The Global Financial Crisis Should Have Been Caught By the Compiler (An Insider’s Perspective)

This is a note about types of probability, and their role in the credit crisis. In particular I cover P, Q and R-probability where the first two are standard and the third is my label. It isn’t the first time I’ve advanced the explanatory power of R-probability, although after my year-long M6 chicanery I feel slightly better placed to do so.

The backdrop this week is the collapse of Silicon Valley Bank obviously—although only time will tell if how relevant that failure is, or subsequent bank failures are, to what I have to say.

Headquarters of Silicon Valley Bank. Photographed by Wikipedia user Coolcaesar on September 4, 2022.

Overview

I’ll get right to my thesis.

1. There are two styles of probability, P and Q, we are taught at school.
2. Most people use R instead, as will be defined, thinking it is P.
3. Structure finance invites an extremely dangerous R-probability calculus, and regulation accidentally makes it worse by shielding the paucity of the numbers.

My behavioral measure-theoretic argument sounds esoteric yet it might appeal to software developers — insofar as my focus is on guarding against the wrong objects being passed to calculations, as distinct from the calculations themselves.

In this note I do not consider the paucity or otherwise of copula and other credit models here. That topic was one I probably talked about as much as anyone in the lead-up to the crisis, 2001 through 2006, when I identified those problems and instigated and led a unique effort on Wall street to advance the state of the art. It is certainly interesting, but I believe what I’ll say here is more important (and simpler).

A brief note on current events

To be honest I wonder more about the portfolios of other banks around the country this week, because we have emerged from a yield-starved period of history where structured finance investments would have looked particularly attractive. SIVB held roughly $100b in Agency Mortgage Backed Securities (MBS), it must be said, though I don’t know which ones. It bought the long end of the govt curve too. The trade went against it. It was forced to sell at a steep loss. The end.

Ratio of available for sale versus hold to maturity assets for Silicon Valley Bank

To describe the demise of any equally unremarkable trading strategy like this is a bit like a doctor signing a death certificate with “cardiac arrest”. Everyone dies that way, pretty much. It tells us little. However there’s a lurking question that is relevant to this post, or might be depending on the constitution of the MBS portfolio.

What framework (model or mental model) did the bank use to make the MBS investment decisions in the first place? What about other banks? We missed the opportunity in the wake of the GFC to consider that type of basic question carefully because of all the chaff in the air. Should we talk about it now?

At the time, the world lunged for sensational journalistic non-explanations peddled by populists who weren’t in the engine room. Greed. Stupidity. Gauss. Even autism for heaven’s sake. This mix of the banal, the asinine and the downright offensive shed almost no light on the peculiarities of structured finance, the epicenter of the crisis. If there was something underlying that was broken in a fundamental way but even mildly technical, it could easily have been missed.

Let me draw your attention away from SIVB and back to 2007.

Minding our P’s and Q’s: quant finance crib notes

I shall quickly review the established dichotomy in financial probability. We are schooled in two very distinct uses of “stuff that adds up to one” that anyone joining the discussion must appreciate. I will use the near-synonym “measure” as a reminder that probability need not be “real” probability.

• The P-measure: Refers to the aspiration of “true objective probability” although as we can’t know that, its more helpful to describe it here as an honest attempt to discern objective probabilities of future events using all available exogenous data without prejudice. (Naturally, there will be a different P-measure for each person, since we see different information.)
• The Q-measure: An extremely useful and different use of probability used to illuminate relative value pricing, and to help summarize information implicit in market prices. Its subjectivity is substantially smaller, though not zero.

The P-measure needs either no introduction at all, or a philosophical treatise I don’t have time for. A brief justification for the utility of Q-measure (moving right along) runs along the lines of basic linear algebra rank considerations, viz:

If a composite good comprising a combination of components isn’t priced as the weighted sum of the prices of its components, and if there are lots of others things priced haphazardly in this fashion, then sooner or later somebody can make money with little or no risk by buying and selling combinations of the composite products. Moreover, this arbitrage will imply a negative price for a component, whether or not it is individually for sale.

Conversely, there’s only one sure fire way to guard against this attack — the notions of coherent pricing and coherent probability are close cousins. Build the former from the latter and you’ll be okay.

