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 March 2005 Appearances vs. Reality A Regression Analysis of Our Five Equity Models
On any given day, economic and financial news floods the airwaves and Internet. The media's desire to keep it coming 24 hours a day, 7 days a week provides incentive to play up any bit of information, no matter how fleeting or inconsequential. It's no wonder investors feel more paralyzed than enabled when it comes to applying this newfound wealth of information. It's also why they've become short-term focused over the past few years. Much of today's information is only valuable today. That's great for short-term traders but much less so for long-term investors. Those who are successful are the ones who can discern short-term market reactions from long-term trends. There's no place that this is more important than in evaluating one's portfolio. Unless you're a day-trader, your investment horizon is probably at least a year or so, and quite likely, much longer. Today's earnings reports or news releases will impact your portfolio just as much as the day-trader's, but the context is much different. Long-term investors know that a day or two does not establish a lasting trend. While it's true that all trends have to start sometime, it's equally true that every little market gyration doesn't necessarily launch one.
Short-term "noise" can actually distort one's long-term perspective. Your portfolio's behavior over the past few days may not be truly indicative of its actual and expected performance.
Our model portfolios are no different. As you may already know, we're currently monitoring five different equity models. (Actually there's a sixth model, too, but it's a balance of both stocks and bonds. With only one year behind, it, it's too new for what we're about to do here.) Three are large cap, one is small cap, and one is multi-cap. Three have a growth bent and two are value. Two employ a bottom-up stock selection process, while the other three are purely quantitative. Historical Performance shows their returns as well as those of their benchmark indexes. Graphs of their cumulative results since inception are also provided. Based on all this data, we obviously have a pretty good understanding of each model and how it should perform. Or do we? Here's were a closer look at the facts that might help separate our short-term perception from long-term reality. The PerceptionEach model is supposed to employ a specific investment style. Serious investors realize their expected returns are greatly determined by their investment style and consistency. For example, an investor with a large cap portfolio will see his or her fortunes rise and fall along with those of the large cap universe. If any small caps or even bonds enter the mix, portfolio returns will begin to stray from the benchmark. There's nothing wrong with this, but it does reduce the portfolio's predictability.We try to make sure that each of our models adheres to its assigned style. This is an effort to enhance predictability and comparability to the appropriate benchmark. Here is a brief description of each:
As you probably noticed, each of these models has a limited universe of potential holdings -- essentially the stocks of its benchmark. Don't confuse this with indexing, though, since no model holds more than 10% of its respective benchmark index. Instead, the goal is to make each model truly comparable to its benchmark while outperforming it on an incremental basis. In other words, we're not looking for home runs, just a lot of singles that will, over time, allow the models to outperform. If you simply look at returns or graphs, it appears the models have, for the most part, been relatively successful. There have been times when they've underperformed (especially P3 and P4) but in general all are tracking their benchmarks. But return isn't the only issue here. What's also important is how it was achieved. If a model is behaving consistently, then there should be a quantifiable relationship between it and the benchmark. That's what we want to examine. Market Match-UpA good place to start is to consider how each model's returns have compared to those of the broader market. This can be based on a number of factors, but the two most important here are style and capitalization.As you'll recall from the model descriptions above, each has a specific style (growth or value) and capitalization (large, small, multicap) objective. To see if they've lived up to them, we constructed a graph with style on the horizontal axis and capitalization on the vertical axis. Since the models are based on the S&P indexes, we used the S&P 500 Large Cap Value, S&P 500 Large Cap Growth, S&P 600 Small Cap Value, and S&P 600 Small Cap Growth to define the limits. It should be pretty obvious that capitalization is continuous, moving from the smallest companies at the bottom of the vertical scale to the largest at the top. Midcaps fall in the middle of the scale.
