*Submitted by Salil Mehta via Statistical Ideas blog*,

It was surely a frightening week in ** global** financial markets. The largest 500 American stocks (S&P) dropped 6%. China's Shanghai Stock Exchange (SSE) doubled this risk, as it dropped 12%. Now there is an overall fear in the markets that we have not seen

**in years**. While these perilous risk statistics should not be something new, the

*surprising**this week provides a renewed opportunity to review crash measures within a broader context, to boldly target your portfolio.*

**jolt** Let's look at the worst weekly loss for the S&P, in each month from 2007 **through August 14** (or right up until last week). Geometrically approximated for symmetry. We see in ** blue** that the distribution of this monthly "worst weekly loss" has generally been similar to the same ranked values from the past couple of years (2103/2014). Now towards the bottom of the chart we can better ascertain that the

**"worst weekly losses", were even worse in the years earlier than this (so 2007 through 2012).**

*more severe* We'll prove out these numerical measurements here, but if you are ** dispassionate** about the mathematics then don't fear. Please just skim what is immediately below -and head straight to the first illustration afterwards- to continue reading. In October 2008, the worst weekly risk was -20% (this makes October the 24th worst month for "worst weekly loss" of 24 months in 2007/2008). Hence it is plotted in

**~98 percentile at the bottom of the vertical axis below. Not perfectly the 100th percentile (0% rank) due to probability math. Also in the same 2007/2008 series, the**

*red***for the "worst weekly loss" statistic was the following month of November. That month saw a -9% change and being 2**

*next worst month*^{nd}worst out of 24 means being ranked about 4% higher on the vertical axis, from where the -20% data is shown:

*2/24 (for second worst of 24) - 1/24 (for worst of 24) **= 1/24 **~4% more favorable rank*

Similarly all of the axis tick marks, for all of the complete 2-year periods shown, are ~4% apart on this inverse distribution axis (i.e., 98%, 94%, 90%, etc.) For 2015, * up until this month of August* there were 7 months, and the worst weekly loss of them was January's -3% change. The lowest

**data shown represents that month (and 7th worst of 7 months is ~93 percentile at the bottom of the vertical axis). To summarize, the worst ~6% of months (100%-94 percentile) in 2007/2008 was about -9% and much worse than for 20015 where it was about -3%.**

*blue* Also for completeness, we see that the ** most favorable** "worst weekly loss" among months in each of the time periods shown below (so

**of the chart) was essentially a non-event at ~0%. We all know that we are no longer at that tail of the distribution!**

*towards the top* So next we identify with a **black star** how last week (5 days though August 21) compares with the 2015 series to date. We see that last week's -6% change for the S&P is** **completely **out of line** with the rest of 2015, and it is ** beyond **anything we've seen since before 2013! Despite this

*multi-year record*blasting across the news, one also

*can***not**fully state that 2015 though is a trend reversion to the risk we experienced during the global financial crisis, since the

**2007/2008 risk statistics are almost all higher than the entire**

*red***statistics shown above.**

*blue* We perform the same exercise again, but for the SSE. We see in ** blue** again the distribution of the "worst weekly losses", and it has generally

*been similar to the distribution of the previous couple years (2103/2014). But unlike with the S&P, 2015 risk statistics are instead closely aligned to the same risk measure from the global financial crisis era 2007/2008 (again, in*

**not****). And this narration stays the same, across the complete collapse risk distribution (i.e., the vertical axis).**

*red* We again show with a **black star** how last week compares with the prior 2015 series to date. We see that last week's -12% change for the SSE is here** **completely **inline **with the rest of 2015 (and also within range of 2007/2008)! Unlike for the U.S., last week's loss in China wasn't their 2015 worst ** nor** 2

^{nd}worst (those even worse months were earlier this summer when the SSE begun to crash). Also, here one

**fully state that 2015 (regardless of how the rest of the year turns out) is a trend reversion to the risk we experienced during the global financial crisis, since the**

*can***2007/2008 risk statistics nicely overlap the**

*red***2015 statistics.**

*blue***colors are also mostly completely more severe than the**

*Both***2009 through 2014 risk statistics shown above! We might see these articles (here, here) for idea generation on future month's SSE risk and whether it might continue to be high.**

*entire* We will further accommodate those unwavering in their false position that there is a ** broad** mathematical relationship between both of these countries' time series (and using China's proximate market burst as a pretext to interpolate back history). See the raw monthly plot below, contrasting both indexes. We see that last week's (still highlighted with a

**) joint losses for the U.S. and China are mostly a shock**

*black star***the 2015 context (**

*within***),**

*blue***but not as much for China.**

*for mostly the U.S.* We see that ** either** the correlation of individual time periods, or of all of these time periods combined (so ignoring the time series colorings), does not exist

**as a routine matter.**With markets, there will always be one-off exceptions (see this Top Article in Pensions & Investments); our goal with this article is to simply present a framework for high-level risk analysis. The overall correlation doesn't exist, even with the one mad,

**shown for October 2008 (-20% SSE and -14% S&P). This data without context**

*worst weekly joint-loss***been considered an outlier. And the 2015-only correlation between China and the U.S. also doesn't exist, even though this year has the most probabilistic**

*should have***potential**for it, as the variance among the SSE is extraordinarily high (this is referred to as the

**in probability language). Lastly, we can collectively respect that the joint losses were more severe in 2007/2008 (**

*sum of squares***) then they are this year (**

*red***+**

*blue***).**

*black star* We are not breaking new theoretical ground in this article since that's not required. The mathematical rigor of these relationships have already been recorded in these articles, sorted by order of consequence: here, here, here, here. It is worth noting that at some point one may want to reallocate to the risky market. Clearly no one should have been 100% stocks a week ago (particularly high ** ?** stocks). Someone was buying stocks a week ago, and many were selling in fear through yesterday (August 21). One might want to instead try that in reverse to make money (buy at a discount and sell at a premium). We should also note that the developed markets was subjected to record-setting wealth annihilation at the end of last week. It is wise to be

**now, with the given**

*carefully attentive***market volatility.**

*global* We can all call attention to nervous economic data, but there are also some core measures (GDP, employment, etc.) showing the U.S. economy is not in chaos. While possible, it is not the expectation that we should expect risk statistics to be worse than the 2007/2008 measures from the global financial crisis. What makes a market is having differing opinions at nearly all times. It is therefore educational for people caught off guard last week to see -** once more**- that markets can drop at a savage

**speed**(as opposed to the overall

**magnitude**), regardless of whatever foggy economic situation we are in (or market participants believe them to be).