January 2004
Correlation and Deviation
| "Statistics: The only science that enables different experts using the same figures to draw different conclusions." |
| --Evan Esar |
ONE OF THE BIGGEST OBJECTIONS to indexing -- usually offered by those who earn commissions through the sale of managed products -- is that it guarantees only mediocre performance. Presumably, if all you're getting is the return of the index, you have no chance to outperform it. Any management fees further reduce your net results.
When we created our large cap and multi-cap models, we wanted them to track their respective indexes, but still have the opportunity to outpace them. In the ideal situation, the models would do no worse than the index in a down market, yet better in an up market.
Obviously one way to test this is to look at performance over various time periods. If successful, the models will do better than their benchmark indexes over time.
Correlation and standard deviation are other important factors. Correlation measures how closely two series move in relation to one another.
Standard deviation quantifies the average variance from the average return. This is usually used as a measure of risk. In other words, the greater the average variance from the mean return, the riskier the series.
While most investors just focus on return, correlation and standard deviation are important as well. Correlation comes into play when the model is used to build a broader portfolio. If, as in the case of quant Portfolios 3 and 4, you want them to represent large cap domestic equities, they should behave like large cap domestic equities. In other words they should be highly correlated with the proxy -- in this case the S&P 500 -- for large cap equities. If not, they won't work correctly in your overall asset allocation.
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Standard deviation is important, too, since there's a direct relation between risk and return. If the model consistently outperforms when stocks are trending up it may only be the result of additional risk. If the model is riskier than the benchmark, it should do better in up markets, but by the same token it will do worse in down markets. Ideally, the standard deviation of the model should be close to that of the benchmark; in the best case scenario, it should actually be less.
Models, Methods, and Measurements
This isn't our first examination of these factors. Back in May 2001 we addressed these issues in regard to P3 and P4. Portfolio 5 was still in development, not being formally introduced until January 2002. For that matter, at the time P3 and P4 had only been around for ten months.So now, with over three years worth of data for the large cap portfolios and almost two full years for the multicap model, it's a good time to revisit performance, correlation, and standard deviation.
To do this, we considered two different time periods: July 2000 - November 2003 for P3 and P4, and January 2002 - November 2003 for all three models. The first includes the greatest amount of data for P3 and P4 while the second is the longest common period for all three models.
We measured cumulative performance, correlation, and standard deviation for each period. In addition, we broke the July 2000 - November 2003 time frame down into seven sub-periods to test how correlation with the index changed over time and with overall market performance.
For the most part, the results were to be expected. Still, there were a few mild surprises.
July 2000 - November 2003
By the time you read this, P3 and P4 will have been around for more than 3-1/2 years. When we compiled our research, monthly data was only available through the end of November 2003 so our results only cover the 41 months from July 2000 - November 2003.Looking first at performance, both models trail the overall benchmark, although P4 has hung right with it. Over the total period, the S&P 500 is off 27.25% while P3 has lost 40.65% and P4 is
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Chart 1 PORTFOLIOS 3 & 4 vs. THE S&P 500 July 2000 - November 2003 |
 Source: A-T Financial/Quantview |
Clearly P4 has mirrored the index much more closely than P3, but to a degree, that's to be expected. P3 is a relatively concentrated portfolio holding 30 stocks with the highest ratings from its quantitative model. P4 is more diversified, utilizing the highest rated stocks from each of the 10 S&P sectors. (For a more detailed description of the underlying models, see the sidebar above or The Starting Point in the Archives.)
The benefits of diversification are often touted, especially in a down market. That's what P3 and P4 faced almost immediately from inception. With its greater diversification, P4 was better enabled to keep up with the index while P4 fell further behind, especially early in the bear market.
In late 2000 and early 2001, P3 focused on tech and other growth stocks, precisely when they fell out of favor. Although P4 tilted towards growth, the requirement that it hold stocks in each sector forced it into value areas as well, helping it keep up with the broad index.
Despite initially losing ground the the S&P 500, P3 has been closing the gap. This has especially been the case in 2003 when stocks turned up. Through November 30, P3 gained more than twice as much as the index, +53.91% vs. +20.27%, respectively.
