Pitfalls of Trend Analysis

Anchoring on linear time trends as a basis to support returns forecasts also appeal to mean reversion logic. This is illustrated in Figure 4.2.

Figure 4.2 S&P 500 Trend Growth

Superimposed on the (log) price history of the index are two trend lines—trend growth from January 1871 to December 2009, which is 4.27 percent annual growth and annual trend growth after World War II (January 1946 to December 2009) at 7.35 percent. These are monthly index values. Trend growth is estimated by regressing ln(Price) = α + β time and computing annual growth in percent using 100∗((1 + β)12 – 1). These are clearly distinct trend estimates, and the conclusions drawn from mean reversion arguments are naturally going to be sensitive to which trend one is referring to. To see how sensitive trend estimates are, consider that the annual trend growth in the S&P was 1.88 percent from 1871 to 1928, 5.86 percent from 1929 to 1989, 6.81 percent from 1929 to 2009 and, as mentioned before, 7.35 percent from 1946 to 2009. This is a rather wide range and, depending on which trend line we use, mean reversion from any given spot index level may be up or down!

For example, historically, when the level of the S&P 500 was as far below the post–World War II trend as it was on, say, April 30, 2009, the subsequent average 10-year return was 13.44 percent. Many market analysts pointed to this statistical artifact to support claims that the index could look forward to a rebound to 7.35 percent growth over the next decade. This is interesting; the index was indeed below trend (note the vertical distance between the index and the post-1946 trend line in Figure 4.2) but it was not below trend associated with the entire history of the S&P. The trend based on the longer series suggests that the S&P is still above trend, not below it, and therefore suggests mean reversion in the opposite direction to a long-run growth rate of 4.27 percent. Strictly speaking, both claims are wrong because there is no single deterministic trend. Instead, it makes more sense (and this is supported by careful statistical analysis) that trend growth in the S&P is variable in the short run and in the long run, tied to fundamentals that determine economic growth.

I would like to drive home this point by demonstrating how sensitive trends can be to the presence of a few outliers. Figure 4.3 shows the sensitivity of total returns to the S&P 500 from 1950 to the present when 10 observations are removed from 15,444 observed daily returns. Perhaps this graph is a better example of how risky markets are, but it serves to also indicate how sensitive trends are to a few extreme returns. The middle line shows that a dollar invested in the index on January 3, 1950, would have grown to $1,356 by April 11, 2011. Had we been able to avoid the 10 worst days in the market, our investment would have grown to $3,457. Had we not been in the market on the 10 best days, our investment would have grown to only $639. These 10 days represent 0.065 of 1 percent of the returns, yet the entire investment turns on these events. It is interesting to note as well that 8 of the 10 best days occurred during the recent credit crisis as did half of the worst days. As suggested before, this may say more about risk than anything else—in particular, risks that have extremely low likelihood but very big impact, a topic we return to when we study systemic risk in Chapter 13. But it also serves to make clear the risks of extrapolating historical trends—a single extreme event can easily obviate any trend and the decisions predicated on that trend.

Figure 4.3 S&P 500 Total Returns Since 1950

Clearly, the shortcomings with extrapolative models are that they rely on the historical record repeating and that they do not require any structural understanding of how fundamentals (for example, earnings and dividends) are related to returns. Yet their popularity among analysts hasn't suffered. In part, that is because extrapolation from perceived linear trends is easier than constructing complex equilibrium models describing market dynamics. Trend extrapolation is a good example of the representative heuristic, that is, when the index level has been high for some time (like since World War II), then agents believe that current high observed returns are normal, and deviations from trend will therefore mean-revert to this notion of normality (Tversky and Kahneman 1974). The mistake agents make in this case is that they base their choices on how well they resemble available data (for example, recent trends) instead of a more comprehensive analysis that combines theory and broader data selection optimally. I have more to say about behavioral finance in Chapter 8.