Beta is a risk statistic which tells us essentially how our portfolio behaves relative to the market. It is a derivative of Modern Portfolio Theory, and more specifically, the Capital Asset Pricing Model (CAPM). The purpose of this post isn't to address whether or not beta is dead; Fama & French can tackle that one. It's really a lot simpler than that.
The market gets 1.0; our portfolio's value is measured using the following formula:
One question that has held firm for CAPM is "what's the market?" Well, that's something that is taken up in the hallowed halls of academia; for our purposes, we have the same question, but not perhaps from such a purist perspective. We want to know what gets the "1.0."
Standard practice seems to be to assign the 1.0 to the portfolio's benchmark. In my opinion, to do anything but this can be of little value. For example, let's say that you use the S&P 500 to represent the broad market, and everything, including your portfolio's benchmark, relative to it. And let's say that as a U.S. small cap manager, your benchmark is the Russell 2000. And so, you first calculate the Russell 2000's beta, which you find to be (for exhibition purposes only; not in reality) 1.53. You next calculate your portfolio's beta relative to the S&P 500, and it turns out to be 1.61. This tells us that both the portfolio's index and the portfolio itself are more volatile than the S&P 500, meaning that they both exhibit more risk. So what? We're interested in the risk you've taken relative to the index your managing against, aren't we? And wouldn't it be more appropriate to set the Russell 2000 to 1.0, and measure the portfolio's beta in comparison to it? We should let the academics try to define "the market." But when we're measuring a portfolio's risk, its market is its benchmark, and risk should be compared against it. This holds for tracking error, information ratio, Jensen's alpha, Modigliani-Modigliani, and any other risk statistic that uses a benchmark in its formula.
p.s., There may be some value to measure a benchmark's beta relative to the S&P 500 (or some other "market"), to exhibit how each behaves relative to it, so I don't want to "pooh-pooh" the idea completely. But from a fundamental, foundational perspective, use the portfolio's benchmark when measuring risk. You're not managing against "the market," other than as represented by the associated benchmark.