The parimutuel or lottery mechanism as Q-measure

Quantitative finance generalizes to infinite dimensions and uses concepts like the “equivalent martingale measure”. It can also quickly establish powerful continuous time tools, but those are not essential to my case. Instead, here’s ashortcut to Q-probability:

• Everybody bets on the horse they think will win.
• The winners split the pot.

The Q-measure is the normalized money invested on each potential future outcome in a lottery with the obvious treatment of ties, nothing more. There are extensions of the idea to continuous space, I cannot help but mention, and here’s a beautiful live example that you might like to marginally improve.

In contrast, here’s a set of Q-probabilities from around 1875. Finance has always trailed gambling, and we’ll return to that when I present my solution to the world’s woes.

Joseph Oller’s Totalizator Wagon, with a hard-to-discern Q-measure

I’ve unbundled Q-measure to simplify. Large tracts of quantitative finance comprise the art of inferring Q-measures from more complex settings where securities are treated as linear combinations of more basic bets (portfolios of horses) — but where it is obviously useful to back into what the Q-probability of a future event is (the price of the horse) whether or not the solution is unique. The horses are called Arrow-Debreu securities.

Examples where Q-inference is useful include equity options markets, mortgage backed securities, capital structure models, warrants and callable bonds. This is all historical happenstance, from a probabilistic perspective, serving only to cloud a simple underlying picture where “dollar-probability” Q is spread thinly across all possible futures of everything in the world.

So ends my quick intro to Q-probability, almost. I should mention though, that although linear algebra is something we all claim to wield, there are traps for the quantitative finance novice who fail to spot the opportunity. For instance, the classic traps sets up a one-period option pricing exercise. An option on a stock will be worth $10 tomorrow (if the stock goes up $10) or zero if the stock drops $10 (the only other possibility, we shall assume). The stock will go up with 90% probability. What is the option worth?

Few get it right.

Mathematical readers looking to ground Q-probability might look to functional analysis and the Riesz Representation Theorem. In statistics and parts of philosophy we have de Finetti and Ramsey’s Dutch Book Philosophy. In gambling we have Joseph Oller and in engineering, Sir George Alfred Julius.

New Zealand led the way in the implementation of the risk neutral measure

Pioneers of quantitative finance (notably Merton, Black, Scholes) considered the infinite dimensional space of hedging strategies. There is a rich literature that tries to relate P-measure and Q-measures (Girsanov), because due to finiteness of wealth, there’s no reason why they should coincide even for rational investors.

The third rail: R-probability

But this note is about R-probability, which you won’t find delineated in the textbooks— at least not the ones I own. It is neologism but my justification for a new category is very simple. People use it. Almost universally they use R, rather than P or Q. And that has very serious ramifications. A definition:

R-Probability. The self-constrained attempt to estimate objective probability of a future event that eschews the use of market prices as exogenous data.

Clearly R-probability, so defined, is similar to P-probability in intent. and technically speaking, it is merely a different choice of “filtration”. As a practical matter they may be numerically similar. Indeed there are many data pipelines running through industrial applications that spit out numbers people don’t bet on or trade. There’s no Q to speak of, so R and P effectively coincide.

(There is no Q-measure to shrink towards at least for now in most mundane operational tasks, though one day there’ll be a prediction web according to a wise man, and people will be Q-ing up to use it).

Groan. Now if there is a proximate market, then R-probabilities and P-probabilities can be materially different. And thus the decision to exclude market information (unconscious, usually) might be described in pejorative terms: a prejudice. Why use all data except market prices? Except market prices. Those three words can build a house of cards, I tell you.

Notice the borderline philosophical stance inherent in R-probabilities, even if it is usually an unthinking one. Market prices are just numbers in the universe and so in principle should be considered exogenous data for a logistic regression model, say, or any other kind, should they not? (That’s a rhetorical question. I’ve lost you if you disagree and won’t read this. Be well, and see you next article.)

R-probabilities emerge all the time when models and mental models and the people who make them consider consciously or unconsciously that this type of information is different, somehow, and thus deliberately or otherwise avoid its use. The most common excuse is that the R-probability under construction is intended to be compared to a Q-probability after it is made, so why include it now?

But it is just an excuse.

A consistent, ingrained behavior

Many years ago, in a blog so obscure I can’t even find it myself, I proposed that the curious habit of R-probability is a useful behavioral description — something to help describe the commercial world I’ve worked in for just over twenty years now. Finance is the study of how humans make financial decisions. It is not restricted to the set of ways that humans should make financial decisions.

