I am hearing reports of a ten-sigma event in Brazil

Those are one year nominal interest rate futures, basis points on the vertical axis, and this was the result of political troubles relating to another impeachment.  The image is courtesy of AJ.

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The equity market was down like -10% on Thursday. It's been pretty crazy.

Econ novice/not in the field. What am I looking at here?

Not in the field either, but what you're looking at is the daily variation of the nominal (one-year) interest rate. See the big red vertical line on the right, which climbs up to 135%? That means that in just one day (yesterday I suppose), the interest rate has more than doubled. That's an enormous variation, which dwarfs the variations during the crisis of 2008 and during the destitution process of Dilma Rousseff one year ago, as the graph also shows (the other big red lines are not half as big). This indicates big febrility, volatility, worries from the investors.

Not really, it means 1y rates went up by 135bps. In this case, from roughly 8.7% to 10.0%+.

Ah, so, it's base point changes, not percent changes. Thanks.

I made the same mistake and I AM (sort of) in the field!

I'm from Brazil. What happened this day the president who is making pre-market reforms, was caught in a film where he asked for bribe, and received it. Now media, people wants his impeachment

This is what happens when a President breaks the law and is caught.

Trouble was that in fact what happened was a newspaper printed a story saying the president was caught on tape asking for a bribe, but the tapes were released the following day, and the literality is not there: what was said was for the businessperson to go talk to his other guy about what he needed, and that other guy asked for the money, therefore not being as bombastic as expected. There are also suspicions it was edited.

Troubles for Mercosur: Argentina, Paraguay and Uruguay. Bolivia too.

Yeah, we just found the economic Higgs boson

In USD terms the equities were down ~16%

The 3x levered etf was down ~45%

Note that market movements aren't normally distributed (they almost always have fat tails). So a 10-sigma event odds isn't 1-in-a-bazilion (1-in-a-Brazilian?). It's more like a once-every-ten-years event.

He realizes that and is being Straussian. If you click on the image, its name is "Mandelbrot."

What does "ten-sigma event" mean in this context? In sciences it would mean confidence level of an observation; i.e. that a statistical fluke explaining the observation would appear only as rarely as a value from the Gaussian distribution ten standard deviations away from the mean. I don't see how this is applicable here.

If it means that this even has a probability of happening lower than a value ten sigmas away from the mean of the Gaussian distribution, I would say that you are using the wrong probability distribution to estimate the likelihood of the event. I'm pretty sure that we have never witnessed a process that draws values from the Gaussian distribution daily experience a "ten sigma event".

it means that the observed event is, like, 10 standard deviations away from the mean.

"Sigma" is simply short-hand lingo for "one unit of standard deviation". When we say some value is "X sigma" it means that the difference between it and the distribution average is X times the distribution's standard deviation. It's true that Gaussian distributions have standard deviations, and are the most common and used distributions. As such, a lot of time "X sigma" is implicitly translated to being "Y% likely", based on the quantile from the Gaussian.

However, many many other classes of known distributions also have standard deviations. Just because we say that this is a "10-sigma" move, doesn't mean that we're implying that the distribution is Gaussian. Most other distributions have "fat-tails", i.e. large sigma draws are much more likely than the Gaussian. For some classes of distributions and large sigmas, this can be orders of magnitude more. So, just because we observe a 10-sigma move, doesn't mean that our assumptions or models are broken. (Unless an explicit part of that model is that the distribution is Gaussian). The only fundamental limit on how unlikely an "X-sigma" draw is Chebyshev's inequality. That says that an "X-sigma" draw can only occur with probability 1/X^2. So, the only fundamental limit, is that a 10-sigma one-day move should not occur more than 2-3 times a year on average.

Finally, when you're talking about "X-sigma" with regards to confidence levels, that doesn't mean anything fundamentally different. What we're explicitly referring to is the distribution of the measured value of the random variable. Here the distribution in question is the standard error of the estimated value. The sigma-value represents the difference between the measured value, and the null-hypothesis threshold (usually zero). If I say something like, "X effect exists at the 6-sigma threshold", that implies that result could only be a false positive if the measurement error drew a 6 standard deviation difference from its mean.

What I had in my mind in the second paragraph of my post was that many distributions don't have standard deviations. For instance any distribution that vanishes like 1/x^n for any n <= 2 is such a distribution. Of course if we want to model a real world phenomenon, all the distributions, strictly speaking, have standard deviations because even the exponential tails are cut off at some limit. However, without knowing this limit value, we can't estimate the real standard deviation of the distribution from data very easily.

