28.8 Expected Return Calculation
28.8.1 Statistical Models
based on statistical assumptions about the behavior of returns (e..g, multivariate normality)
we only need to assume stable distributions (Owen and Rabinovitch 1983)
28.8.1.1 Constant Mean Return Model
The expected normal return is the mean of the real returns
\[ Ra_{it} = R_{it} - \bar{R}_i \]
Assumption:
- returns revert to its mean (very questionable)
The basic mean returns model generally delivers similar findings to more complex models since the variance of abnormal returns is not decreased considerably (S. J. Brown and Warner 1985)
28.8.1.2 Market Model
\[ R_{it} = \alpha_i + \beta R_{mt} + \epsilon_{it} \]
where
\(R_{it}\) = stock return \(i\) in period \(t\)
\(R_{mt}\) = market return
\(\epsilon_{it}\) = zero mean (\(E(e_{it}) = 0\)) error term with its own variance \(\sigma^2\)
Notes:
People typically use S&P 500, CRSP value-weighed or equal-weighted index as the market portfolio.
When \(\beta =0\), the Market Model is the Constant Mean Return Model
better fit of the market-model, the less variance in abnormal return, and the more easy to detect the event’s effect
recommend generalized method of moments to be robust against auto-correlation and heteroskedasticity
28.8.1.3 Fama-French Model
Please note that there is a difference between between just taking the return versus taking the excess return as the dependent variable.
The correct way is to use the excess return for firm and for market (Fama and French 2010, 1917).
- \(\alpha_i\) “is the average return left unexplained by the benchmark model” (i.e., abnormal return)
28.8.1.3.1 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)
\(R_{mt}\) is the market-rate (e.g., S&P 500)
SML: returns on small (size) portfolio minus returns on big portfolio
HML: returns on high (B/M) portfolio minus returns on low portfolio.
28.8.1.3.2 FF4
(A. Sorescu, Warren, and Ertekin 2017, 195) suggest the use of Market Model in marketing for short-term window and Fama-French Model for the long-term window (the statistical properties of this model have not been examined the the daily setting).
\[ \begin{aligned} E(R_{it}|X_t) - r_{ft} = \alpha_i &+ \beta_{1i} (E(R_{mt}|X_t )- r_{ft}) \\ &+ b_{2i} SML_t + b_{3i} HML_t + b_{4i} UMD_t \end{aligned} \]
where
- \(UMD_t\) is the momentum factor (difference between high and low prior return stock portfolios) in day \(t\).
28.8.2 Economic Model
The only difference between CAPM and APT is that APT has multiple factors (including factors beyond the focal company)
Economic models put limits on a statistical model that come from assumed behavior that is derived from theory.