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The Value-At-Risk Fiasco
Submitted by Salil Mehta of Stiatistical Ideas
The Value-At-Risk Fiasco
If you were a looking at a simple portfolio that was a mix of 1 unit S&P 500 and 1 unit Shanghai Stock Exchange (SSE), then you are likely to consider value-at-risk (VAR) to feel cozy with your overall portfolio risk. This measure however is not considered a coherent risk measure that satisfies all of the properties of interest: monotonicity, translation invariance, homogeneity, and subadditivity. We'll explain the first three in a future article, but only focus here on how VAR violates the last of these four properties.
Subadditivity is where the risk associated with multiple holdings, in a portfolio, should not be greater than the sum of the individual holdings' risk. This construes the hallmark of diversification, and yet combined with the inappropriateness of VAR to measure market risk we see subadditivity levels violated. Risk events that should have only happened say one month every 1.5 years have occurred in each of the past three summer months.
VAR (invented at J.P. Morgan well before both the global financial crisis and their entertaining London Whale drubbing) is an expression of the largest possible loss, contained within a specified confidence interval. We can for example explore the history of worst weekly losses in the S&P, for each month starting more than 5 years ago in January 2010 (and through May 2015). A total of 65 months. We can set a probability tolerance of just over 6%, and state that the probability of seeing a loss greater than this VAR should be less than or equal to ~6% (or 1 in 16 months). We can vary this level about, but this is simply a foothold to initiate our analysis.
VAR in this case, for the S&P, would come to a worst weekly loss of 6.0%. Bear in mind that the average worst weekly loss over the 65 months for the S&P was a 1.9% loss. Now we do the same exercise for the SSE, and with the same probability tolerance of ~6% we get a VAR loss of 5.3%. Here the average worst weekly loss over the 65 months for the SSE is a 2.6% loss. Note that the parametric mathematical relationship to estimate the overall VAR from blending two equally volatile stocks (or indexes) does relate to the correlation between those 2 indexes.
VARoverall
= VARindex?[(1/number of indexes) + (1-1/number of indexes)r]
= VARindex?[½ + ½r]
The reason it seems as if the VAR for SSE is set at an easier loss level, versus that for the S&P, is a reflection of the nonparametric nature of equity returns. This turns out to be critical as we go through this article. Also keep in mind that these VAR express a fixed week period, and not any continuous range beyond that. We know the maximum-VAR was higher if we simply augment to the trading week, ending August 21, the following "Black Monday". And most also know by now, that these stock returns are not related to the normal distribution (or elliptical distributions for that matter), and now we also see these fat tails are not even related to one another. For more on the Student t distribution, see here, here.
The number of months where the worst weekly losses should exceed the VAR over 65 months is no more than 3 (roughly an integer <6%*65). For the S&P, the changes have been (in order of severity): -6.6%, -6.8%, and the worst weekly loss before the summer of 2015 was -7.5%.
For the SSE, the changes were: -5.3%, -6.6%, and -6.9%. The 3 months associated with the S&P above, and the 3 months associated here with the SSE, have one month in common (May 2010). The four bolded months of the six months noted (3 S&P and 3 SSE) are part of the worst 3 joint, "worst weekly losses". We show these 3 joint losses below, where again the portfolio constitutes 1 unit S&P and 1 unit SSE (for a portfolio that is 50% in both indexes you would take ½ of every loss and VAR for the purposes of comparison):
-7.5% + (-2.8%) = -10.3%
(-5.2%) + -6.9% = -12.1%
-6.6% + -6.6% = -13.2% (this is May 2010)
(-5.2%) + -6.9% = -12.1%
-6.6% + -6.6% = -13.2% (this is May 2010)
Note values in () are the paired worst weekly loss for the bolded S&P or the SSE worst weekly losses.
Again the portfolio theoretical VAR should have been no greater than the sum of the two holdings' VARs. Or 11.3% (6.0%+5.3%). And empirically we see above that the portfolio VAR comes to smaller than a 10.3% loss.
Does this make everything feel good about your portfolio risk measures? Can you take comfort with the VAR model and diversification, and feel comfortable that you would only see a <10.3% loss level once out of every 16 months? It's attractive, but look at what immediately happened next over the following 3 months of the summer:
June 2015: -0.7% + -14.3% = -15.0%
July 2015: -2.2% + -12.9% = -15.1%
August 2015: -5.9% + -12.3% = -11.9%
In each of these 3 months, the S&P always stayed within VAR yet the overall losses still were always greater than the 10.3% VAR (and all were greater than the theoretical 11.3% VAR for that matter!) A 1 in 16 months event immediately happening 3 months straight is not a quirky <0.02% (6%3) probability situation. It was a case of incorrectly using VAR as the preferred nonparametric risk measure for the market we are modeling (e.g., "extreme" tail risk events). Despite how commonly it is endeared anyway by investors and middling stress testing regulators.
