Aurélien Giroux

All Roads Lead to Disagreement (II): Pricing Disagreement Without Pricing Truth

Sat, 26 Sep 2026 00:00:00 +0200

Two lenders disagree about a group of utilities. One expects an industrial transition that some of the borrowers will struggle to finance. The other expects them to adapt without much damage to their creditors. Both have models, evidence and arguments, and neither finds the other convincing.

“Put your money where your mouth is” sounds like a way to settle it. It leaves rather a lot for the lawyer to write.

They would first have to agree on what counts as an outcome, when it will be observed, and how their forecasts turn into a payment. None of that is obvious. A high default rate might favour one forecast without showing why the defaults happened. A low one might favour the other while hiding an emergency rescue that kept a few borrowers alive. And the fact that each of them is keen to bet tells us nothing about the price at which anyone else would take the other side.

Part I of this essay asked why serious disagreement survives careful inquiry. A companion piece, What Can a Market See?, asked which parts of a disagreement existing markets let an investor act on. This one builds something: a contract that attaches money to a comparison between two forecasts, gives each lender an exposure it can use, and lets anyone inspect the comparison. Choosing the payment, arranging the trade and deciding what its record can prove turn out to be parts of the same design problem.

Several sections have optional mathematical explanations. Open a box for the assumptions and derivations behind the examples; the main argument can be read with all the boxes closed.

First agree on the test

Suppose the lenders pick a list of one hundred utilities and a five-year window. They will compare forecasts of how many default: fewer than five, five to nine, or ten or more. Three bands keep the arithmetic readable; a real contract could use finer ones. Before the window opens, each lender writes down a probability for each band.

That already takes a good deal of agreement. The lenders can hold completely different views about technology and policy, and keep their models to themselves. But they have to recognise the same borrowers, the same horizon and the same three possible answers, and each has to commit to three numbers in a form that cannot be edited once the defaults start arriving.

Before settling on default counts, though, they should ask what the contract is for. A lender worried about a few very large borrowers cares more about the amount lost than about the number of defaults. A lender concentrated in one region may learn little about its own losses from a national count. A statistic can be measured perfectly and still provide poor protection if the payment it triggers does not correspond to those losses. What a lender is exposed to is the reason to look for an observation; the fact that an observation is easy to obtain does not make it the right one.

Then come the questions that sound tedious. Does a distressed restructuring count as a default? What happens if a utility merges, leaves the sector or stops reporting? Who keeps the record, and which version of it is final? Once money depends on the answer, a disagreement about the future can turn into a disagreement about a definition remarkably quickly. The contract has to settle these questions in advance, and say what happens when data are missing or disputed.

Three conditions have to hold together. The disagreement has to matter to somebody’s decision. The observation has to be verifiable. And the forecasts must differ about that observation in a way relevant to the decision. An emissions record can be excellent evidence about emissions and a poor test of a claim that climate policy will lead to defaults. The emissions, the policy and the financial losses are related through an economic explanation that may itself be disputed. A default count asks the narrower question of how many borrowers fail, which two lenders who disagree about the causes might both accept as a basis for payment.

One further choice concerns where the second forecast comes from. It need not be declared by another lender; a benchmark could instead be constructed from market prices. The extraction method would then belong in the contract. Prices reflect both expectations about outcomes and how much money received in each outcome is worth to investors, and a finite set of quotations can be consistent with several probability assignments. An extracted pricing distribution would therefore need an interpretation before it could serve as a forecast of defaults. In our example both forecasts are declared by the lenders themselves. With a market benchmark, the extraction method and its interpretation would instead sit, unstated, inside the phrase “what the market believes”.

What should the contract pay?

“One geometry cannot be more true than another; it can only be more convenient.”

— Henri Poincaré, Science and Hypothesis (1905), Chapter III, p. 50.[1]

Poincaré was writing about the geometry of physical space, but there is a literal geometry here too. A forecast over the three default bands is a point whose three coordinates are probabilities adding up to one, and the lenders’ forecasts are two such points. Measuring how far apart they are requires a rule for distances between forecasts; the mathematics of such rules is called information geometry. There is more than one reasonable rule. Two forecasters might disagree mildly about ordinary outcomes, while another pair agrees almost everywhere and differs sharply about a catastrophe. A geometry that emphasises rare disasters can rank the second pair as further apart; one centred on ordinary predictive performance can reverse the ranking. No geometry is truer than another; once the purpose is fixed, some are more convenient. The lenders’ purpose is what decides.

For this contract, the lenders want to reward how well each forecast anticipated an agreed observation. The comparison should depend on what their probabilities say, not on what the categories are called. If the record is made more detailed, the comparison should change when the lenders disagree about the new detail and stay the same when they agree about it; an example of each follows shortly. And a forecaster who seeks the highest expected reward should do best by declaring the probabilities they actually believe.

The comparison should also work over time. Suppose the contract settles first on a report about refinancing conditions and later on defaults. The second settlement should use forecasts made after the refinancing report is known, so that its information is not counted twice. Each settlement can then simply be added to the earlier ones. I want a payment tied to the relative probability of what occurred, with settlements that add up in this way.

I will use a baseline called relative surprise. I will call the more pessimistic lender Stress and the other Resilience. When the result is known, divide the probability Stress gave it by the probability Resilience gave it. A twofold ratio earns one unit of payment, a fourfold ratio earns two units, and an eightfold ratio earns three. A ratio of one half loses one unit. Equal probabilities produce no comparison payment. The operation that counts these doublings is the base-two logarithm, and one doubling is called a bit; the logarithm also gives intermediate payments for ratios between those values.

Here is the whole contract on one example. Stress gives the low, middle and high bands 20%, 40% and 40%. Resilience gives them 40%, 40% and 20%. They set the unit at €10,000 per doubling. For now there is no fixed entry fee.

Defaults over five years	Stress’s forecast	Resilience’s forecast	Payment to Stress
Fewer than 5	20%	40%	−€10,000
5 to 9	40%	40%	€0
10 or more	40%	20%	+€10,000

If ten or more utilities default, Stress had given that band twice Resilience’s probability, and Resilience pays Stress €10,000. In the low band it is the reverse. In the middle band they agreed, and nothing changes hands. Every amount in the table was fixed before the first default.

We can now see what it would mean to add detail without adding disagreement. Suppose the lenders divide “ten or more” into “ten to fourteen” and “fifteen or more”. They still disagree about the chance of reaching ten defaults. But suppose both say that, if ten or more defaults occur, the two more detailed outcomes are equally likely. Stress therefore divides its 40% into 20% and 20%; Resilience divides its 20% into 10% and 10%. In either new category Stress still assigned twice Resilience’s probability, so the payment remains €10,000. Learning which of these categories occurred gives no further reason to favour one forecast over the other. Later we will examine a different subdivision, in which the lenders also disagree about what happens once defaults reach ten.

Stress is not paid merely for sounding alarmed. Its higher probability of the high band had to be accompanied by a lower probability elsewhere, here in the low band, where the contract can cost it money. Nor does one win establish general forecasting superiority: Resilience also gave the high band a real chance, one in five. The payment records which forecast assigned more probability to what happened. Whether Stress forecasts better across repeated observations is a further question.

Why use a logarithm rather than another function of the probability ratio? Three arguments help explain the choice.

The first concerns how forecasts are evaluated. A scoring rule is a formula chosen in advance that takes a probability forecast and the outcome that occurs, then returns a number—the score. Here higher scores are better. Because a forecast gives a probability to every outcome, a score can reward how much confidence was placed on what happened, not merely whether the most likely category came up. The logarithmic rule takes the logarithm of the probability assigned to the outcome that occurred. In the high band, Stress’s 40% gives a score of about −1.32, while Resilience’s 20% gives about −2.32. Stress’s score is higher by one. Multiplying that difference by €10,000 gives the payment in the table.[2]

The scores can be negative; the difference between them determines who pays. More importantly, a rule needs to give the forecaster a reason to report their genuine uncertainty rather than exaggerate confidence. A scoring rule is proper when reporting one’s actual probabilities maximises the expected score, computed with those same probabilities. It is strictly proper when every other report does worse. The logarithmic rule is strictly proper. It is also essentially the only such rule, provided there are at least three possible outcomes, the rule is smooth, and the score uses the reported forecast only through the probability given to the outcome that occurred, a requirement called locality. For the payment, which subtracts one lender’s score from the other’s, the only freedom left is the choice of unit.[2]

The second argument starts from a budget for a different kind of bet. Suppose Stress pays a stake and receives a non-negative multiple of it when the default band is known. Resilience offers any payment schedule whose expected payout under Resilience’s probabilities is no greater than the stake. Among all such schedules, Stress chooses the one with the highest expected logarithm of the amount returned per unit staked. Logarithms add when successive returns are reinvested, which is why this objective is used to study multiplicative wealth growth. Under this objective, Stress chooses a multiple equal to Stress’s probability divided by Resilience’s: half the stake back in the low band, the stake in the middle, double in the high band.[3] The logarithm of that multiple is relative surprise again. This reinvestment bet and the signed payment in the table are different contracts. A lender maximising expected money would instead stake everything on the high band.

The third argument returns to geometry. A proper scoring rule measures the discrepancy between two forecasts by asking how much expected score a forecaster would lose by reporting the other one. For the logarithmic rule, that loss is the expected relative surprise: 0.2 bits per settlement in the lenders’ example. It is not quite a distance, since swapping the two forecasts can change it, but for nearby forecasts it behaves like one, and what it then measures is how readily observations can tell the two forecasts apart.

Suppose we observe many independent borrowers with the same default probability. Under a 50% forecast, 10,000 borrowers give a standard error of half a percentage point, so the gap between 50% and 51% is two standard errors. Under a 1% forecast, the same one-point gap reaches two standard errors with about 400 borrowers, or about 800 if the standard error is computed at 2%. Relative to sampling noise, a one-point difference is far larger near 1% than near 50%; at 1% it is a doubling.

The Fisher information metric measures small changes in probabilities against this sampling noise. For a single yes-or-no event, its squared distance is the squared change in probability divided by the variance of one observation. The same one-point change is therefore a larger distance near 1% than near 50%. For nearby forecasts, the expected relative surprise is proportional to this squared distance, so the expected payment is larger where observations can distinguish the forecasts more easily. The first box derives this.[4][16]

This geometry also respects the earlier subdivision of the high band, in which Stress’s 40% became 20% and 20% and Resilience’s 20% became 10% and 10%: the extra detail leaves the distance unchanged. Čencov proved that Fisher’s metric is, up to scale, the only way of measuring small differences between forecasts that stays unchanged, for every number of outcomes, whenever outcomes are split in proportions both forecasts share.[4] Other measures of disagreement share Fisher’s geometry for nearby forecasts, but among proper scoring rules only the logarithm has it everywhere, at a fixed scale.

Other questions call for other geometries. Relative surprise requires no distance between different outcomes: it compares the probabilities assigned to the outcome that occurred. Suppose instead that the lenders want a comparison sensitive to how far their predicted default counts differ. A difference of one default might matter less than a difference of twenty. A Wasserstein distance begins by specifying the cost of moving probability from one possible outcome to another, then finds the least total cost of converting one forecast distribution into the other.[14] Where outcomes are actual default counts, that cost could be the difference between the counts. Where outcomes describe losses or electricity shortfalls, it could instead use euros or megawatt-hours.

For the simplest transport distance, there is also a contingent payment whose expected value under Stress’s forecast exceeds its expected value under Resilience’s by exactly that distance. The second box constructs the payment and compares it with the logarithmic one. The choice is between specified financial questions, not between a good mathematical distance and a bad one. How to give beliefs a geometry is a rich subject in its own right; here the alternatives show why the financial purpose belongs in the choice of comparison.

The mathematics: scoring rules and the logarithmic comparison

Forecasts, scores and units. Let $\mathcal Y$ be a finite set of possible outcomes. A forecast $r$ assigns a strictly positive probability $r(y)$ to each $y\in\mathcal Y$, with $\sum_y r(y)=1$. Write $p$ for Stress’s beliefs, $q$ for Resilience’s, and $Y$ for the outcome that will be observed. An expectation $\mathbb E_p$ uses probabilities $p$.

A scoring rule is a function $S(r,y)$: it evaluates the report $r$ when outcome $y$ occurs, with larger values better. Its expected score under belief $p$ is

$$ \overline S(p,r)=\mathbb E_p[S(r,Y)]=\sum_{y\in\mathcal Y}p(y)S(r,y). $$

The rule is proper when $\overline S(p,p)\ge\overline S(p,r)$ for all forecasts $p$ and $r$, and strictly proper when equality requires $r=p$. For a proper rule the regret

$$ R_S(p,r)=\overline S(p,p)-\overline S(p,r) $$

is non-negative. It measures the expected score sacrificed by reporting $r$ instead of $p$.

For the logarithmic rule, $S(r,y)=\log_2 r(y)$. Define its relative entropy in bits by

$$ D(p\|q)=\sum_y p(y)\log_2\frac{p(y)}{q(y)}. $$

Throughout the article, $D$, entropy $H$ and mutual information $I(S;Y)$ use base-two logarithms. Natural logarithms are always written $\ln$. The signed payment to Stress is

$$ X(y)=c\,[S(p,y)-S(q,y)] =c\log_2\frac{p(y)}{q(y)},\qquad c=10{,}000. $$

Here $c$ is measured in euros per bit.

Why base two? It makes one doubling worth one unit, which keeps the payments in the table simple. The same euro claim can be written $X(y)=(c/\ln 2)\ln[p(y)/q(y)]$: another base changes the unit of information, $c$ absorbs the change, and no payment moves.

1. Honest reporting. With belief $p$, the expected logarithmic score of a report $r$ satisfies

$$ \overline S(p,r) =\sum_y p(y)\log_2 r(y) =\sum_y p(y)\log_2 p(y)-D(p\|r). $$

The first term is independent of the report. Jensen’s inequality for the concave logarithm gives

$$ -D(p\|r) =\sum_y p(y)\log_2\frac{r(y)}{p(y)} \le\log_2\!\left(\sum_y p(y)\frac{r(y)}{p(y)}\right) =\log_2 1=0. $$

Equality requires the ratios $r(y)/p(y)$ to be constant. Since both forecasts sum to one, that constant is one. The expected score is therefore uniquely highest at $r=p$. This is Gibbs’ inequality and establishes strict propriety. Subtracting the score of a fixed comparator $q$, then multiplying by $c>0$, leaves that optimum unchanged, so the signed payment rewards honest reporting too.

