## 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

#### 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

#### 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 .

• $$\alpha_i$$ “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

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.

#### 28.8.2.1 Capital Asset Pricing Model (CAPM)

$E(R_i) = R_f + \beta_i (E(R_m) - R_f)$

where

• $$E(R_i)$$ = expected firm return

• $$R_f$$ = risk free rate

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

• $$\beta_i$$ = firm sensitivity

#### 28.8.2.2 Arbitrage Pricing Theory (APT)

$R = R_f + \Lambda f + \epsilon$

where

• $$\epsilon \sim N(0, \Psi)$$

• $$\Lambda$$ = factor loadings

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

• $$\mu$$ = expected risk premium vector

• $$\Omega$$ = factor covariance matrix

### References

———. 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.