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most boring post ever!
Owww, it's too early for my head to hurt like this, and I didn't drink last night. Statistics hurts my brain box q:B
And then the Blac Swan appeared out of nowhere ...
"It was my understanding that there would be no math."
From the Saturday Night Live sketch, Presidential Debate in 1976 (Chevy Chase played Gerald Ford).
https://vimeo.com/65921206
First, we kill all the eggheads.
Math works to describe the real physical world, not the manipulated banskter world.
"Statistics are for losers" --- jerry stovall
After a 10% sampling, I concluded this article was 90% horse shit.
Translation: FIGURES LIE AND LIARS FIGURE.
Duh.
You mooks!
Money for nothing...
https://www.youtube.com/watch?v=lAD6Obi7Cag
Horrible abuse of statistical inference.
Stocks are highly correlated and never subject to genuinely independent co-variant analysis.
Fool is wasting his time. Should not waste ours.
and when things go to shit, bondz and PMz all go to shit and there goes "diversification."
Pointing out the myriad shortcomings of VAR is always good. But I think that the explanation could have been simplified given the audience it hand -
VAR exists as measurement to help Executives, Regulators, Sheeple Investors, and other non professionals sleep better at night, it does NOT actually measure the value that "should" be at risk during a given interval.
Speaking of intervals, only going back to 2010 for both markets (and thus only incorporating data on negative price fluctuation during a bull market, really doesn't tell one anything about the expected price volatility to their long positions in bear market. Had they gone back to 2007-- the numbers would be different, but that would be a more appropriate, but much less frequent use of a very limited statistical (excuse) tool.
4 ounces of coconut rum in your coffee should straighten you right up!
That, or wearing a traditional Scottish kilt on a cold winter's morning.
If not, certainly the most boring post since Lehman.
And then u have a guy with 50 leverage who repackages it for 1% cause it's risk free ..... That's where we are. Thanks for sharing this post, will be happy to see the next ones
article has good points to express, but needs to be broken down for general zh consumption.. its not like the average zh reader is a dummy, so it mostly comes down to translating the jargon and summarizing each sub-theme before and after the details are gone into.
Um... what goes up must come down?
Is that a spade, or a non-mechanized, ferro-lignous excavation device?
It is a fucking shovel.
Well, is it a spade or a shovel?
I need to know if I'll be digging in or cleaning up.
Surely was a long, contrived way to get exactly where common sense already took most of us.
Modeling a manipulated market is more worthless than a 3 dollar bill with Putin's portrait on it. But it provides jobs for fresh MBA graduates, and worthless and pointless research, and worthless, recondite articles on Zero Hedge.
"But it provides jobs for fresh MBA graduates, and worthless and pointless research".
And 'opportunities' for community organizers to become Imperator?
Putin isn't an enemy of the American people.
The American 'government'....??
So JP Morgan's metric was just something they pulled out of their arse?
Well, if the Fed can pull fiat out of its arse, why can't JPM come up with an analysis from the same place?
the FED has decreed with ZIRP/NIRP there is no tradeoff between investment return and risk (for those who are deemed credit worthy and can still invest as they have a net positive worth).
So capitalism has NOW become a RISKLESS exercise in the context of QE/ZIRP, but for the rich.
All the more so that you be super rich. As if you be super rich, you can LEVERAGE that at NO COSt (thanks to your uber gold card status in banksta world) and earn while you sleep thanks to moar riskless scams of HFT services provided to you as a big hitter on the banksta network (and thus mega generator of their commissions/bonuses).
If you be poor you are not invited to the banksta table. You only go to the public sector, where you are told there is no well paid job for you, that your taxes to pay for growing public debt will always be more than what you'll earn. You are a perpetual net negative worth denier. You drown in debt and its not Zirped.
Brave new World. And you don't need probability and complicated math to understand that.
Its guaranteed and poured in stone. Until the system implodes like Icarius's wings. Even the tarmac melts under the mid-day sun of El Nino. Its whats under the carpet that denies that mantra and defies its premises. El Nino of financial world lurks hidden.
