Yesterday, the S&P 500 dropped 59 points, or about 4.71%. The standard deviation of daily returns for the broad market amounts to a bit more than 1%. In statistical terms, it was a four standard deviation event. What exactly does that mean?
Yesterday Was A Very, Very Rare Day
Standard deviation is a measure of how dispersed a collected set of data is. The higher the standard deviation, the more volatile the phenomena. A standard deviation of 1% means that 2/3 of the time, the next sample will fall between +/- 1% of the average value. That is, under normal circumstances you can expect stock market returns to fall within a 2% range. That’s called a one standard deviation event. Two standard deviations in this case would mean a 95% probability of tomorrow’s return falling within a 4% band, etc. A four standard deviation event, like yesterday, implies a 99.994% probability of the daily return falling within an 8% band. In other words, there was only a 0.006% chance of yesterday happening. Furthermore, there’s only a 0.006% chance of it happening on any other day. Basically, that means this was a freak event. It happens occassionally, but it’s so rare that it’s not worth worrying much about. It certainly isn’t likely to happen again tomorrow, or the next day, or the next day after that. The best course of action in this situation is to review your overall asset allocation plan. If it makes sense for your time horizon and risk tolerance, do nothing. Panic is ever the enemy of the individual investor. Whatever you do, don’t make changes to your portfolio out of fear or emotion.
It is said once upon a time long ago, an eastern king gathered all his wise men together and charged them with the task of inventing a sentence that was always true regardless of the current circumstances. After much deliberation, the presented him with the phrase “this too, shall pass.”
Truer words were never spoken.
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9 responses so far ↓
1 Aaron Street // Sep 16, 2008 at 9:58 am
Your math seems sound, but your conclusion from the data has a serious flaw.
By saying that a 4.71% decline only has a 0.0006% chance of happening on a day, assumes that these data points are randomly distributed along a curve. However, our current situation - and yesterday’s decline - weren’t random events along a distribution curve, they were symptoms of acute events (bank failures, bailouts, etc.) that are much more than 0.0006% likely to happen again today and tomorrow.
Put another way, your analysis works if you were to throw a dart at a calendar: then there would be a 0.0006% chance that the date the dart hit would have had a 4.71% decline.
The odds of a 4.71% decline today, tomorrow, or anytime in the near future until these underlying financial problems are solved is much, much greater than your analysis implies.
My two cents.
2 Kyle // Sep 16, 2008 at 10:29 am
True, the conclusion depends on the data points being normally distributed. Historically, daily market returns HAVE been normally distributed even in the face of acute events (like the 1987 crash or 9/11 or yesterday’s failures). Company failures actually aren’t all that rare, so yesterday’s events weren’t exactly abnormal. Furthermore, daily returns are usually if not always completely divorced of the reality of the underlying market fundamentals. Even if there were more bailouts or large bank failures, there’s no reason to believe the market would drop by four standard deviations again in response. It could quite possibly move higher.
3 Aaron Street // Sep 16, 2008 at 11:31 am
Of course the market could react to an AIG failure - or some other major event that might occur in the near future - by going up or down. I’ll concede that market reactions aren’t perfectly predictable.
But your analysis implies that they are perfectly random, which is also not the case.
I’m arguing that the market has never moved by four standard deviations up or down on a day when there weren’t acute crises/successes.
My point is not that people need to panic, just that you should concede that major market declines are more likely to occur in the face of the acute issues of this season, than they are at a time when there are no underlying major financial crises.
4 Andy // Sep 16, 2008 at 12:45 pm
this too, shall pass….Good words for troubled times. The only question when?
5 Luke // Sep 19, 2008 at 1:10 pm
Perhaps in absolute value terms it passed today…? FTSE100 had a 8.4 standard deviation event today! Aaron Street is perfectly correct - you fail statistics… perhap “The (Mis)behaviour of Markets” by B. Mandelbrot might help.
6 Kyle // Sep 19, 2008 at 2:59 pm
Extremely unlikely events happen every day. That they happened in close sequence does not imply it’s not random. Similarly, that daily market returns are not strictly normally distributed (as the above simplistic explanation assumes) does not significantly alter the results. Even taking the flared tails into account, the stated probabilities are relatively close. For your hypothesis to be valid, you’d have to assume causation where there’s no evidence of any being present, which is a logical fallacy of the highest order.
7 Hanks Weekly Hangouts #50 (September 20, 2008) | MiB Smarter Money // Sep 20, 2008 at 4:02 am
[...] 6. AmateurAssetAllocator tells us that Yesterday Was a Four Standard Deviation Event [...]
8 John // Oct 12, 2008 at 9:49 pm
Anyone wish to update their comments? We’re now experiencing fat tail events daily………thoughts??
9 Aaron Street // Oct 14, 2008 at 11:18 pm
Kyle, by your calculations, what has been the standard deviation of the past two weeks?
It’s got to be close to 1:1,000,000,000,000 at this point.
It’s still not time to panic, but the market is definitely not random.
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