Whether it comes from ethics or from theory, the impression is that the days of the contemporary speculator are numbered. For this moody character is reckless: he is one to confuse the stock market and roulette. Economic theory has a hard time understanding him. Nevertheless, a theory of the speculator is possible.

It shall integrate the motives governing actions, the forms taken by anticipation, the specific rationality of individual agents, the general criteria defining rationality, the conditions for conjecture forecast possibility and the conditions for theoretical calculation.

The mathematical expectation for speculators’ gain comprises two parts: on the one hand, the total that dividends are expected to amount to by the date of resale, due to maturity and weighted by inflation, and in the other hand, an exit price that is itself uncertain. Said exit price is therefore expressed as the weighted sum of any dividends taking place between the date of exit and the end of time.

**The zero and the infinite **

These projections in the future, ad infinitum, tailored after the present time, date back to Leibniz’s financial calculations. It could be argued that they are not devoid of a certain boldness. Exploring the validity of the calculation of fundamental value in finance, or even of the alternatives that have been proposed in technical literature, reveals that the case revolves around one crucial element: the trend properties that have been attributed to the propensity of possible futures to tailor themselves after the present time. This constitutes what is known as the “transversality condition”, which suggests that in principle, exit prices tend towards zero. In other words: in order to rationalize, by means of calculation, a purchase taking place today that is slated for resale at a later time, one has to pretend that, well beyond that resale, the value of said property is negligible.

A difficulty arises as soon as one wonders if such trend properties are objective, like a general feature of the economic system, or if they are a function of the specific reasoning of individual agents, or finally if they are epistemic in nature, that is to say that they are axiomatic imperatives for theorists whenever they start reasoning. These are critical little gaps that must be explored.

Let us open Pandora’s box and tinker with the properties of exit prices, starting with the hypothesis that all agents do not envision the future in the same way. To begin with, in the real world, one can hardly assume that an investor will expect exit prices to be infinite. As a matter of fact, that would be a first principle of the subjective probability of reasonable speculators: an agent that would ignore that prices have finitude would be a senseless, reckless agent. We all remember Keynes’ phrase: *“In the long run we are all dead”* (1923). The principle raised in this paper is less severe: it does not forbid considering the distant future, but dismisses the rationality of an agent whose presupposition would imply a trend value of goods unrelated to their current value. The coherence of the standard financial theory would expect exit prices to objectively tend to zero. Well, let us take note that our aforementioned speculator, who now has us in a quandary, is not a conformist. According to investigation on rational bubbles, he is counting on non-zero exit prices.

Recent research in economic history regarding financial institutions has led me to believe that in the course of the last three centuries, procedures, organizations and specialized calculations have had the effect of coordinating and harnessing heterogeneous agents characterized by expectations about exit prices of heterogeneous structures themselves. This historical process has seen many incarnations where each time, depending on the market and at various times in history, recording and calculation techniques, i.e. the state of the art, have had a significant impact. Under these conditions, that have been shaped by centuries, I have reached the conclusion that what characterizes the last decades is a very high degree of technical coordination – which goes hand in hand with the development of theoretical frameworks whose take would precisely be that the average trend for the exit prices of any item should tend to zero. In short, that would effectively end up with the closure of a global financial system characterized by an adjustment based on said objective characteristics. But neither that coordination nor any intellectual framework can impede agents from interacting in ways that are not necessarily consistent with the general trend. One could even question the fact that said coordination and framework should exercise such a restriction at all. Let us now explore what possibilities are opened by a subjective variability of exit prices with two examples.

**The prudent investor and his demon**

Let us first say that a speculator is driven by a demon that lulls him or her with the prospect of perpetual annuities that would also be conveniently constant until the end of time… in short, the *demon of sound and prudent investing*. It is certainly not consistent with the hypothesis of transversality, no arguing about that. However, short of living in a world condemned to perpetual deflation, such reasoning patterns beg for calculation. What is more, they are very commonplace and consist in expecting, with an indefinite horizon, constant dividends. Long before economic agents became familiar with the concept of inflation, a dividend was deemed acceptable when it amounted to one twentieth of the capital, that is to say a ratio of 5%. From the late seventeenth century to the early twentieth century, public credit rates have averaged around 5%. As for the twentieth portion itself, this multiplier is still employed to extrapolate the annual amount of a rent based on the value of a real estate property. The underlying idea is that capital must be more or less redeemed after twenty years, not counting the value of the property itself at that date. But to what extent would such an intuition have remained satisfactory?

Exit prices, fixed annuities and the expected inflation rate are tied. For example, in the case of the twentieth fraction, the exit price is equivalent to nil if inflation reaches 2.9%. In an economic world where investment criteria would be consistent with the demons of prudent investing, adjusted on the ratio of a twentieth, an inflation in the vicinity of 2.9% would provide general conditions comparable to those of a financial world “of today” where all prices would be fixed prices and where agents would behave like standards rational investors. In other words, for example, the real estate market of the Old Regime knew none of the formal requirements of today’s financial markets but was able to function perfectly, even though agents within it intervened according to expectations that would be qualified as unreasonable today, and *precisely because* investors’ expectations took the very forms that are deemed unreasonable by today’s theory.

