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Factor Rotation: Size, Value and Market Correlation

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Of all of the tidbits of financial pornography bandied about in the media by market analysts, none is more hackneyed than "rotation": "Well Lou, I think what we saw this week was a rotation out of the cucumber sector and into bagels." This is a fatuous way of observing that cucumbers got cheaper, and bagels more expensive.

Most rotation is probably random noise, although rarely a few interesting patterns emerge. One of the most fascinating was the divergence of the Dow Jones Industrial Average and the Nasdaq Composite on Monday, April 3, with the former leaping by several hundred points while the latter fell by about the same amount, the pattern then reversing later in the week. Even the thickest of observers realized that the flow of funds into and out of value (DJIA) and tech (Nasdaq) stocks had become a sort of zero sum game, with a good day for one meaning a bad day for the other.

But rotation can be viewed in a much broader context. In order to understand this we're going to have to spend a page or so on the (groan) 3-factor model. These factors are:

  • "Market." This is the market risk premium, defined as the difference in return between the broad market (defined variously as the Wilshire 5000, Russell 3000, or CRSP-All.) and short t bills.
  • "Size." The difference in return between small and large stocks.
  • "Value." The difference in return between value and growth stocks.

A few caveats. First, implicit in this schema is the notion that all 3 of these are simply premiums, which can be positive or negative, and are earned in excess of the risk free (t-bill) rate. Second, the 3 factors are each in fact long-short portfolios. As such, "size" and "value" are impossible to hold in isolation in the real world. For example, to own pure "size" you'd have to own the several thousand stocks in the CRSP 6-10 and short the several hundred stocks in the CRSP 1-5. To own pure "value" you'd have to go long the thousands of stocks with the top third of P/B and short the thousands in the bottom third.

Each factor can be thought of as the flour, water, and yeast in a loaf of bread. The relative proportions of each can be thought of as the basic character of the resultant loaf. (Profuse apologies to James Beard.) And for much the same reason, you don't want to bake with just one.

Let's get our fingers dirty with them for a minute. For the 10-year period (monthly returns) from 1990 to 1999 here are the results of 3-factor analyses for 4 different asset classes and the 3 most commonly used market indexes:

Asset Class/Index
Alpha
Market Loading
Size Loading
Value Loading
R-squared
S&P 500
0.01
1.00
-0.17
0.02
0.99
FF Large Growth
0.08
0.98
-0.19
-0.29
0.98
FF Large Value
-0.12
1.06
0.06
0.68
0.92
FF Small Growth
-0.14
1.09
1.13
-0.32
0.96
FF Small Value
-0.02
1.01
0.93
0.73
0.99
Dow Jones Ind. Avg
0.04
1.03
-0.14
0.22
0.87
Nasdaq Comp.
0.41
1.08
0.52
-0.50
0.91

FF = Fama/French, source = DFA

First, note that the market loadings of each of the indexes are all very close to 1.0. In other words, each of the indexes is fully exposed to market risk and return. Next, note the differences in size loadings, with the large cap indexes having a loading of about zero, and the small cap indexes having a loading of about 1. The Nasdaq is intermediate between the two. Finally, the value loadings top out at about 0.7 for the value indexes, and are about -0.3 for the growth indexes.

The Nasdaq can be considered a "growth index on steroids," with a value loading of -0.5. The R-squareds show how well the returns of each index fit the model, which is very well indeed in almost all cases. The Dow, with only 30 stocks, has the lowest value, but is still quite respectable at 0.87. The alphas tell us how much higher or lower the average monthly return for the index is than is predicted by the model. Most values are very near zero, except for the Nasdaq, which is about 41 bp per month (or 5% per year) higher than predicted by the model.

This laborious preamble is necessary to better understand how real market rotation occurs over decades, because it involves all 3 factors. We'll travel back in time, and plot the returns of $1.00 invested in each of the factors for each decade, starting with the last one:

As you can see, in the 90s the only asset worth owning was the market. The returns of both size and value were negative, which is the same thing as saying that both large cap and growth tilts were favored. No surprise here-large cap growth stocks have been the place to be in recent years.

The 1980s were somewhat different:

Again, "market" had positive returns, but not as dramatic as in the 90s. And unlike the current decade, "value" had significantly positive returns as well. So the growth tilt which did so well would have reduced returns in the 80s.

And finally, the Ghost of Christmas Past, the 70s:

What could be more different than the last decade in the market than an environment where market exposure was a highly negative factor and exposure to small size and value were the only things which saved your bacon? Note particularly the years from 1973 to 1975, where exposure to the value factor nearly made up for the severe market losses of the worst modern bear market.

Finally, consider the Markowitz inputs from 1964 to 1999:

 
Market
Size
Value
Return
SD
Market
1
 
 
5.74%
15.16%
Size
0.26
1
 
2.00%
13.21%
Value
-0.41
-0.24
1
2.96%
12.54

The strong negative correlation between market and value is robust, being present in all 3 decades. If anything, it has grown stronger with time. As might be expected, when these values are fed into a mean-variance optimizer a strong value bias appears. In fact, even when one reduces the return of value it does not disappear from the efficient frontier mix until a return of -1.5% per year is reached. In other words, even if the return of the value factor is zero or slightly negative, you still want exposure to it. The "inclusion threshold" for size is almost exactly zero-you have to believe that its return is positive to use it. (Warning: you cannot toss the above parameters into most off-the-shelf optimizers, as the composition constraints are radically different from the standard case, where their sum must equal unity. In the present case all 3 compositions/loadings can add up to any positive or negative number.)

The strong negative correlation between market and value is robust, being present in all 3 decades. If anything, it has grown stronger with time. As might be expected, when these values are fed into a mean-variance optimizer a strong value bias appears. In fact, even when one reduces the return of value it does not disappear from the efficient frontier mix until a return of -1.5% per year is reached. In other words, even if the return of the value factor is zero or slightly negative, you still want exposure to it. The "inclusion threshold" for size is almost exactly zero-you have to believe that its return is positive to use it. (Warning: you cannot toss the above parameters into most off-the-shelf optimizers, as the composition constraints are radically different from the standard case, where their sum must equal unity. In the present case all 3 compositions/loadings can add up to any positive or negative number.)

So over the long haul, the most important "rotation" is in and out of the 3 major market returns factors. And although we can't predict what they will be over the next decade, it's a lead-pipe cinch that they won't look anything like the last 3.

Copyright 2000, William J. Bernstein 8/02/00