For uniqueness, call a score local when $S(r,y)=s_y(r(y))$: only the reported probability of the realised outcome enters the score. Suppose each $s_y$ is differentiable and the rule is strictly proper throughout the simplex interior, the set of strictly positive probability vectors. At the honest report, maximisation subject to $\sum_y r(y)=1$ requires

$$ p(y)s_y'(p(y))=\lambda(p)\qquad\text{for every }y. $$

With at least three outcomes, vary the probability of one while holding a second fixed and adjusting a third to keep the sum equal to one. The expression on the second outcome stays fixed, so the expression on the first must also stay fixed. Since any two interior forecasts can be linked by such moves, every function $t s_y'(t)$ equals one and the same constant $a$. Integrating yields

$$ s_y(t)=a\ln t+b_y. $$

Here $b_y$ may depend on the outcome but not on the report. Strict propriety requires $a>0$: a negative coefficient reverses the preference and zero makes every report equally good. Thus smooth local strictly proper scores are logarithmic up to positive scale and outcome-dependent additions.[2]

With two outcomes, fixing one probability fixes the other, so this argument no longer applies. For example, the reward version of the Brier score is

$$ S_{\rm B}(r,y)=-\frac12\sum_{z\in\mathcal Y} \big(r(z)-\mathbf 1_{\{z=y\}}\big)^2. $$

Its regret is $\tfrac12\sum_z(p(z)-r(z))^2$. It is strictly proper and bounded. For two outcomes it reduces to $-[1-r(y)]^2$, which is also local; with three or more it generally depends on the other entries of the report as well.

2. A budget for logarithmic growth. Resilience offers every non-negative payout multiple $m(y)$ satisfying $\mathbb E_q[m(Y)]\le1$. Stress chooses among them to maximise $\mathbb E_p[\log_2 m(Y)]$, interpreting $m$ as money returned per unit staked.

Write $m(y)=[p(y)/q(y)]u(y)$. Then

$$ \begin{aligned} \mathbb E_p[\log_2 m(Y)] &=D(p\|q)+\mathbb E_p[\log_2 u(Y)]\\ &\le D(p\|q)+\log_2\mathbb E_p[u(Y)]\\ &=D(p\|q)+\log_2\mathbb E_q[m(Y)]\\ &\le D(p\|q). \end{aligned} $$

Equality is obtained by $u(y)=1$, hence $m(y)=p(y)/q(y)$. For the lenders this is $(1/2,1,2)$. Its expected payout under Resilience is $0.4/2+0.4+0.2\times2=1$, and its expected logarithmic return under Stress is 0.2 bits. In this finite positive setting,

$$ D(p\|q)=\max_{m\ge0,\,\mathbb E_q[m]\le1} \mathbb E_p[\log_2 m]. $$

This is a form of the Donsker–Varadhan variational identity.[3] Expected money has a different optimum: the ratio schedule returns 1.3 units on average under Stress, whereas a feasible schedule paying five units only in the high band returns two units on average under Stress. The latter pays nothing elsewhere and therefore has expected logarithmic return minus infinity.

3. Local geometry and evidence. Fix a strictly positive forecast $p$ and a vector $h$ satisfying $\sum_yh(y)=0$. Let $q=p+\varepsilon h$, where $\varepsilon$ is small enough that all entries remain positive. The Fisher quadratic form at $p$ is

$$ g_p(h,h)=\sum_y\frac{h(y)^2}{p(y)}. $$

For the actual probability change $\delta=\varepsilon h$, put $d^2=g_p(\delta,\delta)$. This is the squared local length of that displacement at $p$; it is not the exact finite distance obtained by integrating a metric along a path. Taylor expansion, with $p$ and $h$ fixed, gives

$$ D(p\|p+\varepsilon h) =\frac{\varepsilon^2}{2\ln 2}\sum_y\frac{h(y)^2}{p(y)} +O(\varepsilon^3) =\frac{d^2}{2\ln2}+O(\varepsilon^3). $$

The expected euro payment is therefore approximately $c d^2/(2\ln2)$.

Why that quadratic form measures statistical distinguishability. For one observation define the evidence in bits in favour of $p$ against $q$ by

$$ \ell(y)=\log_2\frac{p(y)}{q(y)} =\frac{1}{\ln2}\left[-\frac{\delta(y)}{p(y)} +\frac{\delta(y)^2}{2p(y)^2}\right]+O(\varepsilon^3). $$

The linear term has mean zero under $p$, because $\sum_y\delta(y)=0$. Its variance is $d^2/(\ln2)^2$. The quadratic term has mean $d^2/(2\ln2)$. Consequently, to leading order, $n$ independent observations with law $p$ give a cumulative comparison with

$$ \text{mean}\simeq\frac{n d^2}{2\ln2},\qquad \text{standard deviation}\simeq\frac{d\sqrt n}{\ln2}. $$

Their ratio is $d\sqrt n/2$. Within this local approximation, obtaining the same ratio when the separation $d$ is halved requires four times as many observations.

For a yes-or-no event, let $\theta$ be its probability and $\delta$ a small change. The two changes are $\delta$ and $-\delta$, so

$$ d^2=\frac{\delta^2}{\theta}+\frac{\delta^2}{1-\theta} =\frac{\delta^2}{\theta(1-\theta)}. $$

The observed frequency in $n$ independent cases has standard error $\sqrt{\theta(1-\theta)/n}$. A gap of size $|\delta|$ equals two standard errors when $n=4\theta(1-\theta)/\delta^2=4/d^2$. For $\delta=0.01$, this gives 10,000 at $\theta=0.5$, 396 at $\theta=0.01$, and 784 at $\theta=0.02$. The Taylor expansion needs the relative changes $|\delta(y)|/p(y)$ to be small; a change from 1% to 2% does not meet that approximation well.

The relation to likelihood curvature. In a regular model with a scalar parameter $\theta$, write $p_\theta(y)$ for the probability or density of an observation. Assume parameter-independent support and sufficient smoothness and integrability to differentiate the normalisation and expectations. With natural logarithms, the Fisher information in one observation is

$$ I_{\rm F}(\theta) =\mathbb E_\theta\!\left[\big(\partial_\theta\ln p_\theta(Y)\big)^2\right] =-\mathbb E_\theta\!\left[\partial_\theta^2\ln p_\theta(Y)\right]. $$

It is the expected negative curvature of the log-likelihood. A small parameter change $\Delta\theta$ has squared local length $I_{\rm F}(\theta)(\Delta\theta)^2$. For $n$ independent observations from this model and an unbiased estimator $\widehat\theta$ satisfying the Cramér–Rao regularity conditions,

$$ \operatorname{Var}_\theta(\widehat\theta)\ge\frac{1}{nI_{\rm F}(\theta)}. $$

These are two uses of the same local sensitivity: it enters both the discrimination between nearby forecasts and the precision attainable when estimating a parameter.[16]

Refining categories without adding a disagreement. Let each original outcome $y$ be divided into disjoint categories $z\in\mathcal Z_y$, so that observing $z$ also identifies $y$. Specify numbers $\kappa(z\mid y)>0$ adding up to one for each $y$. They are the conditional chances of the detailed outcomes once the original outcome is known. Use the same numbers in both forecasts:

$$ p'(z)=p(y)\kappa(z\mid y),\qquad q'(z)=q(y)\kappa(z\mid y),\qquad z\in\mathcal Z_y. $$

Then $p'(z)/q'(z)=p(y)/q(y)$, so the realised logarithmic comparison is exactly unchanged. For a tangent change $h$, the refined change is $h'(z)=h(y)\kappa(z\mid y)$, and

$$ \sum_{z\in\mathcal Z_y}\frac{h'(z)^2}{p'(z)} =\frac{h(y)^2}{p(y)}\sum_{z\in\mathcal Z_y}\kappa(z\mid y) =\frac{h(y)^2}{p(y)}. $$

Thus the local Fisher length is unchanged too. In the main example, $\kappa=(1/2,1/2)$ inside the high band: Stress’s 40% becomes 20% and 20%, while Resilience’s 20% becomes 10% and 10%. This is different from the later refinement in which the conditional forecasts are 50–50 and 90–10.

Čencov’s theorem identifies Fisher’s metric, up to one overall positive scale, among smooth families of Riemannian metrics defined consistently over all finite outcome sets and invariant under the specified recoverable refinements and relabellings.[4] (A Riemannian metric measures local lengths by a positive quadratic form, as $g_p$ does.)

What the scoring argument adds. Many global discrepancies, including the convex-generator divergences defined in the next box, have this local geometry. To select a score, impose an additional condition. For a smooth proper reward score $S$, define $G(p)=\overline S(p,p)$, its convex expected score under honest reporting. Its regret can be written

$$ R_S(p,r)=G(p)-G(r)-\nabla G(r)\mathbin{\cdot}(p-r). $$

This is a Bregman divergence, the difference between the value of the convex function at $p$ and its tangent-plane prediction from $r$. For nearby forecasts it equals, to second order, one half of the quadratic form given by the Hessian, or matrix of second derivatives, of $G$, evaluated on changes whose coordinates sum to zero; the local metric is that Hessian. Put $F(p)=\sum_y p(y)\ln p(y)$. Its Hessian on those changes is Fisher’s metric. If the Hessian of $G$ equals $a$ times that of $F$ throughout the connected simplex interior, with the same constant $a>0$, then $G-aF$ has zero Hessian and is affine. The score is therefore logarithmic up to scale and outcome-dependent additions.[2] Requiring the metric only at one forecast would not establish this result.

Why contributions from successive observations add. Let $p(y_1,y_2)$ and $q(y_1,y_2)$ be joint forecasts for two observations. Their conditional probabilities describe the second observation after the first is known. The chain rule gives

$$ \log_2\frac{p(y_1,y_2)}{q(y_1,y_2)} =\log_2\frac{p(y_1)}{q(y_1)} +\log_2\frac{p(y_2\mid y_1)}{q(y_2\mid y_1)}. $$

No independence assumption is used. Conversely, let $h$ be a continuous real-valued function of a positive probability ratio. Requiring $h(ab)=h(a)+h(b)$ for every $a,b>0$ and $h(1)=0$ forces $h(a)=C\log_2 a$. To see this, set $g(x)=h(2^x)$. Then $g(x+y)=g(x)+g(y)$, and continuity makes $g$ linear. An increasing reward fixes $C>0$.

The mathematics: other rules, other geometries

Setting. Take strictly positive forecasts $p$ and $q$ on the same finite set of outcomes $\mathcal Y$. Let $Y$ denote the observed outcome and $\eta_y=p(y)/q(y)$ its likelihood ratio. In this box $b>0$ is the number of euros paid per unit of the chosen comparison; to compare payment schedules, we take $b=10{,}000$, as for the logarithmic contract.

Is a positive bargaining interval special to the logarithm? Let Stress’s payment be $b\,g(\eta_Y)$, where $g$ is any real-valued non-decreasing function. Then

$$ \begin{aligned} \mathbb E_p[g(\eta_Y)]-\mathbb E_q[g(\eta_Y)] &=\sum_y q(y)(\eta_y-1)g(\eta_y)\\ &=\sum_y q(y)(\eta_y-1)\big[g(\eta_y)-g(1)\big]\ge0. \end{aligned} $$

The second equality uses $\sum_yq(y)(\eta_y-1)=0$. In the final sum each pair of bracketed factors has the same sign. If $p\ne q$ and $g$ is strictly increasing, the gap is positive.

Suppose both lenders value expected money alone, assess this claim in isolation and settle any fee on the same date as its payment. Stress’s maximum acceptable fee to Resilience is $b\mathbb E_p[g]$; Resilience’s minimum is $b\mathbb E_q[g]$. Their difference is the width of the interval in which both expect to benefit, so any strictly increasing $g$ leaves room to trade when the forecasts differ.

Convex generators. A differentiable convex function $\phi:(0,\infty)\to\mathbb R$ with $\phi(1)=0$ defines an f-divergence by

$$ D_\phi(p\|q)=\sum_yq(y)\phi\!\left(\frac{p(y)}{q(y)}\right). $$

Convexity makes its derivative $g=\phi'$ non-decreasing, so this derivative supplies a payment function for the preceding argument. The divergence is a number comparing two forecasts; the derivative evaluated at the realised ratio supplies the contingent payment. For smooth generators with $\phi''(1)>0$, the local expansion is

$$ D_\phi(p+\delta\|p) =\frac{\phi''(1)}2\sum_y\frac{\delta(y)^2}{p(y)} +o(\|\delta\|^2),\qquad \sum_y\delta(y)=0, $$

which explains why many such divergences have the same Fisher geometry locally, up to scale. For example, $\phi(t)=(\sqrt t-1)^2$ gives the squared Hellinger discrepancy $\sum_y(\sqrt{p(y)}-\sqrt{q(y)})^2$. Its local form is also proportional to Fisher, although its global values differ from relative entropy.

The choice $\phi(t)=(t\ln t-t+1)/\ln2$ gives $g(t)=\log_2 t$. The choice $\phi(t)=(t-1)^2/2$ gives $g(t)=t-1$. On the lenders’ forecasts the second rule pays $(-0.5,0,1)$ units, compared with $(-1,0,1)$ for the base-two logarithm. Its expectation is zero under Resilience and 0.3 units under Stress.

A generator does not by itself supply honest reporting. If Stress reports $r$ and receives $b[r(Y)/q(Y)-1]$, its expectation under its actual belief $p$ is

$$ b\left[\sum_y r(y)\frac{p(y)}{q(y)}-1\right]. $$

This increases as Stress moves its reported probability towards the outcome with the largest $p(y)/q(y)$, here the high band, so honest reporting is not the optimum. The logarithmic rule adds the honesty and additivity properties derived in the first box.

A transport comparison. Number the three categories 0, 1 and 2, in order from fewer than five defaults to ten or more. Specify a distance of one between neighbouring categories and two between the lowest and highest. In symbols, the distance is $|i-j|$.

Let $p_i$ and $q_i$ be the category probabilities. For $m$ categories numbered $0,\ldots,m-1$, define the cumulative probabilities

$$ F_p(k)=\sum_{i=0}^k p_i,\qquad F_q(k)=\sum_{i=0}^k q_i. $$

The Wasserstein-1 distance for this equally spaced category scale is

$$ W_1(p,q)=\sum_{k=0}^{m-2}|F_p(k)-F_q(k)|. $$

Why cumulative probabilities? The difference $F_p(k)-F_q(k)$ is the excess probability one forecast assigns to categories at or below $k$. Converting that forecast into the other requires moving that excess across the boundary between $k$ and $k+1$. Each boundary has unit distance here; summing the absolute excesses gives the least transport cost.[14]

Kantorovich–Rubinstein duality represents the same number as

$$ W_1(p,q)=\max_{u:\,|u(i)-u(j)|\le|i-j|} \left\{\mathbb E_p[u(Y)]-\mathbb E_q[u(Y)]\right\}. $$

Here $u$ assigns a numerical payment to each category and is allowed to change by no more than the assigned distance between categories. A maximising function $u$, fixed before the outcome, gives a cash payment $b\,u(Y)$ whose expected-money bargaining interval has width $bW_1(p,q)$. Adding a constant to $u$ shifts both reservation values equally without changing the width.

