Beta is a measure of the volatility, or systematic risk, of a security or portfolio in comparison to the market as a whole. Beta is used in the capital asset pricing model (CAPM) to calculate the expected return of an investment. A stock with a beta greater than one is more volatile than the market; a stock with a beta less than one is less volatile. For example, a stock with a beta of two is twice as volatile as the market.
A company’s beta can be affected by a number of factors, including the industry it operates in, the size of the company, and its financial stability. A company’s beta is not static and can change over time.
Reference  pointed out that the beta of a stock can change following its addition to or deletion from an index. This lead to beta arbitrage, a trade that is based on the assumption that betas tend to mean regress towards one in the long run.
This paper develops a stylised model for S&P 500 index changes with two beta-based styles: index trackers and beta arbitrageurs who trade in both high and low beta event stocks to exploit mean reversion towards one. Arbitrageurs engage in common or contrarian trading patterns relative to index funds depending on whether historical betas are below or above one. Thus, the overall comovement effect has two distinct components. After index additions, pre-event low beta stocks drive the overall beta increases due to common demand – albeit for different reasons – from indexers and arbitrageurs. By contrast, arbitrageur shorting of high beta additions diminishes or sometimes reverses the beta increases for these stocks driven by indexers. Analogous results hold for index deletions.
This paper presented an interesting form of arbitrage that exploits stock comovements.
 Yixin Liao, Jerry Coakley, Neil Kellard, Index tracking and beta arbitrage effects in comovement, International Review of Financial Analysis, Volume 83, October 2022, 102330
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