Free money? Hahaha, never yours the Oligarchs !
Very understated way of pointing out that VAR as it is currently (and extensively) used in financial risk management is an absolutely useless way of measuring risk. Jamie Dimon convincingly (but probably without realizing it) proved that VAR was a useless metric for risk when he observed earlier this year that the bond market was making moves that should only occur every 3 billion years.
http://www.bloomberg.com/news/articles/2015-04-08/dimon-says-once-in-3-b...
So why is VAR allowed by regulators? Because *any* legitimate measure of risk would show that banks and other financial institutions have excessive risk on their books. When even the emperor admits his nakedness, perhaps the regulators should pay attention.
The markets are based to a large part on faith?? And belief in a system that will give you an edge--- this is just a reach for the holy grail.
The truth is that all these new systems and measures have been "invented" and cannonized during the big bull markets of the last several decades and have little to do with reality. But they have been great for bonuses cause they prove the worth of the mba and phd crowd.
The article was pseudo-mathematical mumbo jumbo.
Economics is pseudo-scientific mumbo jumbo.
Probability & statistics are great; I use them all the time in my trading algorithms. Having siad that, though, there comes a point in time where "tail risk" appears and at that point all bets are off. It's like being at the "event horizon" of a black hole; you may think you have an idea, but in reality nobody has a clue. It's the same phenomena when in Las Vegas you walk past a roulette table where "19" has hit 7 times in a row; once in a 6 billion year event playing 24/7/365. There's a reason it's called "tail risk"; when you are losing your ass(ets), probability isn't much consolation.
www.traderzoo.mobi
The largest possible loss is always 100% - you can always lose ALL of your money in the market.
And if you are trading with leverage - you can potentially lose MANY TIMES the amount of your original investment. Any theory that advocates 6% as the likely maximum loss - is total nonsense.
ALL theories about probable losses are built on certain ASSUMPTIONS. But you know what they say about the word "ASSUME". It makes an ASS out of U and ME!
Do you really think that all of these derivatives transactions will net out? Really, in a world with losses that are real and growing? And a huge number of players who are highly leveraged, with high debts?
They must be joking if they believe in these "Quant Fantasies".
the largest loss is greater than 100% with leverage. then there's the banruptcy court and garnished wages or a 10 for 1 split and a huge tax payer bail out by the tax payer in the form of a compulsory rights issue on people who held no shares.
then of course you can borrow money at negative rates because draghi will do whatevr it takes to keep his gravy trained for job.
it's like a turkey, after having seen all the benevolence of his farmer family, asking what is the possibility of his getting slaughtered right before Thanksgiving
Doesn't this come under the heading:
“If you can't dazzle them with brilliance, baffle them with bullshit.” ? W.C. Fields.
Impossible to determine how much of the above spate presupposes Gaussian distributions of returns and random uncorrelated events, but the references to the t-distribution are ominous and tend to lead one to the conclusion that the post is as much use as a large discharge of pigeon shit on my BMW upholstery.
I had the great pleasure of attending a John Tukey lecture where he told a room full of the world's top statisticians that tails and limit distributions were utter nonsense (did not go over well, but he was THE MAN, so he could say it). He invented the Fast Fourier Transform and coined the digital term "bit," and was great proponent of the Jacknife as opposed to fitting nonsense distributions. He was very practical, probably a result of his experience at Bell Labs putting a more practical spin on his Princeton professorship. Take tails and distribution fitting with grain of salt, ...just passing along the wisdom of Tukey. He was a very cool guy.
Agree. The popular assumption of normally distributed errors/ residuals and the assumption of certain probabilities being associated with events that are so-and-so-many Gaussian sigmas removed from the Gaussian mean are more difficult to eradicate than herpes.
My head hurts now.
Can I sue someone?
zzzzzzz
I'm curious to know if Tyler understood what he posted.
This just in: the morgue lies through their false teeth.
LOL...
1oz Silver American Eagle just €13 @ EurGold
https://www.eurgold.eu/silver/american-eagle-1oz-silver-coin-1-dollar-le...
Its the normal return fallacy, prices don't behave like molicules in a closed container. Humans (& machines programed to trade) display herding behaviors. This give arise to extreme events.
VaR is for Vools.
It works only when you do not need it any child can understand.
In summary: It is the grand casino played 24/7 all around the world and the house always wins and the game is over with the last index falls.
“Past market performance is no guarantee of future outcomes.”
Possibly the greatest cover-your-ass fine-print disclaimer of all time, considering it is used primarily by those who claim to be able to manage their own risk based on past performance.
