Last week SPX gained 7.4%. For prior big up weeks (gaining at least 3%) going back to 1950, what was the maximum drop over the prior 20 weeks? To check this, identified the largest decline from the highest weekly close of the prior 20 weeks to the week before the big up (>+3%).

Since 1950, there there were 191 weeks that gained at least 3%. Regressing their returns vs the maximal decline of the prior 20 weeks shows that the larger the prior drop the bigger the big up-week:

The regression equation is 3% wk = 0.0354 - 0.0970 pr-wk/max20

Predictor Coef SE Coef T P
Constant 0.0354 0.00142 24.93 0.000
pr-wk/max20 -0.0969 0.01119 -8.67 0.000

S = 0.0136929 R-Sq = 28.4% R-Sq(adj) = 28.1%

The recent +7.4% week was preceded by a maximal 20 week decline of -13.8%. The attached regression graph shows this point outlined -near the upper edge of the data cloud. Arguably last week was somewhat atypical, as most big up-weeks of similar magnitude were preceded by larger maximal declines during the prior 20 weeks.

Steve Ellison elaborates:

My theory is that most large short-term gains in the stock indices are driven by short covering panics and hence occur after price declines. When everybody thinks there is a bull market, there are few shorts. Further advances, if they occur, tend to be slow and steady.

I turned Dr. Zussman's query around and asked, "What is the expectation for a big weekly gain when the S&P 500 is down during the past 20 weeks, compared to when it is up in the last 20 weeks?"

I divided S&P 500 futures weekly changes since 1982 into two groups: those that occured after the net change of the previous 20 weeks was positive, and those that occurred after the net change of the previous 20 weeks was negative. The standard deviation of the one-week net change was 3.2% when the previous 20 weeks were negative and 2.0% when the previous 20 weeks were positive. The distribution of one-week net changes was:

.                                                               Percentile
.                      N         90th    80th      60th    Median     40th       20th
Last 20 weeks down      538     3.7%     2.2%     0.9%   0.4%    -0.3%    -2.0%
Last 20 weeks up        1000     2.4%     1.6%     0.6%   0.2%    -0.2%    -1.4%

Large weekly gains (those in the 80th percentile and above) were 40% to 50% larger when the past 20 weeks were down, but large weekly losses were also larger in magnitude by a similar amount.

As of Friday's close, the S&P 500 futures are down 5% from 20 weeks ago.


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