Chapter 5. Case two: many names, one fire

5.1The claim

Knight's chapter claims that gains and losses cancel more completely the more cases are pooled, however unlike those cases may be. The same chapter, in its discussion of fire insurance, records the exception that undoes the claim: losses that are not independent, because one event spreads to many cases. Both passages are read below, with Knight's account of the investor who spreads holdings over many corporations, and then the record Strategy count diversification, in which many differently named strategies lose together on days of sharp moves. The case adds to the second account of the tail, the one that places it inside the pool, a record that measures dependence under stress apart from dependence in calm, and it shows that Knight's own chapter already held the materials for that distinction.

5.2Knight's fire

Fire insurance is Knight's standing example of a business hazard that insurance can meet, and he contrasts it with life insurance. Life insurance, with medical examinations and attention to age, sex, residence, occupation and habits, can select lives whose chances are close to the ideal of a mechanical probability, so close that its problems are worked with formulas from the binomial law, as problems of mechanical chance are. Fire insurance rating, he notes, is a different matter. Only recently had insurers begun to form reasonably homogeneous classes of fire risk and to measure the real probability for a particular building, and even after thousands of classes had been defined, much was still left to judgment.

Knight draws two consequences. The first is that insurance does not cover the whole fire risk. Because of moral hazard and practical difficulties, policies pay only for the physical damage itself, sometimes for only a share of it, and the often larger costs of a halted trade and upset plans stay with the owner. Some uncertainty therefore stays with both insurer and insured, and with it the chance that either side gains or loses.

The second consequence is stated in a single sentence that matters more than its length suggests. The chances of fire, Knight observes, are harder to reckon because one building's risk is tied to its neighbours': flames travel, and losses arrive in bunches. Then he adds the qualification that governs the whole chapter: to the extent that total fire losses can be reckoned beforehand, each owner exposed to them can treat them as a fixed cost, and to that extent nobody profits or loses by them.

The qualification, to the extent that, carries the weight. Pooling turns a contingent loss into a fixed charge only to the degree that the aggregate can be foreseen. Where losses cluster, the aggregate is harder to foresee than the count of cases suggests, because a pool of many buildings that may burn together behaves more like a pool of few. Knight does not develop the point. He records it and moves on.

5.3Knight's investor and Knight's speculator

Later in the chapter Knight returns to pooling in two places that bear on the record. The first is the corporation. Its shares can be divided finely and transferred easily, so an investor can spread holdings over many enterprises. The effect, Knight says, is twofold. The investor gains a further offsetting, since the gains and losses in different companies will largely cancel and make total returns more regular and foreseeable. And the chance of losing a small part of one's resources weighs less, even in proportion, than the chance of losing a large part. Knight calls this second effect diffusion and distinguishes it in principle from consolidation, though he grants that in practice the two usually come together.

The second place is speculation. The standard example is the hedge, by which a miller or spinner passes to a professional speculator the chance of gain or loss from changes in the price of materials between purchase and sale. Even apart from any better judgment the speculator may have, Knight notes, the speculator gains from sheer breadth: a miller deals in the market on one occasion where the speculator deals on hundreds or thousands, so the speculator's errors of judgment must cancel more fully and leave a steadier return.

And then Knight states his strongest form of the claim. The same reasoning applies, he says, to any way of specialising the bearing of uncertainty. Specialising means concentrating; concentrating means pooling; and however unlike the pooled cases are, their gains and losses neutralise one another more the more of them are thrown together.

This is the claim the second record tests directly. Knight's sentence says the number of cases is what matters and their heterogeneity does not. The record's question asks whether the number of differently named strategies is what matters. Knight's fire sentence, a few pages earlier, gives the answer his strong claim leaves out: the number of cases matters only if they do not burn together.

5.4The record: many strategies, one direction

The record Strategy count diversification sets out its scene in three segments. The assumption:

What the decision would need:

And the tags:

The question:

The options:

And the example judgment:

5.5Reading the record with Knight

The record's first option is Knight's strong claim put into practice. Count the cases; the more there are, the more the gains and losses will cancel; recognise diversification by the count. The record's assumption states the fact that defeats it: on days of sharp moves, most of the losses go the same way. In Knight's terms, the strategies are buildings in one district, and the sharp-move days are the fire.

