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) Constant Mean Return Model

The expected normal return is the mean of the real returns

\[ Ra_{it} = R_{it} - \bar{R}_i \]


  • 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) Market Model

\[ R_{it} = \alpha_i + \beta R_{mt} + \epsilon_{it} \]


  • \(R_{it}\) = stock return \(i\) in period \(t\)

  • \(R_{mt}\) = market return

  • \(\epsilon_{it}\) = zero mean (\(E(e_{it}) = 0\)) error term with its own variance \(\sigma^2\)


  • 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 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) FF3

(Fama and French 1993)

\[ \begin{aligned} E(R_{it}|X_t) - r_{ft} = \alpha_i &+ \beta_{1i} (E(R_{mt}|X_t )- r_{ft}) \\ &+ b_{2i} SML_t + b_{3i} HML_t \end{aligned} \]


  • \(r_{ft}\) 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. 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).

(Carhart 1997)

\[ \begin{aligned} E(R_{it}|X_t) - r_{ft} = \alpha_i &+ \beta_{1i} (E(R_{mt}|X_t )- r_{ft}) \\ &+ b_{2i} SML_t + b_{3i} HML_t + b_{4i} UMD_t \end{aligned} \]


  • \(UMD_t\) 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. Capital Asset Pricing Model (CAPM)

\[ E(R_i) = R_f + \beta_i (E(R_m) - R_f) \]


  • \(E(R_i)\) = expected firm return

  • \(R_f\) = risk free rate

  • \(E(R_m - R_f)\) = market risk premium

  • \(\beta_i\) = firm sensitivity Arbitrage Pricing Theory (APT)

\[ R = R_f + \Lambda f + \epsilon \]


  • \(\epsilon \sim N(0, \Psi)\)

  • \(\Lambda\) = factor loadings

  • \(f \sim N(\mu, \Omega)\) = general factor model

    • \(\mu\) = expected risk premium vector

    • \(\Omega\) = factor covariance matrix


———. 1985. “Using Daily Stock Returns: The Case of Event Studies.” Journal of Financial Economics 14 (1): 3–31.
Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” The Journal of Finance 52 (1): 57–82.
Fama, Eugene F, and Kenneth R French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56.
———. 2010. “Luck Versus Skill in the Cross-Section of Mutual Fund Returns.” The Journal of Finance 65 (5): 1915–47.
Owen, Joel, and Ramon Rabinovitch. 1983. “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice.” The Journal of Finance 38 (3): 745–52.
Sorescu, Alina, Nooshin L Warren, and Larisa Ertekin. 2017. “Event Study Methodology in the Marketing Literature: An Overview.” Journal of the Academy of Marketing Science 45: 186–207.