Haute Lumière · The reading

The models keep missing the crash, and nobody changes the models

What finance borrowed from physics, and why a market behaves less like a machine than like a forest.

She has been on the same page for some minutes. The argument is not difficult; it is simply not finished.

THE OLD MACHINE

In the autumn of 2008 the people holding the risk models were not fools. The models had been built by capable quantitative teams, reviewed by committees, blessed by regulators, and backtested against years of price history that they fitted rather well. Mortgage-backed securities had prices, the prices had volatilities, the volatilities had confidence intervals. Then the housing market turned, and the models were not slightly wrong. They were wrong about what kind of thing a market is.

Alan Greenspan, who had spent two decades as the most quoted man in American finance, put the failure into a single sentence afterward: "I made a mistake in presuming that the self-interest of organizations, specifically banks and others, was such that they were best capable of protecting their own assets." It is a more precise admission than it is usually given credit for. He had assumed that prudence at the level of the individual firm would aggregate into stability at the level of the system. Most of the time it does. That is exactly what makes the assumption so durable, and so expensive when it fails.

A model does not fail when its arithmetic is wrong. It fails when the world it assumes is not the world outside the window.

Strip the standard apparatus down and three assumptions are holding it up. Agents optimise, with information good enough to act on. Shocks arrive independently of one another and distribute themselves along a bell curve. Institutions are related to each other but not welded together, so a loss at one can be treated as a loss at one. Each of these is defensible on its own, on a quiet Tuesday, in a market that is not moving. Together they describe a machine: inputs go in, outputs come out, and the relationship between them is stable enough to be estimated.

What stood on Wall Street that September was not a machine. Lehman Brothers did not fail and stop; it failed and propagated. Banks that had no subprime exposure at all found they had exposure to banks that did, and then lending froze between institutions that were all, individually, solvent on paper. The interesting question is therefore not why the models failed, but why they had worked for so long beforehand. The answer is that calm markets really do resemble the assumption, and the resemblance breaks down precisely when a model is most needed. An instrument that is accurate except during the event it exists to measure is not an inaccurate instrument. It is the wrong instrument.

LOOPS AND FLOWS

There is an older vocabulary that fits the object better, and it comes from biology rather than mechanics. A living system is any network whose parts interact strongly enough that the behaviour of the whole cannot be read off from the behaviour of any part. A forest. A coral reef. An ant colony, in which no individual ant holds the plan for the nest and the nest is nevertheless built. The claim of this book is not that markets are poetically like ecosystems. It is that they satisfy the same structural description, and that the description carries mathematics with it.

Six words do most of the work. Stocks are the things that accumulate and sit: money supply, loan portfolios, household debt, accumulated capital, and less tangibly, the stock of expectation that participants hold about what happens next. Flows are the movements between stocks — lending, repayment, investment, wages, interest. A stock changes only through a flow, which sounds obvious until one notices how often a policy is aimed at a stock with no account of the flow that would have to move to change it.

Feedback is what makes the arrangement interesting. A rise in a stock alters a flow, and the altered flow alters the stock again. When the loop amplifies, it is positive: a stock price rises, which draws buyers, which raises the price. When it dampens, it is negative: a central bank raises rates into an overheating economy, borrowing slows, the heat comes off. Neither sign is a moral judgement. Positive feedback builds every boom and also every recovery; negative feedback stabilises and also strangles.

Coupling measures how tightly the parts are bound to one another. Two banks that lend to each other are coupled; two that do not are not. Tight coupling transmits shocks quickly and widely, which is why an interbank market that makes the system efficient on ordinary days makes it fragile on extraordinary ones. Emergence is what appears at the level of the whole and exists nowhere in the parts — a panic, a bubble, a run, a trend. No trader holds a bubble. The bubble is what a great many traders holding ordinary positions add up to.

The sixth word is the one that changes the ambition of policy. Antifragility names systems that do not merely survive shocks but improve under them, in the way that a bone remodels denser after a fracture or a forest clears deadwood in a fire. Some parts of finance already behave this way. Venture capital is built on the assumption that most investments fail and that the failures are the price of information. The question a policymaker inherits from this vocabulary is not how to prevent shocks, which is not available, but what a system that metabolises them would have to look like.