For the three bands, the cumulative probabilities are $(0.2,0.6,1)$ under Stress and $(0.4,0.8,1)$ under Resilience. Thus $W_1=0.2+0.2=0.4$. One optimal schedule is $u=(-1,0,1)$, giving expectations 0.2 under Stress and −0.2 under Resilience. With the chosen coefficient $b=10{,}000$, this happens to reproduce the original logarithmic payments and the €4,000 interval.

Now consider the informative subdivision used later in the article: Stress gives 20% and 20% to ten-to-fourteen and fifteen-or-more defaults, whereas Resilience gives 18% and 2%. Number all four resulting categories 0, 1, 2 and 3, again assigning distance one between neighbours. The calculation is

$$ W_1=0.2+0.2+0.18=0.58, \qquad u=(-1,0,1,2). $$

The transport payment has expectations 0.4 under Stress and −0.18 under Resilience. With the same $b$, the logarithmic schedule is instead approximately $(-1,0,0.152,3.322)$. It pays most in the last category because Stress assigned it 20% and Resilience only 2%. The transport schedule rises by one unit between each neighbouring pair because, in this example, Stress assigns less cumulative probability than Resilience at every boundary. Different cumulative comparisons can make an optimal schedule rise at some boundaries and fall at others.

Assigning the two high categories separate locations was part of this new transport specification. If both retained the old high category’s location 2, their transport distance would remain 0.4. The change to 0.58 therefore comes from the chosen distances between the new categories as well as from the more detailed forecasts.

The two comparisons can serve different purposes. A lender wanting payment to increase with the severity of losses might choose distances derived from those losses; one wanting to reward relative anticipation can choose the logarithmic comparison. The transport payment does not by itself reward honest reporting: on this line the schedule depends only on the signs of the cumulative differences, so many different reports produce the same payments.

From a payment to a price

Under Stress’s forecast, Stress expects to receive €2,000 from this contract: a 40% chance of €10,000 against a 20% chance of paying €10,000. Resilience assesses the opposite side under its own probabilities and also expects to gain €2,000. Each expects to profit because each uses its own probabilities; the contract itself is zero-sum, since whatever one lender receives the other pays.

Suppose each lender cares only about expected money, looks at this contract on its own, and any fixed fee is paid on the same date as the comparison payment. Stress would pay Resilience any fee below €2,000 to enter. Resilience would accept any fee above minus €2,000: a negative fee means Resilience pays Stress, and Resilience would pay up to €2,000 for its side. Every fee strictly between those limits leaves both expecting to gain, before costs. Zero is one possible fee, and nothing in this calculation singles it out. An auction, a dealer or plain haggling has to produce the actual terms.

The interval is €4,000 wide, and the width has an informational interpretation. Stress’s expected relative surprise is the Kullback–Leibler divergence of Stress’s forecast from Resilience’s: the average of the logarithmic probability ratio, taken using Stress’s probabilities. It is 0.2 bits here, worth €2,000 at €10,000 per bit. Reversing the forecasts gives Resilience’s expected relative surprise, also 0.2 bits in this example but generally a different number. The sum of the two divergences is called the Jeffreys divergence. Multiplying it by €10,000 per bit gives the width of the bargaining interval.[3]

The mathematics: expected gains and the bargaining interval

Take strictly positive probabilities $p(y)$ and $q(y)$ on a finite set of outcomes $y$. They are Stress’s and Resilience’s forecasts. Write $D(p\|q)=\sum_y p(y)\log_2[p(y)/q(y)]$, in bits, and $c=10{,}000$ euros per bit. The signed payment to Stress is $X(y)=c\log_2[p(y)/q(y)]$, with Resilience receiving $-X(y)$.

Their expected own-side payments, each evaluated under its own forecast, are

$$ \mathbb E_p[X(Y)]=cD(p\|q),\qquad \mathbb E_q[-X(Y)]=cD(q\|p). $$

Both are non-negative by Gibbs’ inequality and positive when the forecasts differ. For Stress in the example,

$$ \begin{aligned} D(p\|q) &=0.2\log_2\frac{0.2}{0.4} +0.4\log_2\frac{0.4}{0.4} +0.4\log_2\frac{0.4}{0.2}\\ &=-0.2+0+0.4=0.2\text{ bits}. \end{aligned} $$

By the symmetry of these particular forecasts, $D(q\|p)=0.2$ bits too.

Let $f$ be a fixed fee from Stress to Resilience, paid on the same date as $X$. Under expected-money valuation, with this claim assessed in isolation and before costs, Stress strictly prefers the trade when $cD(p\|q)-f>0$. Resilience strictly prefers it when $f+cD(q\|p)>0$. Thus both prefer trade exactly when

$$ -cD(q\|p)At either endpoint one side is indifferent. The interval in the example runs from −€2,000 to €2,000. Define the Jeffreys divergence by $J(p,q)=D(p\|q)+D(q\|p)$. The interval width is $cJ(p,q)$, here 0.4 bits multiplied by €10,000 per bit, or €4,000.

Each limit is a reservation value: the fee at which that lender would be indifferent between entering the trade and declining it. An actual institution’s limits can differ from the €2,000 figures. A lender already heavily exposed to utility failures may value the high-band payment as protection and pay more for it. It may also be unable to deposit the collateral securing its own payment, or lack permission under its mandate to hold the contract at all. Calculating an interval is not evidence that orders already exist within it.

The payment is also protection. If Stress’s loan book loses heavily when defaults are high, the contract offsets part of that loss in exchange for paying out when defaults are low, and it does so while the two lenders go on disagreeing about the transition.

A classic result explains why learning that somebody wants to trade can itself change the calculation. Milgrom and Stokey consider traders whose starting allocation is already efficient: judged by their prior beliefs, no trade could benefit one of them without harming another. The traders agree on how new signals relate to the payoff-relevant outcome. If it then becomes common knowledge that a feasible trade is acceptable to all—each knows this, knows that the others know it, and so on—every trader is indifferent to that trade; with strict risk aversion, the trade is zero.[5] In our expected-money example, both lenders prefer a suitable bet before any new private information arrives. Their starting position therefore does not meet the theorem’s efficiency condition. Real demand could also come from hedging needs, mandates and existing exposures.

Why not use ordinary securities?

The lenders can already act on their views by lending more or less, or by trading bonds. But a bond’s return depends on much more than the number of defaults in the sector: interest rates, the amount recovered after a default, and the fortunes of its particular issuer. The proposed contract instead makes payment depend on the specified comparison of the sector forecasts.

That does not make its cash flow new. If existing securities can reproduce the payment in each of the three outcomes, they replicate it, and complete markets in those outcomes would do exactly that. What replication takes as given is everything the lenders have just had to agree: the forecasts, the observation, the fixing date, the rule and the unit. A dedicated contract organises and authenticates that specification and makes it easier to quote and transfer. Differences of logarithmic scores are not new either; prediction markets built on scoring rules already use them.[6] What is proposed here is the combination of declared forecasts, a financial purpose, a settlement procedure and the governance to run it.

A derivative is useful here because the lender can add a payment conditional on the agreed observation while keeping its loans. Buying a utility’s equity would bring ownership and exposure to the rest of its business; it would not, by itself, give this forecast comparison a separate payment. The same payment clause could also be included in a funded note or in a loan agreement. A swap creates the stated exchange of payments; an option could give a one-sided entitlement in return for a premium. These forms differ in funding, liability and who is permitted to hold them.

Could publishing the forecasts do the same job? Publication allows scrutiny but moves no money; the contract attaches a payment to the comparison.

The liability limit changes the comparison

Our example has a reassuring feature: no payment can exceed €10,000 in either direction. That comes from the forecasts, not from the rule. With finer outcomes, probability ratios can become enormous. If Stress gives some outcome a 10% chance and Resilience gives it one in a million, the payment if it happens is about €166,000. If one side gives an outcome zero probability and the other gives it positive probability, the raw log payment is infinite.

The expected-payout limit in the stake example does not prevent this. It restricts an average under Resilience’s forecast, not the largest amount Resilience could owe. The terms must address that liability separately: restrict the permitted probability ratios, specify a maximum payment and secure it with collateral, or change the payment rule. A symmetric cap, for example, replaces a payment larger than €20,000 with €20,000 and a payment below −€20,000 with −€20,000. Capping the payment can destroy honest reporting. In the four-category example in the next section, a cap of €20,000 in either direction allows Stress to raise its expected payment from about €2,300 to about €4,000 by reporting probabilities different from its actual beliefs.

One alternative keeps the logarithmic rule but sets a minimum probability that either lender may give any outcome. If that minimum is one in a thousand, which requires at most a thousand possible outcomes, no payment can exceed about €99,600 in either direction, and honest reporting remains optimal for any belief that respects the minimum. Another route is a bounded proper rule such as the Brier score, which penalises the squared differences between the reported probabilities and the outcome (one for what happened, zero for everything else). It gives a firm liability limit, at the price of the links to evidence and information that the logarithm provides.[2]

Unbounded liability can arise even when both forecasts give every outcome some probability. With infinitely many possible outcomes, every realised payment can be finite while the expected payment under one of the forecasts is infinite. “We’ll work out the bill when it arrives” is no answer for the institution that has to value and fund the promise today.

A lender might instead want larger payments on very high default counts. Simply multiplying the logarithmic score by a larger weight on those outcomes would reward exaggerating their probabilities; tail-sensitive proper scores need a different construction. Quotes for contracts with different tail weights and maturities could help show which outcomes and horizons participants most want protection against, provided the forecasts, observations and valuation conventions remain comparable. Otherwise a change in the quote may reflect a changed measuring rule, a dealer’s existing positions or financing costs.

What settlement leaves unresolved

Suppose for a moment that the lenders agreed on the probability of each band but disagreed about which utilities would fail. The three-band contract would pay nothing in every outcome: the probabilities entering the payment would be identical. Yet the disagreement could matter greatly to a lender whose loans were concentrated in the borrowers Stress considered vulnerable.

Return to the original probabilities and consider a more detailed default count. Divide the high band into ten to fourteen defaults and fifteen or more, as before, but now let the lenders disagree about that detail. Stress still assigns 40% to the high band, divided equally into 20% and 20%. Resilience assigns 20% in total, divided into 18% and 2%. Thus, conditional on reaching ten defaults, Stress considers the two new categories equally likely, whereas Resilience gives nine tenths of its high-band probability to ten-to-fourteen and only one tenth to fifteen-or-more. This is different from the earlier subdivision on which they agreed.

The original three-band contract is unchanged, because it still uses only the total probabilities of the high band. A contract using the finer record raises Stress’s expected payment from €2,000 to about €4,950. The additional €2,950 reflects their different conditional predictions within the high band. The three-band contract cannot pay for that difference: once defaults reach ten, it pays the same amount whatever the count. The split is exact: Stress’s expected payment on the detailed record equals the part the three-band record captures plus the expected disagreement left inside the bands, both computed with Stress’s probabilities.[3]

The finer comparison also increases the largest possible payment in this example. If fifteen or more utilities fail, Stress assigned that outcome ten times Resilience’s probability, and Resilience owes about €33,200 instead of €10,000. In general, refining the record can raise either side’s largest possible payment but never lower it: the ratio on a coarse band is an average of the ratios on its finer parts, so at least one finer ratio is at least as extreme. A richer record can improve the comparison while making the liability harder to fund.

The disagreement omitted by the settlement record is different from the representational residue of Part I. That residue concerned choices left open while building a forecast, such as which outcomes to distinguish or which similarities to rely on. Here the forecasts are already specified, and we are asking what their probabilities imply for one particular observation. Even a complete record of defaults can leave their causes unresolved, including the role of policy and what would have happened under another financing decision.

The contract also gives Stress a way to assess a possible source of information. Suppose Stress can receive an early report on refinancing conditions before fixing its forecast, while Resilience’s forecast and the other terms stay fixed. Stress can use its own model to compare the expected payment with and without that report. Averaged over the reports and default outcomes Stress considers possible, the improvement equals the report’s mutual information with the default band, multiplied by €10,000 per bit. Mutual information is the average reduction in uncertainty about the band after observing the report.[3]

This values information for the specified contract. Blackwell’s comparison of experiments asks a broader question: would one information source be at least as useful in every decision problem? In the finite setting, this holds when the second source can be reproduced from the first by a random transformation of the observed message, using the same transformation regardless of the unknown state. The transformation can discard or obscure information but cannot consult the state itself. This is called a garbling. A larger mutual-information value for one particular model and question does not establish that stronger ordering.[7]

The mathematics: what settlement records and what a signal adds

Notation and the decomposition. Let $Z$ be a detailed outcome on a finite set, and let $Y=g(Z)$ be the coarser category used for settlement. In the example $Z$ has four default bands and $Y$ combines the two high bands into one. Stress and Resilience specify strictly positive laws $p_Z$ and $q_Z$. Their induced probabilities of $Y$ are $p_Y$ and $q_Y$. Write $D(P\|Q)=\sum_z P(z)\log_2[P(z)/Q(z)]$, in bits, and $c=10{,}000$ euros per bit.

For $y=g(z)$, factor $p_Z(z)=p_Y(y)p(z\mid y)$, and similarly for $q$. Taking the logarithmic ratio and averaging under Stress gives

$$ D(p_Z\|q_Z) =D(p_Y\|q_Y) +\sum_y p_Y(y)D\big(p(\cdot\mid y)\|q(\cdot\mid y)\big). $$

The first term is the expected comparison available to a contract on $Y$. The second is the expected additional comparison available if $Z$ is recorded too. Both are non-negative.

The example. Within the high band Stress gives conditional probabilities $(0.5,0.5)$ to ten-to-fourteen and fifteen-or-more defaults; Resilience gives $(0.9,0.1)$. Their conditional divergence from Stress’s perspective is

$$ 0.5\log_2\frac{0.5}{0.9} +0.5\log_2\frac{0.5}{0.1} \simeq-0.424+1.161=0.737\text{ bits}. $$

Stress reaches the high band with probability 0.4, so its expected additional comparison is $0.4\times0.737\simeq0.295$ bits. Adding the original 0.2 gives about 0.495 bits: €2,000 from the coarse contract and about €2,950 more from the refinement. Evaluating the reversed comparison under Resilience gives €2,000 plus about €1,060. The last category has signed payment $c\log_2(0.2/0.02)\simeq3.322c$, or about €33,200.

