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 July 2008 Lessons from the Regressions A Return Analysis of P3 and P4 vs. the S&P 500
The models were the result of two different regression analyses (for details, see, The Starting Point, and for a primer on regression analysis, click here). P3's focuses on the stocks in the S&P 500 with the top 30 scores without regard to sector or weights. P4's is more structured, and is actually based on ten separate regressions, one for each of the ten S&P 500 market sectors. Its portfolio draws from the top scoring stocks in each of the sectors, which are then weighted like the benchmark index.
Because P3 and P4 were created using regressions, it only makes sense to also evaluate them using regressions. It's nice to report that the results -- particularly in regard to P4 -- are quite promising. Both, however, can stand some improvement.
Sources of Return If all this analysis is correct, stocks periodically selected and rebalanced from the S&P 500 using these formulas should have the potential to outperform the benchmark index over time. But it's important to realize, this is not simply an effort to pick top-performing stocks, but rather outperforming portfolios of stocks. In evaluating the model, we don't just look at the individual component stocks but rather at the portfolio they comprise. As a result, there's more to it than simple individual equity risk and return, there's also correlation.
Correlation measures the extent to which two different data series move in relation to one another. Series are positively correlated if they move up and down at the same time, and are negatively correlated if one tends to rise when the other falls. The degree to which they follow these patterns determines how highly correlated they are. According to Capital Asset Pricing Model (CAPM), by combining risky although non-highly correlated assets, one can construct a portfolio that has less risk than the individual assets but with a potential for higher return. Less risk and more return is the Holy Grail of investing. Because P3 and P4 are evaluated as portfolios rather than individual holdings, the results can be affected by more than just the risk and return of the components. Since P3 doesn't weight the individual stocks or sectors, it will behave a lot more like the sum of the components. By weighting and diversifying across all sectors like the S&P 500, P4 eliminates the opportunity to benefit by overweighting top-performing sectors at the expense of the poorer ones. On the other hand, without a weighting system, P3 can either benefit from concentrating in a handful of sectors, or suffer if the focus is misplaced. According to the CAPM, return can be broken down into two parts: Return associated with market risk and excess return generated by the portfolio's structure, market timing, or specific components. The first is a function of the portfolio's level of market risk or sensitivity to the benchmark index movement. It's referred to as "beta". The second element is considered excess return -- the portion of the portfolio's return that exceeds (or trails) the value expected by considering only the benchmark return and beta. It's generally called "alpha". (For more details on this, click here.)
The portfolio's beta can be altered by adding (or deleting) various holdings with higher or lower levels of market risk. For example, a simple way to lower the overall beta of the portfolio would be to replace all the high beta holdings with lower beta ones. Correlation will have an effect on the overall results, but in general, this should work. You could do the opposite to raise the overall beta. Higher betas make the portfolio more sensitive to movements in the index so a manager could use a high beta in rising markets and a lower one during declines. If successful, this would allow his or her portfolio to outperform the unmanaged benchmark index. Alpha is a harder to come by. It's generally derived from the fundamentals of individual holdings rather than a simple manipulation of market risk. For example, some managers are able to spot stocks of companies that are about to take off because of a new product launch or takeover bid. Others may be able to sniff out turnaround candidates right before the turning point. Some may be able to find the next market leaders before the rest of the Street. In all of these instances, the chosen shares can rise independently of what happens in the market and their particular level of market risk. Managers who consistently build portfolios of assets like this are the ones that are truly adding alpha. Conceivably, P3 and P4 can beat the index by using both alpha and beta. With virtually no restrictions on its holdings, P3 could, at any time, be composed solely of stocks poised to take off or with extremely high (or low) market risk. P4 has to weight its holdings similar to those of the index and while this may impede the search for alpha and manipulation of beta, the model still has plenty of opportunity to beat the index through its individual selections and buy/sell timing.
