# Not Even Wrong

December 11, 2010 6 Comments

**Comments are Welcome – please add a comment.**

I had a bit of a disagreement the other day with a colleague about the interpretation of beta and correlation. He claimed that if a fund has a high beta to a particular index, that meant that it also has a high correlation to the index. He even cited a Wikipedia article to support his point of view. Unfortunately, it’s rather common to believe the whole story is, as the article states, that “correlation measures direction, not magnitude” and “beta takes into account both direction and magnitude.” If that’s most of what you know about beta and correlation, then it’s easy to make the conclusion that my friend made.

**Why do so many people make this mistake?** Beta and Correlation are commonly used terms in traditional investing – mainly by mutual funds – and they’ve been thrown around long enough that people are familiar with the terms, so they think they understand them. “Oh sure, I’ve heard of that before” and therefore, the person thinks to him/herself, I don’t need to know anything more about the topic. In what’s called the Long Only world (think mutual funds), beta is regarded as a risk measure because of a simple relationship between the fund and its benchmark: beta is correlation times the fund’s volatility divided by the benchmark’s volatility. Benchmarks are chosen to closely resemble the fund – for a technology fund, you pick a tech index as the benchmark. For a health-care fund, you pick a health-care index. Obviously. A mutual fund’s benchmark is chosen *because* the two are highly correlated. Mathematically, that means the correlation is close to 1.00. So, in this case (and in this case *only*) beta is basically the ratio of the volatility of the fund to the volatility of the benchmark. If beta is 1.1, that means the fund is 10% more volatile than the benchmark, and if the beta is 0.9, that means it’s 10% less volatile than the benchmark. It all works because of the formula beta = correlation * (vol_fund / vol_index). The problem is that people forget important aspects of the formula – like that correlation is baked into the beta – and they only remember “beta is a relative measure of volatility” or “beta is the slope of the regression line.” Please keep in mind everything I’ve written in this past paragraph is only true for traditional long-only funds, like mutual funds.

**What’s really going on?** First of all, the conference we were speaking at was an Alternatives Industry conference – in other words, hedge funds. Hedge funds usually boast a very low correlation with most market indices. Why is that relevant here? Well, there’s one more important things to realize: correlation is also a measure of how well the data actually fits a straight line. The whole concept of “beta” rests on the assumption that the fund and the index, when plotted on a simple scatter plot, form a straight line. Statistician (look out! – I’m about to get technical) use the measure called R-squared to describe how close to a straight line their relationship actually is. Here’s the important thing to remember: R-square is, in fact, the square of the correlation (for a one-variable simple linear regressions). Why is that important? Well, while it’s true that you can have a high beta even if you have a low correlation (because the fund’s volatility is high enough to make correlation * fund volatility / index volatility a big number), **a low correlation has a much more important effect: it renders the entire analysis inappropriate.**

*What*? That’s right. The entire use of beta, a *linear* measure, is inappropriate if you don’t have a good linear fit. See my previous post for some pretty pictures showing low R-square values, and just keep in mind that correlation is the square root of R-square. In other words, if correlation is low – like 0.2 (a typical hedge fund value) – then R-square is 0.04, and a regression with an R-square of 0.04 is not worth talking about. A straight line doesn’t describe the data and it’s just *wrong* to talk about the fund’s beta. The problem is that the math still gives a value for beta – it could be 1.3 or 0.7 or any other number. But regardless of what the value is, it’s actually wrong to use beta in the first place! Why? Because the fund’s behavior is inconsistent with the beta model, so beta is irrelevant and inappropriate.

But people don’t get this. **It’s like using a particular medical test even in a circumstance where the test is invalid, but still talking about the results. It’s voodoo. It’s nonsense. It’s dangerous. **And it happens all the time in this industry.

So the statement that a high beta means you have a high correlation is not only false (because you can have a high beta and low correlation), it’s also very misleading. In fact, this falls into the category of “not even wrong” where the statement’s assumptions are so wrong that the statement itself is *beyond* wrong, it’s “**not even wrong.**“

万年筆

Right after reading this post, I came across an interesting analysis by FactSet that aims at finding “the most appropriate look back period” in the Beta analysis and this should “be that look back period that maximizes the R-sqr”.

It is worth reading it:

http://www.factset.com/blogs/takingrisk/2010/03/introducing-betamax-a-new-measure-of-covariance-stationarity-part-2

In light of this article, it seems it is necessary to both find the right index that maximizes the R-sqr and to revise the time period of the calculation. However this introduces a lot of moving parts that make historical comparison difficult. Thoughts?

Toward the end of his post, he states “…this implies that the failure of the market to be covariance stationary is partly due to the time varying nature of volatility, and partly due to a time varying nature of the correlation structure.” I think he went through a lot of math / analysis to demonstrate what we already know simply for the non-stationary nature of correlation and volatility. I think he makes some very good points, and the blog is worth reading. In fact, I’ve added it to my list of things to keep up-to-date on, so thanks for the intro.

Another aspect of it is the market relevance of the time frames. Although mathematically 60 months may be what maximizes the stability of beta, 60 months is 5 years. The stock market can go through 2 or 3 difference economic cycles in that timeframe, making the analysis a bit weak. A shorter timeframe is almost certainly warranted from simple economic concerns, even if the math suggests otherwise. It’s the balance of the needs of the math with the practicalities of the market that’s very interesting to me.

your article is exactly correct. however in general when it comes to risk management im always leary of correlation because it is extremely unstable thats why ive never been a believer in var as its too easy to manipulate the correlation matrix to greatly reduce var estimates and hence have the excuse to increase leverage. this folly has led to the downfall of many trading desks

Way to go Damian, what people don’t know(but think they do) is astounding. However, it makes opportunities for the rest of us.

Great stuff. Appeals to authority are tough, especially in a public forum. I’d like to think he wasn’t holding the reigns to anything important, but since he was a speaker…. You can’t save everyone.