# How to convert Jensen's Alpha from monthly to quarterly observations

I am being puzzled while calculating jensen's alpha for single stocks. I have monthly returns data and have calculated alpha for each stock on a monthly basis (used 36-month rolling window for beta estimation). Now, I need to convert my monthly alphas into quarterly and yearly observations. What is the best way to do it?

I thought of averaging over 3 and 12 months, respectively, but not sure if it's correct.

Would be glad if anyone could help!

• Can you please define Jensen's alpha, in a mathematical formula? – Gordon Jun 17 '16 at 20:14
• @Gordon yes, sure, it's an excess return over a benchmark: Ri-Rf-beta*(Rm-Rf), where Ri, Rf, Rm are stock return, risk-free rate and market return, respectively. – Daria Diachenko Jun 22 '16 at 13:12
• Thanks for clarification. Is the answer below helpful for you? – Gordon Jun 22 '16 at 13:14
• @Gordon yes, it is, I used the same approach, however it causes further questions – Daria Diachenko Jun 22 '16 at 13:20

## 1 Answer

You can do a couple of things:

The easiest way to calculate your quarterly Jensen's alphas is achieved by calculating quarterly returns and then applying the regression method you have done before.

Alternatively, your Jensen's alpha represents the abnormal monthly return over a benchmark. Therefore, your quarterly Jensen's alpha can be calculated by annualizing your returns : $$r_q = (1+r_{m1})\cdot(1+r_{m2})\cdot(1+r_{m3})-1$$.

• The problem with the first approach is that when estimating alpha I need to use rolling windows to estimate betas. Thus, when using same length (3-years) rolling windows, I will estimate beta for each quarter relying on 12 observations versus 36 when using monthly. To get a consistent number of observations for beta estimation I would need to go back into historical returns for at least 6 years, which is more likely to produce an erroneous beta. That's why I thought monthly alpha estimators make much more sense. – Daria Diachenko Jun 22 '16 at 13:16
• As of now, indeed, I went for the second option and calculated the cumulative quarterly returns. Does it mean though that the stocks are assumed to be reinvested each month? – Daria Diachenko Jun 22 '16 at 13:18
• Also, are you sure that those quarterly returns are annualised? I thought this formula only represent the cumulative quarterly returns.. that can be put into the regression right away. I also needed to calculate alpha on yearly basis and I used the same cumulative formula but across 12 month... – Daria Diachenko Jun 22 '16 at 13:22