33.13 Expected Return Calculation
Expected return models are essential for estimating abnormal returns in event studies. These models help separate normal stock price movements from those caused by specific events.
33.13.1 Statistical Models for Expected Returns
Statistical models rely on assumptions about the behavior of returns, often assuming stable distributions (Owen and Rabinovitch 1983). These models do not impose economic constraints but instead focus on statistical properties of returns.
33.13.1.1 Constant Mean Return Model
The simplest statistical model assumes that a stock’s expected return is simply its historical mean return:
\[ Ra_{it} = R_{it} - \bar{R}_i \]
where:
- \(R_{it}\) = observed return of stock \(i\) in period \(t\)
- \(\bar{R}_i\) = mean return of stock \(i\) over the estimation period
- \(Ra_{it}\) = abnormal return in period \(t\) (i.e., deviation from historical average)
Assumptions:
- Returns revert to their mean over time (i.e., they follow a stable mean-reverting process).
- This assumption is questionable, as market conditions evolve dynamically.
Empirical Note:
The constant mean return model typically delivers similar results to more complex models since the variance of abnormal returns is not substantially reduced when using more sophisticated statistical approaches (S. J. Brown and Warner 1985).
33.13.1.2 Market Model
A widely used alternative to the constant mean model is the market model, which assumes that stock returns are linearly related to market returns:
\[ R_{it} = \alpha_i + \beta_i R_{mt} + \epsilon_{it} \]
where:
- \(R_{it}\) = return of stock \(i\) in period \(t\)
- \(R_{mt}\) = market return in period \(t\) (e.g., S&P 500 index)
- \(\alpha_i\) = stock-specific intercept (capturing average return not explained by the market)
- \(\beta_i\) = systematic risk (market beta) of stock \(i\)
- \(\epsilon_{it}\) = zero-mean error term with variance \(\sigma^2\), capturing idiosyncratic risk
Notes on Implementation:
- The market return (\(R_{mt}\)) is typically proxied using:
- S&P 500 index
- CRSP value-weighted index
- CRSP equal-weighted index
- If \(\beta_i = 0\), the market model reduces to the constant mean return model.
Key Insight:
The better the fit of the market model, the lower the variance of abnormal returns, making it easier to detect event effects.
Robust Estimation:
- To account for heteroskedasticity and autocorrelation, it is recommended to use the Generalized Method of Moments (GMM) for estimation.
33.13.1.3 Fama-French Multifactor Models
The Fama-French family of models extends the market model by incorporating additional factors that capture systematic risks beyond market exposure.
Key Considerations:
- There is a distinction between using total return and excess return as the dependent variable.
- The correct specification involves excess returns for both individual stocks and the market portfolio (Fama and French 2010, 1917).
Interpretation of \(\alpha_i\):
\(\alpha_i\) represents the abnormal return, i.e., the return that is unexplained by the model.
33.13.1.3.1 Fama-French Three-Factor Model (FF3)
\[ \begin{aligned} E(R_{it}|X_t) - r_{ft} &= \alpha_i + \beta_{1i} (E(R_{mt}|X_t )- r_{ft}) \\ &+ b_{2i} SML_t + b_{3i} HML_t \end{aligned} \]
where:
- \(r_{ft}\) = risk-free rate (e.g., 3-month Treasury bill)
- \(R_{mt}\) = market return (e.g., S&P 500)
- \(SML_t\) = size factor (returns on small-cap stocks minus large-cap stocks)
- \(HML_t\) = value factor (returns on high book-to-market stocks minus low book-to-market stocks)
33.13.1.3.2 Fama-French Four-Factor Model (FF4)
(Carhart 1997) extends FF3 by adding a momentum factor:
\[ \begin{aligned} E(R_{it}|X_t) - r_{ft} &= \alpha_i + \beta_{1i} (E(R_{mt}|X_t )- r_{ft}) \\ &+ b_{2i} SML_t + b_{3i} HML_t + b_{4i} UMD_t \end{aligned} \]
where:
- \(UMD_t\) = momentum factor (returns of high past-return stocks minus low past-return stocks)
Practical Application in Marketing:
(A. Sorescu, Warren, and Ertekin 2017, 195) recommends:
Market Model for short-term event windows.
Fama-French Model for long-term windows.
However, the statistical properties of the FF model for daily event studies remain untested.
33.13.2 Economic Models for Expected Returns
Economic models impose theoretical constraints on expected returns based on equilibrium asset pricing theory. The two most widely used models are:
33.13.2.1 Capital Asset Pricing Model (CAPM)
CAPM is derived from modern portfolio theory and assumes that expected returns are determined solely by market risk:
\[ E(R_i) = R_f + \beta_i (E(R_m) - R_f) \]
where:
- \(E(R_i)\) = expected return of stock \(i\)
- \(R_f\) = risk-free rate
- \(E(R_m) - R_f\) = market risk premium (excess return of market portfolio)
- \(\beta_i\) = firm-specific market beta (systematic risk measure)
Key Assumption:
Investors hold the market portfolio, and only systematic risk (beta) matters.
33.13.2.2 Arbitrage Pricing Theory (APT)
APT generalizes CAPM by allowing multiple risk factors to drive expected returns:
\[ R = R_f + \Lambda f + \epsilon \]
where:
- \(\Lambda\) = factor loadings (sensitivities to risk factors)
- \(f \sim N(\mu, \Omega)\) = vector of risk factors
- \(\mu\) = expected risk premiums
- \(\Omega\) = factor covariance matrix
- \(\epsilon \sim N(0, \Psi)\) = idiosyncratic error term
APT vs. CAPM:
CAPM assumes a single factor (market risk).
APT allows multiple systematic factors, making it more flexible for empirical applications.
Summary: Model Comparison
| Model | Key Assumptions | Factors Considered | Best Use Case |
|---|---|---|---|
| Constant Mean Return | Mean-reverting returns | None | Simple event studies |
| Market Model | Linear relationship with market returns | Market factor (\(R_m\)) | Short-term studies |
| Fama-French (FF3) | Size & value factors matter | Market, Size (SML), Value (HML) | Medium- to long-term studies |
| Fama-French (FF4) | Momentum also matters | Market, Size, Value, Momentum | Momentum-driven strategies |
| CAPM | Market is the only risk factor | Market factor (\(R_m\)) | Classic asset pricing |
| APT | Multiple systematic risks matter | Market + Other Factors | Flexible risk modeling |