Although perhaps less obvious, style moves in a similar fashion along the horizontal axis. We've pointed out before that "value" and "growth" are not two mutually exclusive concepts. Instead, they anchor the ends of a style spectrum moving from deep value to relative value and core, then on to growth-at-a-reasonable-price and ultimately to aggressive growth. This is represented by along the horizontal axis on Chart 1. From a style standpoint, their results are about what you'd expect. P1 and P2 fall in the value half of the chart, although P2 is actually in the "core" area. P3, the most aggressive growth model falls far to the right, just where it should. P4 is a more moderate growth model and lies in the center of the growth section. P5 actually has no specific growth or value requirement and not surprisingly, falls in the "core" area. It's also not surprising that P5 would fall in the midcap range on the vertical scale. As we've previously pointed out, P5 has almost always maintained a sizeable midcap component, so it's no wonder it maps as a midcap portfolio. It is more noticeable that P2 (which is supposed to be small cap), P3 and to a lesser extent P4 (both large cap) also plot as midcap portfolios. Only P1, falls as it should in the large cap range. If you dig beneath the surface, however, it becomes a little more understandable. Although created as a small cap model, P2 was the victim of it's own success. By March 2004, the appreciation of its holdings turned it into a midcap portfolio. It was necessary to turn over 50% of its holdings in order to bring the average market cap back into the small cap range. This wasn't the first time this happened, so a midcap average isn't all that unreasonable. P3 has always been the most aggressive model. As such, it's usually relied on the smaller, more speculative stocks of the S&P 500. Throughout its history it's always had a major representation in the Tech sector with many holdings falling in the lower end of the capitalization scale following the demise of the tech bubble. P4 is more diversified but still has a growth tilt. As a result, it hasn't focused on the largest of the large, either. Relative to P3 you'd expect its average capitalization to be larger, and it is. You'd also expect it to be smaller than the large value stocks of P1, and again, it is. So all in all, Chart 1 is a fairly good representation of the style and capitalization composition of the five equity models. Arguably, the restrictions placed on the models' universe of potential holdings has achieved its goal of keeping them within the realm of their respective benchmarks.
Reading the LineNext we wanted to quantify their relationships with the benchmarks. To do this, we ran a regression analysis comparing the each model (the dependent variable) against its benchmark (the independent variable). Although this may bring back less than pleasant memories of 8th grade algebra, it's actually informative -- and relax, you don't have to do the calculations.Here's how it works: For each model, we made a graph with the index returns running across the horizontal axis and the model returns on the vertical. Each combination of rolling twelve-month monthly returns resulted in a point on the graph. The regression process then found the straight line that came closest to hitting all the data points. Chart 2 shows the results for P1. As the oldest model, it had the greatest number of data points. Each box on Chart 2 represents a rolling twelve-month period. The size of the boxes reveals their relative age, the oldest being the smallest and the most recent the largest. The diagonal line is the "best fit" or regression line. As you'll notice from Chart 2, many monthly data points plot on, or very close to it. That's good, since it implies there's a relatively meaningful relation between P1 and the S&P 500. We ran similar regressions for the other four equity models as well. The results of all are given on Chart 3. The first column shows the model (y-variable) and benchmark (x-variable). The second column shows their relationship in the resulting regression equation. In interpreting the regression equations, it again helps to recall some of that 8th grade algebra. As you probably remember, by substituting values for all of the terms on the right hand side of the equation, you can come up with a value for the term on the left, in this case the model return. The regression equation can therefore be interpreted as a historically based prediction of model returns. The terms on the right side of the equations actually give you more information about the models. As you'll notice from Chart 3, each has the same general form: y = α + (β * x) This is actually the equation of the best fit regression line. While model return (y) and benchmark return (x) are the variables, α (alpha) and β (beta) are constants. As you'll notice on Chart 3, they always appear as numbers rather than letters in the regression equations. Alpha is the value of the model when the benchmark index has a return of zero. In other words, if x = 0, the final term in the regression equation also equals 0 since anything multiplied by 0 is always 0. When that's the case, y = α. For example, consider the values for P1 on Chart 3. Alpha is.0028 so if the S&P 500 (the benchmark) has a return of 0% (x = 0), P1 should return .28%. Beta is a measure of how sensitive the model is to a change in the benchmark. Again consider P1 on Chart 3. Here β = 1.2363 so if the S&P 500 (the benchmark) goes up 1% (x = .01), P1 will gain approximately 1.24%. High betas increase returns in up markets but they also magnify losses when the benchmark declines.
