**Long-term event studies and the calender-time portfolio approach**

Long-term event studies aim at identifying if certain events affect asset prices over long periods of time, i.e., several months or years. Thus, they can be interpreted as tests of performance persistence. Prominent examples include examinations of long-run stock price performance after initial or seasoned public offerings of equity (see, Ritter, 1991; Loughran and Ritter, 1995, among others), or the long-run performance of mergers and acquisitions (see, e.g., Franks, Harris and Titman, 1991; Rau and Vermaelen, 1998).

Results allow academics and investors to draw inferences about the value of particular long-run investments. This way, one can provide empirically sound insights with respect to, for example, the value and certification abilities of reputable versus less reputable underwriters (see, e.g., Carter, Dark and Singh, 1998) or the realization of synergies in M&As.Among many other examples, one can further examine whether it pays to mimic the investment behavior of firm insiders or to purchase stocks recommended by all-star analysts.

For long-term event studies, two alternative methodologies to calculate risk-adjusted post-event performance exist. Besides the buy-and-hold abnormal return (BHAR) approach (described in the event study blueprint), scholars have developed the method of calendar-time portfolios (CTIME). The method is also known as the Jensen’s alpha approach. The basic idea of the approach is to construct a portfolio of firms for which the event of interest occurred and define the abnormal return earned by this portfolio as the portfolio’s excess return (i.e., the return over the risk-free rate) that cannot be explained by risk-factor models used to predict expected returns (the CAPM and the three- and four-factor models). This is what gives the CTIME approach its alternative name “alpha approach”: if the aforementioned models for expected return estimation are correct (i.e., capture all factors necessary to explain expected returns), the intercept – often denoted as alpha – in a time-series regression on realized returns that empirically tests these models should be statistically insignificant and close to zero. This means that a portfolio (i.e., a trading strategy) generating statistically significant alphas larger than zero creates positive abnormal returns, or put differently, beats the market. Thus, the CTIME approach, similar to the BHAR approach, resembles the investors’ actual investment experience. The period of months over which each stock of an event firm is included in the rolling portfolio (after the event) can be chosen freely. Usually, 12 to 60 months are considered. The length of the period event firms remain in the portfolio after the event corresponds to the performance persistence one wants take into account.

Since Jaffe (1974) and Mandelker (1974), the method of calendar-time portfolios has been used in many studies such as Barber and Odean (2000 and 2001), Brav and Gompers (1997), Jeng, Metrick and Zeckhauser (2003), or Mitchell and Stafford (2000). It is a standard method in fund research and it is also frequently used in behavioral and corporate finance. More recently, the approach has also been used in other fields like marketing (see, e.g., Moorman et al., 2012). The methodology of calendar-time portfolios is an approach of simulating portfolios, or trading strategies, which involves two general steps: the calculation of average excess returns of the rolling portfolio of event firms (step 1) and the time-series regression of the excess returns on a number of risk factors depending on the model chosen to predict returns (step 2).

In the following, we present the two steps in detail. Event Study Metrics computes both steps at once (i.e., without demanding further input from the user) after the user has made the relevant settings (see settings). However, we explain the two steps separately to enhance the understanding for the methodology. For additional literature on long-term event studies, please refer to Kothari and Warner (2008).

### Settings

To start the calendar-time portfolios calculation in the Event Study Metrics software, you have to do the following. After the list of events and event firms as well as the necessary stock return data has been imported, simply click on the tab ‘BHAR/CTIME’ and select ‘Calendar-Time Portfolio’. Next, select the ‘Normal Return Model’ (either one-, three- or four-factor model) and ‘Estimation Method’ (either OLS or WLS). These two settings are motivated and explained in detail in step 2. Finally, define the inclusion and exclusion date of each event firm’s stock in the rolling portfolio of event firms, see step 1.

## Step 1

When the settings are made and the data is available, one has to define the length of the period over which the stock of an event firm is included in the rolling portfolio. This is done by defining the inclusion and exclusion date for each asset. In particular, the inclusion date refers to the month after the event date (i.e., after the day the event of interest occurred for a certain firm). It is the (event-firm specific) month in which the firm’s stock is added to the portfolio. The exclusion date is the month after the date of the event (for each event firm) after which the stock of the event firm is excluded from the portfolio.

That is, if the inclusion date is set to 1 and the exclusion date is set to 12, the stock of a certain event firm is part of the portfolio for one year, i.e., it is added to the portfolio in the month after the event date and it is excluded from the portfolio of event firms after 12 months. This is why the calendar-time portfolios approach is a rolling-window approach and why the portfolio is denoted rolling portfolio. It simply means that assets of event firms are part of the portfolio for a certain period of time only. This period can be chosen freely and depends on the time frame and persistence the user wants to inspect.

Suppose one wants to assess abnormal returns over a three-year period. For each month in calendar-time the portfolio is constructed by all firms that had an event in the three years (36 months) prior to the calendar month (month in which the event of an event firm was announced or took place). As a consequence of the described methodology, the rolling portfolio can, and usually does, consist of different event firms each month. By default, the Event Study Metrics software constructs equal-weighted portfolios and calculates the excess return of each monthly portfolio hence as the simple (non-weighted) average of each asset’s excess return over the monthly risk-free rate (i.e., the one-month Treasury rate in case of a study with U.S. data). You may select the ‘Use Weights’ option (in the left-hand corner at the top) to form value-weighted portfolios; these are usually based on the market capitalization of the event firms.