Today, he who plans to invest a sum in the hopes of ending up with consistent and sustainable dividends despite a persistent inflation is merely giving in to the sirens of prudent investing – even without delving into the intricacies of financial theory or in the kind of commercial refinements in which private banking is known to excel. Whereas an expert would be indignant at having to settle for such gross adjustment. One is compelled to acknowledge that such an equalization is by no means absurd as long as inflation is kept moderate. Today as it was in past, it is consistent with the most common approaches of laymen to financial investment.

**The demon of air castles (Perrette 0, Pareto 1)**

Let us proceed to a second example to ponder, taken from one of La Fontaine’s famous fables, “The Milkmaid and the pot of milk.” It has widely been commented, often by highlighting the irony behind a young maid’s propensity to get carried away by ever inflating hopes. But we must not forget that, precisely, the maid Perrette is speculating, calculating, and losing herself in conjecture and that therefore every stage of her mind’s journey should be scrutinized through the calculation of probabilities. We shall remember that Perrette *“so attired/Counted already in her thought/The price she got for her milk; used the money/Bought a hundred eggs; had a triple brood of chickens,”* she would then proceed to sell the chickens for a pig, would fatten it to resale it and buy a cow, etc. All until *“Perrette, as she thought this, leaps also, carried away/The milk falls; goodbye calf, cow, pig, brood.”*

It could be argued that the milkmaid, at each and every stage of the expected growth, is elaborating upon the hypothesis that she can indeed hope to increase the wealth of her homestead by a constant factor.

Thus the milkmaid, neglecting any dividends as she is, is continuously envisioning each future price, as a next step, exclusively on the side of hope, where all the possible rates are conveniently superior to the present rate. A law of probability that along those very lines would link the present price to the price immediately following would meet the criteria that are characteristic of the Pareto principle (a famous mathematical law of probability that states that 80% of any given effects can come from 20% of cases, empirically). Now we shall remember that an exit price is the projected hope of a future price, adjusted by inflation. In line with the journey of the milkmaid’s mindset, let us analyze a price, step by step, where an initial price is assumed as a given, and where the subsequent price relates to the previous one according to a probabilistic dependence between events, itself a function of a power law whose parameters are assumed to be constant.

In this rudimentary formulation, the expected exit price is equal to the current price minus the objective inflation but amplified by a probabilistic factor of personal enthusiasm (this is the coefficient of the Pareto principle: it is constant here, but it could be made variable). One readily understands the projection embraced by Perrette the milkmaid’s as she indulges in daydreaming: the propensity of her speculation to tend towards an infinite horizon for the exit price.

However such a trivial result is specific to geometric progressions: here, there are only three trends. They are a function of the common ratio of the geometric sequence – whether it is greater than, equal to, or less than 1. In other words, depending on the prevailing outcome, i.e. whether it turns out to be the growth that was projected as a probability, or at the contrary overall weariness affecting economic activity, a speculator driven by the demon of air castles will either be carried away by boundless enthusiasm, remain sober, or fall into despondency. The same investing patterns, the same integration of environment parameters, and minute variations in the outcome of expected growth can dramatically tip over a speculator’s perspective on things, from hoping it all to expecting nothing anymore.

Such a conclusion was a priori far from obvious: should my state of mind lead me into anticipating the future according to Pareto patterns, this expectation would abruptly switch from enthusiasm to despondency, or conversely – as long as my take on the relation between the near future and the present is only slightly more optimistic, or only slightly more pessimistic, than what the actual state of the outside world would allow anyone to consider. Thus, nothing is more consistent with calculations that this model, yet it leads to the archetype of things unreasonable: a succession of sharp ups and downs, based on gut feelings and needing but a tiny bump in the road to disturb one’s expectations. We now benefit from a new interpretation of the fable’s punchline, a punchline that is still widely regarded as enlightening even though its actual power is lost on us without the framework of probabilistic analysis. Along the way, Perrette the milkmaid gives in to the lull of her own Pareto-shaped demons. She dreams, as if from stage to stage and in all probability, her wealth was already growing favorably, according to the dream. Then she stumbles. And sees the milk, spilled. At which point she must revise her expectations, shifting from high hopes to nothing in the blink of an eye.

**Under volatility lies rationality**

Isn’t this all consistent with the throes of speculation, so often stigmatized but never *analyzed* in the strictest sense of the term? Doesn’t this constitute yet another lead to further understand the much decried mindsets that induce high volatility in financial markets? To reason in that manner, i.e. distinguishing the objective trend upon which standard financial calculation functions from an array of novel hypotheses, that have been presented in simple form here and probably outside of the usual standards of financial calculation so far, opens a whole new dimension for research. We have full scope in our pursuit to develop a model for exit price, insofar as it remains positive and bounded.

As long as the logic of the financial system remains characterized by an overall condition of transversality, there is no reason to inhibit the singular rational action of an economic agent through the implicit assumption that tendentially, an exit price zeroes in the zero mark. “Nothing great was ever achieved without a little enthusiasm,” wrote Voltaire in 1761. This portion of enthusiasm is the product of the proximity between a given agent’s own subjective probability and objective chances – chances that could be measured and computed. Getting speculators to be inspired by “demons” or “sirens” with fairly simple appetites (here a constant hope, there, Pareto’s probabilistic dependence) would empower us with the means to account for the evolution of their shifting expectations, and ergo, of the logic underlying their actions. After all, ever-changing and sometimes whimsical maverick speculators are nevertheless reasonable.

## References

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