The New Regressions The regressions are based on the monthly returns of the models and the S&P 500. Charts 1 and 2 plot the returns of the index on the horizontal axis and those of the model on the vertical. Each pair of returns creates a point (blue bubble) on the chart. Although you can't count them on the charts, there are ninety-five separate points, one for each month between July 2000 and May 2008. If the models and the S&P 500 were perfectly correlated, all the dots would fall along a straight line, but that only happens in textbooks. In reality, the pairs are more dispersed. The lower the correlation, the greater the spread. The regression analysis finds the line that comes the closest to connecting all the dots. It's based on a technique called the "least squares" method which essentially searches for the single line that minimizes the distance between the line and the dots. This is referred to as the "best-fit" or "regression line". For any two series of numbers, you can always find a best-fit line. The problem is, it may be so far away from so many pairs of data points that the statistical information it conveys is essentially meaningless. Statisticians look at the "coefficient of determination" (generally called the "r-squared") to determine if the results of the regression are statistically significant. If the average distance between the data points and the regression line is small, the r-squared value is high, and vice-versa. R-squared runs from 0 (an extremely poor relation) to 100 (perfect correlation). The higher the r-squared, the stronger the relation. Those over 65-70 are generally considered statistically significant. Drawing from the regression equation, the slope of the best-fit line is the beta term while the y-intercept is the alpha value. You remember these terms from seventh grade algebra, don't you? The slope is simply the change in y-value (vertical distance) divided by the change in the x-value (horizontal distance) between any two points on the line. The y-intercept is the point where the regression line crosses the vertical axis. We ran three regressions for each model. The first, appearing on Charts 1 and 2, covered the entire period from July 2000 through May 2008. The second covered the period from inception through June 2003 when we made the changes to the models. The final covered the remainder of the measurement period. The last two regressions were run to gauge the effect of the 2003 changes.
Few Surprises Starting with the overall analyses illustrated in Charts 1 and 2, it's pretty obvious that both regressions were statistically meaningful. This is confirmed by the actual data on the first line on Chart 3 showing showing both r-squared values over 70. This is what you'd expect since both models can only utilize stocks from the benchmark index. If they're comprised of stocks in the index, they ought to behave similarly, and apparently they do. Secondly, P4's r-squared is higher than P3's; again what you'd expect. Charts 1 and 2 suggest this would be the case because the data points are clustered a little more tightly around the P4 regression line than the P3 line. Intuitively this makes sense given that P4 has to be diversified across all ten sectors and also must be weighted like the index. P3 doesn't have these limits and is freer to diverge from the index. That's captured by its lower r-squared. This relation holds in all three measurement periods, and there's no reason to think it would be otherwise.
P3 has historically tended to concentrate in only a few of the ten sectors, favoring technology, consumer discretionary, and financials. These are some of the more cyclical sectors which would lead you to think it would face more market risk than P4. That proves to be the case as P3 consistently has a higher beta in all three measurement periods. The overall difference is quite notable (1.76 vs 1.25, respectively). You'd think this would lead to some major performance differences and indeed, it does as we'll see in a moment. Alpha is another story. The overall alpha for both models is virtually nonexistent. P3's is slightly negative (-0.0029%) while P4's is an even smaller positive amount (0.0005%). Clearly, neither is generating any meaningful alpha, meaning any excess return -- either positive or negative -- is a function of beta. We didn't anticipate this. The intent of the models was to track and incrementally exceed the return of the benchmark. We expected excess return would come from both beta and alpha. We expected the periodic rebalancing of each model to provide an opportunity to generate alpha from prescient stock picking and market timing. In fact, in constraining P4, we believed its excess return would have to flow from alpha. Evidently that was wrong. What's more puzzling is the fact that there's nothing in the model's algorithms to inhibit alpha. In other words, there's nothing limiting the focus to beta alone so there's no obvious reason to think alpha wouldn't play a role, yet it doesn't. And it hasn't throughout the entire period. The July 2003 changes had virtually no affect on alpha for either model. You can see this when comparing all three time periods on Chart 3. From inception, both P3 and P4 were slightly negative. Both improved slightly after the change, yet neither meaningfully. Betas for both came down after the 2003 adjustments, but it's not clear whether that was the result the changes or changing market conditions. Up until that point, many of the model holdings -- particularly in P3 -- were heavily influenced by momentum from the 1990s. In other words, they focused on some of the riskiest shares available. As the internet bubble burst, lower risk stocks returned to favor, and betas came down. The 2003 changes may have precipitated this move or perhaps simply helped it along. Although it may be tempting to infer, it's not possible to determine causality from simple regressions. On the other hand, there may be reason to suspect the 2003 modifications did have an effect on P4's r-squared. Prior to the change, its tie to the index was a remarkably high 85, but afterwards it fell to 71. (P3's r-squared also declined, but nowhere near as sharply.) This, too, is puzzling.