A signal received before reporting. Let $S$ be a finite-valued signal with possible values $s$, and let Stress specify a joint law $p_{S,Y}$. Before receiving it, Stress forecasts $p_Y(y)$; after receiving $s$, Stress can report $p(y\mid s)$. Assume positive conditional probabilities, keep Resilience’s forecast $q_Y$ and all payment terms fixed, and average under Stress’s joint law. Then

$$ \begin{aligned} c\,\mathbb E_p\!\left[\log_2\frac{p(Y\mid S)}{q_Y(Y)}\right] &=c\,\mathbb E_p\!\left[\log_2\frac{p(Y\mid S)}{p_Y(Y)}\right] +c\,\mathbb E_p\!\left[\log_2\frac{p_Y(Y)}{q_Y(Y)}\right]\\ &=cI(S;Y)+cD(p_Y\|q_Y). \end{aligned} $$

The signal’s mutual information, in bits, is

$$ I(S;Y)=\sum_{s,y}p(s,y)\log_2\frac{p(s,y)}{p_S(s)p_Y(y)}. $$

It is the relative entropy between Stress’s joint law and the law making the signal and outcome independent while retaining their marginal probabilities. Equivalently,

$$ \begin{aligned} H(Y)&=-\sum_y p_Y(y)\log_2 p_Y(y),\\ H(Y\mid S)&=\sum_s p_S(s)\left[-\sum_y p(y\mid s)\log_2 p(y\mid s)\right],\\ I(S;Y)&=H(Y)-H(Y\mid S). \end{aligned} $$

Since $H(Y\mid S)\ge0$, the expected-payoff improvement $cI(S;Y)$ cannot exceed $cH(Y)$. Stress’s probabilities $(0.2,0.4,0.4)$ give $H(Y)\simeq1.52$ bits, so no signal can raise Stress’s expected payment by more than about €15,200.

A quote is not a consensus probability

Once the contract trades, someone will want to infer the probability “the market” gives to the adverse outcome. A single quote does not generally supply it. Even in a model that prices by expected payments alone, the value of our signed contract depends on the high-band probability minus the low-band probability. Many three-band forecasts have the same difference. The quote therefore need not identify the whole forecast, even before considering the other reasons an institution might value the payment.

A larger collection of quotes can constrain the interpretation more tightly, but it still requires an economic model. The absence of arbitrage requires consistent prices for identical payments. It does not choose whose probabilities become a public forecast. That depends on the participants, their capacity to bear risk, their resources and the arrangement through which they trade.

One specific economy shows what such a model looks like. Suppose participants can trade a payment for every possible outcome, and each has exponential utility, meaning that their tolerance for risk, measured in euros, does not change with their wealth. Competitive trading then pools the probabilities they trade on in a particular way: it takes a weighted average of the logarithms of their probabilities, weighting each participant by risk tolerance, converts the result back from logarithms, and rescales it so that it sums to one. Prices also depend on how total resources vary across outcomes, so even here the pooled forecast and the prices are two different objects.[8]

The choice of economy matters. Give two participants equal risk tolerance and event probabilities of 80% and 99%. Their ordinary average is about 90%. This economy instead averages the logarithms of their odds, 4 to 1 and 99 to 1, with equal weight, and converting that average back to a probability gives about 95%. If aggregate resources are the same whether or not the event happens, 95% is also the event’s normalised state price: the price of one euro paid if the event happens, divided by the price of one euro paid for certain. Change the risk-sharing arrangement and the pooling rule can change too.[8]

The mathematics: where the geometric pool comes from

Consider finitely many outcomes $\omega$ and participants indexed by $i$. Participant $i$ has a strictly positive probability law $p_i(\omega)$ and utility $u_i(x)=-e^{-x/\tau_i}$, with constant absolute risk tolerance $\tau_i>0$, measured in euros. Its final consumption or wealth in outcome $\omega$ is $x_i(\omega)$. All outcome-contingent transfers are tradable and there are no further position constraints.

Let $\pi(\omega)>0$ be today’s price of one euro paid only in outcome $\omega$, and $w_i$ participant $i$’s initial budget at those prices. The participant maximises expected utility subject to $\sum_\omega\pi(\omega)x_i(\omega)\le w_i$. For an interior optimum, a positive budget multiplier $\lambda_i$ gives

$$ \frac{p_i(\omega)}{\tau_i}e^{-x_i(\omega)/\tau_i} =\lambda_i\pi(\omega), $$

and hence, taking natural logarithms,

$$ x_i(\omega)=\tau_i\left[\ln p_i(\omega)-\ln\pi(\omega) -\ln(\lambda_i\tau_i)\right]. $$

Write $e(\omega)$ for total resources in that outcome and $T=\sum_i\tau_i$ for aggregate risk tolerance. Market clearing requires $\sum_i x_i(\omega)=e(\omega)$. Summing the preceding equation yields

$$ \pi(\omega)\propto \left[\prod_i p_i(\omega)^{\tau_i/T}\right]e^{-e(\omega)/T}. $$

The normalised geometric pool is the probability law

$$ \overline p(\omega)= \frac{\prod_i p_i(\omega)^{\tau_i/T}} {\sum_{\omega'}\prod_i p_i(\omega')^{\tau_i/T}}. $$

State prices contain an additional resource factor: payments in outcomes with fewer total resources are valued more highly. Their normalisation is $\pi(\omega)/\sum_{\omega'}\pi(\omega')$, and it coincides with $\overline p$ only if $e(\omega)$ is constant. Budgets determine the multipliers and the individual allocations; with beliefs, risk tolerances and aggregate resources fixed, they do not change this pool.[8]

For two equally risk-tolerant participants giving a binary event probabilities 80% and 99%,

$$ \overline p(\text{event})= \frac{\sqrt{0.8\times0.99}} {\sqrt{0.8\times0.99}+\sqrt{0.2\times0.01}} \simeq0.952. $$

Their event odds are $0.8/0.2=4$ and $0.99/0.01=99$. Equal weighting of log odds gives pooled odds $\sqrt{4\times99}$; dividing those odds by one plus themselves gives the same probability.

Quotes at several maturities could also be used to estimate how quickly participants expect a disagreement to diminish, but only through a dynamic model linking future forecast differences to the quoted payments, and the estimate would depend on that model as much as on the quotes.

Can a record of payments become evidence?

A five-year contract gives one five-year observation. Marking it to market every day does not turn it into thousands of independent tests. Learning faster requires genuinely new observations, with the forecasts committed before each one arrives.

There is a rigorous way to turn sequential forecast comparisons into an evidence record. Return to the bet that pays a non-negative multiple of its stake, rather than the signed contract in the table. Start an account with one euro. Before each observation, choose a bet whose expected payout under Resilience’s forecast, given the record so far, is no greater than the account balance being staked. Reinvest the payout, add no outside money, and never allow a negative balance. Under Resilience’s model, the expected next balance is then at most the present balance. A sufficiently large increase can be evidence against that model.[9]

Ville’s inequality bounds how often this procedure could produce a misleadingly large balance. If Resilience’s model generates the observations, the probability that the account ever reaches €100 is at most one in a hundred. The bound already includes the possibility of checking after every observation and stopping when that balance is reached. It is not a 99% posterior probability that Resilience is wrong. Independent observations are unnecessary: the forecast for each new observation must condition on the available past, and the bet must satisfy the expected-payout restriction under that conditional forecast. This allows, for example, a forecast of this year’s defaults to depend on last year’s economy.[9]

The bookkeeping is part of the result. Forecasts must come before outcomes, and the rules deciding which contracts count must come before the results. Losses stay in the account. Restarting after a bad year, or publishing only the contracts that went well, changes what is being tested. A proprietary method can choose the bets, but an auditor has to be able to check that the complete account followed the declared rules. A profitable trading book built with arbitrary prices and leverage is not an account of this kind, however good its returns.

How long would this take for our lenders? Imagine a fresh, statistically identical cohort at each five-year renewal, with the same forecasts and independent results. This is a deliberately simple, stationary experiment. The evidence account is a separate, notional ledger that reinvests its balance: with likelihood-ratio bets, each high-band outcome doubles it and each low-band outcome halves it, while the signed contract simply pays €10,000 one way or the other. The account first exceeds €100 when Stress’s cumulative comparison payments reach €70,000. Under Stress’s forecast, that takes 35 renewals on average: 175 years. The median is 29 renewals, and in about one case in nine it takes more than 60, or over three centuries. Under Resilience’s forecast, the chance of ever crossing is 1 in 128.

The 175 years is the expected wait for one evidence threshold in a sparse experiment. The contract itself pays after five years, and its protection can affect Stress’s decisions from the day the terms are agreed. Each observation contributes 0.2 bits of evidence on average under Stress, the same 0.2 bits that set Stress’s expected payment. Raising the monetary unit enlarges the transfers; it does nothing to speed up the evidence.

A well-kept account can produce strong evidence against Resilience’s forecast on the observations used for settlement. It cannot, however, tell apart explanations that give those observations the same probabilities. The rate at which genuinely informative observations arrive remains part of the investment problem.

The mathematics: Ville's inequality and the 175 years

The account. Measure the account balance in units of its initial one-euro stake and put $K_0=1$. Before observation $t$, let $q_t(y)$ be Resilience’s conditional probability of outcome $y$, given everything observed so far. Choose non-negative multiples $m_t(y)$, using only that past information, with $\sum_y q_t(y)m_t(y)\le1$. After observing $y_t$, set $K_t=K_{t-1}m_t(y_t)$. Under the law $Q$ specified by Resilience’s conditional forecasts,

$$ \mathbb E_Q[K_t\mid\text{past}]\le K_{t-1}. $$

Such a process is called a non-negative supermartingale: given its past, its next balance has expectation no greater than its current balance.

Ville’s inequality. Under the reference law $Q$, for any chosen error bound $0<\alpha<1$,

$$ \Pr_Q\big(K_t \ge 1/\alpha \text{ for some } t\big) \;\le\; \alpha . $$

Proof. Let $\tau$ be the first time the account reaches $1/\alpha$, and fix a horizon $n$. Stopping at the earlier of $\tau$ and $n$ cannot raise the expected balance above its starting value of 1 (the optional stopping theorem), and on the event $\tau \le n$ the balance is at least $1/\alpha$. So $\Pr(\tau \le n)/\alpha \le 1$ for every $n$, and letting $n$ grow gives the result. With $\alpha = 1/100$, the account reaches €100 with probability at most 1%.

The lenders. For independent renewals with fixed forecasts $p=(0.2,0.4,0.4)$ and $q=(0.4,0.4,0.2)$, use $m(y)=p(y)/q(y)$. Let $c=10{,}000$ euros per bit and let $L_n$ be the cumulative signed comparison payment to Stress through $n$ renewals. Then $L_n=c\log_2K_n$. Each renewal adds −1, 0 or +1 to $\log_2K_n$. The account crosses 100 when $\log_2K_n\ge\log_2 100\simeq6.64$. Its logarithm is an integer in this example, so the first crossing is at seven bits, a balance of 128 times the initial stake and a net signed payment of €70,000.

Under Stress’s law $P$, the increments −1, 0, +1 have probabilities 0.2, 0.4, 0.4 and mean 0.2. Let $S_n=\log_2K_n$, and let $T$ be its first hitting time of seven. To apply Wald’s identity at this possibly unbounded time, first show that $\mathbb E_P[T]$ is finite. The event $T>n$ implies $S_n<7$. For $n\ge70$, bounded independent increments give the Hoeffding bound $\Pr_P(T>n)\le e^{-n/200}$. Summing these tail bounds establishes finite expectation. Since the walk reaches exactly seven at $T$, Wald’s identity now gives

$$ 7=0.2\,\mathbb E[T],\qquad \mathbb E[T]=35. $$

The median and tail probability can also be obtained without simulation. Under Stress’s law, let $v_n(j)$ be the probability of being at $j<7$ after $n$ renewals without having reached seven. Start with $v_0(0)=1$, use

$$ v_{n+1}(j)=0.2v_n(j+1)+0.4v_n(j)+0.4v_n(j-1),\quad j<7, $$

and discard mass absorbed at seven. Then $\Pr(T>n)=\sum_{j<7}v_n(j)$. This deterministic recursion gives median 29 and $\Pr(T>60)=0.11329248$.

Under Resilience’s law, the increments −1, 0, +1 have probabilities 0.4, 0.4, 0.2. Ignoring the zero increments leaves an upward probability of one third and a downward probability of two thirds. The probability that this downward-drifting walk ever reaches seven is $[(1/3)/(2/3)]^7=1/128$, about 0.78%, consistent with Ville’s upper bound of 1%.

What “eventually” means

“But this long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task if in tempestuous seasons they can only tell us that when the storm is long past the ocean is flat again.”

— John Maynard Keynes, A Tract on Monetary Reform (1923), Chapter III, p. 80.[10]

Suppose the renewals continue without a final date. For this next calculation, each lender specifies all its forecasts in advance and treats the renewals as independent. The probabilities may change from one renewal to the next, but they do not depend on results already observed, and every possible outcome has positive probability under both forecasts.

There are two questions. Could a rule given the complete infinite history identify which of the two probability models generated it, with probability one of being correct under either model? And do the expected net comparison payments, added up without discounting, remain finite? Kakutani’s theorem answers the first question; the expected-payment calculation supplies the second. Together they give three cases.[11]

No error-free identification, finite expected net total. Even with the complete infinite history, no rule can select between the two models with probability one of being correct under each. The cumulative signed payments converge to a finite net amount, and each lender’s expected own-side total is finite. The forecasts may differ at every renewal; the box gives an example in which those differences decrease sufficiently quickly.

No error-free identification, an infinite expected net total on at least one side. The statistical conclusion is the same, and the net total along almost every history is still finite. But rare, very large transfers make at least one lender’s expected net total infinite. This concerns undiscounted payments. In the box’s example, ordinary geometric discounting makes the expected discounted payments finite.

Identification in the infinite-history limit. Here the complete history does admit a rule that identifies the generating model with probability one under either law. As observations accumulate, the logarithmic comparison tends to plus infinity under Stress’s law and minus infinity under Resilience’s. The lenders’ unchanged forecasts, repeated independently, give this case. All three cases assume that one of the two forecasts is the true law.

The first two cases still allow strong evidence at an error rate chosen in advance; what they rule out is certainty. And even in the third case, nothing guarantees a verdict by any given date: every finite run of results remains possible under both forecasts. Our two parties did not need one. They agreed a five-year payment and can decide whether to renew it when the time comes.

The contract can settle long before the argument does.

The mathematics: infinite histories and expected net payments

Setup. At renewal $k$, let $p_k$ and $q_k$ be Stress’s and Resilience’s strictly positive probability vectors on a finite outcome set. All these vectors are specified before any results are observed. Renewals are independent under both models, so the laws of the complete history are the product measures $P=\bigotimes_{k\ge1}p_k$ and $Q=\bigotimes_{k\ge1}q_k$. Adaptive conditional forecasts need not produce product laws; the preceding supermartingale argument allows such adaptation, whereas the product theorem here does not automatically extend to it.

