Tail event
You asked me what’s my pleasure:
A movie or a measure?
I’ll have a cup of tea
And tell you of my dreaming.
- —Blondie, Dreaming (1979)
Tail event
/teɪl ɪˈvɛnt/ (n.)
Statistics: Of a range of possible independent events, one whose frequency is three or more standard deviations from the mean. An event with a low probability.
Work life: An unwanted outcome you didn’t expect, to which you weren’t paying attention, and, therefore, for which you don’t think you should be blamed.
The randomly distributed marketplace
Amarket, in the abstract, looks like what Nancy Cartwright calls a “nomological machine”. A simplified model of the real world having defined boundaries and simplified operating conditions: a finite trading day, a limited number of market participants and a defined set of fungible financial instruments with which one participants engage in a limited range of transactions, whose outcomes deterministically set observable prices for that set of traded instruments, bearing numerical relationships with previous traded prices for the same instrument (in that they will be higher, lower, or the same).
The world these instruments represent is intractable. It does not have boundaries, even similar “instruments” are not fungible, the range of possible events that can occur to them in undefined.
“A portfolio of asset-backed securities cannot,” a commodities trader would say, “suffer water damage. They do not rust.”
Not having to deal with rust, water damage, and manufacturing defect simplifies the business of investing. The effects of these events are supposed to play out in the information layer , and translate efficiently into the prices at which related instruments trade. If an oil company’s tanker is wrecked, it's share price declines.
It is tempting to infer information from price: to put a drop in the market to “unexpectedly soft non-farm payroll data”. Many people make a living reading tea-leaves in this way.
From this price information we can derive a relationship between transactions — price went up, price stayed the same, price went down — and a trend. A trend is a stab at extracting a signal from the noise.
The signal depends on a theory of the game, Otherwise the “relationship” between the two discrete transactions is arbitrary. Without a theory, everything is noise.
The theory-dependence of signal
If events are truly “independent” — in a first order sense, they are: the participants in the later trade do not know who or where the participants in the earlier even are, let alone what their motivations for trading were — then any “trend” we draw between them beyond their distribution is, more or less, meaningless. All that is left is mathematics.
But we have a theory, so we draw the line all the same. We assume the market is homogeneous, that all participants have similar price information — those who have more are forbidden to trade — and that all are propelled by the same rationale: you don’t sell things you expect to do well, and you don’t buy things you expect to do badly.
Private narratives wash out
Given these assumptions, across the market investors’ private motivations, opinions, theories and idiosyncrasies cancel out — they are like the Brownian motion of molecules in a nice hot cup of tea. They are reversions to the entropic mean; baseline white noise — so we can disregard them. Which is just as well for the complexity of our models. Until it isn’t.
Put another way: although the “interconnectedness” of similar transactions means they do not have the quality of independence that normal distributions require, most of the time it’s close enough: the information is chaotic — as traders say, “noisy” — in the immediate term, here the dissimilarities between trader motivations are most pronounced, but over a large aggregation of trades and a longer period a “signal” emerges. This is what Black-Scholes, volatility and convexity models track: as long as all traders all use the same aggregated market information — and the market works hard to ensure they do — a “normal” probabilistic model[1] works fairly well. It’s not a bad model.
We treat professional market participants as a largely homogenous group from which emerges, over time, a signal. Almost like, you know, like an invisible hand is guiding the market.
This is good: it gets our model out of the gate. If investors were not broadly homogeneous, our probability models would not work. “The average height of every item in this shed” is not a particularly useful calculation. Which way the causal arrow flows — whether signal drives theory or theory determines what counts as a signal — is an open question.
But there is a second-order sense in which the earlier and later trades are related, in practice: the later participants know about the earlier trade and its price — it is part of that universal corpus of market information, deemed known by all, it informs price formation process: all can thereby infer the trend from prior trades — and use this abstract information to form their bid or ask.
Nomological machines never quite work in the real world
When you bounce a ball, friction, energy loss, structural imperfections, impurities in the rubber and environmental interference frustrate the conditions needed to satisfy the “nomological machine”: the required conditions for Newton’s laws to hold are not present so, when our bouncing ball never quite conserves momentum, we let it pass. It is close enough and usually no one is counting in any case.
