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Calculating Buy-and-Hold Abnormal Returns with Simple Returns

Article Quant Q&A · Author: elbarto
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Summary

The document addresses how to calculate buy-and-hold abnormal returns (BHAR) over an event window. The stated calculation compounds the stock’s returns across the period and subtracts the compounded market returns, comparing the wealth growth of the investment with that of the market benchmark.

The question includes price and return data and asks whether the displayed period factors should be multiplied across the window. The answer clarifies the key input convention: BHAR uses simple returns, rather than logarithmic returns. The document does not provide a worked numerical BHAR result or further guidance on defining the event window, choosing a benchmark, or interpreting statistical significance. It therefore serves as a concise reminder about return type when implementing a compounded event-study measure.

Key ideas

  • BHAR compares compounded asset performance with compounded benchmark performance over an event window.
  • The return factors for each period are multiplied to represent buy-and-hold growth.
  • Use simple returns in the BHAR calculation rather than logarithmic returns.
  • The document does not address benchmark selection or statistical inference for event studies.

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EquitiesEvent-drivenStatistics
Full text
# Answer by nbbo2 (score 1)


# Computing Buy-and-hold abnormal returns (BHARs) $= \prod_{t=\tau_1}^{\tau_2}(1+R_{i,t}) - \prod_{t=\tau_1}^{\tau_2}(1+R_{m,t})$












I am doing an event study and wanted to know if was going about this correctly$$ \text{BHAR}_{i(\tau_1,\tau_2)}\quad=\quad\prod_{t=\tau_1}^{\tau_2}(1+R_{i,t})~-~\prod_{t=\tau_1}^{\tau_2}(1+R_{m,t}) $$

$$ \begin{array}{|c|c|c|c|c|} \hline \textbf{Date} & \begin{array}{c} \text{Price of} \\ \text{Stock}~i \end{array} & \text{LOG RET} & 1+R_{i,t} & 1+R_{m,t} \\ \hline \text{2015-01-01} & 100 & \text{--} & \text{--} & \text{--} \\ \text{2015-02-01} & 101 & \phantom{-}0.99503 & 1.99503 & 1.004987\phantom{0} \\ \text{2015-03-01} & 102 & \phantom{-}0.00985 & 1.00985 & 1.0039722 \\ \text{2015-04-01} & 103 & \phantom{-}0.00975 & 1.00975 & 0.9990084 \\ \text{2015-05-01} & 104 & \phantom{-}0.01445 & 1.01445 & 1.005934\phantom{0} \\ \text{2015-06-01} & 104 & -0.0047\phantom{0} & 0.99520 & 1.00491\phantom{00} \\ \hline \end{array} $$

Then if I want to calculate the 4-day $\text{BHAR}$ from the 2015-02-01 to 2015-06-01, would it simply be: $$ \begin{array}{cr} & (1.9950)_{\text{Day0}} \times (1.0098)_{\text{Day1}} \times (1.00975)_{\text{Day2}} \times (1.01445)_{\text{Day3}} \times (0.9952)_{\text{Day4}} \\ - & (1.0049)_{\text{Day0}} \times (1.0039)_{\text{Day1}} \times (0.9990)_{\text{Day2}} \times (1.00593)_{\text{Day3}} \times (1.00491)_{\text{Day4}} \end{array}? $$

## Answer by nbbo2 (score 1)

https://quant.stackexchange.com/a/38287

Summarizing the discussion in the comments: BHAR are computed using simple returns, not logarithmic returns.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.