How well does it work? Certainly not perfectly. Again using P1 as the example, in January 2005 the S&P 500 was down 2.53% (x = -.0253). Plugging this into P1's regression equation, you get: So P1 should have declined 2.86%. Actually, it fell only 2.20% so while in the ballpark, the estimate was off. That's why all the boxes in Chart 2 don't fall squarely on the regression line.y = .0028 + (1.2363 * -.0253) There's a statistical means of measuring the strength of the regression relation; it's called the coefficient of determination commonly represented by R2. It's the percentage change of the dependent variable (in this example, P1 return) explained by the change in the independent variable (S&P 500 return). According to Chart 3, it's .6396, or about 64%. So changes in the S&P 500 only account for about 2/3 of P1's return. That's why in any given month the actual monthly returns are likely to stray from the calculated values. There's also a way to test the statistical significance of the two constants, alpha and beta. Like the coefficient of determination, it can be applied to the regression equation for each of the models. It's called the t-test and the details of how it's calculated aren't necessary here. What is important is that in general, any value over 2.6 indicates the constant is significant. We ran the t-test for both alpha and beta in all three models. The results appear in the final two columns of Chart 3. Reading the Alpha and the BetaNow with all this background, let's see what it says about the five equity models.The t-test shows that none of the alpha values are significant. This comes as no surprise since all are extremely small and have very little impact on any of the regression equations. On the other hand, the beta constant is significant in all five equations. Recall that this constant quantifies the sensitivity of the model to movements in the benchmark index.
The only surprise here is the beta value for P1. The betas for the other models are essentially what you'd expect. Like the small cap indexes, P2's (0.8059) is less than that of the S&P 500 (1.00). P3 and 4 are both growth models and their betas would be expected to be higher than that of the index. In addition, P3 has tended toward more aggressive sectors than P4, so it stands to reason that its beta would be higher (1.7626 vs. 1.1293, respectively). P5 is a broad market model so over the long-term, you'd expect it's beta to be similar to that of the index. At 0.9860 it is. However, as a large cap value portfolio, P1's beta would normally be expected to be less than that of the index. Instead, it's 1.2363, higher than growth model P4. Equally puzzling is the fact that it has the highest t-statistic (16.9861). This is surprising from a quantitative perspective as well. Every two months we calculate portfolio statistics for each model and report them in the Historical Performance summary. These figures are based on the stocks in each model as of the reporting date. Chart 4 compares the December 31, 2004 portfolio betas with those from the regressions. They are in close agreement in all instances except P1.
The data underlying each calculation may explain this discrepancy. The betas on the Historical Performance page are based on the stocks in the actual portfolio at that specific time. The regression betas are derived from the historical performance of the model. In P1's case, the portfolio beta is based on those of the stocks in the model on December 31, 2004 while the regression beta draws on information going all the way back to July 1, 1991. P1 hasn't always behaved like a large cap value portfolio. Chart 5 plots rolling twelve-month period returns with style on the horizontal axis and market caps on the vertical. The larger the boxes, the more recent the returns. The red box shows the average as it appears on Chart 1. Back in its earliest days (the smallest boxes in Chart 5), P1 looked like a small cap growth portfolio. Later it moved to the middle of the chart as mid cap core, then to the upper lefthand quadrant as large cap value, and currently it's in the upper right as large cap growth. The changes in attribution are, to a great extent, due to the buy-and-hold nature of the portfolio. That's why it's most recently been moving from value to growth as its value holdings hit their stride. But by the same token, it stands to reason that P1's beta has changed along with its attribution. When the it trades as a growth portfolio, beta increases. When it's small cap or value, beta decreases. The present holdings -- as measured by portfolio beta -- render an average that's close to the index. The historical beta that encompasses 162 months -- when P1 often looked like a growth portfolio -- is naturally higher. So yes, P1's regression and portfolio betas differ, but they aren't necessarily incompatible. They simply measure two different things. Interpreting the R2sSo how meaningful are the regression equations themselves? That's where R2, the coefficient of determination, comes into play.As explained above, it represents the percentage change in the model explained by the change in the index. The greater the value of R2, the stronger the relationship with the index. Looking at Chart 3, four of the five R2s are moderate to strong. Actually, it's the two extreme values that are most interesting.