"Standard deviation" is a property that can be computed for any set of numbers, and does not necessarily refer to any underlying probability distribution.

Yes, but saying "today's even was 10-sigma" is comparing the movement today to the historical data, which only makes sense if you're claiming they are all drawn from some common probability distribution. In contrast, if the standard deviation of historical rainfall in the Sahara was 0.01 inches per day, it wouldn't make any sense to say "there was a 100-sigma rainfall in the Amazon yesterday" as a short-hand for 1 inch of rain, because they obviously aren't drawn from a common distribution.

The issue is that the probability distribution is non-Gaussian, not that there's no distribution at all.

You can indeed calculate this number for any set of data. However, for many probability distributions from which we can draw these sets of data the number is completely meaningless. For example, if the distribution has a tail that vanishes like 1/x for large x (up to some cutoff value perhaps), a single event can, if it occurs at the tail, change the computed value completely. The fact that a "10 sigma event" can occur within the lifetime of the human species tells me that distribution where these values come from is indeed one with heavy tails.

Right. The Gaussian is based on the function e to negative x *squared* which will obviously get vanishingly small very quickly. I punched it into R out of curiosity and the norm function wouldn't even calculate that far out. For 8 sigma, I get 6.66x10^-16. The commonly heard "six sigma" is 9.87x10^-10 which is about 1 in a billion (for a single tail).

It means everyone has an amp that goes to ten, but Tyler has one that goes to eleven.

What is represented by the green marks?

The bars are green when there's a drop (note that they're always below the 0). They are red when there's an increase

' I would say that you are using the wrong probability distribution'

Generally people use the gaussian informally/wrongly simply to indicate the (arbitrary) extremeness of a move. Sometimes reality takes a back seat to mathematical tractability and/or a 'tidy' explanation.

Most learned decades ago that returns are not (log)normal (hopefully)

Thiago, Thiago, where are you when we need you?

Busy being impeached, drinking a caipirinha while dancing samba on the beach :D

Seriously!! One of my favorite posters. Has single handedly started to get me interested in Brazil over the past year. Would love to hear his two cents

1 - Evidently, there is some corruption around, but it is basically under control. It is not like, say, Clinton selling pardons, other Clinton's lectures or Lewis Libby's obstruction of justice.
2 - President Temer has denied any corrupt dealings and, after listening to the records, said they could prove nothing. "The labor gave birth to a mouse", he said to his aides.
3- Only two minor-to-medium political parties out of the 30+ parties have left Mr. Temer's ruling coalition so far.
4 - Anti-Temer street protests on Sunday have failed.
5 - Brazil's popularios remains calm and united under the correct guidance of its legitimate leaders.

Correction: Brazil's population remains calm...

Thiago, you are seriously downplaying the level of corruption that permeates all through Brazilian business and politics. Americans can't even begin to imagine the corruption of Brazil. I lived there for 14 years, so I have a pretty good idea. Americans should think of countries like Nigeria or Venezuela.

This is wild exaggeration and anti-Brazilian propaganda. There is nothing in Brazilian history like America's Jim Crow or oppression of the Filipinos or Hiroshima. Brazil has never waged a war of aggression or sought territorial aggrandizement of any nature. Brazil's moral ground is the highest possible one. As for corruption, most Brazilians are honest, we demand it from all. Corrupt people are prosecuted and jailed. The president and all living former presidents are under federal investigation. More than one-third of the Congressmen are being investigated. Some former Representatives and Senators are already jailed. Three former treasurers of former ruling party PT are jailed alredy. The better part of Brazil's GDP is behind bars. Brazil is undertaking the most thorough political investigation since the Stalinist Purgues. No evil-doer is safe. It is not like America, where no one can trust the leaders because half the country is sure he is taking everything not nailed to the floor (I still remember what people said about Clinton 1, Bush 2 and Clinton 1.5 - not to mention Trump). In Brazil, as the title of a movie recently released has it, The Law is For Everyone. For me, for thee, for President Temer, for multibillionaire Marcelo Odebrecht, for everyone.

Wow, what a silly reply. I thought you must be writing some kind of satirical parody.

Maluf stole over a billion dollars from the city of Sao Paulo; he sent it to Switzerland; the Swiss banks sent all the evidence to the Brazilian Attorney General. Maluf never served time for this and was subsequently the most voted for congressman in the next election. And I know this for a fact because my colleague and I were the translators of the AG's 150-page report for the international community.

