Having once sketched out my ideas on risk, and discussed some of the implications of Daniel Bernoulli’s work, I have decided to publish my Master’s thesis from the Université d’Angers in France, on, “The Nature of Risk”. In part, it is a genealogy of sorts of a school of risk theory. Standard risk theory treats sees risk as a measurable distribution and a forecasting problem. Through this genealogy, I argue that it is neither. Drawing on Knight, Keynes, von Mises, Bernoulli, Shannon and Kelly, it shows that risk is subjective, that parts of it cannot be quantified, and that because wealth compounds multiplicatively, loss aversion is rationally obligatory rather than a behavioural bias. Lose 50% and you need 100% to recover. The geometric mean follows as the decision criterion.

Why loss aversion is rational

Behavioural finance treats loss aversion as a bias. Kahneman and Tversky found that people feel losses more sharply than equivalent gains, and the finding is usually presented as a departure from rationality that a disciplined investor should correct. I argue the opposite. Wealth compounds multiplicatively, not additively. A 50% loss requires a 100% gain to recover. A 90% loss requires a 900% gain. The arithmetic mean of a return series tells you nothing about the terminal wealth of an investor who actually lived through it, because the compounding path is what determines where you end up. Once that is granted, weighting losses more heavily than gains is not a psychological distortion. It is the correct response to the shape of the wealth curve. Daniel Bernoulli saw this in 1738 and called it “nature’s admonishment to avoid the dice”.

This is why I build cost of equity on downside and expected-shortfall beta rather than on variance. Variance treats a gain and a loss of the same magnitude as the same event. The investor’s wealth does not.

Abstract

This treatise presents a pure theory of the nature of risk, arguing that aspects of risks are not quantifiable, that risk is subjective, and that losses impact portfolios more strongly than gains, leading to rationally obligatory loss aversion. Risk is built out as being involved not simply with probability distributions, but the weighing of evidence and judgement between ideas. This is done by reimagining probability as an extension of logic. Given the multiplicative nature of wealth, loss aversion is not only necessary to avoid gambler’s ruin, but to grow it as well. The geometric mean is, therefore, the more appropriate decision criterion to build wealth and avoid the assumption of debilitating risks.

Keywords: measurability, uncertainty, truth, pre-Pascalian probability theory, subjectivity, Bayesian, logical probability, downside risk, losses, loss aversion

Part I: Is Risk Measurable?

The first half of the thesis takes on the Knightian consensus, which holds that risk is measurable and uncertainty is not.

It begins before Pascal. Drawing on James Franklin’s work on pre-Pascalian probability, I trace probability back to rhetoric, Aristotelian logic, and the medieval law of evidence, where probability meant the weight of an argument rather than a frequency. The mathematization of probability in the seventeenth century gained precision and lost that older sense of judgement between competing claims.

Frank Knight’s theory of risk and uncertainty is then examined on its own terms, along with its limits. Knight requires risk to be knowable to a degree that admits no error rate, which is a demanding condition that few real estimates meet.

Ludwig von Mises supplies the distinction between case probability and class probability, and the argument that human action is reflexive in a way that frequentist methods cannot capture.

John Maynard Keynes supplies the logical interpretation: probability as a relation between propositions, the weight of an argument as distinct from its probability, and the claim that many probabilities are not numerically comparable at all.

The conclusion of Part I is that risk is an epistemic category, not an objective one. Even if risk were objective, our models of it are not, and they generate uncertainty of their own.

Part II: The Prospect of Ruin

The second half asks what an investor should do given that conclusion.

It starts with Pascal and Fermat on the problem of points, and with the weaknesses of expected-value maximisation that follow from it. The St Petersburg paradox is the sharpest of these: a wager with infinite expected value that nobody will pay much to enter.

Daniel Bernoulli’s 1738 resolution is the centre of the thesis. Utility derives from growth, the wealth curve is concave, and the appropriate criterion is the geometric rather than the arithmetic mean. From that single move fall out an explanation of why diversification works, why insurance can increase expected wealth rather than merely transfer risk, and a partial deconstruction of the equity premium puzzle.

Bernoulli’s model also contains an error, which I set out: his treatment of the entry fee implies that the bet cannot produce a loss.

Claude Shannon’s information theory and John Kelly’s 1956 criterion complete the line. Kelly’s result confirms Bernoulli’s: the way to grow wealth is to maximise the geometric mean. It also delivers a harder lesson. Without an edge, there is no rational reason to bet at all.

The Nature of Risk – Joseph Noko (2022)Download

How to Cite It

Noko, J. (2022). The Nature of Risk. Master’s thesis, Université d’Angers.