for five days and the ARs were 4, 6, 8, and 5, you would add up these numbers together to get a CAR of 23.
Wrapping Up
Abnormal return is an important rate used to compare the riskadjusted performance of a security or portfolio against the overall market performance. In other words, it is the difference between the expected and actual returns.