For outcomes $Y_1,\ldots,Y_n$, define the likelihood ratio and cumulative signed payment

$$ R_n=\prod_{k=1}^n\frac{p_k(Y_k)}{q_k(Y_k)},\qquad L_n=c\log_2R_n, $$

where $c>0$ is the euro amount per bit. Write $D$ for relative entropy in bits. The Hellinger affinity

$$ \rho_k=\sum_y\sqrt{p_k(y)q_k(y)} $$

lies in $(0,1]$, with value one exactly when the two forecasts agree.

Kakutani’s distinction. The complete-history laws are either equivalent, meaning that they give probability zero to the same events, or mutually singular, meaning that an event has probability one under one law and zero under the other. Equivalence holds exactly when

$$ \prod_{k\ge1}\rho_k>0, \quad\text{equivalently}\quad \sum_{k\ge1}(1-\rho_k)<\infty. $$

If the product is zero, the laws are singular. In the equivalent case $R_n$ converges under either law to a finite, strictly positive limit, so $L_n$ converges to a finite net total. In the singular case $R_n\to\infty$ under $P$ and $R_n\to0$ under $Q$, almost surely; the signed logarithmic total diverges in the corresponding direction.[11]

Unchanged forecasts. For the lenders,

$$ \rho=\sqrt{0.2\times0.4}+0.4+\sqrt{0.4\times0.2} \simeq0.966<1. $$

Thus $\rho^n\to0$, and their product laws are singular. The likelihood-ratio account grows without bound almost surely under Stress’s model and tends to zero almost surely under Resilience’s.

Decreasing differences and finite expected totals. Fix a strictly positive vector $p$ and a nonzero vector $h$ whose coordinates sum to zero. Assume $p(y)+h(y)>0$ for each outcome, and set $p_k=p$, $q_k=p+h/k$. Each forecast is then strictly positive. Taylor expansion gives $1-\rho_k$, $D(p_k\|q_k)$ and $D(q_k\|p_k)$ of order $1/k^2$. The series are finite: the complete-history laws are equivalent and both expected net totals are finite.

Equivalent histories with an infinite undiscounted expectation. For $k\ge1$, give each renewal a rare event and its complement. Stress assigns the rare event probability $a_k=1/(k+1)^2$, while Resilience assigns $b_k=a_ke^{-k}$. Thus $p_k=(a_k,1-a_k)$ and $q_k=(b_k,1-b_k)$. Their entries are positive and $1-\rho_k$ is of order $1/k^2$, so the history laws are equivalent. However,

$$ \begin{aligned} D(p_k\|q_k) &=a_k\log_2\frac{a_k}{b_k} +(1-a_k)\log_2\frac{1-a_k}{1-b_k}\\ &=\frac{k}{(k+1)^2\ln2}+O(k^{-2}). \end{aligned} $$

The leading term is of order $1/k$, whose sum diverges. Under Stress, only finitely many rare events occur almost surely because $\sum_k a_k<\infty$; the same is true under Resilience. Each realised rare-event payment is finite, and the common-outcome payments have a summable magnitude of order $1/k^2$. The net total is therefore finite almost surely, even though Stress assigns it an infinite expectation.

The separate expectation question. At each finite horizon,

$$ \mathbb E_P[L_n]=c\sum_{k=1}^nD(p_k\|q_k),\qquad \mathbb E_Q[-L_n]=c\sum_{k=1}^nD(q_k\|p_k). $$

Finite total Jeffreys divergence, the sum of both directional series, implies equivalent histories and finite own-side expectations of the terminal net amount. Equivalence alone does not imply finite expectations, as the rare-event example shows.

In that example, discounting renewal $k$’s payment by $\beta^k$, for $0<\beta<1$, makes the expected absolute discounted payments under Stress of order $\beta^k/k$, a summable series. An infinite undiscounted expectation therefore does not imply an infinite present value.

Who writes the test?

“[A]s a rule, regulation is acquired by the industry and is designed and operated primarily for its benefit.”

— George J. Stigler, “The Theory of Economic Regulation” (1971), p. 3, excerpt.[12]

Our lenders needed a default definition before they needed a logarithm. Once that definition moves payments, spreads or capital requirements, people have reasons to influence it. Excluding a particular restructuring, moving the observation date or changing how mergers are treated can shift money without changing anyone’s view of the transition. Stigler argued that an industry tends to acquire the regulation written for it. The definitions behind a contract like this one face the same pressure, and even the technical appendix can become a site of negotiation.

A proper score makes honesty optimal when the expected score is all the forecaster cares about. In a market, a declared forecast can also move the entry price, the collateral required or the value of another position the forecaster holds, and those effects belong in the incentive calculation too. Research on market scoring rules shows that honest reporting is optimal for a trader who considers only the current trade, but not always for one who plans later trades; it depends on how the traders’ information is related.[6] A good score therefore has to come with a price-formation procedure and controls against these outside incentives: depending on the design, disclosure, position limits or penalties.

The observed defaults and their reporting can also respond to the contract. A lender might delay recognising distress until after the fixing date, or change refinancing decisions in ways that alter which utilities default. Some responses could improve the underlying situation; others could improve only the recorded result. Goodhart’s warning concerns a statistical regularity breaking down when it is used for control.[13]

So the rulebook needs a stated purpose and a revision history. Whom is the protection meant to serve? Which exposures does the chosen count leave out? Why trust this data provider, and what happens when its methods change? Each amendment should keep the old definition, the new one, the reason for the change and the treatment of contracts already outstanding. An institution that asks lenders to commit to forecasts in advance cannot keep the right to change its own test after seeing the results.

Access matters too. A bank with a technical team can argue for its preferred default definition in detail. The beneficiaries of a pension fund, whose protection depends on that definition, rarely have the time or expertise to argue back. A consultation that is formally open is not enough: the people who bear the unresolved risks need a practical way to contest how their exposure has been written down.

Where it could be used first

None of this needs a public exchange to begin. Take a bank that publishes a view of the transition and also lends to the sector. Its published view and the assumptions in its lending can be compared, but only once both are translated into forecasts of something observable: defaults, investment, or another agreed record. A contract can then attach money to the comparison. It would not show whether the published view was sincere. And a payment between two parties does not by itself benefit anyone else; that requires useful insurance, a better allocation of capital, or investment that happens on better terms because of the contract.[5]

The utilities also offer observations closer to the physical transition itself: electricity delivered, fuel consumed and emissions under an agreed accounting boundary. Jean-Marc Jancovici has emphasised that energy’s physical importance is not measured by its share of monetary expenditure.[15] A disagreement about whether these utilities can maintain output while reducing fuel use could be written as competing forecasts of those quantities. An authenticated annual record could then test that particular claim before a five-year default count arrives.

The activity covered by the measurement matters. A fall in emissions can result from closing a plant rather than improving its technology, so the record may need to specify electricity output, which plants are included and how that coverage changes. Physical quantities provide evidence against which forecasts can be assessed; the relationship between those quantities and cash flows still needs an economic explanation. A payment for forecast performance also differs from charging an emitter for environmental damage. The promising connection is specific: make financial commitments depend on declared, verifiable predictions about energy, materials and production, while separately identifying who bears the social costs.

Adoption could proceed in stages: internal diagnostics first, then bilateral contracts, and a public venue only if demand justifies one. At each stage the question is what changes in actual decisions. Does the new quote explain lending terms, collateral or investment beyond what ordinary prices already explain? Do the data and dispute procedures hold up when used by parties with every reason to challenge them? Those are questions about institutions, and the mathematics of the payment rule cannot answer them.

What has actually been settled?

Five years on, eleven of the hundred utilities have defaulted. Under the original three-band contract, Resilience pays Stress €10,000.

Resilience still attributes most of the failures to interest rates rather than the industrial transition. The payment need not persuade Resilience otherwise: both lenders had allowed for the high-default outcome, and their agreement was about how their different forecasts would be compared. What matters now is that the obligation has been honoured. Stress received money in the outcome to which it had assigned more weight, on terms that Resilience was willing to accept before either knew the result.

To judge the value of that arrangement, however, we should return to the day it was signed. A lender exposed to utility failures could use the high-band payment to offset part of its losses, accepting the cost of the opposite payment when defaults stayed low. At a suitable price and scale, that protection could change which loans it was prepared to keep or make. A counterparty with a different forecast or a different balance sheet could take the other side. Their disagreement would then help determine how the risk was carried, while research into the transition continued. The contract could be useful years before the evidence became conclusive.

The same construction makes the comparison itself available for scrutiny. Instead of looking at two lending books and trying to infer what their owners believe, a third party can examine the forecasts, the observation on which they differ, and the terms on which that difference is being exchanged. Further research can improve those forecasts; better evidence can reveal a distinction the original settlement missed; a new contract can make that distinction available to investors. Each improvement has an identifiable consequence for what can be priced, held or transferred.

This is the constructive promise of contracts like this one: giving a financially consequential difference in understanding an exposure of its own, with terms that others can inspect and, where there is demand, trade. Whether a particular design is worth adopting depends on the decisions it improves and the protection it provides relative to the alternatives. The wider ambition is a financial system in which the work of understanding an uncertain future has more ways to affect the commitments made to it. Better understanding should be able to change what gets financed before it changes everybody’s mind.

References

[1] Henri Poincaré, Science and Hypothesis, translated by W. J. Greenstreet (Walter Scott Publishing, 1905), Chapter III, p. 50.

[2] José M. Bernardo, “Expected Information as Expected Utility”, The Annals of Statistics 7(3), 686–690 (1979), especially pp. 688–689; Tilmann Gneiting and Adrian E. Raftery, “Strictly Proper Scoring Rules, Prediction, and Estimation”, Journal of the American Statistical Association 102(477), 359–378 (2007).

[3] Yury Polyanskiy and Yihong Wu, Information Theory, MIT 6.441 course notes (2016), Chapters 2–3.

[4] N. N. Čencov, Statistical Decision Rules and Optimal Inference (American Mathematical Society, 1982); Hiroshi Nagaoka, “The Fisher Metric as a Metric on the Cotangent Bundle”, Information Geometry (2024), Theorem 7.1, DOI: 10.1007/s41884-023-00126-9.

[5] Paul Milgrom and Nancy Stokey, “Information, Trade and Common Knowledge”, Journal of Economic Theory 26(1), 17–27 (1982); Jack Hirshleifer, “The Private and Social Value of Information and the Reward to Inventive Activity”, The American Economic Review 61(4), 561–574 (1971), especially p. 561.

[6] Robin Hanson, “Combinatorial Information Market Design”, Information Systems Frontiers 5(1), 107–119 (2003); Yiling Chen, Stanko Dimitrov, Rahul Sami, Daniel M. Reeves, David M. Pennock, Robin Hanson, Lance Fortnow and Rica Gonen, “Gaming Prediction Markets: Equilibrium Strategies with a Market Maker”, Algorithmica 58(4), 930–969 (2010; online publication 2009).

[7] David Blackwell, “Equivalent Comparisons of Experiments”, The Annals of Mathematical Statistics 24(2), 265–272 (1953); Henrique de Oliveira, “Blackwell’s Informativeness Theorem Using Diagrams”, Games and Economic Behavior 109, 126–131 (2018).

[8] Michail Anthropelos and Constantinos Kardaras, “Equilibrium in Risk-Sharing Games”, Finance and Stochastics 21, 815–865 (2017), especially Sections 2.3–2.6 and Theorem 2.2.

[9] Glenn Shafer, Alexander Shen, Nikolai Vereshchagin and Vladimir Vovk, “Test Martingales, Bayes Factors and p-Values”, Statistical Science 26(1), 84–101 (2011).

[10] John Maynard Keynes, A Tract on Monetary Reform (Macmillan, 1923), Chapter III, p. 80.

[11] Shizuo Kakutani, “On Equivalence of Infinite Product Measures”, Annals of Mathematics 49(1), 214–224 (1948).

[12] George J. Stigler, “The Theory of Economic Regulation”, The Bell Journal of Economics and Management Science 2(1), 3–21 (1971), p. 3.

[13] C. A. E. Goodhart, “The ECB and the Conduct of Monetary Policy: Goodhart’s Law and Lessons from the Euro Area”, Journal of Common Market Studies 44(4), 757–778 (2006), DOI: 10.1111/j.1468-5965.2006.00661.x; Donald MacKenzie, An Engine, Not a Camera: How Financial Models Shape Markets (MIT Press, 2006).

[14] Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning 11(5–6), 355–607 (2019); author version arXiv:1803.00567v4, especially Section 2.6, Remark 2.30, equations (2.37)–(2.38), Section 6.1, Proposition 6.1 and equations (6.1)–(6.2), and Sections 8.1–8.2.1.

[15] Jean-Marc Jancovici, “L’énergie, de quoi s’agit-il exactement ?”, author website, first published 1 August 2011; accessed 26 September 2026.

[16] John Duchi, Statistics 311 / Electrical Engineering 377: Information Theory and Statistics, Stanford University lecture notes, Chapter 8, Sections 8.1–8.3.

What Can a Market See?

Fri, 25 Sep 2026 00:00:00 +0200

Suppose you really do understand something the market does not. Why should that make you richer?

“Trade on it” is the obvious answer. But an insight does not arrive with a suitable security attached. You may anticipate a technological change while finding that the companies best placed to benefit are already expensive. You may understand a risk that your fund is forbidden to hedge, or distinguish two possible futures that every available investment pays exactly the same way. In each case there is work to do between understanding the future and finding a worthwhile position.

There is a question about the financial system here, as well as about the individual investor. Markets are often expected to reward sound judgment and withdraw capital from people who are repeatedly wrong. Friedman argued that unless businessmen behaved approximately as return maximisers, “it seems unlikely that they would remain in business for long.”[1] His claim concerned the behaviour that competition selects, rather than the calculations businessmen consciously perform. The related expectation that competition favours more accurate beliefs is known as the market selection hypothesis.[2] It depends on what people can do with their judgments. A comparison conducted through investment returns will recognise the differences expressed in those investments; a difference excluded from every permissible trade has no such opportunity to prove itself.

What, then, can a market actually see? Here, see means more than “appear in today’s price”: a distinction becomes financially visible when it can change a valuation, a feasible position, or a later decision. To understand that visibility, we have to look at the securities, the rules under which they can be held, and the models through which investors interpret the evidence. An investor may need a better model, permission to take a position, a different security, or a more effective way to manage one. We also have to ask how much of the evidence a good decision really needs.

Which part of a forecast can you trade on?