THE SIMPLEST CURVE

Almost every sentence in the paragraphs above can be written down. That is the point of writing them down: a metaphor that cannot be made arithmetic stays a metaphor, and no one can test it. The smallest honest starting point is the logistic equation, which describes anything that grows quickly while it is small and slowly as it approaches a limit.

dX/dt = rX(1 − X/K). Four symbols. X is the current size of the thing — assets, population, adoption. dX/dt is how fast it is changing right now. r is the intrinsic growth rate, the speed it would grow at with nothing in the way. K is the carrying capacity, the ceiling imposed by saturation, resources, or the size of the available market. The whole of the interesting behaviour lives in the bracket. When X is far below K, the bracket is nearly one and growth is nearly exponential. As X approaches K, the bracket approaches zero and growth stalls.

Put money in it. A company holds a million dollars in assets, grows at twenty percent a year unimpeded, and operates in a market that can support ten million. In the first year the calculation is 0.2 × 1,000,000 × (1 − 1,000,000/10,000,000), which is $180,000. Update and repeat: assets of $1,180,000 produce 0.2 × 1,180,000 × (1 − 1,180,000/10,000,000), which is $208,152. The growth rate accelerates, and then — this is the part the founder never puts in the deck — it turns over and decelerates towards zero without anything having gone wrong.

Run the same equation on rabbits, which is where it came from. A meadow holds a hundred, the intrinsic rate is 0.5, and twenty rabbits are present. The first year's growth is 0.5 × 20 × (1 − 20/100), which is 8. Same arithmetic, same shape, entirely different subject matter. This transferability is the first quiet argument for the whole approach: a differential equation does not know whether its X is a rabbit or a balance sheet, and the behaviours it produces — saturation, overshoot, collapse — turn up wherever the structure turns up.

Two systems with the same equation have the same future, however little they have in common.

One term of negative feedback was enough to convert unlimited exponential growth into a curve with a ceiling. That is the general method. Complexity in these models does not come from complicated equations; it comes from simple equations wired into each other, where X is a function of Y and Y is a function of X, and the pair produces behaviour that neither would produce alone. A model is built the way a circuit is built, and the surprises arrive from the wiring.

The plant was here before the desk was. Nothing in this room grows on a schedule anyone set.

THE FAT TAIL

Take daily price changes for a large listed company and plot them. Over a week the line looks like noise. Over several years a shape appears, and it is not the shape most finance courses teach. The bell curve says that moves far from the average are vanishingly rare — a five-standard-deviation day should not be seen in a working lifetime. Real market data produces such days regularly enough that traders have slang for them.

The technical name is fat tails, and the mathematics that fits them is not Gaussian. The Lévy stable distributions are described by a characteristic exponent, written α, that controls how heavy the extremes are. A normal distribution corresponds to α = 2. Many financial assets come in nearer 1.5 to 1.8. The gap between 1.8 and 2 looks like a rounding error and is not: it is the difference between an extreme event being a once-in-a-career curiosity and a recurring feature of ordinary operation.

The same signature appears elsewhere in the system. Trading volumes, the sizes of firms, the number of connections a bank has to other banks — these tend to follow power laws, where the probability of finding something of size x falls off as x raised to a negative exponent. Power-law systems have no meaningful typical member. The largest city is not somewhat larger than the second; the largest bank is not somewhat more connected than the next. Averages, which are the workhorse of ordinary statistics, describe almost nothing in such a population.

Econophysics — the import of statistical mechanics into finance — offers a mechanism as well as a description, and the mechanism is worth more than the fit. Self-organised criticality is best told through a sandpile. Grains are added one at a time. Most do nothing. Some cause a small slide. Occasionally one causes a collapse down the whole face, and the grain responsible is indistinguishable from every grain that did nothing. The pile organises itself towards the critical state, where it is perpetually one grain from an avalanche of any size.

A market that has been calm for a long time is accumulating grains. Positions get larger, leverage ratios drift up, correlations tighten because everyone has arrived at the same trade by different routes. Nothing looks wrong, because nothing is wrong in any individual position. Then a piece of news that would have been absorbed without comment three years earlier arrives, and the slide takes the whole face. This reframes the search for a cause. Asking what caused a flash crash is like asking which grain caused the avalanche: answerable, and almost entirely uninformative about the next one.

THE DOMINO

Systemic risk has a definition worth keeping precise: the possibility that the failure of one institution triggers the failure of others, so that a loss the system could easily absorb becomes a loss it cannot. The word systemic points at the connections rather than the institutions. Two banks, each perfectly capitalised, can constitute a fragile system if the connection between them is arranged badly.

