Calculating Buy-and-Hold Abnormal Returns from Periodic Returns
Summary
Buy-and-hold abnormal return (BHAR) compares an asset’s compounded return over a holding period with the benchmark’s compounded return over the same period. The document clarifies that the asset and benchmark returns should be built from consecutive period-to-period simple returns, then compounded separately before taking their difference. This corresponds to chaining each month’s return from the preceding month, rather than repeatedly comparing every month’s price with the first month’s price.
The answer restates the BHAR formula and identifies the consecutive-return calculation as the appropriate interpretation. It offers no worked numerical example or discussion of alternative benchmarks, sample selection, delisting returns, or statistical inference. The result is a concise clarification of the compounding convention, not a comprehensive guide to measuring long-run IPO performance; applying it still requires consistently defined holding periods and benchmark returns.
Key ideas
- BHAR is the difference between compounded asset and benchmark returns over the same horizon.
- Each period’s simple return is calculated relative to the preceding period.
- The period returns are compounded separately for the asset and its benchmark.
- The document clarifies the return calculation but does not cover benchmark choice or statistical inference.
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Full text
# How to calculate the BHAR (Buy-and-Hold Abnormal Returns)?
# How to calculate the BHAR (Buy-and-Hold Abnormal Returns)?
I am doing my research related to IPOs long term performance. For the $\text{BHAR}$ (Buy-and-Hold Abnormal Returns) formula, I just want to clarify the formula is that always compare with the first month trading price, or is compared with last month trading price?
The following two ways are listed
- Always compare with First Month Trading Price $\bigg[ \big(\frac{\text{Month}2}{\text{Month}1} \big) \times \big(\frac{\text{Month}3}{\text{Month}1} \big) \times \big(\frac{\text{Month}4}{\text{Month}1} \big) \bigg] - \bigg[ \big(\frac{\text{indexMonth}2}{\text{indexMonth}1} \big) \times \big(\frac{\text{indexMonth}3}{\text{indexMonth}1} \big) \times \big(\frac{\text{indexMonth}4}{\text{indexMonth}1} \big) \bigg] $
- Compare with the last Month Trading Price $\bigg[ \big(\frac{\text{Month}2}{\text{Month}1} \big) \times \big(\frac{\text{Month}3}{\text{Month}2} \big) \times \big(\frac{\text{Month}4}{\text{Month}3} \big) \bigg] - \bigg[ \big(\frac{\text{indexMonth}2}{\text{indexMonth}1} \big) \times \big(\frac{\text{indexMonth}3}{\text{indexMonth}2} \big) \times \big(\frac{\text{indexMonth}4}{\text{indexMonth}3} \big) \bigg] $
I calculated $1 + R_{i,t}$ using above two methods.
In the table, the `1 + Rit (a)` indicates first method, and `1 + Rit (b)` indicates second method.
Which one is correct in the $\text{BHAR}$ formula?
Reference Formula
$$\boxed{\text{BHAR}_{i,T} = \prod_{t=1}^{T}(1+R_{i,t}) - \prod_{t=1}^{T}(1+R_{m,t})}$$
where $\text{BHAR}_{i,T}$ is the abnormal return of the asset $i$ over the period $T$, $R_{i,t}$ is the month $t$ simple return of the asset $i$, and $R_{m,t}$ is the month $t$ simple return of the benchmark portfolio or index $m$.
## Answer by Alper (score 2, accepted)
https://quant.stackexchange.com/a/68474
Welcome to Quantitative Finance!
I reckon the BHAR (Buy-and-Hold Abnormal Returns) formula you are referring to is
$$\text{BHAR}_{i,h} = \prod_{t=1}^{h}(1+R_{i,t}) - \prod_{t=1}^{h}(1+R_{m,t})$$
where $\text{BHAR}_{i,h}$ is the abnormal return of the asset $i$ over the period $h$, $R_{i,t}$ is the month $t$ simple return of the asset $i$, and $R_{m,t}$ is the month $t$ simple return of the benchmark portfolio or index $m$, and you are specifically enquiring about the first part of the right-hand side of the above equation.
Then, the answer to your question is the second formula in your post (the one in the paragraph starting with “or compared”) which, curiously, seems to be related rather to the first set of calculations (those in the column whose heading is “1+Rit (a)”) in the table in your post.
On another note, if you are planning to ask (or answer) more questions including formulas on the Internet, I recommend that you learn MathJax, which is basically LaTeX for the Internet. You are more likely to get (better) answers if the formulas, if any, in your questions are easily comprehensible. You can start with this YouTube video if you like. The MathType software mentioned in that video is a bit like a tricycle: it helps an absolute beginner start using MathJax immediately. And once you learn MathJax further, you may find this practical but rich and detailed reference on Math Stack Exchange to be quite helpful.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.