"So notorious is Maluf's reputation that in Brazil the verb 'malufar' was created, meaning 'to steal public money' "

I guess you never heard of Tammany Hall and other American political-criminal organizations.
The extant of Mr. Maluf's malfeances are politically exagerated, there is no reason to believe he has stolen more then a few dozen million dollars. Again, such criminal activities are deeply uncommon in Brazil and usually fiercely punished by the authorities as people much richer and much more influentil than Mr. Mluf have found out.

My grandmother used to talk about Tammany Hall.
I don't know what Brazil you are talking about, but it sounds like one that must exist in a parallel universe, if at all.
The extant of Maluf's crimes cannot be exaggerated. The Swiss banks pierced the veil of bank secrecy and sent 40 boxes of evidence to the Brazilian Attorney General. He even sent as much as $400 million in a single day to Switzerland. This is all a matter of public record.
Why are you so interested in spreading obfuscation about Maluf?

I just did a google search and it came up with an article that today the Brazilian Supreme Court has sentenced Maluf to over 7 years in prison for money-laundering. I suppose he will appeal the conviction until the day he dies in luxury: http://plus55.com/brazil-politics/2017/05/deputy-paulo-maluf-prison-money-laundering

There is no obruscation. Again, it has been shown the Brazilian system fights corruption and dles not spare even knemof the most important politicians - again, lots of top politicians, public workers and businesspeople are behind bars. There are no Clintons and Bushes in Brazil. Law is for everyone.
There is no good reason to believe Mr. Maluf has stolen mire than a few millions. Compare and constrat with former governor Cabral, whose criminal group stole at least a half billion dollars. He has veen jailed. Yesterday, two former governors whose criminal groups is said to have stolen almost one billion dollars were arrested. We behave in a implacable way against corruption.

Driving his Honda Civic to his accountant office job in Dayton OH

I do not own a Honda, I do not work at an accountant office and I do not work in Ohio. I do doubt anything interesring ever happened in Dayton.

Ha!

https://en.wikipedia.org/wiki/Wright_brothers

IT IS A LIE! The airplane was invented by Brazilian Santos Dumont! He demonatrated his invention for all to see and donated his plans for the well-being of Mankind. Meanwhile the Wright Brothers kept hiding the invention they intended to sell to foreign powers as a murderous weapon until they could steal Dumont's idea!

Such a fraud is even more interesting than the invention of the airplane, so you lose, sir.

No, it is not. Swindlers in America are a dime a dozen. Adamski, Joseph Smith, Ponzi, Madoff, P. T. Barnum, the Wright Brothers.

I'm not Thiago but I was also born in Brazil. What is happening here is the unravelling of =~ 80 years of institutionalized corruption. Basically, the whole system is corrupt but pieces are falling at random intervals. The current President (Temer) was part of the ticket with Dilma (who was impeached) so he obviously was part of the same corruption schemes. It would have been better for the country that all of them fell at once, but logistically it doesn't work like that. Now, in the meantime, Temer was trying to buy time (and connections) by imposing some desperetely needed reforms to a country that has one of the most left-leaning constitutions in the world. The market reacted well to that, and now that Temer is about to fall, the market is reacting bad to that. This cycle will continue until a new crop of politicians takes power. However, the old rulers will not give up easily. Lula (Dilma's predecessor who is also being investigated and will eventually go to jail) is promoting a movement to call for new elections. That way he can run and become president again, which gets him some judicial privileges which could postpone his arrest.

So yeah, this is not close to the end. Only when the next generation takes over things will improve (according to current trends, the chosen one should be Joao Doria, current mayor of Sao Paulo).

Stipulating to all of that, what is special about today? Everything I know that seems reasonably probable today was reasonably probably last week,.

You never see this kind of corruption in Uruguay.

In a just world, Uruguay would just take over Brazil already.

There is no Uruguay, it is Brazil's Cisplatine Province.

There is no Brazil, just a colony of Portugal.

Its strikes me that sigma as a measure of volatility is losing its significance for various reason. HFT is very good at lower volatility 90% of the time and massively raising volatility a small amount of the time. So he probabilities assocated with a sigma event are not very accurate anymore.

I am hearing reports of a ten-sigma event in Brazil

Isn't that supposed to be an event with one chance in a trillion trillion of happening? Someone is using the wrong statistical model. Again.

All models are false. A 10 sigma event proves that a Gaussian model is false; thus it proves nothing.