Consider a small, deliberately restricted market. You can put money into a fixed basket of loans or leave it in cash. You cannot buy the loans separately, and you must hold your investment until repayment. For now, suppose your only concern is the wealth those investments will deliver. Two analysts disagree about which borrowers are likely to default. Yet they give exactly the same probabilities to every possible total repayment from the basket. Their disagreement concerns the identities behind the losses, not the distribution of the amount you receive. Under either forecast, every permitted mixture of the basket and cash has the same prospects. Paying to find out which of these two analysts is right would not improve this particular investment choice.

Now let the investor buy the loans separately. The identities become useful: the investor can avoid one borrower and lend to another. The evidence has not changed. The available decision has.

There is another way to make the identities matter. Suppose the investor also owns a business whose income falls when a particular borrower gets into trouble. The investor now cares about the basket’s repayment alongside that business income: do both fall together, or does one provide money when the other is doing badly? Agreement about the basket alone no longer settles its usefulness in the investor’s overall finances. An investment can look quite different once we know what it is supposed to pay for.

This is why the financial value of a forecast depends on the decision, the securities and the investor’s other commitments. It is also why agreement on a price is weaker than agreement about the future. One buyer expects generous cash flows; another accepts a less optimistic forecast because the payment would be especially useful when other investments disappoint. They may accept the same price for different reasons. Even prices across many different payoffs must be interpreted through those reasons before they can be read as probabilities of what will happen. Paying more for money received in a crisis, for example, can reflect how badly that money is needed as well as how likely the crisis is.

The holding period belongs in the example too. Allow sales before repayment, and a report about an individual borrower might become useful because it changes the price at which the basket can be sold. Two forecasts can agree about final repayment and differ about what an interim report will reveal. Once investors can respond to that report, we need to compare the strategies it permits, rather than just the investments held at the start.

When the rules prevent the trade

“Worldly wisdom teaches that it is better for reputation to fail conventionally than to succeed unconventionally.”

— John Maynard Keynes, The General Theory of Employment, Interest and Money (1936), Chapter 12, Section V, p. 158.[3]

The basket example is a case in which a particular forecast difference has no use among the available choices. Institutional restrictions can produce a more frustrating case: the analyst can describe a useful response, but cannot obtain permission to carry it out. Keynes was describing the professional risks of departing from convention. A formal approval rule can turn that pressure into a barrier to action.

Imagine a fund with a rule that permits the cost of a protective hedge only when an approved risk model records a corresponding benefit. An analyst thinks a change in industrial policy could sharply reduce the value of some holdings and proposes protection against that loss. The committee takes the concern seriously. But the policy change is absent from the model’s approved scenarios, so the required assessment counts the cost of protection while giving no credit to the loss it could avoid. Under the rule, the hedge fails its approval test. The analyst has run into a circular requirement: the model being challenged must supply the recognised justification for acting on the challenge. Repeating the argument more accurately does not, by itself, change what the approval procedure accepts. The institution needs a way to assess the alternative scenario, revise the model, or authorise a decision under acknowledged disagreement.

An institution can become paralysed in this way: a credible assessment has no route to an authorised action. Apply the same rule to all the institutions that might take the other side of the prevailing view, and their inactivity begins to tell us about a shared restriction. It cannot be treated as independent confirmation that their concern was mistaken.

A mispricing sustained by a constraint that also binds you cannot be harvested by you.

An investor proposing to exploit the discrepancy must therefore explain what makes the relevant position available to them. Otherwise the obstacle invoked to explain everyone else’s inactivity blocks their own trade as well.

Funding can have a similar effect even after a position has been approved. Shleifer and Vishny study investors who depend on other people’s capital: losses can lead clients to withdraw precisely when an apparent discrepancy widens and the trade needs continued support.[4] Permission, capital and the capacity to wait all affect which views reach the market through positions.

There is also a distinction between preventing a particular investor from trading and omitting the relevant trade altogether. If the loan basket is the only credit exposure anyone can hold, no amount of competition in that basket creates a borrower-specific position.

Blume and Easley’s “If You’re So Smart, Why Aren’t You Rich?” examines long-run market selection. In the complete-market economies they study, survival depends on beliefs and on how heavily consumers discount future consumption; within their assumed class of preferences, differences in risk aversion do not determine who survives.[2] Incomplete markets allow a revealing counterexample. Two states have identical payoffs under every available asset. A trader who badly misjudges their relative probabilities, but gets their combined probability right, drives out another whose forecast is closer overall but slightly wrong about that combined probability. The example rewards accuracy about the distinctions the investments can exploit, rather than accuracy about the full state of the world.[2]

Some restrictions protect solvency or make obligations enforceable. Others may exclude an exposure that would improve allocation or risk management. The question is what a particular restriction accomplishes and what it prevents. Calling every rule a distortion would miss the first half of that question; appealing to market efficiency without examining the rules would miss the second.

What the model leaves out

A useful relationship can disappear earlier still, before an analyst considers a trade. Suppose a portfolio system groups firms by industry and estimates their dependence from recent returns. Two holdings in different industries look comfortably diversified. But both must refinance in the same month, and both depend on the same lender. The relevant documents are public; the system never compares those fields. Putting the deadlines and lender identities together suggests a common vulnerability. If the lender becomes unable to renew credit at that time, both borrowers may struggle simultaneously, despite the different products they sell. That possibility changes the reason for holding the two loans together. It may justify reducing one exposure, finding protection, or investigating a borrower with a different source of finance. The analyst has to assess the mechanism and the price of each response, but at least the research now has a specific question to answer.

Notice where the improvement came from. The lender names and dates were already in the records. Comparing them brought a relationship into the analysis that the earlier model had ignored. No new observation was created, but an investor with limited time and computing resources can make better use of the observations already available.

In information theory, processing a dataset cannot increase its Shannon information about a target; an invertible change of variables preserves that information.[5] Yet a transformation can make a relationship easier to detect, estimate or use. Sorting debt commitments by repayment date may make a concentration apparent that was very hard to see in separate company files. A different learning method may then exploit it.

Xu and co-authors study this distinction by restricting the prediction methods available to the observer. What they call usable information can increase after processing: a limited method may learn from a representation of the data that it could not exploit in the original form.[6] Whether the proposed dependence really helps predict losses still has to be assessed against evidence.

There is also a scarcity problem before any statistical test is run. Herbert Simon observed that “a wealth of information creates a poverty of attention.”[7] Once evidence is abundant, research has to decide which relationships deserve examination, which sources deserve comparison, and when the expected value of another investigation is lower than the value of acting. That allocation will matter increasingly wherever the raw material for investment research is plentiful and the capacity to examine it remains finite.

Changing a modelling assumption is another matter. Replacing independence with common exposure to a lender changes the probabilities assigned to joint losses. So does choosing a fresh “uniform” distribution after changing variables. Equal probability for equal-sized intervals of euros per dollar gives a different forecast from equal probability for equal-sized intervals of dollars per euro: intervals evenly spaced on one scale are uneven on the other. Re-expressing the same forecast in the reciprocal rate preserves its probabilities. Imposing uniformity again changes them, so the new model needs its own justification from the economic argument and the evidence.

What should justify a model, then? Start by separating the reasons for accepting or rejecting it. A cash-flow projection that spends money twice violates an accounting identity. A dependence estimate rests on observations and carries sampling uncertainty. A scenario can respect both and still depend on a disputed assumption about policy. A mandate prohibiting a trade is different again: it restricts the investor’s conduct, not what the economy can do. Lumping all four together as “constraints” makes it easy to mistake an institutional choice for empirical evidence, or an estimate for a physical necessity.

The distinction also helps when several models agree. If every model in a collection ignores the common lender, their forecasts may be reassuringly close. Changing parameter values inside that collection does little to investigate the shared omission. A more revealing test introduces the missing relationship and asks what follows, how much the evidence supports it, and whether the investment decision changes. That is more informative than another decimal place in the consensus forecast.

How much do we need to know?

Nobody can investigate every relationship before making an investment. The practical question is which unresolved ones could change the decision.

Suppose two investment proposals have already been compared on the same basis, including their costs and the investor’s treatment of risk. Under every model still being taken seriously, proposal A is preferable to B. The estimated advantage varies from €80,000 to €140,000. Learning which model is closer to the truth could improve the valuation, but it would leave the choice between these two proposals unchanged. Now suppose the range runs from a €20,000 disadvantage to a €30,000 advantage. Further work on the assumption responsible for that reversal has a different purpose: it could change which proposal should be chosen.

What matters is comparing the uncertainty with the margin between the alternatives, in terms that the decision actually uses. A large disagreement about something else may have no effect on the ranking. A small disagreement about a finely balanced choice may decide it. And selecting a portfolio is a different task from reporting its value accurately: the first comparison could support a clear choice while leaving too much uncertainty for a valuation required to be accurate within a few thousand euros. This is the decision-based view of information developed by Ronald Howard: uncertainty matters through its consequences for the choices we make.[8]

There are several ways to make the decision less vulnerable. An investor worried about a borrower’s dependence on one lender could investigate the refinancing arrangements. They could instead reduce the holding, or buy protection on terms that cover the relevant loss. Research improves the assessment of what may happen. Reducing or hedging the position changes what happens to the investor’s money if it does.

For a more complicated portfolio, consider two small revisions to its forecast. One raises each borrower’s default probability. Another keeps those individual probabilities unchanged but makes joint failures more likely around the two borrowers’ shared refinancing date. With enough separate cases, default records can help assess the first revision. The second calls for evidence about joint failures under the relevant refinancing conditions. If those conditions rarely appear in the record, a change that matters greatly to the portfolio may remain difficult to distinguish statistically.

Information geometry measures how strongly each small change alters what the observations are likely to show, so that it can be compared with the change’s effect on portfolio losses.[5] In this example, the joint-failure revision may barely show in the data yet matter greatly to the portfolio. Investigating the shared lender addresses the first problem; reducing the paired exposure addresses the second.

The same issue can be viewed as a problem of communication. Suppose an analyst holds a detailed report but can send the investor only a short message. What should the message preserve? Reproducing the report’s contents faithfully is one objective. Preserving the investment choices that would have been made after reading it is another. Erkip and Cover study this question for repeated investment. An investor receives a description of information correlated with asset returns, limited to a fixed average number of bits per investment period; the description is chosen to maximise the improvement in long-run wealth growth. Their result characterises the trade-off between the amount communicated and that growth improvement, assuming the probabilistic relationship between the information and market returns is already known.[9]

For a simple illustration, return to an investor choosing between cash and a fixed loan basket. Suppose the complete report affects that choice only by identifying whether a particular refinancing warning applies. Within each of those two categories, further details never change the preferred investment. A message stating whether the warning applies preserves the choice, although it leaves out almost all of the report. A summary that keeps the company descriptions and omits the warning can be more comprehensive and less useful.

A short summary is adequate when it preserves what the decision needs. Change the investment menu, the investor’s obligations or the opportunity to trade later, and yesterday’s adequate summary may be inadequate. The original records should remain available for the questions that have not yet occurred to us.

When a useful contract is missing

Sometimes research improves a decision that the existing securities already allow. Sometimes it identifies an exposure that none of them provides. The second situation calls for a different response: can a suitable contract be created? Return to the two forecasts that agree about total loan repayments but disagree about which borrowers will fail. A payment based on total repayment cannot distinguish them. A payment based on an agreed record of individual defaults potentially can. The parties could fix their forecasts in advance and agree on a rule for comparing them with that record, with the payment depending on the comparison. They would also need to specify what counts as a default, who supplies the record, when it becomes final, how revisions are handled, and how the promised payment is secured. “Which forecast was better?” becomes a question about an identified observation and an agreed scoring rule.

The rule determines what is rewarded. A comparison designed around ordinary outcomes need not give the same emphasis to rare disasters as a tail-sensitive contract. Nor will every participant value the resulting payments alike. One may want to back a view; another may want protection against an outcome in which both their forecast and their existing portfolio perform badly. Whether those motives produce a trade depends on the price and the participants’ exposures. A forecast difference is a reason to examine a contract, not evidence that there is already demand for it.

Some such payments could already be reproduced by combining existing securities. In that case, a standard contract might improve access or make the exposure simpler to transact. Where the available securities cannot reproduce the payment, a new claim can add an exposure. Either way, the new contract remains subject to the institution’s approval rules, so introducing it does not remove the restriction that blocked the original trade.

Even a well-designed contract answers only the question its observation can resolve. A record of defaults can compare default predictions while leaving open the causal role of policy, the consequences of another financing decision, or the prospects of a different group of firms.

A payment between counterparties also needs a separate economic justification at the level of the system: useful insurance, better allocation or an investment made possible. Hirshleifer’s analysis draws the distinction sharply: better information can yield a private gain even when the information has no social usefulness.[10]

What this changes for portfolio management

For many investors, the immediate opportunity is to use the markets that already exist. The common-refinancing example might lead to a better diversified portfolio without a new security, a regulatory reform or a private dataset. Other relationships could support trend-following, a valuation trade, or a strategy that earns a return for carrying an exposure over time. The task is to find relationships whose consequences can be used in the actual portfolio, and to learn them before the opportunity disappears.

A collection of forecasting models can help, but an overall league table can be a poor guide to their contributions. One model may estimate sales well and another the cost of inputs. Neither provides a satisfactory forecast of profit on its own; together they may support one, provided their assumptions about volumes, dates and prices fit. A model useful in stable markets may become misleading during a funding shock. The portfolio manager needs to know where each forecast helps, what it leaves unresolved and how the sources relate to one another. Several reports repeating the same original forecast add less independent evidence than their number suggests.

There is also a choice about what to observe and what to trade. For a manufacturer exposed to an input cost, the commodity price may be a more direct observation of that cost than the manufacturer’s share price, which also reflects its debts, customers and other activities. But the share may be the investment through which the manager wants to express the resulting view. The learning problem must connect the observation to the payoff. Selecting a statistically predictable series and only later looking for a financial use can leave an impressive forecast with no worthwhile trade attached.

Then there is timing. A small forecast advantage expected to last for months is different from one likely to disappear tomorrow, especially when entering and leaving the position both cost money. Gârleanu and Pedersen examine this problem with signals that decay at different speeds. In their model, the optimal trade depends on the existing portfolio and on where desired holdings are expected to move, with more persistent signals receiving greater weight in the portfolio towards which the investor adjusts.[11] The forecast and the trading policy have to be evaluated together.

The same is true of the choice to wait for evidence. Keeping the present portfolio while investigating a relationship is already an investment decision: it preserves some exposures and forgoes others. A new disclosure may make the choice easier, but a competing investor may act first, or the price may move before the disclosure. A research procedure has to compare those consequences with the risks of acting now. Much of the raw material can be freely available; the difficulty is discovering and testing useful relationships quickly enough, combining them coherently, and choosing positions that remain attractive after costs.