Here is the arrangement, small enough to hold in the head. Bank A has assets of $100 million, liabilities of $90 million, and therefore equity of $10 million. Bank B has assets of $80 million, liabilities of $70 million, equity of $10 million. Both are well inside their regulatory limits, and any examiner looking at either balance sheet in isolation would pass it. Bank A has also lent Bank B $10 million, which sits as an asset on A's books and a liability on B's.

Now a single borrower of Bank A defaults on a $2 million loan. Assets fall to $98 million, liabilities stay at $90 million, equity drops to $8 million. A twenty percent reduction in equity is painful and survivable. But Bank A, needing to restore its position, calls in what it is owed — which means demanding repayment from Bank B. Bank B now has to find $10 million, which it does the only way it can, by selling assets or calling in its own loans. If it sells into a falling market it realises losses, and its own equity falls.

A capital buffer measured one institution at a time tells you nothing about the system those institutions form.

Follow the chain one link further and the arithmetic stops being reassuring. Bank B's forced sales depress the prices of exactly the assets that Bank C holds, so Bank C marks down, so Bank C's equity falls, so Bank C calls in loans. Each step is prudent. Each institution is doing what a well-run institution should do when its capital position deteriorates, and the sum of those individually correct decisions is a spiral. The 2008 crisis is this diagram at scale, with more nodes and a great deal more leverage.

This is why network analysis has become a core instrument rather than an academic curiosity. Represent each institution as a node, each exposure as a weighted edge, and structural questions become computable. Which nodes have the highest degree centrality, and are therefore most exposed to shocks arriving from elsewhere? Where are the bridges connecting otherwise separate clusters, along which a local failure becomes a general one? Acemoglu, Ozdaglar and Tahbaz-Salehi, and before them Allen and Gale on financial contagion, gave this work its formal spine. The practical finding is uncomfortable and important: the same interconnection that distributes small shocks efficiently also transmits large ones, and the crossover between the two regimes is sharp.

MANY MINDS

Traditional macro models represent a population by its average — a representative household, a representative firm. The convenience is real and the cost is specific: averaging away the differences between participants averages away the mechanism. Panics happen because participants differ. If everyone held identical beliefs there would be no trade, since every buyer needs a seller who reads the same facts differently.

Agent-based modelling keeps the differences. Each participant is represented separately, with its own capital, its own risk tolerance, its own rule for deciding. Then the population is set running and the modeller watches what the interactions produce. Nothing about the aggregate is specified in advance; the aggregate is the output. Farmer and Foley argued in Nature in 2009 that economics needed this method for precisely the events the standard apparatus had just missed.

The rules can be simple and still generate a great deal. Give a population three types. Fundamentalists price on earnings and cash flow, and buy what looks cheap against those. Chartists price on the recent path, and buy what has been going up. Noise traders act on rumour, sentiment and impulse. Each agent computes an expected return adjusted for its own appetite: expected return equals the market return minus its risk tolerance multiplied by perceived volatility. An agent with a tolerance parameter of 0.2 looking at a market return of 0.08 and a volatility of 0.15 arrives at 0.08 − 0.2 × 0.15, or five percent, and trades if that clears its personal threshold.

Now let news arrive — a rate rise, say. The fundamentalists reprice and sell the assets most sensitive to borrowing costs. The chartists observe the resulting decline and read it as a trend, and sell into it. The noise traders see two groups selling and sell harder. Nobody in this population is irrational, and nothing in the rules mentioned crashes, and the output is a crash. That is emergence produced in a laboratory rather than asserted in a paragraph.

The method has an equivalent in a much smaller setting. A Saturday farmers' market contains an apple grower worried about a new orchard, a cheesemaker building relationships with restaurants, teenagers optimising for price, a parent optimising for nutrition per dollar, and a baker who sells out by noon on reputation alone. Ask what the market will do next season and no average captures it, because the answer depends on who responds to what. Kirman made a version of this point with ants choosing between two identical food sources, which is one of the most cited results in the field.

The discipline that keeps this from becoming a video game is calibration. Agent rules must be grounded in observed behaviour, parameters fitted against real data, and the model made to reproduce historical episodes it was not fitted on before any weight is placed on what it says about the future. A simulation that can be tuned to produce any outcome has told you about the tuning.

A calculation put down mid-line, late in the afternoon. The number will still be there when she comes back to it.

REFLEXIVE GROUND

There is a complication in all of this that has no counterpart in physics, and it deserves naming before the practical chapters. Sand does not read the sandpile literature. Markets do. A model that becomes widely believed changes the behaviour it was built to describe, and the change can run in either direction.