This is the sense in which, as Nancy Cartwright puts it, the laws of physics lie. They don’t represent what happens in the real world.
The same applies to the statistical techniques we use to measure market behaviour. Much of the non-homogenous behaviour cancels itself out. Where it doesn’t — where it creates a persistent variance from how a normal distribution would behave over time, we can model that, too, with measures like volatility. We use probabilistic — that is, independence-assuming —techniques to model these second-order corrections like volatility, too.
Why do we assume independence and homogeneity of events? Because otherwise, we could not predict at all. A human being with free will and moral agency does not obey laws of probability. She can put a coin down heads up every time. She can go out of her way to deliberately frustrate any prediction or suggestion her husband another person makes.
“Oh, you predicted heads? Well, I say tails.”
It’s not just that individual humans can do that: they like doing that. Likewise, you can’t draw models that predict the behaviour of dissimilar objects. Statistical rules require homogeneity. The odds of rolling a six hold true for fair dice, but not for carpet slippers or fish.
But this is the magic, so claimed, of big data. All those idiosyncrasies cancel themselves out and leave us with a set of basically homogenous participants. You night not like rice pudding or lentils but over a whole population, a fairly reliable proportion of the population does. We can ignore individuals. The variances they represent are noise. It is our dystopian lot that our institutions and social systems increasingly are configured to ignore us.
BRIAN: “You’re all individuals!”
CROWD: “Yes! We’re all individuals!”
BRIAN: “You’re all different!”
CROWD: “Yes! We’re all different!”
(small voice at the back): “I’m not.”
- —Monty Python’s Life of Brian
Our agency and our idiosyncrasies average out. We all want to eat, be warm and dry and have rewarding careers. That we all go about this in subtly different ways doesn’t, to a data aggregator, much matter. Until it does.
For there is a third order of dissimilarities. In times of market stress, other people’s behaviour directly and directionally affects you and your transactions, and your behaviour affects theirs. This is not the irrationality of panic — if each decision were irrational, the effect would be random and the Brownian cancellation effect would come into play and everything would be fine — but an instinctive imitation of whatever it is the surrounding community is doing. THOSE GUYS ARE RUNNING AWAY. I DO NOT KNOW WHY BUT I MUST PRESUME THEY HAVE A REASON. THEREFORE I AM RUNNING AWAY.
This is “memesis”. Most of the time, thanks to the Dunning-Krueger-by-proxy[2] effect or otherwise, we presume the perspective we can bring to the information we have gives us an edge over the crowd, and we are happy to make our own decisions, whose individual variances boil off into Brownian randomness that can be neatly fitted to a standard deviation from the mean. But there are moments — by nature unexpected — when that confidence vanishes. Suddenly our conscious models, theories and nomological machines are less valuable than the tacit information we gather from the changed behaviour of everyone around us. There is something important we don’t know. It is better to mimic the behaviour of those around us. We presume they know — or that they are imitating the behaviour of someone else who knows.
This is the extraordinary behaviour of fish when a shark bursts through the school. This is the bewitching murmuration of starlings over a twilight meadow. In an instant that entropic, Brownian normalcy disappears and every particle darts the same way at once, as if by magic.
We are mesmerised but not surprised to see starlings perform their aerial magic. We would be gobsmacked if a cup of tea did this.
When the planet has unexpectedly gone into lockdown as a result of a global pandemic, buying habits for toilet paper and, oddly, lentils suddenly change. The fact that there are only three tins of lentils left on the shelf leads you to grab them. The fact that there are none leads to a nationwide run on tinned pulses people don’t, in normal times, much like. The Contrarian household still groans under the weight of tinned borloiti beans years after the last new variant.
There are not just these “cry fire in a crowded theatre” effects whereupon everyone stampedes for the exits at once, but second-order effects. You might not wish to head for the exit: you might be strong-willed enough to rise above the madding crowd — but you might still have no choice. You are not independent when your asthma inhaler is in your spouse’s rucksack.
