# Efficient algorithm for calculating Beta coefficient

I'm using Python/Pandas. Using naive nested for-loops to do Beta calculation for all ~5k stocks by ~5k days (moving window ~250 days) is unbearably slow. Is there any fast and elegant way to accomplish this goal?

Edit: Simply using Numpy instead of Pandas for all the intermediate steps, would speed up the whole process by >10X.

• You mention 'naive' nested for-loops, what do you suspect would be better? What have you tried? Commented May 23, 2018 at 8:15
• What method are you using to calculate Beta? Commented May 23, 2018 at 14:14
• Consider - in addition to other proposed solution - trying pandas.iteritems() and pandas.iterrows(). These by-column and by-row iterators are pretty awesome. Commented May 27, 2018 at 20:24

I don’t know how naïve your nested loops are, but I assume you are using the OLS calculation $\beta = (X’X)^{-1}X’Y$, where $X$ contains the index returns and $Y$ contains the security returns.

If you have data for all time periods for all securities, then $(X’X)^{-1}$ will not change for each security. The best solution would be to use numpy to calculate the matrix multiplication directly for all securities. Alternatively, you can calculate $(X’X)^{-1}$ before entering the loop, and then calculate $\beta$ for every individual security.

If you don’t have data for all time periods, then there are speedups, but it gets more complicated.

• Thanks, this method does reduce the amount of calculations. Can it handle empty values well? Commented May 29, 2018 at 2:57
• This doesn't really handle empty values, that's why I mentioned missing values in my response. If you have empty values, a good first step for speedups would be to separate your securities into those with complete data and those with incomplete data. If you're lucky, the number of securities with complete data will completely outnumber those with incomplete data. Commented May 30, 2018 at 16:34

You might find this code snippet helpful. It's the vectorized beta calculation used by Zipline, an open source backtester written in python.

It is computed over a lookback window, with data for all assets over that time period. As Tim mentioned above, this can be efficiently computed using numpy and matrix multiplication.