## Expected Return Models

Expected return models are widely used in Finance research. In the context of event studies, expected return models predict hypothetical returns that are then deducted from the actual stock returns to arrive at 'abnormal returns'. Expected return models can be grouped in statistical (models 1-5 below) and economic models (models 6 and 7). The following models are implemented in this website's abnormal effect calculators:

1. Market Model (Abbr.: mm): The 'market model' considers the focal firm's individual CAPM risk by multiplying the market return with the firm individual $\beta$ factor: $\alpha_i+\beta_i R_{m,t}$. Although the 'market model' is widely accepted as the standard model, there is also some criticism. The model assumes that the risk-free interest rate included in the $\alpha$ factor is constant, which conflicts with the presumption that market returns vary over time.
2. Market Adjusted Model (Abbr.: mam): Using the actual market return is the simplest way to 'control' for potential effects of the event on the general market, yet it does not adjust for basic CAPM risk and thus abstracts from the focal firm's distinct systematic risk profile.
3. Comparison Period Mean Adjusted Model (Abbr.: cpmam):
4. Market Model with Scholes-Williams beta estimation (Abbr.: mm-sw):
5. Market Model with GARCH and EGARCH error estimation (Abbr.: garch / egarch):
6. Fama-French 3 Factor Model (Abbr.: ff3f): 'Multi-factor models', as the Fama-French three-factor model or APT suggest to mitigate the issues related with the CAPM model and thus provide better estimates for benchmark returns in event studies.
7. Fama-French-Momentum 4 Factor Model (Abbr.: ffm4f): 'Multi-factor models', as the Fama-French three-factor model or APT suggest to mitigate the issues related with the CAPM model and thus provide better estimates for benchmark returns in event studies.
Other expected return models, which are currently not yet available on this website are:
• Matched firm model: $R_{mf,t}$: Instead of referring to the market, scholars may also turn to the performance of a comparable firm's stock when seeking a proxy for a distinct firm's expected returns.
• $R_{f,t}+\alpha_i+\beta_i (R_{m,t}-R_{f,t})$: The 'CAPM model' includes the specific-risk free rate in the estimation and therefore represents a more granular approach than the market model. This prediction model has been criticized: The $\beta$ factor only provides an imperfect correlation to market risks, it was found inferior to e.g., the Fama-French three-factor model. And also the $\alpha$ parameter as a post-event estimator is deemed biased for post-event predictions.

Choosing from the expected return models is generally related to sample selection biases (e.g., Ahern, 2009). Depending on the specific sampling bias, each of the above models may imply slightly different biased results. For example, if a study's sample largely consists of small firms, the CAPM model was found to predict too low returns (Banz, 1981), leading to inflated abnormal returns in the event study. Multi-factor models try to circumvent this problem by considering the factors that drive the biased results.

Since abnormal return biases tend to be small, most scholars interested in the economic substance of individual event types, rather than methodological discussions, still use the 'market model'. Respective meta-research (Holler, 2014) found that in its sample of 400 reviewed event studies, 79.1% of the studies used the 'market model', 13.3% the 'market adjusted return model', 3.3% the 'constant mean return model', 3.6% 'multi-factor models', and only 0.7% the CAPM model.

Formulas of the expected return models available in this website's abnormal effect calculators

[1] Market Model (Abbr.: mm)

In the market model we assume that the return follows a single factor market model

$$R_{it} = \alpha_i + \beta_i \cdot R_{mt} + \varepsilon_{it},$$

where $R_{it}$ is the return of the stock of observation $i$ (e.g. firm) on day $t$, $R_{mt}$ is the return of the reference market on day $t$, $\varepsilon_{it}$ is the error term (a random variable) with expectation zero and finite variance. It is assumed that $\varepsilon_{it}$ is uncorrelated to the market return $R_{mt}$ and firm return $R_{jt}$ with $i \neq j$, not autocorrelated, and homoskedastic. The regression coefficient $\beta_i$ is a measure of the sensitivity of $R_{it}$ on the reference market. The abnormal return is then calculated as follows:

$$AR_{it} = R_{it} - (\alpha_i + \beta_i \cdot R_{mt}).$$

[2] Market Adjusted Model (Abbr.: mam)

In the market adjusted model, the observed return of the reference market on day $t$ $R_{mt}$ is substracted from the return $R_{it}$ of the observation $i$ on day $t$. We get for the abnormal return:

$$AR_{it} = R_{it} - R_{mt}.$$

[3] Comparison Period Mean Adjusted Model (Abbr.: cpmam)

In the comparison period mean model the abnormal return in the event window is the return of observation $i$ on day $t$ minus the average return of the observation $i$ in the estimation window:

$$AR_{it} = R_{it} - \bar{R}_{i},$$

where $\bar{R}_{i} = \frac{1}{T_1 - T_0}\sum\limits_{t\in [T_0, T_1]}R_{it}.$

[4] Market Model with Scholes-Williams beta estimation (Abbr.: mm-sw)

For non-synchronus trading you may choose the market model with Scholes-Williams beta estimation. The betas are defined as

$$\beta^{SW}_i = \frac{\beta_i^- + \beta_i + \beta^+_i}{1 + 2 \cdot \rho_M},$$

where $\beta^-_i$ is the regression coefficient of $R_{it}$ on $R_{m,t-1}$, $\beta^+_i$ is the regression coefficient of $R_{it}$ on $R_{m,t+1}$, and $\rho_M$ is the first-order autocorrelation of $R_m$. The intercept $\alpha^{SW}_t$ is estimated through the sample mean

$$\alpha^{SW}_i = \bar{R}_{i, EST} - \beta^{SW}_i \cdot \bar{R}_{M, EST},$$

where $\bar{R}_{i, EST}$ is the mean of returns of observation $i$ in the estimation window and $\bar{R}_{M, EST}$ is the mean of the returns of the reference market in the estimation window.

[5] Market Model with GARCH and EGARCH error estimation (Abbr.: garch / egarch)

If you choose the GARCH option on our EST API interface a single factor market model with GARCH(1, 1) errors is estimated, namely

$$R_{it} = \alpha_i + \beta_i \cdot R_{mt} + \varepsilon_{it}.$$

The conditional variance (Bollerslev (1986)) may be written as:

$$\sigma^2_{t} = \omega + \gamma_1 \cdot \varepsilon^2_{t-1} + \delta_1 \cdot \sigma^2_{t-1}$$

with $\sigma^2_t$ denoting the conditional variance, $\omega$ the intercept, and $\varepsilon^2_t$ the residuals from the mean filtration process. Parameters are estimated by maximum likelihood (a non-linear solver is used for the optimization problem).

[6] Fama-French 3 Factor Model (Abbr.: ff3f)

coming soon

[7] Fama-French-Momentum 4 Factor Model (Abbr.: ffm4f)

coming soon

References and further studies

K. Ahern 2009. 16 (3): 466-482
S. Armitage 1995. 8 (4): 25-52
R.W. Banz 1981. 9 (1): 3-18
S. Brown; J. Warner 1980. 8 (3): 205-258
J. Cable; K. Holland 1999. 5 (4): 331-341
R. Chandra; B. Balachandran 1990. 6 611-640
D.W. Collins; W.T. Dent 1984. 22 48-84
A. Coutts; T.C. Mills; J. Roberts 1995. 2 163-165
P. Dodd 1980. 8 (2): 105-137
P. Draper; K. Paudyal 1995. 22 157-177
D. Oler; J. Harrison; M. Allen 2007.
N. Park 2004. 25 (7): 655-668
J.N. Patell 1976. 14 246-276
N. Strong 1992. 19 533-553