As you'd expect given their concentration and focus on growth stocks, both P3 and P4 are "riskier" than the S&P 500 -- at least according to their standard deviations. For the period July 2000 - November 2003, the index's daily standard deviation was 0.0137 while P3's was 0.0263 and P4's was 0.0178.
Like their returns, P4's standard deviation was much closer to the benchmark than P3's. Again, P4's greater diversification reduced its risk relative to P3.
So how much did daily returns differ from those of the index? That's where correlation comes in.
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Chart 2 RETURN CORRELATIONS July 2000 - November 2003 |
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Chart 3 RETURN CORRELATIONS July 2000 - November 2003 |
 Source: Quantview |
The correlation coefficient is a measure of the strength of the relation between two series, in this case the S&P 500 and P3 or P4. If they were perfectly correlated -- moved identically step for step -- they would have a correlation coefficient of +1. Any value less than that indicates less than perfect correlation with weaker relations as the coefficient approaches 0.
An easy way to get a gauge of correlation is through a scatter plot. This is a graph that plots the return of one series on the x-axis and that of the other on the y-axis.
If the two series are perfectly correlated, all the points will fall on a 45º line extending through the origin, (0,0). The greater the dispersion of points, the weaker the correlation.
Even when the points are widely dispersed, linear regression analysis, a statistical process, can determine the best-fit line for the relation. Few (if any) natural series are perfectly correlated, so most will require this calculation.
Charts 2 and 3 show the scatter plots and linear regressions for P3 and P4 relative to the S&P 500. Consistent with the difference in returns and greater standard deviations, the points on the Chart 3 are more closely bunched around the regression line, indicating that P4 is more highly correlated with the index. Even so, there is still a considerable degree of dispersion and a number of outliers.
Notice that each chart shows the regression formula of the best-fit line. In each instance, "y" represents the portfolio value and the equation shows how its best estimate can be determined for each value of "x", the S&P 500.
The "R2" value is the square of the correlation coefficient and is known as the "coefficient of determination". In essence, if viewed as a percentage, it tells what portion of the portfolio's performance is determined by that of the index. The higher the value, the more closely related the series.
According to Chart 2, about 63% of P3's performance is explained by that of the S&P 500, while the figure rises to almost 80% for P4. This again quantitatively demonstrates why P4 tends to more closely mirror the index.
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Chart 4 CORRELATIONS OVER 6-MONTH* PERIODS July 2000 - November 2003 |
 *Final period is 5 months, July 2003 - November 2003 Source: Quantview |
When we first looked at the models' correlations in May 2001, both R2 were lower, 57% and 72%, respectively explained by the index. Those figures were based on only 10 months of data, so you can't really conclude that correlation has improved.
But how has it changed? Correlations aren't static and can change over time. Most specifically, we were interested to see if they strengthened when the market turned earlier this year.
To examine this, we calculated the correlations for six month periods with the first ending December 31, 2000. (The last period was only 5 months, July 2003 - November 2003.) We then compared them to the S&P's return over the same periods. Results are shown in Chart 4.
Notice any trends? Probably not, because there aren't any.
In six of the seven periods, P4 has a greater correlation with the index than P3. That was to be expected given the overall results.
We thought correlations might increase when the market turned in late 2002, but they really didn't. The standard deviations that worked against the models in the down market worked in their favor when the market turned. Correlations strengthened only slightly.
That's actually a good thing. While the models may not be able to outperform the index under all circumstances, it is still important for the asset allocation process that they be consistent regardless of market conditions. According to Chart 4, they've succeeded in this regard.
January 2002 - November 2003
Portfolio 5 didn't officially come into existence until January 1, 2002. Over its first 23 months, it's seen a roughly equal combination of up and down markets. Throughout the period, it's traded both above and below the index, but essentially right with it.Chart 5 shows the performance of all three models relative to the S&P 500. Technically, the S&P 500 is the benchmark for multicap P5, but given the fact that S&P cap weights its indexes, the 500 has always traded similarly to the 1500 so this comparison should suffice for our purposes here.