Initially P4 was reformulated only once a year. When it was, it was balanced to the sector weights of the benchmark index. One might think (as we did when creating the model) that tracking error would probably grow as the year progressed because various sectors and/or stocks would fall behind and rebalancing would not occur until the end of the year. In addition, any time S&P rebalanced the index or added and removed constituents, the model's differences would grow. By rebalancing semi-annually -- the 2003 change -- one might think the tracking error problem would be reduced, not magnified. But the opposite has occurred. P3's r-squared with the index has gone down. This could actually be a good thing. Perhaps now the model's algorithm is focusing on top performing stocks twice a year, rather than just once. As a result, performance gets a little goose every time the model is rebalanced. Returns improve but r-squared drops because the model is now performing differently (better) than the index. We've got the evidence supporting the decline in r-squared, what about returns?
The Real Test The performance patterns are actually fairly consistent across all periods for the three series: P4 is on top, followed by the benchmark, and P3 brings up the rear. That doesn't fit well with the earlier analysis of beta: The higher beta of P3 should result in greater losses than P4 and the index, but it should also lead to greater gains on the upside. While that's the case relative to the index, it's clearly not the case in relation to P4. Could this be the result of P4's stock selection process? That wouldn't seem likely given that P4's alpha is so similar to P3's. Perhaps it could be a function of market volatility. Given its greater diversification and lower reliance on momentum, P4 may be better able to benefit from volatile market conditions. Its lead over the benchmark may come from its higher beta. Overall, all three series have remained in negative territory since inception. P4 poked into positive territory for three days in mid-May. At its peak, it was 0.39% in the black. The S&P 500 got no closer than -1.92% while P3 was left in the distance, -40.71% at its best. Eight years into this experiment there are still more questions than answers. Arguably, the models are behaving roughly as expected. The lack of alpha is something of a surprise, but perhaps time will shed more light on that. The relative beta levels are what would be anticipated, given P3's sector concentration and P4's diversification. To some extent, the jury's still out on the 2003 changes given that they coincided with the market turn. Intuitively, it would seem they were on target, but here again, time will tell. The ultimate goal was for both models to exceed the benchmark incrementally over time. P4 has done that, but P3 hasn't. Admittedly, their inception date was a major disadvantage, but at least P4 has been able to finally move ahead of the index and back towards positive territory. On that count, P4 would seem to be the success and P3 the failure, yet look again at Chart 4: P3 has kept pace with P4 over the past five years, so maybe the mid-course correction is paying off. For the most part, the foregoing regression analysis confirms a great deal of what we've observed in the past about the models' performance. The lack of alpha is the only real surprise, and is worthy of further analysis. We'll come back to that in the following months. Search this site! Just enter you key word or words: Get current quotes or follow your own custom portfolio,
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HE MAIN IDEA BEHIND Portfolios 3 and 4 was to create purely quantitative models designed to outperform the S&P 500 over the long-term. The concept of "purely quantitative" is critical, because once the models were designed and taken out of sample, no additional changes would be made for at least three years. In the interim, there would be no human decision-making, trading, or weighting. It was all purely quantitative. Every three years we've reviewed the results, and in 