## Step 2

Once the excess returns of the monthly portfolios have been calculated, these excess returns are regressed on risk factors in a multivariate time-series regression in order to determine the alpha of this regression, i.e., the abnormal return of the monthly portfolio of event firms. While the portfolio excess return is the dependent (or left-hand-side) variable of the regression equation, the chosen risk factors are the independent (or right-hand-side) variables. We provide the relevant formulas in our methodology section. The choice of risk factors depends on the (user’s) choice of the underlying model to predict expected returns. The literature has proposed to use one of the following models: the one-factor model (Jensen, 1968), which is an empirical version of the CAPM in excess returns, the three-factor model (Fama and French, 1993), or the four-factor model (Carhart, 1997).

The one-factor model uses the beta factor as the only risk factor. The three-factor model uses the beta, the SMB factor and the HML factor. The SMB factor, standing for “small minus big”, accounts for return differences between small and big stocks, while the HML factor, standing for “high minus low”, accounts for return differences between stocks with high and stocks with low book-to-market (BTM) ratios (using the BTM of equity). Finally, the four-factor model adds a momentum factor abbreviated MOM (monthly momentum). This factor captures a stock price’s tendency to continue rising if it has been rising in the near past and to continue declining if it has been declining.

The one-factor model (based on the CAPM) is the only risk-factor model based on a theory of expected returns. The three- and four-factor models are just empirically motivated. Fama and French (1993) and Carhart (1997) as well as several studies thereafter have documented that multi-factor models perform considerably better in explaining expected stock returns. As a consequence, many studies simply run regressions for all of the three risk-factor models and report the results for each model (for robustness).

Event Study Metrics allows choosing all of the aforementioned models. This is done by selecting ‘1 factor Model’ or ‘3 Factor Model’ or ‘4 Factor Model’ in the dropdown menu labeled ‘Normal Return Model’. Regressions with all models can be run consecutively. According to the chosen normal return model, the software runs a multivariate time-series regression on the monthly portfolio excess returns determined in the first step and provides an output with the regression coefficients of the risk factors and, most important, the alpha of the regression (shown in the right-hand corner at the bottom). Besides the number of observations and the R-squared of the regression, the software further shows the corresponding test-statistics for each regression coefficient and the intercept. The regular t-statistic and the heteroskedasticity-consistent t-statistic developed by White (1980) are reported. Remember that accounting for heteroskedasticity is important in time-series regressions. In this regard, further take into account that since the number of securities included in the portfolio of event firms varies over time, the error terms in calendar-time portfolio regressions can be heteroskedastic as shown in Lyon, Barber and Tsai (1999).

The authors propose to use a weighted least squares (WLS) regression, instead of an ordinary least squares (OLS) regression, where the weighting factor is based on the number of assets in the (monthly) portfolio. Event Study Metrics provides both estimation methods. Whether the multivariate regressions of the second step are run using OLS or WLS can be selected in the Estimation Method dropdown menu. By default, the OLS regression method is used.

### Calculating long-short portfolios

Long-short portfolios can be calculated with the Event Study metrics software by using the ‘Use Weights’ function. Simply weight stocks with -1 if you want to include them in the short portfolio and weight stocks with 1 to include them in the long portfolio .

### BHAR vs. CTIME

Although, of course, one can calculate both buy-and-hold returns and calendar-time portfolios, the question arises which approach should be preferred. In this regard, the major criterion is the reliability of the two approaches. Several financial economists, including Nobel laureate Eugene Fama, advocate the CTIME approach. Fama (1998) argues that the BHAR approach does not adequately control for cross-sectional correlation among individual firms. This can lead to overstated test statistics as shown by Mitchell and Stafford (2000) and may hence produce less reliable results. An important statistical advantage of using the calendar-time portfolios approach is that it uses a time-series of portfolio returns.

As a result, the portfolio variance includes the cross-correlations of firm abnormal returns and the problem of cross-sectional dependence is eliminated (see Lyon, Barber and Tsai, 1999). Further, Mitchell and Stafford (2000) provide evidence that the distribution of the estimator in the CTIME approach is well-approximated by the normal distribution. This favors robust statistical inference. Another disadvantage of the BHARs approach, mentioned in Eckbo, Masulis and Norli (2000), is that it is not a feasible portfolio strategy because the total number of securities is unknown in advance. Prominent criticism against the CTIME approach is brought forward by Loughran and Ritter (2000) who argue that this approach is potentially biased to find results consistent with market efficiency as it does not put enough weight on managers’ timing decisions of corporate events.

In fact, the logic of timing certain corporate events, such as debt or equity issues or dividend payments (see Baker, 2009; Baker and Wurgler, 2002 and 2004), implies that corporate managers endogenously time corporate events and partly capitalize misvaluations varying over time in the capital markets. Yet, as the CTIME approach weights each time period equally, it can have a lower power of detecting abnormal returns in case firms time their corporate actions. To mitigate this potential problem, Fama (1998) suggests weighting calendar months with their statistical precision which depends on the sample size of each monthly portfolio.

### References

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Baker, M. and J. Wurgler (2002): Market Timing and Capital Structure, *Journal of Finance* 57, 1-32.

Baker, M. and J. Wurgler (2004): A Catering Theory of Dividends, *Journal of Finance* 59, 1125-1165.

Barber, B.M. and T. Odean (2000): Trading is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors, *Journal of Finance* 55, 773-806.

Barber, B.M. and T. Odean (2001): Boys Will Be Boys: Gender, Overconfidence, and Common Stock Investment, *Quarterly Journal of Economics* 116, 261-292.

Brav, A. And P. Gompers (1997): Myth or Reality? The Long-Run Underperformance of Initial Public Offerings: Evidence from Venture and Non-Venture-Backed Companies, *Journal of Finance* 52, 1791-1821.

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Kothari, S.P. and Warner, J.B. (2008): Econometrics of Event Studies, in: Eckbo, B.E. (ed.), *Handbook of Corporate Finance: Empirical Corporate Finance*, Vol. 1, Elsevier/North-Holland, 3-36.

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