Expected-return models

An expected (normal) return model estimates the return a security would have earned if the event had not occurred. The abnormal return is the actual return minus this expected return:

$$AR_{it} = R_{it} - E(R_{it} \mid X_t).$$

Here $R_{it}$ is the realized return of observation $i$ (for example a firm) on day $t$, and $E(R_{it} \mid X_t)$ is the normal return predicted by the chosen model given the conditioning information $X_t$ (typically the market return or a set of factor returns). Every model on this page is simply a different way to estimate $E(R_{it})$: from a constant historical mean, to a single-factor market regression, to multi-factor regressions. For the regression-based models the abnormal return is the regression residual (MacKinlay (1997)).

How these models are used: estimation window and event window

All of these models share one mechanic. Their parameters (for example $\alpha_i$ and $\beta_i$) are estimated by ordinary least squares over a pre-event estimation window, typically about 120 to 250 trading days, that does not overlap the event window in which abnormal returns are measured. The estimated parameters are then held fixed and applied to the event window to generate predicted (normal) returns (Campbell, Lo & MacKinlay (1997), ch. 4).

The market model is the standard workhorse for a concrete reason: by removing the portion of a stock's return that is explained by market movements, it reduces the variance of the abnormal return and thereby increases the power to detect event effects. The size of this benefit scales with the regression $R^2$, which is typically around 20 to 40 percent for daily single-stock returns and around 30 to 50 percent once size and value factors are added (MacKinlay (1997); Campbell, Lo & MacKinlay (1997)). That said, for short daily windows the simpler constant-mean and market models perform about as well as more elaborate models, because abnormal returns are dominated by the event-period return rather than by the choice of benchmark (Brown & Warner (1985)). Unless otherwise stated, all returns below are simple discrete returns.

The models implemented on this website

Expected return models are widely used in finance research. Following the canonical taxonomy of MacKinlay (1997), normal-return models fall into two classes. Statistical models rest on statistical assumptions about return behavior rather than on an economic argument; these include the constant-mean-return model, the market model, and the multifactor (Fama-French and Carhart) models. Economic models, by contrast, impose restrictions derived from equilibrium or arbitrage theory; the leading examples are the CAPM and the Arbitrage Pricing Theory (APT). The models below are statistical models implemented in this website's event study research apps; the CAPM is discussed as a related economic model.

1. Market Model (Abbr.: mm): the market model relates a stock's return to the market return through the firm's statistical sensitivity to the market, $\alpha_i + \beta_i R_{m,t}$. Here $\alpha_i$ captures the stock's average return not explained by market movements, and $\beta_i$ is the firm's sensitivity to the market return (a market-model slope, not the CAPM equilibrium beta). The model is widely accepted as the standard, but it rests on two assumptions that are worth keeping in mind: (a) $\alpha_i$ and $\beta_i$ are stable across the estimation and event windows, and (b) the single market factor omits size, value, and momentum risk.

2. Market Adjusted Model (Abbr.: mam): subtracting the actual market return is the simplest way to control for market-wide effects, but it does not adjust for the firm's distinct systematic risk. It is exactly the market model with the restriction $\alpha_i = 0$ and $\beta_i = 1$ imposed, so no parameters are estimated (MacKinlay (1997)).

3. Comparison Period Mean Adjusted Model (Abbr.: cpmam), also called the constant mean return model (MacKinlay (1997)) or mean-adjusted model: here the abnormal return is simply the firm's event-window return minus its own average (mean) return over the estimation window. The model needs no market index, which helps when a reliable market proxy is unavailable, but in turn it does not control for market-wide movements or systematic risk.

4. Market Model with Scholes-Williams beta estimation (Abbr.: mm-sw): a market-model variant that corrects the firm's $\beta$ for non-synchronous (infrequent) trading by estimating it from the lagged, contemporaneous, and lead market return. This reduces the bias that thin trading introduces, which is especially relevant for less liquid stocks (Scholes & Williams (1977)).

5. Market Model with GARCH and EGARCH error estimation (Abbr.: garch / egarch): a single-factor market model whose error term follows a $GARCH(1,1)$ process, so it captures time-varying volatility (volatility clustering) instead of assuming a constant variance (Bollerslev (1986)). The EGARCH variant additionally allows for asymmetry, the stronger reaction of volatility to negative shocks, which yields more reliable test statistics for such returns (Nelson (1991)).

6. Fama-French 3 Factor Model (Abbr.: ffm3): a multifactor model that adds a size factor (SMB, small minus big) and a value factor (HML, high minus low book-to-market) to the market factor, mitigating the single-factor model's omission of size and value risk and providing better benchmark returns in event studies (Fama & French (1993)).

