28.5 Long-run event studies


# example by the package's author
SAR <-
    sar(event = demo_returns$EON,
        control = demo_returns$RWE,
        logret = FALSE)
#> [1] 0.006870196

28.5.1 Buy and Hold Abnormal Returns (BHAR)

Use a portfolio of stocks that are close matches of the current firm over the same period as benchmark, and see the difference between the firm return and that of the portfolio.

  • More technical note is that it measures returns from buying stocks in event-experiencing firms and shorting stocks in similar non-event firms within the same time.
  • Because of high cross-sectional correlations, BHARs’ t-stat can be inflated, but its rank order is not affected (Markovitch and Golder 2008; A. B. Sorescu, Chandy, and Prabhu 2007)

To construct the portfolio, use similar

  • size
  • book-to-market
  • momentum

Matching Procedure (Barber and Lyon 1997):

  1. Each year from July to June, all common stocks in the CRSP database are categorized into ten groups (deciles) based on their market capitalization from the previous June.

  2. Within these deciles, firms are further sorted into five groups (quintiles) based on their book-to-market ratios as of December of the previous year or earlier, considering possible delays in financial statement reporting.

  3. Benchmark portfolios are designed to exclude firms with specific events but include all firms that can be classified into the characteristic-based portfolios.

Similarly, Wiles et al. (2010) uses the following matching procedure:

  1. All firms in the same two-digit SIC code with market values of 50% to 150% of the focal firms are selected
  2. From this list, the 10 firms with the most comparable book-to-market ratios are chosen to serve as the matched portfolio (the matched portfolio can have less than 10 firms).


\[ AR_{it} = R_{it} - E(R_{it}|X_t) \]

Cumulative Abnormal Return (CAR):

\[ CAR_{it} = \sum_{t=1}^T (R_{it} - E(R_{it})) \]

Buy-and-Hold Abnormal Return (BHAR)

\[ BHAR_{t = 1}^T = \Pi_{t=1}^T(1 + R_{it}) - \Pi_{t = 1}^T (1 + E(R_{it})) \]

where as CAR is the arithmetic sum, BHAR is the geometric sum.

  • In short-term event studies, differences between CAR and BHAR are often minimal. However, in long-term studies, this difference could significantly skew results. (Barber and Lyon 1997) shows that while BHAR is usually slightly lower than annual CAR, but it dramatically surpasses CAR when annual BHAR exceeds 28%.

To calculate the long-run return (\(\Pi_{t=1}^T (1 + E(R_{it}))\)) of the benchmark portfolio, we can:

  1. With annual rebalance: In each period, each portfolio is re-balanced and then compound mean stock returns in a portfolio over a given period:

\[ \Pi_{t = 1}^T (1 + E(R_{it})) = \Pi_{t}^T (1 + \sum_{i = s}^{n_t}w_{it} R_{it}) \]

where \(n_t\) is the number of firms in period \(t\), and \(w_{it}\) is (1) \(1/n_t\) or (2) value-weight of firm \(i\) in period \(t\).

To avoid favoring recent events, in cross-sectional event studies, researchers usually treat all events equally when studying their impact on the stock market over time. This approach helps identify any abnormal changes in stock prices, especially when dealing with a series of unplanned events.

Potential problems:

  • Solution first: Form benchmark portfolios that will never change constituent firms (Mitchell and Stafford 2000), because of these problems:

    • Newly public companies often perform worse than a balanced market index (Ritter 1991), and this, over time, might distort long-term return expectations due to the inclusion of these new companies (a phenomenon called “new listing bias” identified by Barber and Lyon (1997)).

    • Regularly rebalancing an equal-weight portfolio can lead to overestimated long-term returns and potentially skew buy-and-hold abnormal returns (BHARs) negatively due to constant selling of winning stocks and buying of underperformers (i.e., “rebalancing bias” (Barber and Lyon 1997)).

    • Value-weight portfolios, which favor larger market cap stocks, can be viewed as an active investment strategy that keeps buying winning stocks and selling underperformers. Over time, this approach tends to positively distort BHARs.