If you are long “on margin” you might wish to ride out a sudden correction by meeting your margin calls. In most dislocations this is the obvious and — if you can manage it, correct — thing to do. You might, per your own books and records, be solvent, well-capitalised and in good standing with your banks, so why not?
But meeting the margin call means drawing on your standby revolving credit facility — you don’t keeps a yard of spare cash off the table for emergencies, right? — but it turns out your bank is, like everyone else, suffering a liquidity squeeze. It evokes some obscure market conditions CP buried in the docs and suspends drawdowns on the RCF as a result. This is nothing to do with you: the bank is managing its own cash position. It needs the money more than you.
At the same time your margin lenders — usually so patient with you, generally genial, good for a knees-up at Ascot and tolerant of peripheral looseness in your margin operations — have had a sense of humour failure. They are apologetic, but they are shipping a shower of grief from the head of risk and have been told to tell you that you today there is no flex. Today the money must be there on time without fail — and for good measure they are jacking up your IM.
You say this is absurd, that everything is fine, but appeals to their better nature and your solid, five-year track record fall upon deaf ears. Today they don’t know what to believe. Normal conditions of trust and amity are suspended. This could be the final round of the prisoner’s dilemma.[3] Anything they can’t see unaided with their own naked eyes could be fake news. The one thing they can see is that everyone else is running for the door.
The value of all that near-perfect market information evaporates and other information, which the market doesn’t have, but until now took for granted — such as the essential viability of systemically important financial institutions and the strength of the commercial imperative — is suddenly much more important. All at once, no-one fancies “taking a view” on anyone’s credit.
Cash is King, Queen, Jack and Ace. There are dazed people in sharp suits wandering around Canary Wharf clutching Iron Mountain boxes.
All indicators are going one way, across the board, in all markets and all asset classes.
Now we find our model has stopped being largely right, or broadly right, or even vaguely right. It is flat-out wrong.
Twenty-five sigma events
If a coin lands tails a hundred times in a row it is either a unique moment in the life of the cosmos or a dicky coin.
If you are the CFO of a bulge bracket Vampire Squid you will earn limited sympathy if you blame your losses on a statistical model, but absolutely none if you blame it on the misbehaviour of the universe. Do not say things like:
“We were seeing things that were 25-standard deviation moves, several days in a row”
Twenty-five sigma events do not happen once, let alone several days in a row. Your model did not work.
This is a tail event. This is what all the meaningful terms in your legal agreements are designed to protect you against.
See also
References
You asked me what’s my pleasure:
A movie or a measure?
I’ll have a cup of tea
And tell you of my dreaming.
- —Blondie, Dreaming (1979)
Tail event
(n.)
- Statistics: Of a range of possible independent events, one whose frequency is three or more standard deviations from the mean. An event with a low probability.
- Work life: An unwanted outcome you didn’t expect, to which you weren’t paying attention, and, therefore, for which you don’t think you should be blamed.
The randomly distributed marketplace
Amarket, in the abstract, looks like a nomological machine. There is a bounded environment, a finite trading day, a limited number of market participants and a defined set of financial instruments with which one can engage in a limited range of transactions, whose outcomes will set the price for the traded instrument, which can be easily compared with the last traded price for that instrument (in that it will be higher, lower, or the same).
From this information we can derive a relationship between transactions — price went up, price stayed the same, price went down — and a trend. A trend is a stab at extracting a signal from the noise.
The signal depends on a theory of the game, Otherwise the “relationship” between the two discrete transactions is arbitrary. Without a theory, everything is noise.
The theory-dependence of signal
If given events are truly “independent” — in a first order sense, they are: the participants in the later trade do not know who or where the participants in the earlier even are, let alone what their motivations for trading were — then a “trend” we draw between them is, more or less, meaningless. All that is left is mathematics.
But we have a theory, so draw the line all the same. We make assumptions about the homogeneity of all market participants: we assume all have similar price information, and that all are propelled by the same essential economic rationalism: you don’t sell things you expect to do well, and you don’t buy things you expect to do badly.