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Chart 5 PORTFOLIOS 3, 4, & 5 vs. THE S&P 500 January 2002 - November 2003 |
 Source: A-T Financial/Quantview |
The first thing that jumps out from Chart 5 is the fact that all three models have outperformed the S&P 500. As you'd expect, P3 with its higher standard deviation and lower correlation leads the pack while P4 and P5 are right behind.
For the overall period, only P3 is in the black, up 6.0%. P4 is down 2.6% and P5 is off 1.8%. The benchmark is off 7.8%.
The standard deviations for P3 and P4 are down only slightly from the longer period, 0.0201 vs. 0.0263 and 0.0151 vs. 0.0178, respectively. This is the type of consistency you'd hope to find in well-constructed models.
For its first 23 months, P5's standard deviation is actually a little lower than that of the benchmark, 0.0139 vs. 0.0141. This is not a statistically significant difference, but does attest to the similarities of the two series.
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Chart 6 RETURN CORRELATIONS January 2002 - November 2003 |
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Chart 7 RETURN CORRELATIONS January 2002 - November 2003 |
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Chart 8 RETURN CORRELATIONS January 2002 - November 2003 |
 Source: Quantview |
This is also evident in the correlations. Charts 6-8 show the scatter plots and regression lines for all three models vs. the S&P 500 for the period January 2002 - November 2003.
As we had thoughT would happen in a more positive market, correlations are up for P3 and P4. You can clearly see this from the scatter plots for P3 (Charts 2 and 6) and P4 (Charts 3 and 7). Notice how the points are bunched much closer to the best-fit line in Charts 6 and 7, indicating a higher correlation.
This is also reflected by the R2s: 0.7349 vs. 0.6267 for P3 and 0.9122 vs. 0.7965 for P4. Again in translation, for the January 2002 - November 2003 period, more of the models' movement is explained by that of the index than in the previous period.
Portfolio 5 also has a high R2, 0.8716. There are few outliers on Chart 8, showing a close fit with P5's regression line. This is to be expected given how closely P5 has tracked the benchmark.
Over the past two years, all three models have done exactly what you would have wanted -- they were highly correlated with their benchmark yet provided more return. However, as the correlation figures indicate, three-quarters or more of that return is the result of an up market.
Take Aways
There are a number of things to take away from this, most of them good. While there's a greater body of data for P3 and P4, it's still possible to include some reasonable conclusions about P5, too.The models are not exceptionally risky relative to the benchmark. Although P3 and P4 have standard deviations that are greater than that of the index, their relative performance is commensurate with their increased risk. P5's standard deviation is right in line with the benchmark and at least so far, has managed to outpace the index.
The models are consistent. While correlations have shown minor variations over the various measurement periods, they have remained relatively steady. This is particularly true in the case of P3 and P4. This doesn't mean they will always outperform the index -- indeed, the results show otherwise -- but consistency relative to the index is important for another reason:
The models can serve as reliable components in an asset allocation model. The majority of a long-term investor's return comes from his or her asset allocation. It's important that one's actual holdings truly act like the series that they represent in the asset allocation model. The correlations of all three models are relatively high, making them good candidates for the asset allocation process. Their statistical consistency further recommends them in this role.
The models won't always outperform the index -- at least not yet. The historical data shows they have had periods -- particularly P3 -- of significant over and underperformance. P5 has held up well, but with just 23 months of data, it's too early to come to any conclusions.
As the page title suggests, all three models are still works in progress. We've tried to minimize the possibility of ad hoc changes, but did make some alterations to the quantitative models for P3 and P4 in July 2003. As a rule of thumb, we'd like to allow at least three years between any revisions to observe them across a full market cycle. In other words, don't look for any major changes any time soon.
In the meantime, the models are working pretty much as anticipated. They've tracked their benchmarks and do offer the opportunity to outperform. They may not be as consistent as pure indexing, but they have demonstrated a greater upside potential. What remains, is to limit the downside.
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