## Regression to Trend: A Perspective on Long-Term Market Performance

### April 2nd, 2014

### by Doug Short

**Quick take** : At the end of March the inflation-adjusted S&P 500 index price was 80% above its long-term trend, up from 76% above trend the previous month.

About the only certainty in the stock market is that, over the long haul, over performance turns into under performance and vice versa. Is there a pattern to this movement? Let's apply some simple regression analysis (see footnote below) to the question.

Below is a chart of the S&P Composite stretching back to 1871 based on the real (inflation-adjusted) monthly average of daily closes. I've using a semi-log scale to equalize vertical distances for the same percentage change regardless of the index price range.

The regression trendline drawn through the data clarifies the secular pattern of variance from the trend — those multi-year periods when the market trades above and below trend. That regression slope, incidentally, represents an annualized growth rate of 1.75%.

The peak in 2000 marked an unprecedented 150% overshooting of the trend — nearly double the overshoot in 1929. The index had been above trend for two decades, with one exception: it dipped about 12% below trend briefly in March of 2009. But at the beginning of April 2014, it is 80% above trend, up from 76% above trend the month before. In sharp contrast, the major troughs of the past saw declines in excess of 50% below the trend. If the current S&P 500 were sitting squarely on the regression, it would be around the 1033 level. If the index should decline over the next few years to a level comparable to previous major bottoms, it would fall to the vicinity of 500.

Incidentally, the standard deviation for prices above and below trend is 40.56%. Here is a close-up of the regression values with the regression itself shown as a mean of zero. I've highlighted the standard deviations. We can see that the early 20th century real price peaks occurred at around the second deviation. Troughs prior to 2009 have been more than a standard deviation below trend. The peak in 2000 was well north of 3 deviations, and the 2007 peak was above the two deviations -- a level we now just fractionally below.

**Footnote on Calculating Regression**: The regression on the Excel chart above is an exponential regression to match the logarithmic vertical axis. I used the Excel Growth function to draw the line. The percentages above and below the regression are the calculated as the real average of daily closes for the month in question divided by the Growth function value for that month minus 1. For example, the monthly average of daily closes last month was 1863.52. The Growth function value for the month was 1,037.50. Thus, the former divided by the latter minus 1 equals 79.62%, which I rounded to 80%.*
Footnote on the S&P Composite : For readers unfamiliar with this index, see this article for some background information.*