Soros called this reflexivity and built a career on it. Participants act on their understanding of the market; their actions become the market; the market then furnishes the evidence on which the next understanding is built. Believing that house prices always rise causes lending against that belief, which supplies the money that makes prices rise, which confirms the belief. The reasoning is circular and the money is not, and the circularity can run for years before the underlying capacity to repay asserts itself.

Minsky built the same observation into a full account of the cycle, and the shape of it has become hard to unsee. A long period of stability persuades participants that the environment is safe. Safe environments justify more debt against the same income. More debt raises returns while conditions hold, which further confirms the safety, which justifies more debt. Stability, in this account, is not the opposite of crisis. Stability is what produces the conditions for crisis, on a schedule set by how long it lasts.

A market that has been calm for a decade is not a safe market. It is a market in which nothing has been tested for a decade.

Reflexivity closes off one hope that people reach for when they first meet complexity mathematics: that a sufficiently good model would let policy see the crash coming and step aside. It would not, because publishing the model changes the population. This is not a counsel of despair, and treating it as one is the error to avoid. It is a specification change. It rules out prediction as the goal and puts something else in its place — structures that hold up across many futures rather than forecasts that are right about one.

Kindleberger and Aliber, cataloguing four centuries of manias and panics, found the same sequence recurring in tulips, railways, Florida land and dot-com equities, under different technologies and different regulatory regimes. What recurs is not a price level or a trigger. What recurs is the structure: credit expansion, a plausible new thing that justifies it, reflexive confirmation, a peak at which the marginal buyer has been used up, and a reversal that runs faster than the ascent. The pattern is robust precisely because it lives in the wiring rather than in the subject matter.

THE GARDENER

If prediction is off the table, what does a policymaker actually do on Monday morning? The answer this book gives is a practice rather than a doctrine, and it has five steps that hold whether the object is a national economy or a pension fund.

First, state the objective in terms that could be measured. Stable growth, inflation within a band, a retirement income that survives a bad decade — a goal that cannot be checked cannot be adapted towards. Second, identify the variables that actually move it and the relationships between them. A pension fund's outcome depends on rates, inflation, volatility and longevity, and the interactions between those four matter more than any one of them in isolation.

Third, build the monitoring before the intervention. This is the step most often skipped, and skipping it converts the whole approach into ordinary policymaking with better vocabulary. For an institution it means economic data collected on a short cycle, stress tests run with feedback loops written in rather than as single isolated shocks, and early-warning indicators watching credit spreads, volatility and trading volumes for the signatures that precede an amplifying loop. Modelling a twenty percent fall in house prices is a stress test. Modelling a twenty percent fall that triggers defaults that produce bank losses that force sales that depress prices further is a stress test of the system.

Fourth, design strategies that can be adjusted rather than strategies that must be right. A rate path announced as a commitment and a rate path announced as a response to named indicators are different instruments even when they produce the same first move. Fifth, iterate in public. Evaluate what the intervention did, distinguish what it did from what would have happened anyway, and change it. The willingness to revise is the whole mechanism, and a policy regime that treats revision as an admission of failure has removed its own steering.

The image the book returns to is a gardener rather than an engineer. An engineer specifies a machine, builds it to specification, and expects the specification to hold. A gardener observes, intervenes lightly, observes the response, and intervenes again, holding a clear intention about the garden and very little illusion about control over any particular plant. The gardener's advantage is not superior foresight. It is a shorter loop between acting and finding out.

The measure of a policy is not whether it was right. It is how fast the system tells you it was not.

Two structural commitments follow, and they are the ones that distinguish this approach from responsiveness for its own sake. Promote diversity: a population of institutions all following the same risk model is a single institution wearing many names, and it will turn at the same moment. Keep coupling loose where the efficiency gain is small: interconnection is worth paying for, and it should be bought deliberately rather than accumulated by default.

AT YOUR DESK

None of this is confined to central banks, and the arithmetic scales all the way down to one household. Take a portfolio of ten thousand dollars, split sixty percent into a stable established company at a hundred dollars a share and forty percent into a volatile growth company at fifty. That is sixty shares of the first and eighty of the second. Regulation changes, the growth company falls to thirty dollars, and a rule-based rebalancing sells twenty of those shares for six hundred dollars and buys six shares of the stable one. The holding becomes sixty-six and sixty.