That’s an easier sell these days since behavioral finance has had a nice run, even if I’m talking also about something a little different from the usual kinds of well-analyzed individual foibles. I’m interested in how entrenched modes of communication amongst entire groups and organizations (like ratings agencies) influence events.

And unlike mildly fluffy assertions like “irrational exuberance” that are open to the critique of hindsight, there isn’t any empirical debate about whether people and organizations use R-probability, or did last year, or will next year. They do. They did. They will. It’s written in the docs. It is right there in the code. So I’m saying that what I propose is behavioral, but don’t paint me with the same skeptical brush.

Going back to the time of the crisis, only a handful of people gave the explanatory power of R-probability any serious consideration (but thanks Bjorn). Today though, there is at least one demographic reason you should consider the matter more pressing. R-probability is the default for the vast and growing new population constituting the machine learning and data science community.

R-probability is also still the cornerstone of actuarial finance, I must add, an entire field that cannot make complete sense as a result. An entire field that thinks they are computing P-probability, when clearly they are not. An entire slew of people who are just about to un-follow me.

R-probability is a widespread affliction even in the quant community too, despite their schooling in the Q-measure and the near-efficiency of markets. I put this down to an overly sharp line being drawn, sometimes, between Q-quants and P-quants.

The horrendous track record of R-probability

A prejudice against the use of a particular type of exogenous data doesn’t sound like it will be good statistics, does it? And lo, the data suggests it isn’t good statistics … or good finance. I tie together some seemingly unrelated pieces of evidence:

1. M6. The under-performance of 97.6% of competitors in the M6 Financial Forecasting contest is explained succinctly by the observation that they were creating R-measures.
2. GFC. When institutional investors outsourced their homework to rating agencies on the matter of subordinated, pooled combinatorial cashflows (CDOs, CSOs, MBS…) it led to a reckless algebraic manipulation of R-probabilities with no foundational justification.
3. Handicapping. Teams that have tried to beat the racetrack using P-probability (i.e. explicitly including Q in the construction of P) have been more successful than teams that use R-probability.

The first example is clean cut and you can read about my little stunt here. The machine learning, forecasting and data science communities were challenged to compute P-quintile probabilities for stock returns. They chose, almost universally, to compute R-quintile probabilities instead — and so I beat up on the vast majority of them effortlessly.

The third example is mostly just here as an analogy, useful for some readers I hope (exactas versus CDOs) but the assertion is based solely on my own anecdotal evidence. I would not want my case to rest on it.

The non-theory of R-probability

I’ve asserted the obvious: R-probability is the norm in almost all quantitative fields … because only crazy people like me make any real effort to rip up their model pipelines and lay down market-like things in their stead.

The use of R-probability happens to be much more dangerous in finance because, due to the presence of powerful lurking prediction machines in the form of markets, the self-imposed accuracy degradation inherent in R-probability creation tends to be larger (because the information deliberately forgone is also larger, obviously).

Structured finance is a bit of a mess. Structured finance companies appear in other structured finance companies and so on to a depth of 11 nested levels (if memory serves, somebody from Intex might be able to confirm that’s the depth record). So R-probability is a potentially dangerous contaminant. And once something is R and not P or Q, you can’t put it in any financial decision making process. Not in a safe manner.

People use R-probability anyway, naturally. But worse, the usage is institutionalized: mandated. People and organizations constantly receive and transmit R-probabilities — in particular ratings.

I should clarify that although financial ratings issued by rating agencies involve letters of the alphabet they are probabilities in our discussion. The rating agencies might claim otherwise, or try to finesse the issue. But it’s irrelevant what’s in the legal boilerplate.

Ratings must be characterized as (quantized) probability estimates in the context of structured finance because rating agencies themselves use them that way. In the rating of structure finance vehicles (such as Collateralized Debt Obligations) those ratings are, in point of fact, converted to probabilities and sent back into a model.

The R-probability calculus is algebraically plausible (just as ‘cat’*3 is plausibly ‘catcatcat’ in Python) but actually there isn’t a financially justified thing to do with the result — so that’s where the compiler or type-checking comes in. The “program” — that big computer simulation we are all living in — should not have been allowed to run or if it did, it should have thrown a type exception.

