# 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

1. Always compare with First Month Trading Price

$$\bigg[ \big(\frac{\text{Month}2}{\text{Month}1} \big) \times \big(\frac{\text{Month}3}{\text{Month}1} \big) \times \big(\frac{\text{Month}4}{\text{Month}1} \big) \bigg] - \bigg[ \big(\frac{\text{indexMonth}2}{\text{indexMonth}1} \big) \times \big(\frac{\text{indexMonth}3}{\text{indexMonth}1} \big) \times \big(\frac{\text{indexMonth}4}{\text{indexMonth}1} \big) \bigg]$$

2. 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 $$m$$.

$$\text{BHAR}_{i,h} = \prod_{t=1}^{h}(1+R_{i,t}) - \prod_{t=1}^{h}(1+R_{m,t})$$
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.