What makes the record more than a restatement of Knight's fire is its first clause. Correlation of returns is usually low. In calm periods the strategies do behave like unlike cases, and their gains and losses do offset. A pool measured in calm would pass Knight's test. The dependence appears only in the periods when offsetting is most needed. Knight's fire spreads whenever there is a fire; the record's losses converge only under a particular condition, and the condition is the one in which the pool's purpose is tested.

The example judgment names three factors on which separate strategies may depend: liquidity, direction and volatility. Each is a channel by which unlike cases become like ones under stress. Strategies that need to trade in the same markets depend on the same liquidity, and if that liquidity disappears, they all suffer at once; Bookstaber's observation that liquidity vanishes when it is most needed is a description of this channel. Strategies that hold positions leaning the same way depend on the same direction, whatever they are called. Strategies that earn from calm depend on volatility staying low. The record's judgment is that where the dependence is shared, the risks are to be added, not pooled, whatever the strategies are named.

Daníelsson's note goes further than the record. When many firms use the same risk models under regulation, their use changes the market and risk becomes endogenous. The second account takes one more step, which the note does not: a shared model can make a shared loss. The record says nothing about why its strategies share exposure. It does not say whether they were built with the same model, run by managers rewarded in the same way, or simply exposed to the same market. Rajan's herding is one candidate cause, Daníelsson's endogeneity another. The record leaves the cause open and specifies only the test.

5.6Names and classes

The record's question turns on names. Its phrase, differently named strategies, is the Korean "서로 다른 전략의 이름", literally the names of strategies that differ from one another. The record does not deny that the strategies differ. It denies that their difference in name is a difference in the class of risk they belong to.

This bears on the older dispute about Knight. Langlois and Cosgel (1993) read Knight's uncertainty as the condition in which instances cannot be sorted into classes of like cases. The second record presents the reverse difficulty: instances that appear to belong to many classes, and are labelled so, turn out under stress to belong to one. The trouble is not that no class can be formed. It is that the classes formed in calm are the wrong classes for the event that matters. The record's proposed next check, read in Section 6, is an attempt to form the right class by sorting periods rather than strategies.

Knight's account of the investor who spreads holdings across corporations has the same shape. The investor's offsetting depends on the companies' fortunes being, in large measure, independent. Knight says they will tend to cancel in large measure, and the qualification is honest: in large measure, not wholly. The second record describes a portfolio in which, on the days that matter, the measure is small.

5.7Diffusion without consolidation

Knight's distinction between consolidation and diffusion acquires a sharper edge against the record. Diffusion, in his sense, is the gain from spreading a given loss across more holders, so that each bears a smaller share: a loss falling lightly on many is, other things equal, better than the same loss falling heavily on one. Consolidation is the reduction of the loss's uncertainty by pooling it with other, independent losses. Knight remarks that in practice diffusion perhaps always comes with consolidation, though the two are separable in logic.

The record describes a portfolio that may have the first without the second. Many strategies, each with its share of the capital, spread the exposure across many lines; that is diffusion of a sort. But if the lines lose together, the spreading does not reduce the uncertainty of the total. The loss is distributed across many names and arrives at once. The record's example judgment, to aggregate the risks of strategies that share factors, is a refusal to count diffusion as consolidation.

The record's second scenario segment adds one more element: portfolio-level collateral and liquidation requirements. A portfolio that holds its strategies under one account faces collateral demands on the total, not on each strategy. Whatever the strategies' names, the lender sees one borrower. Under stress, the demand for collateral on the whole may force the sale of positions in strategies that were not themselves losing, which spreads the loss across lines that were not otherwise exposed. The record does not describe this mechanism in detail; it lists the requirement as evidence the decision would need. Its presence in the list shows that the record's authors treated the portfolio's structure, not only its strategies' returns, as a source of shared loss.

5.8What the case adds

For the account that places the tail inside the pool, this case supplies a record in which the dependence is conditional: absent in calm, present under stress. Knight's chapter contains both halves of the matter, his strong claim that heterogeneous cases offset more as they are multiplied and his fire, whose losses spread and cluster. It does not join them. The record joins them in a single decision, refusing to count names as classes and proposing to measure dependence under stress separately. The record supports the second account's diagnosis, that pooling can manufacture its own tail, only in the weak sense that it describes the shared loss; it does not say what produced the sharing, and so does not decide between the herding Rajan (2005) describes, the endogenous model risk Daníelsson (2002) describes, or a common market exposure that neither needs.