There isn’t a theoretical stance that can be taken to the contrary. There isn’t a page you can open in the textbook where it says R-probabilities are okay — not as inputs to a portfolio optimization, not as inputs to a relative pricing tool, nada. Only a brave soul will suggest a place you can put them, or how they may relate to a financial decision making. Because they don’t — or not directly. Any use of them involves a reverse engineering of P-probability, or can be shown to be equivalent to one. Put it this way:

Financial theory exists for the P-measure and the Q-measure, only.

Of course there are heuristics for trading of the sort “if R-probability for X is greater than Q-probability for X then bet on X” and plenty of people have successfully made money that way. This is a bit like saying that plenty of people can make money backing the right horse so let’s not worry about R-measure versus P-measure. My retort is that few of those people can construct a robust trifecta portfolio — and that’s the relevant point of comparison to structured finance.

So, let me put you back on your heals if you think the burden lies with me to construct an elaborate reconstruction of the mechanics of the financial system using R-probabilities and how it did collapse or how it will collapse — because a priori there is no reason to believe the use of R-probability won’t be disastrous. I also don’t know exactly what will happen if you put beer in you car instead of fuel but it probably ain’t good.

S-Probability?

I ought to know the skinny because — small plot twist — I’m the guy who devised some of the rating agency models for Moodys and S&P (capital models for Credit Derivative Product Companies at least, a term for long-lived CDO-squared-like companies) even though I never worked for those organizations.

Scenario A

But why, you might ask, didn’t something go wrong with R-probability well before rating agencies started applying R-probabilities to structure finance companies? After all, rating agencies have used R-probability for a long time, and messaging has sort-of worked:

• A rating agency determines the R-probability of IBM defaulting in the next five years.
• Silly Valley Bank adds a “grain of salt”.

Here the “grain of salt” might reference the difference between actuarial probability (definitely in the R-measure) and various Q-measures inferred from, say, IBM’s bond or credit default swap prices (i.e. premiums for insurance on IBM’s bonds).

Things are mostly okay here, and the world survived for quite a while. The grain of salt addition might comprise the taking of a convex combination R-probability and Q-probability, as an attempt to “put back in” the information the rating agencies decided to deliberately ignore. Or the grain of salt, or some of it, might even be nutritious, since it might represent risk premium to be scooped up in a diversified portfolio of bonds. I take no view.

Scenario B

But … let’s move to the other half of the rating agencies revenue (yes, half came from structured finance ratings, roughly).

• A rating agency sends a number to Silly Valley Bank. This time it implies a probability that three out of seven companies will default in the next five years.
• Silly Valley Bank adds … how much salt exactly?

Time for another definition:

S-Probabilities are the numerical result of performing a naive multiplicative calculus involving R-probability, such as computation of a binomial probability.

If you start with a collection of probabilities that eschew market information and start multiplying them together you are not necessarily going to come up with a number that is easily relatable to other R-probabilities, such as that from the first example, or easily relatable to reality of any kind, for that matter.

Indeed it is straightforward to create examples where some reasonable combinatorial Q-influenced P-probability, such as the likelihood of at least ten defaults in a basket of a hundred companies say, is materially greater than zero whereas the corresponding naive estimate using R-probabilities is astronomically small. After all, if the ratio of P-probability to R-probability is a factor of two, that will get multiplied together many times.

If you then take CDOs and put them into another CDO then you multiply again and, well, you get the joke. If eschewing information deliberately leads to even slightly uncalibrated probabilities (which is surely does), then multiplication of fifty of them in a CDO and repeating the process sure as heck will put you in bananas land. My financial example is stylized but it captures the essence of the problem in structured finance.

Things are worse, because there are also R-correlations — sometimes explicitly used or worse implicitly arising in de-factor correlation models (usually ones where the correlation number can’t even be changed!). Put R-probabilities and R-correlations in a calculator and you get probabilities provided by ratings agencies.

Sound good?

It is hard not to interpret these as willfully bad, assuming you are willing to ascribe any intelligence at all to the market (which, I am delighted to report, at least 4% of data scientists do). Things are even worse than that, actually, because now the same strange desire to avoid market intelligence must be applied to the final multiplied R-probabilities. There are products traded in the over-the-counter markets which could serve as a check and balance at this stage— but we shall sail on and willfully ignore that too (to do otherwise might explode everyone’s actuarial heads).

Let us then return then, to the recipient of this R-probability spaghetti: Silly Valley Bank.

