The Tail

The long calm can conceal the very risk for which a premium is being paid. A venture exposed to a rare catastrophe may produce years of ordinary gains while the obligation that could undo them remains uncalled; its record then shows what happened while leaving the most consequential part of what could happen almost untouched. This is the difficulty the tail brings to the price of risk. A small probability can carry a large share of the expected loss, and a loss that arrives when everyone is already badly off can matter still more to the price people will pay to escape it. The problem has several layers. There is the severity of the event, the reliability of the probability assigned to it, and the possibility that the arrangements made to bear the loss will amplify it. The earlier plates described risk that could be pooled, spread or priced. The tail asks how much of that achievement survives when the observations are sparse and the institutions meant to absorb the shock are struck together.

A fat tail gives extreme outcomes more weight than a thin-tailed benchmark would allow, without making every extreme certain or every distribution the same. The normal distribution places rapidly diminishing probability far from its centre. A power-law tail diminishes more slowly: if the probability of exceeding a sufficiently large loss is proportional to the inverse cube of that loss, doubling the threshold divides the exceedance probability by eight. That is a statement about the shape of a specified model, and the exponent must be estimated rather than supplied by the adjective fat. In his survey of financial returns, Rama Cont describes substantial departures from normality, particularly at short horizons, together with volatility clustering and difficult tail estimation. The evidence does not entitle us to say that every economic quantity follows one power law, or that its variance must be infinite. A distribution can have finite variance and still place much more weight on large losses than a normal approximation with the same variance. Nor does a heavy tail by itself tell us which institutions will fail together. The magnitude of each loss and the dependence among losses are separate parts of the danger.¹

A catastrophe missing from a record can have been possible throughout it, and the absence of a loss supplies less reassurance than the smooth record invites. Suppose, purely for illustration, that a particular disaster has an independent one-per-cent probability in each year. The probability of passing a century without it is 0.99 raised to the hundredth power, about thirty-seven per cent. A century of survival is therefore compatible with the assumed danger; it does not establish that the annual probability was zero. In practice the probability itself is uncertain, conditions change, and successive years need not be independent, so the illustration gives a reason to examine the assumptions rather than a frequency to carry into the world. Taleb's black swan adds a further difficulty: an event outside an observer's regular expectations, with a large impact, subsequently made to look intelligible and foreseeable. It concerns the limits of the observer's knowledge. A severe loss already included in a probability model can be a tail event without being a black swan to its author. An event can surprise a buyer while having been anticipated by a seller. The distinction matters because adding probability to a known disastrous state and discovering that the map omitted a state are different repairs.²

The price of catastrophe depends on what is lost and on the condition of the person who must bear the loss. The earlier Systematic plate gave the governing intuition: a payment is especially valuable when resources are scarce and their marginal use is great. A rare collapse in consumption can therefore carry an importance for asset prices far larger than its frequency suggests. In Barro's extension of the rare-disaster approach, a representative consumer faces ordinary fluctuations alongside a small probability of a large contraction; claims that fare badly in that contraction must offer compensation relative to claims that preserve resources there. The mechanism gives a reason why the measured variability of a relatively tranquil sample may understate the danger relevant to the premium. Its force depends on the disaster's probability and size, preferences, and the comparative fate of the supposedly safe asset. If both claims fail together, the comparison changes. A disaster model can thus help explain a large premium without establishing that every observed premium is adequate or inadequate. The same word catastrophe cannot settle the price on both sides of every trade.³

The catastrophe can also be enlarged by the way the financial system has arranged to meet it. Borrowing makes a small fall in asset values a much larger fall in the owner's equity. If lenders then demand more collateral, or decline to renew short-term funding, the owner may have to sell into the falling market. Others holding the same assets suffer the resulting markdown and may be forced to sell in turn. Brunnermeier's account of the 2007–2008 crisis traces these loss and margin spirals alongside runs and the network of counterparties: the attempt by each participant to become safer can tighten the constraint on the rest. Here the dependence among losses is partly produced by shared holdings, funding and responses to distress. A portfolio divided among many names can remain exposed to one financing condition. Pooling still helps where losses are sufficiently separate; its reach is limited where the common shock remains, and a chain of promises is only as useful as the resources available to honour it. An account of systemic risk must therefore examine who will be required to pay, sell or refinance at the same time.⁴

A record of small premiums collected in quiet years is consequently evidence of receipts, while the quality of the risk-bearing requires a further account of the promises outstanding. Imagine a seller who receives a regular payment for an obligation that becomes costly only in a severe downturn. Most observations will show the payment and none of the large settlement. The same pattern can arise from a well-capitalized insurer charging adequately for the exposure or from a seller who has underestimated it and retained too little capacity to pay. A short return history alone need not distinguish them. The contract, the concentration of exposures, the capital behind it and the funding required before final settlement provide information the winning years cannot supply. Even a correctly estimated loss threshold has a limited meaning: a ninety-nine-per-cent value-at-risk quantile describes a boundary within a distribution, while the size of losses beyond that boundary remains another question. It is neither a maximum possible loss nor a promise that the next hundred observations will contain exactly one breach. Measures of the loss beyond the threshold and scenarios outside the fitted record can ask further questions, although their answers remain conditional on the events and relationships they include.⁵

The tail leaves the premium with a demand for an account of what has actually been borne. A quiet interval can be fortunate without being fraudulent; a large premium can compensate a real danger without proving that the danger was measured well; a model can describe the centre accurately while leaving its most consequential extrapolation uncertain. The question is sharpened by following the loss beyond the contract's signature. If the promised protection fails in the state for which it was bought, the buyer has retained a risk that the receipt seemed to transfer. If the settlement can be made only by selling what everyone else is selling, the transfer has entered the machinery of the catastrophe. And if the obligation can be met, someone has performed a genuine service even though the calm years made that service almost invisible. To price a calamity that has not happened is to reason about possibilities with evidence that cannot contain them all, then to decide where the resulting burden will lie. The next plate takes up the contract that makes this division explicit: The Hedge, and the price of a payment that arrives on the other side of another person's loss.