Let's say we want to compute beta to S&P500 of a portfolio, using 3 years of weekly returns, as of today. We would take each stock in the portfolio and regress the weekly returns of that stock against the weekly returns of the S&P500 over the 3 years, right?
But let's say the current portfolio consists of exactly the same stocks and weights as the current holdings of S&P500. Beta should be exactly 1, right? Yet when we regress the returns of the current portfolio holdings against S&P500 historical returns, we might get a different value, because the composition of S&P500 changed over time.
So would it not be more correct to regress the returns of the current portfolio holdings against the historical returns of the current basket of S&P500, instead of regressing against the actual historical returns of S&P500?