Although all of us are seeking alpha, many also pay attention to beta when deciding whether a particular stock might be a worthy addition to our portfolio. But although values for beta are readily available from a number of sources, I don't believe it's widely appreciated that the significance of these numbers varies wildly from one stock or fund to another even if the values come from the same source and are calculated by the same method using data over the same period of time.
My focus here is on the reliability of beta as a measure of volatility of a particular stock relative to the broad market, which is often interpreted as one measure of risk. For example, it is often implied that a stock with a low beta has exhibited smaller changes in value (up and down) than the market as a whole (and the implied assumption is that it will continue to behave this way in the future). The purpose of this essay is to demonstrate that beta alone is insufficient information, and that beta often reflects no such behavior.
Let's look at the calculation of beta, and I'll emphasize the graphical representation because it really helps in the understanding of the significance of this metric but is usually not available unless you do the analysis yourself. To begin we need to choose the period of time over which the calculation will be performed and also the intervals within that period. For the examples here I have chosen to use monthly intervals over a ten-year period.
So the raw data needed are the monthly changes, expressed as a percentage, in both the price of the stock being studied and the value of an appropriate benchmark. Once the raw data is in a spreadsheet, beta can be calculated by several different formulas but it's most instructive to first create a graph with the changes in the benchmark on the x-axis and the corresponding changes in the stock or fund being compared on the y-axis. If the best straight line (a linear regression line) is drawn through all the data points, the slope of that line is equal to beta.
Look at such a graph for a simple example. SPY is a fund designed to track the S&P 500 index, so a graph of changes in SPY vs. changes in the S&P 500 is expected to give points falling along a nice straight line with a slope very close to 1.00. The graph below shows this to be true. The slope of the regression line, beta, is 0.99, but more importantly the fact that all the data points are close to the regression line means that changes in the S&P 500 in any month can be used to predict the change in SPY in that month with a high degree of confidence.
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Compare that with the behavior of an individual stock, United Technologies (UTX), shown below.
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Here the calculated beta is a little higher, 1.06, but now there is much more scatter in the data points, each of which represents changes in an individual month. The result is that even though beta is close to 1.0, this doesn't mean that month-to-month changes in UTX followed the S&P 500. In July 2007 for example, the S&P was down 3.2% while UTX gained 2.9%. And in December 2011 the S&P gained 0.9% while UTX lost 4.6%.
Clearly beta alone simply doesn't tell you anything useful in this situation. And in general, this kind of data for individual stocks will always show significant scatter around a regression line. In the language of CAPM, this scatter represents "unsystematic risk" or "specific risk." In the case of SPY there is no unsystematic risk and the up and down movement is purely due to systematic, or market risk.
A measure of scatter is given by another statistic called the coefficient of determination, better known as r-squared, and is also easily calculated with a spreadsheet formula. R-squared has a very useful interpretation. It is the fraction of the variation seen in the stock being studied that can be predicted, or attributed to, the variation in the benchmark index, i.e., due to market risk. For the graph with SPY, r-squared is 0.99. In other words, 99% of the month to month change in SPY was explained by the change in the S&P index.
But r-squared for the graph of UTX is only 0.64, reflecting the greater scatter about the regression line and the fact that more than one-third of the month to month change in UTX cannot be explained or correlated with the change in the S&P index.
Some people may also be tempted to interpret beta as a measure of the change in value of a stock, relative to a benchmark over the period being studied. After all, if beta is 1.0 doesn't that mean the stock should have performed just as well as the benchmark? Again, the answer is yes only if r-squared is close to 1.0, as was seen when looking at SPY. When r-squared is smaller, beta tells you nothing about the performance of the stock. Over the ten-year period from July 2003 through July 2013, the S&P 500 gained 73% while UTX gained 196%, again with a calculated beta of 1.06.
Consider one more example - a stock with a low beta. Colgate-Palmolive (CL) is in the defensive household and personal products industry and its low beta is usually interpreted as reflecting much lower swings, up and down, than seen in the benchmark S&P 500. Beta for CL calculated over the same 10-year period is only 0.42, but look at the graph of actual month to month variation.
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The calculated value of r-squared is only 0.16, reflecting a scatter so great that the value of beta is meaningless at best and quite misleading at worst. It is also worth pointing out again that (because of the low r-squared) low beta does not mean underperforming in a rising market. In fact CL outperformed the S&P 500 (107% to 73%) in these ten years.
Actually it shouldn't be surprising that r-squared is usually relatively low for individual stocks and that there is corresponding scatter in this kind of graph. Again, r-squared is the fraction of change in the price of a stock that is explained by movement in the market as a whole. But of course there are many reasons for changes in a particular stock that have nothing to do with the market. Examples are earnings reports, dividend changes, analyst recommendations, news specific to the company, etc. All these factors can change a stock price independent of what happens in the market and are responsible for the scatter in the data points shown in the graphs above.
This also implies that the situation for mutual funds is different, because they are made up of many individual stocks, and indeed r-squared for mutual funds tends to be higher than for most stocks and the value of beta is thus correspondingly more useful. (Of course this depends on the type of fund and the benchmark it is compared to). It is interesting that Morningstar reports both beta and r-squared for mutual funds, and for 3, 5, 10 and 15-year periods, but only a single value of beta and no value for r-squared for individual stocks. As stated before, I believe such values for beta are useless and potentially misleading.
Finally, what about beta as a measure of volatility or "bounciness" of a stock relative to an index? Sorry, as long as r-squared is not close to 1.0 it's no good for that either. The most direct measure of bounciness is the standard deviation of daily (or monthly) changes. So bounciness compared to an index is logically given by the ratio of their standard deviations. Interestingly, it can be shown that this ratio is mathematically equal to beta divided by r (not r-squared). For example, using values shown above, this ratio for the low beta stock CL is 0.42 divided by the square root of 0.16, which equals 1.05. In other words, CL actually showed slightly more volatility of monthly changes over this ten-year period than the benchmark S&P 500. Of course this ratio doesn't tell you whether the stock moves in tandem with the benchmark - this will be true only if r is close to 1.0.
For those who wish to use a spreadsheet to construct their own graphs and calculate beta and r-squared here is a quick summary of the procedure.
- Obtain historical closing prices for a stock and an appropriate index used for comparison. Choose a time period, often five or ten years, and use daily or monthly intervals.
- Add columns where the percent change between intervals is calculated.
- Select the two percent change columns and insert a scatter chart. The index should be on the x-axis. Chart tools can be used to add a linear regression line if you like.
- There are several different ways to obtain values for beta and r-squared. In Excel, a simple way is to apply the SLOPE function for beta and the RSQ function for r-squared to the two percent change columns.
In conclusion, here are my take-home messages:
- Don't trust any value for beta unless you have a corresponding value for r-squared.
- Understand that beta tells you less and less the farther r-squared is from 1.0.
- Understand that beta does not reflect past performance unless r-squared is close to 1.0. Low beta stocks can outperform the market.
- Don't use beta as a measure of relative volatility. Use beta/r instead.