# Beta to the Max

September 27, 2010 1 Comment

Last week, I was having a really hard time explaining myself until I realized that I was trying to explain *in words* how to take advantage of the fact that humans perceive images faster and better than we perceive words or numbers. This has a lot to do with our evolutionary history and with behavioral finance. My example was the interpretation of a fund’s beta – and my point was that it’s really really hard to get the right interpretation from the numbers without the right picture. In this case, the right picture makes a BIG difference. It was really frustrating trying to give an example of what I meant until I realized that I was suffering from the same thing I was trying to explain! Wow, I guess I really can be that thick. I should have been *showing* her the pictures rather than talking about them. So that’s exactly what I then did.

If the R-squared is high, like 0.8 or 0.9, it's a good fit. If the R-squared is low, like 0.3 or 0.4, it's supposed to be interpreted as a poor fit. What people fail to realize is that a low R-squared means that the associated beta is meaningless – and the relationship of "index up by 1% means fund up by beta%" is WRONG. While these words may make sense and each of us can understand them and remember them when we need to, the pictures that go along with beta and R-squared are a much better way to convey and quickly interpret the information. Why don't more risk systems show the relationship that goes along with beta and R-squared? I have an opinion that I'll suspensefully hold off until the end of this post, as if you can't scroll down the screen if you really want to read that part now.”]

Enough words – let’s look at some pictures. Figure 1 shows a scatter plot of the returns of a fund against the returns of an index. Each blue point represents a particular month’s returns for the fund and the index, showing how each of them performed in relation to each other. As is visible in this plot, when the index is up the fund is up, and vice-versa. Clearly, there is a strong relationship between them. That relationship is captured neatly in the equation: . In this case, alpha is 0.46 and beta is 0.68. Mathematically, beta is the slope of the green line in the plot, and its interpretation is “this is how strongly the fund responds to changes in the index.” Note that the equation represents a straight line, and only a straight line, and calculating beta assumes that that’s just the way it is. In this case, the fund and the index have a strong relationship which is visibly obvious. The mathematical version of that is the R-squared, which is a high 0.91, telling us that a linear fit is a good approximation to the actual data. In English, this means that you can reliably estimate how the fund will perform by multiplying beta times the index’s return.

Let’s take a look at some other funds. In Figure 2 you’ll see three funds, each with similar betas (of about 0.7 or 0.8), but with decreasing R-squares. Something funny can happen with these calculations – a fund can still have a high beta, like 0.7, even if the relationship isn’t described by a straight line all the well. The question is – if this happens, how do I interpret that beta number? By the time the R-square is 0.64, you’ve got to be questioning whether that data is really described by a straight line or not. And if you decide it’s not, then the interpretation of the beta is “do not use.”

NB: when looking at plots like this, it’s really really important to have the same limits on the x-axis and the y-axis. For this post, I’m using -4.0 to +4.0. Most graphing packages choose the limits for you and won’t pick the same numbers up-down as they do left-right. But for beta, which is the slope the the line, it’s really important to do it this way. This is one of those dirty little details that can make a big difference.

Now take a look at Figure 3 – this shows three more funds, again with betas of about 0.7. But those betas aren’t worth anything – they don’t describe any sort of relationship between the fund and the index. The notion that you can sort-of predict what the fund will do as “beta times the index” is nonsense. In the rightmost pane of Figure 3, when the index has been up about 1%, the fund has indeed been up – about 3% (but NOT the 0.7% predicted by beta times index return). It gets worse – the fund has also been down as much as 4% when the index was up about 1%. There’s just no relationship between them. Yet, the math still gives a beta of 0.68. The math will always give a beta – even if the beta is meaningless. **My whole point in this post is that seeing these images is much more useful than getting the beta and R-squared as numbers for the simple fact that the plots force you to look and see if there is a relationship**. And it’s obvious when that straight line isn’t describing anything based in reality. Just look at that rightmost panel in Figure 3. **Do the numbers** “beta=0.68, R-squared=0.28” actually

**tell you there’s a problem? Nope**[now before every quant starts complaining “but that’s what R-squared is

*FOR*,” my point is not that quants are hiding something – it’s that the

*picture*is much more easily interpreted than the

*number*]. Now,

**does the**That’s why I always want the plot.

*picture*tell you there’s a problem? Yup.So why don’t most risk systems show this? Well, it doesn’t exactly make it easier to sell your software if the customer thinks your models don’t work. And revealing the R-squared *graphically* would actually *show* the customer how bad the model or choice of index is. For many equity funds, beta is *the* *singular* risk measure that (they think) matters, so it would be a Bad Thing to reveal flaws in the approach. To this, I say nonsense. It’s much better to educate the potential customer about which models are appropriate and why. Doing otherwise is too close to snake oil for my taste.

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