Calculating beta to market

Question

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?

Answer 1

In a word, yes. That's a correct and valid view to take but, as you'll always find in finance, it really depends on context and the question that you're trying to answer. This is the case in markets but more broadly in business and something that academically minded scientists/engineers struggle often understand and appreciate fully. This boils down to the fact that the words we use in business (and generally outside of the scientific context) are often, if not the majority of the time, ambiguous.

Reading into the context from your question I might guess that you're coming from a portfolio/risk management perspective and what you're really interested in measuring is an estimate of your forward looking beta. Hence any methodological change that improves upon linear regression is welcome. Bear in mind that you can go further than simply updating your market (SPX) returns from realised index to current basket back-test. e.g. GARCH based models for forecasting the volatility (and hence beta) of your portfolio stocks.

On the other hand, the objective may be an ex-post analysis of a portfolio relative to the market. In that case you most likely want to look at realised index rather than the current basket, or perhaps back-test previous market baskets for each time point. Then there could be scenario analyses through stress periods when we might elect to use a combination the above mentioned methods.