Private narratives wash out
Each investor’s private motivations, and opinions, may be nuanced and personal — how is the rest of its portfolio positioned, what local risks is it especially sensitive to — but these idiosyncrasies cancel out in a large sample — they are like the Brownian motion of molecules in a nice hot cup of tea. They are reversions to the entropic mean; baseline white noise — so we can disregard them. Which is just as well for the complexity of our models. Until it isn’t.
Put another way: although the “interconnectedness” of similar transactions means they do not have the quality of independence that normal distributions require, most of the time it’s close enough: the information is chaotic — as traders say, “noisy” — in the immediate term, here the dissimilarities between trader motivations are most pronounced, but over a large aggregation of trades and a longer period a “signal” emerges. This is what Black-Scholes, volatility and convexity models track: as long as all traders all use the same aggregated market information — and the market works hard to ensure they do — a “normal” probabilistic model[4] works fairly well. It’s not a bad model.
We treat professional market participants as a largely homogenous group from which emerges, over time, a signal. Almost like, you know, like an invisible hand is guiding the market.
This is good: it gets our model out of the gate. If investors were not broadly homogeneous, our probability models would not work. “The average height of every item in this shed” is not a particularly useful calculation. Which way the causal arrow flows — whether signal drives theory or theory determines what counts as a signal — is an open question.
But there is a second-order sense in which the earlier and later trades are related, in practice: the later participants know about the earlier trade and its price — it is part of that universal corpus of market information, deemed known by all, it informs price formation process: all can thereby infer the trend from prior trades — and use this abstract information to form their bid or ask.
Nomological machines never quite work in the real world
When you bounce a ball, friction, energy loss, structural imperfections, impurities in the rubber and environmental interference frustrate the conditions needed to satisfy the “nomological machine”: the required conditions for Newton’s laws to hold are not present so, when our bouncing ball never quite conserves momentum, we let it pass. It is close enough and usually no one is counting in any case.
This is the sense in which, as Nancy Cartwright puts it, the laws of physics lie. They don’t represent what happens in the real world.
The same applies to the statistical techniques we use to measure market behaviour. Much of the non-homogenous behaviour cancels itself out. Where it doesn’t — where it creates a persistent variance from how a normal distribution would behave over time, we can model that, too, with measures like volatility. We use probabilistic — that is, independence-assuming —techniques to model these second-order corrections like volatility, too.
Why do we assume independence and homogeneity of events? Because otherwise, we could not predict at all. A human being with free will and moral agency does not obey laws of probability. She can put a coin down heads up every time. She can go out of her way to deliberately frustrate any prediction or suggestion her husband another person makes.
“Oh, you predicted heads? Well, I say tails.”
It’s not just that individual humans can do that: they like doing that. Likewise, you can’t draw models that predict the behaviour of dissimilar objects. Statistical rules require homogeneity. The odds of rolling a six hold true for fair dice, but not for carpet slippers or fish.
But this is the magic, so claimed, of big data. All those idiosyncrasies cancel themselves out and leave us with a set of basically homogenous participants. You night not like rice pudding or lentils but over a whole population, a fairly reliable proportion of the population does. We can ignore individuals. The variances they represent are noise. It is our dystopian lot that our institutions and social systems increasingly are configured to ignore us.
BRIAN: “You’re all individuals!”
CROWD: “Yes! We’re all individuals!”
BRIAN: “You’re all different!”
CROWD: “Yes! We’re all different!”
(small voice at the back): “I’m not.”
- —Monty Python’s Life of Brian
Our agency and our idiosyncrasies average out. We all want to eat, be warm and dry and have rewarding careers. That we all go about this in subtly different ways doesn’t, to a data aggregator, much matter. Until it does.
For there is a third order of dissimilarities. In times of market stress, other people’s behaviour directly and directionally affects you and your transactions, and your behaviour affects theirs. This is not the irrationality of panic — if each decision were irrational, the effect would be random and the Brownian cancellation effect would come into play and everything would be fine — but an instinctive imitation of whatever it is the surrounding community is doing. THOSE GUYS ARE RUNNING AWAY. I DO NOT KNOW WHY BUT I MUST PRESUME THEY HAVE A REASON. THEREFORE I AM RUNNING AWAY.
