Probabilities for the S&P 500 weekly return

The previous post highlighted the relative rarity of 2016's week 1 return in the context of history. The following plot shows this in another way. It shows the probability density function (Wikipedia) for the S&P 500 weekly return based on history:

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:

If you are unfamiliar with probability density functions, here is the same data shown as a histogram. A histogram sorts the data and organizes it into buckets with the height of each bucket, in this plot, a count:

And a zooming-in of the nasty left tail shows the location of the first week of 2016: