# How to calculate the BHAR (Buy-and-Hold Abnormal Returns)?

I am doing my research related to IPOs long term performance. For the BHAR formula, I just want to clarify the formula is that always compare with the first trading day price, or is compared with last month trading price?

For example, (always compare with month 1) [(Month 2/Month 1) x (Month 3/Month 1) x (Month 4/Month 1)] - [(index Month 2/index Month 1) x (index Month 3/index Month 1) x (index Month 4/index Month 1)]

or compared with last month trading price [(Month 2/Month 1) x (Month 3/Month 2) x (Month 4/Month 3)] - [(index Month 2/index Month 1) x (index Month 3/index Month 2) x (index Month 4/index Month 3)]

simply, I calculated 1+Rit (a) and (b), which one is the correct one used in BHAR formula?

BHARi(t, T) = Πt = 1 to T (1 + Ri,t) - Π t = 1 to T (1 + RB,t)

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