This is “memesis”. Most of the time, thanks to the Dunning-Krueger-by-proxy[5] effect or otherwise, we presume the perspective we can bring to the information we have gives us an edge over the crowd, and we are happy to make our own decisions, whose individual variances boil off into Brownian randomness that can be neatly fitted to a standard deviation from the mean. But there are moments — by nature unexpected — when that confidence vanishes. Suddenly our conscious models, theories and nomological machines are less valuable than the tacit information we gather from the changed behaviour of everyone around us. There is something important we don’t know. It is better to mimic the behaviour of those around us. We presume they know — or that they are imitating the behaviour of someone else who knows.
This is the extraordinary behaviour of fish when a shark bursts through the school. This is the bewitching murmuration of starlings over a twilight meadow. In an instant that entropic, Brownian normalcy disappears and every particle darts the same way at once, as if by magic.
We are mesmerised but not surprised to see starlings perform their aerial magic. We would be gobsmacked if a cup of tea did this.
When the planet has unexpectedly gone into lockdown as a result of a global pandemic, buying habits for toilet paper and, oddly, lentils suddenly change. The fact that there are only three tins of lentils left on the shelf leads you to grab them. The fact that there are none leads to a nationwide run on tinned pulses people don’t, in normal times, much like. The Contrarian household still groans under the weight of tinned borloiti beans years after the last new variant.
There are not just these “cry fire in a crowded theatre” effects whereupon everyone stampedes for the exits at once, but second-order effects. You might not wish to head for the exit: you might be strong-willed enough to rise above the madding crowd — but you might still have no choice. You are not independent when your asthma inhaler is in your spouse’s rucksack.
If you are long “on margin” you might wish to ride out a sudden correction by meeting your margin calls. In most dislocations this is the obvious and — if you can manage it, correct — thing to do. You might, per your own books and records, be solvent, well-capitalised and in good standing with your banks, so why not?
But meeting the margin call means drawing on your standby revolving credit facility — you don’t keeps a yard of spare cash off the table for emergencies, right? — but it turns out your bank is, like everyone else, suffering a liquidity squeeze. It evokes some obscure market conditions CP buried in the docs and suspends drawdowns on the RCF as a result. This is nothing to do with you: the bank is managing its own cash position. It needs the money more than you.
At the same time your margin lenders — usually so patient with you, generally genial, good for a knees-up at Ascot and tolerant of peripheral looseness in your margin operations — have had a sense of humour failure. They are apologetic, but they are shipping a shower of grief from risk management and have been told to tell you that you today there is no flex, today the money must be there without fail —and for good measure they are jacking up your IM. You tell them this is absurd, that everything is fine, but today they don’t know what to believe. Normal conditions of trust are suspended. This could be the final round. Anything they can’t see unaided with their own naked eyes could be fake news. The one thing they can see is that everyone else is running for the door.
The value of all that near-perfect market information evaporates and other information, which the market doesn’t have, but until now took for granted — such as the essential viability of systemically important financial institutions — is suddenly much more important. All at once, no-one fancies “taking a view” on anyone’s credit.
Cash is King, Queen, Jack and Ace. There are people on the TV in sharp suits wandering dazedly around outside their buildings clutching Iron Mountain boxes.
All indicators are going one way, across the board, in all markets and all asset classes.
Now we find the model we were using has stopped being largely right, or broadly right, or even vaguely right. It is flat-out wrong.
Advice. If you are the CFO of a bulge bracket Vampire Squid you will find there is limited tolerance for blaming your performance on a statistical model an absolutely none for blaming it on the essential misbehaviour of the universe, as if your model was right but the atoms and molecules were wrong. Do not say things like:
“We were seeing things that were 25-standard deviation moves, several days in a row”
This was not a twenty six sigma event. Your model did not work.
This is a tail event. This is what all the meaningful terms in your legal agreements are designed to protect you against.