Upon receipt of the multiplied R-probabilities, and in order to make an investment decision — which must be performed in the P-measure because nothing else makes any sense — the bank is faced with a quandary. Should it kludge the message it has just received from the rating agency back into its “add a grain of salt” relative-investment methodology? How big should that grain be? Is it even obvious how to turn the S-probability is has received from the agency into either a more benign R-probability, or a P-probability?

This discussion is confusing, even to students of finance because everything is mislabelled. They’ve been taught that the rating agency pseudo-probabilities are real-world P-probabilities and they equate that with actuarial probability. False. Dead wrong as I’ve noted. Actuarial science might be intellectually bankrupt for precisely the same reason. Think about it.

A possible solution — really the only one — is for Silly Valley Bank to back all the way into the internal workings of the rating agency CDO models and change the R-probability inputs back into P-probability inputs (using the Q-probabilities on hand as something to shrink towards, one would think). Not hard in principle, but almost nobody does it. If they did go to that trouble, what would be the point of the rating agency, exactly?

What really tends to happen, in practice, is that multiplied R-probabilities get mentally (or mechanically) compared to other R-probabilities — and that really does drive investment decisions. Institutions look at a AA rating for IBM and a AA rating for a tranche of this-or-that and make some, but not always a huge distinction, between the two.

That is, in my view, another type exception because putting S-probabilities and more vanilla R-probabilities into the same function is so dangerous you might as well just go full Dan Aykroyd and cross the streams. I grant you that something will come out, but its going to be rather unpredictable.

Implied correlations and volatilities

To return to my three examples of the paucity of R-probabilities, let’s look at some key information they leave out.

1. M6 participants ignored implied volatilities.
2. Ratings ignore implied correlations and implied default probabilities.
3. Punters (racegoers) ignore the market in the first handicapping phase.

We are not talking about small discrepancies here, certainly not in the most important second example. Do you imagine, for example, that it could really happen that a rating agency would use a correlation of 3–5% while the inter-dealer market implied 20-30%? Well, that did happen.

So you see the problem isn’t with the normal copula model. It isn’t inherently benign. Despite the name, you can use it to generate a disaster far worse than the actual FGC if you want (so leave Karl Friedrich alone). The problem is the number that goes in, obviously. But the underlying problem there, as I have stressed, is that the numbers that go in are too often R-numbers.

Now to the analogy between the multiplicative problems that arise:

1. M6 Not applicable.
2. Ratings of structured finance products (combinatorial securities) are almost meaningless.
3. Exacta, quinella and trifecta pricing can be wildly uncalibrated, forcing a resort to very defensive investment strategies.

By the way I’m not making any argument against structured finance here. There isn’t anything inherently evil in either of the only two organic ingredients that go into creating a structured finance investment: namely diversification and subordination. You could sell it at WholeFoods.

A radical solution

Now I shall stop complaining and propose a solution —albeit one many of you might instinctively recoil from.

If one accepts that R-probability is pretty much the root of all evil, then the solution is obvious: use P-probability instead. But as we follow that train of thought things get more controversial, because the tenet of my argument is that the best way to improve P-probability is to ensure there’s a copious supply of Q-probability in the vicinity (the latter informs the former.)

So:

Pass a law that allows people bet on absolutely anything that informs, improves, or even mocks systemically important R-probabilities.

After all, that is what is means to make more Q-probability and I lack the energy to dance around everyone’s sensibilities in this wording (or pretense about the distinction between investing, gambling, prediction markets or what-have-you … they are obviously mathematically equivalent).

If you want better P, you need more Q. If you want more Q, you need to manufacture markets, or more things for people to wager on.

It really is that simple and in going down this path one will, over time, illuminate, and fix, really bad R-probabilities. For example you might allow people to bet on the number of companies that will default, rather than only allowing a tiny, tiny proportion of humanity (likely not the smartest) to create a cumbersome version of the same game in a very roundabout and expensive manner involving the establishment of companies in the Cayman Islands and the acquisition of broker-dealer licenses, et cetera.

It’s my assertion that even a tiny betting market (say on the scale of the Iowa Electronic markets) revealing and creating an independent source of implied correlations would force a dramatic improvement in rating agency models — lest they would be humiliated by graduate students looking to win beer money. Only the absurd protectionism of R-probability allows it to survive.