7. Carhart Four-Factor Model (Abbr.: ffm4): the Fama-French three factors plus a momentum factor (UMD = Up Minus Down = WML = MOM, the momentum factor: past 12-month winners minus losers) (Carhart (1997); momentum originates with Jegadeesh & Titman (1993)).

8. Fama-French Five-Factor Model (Abbr.: ffm5): the three-factor model plus a profitability factor (RMW, robust minus weak) and an investment factor (CMA, conservative minus aggressive). Note that the five-factor model contains no momentum factor (Fama & French (2015)).

Each of SMB, HML, RMW, CMA and UMD is a realized long-short portfolio return (a factor return), not a firm characteristic: SMB is the size premium (small minus big), HML the value premium (high versus low book-to-market), RMW the profitability premium (robust minus weak), CMA the investment premium (conservative minus aggressive), and UMD the momentum premium (past winners minus losers).

Related models and benchmarks

Beyond the eight models above, several related approaches are standard in the literature and partly available in our research apps:

• Matched-firm and characteristic-matched benchmarks: instead of referring to the market, scholars may use the return of a comparable firm, $R_{mf,t}$, or a portfolio of firms matched on size and book-to-market, as the proxy for expected returns. Size-decile and beta-decile matched benchmarks are common long-horizon benchmarks.
• Time-varying-beta models: rolling-window estimation and DCC-GARCH (where $\beta_t = \mathrm{Cov}_t(R_i, R_m) / \mathrm{Var}_t(R_m)$) relax the market model's assumption of a constant $\beta$ across the estimation and event windows.
• Long-horizon designs (BHAR): over long horizons the benchmark choice becomes decisive, and abnormal performance is usually measured as the buy-and-hold abnormal return, which compounds rather than sums returns (Kothari & Warner (2007)): $$BHAR_i = \prod_t (1 + R_{it}) - \prod_t (1 + R_{\text{bench},t}).$$

CAPM (related economic model)

The CAPM is the textbook example of an economic normal-return model. Its abnormal return is

$$AR_{it} = R_{it} - \left[ R_{ft} + \beta_i (R_{mt} - R_{ft}) \right],$$

where $R_{ft}$ is the risk-free rate. The CAPM is the market model with the intercept restricted by the equilibrium condition (the market model, by contrast, estimates a free intercept $\alpha_i$ and imposes no cross-sectional restriction). Although common in event studies of the 1970s, the CAPM's use has almost ceased: deviations from its restrictions, such as the size effect (Banz (1981)), can make inferences sensitive to the validity of those restrictions, whereas the market model sidesteps this (MacKinlay (1997)). The CAPM is presented here as a related model and is not currently offered in our calculators.

Which model should I use?

The choice of expected return model depends on data availability and the structure of the sample. The table below maps a situation to a sensible default.

This mirrors what researchers actually do. In a sample of 400 reviewed event studies, Holler (2014) found the following usage shares:

Worked example: one event day, three models

Suppose that estimation over the pre-event window yields $\alpha_i = 0.0002$ (0.02% per day) and $\beta_i = 1.20$ for a given stock. On the event day the market returns $R_{mt} = -1.0\%$ and the stock actually returns $R_{it} = +2.0\%$. The three simplest models then give different abnormal returns for the same day:

• Market model: expected return $= 0.0002 + 1.20 \times (-0.010) = -1.18\%$, so $AR = 2.0\% - (-1.18\%) = +3.18\%$.
• Market-adjusted: $AR = 2.0\% - (-1.0\%) = +3.0\%$.
• Constant mean (assume estimation-window mean $= 0.05\%$): $AR = 2.0\% - 0.05\% = +1.95\%$.

The same realized return produces materially different abnormal returns depending on how the market move is treated. This is precisely why model choice matters, especially over long horizons.

Model choice and sample-selection bias

Choosing among expected return models is generally related to sample-selection biases (for example 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 was found to predict too-low returns (Banz (1981)), leading to inflated abnormal returns in the event study. Multifactor models try to circumvent this problem by explicitly modeling the factors that drive the biased results.

Because abnormal-return biases tend to be small over short windows, most scholars interested in the economic substance of individual event types, rather than methodological discussions, still use the market model, consistent with the usage shares reported above (Holler (2014)).

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 subtracted from the return $R_{it}$ of the observation $i$ on day $t$. We get for the abnormal return:

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

This is the market model of section [1] with the restriction $\alpha_i = 0$ and $\beta_i = 1$ imposed, so no parameters are estimated.

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

In the comparison period mean model, also called the constant mean return 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 $\bar{R}_{i} = \frac{1}{T_1 - T_0 + 1}\sum\limits_{t\in [T_0, T_1]}R_{it}$, an average over the $(T_1 - T_0 + 1)$ observations in the inclusive estimation window $[T_0, T_1]$.