The move is small and the principle is not. A fixed allocation is a bet placed once on a market that keeps moving. A rule that rebalances is a negative feedback loop installed deliberately in a personal system, and it does the one thing a person under stress reliably fails to do: it sells what has risen and buys what has fallen, at the moment when every instinct argues the other way.

The optimisation that sits underneath is Markowitz's, and it is worth doing once by hand. Two assets: one with an expected return of fifteen percent and a standard deviation of twenty, one with five percent and a standard deviation of three. A target of ten percent lands at roughly sixty percent in the stable asset and forty in the volatile one. The number is useful. What the calculation cannot tell you is that the volatility estimate was taken from a period that did not contain a crisis, and that the correlation between the two assets — the thing the whole calculation depends on — tends to move towards one exactly when diversification is being relied upon.

So the household version of adaptive policy has the same five steps in smaller clothes. State the objective. Name what moves it. Watch a short list of indicators on a schedule rather than on impulse, since a portfolio checked daily produces anxiety and a portfolio checked never produces drift. Set rules in advance for what a given move triggers, because a rule written calmly executes under conditions where judgement will not. Then review the rules annually, which is iteration.

One further step belongs here and is usually missed. Diversification is normally described across asset classes, and that is the shallow version. The deeper version diversifies across feedback loops. Holding six companies that all depend on one commodity, or on one interest-rate environment, or on one regulatory decision, is one position held six times. The question is not how many things are owned. It is how many distinct ways the holdings can be wrong at once, and honest answers are usually smaller than the number of line items.

The reason to do this personally is not that a household can hedge a systemic event. It cannot. It is that the same shape of thinking — loops, coupling, buffers, rules written before they are needed — is what resilience means at any scale, and it is cheaper to learn on ten thousand dollars than on a balance sheet.

WHAT HOLDS

It would be easy to read all of this as a case against models, and that reading is wrong in a way worth correcting directly. The argument is not that mathematics fails in finance. It is that a particular mathematics, built for independent draws from a stable distribution, was applied to a system that supplies neither, and that better mathematics exists. Differential equations, network analysis, stochastic processes with heavy tails, agent-based simulation — these are more demanding than what they replace, not less.

What changes is the standing of the output. A model of a complex adaptive system is an instrument for exploring what could happen, not an oracle producing what will. Run it a thousand times with varied initial conditions and it returns a distribution of futures and, more valuably, a map of which structural features produce the bad ones. That is a different product from a point forecast, and it is more useful, because the structural features are what policy can actually reach.

This has an ethical edge that the book does not leave implicit. These models allocate credit, price insurance, set capital requirements and determine who is lent to. A model that is confidently wrong about the tail of a distribution is not an academic error; it is foreclosures. Anyone deploying one owes an account of what it assumes, where it has been tested, and what it cannot see — and a model whose limitations are undocumented should be treated as undocumented rather than as unlimited.

An instrument that has never been wrong in a way anyone noticed is an instrument nobody has checked.

There is a closing image the book keeps returning to, and it earns its place. Prosperity is not a quantity sitting in an account; it is something more like a reef — built by many small organisms, dependent on flows it does not control, capable of extraordinary productivity and of collapse when conditions cross a threshold no single polyp could perceive. A reef is not managed by issuing instructions to polyps. It is managed by attending to temperature, nutrient flow, diversity and the health of the connections.

The practical inheritance is smaller than the imagery and more useful. Financial systems are living systems, and living systems are governed by tending rather than by command. Watch the loops. Keep the buffers. Preserve the diversity that lets the whole population not be wrong at once. Shorten the interval between acting and learning what the action did. None of that predicts the next crisis, and none of it is meant to. It builds something that bends and comes back, which is the only kind of stability a system like this one has ever offered.

Free to read

Free to read, and free to hear. Every chapter of every book in this house, and every narration of it, is open to anybody. No account, no card, nothing to cancel.

What is in it

• Introduction: The Evolving Landscape of Financial Systemsopen this in the house search
• Complexity and Emergence in Finance: Beyond Traditional Modelsopen this in the house search
• Econophysics and the Mathematics of Adaptive Systemsopen this in the house search
• Feedback Loops and Systemic Risk: Understanding Interconnectionsopen this in the house search
• Agent-Based Modeling: Simulating Heterogeneous Behavioropen this in the house search

A market is not a machine that broke. It is a body that was measured while it slept.

Interconnection is not a safety feature. It is the medium through which failure travels.

The rare event is only rare inside the distribution somebody chose to assume.

Policy that cannot learn is a forecast in a uniform.

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