- The market is generally diversified. If you carve out all the personal idiosyncrasies — stochastic modelling requires — that might explain why one rational person is prepared to sell what another is prepared to buy, it stands to reason that a buyer’s gain is a seller’s loss. In a diversified market, a sudden collapse in value for some traders means an appreciation for others, and all kinds of other effects. See: Brownian motion
- No individual participant, or group of participants with correlated interests dominate the market
- Information about the market, and any “crowded” positions in the market, is widely held. Of course individual positions are private, proprietary and confidential, so this last condition is usually satisfied by the general assumption of liquidity: in a sufficiently deep market, no-one is big enough to have such a concentrated position, so we can take it as a given that no-one does. (In some markets there are materiality thresholds over which positions must be reported too.)
But in the modern market, where scale and leverage are so important, these are not always safe assumptions. Lenders only know what they know.
Derivatives trading
In the context of trading derivatives, things that (a) you didn’t reasonably expect and that . (b) bugger up your contract.
Credit defaults
A swap being a private, bilateral affair, the most obvious category of tail events is “things which mean your counterparty cannot, or will not, or has not, performed its end of the deal”.
Straight out refusal to — repudiation — is rare, at least without the cloak of some kind of dispute as to whether the party was under such an obligation in the first place.
Inability is the main player here: generally captured by insolvency, and correlative defaults under other agreements.
Much of financial services being a play on leverage — the name of the game being to earn more, with other people’s money, than it costs you to borrow it — many market participants flirt with various formulations of insolvency as a basic business model, so there tend to be some pushback on the parameters of these correlative failures and “ostensible inabilities” to perform. Much of a negotiator’s life is spent haggling about them.
Where refusal or inability to perform cannot be proven, actual failure to pay or deliver ends all arguments. If you actually haven’t performed, it no longer matters why.
There is therefore a sort of hierarchy of these events. Actual default is the safest, and most common, default trigger. Bankruptcy is the next — though there is more looseness around some of its limbs, an administrator actually being appointed, or a petition actually being filmed is clean, public and unlikely to prompt many arguments. Default Under Specified Transaction — that transaction being one to which you are directly a party,
The remaining events are sketchy and unpopular, depending as they do on private information you most likely won’t have about thresholds you can’t easily calculate. We may argue till we are hoarse about Cross Default. We will not invoke it.
Externalities
There are a category of events which make it impossible even for a solvent counterparty to perform. Change in law, for example — it is not beyond possibility that certain kinds of swaps might be restricted or outlawed altogether[6] or Tax events that make the transaction uneconomic as originally envisaged.
Secondary events of this kind — things that limit a dealer’s ability to hedge, or materially increase its costs of doing so, tend not to be Termination Events partly this reflects a fact not often stated, but nonetheless true: there is a price at which the parties will agree to terminate any swap. Just because a party doesn’t have an economic option to terminate the trade doesn’t mean it can’t terminate the trade. It always has an “at market” option. In liquid markets during times of fair weather this is a source of great comfort; in illiquid markets and at times of stress, less so. A dealer will say, “I will always show you a price. You just might not mind the price, is all.”
Customers have less incentive to break trades if it means realising
See also
References
- ↑ I am working hard not to use the intimidating term stochastic” here by the way.
- ↑ I just made this up but it seems, for reasons I cannot now articulate, like a good and possibly profound idea. Possibly that reason is that I suffer from Dunning-Krueger-by-Proxy Syndrome
- ↑ According to game theory it is rational to cooperate in non-zero sum games as long as you expected them to repeat. If you expect them not to repeat, it is rational to defect. This is the traitor’s dilemma.
- ↑ I am working hard not to use the intimidating term stochastic” here by the way.
- ↑ I just made this up but it seems, for reasons I cannot now articulate, like a good and possibly profound idea. Possibly that reason is that I suffer from Dunning-Krueger-by-Proxy Syndrome
- ↑ Not long ago the European Union proposed restricting the carbon market to “end users” to discourage financial speculation, for example. This would have rendered certain forward contracts in Allowances involving delivery to non-users illegal.