That is the hidden downside of the tangle of securities laws, anti-gambling regulations and the like — all of which is well intentioned and intended to save society from senior citizen zombie death in poker machine rooms (the tragedy anyone who has lived in Australia is familiar with).

But it’s up to the social scientists to survey the downside of unfettered Q-probability creation, as they know more than me. It is my role to point out the massive downside of not allowing it unilaterally, or discouraging Q-innovation by forcing companies to run the no-action gauntlet.

The downside of puritanical probability is a future financial crisis, almost surely. The intuition comes from civil engineering.

AI-artists’ impression of why inhibiting the creation of Q-probability leads to a financial system where a small number of enormous markets have no probabilistic support.

Here’s an artists impression of the financial system that arises when you adopt the strategy of allowing Q “dollar probability” to exist for only a very sparse collection of approved bets (equities, mortgage backed securities and things deemed societally acceptable like companies in the Caribbean that build or sell absolutely nothing).

This creates a pretty lame Q-probabilistic map of the future. The tall structures which arise (I mean the large markets) don’t do a whole lot to support or inform each other. That isn’t their design. So there is only flimsy, R-probabilty scaffolding.

That’s the cost of the bureaucratic approach — the one that tries to limit the number of market possibilities to a manageable number and then police the R-probabilities around them with layers of process (value at R-probability risk, review of R-probability models, and the like).

None of it makes anyone safer by the way. It just forces the creators of the R-probability models to spend so much time wrapping each model in an onion of pointless documentation and check-boxes that they never get around to making a better model. So R-probability gets even worse. When it isn’t the right thing to begin with.

A better approach simply unshackles creative minds who are free to create the best possible P-probabilities. And some of those people are smarter than the regulators, so they know that doing this should involve creating more markets not fewer. They know they can perform the equivalent of my M6 stunt by first creating a new market and then drawing a sharp contrast between the Q-measure so created and actuarial nonsense.

That is how you encourage scrutiny of every possible weakness in the economy, and how you expose every financial abomination before it blows something up. More Q, less R.

The result will sill involve some tall timber as in my picture but also a larger number of small but strong supporting cross-structures. A fine mesh-like universe of Q-probabilities can arise, even if the notionals are small, in the style that both nature and material science would approve. That kind of economy, where information propagates freely and is not deliberately obfuscated by things like structured credit ratings, might not collapse the same way it did last time (though of course, there are other ways it might).

My call to “let everyone bet on anything” is a crude Australian way of putting it, I admit. Yet it echos the more refined Italian exasperation of de Finetti who declared that probability does not exist … if people aren’t wagering. I’ve pointed to the paucity of probabilities that inhabit the vicinity of all major markets, and can bring them down.

De Finetti’s concern was a little different, but anyone who is serious about the robustness of the financial system needs to get over their puritanical predispositions and, in the interest of avoiding trillion dollar losses and disastrous transfers of wealth in periodic collapses, stop worrying about whether somebody contributing to superior prediction of a crucial number will lose a hundred bucks.

There is too much at stake to allow the conceit of central planning to be married with the conceit of modeling itself — that is, the creation of R-probability in a chain of quantitative calculations orchestrated by a mediocre human or team, as compared with the creation of Q-probability as the output of fierce competition with no barriers to entry. The rest can be put to music.

Capitalism’s about profit and loss. You bail out the losers there’s no end to the cost. The lesson I’ve learned is how little we know. The world is complex, not some circular flow. The economy’s not a class you can master in college. To think otherwise is the pretense of knowledge.

(Keynes v Hayek Round II)

And in keeping with that source: we need probability by the many, not by the few.

Summarizing thoughts

The collapse of structured finance can be characterized as an avoidable run-time pathology in a weakly typed computer program. The following should never be cast from one to the other:

• P-probability. Use anything you can find.
• Q-probability. Use prices only.
• R-probability: Discards really useful market information.

Do not complain to me about this taxonomy or its incremental complexity compared to P union Q unless:

• You can derive a coherent theory for investment using R-probability — I don’t think you can.
• You can present a careful justification for R-probability messages in any combinatorial setting.
• You think the union of Q-probability and P-probability users constitutes more than 10% of the population.

Financial textbooks have let us down — though only by omission. The theory, while not wrong in any way, conditions us to expect a false partition into P and Q categories. That simply cannot explain how people acted, and how they will.