Temer may well be innocent, and so well may be Lula, although he has been charged. I gather there are now some possible charges against Dilma, but it should be kept in mind that although the driving force for her impeachment was the general state of corruption in her government, which had also been going on in previous governments, including those prior to Lula's, she was not impeached because of any personal corruption on her part, and it looks if anything like Temer and many members of his cabinet are/were more corrupt personally than she was/is.

She was impeached for something that was not illegal when she did it, but was made illegal after the fact, something that many of her predecessors had done, and which was not corruption, per se, although arguably bad fiscal practice. She was impeached for misestimating and stating details of fiscal policy that ex post turned out to be different than what it was claimed they would be ex ante. There are many in Brazil, although not our Brazilian commentators here apparently, who think that impeaching her on the basis of something that was only made illegal after she did it was extremely unfair, and that what was involved was essentially a parliamentary coup. She is not personally popular, so the unhappiness about this shows up in the popularity of Lula, who might well win an election if indeed one were to happen in the near future.

Note that this is not a comment on the pros or cons of Temer's proposals, which are not nearly as free market as some here would like, if undoing some things done under Lula and Dilma. More a reminder that most people outside of Brazil have a serious misconception about what went down with Dilma's impeachment and how it is widely viewed in Brazil.

She knew very well her Treasure (Fazenda) Minister managed to miss nine out of eight inflation, GDP and revenue forecasts. She knew, even as she was running for re-election on her and Lula's record of prosperity and low unemployment rates, that the rhing could hold umril the elections, but no further. (To be fair, Fernando Henrique, who got re-elected in 1998, also knew he could not keep the supervalued Real for much more time and Sarney had to know the prices freeze would only hold until the Constitution Convention, which his party taking most seats, was elected - Dilma's trick is an old one). And it is hard to argue that the creative account her people was engaged in did not break the Fiscal Responsability Law. Other governments did such things, but with more restrain. But, yeah, like the Nazis or Ceaucescu or the Gang of Four or Nixon, she was punished for losing the war, not because (all) her enemies were better people than she is (President Lula's administration had its share of scandals, but the economy was booming, so who cares? ). The same MAY happen to President Temer unless he be able to show that he ismpart of the solution, not of the problem. If you do not stand the hear, leave the kitchen.

But, Thiago, is it not true that this law was passed ex post, that she was impeached for breaking a law that was not a law when she broke it?

There is (was) lively discussion regarding it. The Public Ministry (something like a Public Prosecutor Office) declined to consider Mrs. Rousseff's actions a crime. However, the TCU (the All-Union budget-overseeing court) dexided otherwise, though. They decided that Mrs. Rousseff's actions were in violation of LC101, Article 36 from 2000, which says no state bank can lend money to the instance of power (the federal executive power in this case) that controls it. To prevent a deficit from showing at the accounting, the first in years, Mrs. Rousseff's government used state bank's money in lieu of taxes revenues. She was not the first to do that, but she was the greatest offender.

1 - Was there legal basis for impeaching her?
Just enough to make it not a coup tout court.

2 - If things were going well as they seemed to go under Rousseff's sponsor Lula, would she have ben able to get away with it?
Yes, for sure.

3 - Did people who voted against her or protested against her really cared about those tedious accounting gimmicks?
Mostly, no. As Al Capone, she was charged with what her enemies thought they could make stick. Although there is good reason to think she was part of Petrobras' woes - she was on the company's board when billions and billions were stolen - and new information provided by criminals seeking lighter sentences seems to indicate she got hundreds of unregistered millions dollars for her campaigns), those investigations would take too much time to be at a point they could use against her, but were already at a point they could taint her through her close association with known criminal (three treasurers of her party are or were at jail).
4 - Would it be more decent to have waited the corruption investigations?
By all means and it was my favorite option. No one asked me then, though. A caveat is, it is not clear the country would have survive months of radical polarization and a president everyone (even her supporters) thought was not up to the task and most people thought was morally compromised. She got the worst popular support rate numbers I have ever heard of.
5 - In conclusion, Brazil is a land of contrasts and Mrs. Rousseff was the perfect Rorschach Test: right wing people hated her and wanted tooverthrow her, left wing people despised her (because she is no Lula) and wanted to keep her (and them themselves) in charge. Non-aligned people looked from the government to the opposition and back and asked themselves what they have done to deserve such a dilemma.

Sorry, "extant" should be "extent."

Okay... I'm Brazilian too. If you guys from comment section want to understand Brazil just read Thiago's claims and make it the opposite, Glad to help.

Thanks for chiming in, Zamba. I don't know what that guy's story is, but something is motivating him to lie.

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