Tail event

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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.)

1. 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.
2. 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 participants engage in a limited range of transactions, whose outcomes deterministically set observable prices for that set of traded instruments, which prices bear quantifiable relationships with previously traded prices for the same instruments (in that they will be higher, lower, or the same).

But the real world these instruments represent is intractable. It does not have boundaries. It is inchoate, and our knowledge about it less so: The “instruments” of the real world are not “fungible” — 2 shipments of the same commodity have indefinable idiosyncratic impurities and characteristics — abd the range of possible events that can occur to physical commodities in undefined in a way that the range of events that might occur to financial instruments is not.

“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 or manufacturing defect simplifies the business of investing in the financial instruments. 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, its share price declines.

It is tempting to infer information from price: to put a drop in the market down to “soft non-farm payroll data”. People make a living reading tea-leaves in this way.

From price information we can derive a relationship between transactions — the price went up, stayed the same, 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” then any “trend” we draw between them beyond their distribution is, more or less, meaningless. In a first order sense, market events are independent: the participants in the later trade do not know who or where the participants in the earlier are, let alone what were their motivations for trading. All we have is a theory and some mathematics. But we draw the line all the same.

We make assumptions: the market is homogeneous; all participants have similar price information; all are propelled by the same rationale. No trader sells things she expects to do well, nor buys things she expects to do badly.

Private narratives wash out

Given these assumptions, individual investors’ private motivations, opinions, theories and idiosyncrasies cancel each other out, so we can disregard them. They are like the Brownian motion of molecules in a nice hot cup of tea: reversions to the entropic mean; baseline white noise. This is just as well, because otherwise our models would not work. We can ignore individual sentiments because they don’t matter. Until they do.

Put another way: although the “interconnectedness” of similar transactions means they do not have the quality of independence that a normal distribution requires, most of the time they pretty much do: information is chaotic 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 works fairly well. It’s not a bad model.

So 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 gets our model out of the gate. If investors were not broadly homogeneous, our 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 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. 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

• Nomological machine
• The purpose of an ISDA
• The map and the territory