The green area highlights areas where the weekly return is positive; the red area highlights areas where the weekly return is less than -5 percent; and the blue area highlights the in-between areas.

With this probability density function, we deduce probabilities for the S&P 500 in any given week:

- Probability of S&P 500 returning less than -10 percent in a week is 0.1 percent.
- Probability of S&P 500 returning less than -5 percent in a week is 1.4 percent.
- Probability of S&P 500 returning less than -2 percent in a week is 12.5 percent.
- Probability of S&P 500 returning less than -1.5 percent in a week is 17.9 percent.
- Probability of S&P 500 returning less than -1 percent in a week is 25.3 percent.
- Probability of S&P 500 returning less than -0.5 percent in a week is 33.2 percent.
- Probability of S&P 500 returning less than 0 percent in a week is 43.5 percent.
- Probability of S&P 500 returning less than 0.5 percent in a week is 54.9 percent.
- Probability of S&P 500 returning less than 1 percent in a week is 66.5 percent.
- Probability of S&P 500 returning less than 1.5 percent in a week is 77.9 percent.
- Probability of S&P 500 returning less than 2 percent in a week is 85.1 percent.
- Probability of S&P 500 returning less than 5 percent in a week is 98.6 percent.
- Probability of S&P 500 returning less than 10 percent in a week is 99.9 percent.

The second bullet agrees with the observation of the previous post, that the S&P 500 losing more than 5 percent in a week is about 1 in 70, or about 1.4 percent.

According to the seventh bullet, the S&P 500 suffers a down-week 43.5 percent of the time. Consequently, the S&P 500 encounters an up-week 56.5 percent of the time. This fits the conclusion of an earlier post about daily returns -- that the market closes up on any given day about 53.7 percent of the time (though that post was based on a different set of data).

*Weekly*returns show positive closes with a slightly higher up-probability than daily returns because of compounding. In fact, this logic works sequentially for all types of returns -- thus monthly returns have higher up-probabilities than weekly returns, annual returns have higher up-probabilities than monthly returns, five-year returns have higher up-probabilities than annual returns, and so on.
I zoomed in on the red area, where life is unpleasant:

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