  1. Without annual rebalance: Compounding the returns of the securities comprising the portfolio, followed by calculating the average across all securities

\[ \Pi_{t = s}^{T} (1 + E(R_{it})) = \sum_{i=s}^{n_t} (w_{is} \Pi_{t=1}^T (1 + R_{it})) \]

where \(t\) is the investment period, \(R_{it}\) is the return on security \(i\), \(n_i\) is the number of securities, \(w_{it}\) is either \(1/n_s\) or value-weight factor of security \(i\) at initial period \(s\). This portfolio’s profits come from a simple investment where all the included stocks are given equal importance, or weighted according to their market value, as they were in a specific past period (period s). This means that it doesn’t consider any stocks that were listed after this period, nor does it adjust the portfolio each month. However, one problem with this method is that the value assigned to each stock, based on its market size, needs to be corrected. This is to make sure that recent stocks don’t end up having too much influence.

Fortunately, on WRDS, it will give you all types of BHAR (2x2) (equal-weighted vs. value-weighted and with annual rebalance and without annual rebalance)

  • “MINWIN” is the smallest number of months a company trades after an event to be included in the study.
  • “MAXWIN” is the most months that the study considers in its calculations.

    • Companies aren’t excluded if they have less than MAXWIN months, unless they also have fewer than MINWIN months.
  • The term “MONTH” signifies chosen months (typically 12, 24, or 36) used to work out BHAR.

    • If monthly returns are missing during the set period, matching portfolio returns fill in the gaps.

28.5.2 Long-term Cumulative Abnormal Returns (LCARs)

Formula for LCARs during the \((1,T)\) postevent horizon (A. B. Sorescu, Chandy, and Prabhu 2007)

\[ LCAR_{pT} = \sum_{t = 1}^{t = T} (R_{it} - R_{pt}) \]

where \(R_{it}\) is the rate of return of stock \(i\) in month \(t\)

\(R_{pt}\) is the rate of return on the counterfactual portfolio in month \(t\)

28.5.3 Calendar-time Portfolio Abnormal Returns (CTARs)

This section follows strictly the procedure in (Wiles et al. 2010)

A portfolio for every day in calendar time (including all securities which experience an event that time).

For each portfolio, the securities and their returns are equally weighted

  1. For all portfolios, the average abnormal return are calculated as

\[ AAR_{Pt} = \frac{\sum_{i=1}^S AR_i}{S} \]


  • \(S\) is the number of securities in portfolio \(P\)
  • \(AR_i\) is the abnormal return for the stock \(i\) in the portfolio
  1. For every portfolio \(P\), a time series estimate of \(\sigma(AAR_{Pt})\) is calculated for the preceding \(k\) days, assuming that the \(AAR_{Pt}\) are independent over time.
  2. Each portfolio’s average abnormal return is standardized

\[ SAAR_{Pt} = \frac{AAR_{Pt}}{SD(AAR_{Pt})} \]

  1. Average standardized residual across all portfolio’s in calendar time

\[ ASAAR = \frac{1}{n}\sum_{i=1}^{255} SAAR_{Pt} \times D_t \]


  • \(D_t = 1\) when there is at least one security in portfolio \(t\)

  • \(D_t = 0\) when there are no security in portfolio \(t\)

  • \(n\) is the number of days in which the portfolio have at least one security \(n = \sum_{i = 1}^{255}D_t\)

  1. The cumulative average standardized average abnormal returns is

\[ CASSAR_{S_1, S_2} = \sum_{i=S_1}^{S_2} ASAAR \]

If the ASAAR are independent over time, then standard deviation for the above estimate is \(\sqrt{S_2 - S_1 + 1}\)

then, the test statistics is

\[ t = \frac{CASAAR_{S_1,S_2}}{\sqrt{S_2 - S_1 + 1}} \]


  • Cannot examine individual stock difference, can only see the difference at the portfolio level.

    • One can construct multiple portfolios (based on the metrics of interest) so that firms in the same portfolio shares that same characteristics. Then, one can compare the intercepts in each portfolio.
  • Low power (Loughran and Ritter 2000), type II error is likely.


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