P2 has the lowest value. It's R2 of .4109 indicates that only 41% of its returns are explained by changes in the S&P 600. We've always used the 600 as P2's benchmark and have tried to keep the portfolios' analytics close to those of the index. In an effort to see if these results were simply a quirk of the S&P 600, we also ran a regression against the Russell 2000, another widely used small cap index. Not too surprisingly, the R2 for the resulting equation was even lower, 0.3668. The obvious explanation for the low R2 is that P2 doesn't act like the S&P 600. This is indeed borne out by the historical chart of the model's return. As you'll notice from Chart 6, P2 has traded all over the board. At some point in its history, it's behaved like each style and capitalization. This is a problem eventually encountered by all small cap value portfolios. Unless they're traded rapidly or comprised totally of stocks that remain dead in the water, their stocks will begin to move towards fair value and in the process, move towards growth while their market cap grows. The patterns on Chart 6 clearly demonstrate this. The smallest triangles (oldest attributions) start in the lower left quadrant (small cap value) but eventually move to the right (growth) and up (mid to large cap). After considerable turnover in 2004, the most recent attributions (largest triangles) are now back in the small cap value area. Overall, however, the average (red triangle) is almost right in the center of the graph. So it's no wonder P2 doesn't have a strong relation with the Smallcap 600. Once again, the regression process has correctly quantified the model's behavior.
Better than GoodP5 lies at the other extreme. As a broad market model, drawing from all equity styles and capitalizations, its regression equation was based on the S&P Super Composite 1500. Its R2 is a very strong 0.8747. But could it be stronger? This isn't just a bizarre question of greed, it actually stems from an oddity in P5. Back in November 2003 we noted that P5 has always relied heavily on mid caps. In fact, including its backtest, P5 has had a midcap component since October 1997. That's why ends up looking like a mid cap core portfolio on Chart 1. Given all this, doesn't it seem logical that a regression with mid cap indexes might yield a stronger relation? It's at least worth exploring. Indeed, when we ran a regression with the the S&P Mid Cap 400, the R2 jumped to 0.9576. The intercept (alpha) values although small (0.0044) was also statistically significant with a t-value of -2.7994. The results are graphically depicted on Chart 8. As you'd expect, all data points are closely clustered around the regression line.
So far, so good, but is there an even potentially stronger relationship? For example, Chart 7 suggests that P5's historical returns have tended towards growth. What if a regression was run with the S&P 400 Growth Index? To check this out, we ran regressions against both the S&P 400 Growth and Value Indexes, but R2s (0.9467 and 0.8953, respectively) didn't improve. These were good results, just not as good as those from the overall index. Details appear on Chart 9 Apparently the historical returns bounced around too much to be closely associated with just one style index. Chart 7 would seem to support this. P5 is, after all, a broad market model. Since that's the case, perhaps the best relation would be with all capitalization and style benchmarks. This can be examined by using nine independent variables, the style and capitalization indexes populating the Morningstar stylebox. These are the potential universe of holdings for P5. (For details, please see Think Inside the Box.) The resulting equation appears on the last row of Chart 9.
The R2 is remarkably high, 0.9778. Unfortunately, there's a problem with this as well as with the S&P 400 regression: Data snooping. Good Statistics vs. Bad StatisticsData snooping is the discovery of seemingly significant but ultimately spurious relationships within data sets. It's bad statistics. Good statistical procedure starts with a hypothesis and then examines the data to see if it's supported. Bad statistical procedure -- data snooping -- looks at patterns in the data to come up with a hypothesis. It's essentially circular reasoning.Unfortunately, the results on Chart 9 are nothing more than data snooping. They're the product of examining the data after the fact and coming up with more precise benchmarks. P5 wasn't designed to look like mid caps or all nine components of the Morningstar stylebox, it's supposed to track (and exceed) the broad market. The fact that it closely tracked various parts of the market is spurious and likely to change over time. On the other hand, the original regressions shown on Chart 3 actually test the models' actual results against their benchmarks. They're the result of valid statistical procedures. For the most part, they support the fact that the models are behaving as anticipated. Unlike the spurious results on Chart 9, they provide some insight into long-term historical performance without being clouded by short-term noise. Search this site! Just enter you key word or words:
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NE OF THE MOST DIFFICULT tasks facing any investor is the need to make long-term decisions based on short-term information. Oddly enough, the increased availability of financial data has actually made this process even more daunting.
