To identify abnormal effects in an event study, there need to be a measure of the unobservable normal returns. The definition of normal returns is the expected return without conditioning on the event taking place (MacKinlay, 1997).
The approaches available to calculate the normal return for a given security can loosely be grouped into two categories – statistical and economic (MacKinlay, 1997). Statistical models follow from statistical assumptions and do not depend on economic arguments in contrast to the latter model.
In this section we present various models that are used to estimate normal returns in the event window.
4.2.1 Constant mean return model
One of the simplest models is the constant mean return model which assumes, as its names implies, mean return of a given security to be constant trough time.
Even though this model is simple, Brown and Warner (1980, 1985) find that the model often perform similar to the more sophisticated models described below.
This could be because the variance of the abnormal return is frequently not
reduced by much by choosing the more sophisticated models (MacKinlay, 1997).
The normal return for security i at time t, 𝑅𝑖𝑡, equals the mean return for security i at time t, 𝜇𝑖,plus a disturbance term, 𝜀𝑖𝑡, where 𝐸[𝜀𝑖𝑡] = 0 and 𝑣𝑎𝑟[𝜀𝑖𝑡] = 𝜎𝜀2.
𝑅𝑖𝑡 = 𝜇𝑖 + 𝜀𝑖𝑡
22 4.2.2 The market model
The market model is a statistical model which relates the return of any given security to the return of the market portfolio. The stock return, 𝑅𝑖𝑡, during period t, is expressed mathematically as
𝑅𝑖𝑡 = 𝛼𝑖 + 𝛽𝑖𝑅𝑚+ 𝜀𝑖𝑡
Where 𝑅𝑚 is the market’s rate of return during the period and 𝜀𝑖𝑡 is the return resulting from firm-specific events. 𝛼𝑖 is the average rate of return security i would realize in a period with zero market return. Thus, the return of any asset provides a decomposition of 𝑅𝑡 into market a firm-specific return (Bodie et al., 2014).
The market model is a flexible tool, because it can be generalized to include richer models of benchmark returns.
4.2.3 The Capital Asset Pricing Model
The CAPM model, was developed almost simultaneously by Sharpe (1963,1964) and Treynor (1961) (cited in Copeland et al. (2014, p. 145)), and has further developed to be one of the most recognized models in economics and finance. It assumes that the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio (Copeland et al., 2014, p. 145).
Compared to the market model presented above, CAPM implies that 𝛼𝑖 should equal 𝑟𝑓(1 − 𝛽). This makes the fitted security market line (SML) of CAPM steeper than for the market model (Bodie et al., 2014, p. 359).
Furthermore, the CAPM is developed in a hypothetical world, with the following assumptions (Copeland et al., 2014, p. 145-146):
1. Investors are risk-averse individuals who maximize the expected utility of their wealth.
2. Investors are price takers and have homogenous expectations about asset returns that have a joint normal distribution.
3. There exists a risk-free rate asset such that investors may borrow or lend unlimited amounts at a risk-free rate.
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4. The quantities of assets are fixed. Also, all assets are marketable and perfectly divisible-
5. Asset markets are frictionless, and information is costless and simultaneously available to all investors.
6. There are no market imperfections such as taxes, regulations, or restrictions on short selling
The investors will hold a combination of the risk-free asset and the market
portfolio, depending on their risk aversion. As the portfolio is perfectly diversified the only risk involved is systematic risk. The relationship between the expected return of a security, the beta and the risk premium are given as:
𝐸[𝑅𝑖] = 𝑟𝑓+ 𝛽𝑖[𝑟𝑚− 𝑟𝑓]
The CAPM has strong assumptions and followingly has received a lot of
criticism, e.g. for not doing a good job explaining the variance in returns for small firms (Fama and French, 1996). Thus, the use of the CAPM in event studies has almost ceased (MacKinlay, 1997).
4.2.4 The Arbitrage Pricing Theory
Like the CAPM, the APT, developed by Stephen Ross in 1976, predicts a linear relationship between expected returns and risk, but the path it takes to the Security Market Line is different. 1) security returns can be described by a factor model; 2) there are sufficient securities to diversify away idiosyncratic risk; and 3) well-functioning security markets do not allow for the persistence of arbitrage opportunities (Bodie et al., 2014, pp. 327). The model is given as:
𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖1𝐹1+. . . + 𝛽𝑖𝑛𝐹𝑛
Some of the downsides with the APT model is that finding the right factors has proven difficult and time consuming (Bodie et al., 2014), and in general the
additional factors to the market factor has little explanatory power. Thus, the gains from using an APT model versus the market model are small (Macklin, 1997).
24 4.2.5 Fama-French Three Factor Model
Fama and French’s three factor model (FF3) is among the most recognized APT models. The FF3 is a multi-factor model that can be used to measure normal returns of a stock while capturing more of the systematic risk that cannot be smoothen out by diversification, than what can be done in a single factor model.
The model can be written as Bodie et al. (2014, p. 340) did:
𝑅𝑖𝑡 = 𝛼𝑖 + 𝛽𝑖𝑀𝑅𝑀𝑡+ 𝛽𝑖𝑆𝑀𝐵𝑆𝑀𝐵𝑡+ 𝛽𝑖𝐻𝑀𝐿𝐻𝑀𝐿𝑡+ ℯ𝑖𝑡 where
SMB = Small Minus Big, i.e. the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks.
HML = High Minus Low, i.e. the return of a portfolio of stocks with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio.
Hence, there are two firm-characteristic variables in the model that are chosen because observations have shown that firm size and book-to-market ratios predicts deviations of average stock returns from what is found using the CAPM (Bodie et al., 2014, p. 240-241). Fama and French (1996) point out that firms with low earnings tend to have high book-to-market ratios with positive slope on HML, and vice versa for firms with low book-to-market rations with negative slopes on HML. This implies that SMB and HML can be used to proxy size and financial distress or business cycle risk (Bodie et al., 2014), where SMB mimics the risk factor related to size and HML mimics the risk factor related to book-to-market equity (Fama and French, 1993, p. 9).
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