Whoever came up with it first should get an award for best con artist of all time.
Eat your heart out Ponzi.
What the fuck?
VAR has been known to be shit for at least 2 decades. As a tutor in Econometrics (Quant Economic Policy/Applied Econometrics/Financial Econometrics) in the 90s, part of the material I taught was why VAR is not even useful as a rule of thumb - and especially not if the distribution of returns is assumed Gaussian: that shit makes the maths easier to teach (and easier to examine) but has no practically-relevant real-world implementation. Fat tails have been a thing since the late 80s at the latest, so anybody who thinks that writing about them is revelatory, is a fuckwit.
Wanna know how behind the times this article is? Even MBA courses - whose content is roughly at a level that is comprehensible to the mid-performing second-year undergraduate - teach it as a historical relic that should not be deployed operationally. Wharton teaches that in its Foundation unit on FE&RM - that is, in the material you have to master before you get to enrol.
So this article sounds the alarm on a thing that has had no operational usefulness for almost half my life... it has the time-relevance of a health warning about Thalidomide.
It's pretty clear that the writer thinks that he has found the financial sector's methodological Achilles' Heel - and in doing so betrays an ignorance of even the basic pedagogical models in financial econometrrics, financial engineering and risk management.
This shit is what happens when engineers, mathematicians, statisticians and CompSci majors arbitrage across to finance: it is a very popular path (and 'quants' mystify the C-suite types, most of whom are innumerate MBAs). But it leads to fucking disaster because they do not understand how financial markets work - and once they are in a role they don't have time to learn anything except the veneer.
It would be as silly as me thinking that because I have first-rate quant training, I could design bridge-building software without the faintest understanding of engineering - maybe I'll do a 4-week 'professional' course once I start in the bridge-building game. (In fact designing bridge-building software would be easier, because the feedback mechanisms are far more predictable than human behaviour).
It's like fucktards who think that 'supply-demand analysis is far too simplistic' is a valid critique of economics as a discipline - as if economists literally believe the first-year pedagocical model ... the straight-line static S/D 'X-diagram'. Every well-trained economist knows that demand (and supply) schedules can't be linear, that tatonnement takes place over time, and that equilibrium is an attractor at t < T (and even then only when all shocks have ceased).
hahaha...well said...i was just thinking that an assumption that a the height of a big bell curve of best fit is a repeatable constant, or the height of a lot of little bell curves in the left or right hand tails or edges ot the curve can predict risk is a bit like assuming everyone must wear a certain size shoe at a certain age and sex. shoes manufactured to this premise will only fit a single individual by pur coincidence (rather alike a stopped clock is right twice a day).
mind you, there is opportunity here, as most tradeable options are priced using some variant (deviant?) of estimated probability based on past (co-)incidence ad plot a whole bunch of time and distance (and by inference, VaR's) via volatility "surfaces". If you come up with arbitrage opportunities using the flaws in VaR and market pricing of risk, you will be a weathy man (though probably worn out like bitcoin miners)
anyway, well said.
Since nodes on vol surfaces are limited by the intervals of moneyness and maturity, and every other tenor or moneyness is interpolated, there is probably a basis difference between actual and the interpolated value. I'm sure someone in Susquehanna is looking into it as I type. Statistical arbitrage.
I think Taleb has shown that since price movements are not normally distributed, but everyone prices options as if they were (using Black Scholes or similar), the only real option strategy is to buy lots an lots of very cheap, deep out of the money puts (at a strike of 0.5 or less), and wait for the inevitable black swan, while praying you stay liquid long enough to collect.
You would be surprised at how many banking institutions still do their regulatory capital measurements with VaR. Although, they tend to use the 97.3 confidence interval, and the Basel II weighting for internal model risk measures is (I think, from memory) around 2.3X.
Some institutions use greeks calculated by the front office to do Delta-Gamma VaR. But that's OK, you can always trust numbers from the trading desks. It's not like their bonus depends on profits after the cost of capital (VaR) is deducted. (ha ha).
Other institutions have their risk department build their own pricing models independent of the front office, and they use these to calculate HSVaR. Options are still priced with some variant of Black-Scholes, even though it is well accepted that financial asset price movements are not normally distributed, and volatility is an observation, not a forward estimate.
At any rate, you can always change your model to fit your backtest. As soon as something unexpected happens, you're fucked. Everyone has a plan until they get punched in the face.
should change the website name to "statisticals ideaLs"