This is the portfolio-management application I want to develop. The aim is higher net investment performance against methods already capable of forecasting, allocating and trading under the same mandate, compared on the same evidence dates, investment opportunities and costs. Serious comparators include strong forecasting-and-allocation systems, factor and characteristic strategies, and portfolio policies that already account for trading costs. The complete record matters, including research and trades that fail. What to observe, which models to build or revise, and what to hold have to be decided together, because each changes the value of the others. A hedge can make one uncertainty less urgent to resolve; a new relationship can change which data deserve attention; a trade can make later trades cheaper or harder. A good forecast fed into an unchanged allocation rule leaves those interactions unattended.

What can the market reward?

Suppose, then, that you really do understand something the market does not. The loan basket still pays on total repayments, however much you know about the borrowers; the committee still asks its approved model to justify the hedge, however clearly you have identified the model’s omission. Turning the insight into a better decision requires finding, or making possible, the response it calls for. More observations will not create a missing contractual right; a new contract will not repair an unsupported forecast.

Fama’s familiar definition sets a demanding standard for the prices of existing securities:

“A market in which prices always ‘fully reflect’ available information is called ‘efficient.’”

— Eugene F. Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work” (1970), p. 383.[12]

That definition does not interpret itself. Fama insisted that a test of efficiency requires a model of price formation; he later made the resulting joint-hypothesis problem explicit. An abnormal return can reflect inefficient pricing, a bad model of expected returns, or some combination of the two.[12] The problem is different from the one developed here. Even when a security exists and trades, its price cannot tell us by itself whether our model for interpreting that price is adequate.

Our restricted loan market raises another question. It could use all information relevant to total repayment and still leave the borrower-by-borrower disagreement financially silent. An incomplete opportunity set does not imply that the securities which do trade are inefficiently priced, just as efficient pricing of the basket does not make borrower-specific positions exist. The efficiency of existing prices and the adequacy of the available ways to express knowledge are different achievements.

Investors have reason to seek better models and faster learning wherever these improve the use of existing opportunities. They also have reason to ask whether the opportunities themselves exclude a consequential part of what they know. A market may organise competition among its permitted positions extremely well while never testing the understanding those positions leave out. Before treating financial success as a verdict on knowledge, we should establish which knowledge was allowed to enter the contest.

References

[1] Milton Friedman, “The Methodology of Positive Economics”, in Essays in Positive Economics (University of Chicago Press, 1953), pp. 3–43, especially pp. 21–22.

[2] Lawrence Blume and David Easley, “If You’re So Smart, Why Aren’t You Rich? Belief Selection in Complete and Incomplete Markets”, Econometrica 74(4), 929–966 (2006), DOI: 10.1111/j.1468-0262.2006.00691.x, especially Section 5.

[3] John Maynard Keynes, The General Theory of Employment, Interest and Money (Macmillan, 1936), Chapter 12, Section V, p. 158.

[4] Andrei Shleifer and Robert W. Vishny, “The Limits of Arbitrage”, The Journal of Finance 52(1), 35–55 (1997), DOI: 10.1111/j.1540-6261.1997.tb03807.x.

[5] Marc Mézard and Andrea Montanari, Information, Physics, and Computation, author draft (2007), Chapter 1, Proposition 1.12; Korbinian Strimmer, Statistics 2: Statistical Learning with Likelihood and Bayes, Sections 4.2 and 5.

[6] Yilun Xu, Shengjia Zhao, Jiaming Song, Russell Stewart and Stefano Ermon, “A Theory of Usable Information under Computational Constraints”, International Conference on Learning Representations (2020), Sections 2–3.2, arXiv:2002.10689.

[7] Herbert A. Simon, “Designing Organizations for an Information-Rich World”, in Martin Greenberger, ed., Computers, Communications, and the Public Interest (Johns Hopkins Press, 1971), pp. 40–41.

[8] Ronald A. Howard, “Information Value Theory”, IEEE Transactions on Systems Science and Cybernetics 2(1), 22–26 (1966), DOI: 10.1109/TSSC.1966.300074.

[9] Elza Erkip and Thomas M. Cover, “The Efficiency of Investment Information”, IEEE Transactions on Information Theory 44(3), 1026–1040 (1998), DOI: 10.1109/18.669153.

[10] Jack Hirshleifer, “The Private and Social Value of Information and the Reward to Inventive Activity”, The American Economic Review 61(4), 561–574 (1971), especially p. 561.

[11] Nicolae Gârleanu and Lasse Heje Pedersen, “Dynamic Trading with Predictable Returns and Transaction Costs”, The Journal of Finance 68(6), 2309–2340 (2013), DOI: 10.1111/jofi.12080.

[12] Eugene F. Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work”, The Journal of Finance 25(2), 383–417 (1970), especially pp. 383–384, DOI: 10.1111/j.1540-6261.1970.tb00518.x; Eugene F. Fama, “Efficient Capital Markets: II”, The Journal of Finance 46(5), 1575–1617 (1991), especially p. 1575, DOI: 10.1111/j.1540-6261.1991.tb04636.x.

All Roads Lead to Disagreement (I): Why Two Careful Analysts Can Still Disagree

Mon, 21 Sep 2026 00:00:00 +0200

“About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know.”

— John Maynard Keynes, “The General Theory of Employment” (1937), p. 214.

Modern finance is technically magnificent and epistemically credulous.

The technical part is not in doubt. Calibration, hedging, optimisation and stress testing are real achievements, and they work precisely because nobody has to reopen the construction of the problem at every decision. But something is always settled before the valuation begins: which futures count as distinct, what evidence bears on them, which probabilistic model is adequate enough to use, and how much authority to give the current price. When those four choices are stable and widely shared, the machinery runs, and it runs extraordinarily well. That is exactly why the choices are easy to forget.

For long-dated assets exposed to structural change, the arrangement becomes difficult to defend. Climate transition, infrastructure exposed to contested policy, geopolitical fragmentation, technological obsolescence: each requires a commitment today to a future whose relevant features are themselves disputed. And the dispute is usually not about the arithmetic. Two analysts can agree on every number in front of them and still disagree about which mechanisms to represent, which relationships will carry forward, and which evidence deserves authority.

I did not arrive at epistemology by preference. I set out to build a valuation I could defend. Defending it required an account of the beliefs it rested on, and examining those beliefs required an account of how the problem had been posed in the first place. Epistemology arrived as a forced move.

Finance already has names for much of this difficulty. A scenario is a possible path. A stress test measures an exposure along it. Model uncertainty allows competing specifications. A market price gives the terms on which a claim trades. What none of them supplies is a way of comparing two serious accounts of the same financially consequential future: what could count as evidence between them, and what should follow when the comparison does not resolve.

I need a short name for “a serious account of the future”, and will use probability picture: a belief expressed as a probability law, or a probabilistic model that supplies one. In the Cox–Jaynes tradition, probabilities represent degrees of plausibility conditional on stated propositions and information, subject to demanding consistency requirements.[1] “Belief” here therefore means something much more disciplined than an opinion. But notice the order of operations. Probability theory can discipline a belief only after someone has decided what counts as an event and what counts as evidence. Those decisions come first, and the theory does not make them.

Risk, Knightian uncertainty and radical uncertainty

“There is a fundamental distinction between the reward for taking a known risk and that for assuming a risk whose value itself is not known.”

— Frank H. Knight, Risk, Uncertainty and Profit (1921), p. 44.[2]

Three different things get called uncertainty, and the difference between them decides what kind of work a model can be asked to do.

Risk, in the standard sense, needs two things: outcomes described well enough for probabilities to have definite objects, and enough information about those outcomes for estimation and revision to mean something. Historical frequencies are one source of that information. Structural modelling, market observations and expert judgment are others. Given a reasonably stable description, expected utility, risk-neutral valuation and the familiar risk measures do useful work, and none of them requires anyone to know the true probabilities. Much of treasury practice, short-dated option trading and ordinary risk management lives here, in domains where evidence, convention and experience supply enough regularity to act on.

Knightian uncertainty is a difficulty inside an accepted description. We may know the relevant outcomes perfectly well and still have no defensible way to assign reliable probabilities to them. Ellsberg’s comparison of bets with known and unknown probabilities makes the point sharply: an unchanged set of possible payoffs does not imply an unchanged decision problem. When Keynes wrote “We simply do not know”, he had in mind the prospect of a European war, or the rate of interest twenty years hence.[2] Finance meets related difficulties under other names: model risk, regime dependence, uncertain correlations. Caballero and Krishnamurthy give the point an explicitly financial form: in their model of a flight-to-quality episode, agents are uncertain how their own shocks relate to system-wide shocks, and each agent’s individually conservative response leaves the economy collectively exposed.[3]

Radical uncertainty goes further: even the relevant outcomes, and the way they are organised, may be unsettled. A new technology, an institutional reaction or a political rupture can introduce a distinction that an established model never represented. A model of technological improvement may describe the incumbent technology with considerable sophistication and have no way at all to represent its displacement by another. Enlarging the family of models is then necessary, but the enlargement has to be chosen, and nothing in the old model guarantees that it supplies the right alternatives. Aydogan and co-authors draw the parallel distinction: uncertainty within a model, uncertainty among models, and uncertainty about whether the family contains an adequate model at all.[3] It is the third that concerns me here.

Scenarios are how the industry makes such questions tractable. They reduce an open future to conditional worlds that can be investigated, compared and revised, and their usefulness depends on the restriction they impose. Draw the family too narrowly and it excludes material possibilities. Draw it broadly enough that almost any decision can be defended and it stops discriminating between decisions at all.

That boundary is also institutional. Scientific assessments, central banks, regulators, accounting rules and market conventions between them help determine which observations and distinctions become usable in financial practice. A scenario need not exhaust the future to be useful. But neither its publication nor its inclusion in an official exercise supplies a probability that was absent from its construction.

Climate transition concentrates all three difficulties at once. Physical and engineering constraints coexist with political choices, technological change, and consequences that fall very unevenly across sectors and across the capital structure. For a long-lived commitment that is difficult to reverse, the two errors are not symmetric: underestimating transition risk can lock in an exposure before the repricing arrives, whereas overestimating it takes the more revisable form of an opportunity cost. Climate science, macroeconomic scenarios and financial evidence all constrain the problem, and none of them provides repeated observations of the complete transformation.[4] The practical task is to use that knowledge before the commitments have become irreversible, rather than to withhold judgment until the historical record has caught up with the question.

Bayesian discipline

“[Probability theory is] the optimal processing of incomplete information.”

— Edwin T. Jaynes, Probability Theory: The Logic of Science (2003), p. xiii.[1]

The obvious objection to all of this is that models should face the data. They should. The objection only becomes interesting when you ask which claims the available observations are actually capable of testing.

George Frankfurter reports being told by the editor of a leading finance journal: “We in finance do not care much about philosophy. We care about data and what data can tell us.”[5] I have some sympathy for the impatience. But the remark skips a prior question. Before data can tell us anything about the decision in front of us, we have to establish what they measure, why the observations are comparable, and which conclusions the comparison warrants. In familiar settings those judgments can reasonably stay in the background. Under structural change they become part of the dispute. The editor has not dispensed with an epistemology. He has declined to examine the one he is using.

Plenty of evidence still bites. Historical defaults constrain a credit mechanism. Engineering rules out an impossible production path. An accounting identity exposes an investment scenario whose resources do not add up. None of this requires a completed climate transition, and all of it is worth doing.

What the record cannot supply is the repeated full experiment needed to validate every long-horizon implication of the model in the usual way. Asking a unique transformation to pass the evidential test of a recurring short-horizon process is asking the world for evidence it cannot produce — and then mistaking the impossibility for methodological virtue. Backtesting remains valuable wherever history is informative. It cannot warrant an extrapolation whose structural continuity it has not tested.

Tests have to be matched to the claims they support. A transition path that depends on an infeasible reallocation of labour or investment should fail its diagnostic even when its financial outputs look entirely familiar. A sound accounting identity, in the other direction, cannot establish the political likelihood of the policy path built around it. Rival specifications have to be examined, and the rules for revision fixed before the inconvenient observation arrives rather than after. That includes a distinction which is easy to lose: an error in a parameter is not the same thing as evidence that the proposed mechanism, or the whole family of models, is inadequate. Adding observations to the wrong exercise will not turn it into the right one.

Bayesian reasoning belongs to this larger practice, though the financial literature often invokes it in a much narrower sense. Give me a prior, a likelihood and a fixed family of hypotheses, and Bayes’ rule determines exactly how the probabilities change when evidence arrives. It will also do that impeccably inside a model that omits the relevant possibility, treats an unsuitable data source as decisive, or owes its form to estimation convenience. The broader Bayesian project begins before the prior and continues after the posterior: what information should enter, which families deserve comparison, how their implications should be checked, and how to act when several pictures survive the check. Robust and ambiguity-sensitive methods contribute to that examination, but they too require reasons for the models and the departures they admit.[1,6]

Acting may require a single operative picture even when inference has not selected one uniquely. There are at least three ways to get one. You can average across a stated family. You can protect against its most adverse member. You can complete a partial description by maximum entropy, or by minimum relative entropy to a reference. These solve different decision problems and they need not agree.

Entropy-based completion is appropriate when the aim is to introduce as little unsupported structure as possible, relative to declared outcomes, constraints and a reference — formulating a prior, say, or completing an assessment once other evidence has been incorporated. What it contributes is a selection rule. The evidential basis remains the stated constraints and the reference, which is why naming the reference, and the assumptions made about dependence, is part of explaining what the principle has actually selected.[7]

Coherence has an organisational dimension too, and this is where it becomes uncomfortable in practice. An institution can be perfectly protected against a Dutch book — a combination of accepted bets that guarantees a loss — while running incompatible transition assumptions in disclosure, in investment and in risk management. Different purposes can justify different pictures. Departmental separation is not itself one of those justifications. Nor is revising a model in response to evidence the same thing as changing the rules of revision to accommodate an unwelcome result. The remedy is unglamorous: state which account governs which decision, and be able to defend its grounds when someone examines them again.[8]

“Since all models are wrong the scientist cannot obtain a ‘correct’ one by excessive elaboration,” wrote Box.[8] More parameters and more simulations examine a representation more thoroughly; they do not establish that it is adequate. A Bayesian analysis has to use the evidence as fully as it can within the models proposed, and at the same time examine where those models’ assumptions came from: data, structural knowledge, institutional purpose, or a judgment nobody has resolved. This discipline reaches much further than updating. It is still not self-grounding.[8] Its limit is reached at the point where the available evidence and the applicable principles still permit more than one consequential representation.

That point is where the rest of this essay begins.

The representational residue

“My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this: probability does not exist.”