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

For non-synchronous trading, you may choose the market model with Scholes-Williams beta estimation. Under thin or infrequent trading a stock's last recorded price often predates the market close, so a contemporaneous OLS beta is biased toward zero through an errors-in-variables (stale-price) effect. The Scholes-Williams estimator corrects this by combining lagged, contemporaneous, and lead market regressions:

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

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 $\alpha^{SW}_i$ 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 (Scholes & Williams (1977)).

[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

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

with the abnormal return defined, as in section [1], by $AR_{it} = R_{it} - (\alpha_i + \beta_i \cdot R_{mt})$. 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 $\varepsilon^2_{t-1}$ the lagged squared residual (the ARCH term) from the mean filtration process. (The conventional GARCH labels $\alpha_1$ and $\beta_1$ are written here as $\gamma_1$ and $\delta_1$ to avoid a clash with the market-model $\alpha$ and $\beta$.) Parameters are estimated by maximum likelihood (a non-linear solver is used for the optimization problem).

The EGARCH variant (Nelson (1991)) models the logarithm of the conditional variance, which guarantees a positive variance without parameter constraints and adds an asymmetry (leverage) term:

$$\ln(\sigma^2_t) = \omega + \alpha \left( |z_{t-1}| - E|z| \right) + \gamma \, z_{t-1} + \beta \ln(\sigma^2_{t-1}),$$

where $z_t = \varepsilon_t / \sigma_t$ is the standardized residual and $\gamma$ is the leverage parameter: $\gamma 0$ means negative shocks raise volatility more than positive shocks of equal size. In both variants the practical payoff is a time-varying conditional $\sigma_t$ that feeds standardized test statistics on the abnormal returns.

[6] Fama-French 3-Factor Model (Abbr.: ffm3)

In an event study the Fama-French three-factor model is estimated as a time-series regression of the firm's excess return on the factor returns, with an intercept and an error term:

$$R_{it} - R_{ft} = \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t + \varepsilon_{it},$$

where $R_{ft}$ is the risk-free rate, $R_{mt} - R_{ft}$ is the market excess return, $SMB_t$ is the size factor (small minus big), and $HML_t$ is the value factor (high minus low book-to-market). The coefficients $b_{i,M}$, $s_i$ and $h_i$ are the factor sensitivities (loadings) of stock $i$. The abnormal return is the regression residual:

$$AR_{it} = (R_{it} - R_{ft}) - \left[ \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t \right].$$

(Fama & French (1993).)

[7] Carhart Four-Factor Model (Abbr.: ffm4)

The Carhart model adds a momentum factor to the three-factor model:

$$R_{it} - R_{ft} = \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t + u_i \, UMD_t + \varepsilon_{it},$$

where $UMD_t$ (UMD = Up Minus Down = WML = MOM, the momentum factor) is the return of past 12-month winners minus losers, and $u_i$ its loading. The abnormal return is again the residual:

$$AR_{it} = (R_{it} - R_{ft}) - \left[ \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t + u_i \, UMD_t \right].$$

(Carhart (1997); momentum from Jegadeesh & Titman (1993).)

[8] Fama-French Five-Factor Model (Abbr.: ffm5)

The five-factor model adds a profitability factor and an investment factor to the three-factor model (it contains no momentum factor):

$$R_{it} - R_{ft} = \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t + r_i \, RMW_t + c_i \, CMA_t + \varepsilon_{it},$$

where $RMW_t$ is the profitability factor (robust minus weak) and $CMA_t$ is the investment factor (conservative minus aggressive), with loadings $r_i$ and $c_i$. The abnormal return is the residual:

$$AR_{it} = (R_{it} - R_{ft}) - \left[ \alpha_i + b_{i,M}(R_{mt}-R_{ft}) + s_i \, SMB_t + h_i \, HML_t + r_i \, RMW_t + c_i \, CMA_t \right].$$

(Fama & French (2015).)

Factor data and return convention

To run the ffm3, ffm4 and ffm5 models you need time series for the SMB, HML, RMW, CMA and momentum (MOM) factors. These are published free, with regional coverage, in the Kenneth R. French Data Library. Returns in all formulas above are simple discrete returns unless a log-return option is selected.

From model to significance

Model choice sets not only the abnormal-return point estimate but also its standard error: each model produces a residual standard deviation $\sigma$ that drives the t-statistics on the abnormal returns, and our calculators apply a forecast-error correction (with HAC / Newey-West options) when computing it. See the test statistics page for how each model's residual variance feeds the significance tests.