**Tom McClellan – McClellan Market Report**

The share price of Apple Corp surprised a lot of people this week with a one-day drop of 37 points, or -6.43%. Part of why that was a surprise is that over the past year, the standard deviation of Apple’s daily price change has been 1.8%, and so this was a down move which was greater than 3 standard deviations.

Those who have taken a course in statistics may recall “The 68-95-99.7% Rule“. Simply stated, the rule indicates that for a normal distribution:

- 68% of all observations should fall within 1 standard deviation of the mean
- 95% of all observations should fall within 2 standard deviations of the mean
- 99.7% of all observations should fall within 3 standard deviations of the mean

So if Apple’s daily share price change followed the rules for a “normal” distribution, and if one were to know that the standard deviation of daily returns is 1.8%, then there should be a 99.7% chance that any single day’s change would be within +5.4% and -5.4%. Saying it another way, there should be only a 0.3% chance of a daily percent change being greater than 5.4% up or down. That’s roughly one instance out of every 333 times, and so in a year with only 252 trading days, it should almost never happen.

But just in 2012, we have already seen 5 different days when Apple’s share price rose or fell by more than 5.4%, something which the whole notion of a “normal” distribution says just should not happen. This is just one example of what some market analysts are referring to when they talk about the concept of “fat tails”.

The “tails” of a bell curve distribution should see diminishing numbers of observations the farther you get away from the mean value. But when a distribution has “fat tails”, that means there are more observations than theory would suggest out near the tails. To help see what that means for Apple’s share price movements, here is a chart showing where they have fallen over the past 24 years:

What we find is a whole lot of observations well outside of that standard deviation value of 1.8% that I mentioned above. Standard deviation of daily returns is often used as a measure of the inherent volatility of a stock price or other data, but as most traders know, volatility is not constant. And that leads to the final point, which is that the figure for the 1-year standard deviation of daily price changes varies dramatically over time.

Since 1988, Apple’s 1-year standard deviation has been as low as 1.3% and as high as 4.5%. In fact, the past 3 years when Apple’s product innovations have brought so much of a big gain in its shares has seen unusually low volatility in comparison to the rest of its history. And if Apple is starting to transition into a corrective period as the comparison to RCA’s share price pattern suggests is coming, then we can figure on volatility increasing, and on having big up and down days becoming much more common occurrences. We have been updating that comparison of Apple’s share price now to RCA’s price pattern from the 1920s and 1930s with subscribers to our twice monthly *McClellan Market Report* and our *Daily Edition*.

Related Charts

Nov 08, 2012 Apple Walking In RCA’s Footsteps |
Aug 06, 2010 Correlations May Not Be What They Seem |
Nov 29, 2012 Why Increased Revenues Won’t Solve Fiscal Cliff |

I feel like I have the dumb today. Can someone help clarify for me? I would think the way to interpret Tom’s second chart is that the vertical bars represent the number of occurences for a given percentage change in the price of a share of AAPL.

For example, it looks like the chart is saying there have been 937 times in AAPL’s stock price has moved between 0 and -1%.

(A) Is that the correct way to read this chart?

Then, in terms of understanding the 68-95-99.7% Rule, I’m confused about the use of standard deviation of change vs percentage stock price change. I’m used to using standard deviation relative to a mean.

That is, to know the expected range of a 3-standard deviation event, don’t you need to know the mean _and_ the std dev? A “one standard deviation event” is something that falls within (the mean minus one times the std dev) and (the mean plus one standard deviation)?

I think the point (and the math) is close enough because I am calculating the trailing 12-month daily price change (from open to close) mean at about -0.05% with a std dev of 1.41%. So, a 3-std dev event should cover -4.29% to +4.19% and there are a significant number of events (a few hundred) outside of that range.

hahaha, having “the dumb today”, that’s funny. anyway, i think the histogram is the “daily” deviation of the stock price. for example, if the stock price is $1000 at the beginning of the day then the assumption is that the stock price will be $1000 at the end of the day. therefore, with this assumption, the mean of the stock price is $1000 and the daily fluctuations can be readjusted around zero percent, so in the histogram there were 937/977 times in the stocks history where the daily fluctuation around that days beginning stock price was around zero. basically, to examine the fluctuations around a number you can shift and plot the deviations around zero, but in general you are right about knowing both the mean and standard deviation and knowing what assumptions you are using. well, this is my interpretation, but i may have the dumb today as well