— Bruno de Finetti, Theory of Probability (1974), Preface.[9]

Why quote something so theatrical? Because probabilistic language encourages a tempting picture, and the theatre is the quickest way to break it open. The picture is this: once a model is coherent, carefully estimated and repeatedly revised, it becomes natural to speak as though there were a probability already sitting behind the evidence, waiting to be uncovered more accurately. De Finetti’s own position is far stronger than anything I need (he rejected objective probability, not probability theory), but his provocation does the job. A probability is always a probability of something, conditional on some information, expressed in some representation of uncertainty. Those three ingredients have to be fixed before the calculus can operate at all.

The representational residue is what remains when logic, the available evidence and the applicable technical principles do not uniquely determine them. It can concern which states to distinguish, which features define similarity, which family of explanations to examine, or how to treat dependence. I want to be precise about the scope of the term: it names what remains genuinely unforced after the relevant discipline has done its work. It is not a licence to call every judgment a modeller makes a residue.[9]

Three independent arguments arrive at the same place.

The first route is the representation of ignorance. Bertrand’s chord problem is the classic demonstration. Ask for the probability that a random chord of a circle is longer than the side of the inscribed equilateral triangle, and the answer depends on what “random” is taken to mean: pick two endpoints uniformly on the circumference and you get one third; pick a midpoint uniformly along a radius and you get one half; pick a midpoint uniformly inside the disc and you get one quarter. Same geometrical event, three defensible answers.

Finance encounters the same issue in an entirely ordinary place. An analyst with no view on a discount-rate gap represents that ignorance by spreading it uniformly. A colleague, equally without a view, spreads it uniformly over the associated valuation multiple instead. The set of possible valuations is identical. The probability weights on them are not. A change of coordinates that carries the law along with it would change nothing; choosing uniformity afresh in the new coordinates introduces a different law. Jaynes showed that a symmetry genuinely supplied by the problem can sometimes settle the choice. Where no such argument is available, “uninformative” is not an adequate description of what the analyst has assumed; it is a refusal to describe it.[1,10]

Similarity poses a related problem inside the same route. Goodman’s “grue” and “bleen” show how descriptions that agree with every observation already made can imply different continuations; Watanabe shows how similarity judgments depend on the features a language makes available in the first place.[11] The financial version is mundane. One credit model groups firms by current industry. A second groups them by technological dependence. A third groups them by exposure to future policy. Two firms can be far apart under one classification and close under another, and the three models will not learn the same thing from the same default histories. Each classification can be perfectly coherent, and evidence can favour one over the others. Until it does, the classification remains one of the premises through which the data acquire predictive meaning at all.

The second route concerns the problems a method is equipped to solve. The no-free-lunch results deny any general advantage when methods are compared over unrestricted problem classes under the stipulated averaging assumptions.[12] The practical reading is that a successful method works because it exploits structure in the problems it actually encounters, so its record establishes competence within a domain. Take a credit model trained under gradual policy adjustment and apply it to an abrupt prohibition or a technological displacement, and a further structural judgment is required about which relationships survive the break. However impressive the record, the record cannot supply that judgment.

The third route concerns the inputs to inference itself. Cox–Jaynes consistency constrains probability assignments once the events or propositions at issue, and the information being conditioned on, have been specified — once, not before. Entropy-based selection likewise requires evidence constraints, a reference, and assumptions about how separate systems compose.[1,7] That last requirement is not a formality. If two sectors are independent, their information combines one way; if they share a policy exposure or a supply-chain dependence, the same marginal observations can support materially different joint losses. A rule can be uniquely justified for a stated inferential problem without selecting every feature of that problem for itself.[7]

None of this shows that all representations are equally defensible. Evidence can eliminate some. A symmetry can determine a measure. Stronger structural knowledge can settle an apparent ambiguity outright. What the three arguments share is narrower and more awkward: the success of a calculation within its premises does not establish that all of those premises were uniquely required. Enlarging the model family can improve the account without automatically settling the reasons for the enlargement. And the choice that remains is logical before it is institutional: a committee may have to make it, but the committee’s need to decide is not what created the underdetermination.

Rationality does not supply the missing common starting point either. “It has become an article of faith among economists that differences in beliefs among rational individuals must be explained by different information,” writes Stephen Morris.[13] The results usually invoked for that faith are more conditional than the faith is. Aumann’s agreement theorem depends on a common prior and common knowledge of the relevant posterior probabilities.[13] Blackwell and Dubins show that forecasts can merge as evidence accumulates when the forecasters’ probability laws over complete future histories satisfy an absolute-continuity condition, roughly when one does not rule out whole histories that the other still treats as possible.[13] Both conclusions inherit their assumptions. Rationality alone supplies neither a common starting point nor a timetable for agreement.

So continued disagreement is not, by itself, evidence of a residue. It calls first for the ordinary explanations: differences in information, differences in which histories each model permits at all, differences in updating, and plain error. Where those have been examined and the evidence still permits different consequential representations, the disagreement is not a failure of the inquiry. It is an outcome of it.

The residue also changes what institutions do with the resulting plurality, and this is where the consequences stop being methodological and start being financial. Comparing two probability pictures requires a purpose: a measure centred on ordinary predictive accuracy and one designed to give exceptional weight to tail events can rank the same disagreements differently. Selecting one picture for action then raises a second question, because averaging, protection against the adverse alternative, and entropic completion can produce different budgets and different capital requirements from the same inputs. The institution’s mandate and its reasons have to determine which procedure it uses. Mathematics can establish what follows from the choice; but where the accepted premises leave the choice open, authority over it is part of the financial problem rather than a preliminary to it. It decides whose exposure appears in the assessment, which risk receives protection, and which consequences are left outside it.

Prices and beliefs

At this point the market price looks like a way out of the analysts’ disagreements. It is public, it is transactable, and it is disciplined by other people’s capital — three things no spreadsheet can claim. Institutions need common prices to coordinate their positions, and a model claiming to value currently traded claims cannot simply ignore what they trade at. It is worth remembering that Bachelier began mathematical finance with conjectures about probability, not with a presumption that quotations disclosed the true law.[14] The difference between taking prices seriously and deferring to them matters most in exactly the case at issue here: when the price is observable and the reasons behind it are not.

Beliefs enter a price alongside wealth, risk bearing, endowments, funding conditions, mandates, hedging needs and horizons, and different combinations of these can sustain the same quotation. A deep order book establishes that a claim can be traded under prevailing conditions. It does not establish that the participants were able to express every relevant view, or that the marginal investor’s horizon is anything like the horizon of the institution using the price. Even a sufficiently rich option surface reveals a valuation object (a state-price picture, under some extraction method) rather than a purified physical forecast.[14] State prices reflect both what people think will happen and how much a unit of money is worth to them when it does, which is why they cannot be read off directly as physical probabilities. And with finitely many liquid claims, the quotations can leave several valuation pictures compatible with them, so recovering one requires further restrictions or an explicitly stated selection rule.

This is not a merely hypothetical concern in climate finance. Stroebel and Wurgler surveyed 861 finance academics, practitioners and public-sector respondents, and found them at least twenty times more likely to say that markets underestimated climate risk than that they overestimated it.[15] The survey establishes widespread disagreement with prevailing pricing. It does not tell us whether the respondents were right, or why the gap arose. A model that reproduces market prices has established market consistency, which is a real property and a different one from the adequacy of the implied assessment for a long-horizon decision. Any discrepancy between physical probabilities and pricing weights still has to be interpreted through the valuation assumptions before anyone is entitled to call it mispricing.

“The peculiar character of the problem of a rational economic order is determined precisely by the fact that the knowledge of the circumstances of which we must make use never exists in concentrated or integrated form, but solely as the dispersed bits of incomplete and frequently contradictory knowledge which all the separate individuals possess.”

— Friedrich A. Hayek, “The Use of Knowledge in Society” (1945), p. 519.[14]

Hayek’s account gives prices a substantial authority, but a different one from the authority usually claimed for them. They allow people with dispersed and incomplete knowledge to adjust their plans without requiring any one of them to understand the whole. That achievement depends on making coordination possible despite differences in knowledge. It does not turn the resulting price into a complete register of what is known. Grossman and Stiglitz locate a limit inside the economics of information itself: in their costly-information model, a price that revealed everything would remove the reward that sustains information acquisition in the first place.[16] Some knowledge therefore stays private for reasons internal to market equilibrium, not in spite of it.

The standard reply is that a well-founded objection would already have been traded upon. That reply assumes the disagreement can be expressed in a position the market permits.

A mispricing sustained by a constraint that also binds you cannot be harvested by you.

Short-sale bans, funding limits and performance-sensitive capital are the familiar cases, and Shleifer and Vishny worked through what they do to the arbitrageur.[17] But the principle is broader than any list of frictions. A financial system also determines which distinctions between probability pictures can be expressed at all, through admissible portfolios, contracts or hedges. If a more accurate belief differs from the prevailing one only along a dimension that no admissible position can load on, that accuracy is financially silent. The market can be efficient on the space it is allowed to trade and still have no selection mechanism whatsoever on the dimensions it excludes. The relevant failure is then not a distortion inside a market. It is a limit on what the market itself can express.

Prices also help determine the future against which their apparent prescience will later be judged. Merton’s example is the solvent bank destroyed by withdrawals prompted by a rumour of insolvency. The everyday version is slower: financing conditions can prolong a technology, accelerate its deployment, or create a constituency resistant to its replacement, and restrictive finance can weaken the very activity whose prospects the price purported to assess. MacKenzie’s An Engine, Not a Camera is the standard reference for this performative possibility.[18] Whatever the later evidence looks like, it bears some imprint of the earlier beliefs and valuations.

Disagreement in financial decisions

None of this requires a new security. The analysis already has financial uses inside an institution as it stands.

An institution may discover that its own portfolio is difficult to reconcile with the transition assumptions it publishes, or that two of its scenario programmes imply incompatible labour, investment and default paths. That is a real diagnostic, and it establishes a material discrepancy without anyone having to put a price on it. Putting a price on it — a premium wedge — requires a further translation: the competing pictures must be used to value the same financial claim. Even then, the prices at which each side would individually be willing to trade, their reservation values, are distinct from the price at which a transaction might actually occur.

An investor may also express a view through existing assets, through underwriting or through a hedge. A relative-value position can rest on a perfectly well-founded epistemic thesis and still lose money, and still be exposed to risks that have nothing to do with the disagreement that motivated it. Arbitrage requires the much stronger case of claims whose quoted terms can be combined to guarantee a gain. The usefulness of better reasoning is not confined to that case, just as it is not contingent on anyone creating a market dedicated to disagreement. It can improve ordinary investment and risk decisions while making the grounds of those decisions explicit.

The distinctive question for a contract written on a disagreement is whether the comparison itself can be specified publicly. A counterparty may declare its own probability picture; alternatively, a comparator may be obtained from conventional markets under a stated extraction method. The participant’s decision determines which aspect of the difference matters. The settlement observation determines which part of it can be tested. The source, timing and treatment of revisions all have to be fixed before the evidence arrives. And where a public observation fails to distinguish an important part of the dispute, that part simply stays outside the contract: pricing the observable component leaves the rest with the parties.

Such a contract could support either a position on a view or protection against the outcomes on which that view performs badly. What it would add is the public specification and administration of the comparison, rather than an intrinsically unreplicable cash flow; sufficiently complete state-contingent markets could reproduce the payment once its terms were known. Whether standardising it would attract genuine demand is a separate empirical question. In either case an exchange can decide whether a declaration supports the agreed settlement; it cannot certify the quality of a participant’s entire account of the future. A dedicated market would not create responsibility for the underlying representation. The institution already carries that responsibility the moment it chooses a representation for a decision.[19]

Conclusion

“The recognition of the insuperable limits to his knowledge ought indeed to teach the student of society a lesson of humility.”

— Friedrich A. Hayek, “The Pretence of Knowledge”, Nobel Memorial Lecture (1974), excerpt.[20]

When evidence and accepted principles leave more than one consequential probability picture standing, finance is in an awkward position. It cannot demand certainty from a forward-looking model while accepting the assumptions of current practice without examination. Both organise an unsettled future. Both owe an account of the evidence and the conventions they rely on. A rigorous inquiry may reject many representations, justify one for a particular use, and still leave consequential alternatives unresolved, and when it does, the disagreement that remains belongs to the analysis rather than counting against it. A price can coordinate a transaction in its presence. That accomplishment does not relieve an institution of judging the beliefs on which its decisions depend.

The obligations that follow are concrete. An institution should state the picture that governs the decision, explain which constraints support it, distinguish a deliberate selection from an empirical finding, and retain the material alternatives that have not been refuted. It can then identify which consequences are robust, which depend on an unresolved assumption, and what evidence would justify revision. Where the choice determines a valuation, a capital requirement, or whose losses enter a stress test, the reasons for it are part of financial accountability. Adopting the conventional picture without comment is not an escape from this; convention is one of the choices under examination.

Climate transition makes the stakes particularly clear, but the argument extends to every consequential commitment whose future the available record cannot uniquely determine. The practical task is to make the probability pictures explicit, isolate the disagreement that actually matters, establish what evidence could test it, and give it an appropriate financial consequence while keeping the unresolved part visible.

Hayek’s closing appeal to humility was a warning against turning limited knowledge into a warrant for far-reaching control. In finance the same discipline requires that the authority claimed for a model, a price or a contract stay commensurate with what it can actually establish. Finance can coordinate under uncertainty while saying clearly what its models assume, what its evidence supports, what its contracts can observe, and what remains a matter of judgment. Carveth Read’s standard remains apt: “It is better to be vaguely right than exactly wrong.”[20] For a financial system facing risks on a planetary scale, that is the difference between coordination disguised as knowledge and knowledge that has earned the right to coordinate.

References

[1] Richard T. Cox, “Probability, Frequency and Reasonable Expectation”, American Journal of Physics 14, 1–13 (1946); Edwin T. Jaynes, Probability Theory: The Logic of Science (2003).

[2] Frank H. Knight, Risk, Uncertainty and Profit (1921); Daniel Ellsberg, “Risk, Ambiguity, and the Savage Axioms”, The Quarterly Journal of Economics 75, 643–669 (1961); John Maynard Keynes, “The General Theory of Employment”, The Quarterly Journal of Economics 51, 209–223 (1937), quotation p. 214.

[3] Ricardo J. Caballero and Arvind Krishnamurthy, “Collective Risk Management in a Flight to Quality Episode”, The Journal of Finance 63, 2195–2230 (2008); Ilke Aydogan, Loïc Berger, Valentina Bosetti and Ning Liu, “Three Layers of Uncertainty”, Journal of the European Economic Association 21, 2209–2236 (2023); John Kay and Mervyn King, Radical Uncertainty (2020); Francis X. Diebold, Neil A. Doherty and Richard J. Herring, eds., The Known, the Unknown, and the Unknowable in Financial Risk Management (2010).

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