Beyond Risk Management

I recently finished economist Frank H. Knight’s 1921 book, “Risk, Uncertainty, and Profit.” Even though it was written more than a century ago, its relevance today, in my view, remains significant. Many of its core insights on business and decision-making are sometimes less emphasized in modern discussions. This article is not a book review, but an attempt to extract and apply the ideas that challenge today’s common framing of economics and risk management.

At the center of Knight’s work is a clear distinction between risk and uncertainty and their role in generating profit. In Knight’s framework, risk refers to situations in which outcomes are random yet measurable. In Knight’s framework, uncertainty arises when a lack of knowledge prevents the assignment of predetermined probabilities. That distinction matters because it can be understood to underpin “pure profit,” the excess return that cannot be competed away. Unlike accounting profit, which is simply revenue minus costs, pure profit can be understood as arising from the fact that the future is not fully knowable.

Knight’s argument is direct: Without uncertainty, there is no profit, no incentive to act and no market to speak of. By following that logic, he not only explains where profit comes from but also offers a basis for re-examining how businesses operate and manage uncertainty.

ON RISK

A definition of risk directly from the 2017 book “Financial Markets Theory: Equilibrium, Efficiency, and Information,” which I reference here, is one I believe Knight would still very much recognize:

“At the initial time t=0, the agent is assumed to know the probability space and, therefore, the set of all possible elementary events and the probability measure. At time t=1, the state of the world is fully revealed, and the agent discovers the realized elementary event. When an agent cannot observe the state of the world at the initial time but has complete knowledge of the probability space, we say the situation is risky. This situation differs from the case of an agent who does not perfectly know the probability of events: in this case, we say the agent faces an uncertain situation. In our analysis, risk refers to a setting of known probabilities, while uncertainty refers to a setting with unknown probabilities.”¹

From this definition, the distinction between risk and uncertainty depends heavily on how probability is understood.

ON PROBABILITY

Here are three definitions of probability from Knight for consideration.²

1. “A priori probability. Absolutely homogeneous classification of instances, completely identical except for really indeterminate factors.” Russian mathematician Andrey Kolmogorov later formalized the mathematical foundations of this type of probability. It is based on the combinations and permutations of equally probable outcomes in a situation, which can be determined mathematically without running experiments or trials. It is the type of probability defined in the risk definition above. Think of casino games like blackjack or roulette.
2. “Statistical probability. Empirical evaluation of the frequency of association between predicates that are not analyzable into varying combinations of equally probable alternatives. The main distinguishing characteristic of this type is that it rests on empirical classification of instances.” To simplify, this probability cannot be determined by the combinatorics of the situation, like a priori probability. Furthermore, it is impossible to eliminate all factors that are not strictly indeterminate. For example, suppose trying to determine the probability of success by offering a new insurance product through a new distribution channel.
3. “Estimates. The distinction here is that there is no valid basis of any kind for classifying instances. The form of probability is involved in the greatest logical difficulties of all, and no very satisfactory discussion of it can be given, but its distinction from other types must be emphasized, and some of its complicated relations indicated.” Consider determining the probability of different climate change outcomes.

These definitions of probability coalesce around the idea of how much knowledge about a future event can be gained with certainty. Uncertainty generally emerges when that knowledge is incomplete or constrained. Although fields such as computation theory, behavioral economics and complex systems were formalized decades later, Knight anticipates many of their central insights—particularly the limits of knowledge, the role of uncertainty and the behavior of complex economic systems. He arrives at these ideas independently, outlining a framework that parallels conclusions reached much later. A complex system, in this context, consists of many interacting components whose collective behavior cannot fully be understood by analyzing the parts in isolation.

RISK-ONLY WORLD

Knight, in many instances, imagined a “risk-only” world—one without true uncertainty—which he equated with perfect competition. In his framework, risk exists only when outcomes have known probabilities (a priori). If probabilities can’t be defined, then we are dealing with uncertainty.³

The way I see it, a world without uncertainty would be almost unrecognizable. Randomness could still exist, but there would be no real surprises.⁴ Kind of like a dice game where all possible outcomes are known, not which one will occur. Businesses operating at scale would know event frequencies with precision. In practice, casinos come closest to this setup: While individual outcomes are random, the business relies on predictable long-run averages. The house edge—typically around 0.5% to 5% for major table games—creates a small but persistent expected advantage, which, over large volumes of play, is sufficient to sustain the industry.

In a stylized theoretical setting, I see four possible developments:

1. Competition would be highly efficient because future outcomes are predictable.
2. With full information, everyone might act optimally and rationally.
3. Prices could equal costs.
4. Supply would tend to always match demand.

The result is more extreme than just the absence of profit—there would not even be accounting profit. In this setting, the need for specialization, management and markets would be radically reduced because decision-making would no longer depend on judgment under incomplete information. Everyone already knows what to do, and the system effectively runs itself. Opportunities disappear, incentives vanish and the economy becomes static.⁵

Knight’s risk-only world closely mirrors economist Léon Walras’ concept of general equilibrium, in which perfect competition and complete information eliminate profit and leave prices to fully coordinate economic activity. Mathematician John von Neumann later provided a rigorous mathematical formulation of these ideas using linear models and fixed-point methods, helping lay the foundation for modern general equilibrium theory. In this context, linear systems—where combined inputs produce proportional, additive outputs—offer a tractable way to model economic relationships. Much of modern neoclassical economics, actuarial science and financial modeling builds on these foundations.

WHAT GLITTERS IS NOT GOLD

Let’s consider modeling the economy as a linear system, with current economic thinking and risk management in play. Economist Robert Solow, in an episode of his “People I (Mostly) Admire” podcast, argues that economists often design their models as mathematical exercises rather than as explanations of how the economy actually works. Much of this stems from reliance on Dynamic Stochastic General Equilibrium (DSGE) models.

Solow appears skeptical of DSGE frameworks because they assume a single representative agent whose behavior drives the entire economy. That simplification can be useful analytically, but it also abstracts away much of the complexity that characterizes real-world markets. In that sense, the theory’s centerpiece can appear to be at odds with the decentralized nature of real-world markets. The deeper issue is the fixation on mathematically proving economic theories. Proof becomes far more tractable when systems are linear rather than the complex, adaptive systems we inhabit.

As British statistician George Box famously put it, “All models are wrong, but some are useful.” To me, there is a meaningful difference between a model that simplifies reality to highlight its essential features and one whose assumptions are so restrictive that its connection to observed behavior is harder to defend. Knight’s extended treatment of perfect competition in a risk-only world makes that gap hard to ignore. Here’s one perspective, as argued in the 2025 book “Entropy Economics,” to consider: “Equilibrium is nothing more than a figment of economic models and the imagination of their creators.”⁶ Whether or not one accepts this, it illustrates the broader concern that equilibrium models can obscure the roles of uncertainty, adaptation and institutional context.

What makes this potentially frustrating for someone like me is how mathematics is often taught. It is presented as a set of procedures to follow, a matter of plugging values into equations and moving on. Mathematics is a powerful language for describing systems, each defined by its own assumptions and axioms. Those assumptions shape entirely different worlds. Yet there is rarely an emphasis on reading and questioning the assumptions and underlying models themselves. That gap is part of what I believe makes Knight’s work so compelling—he forces the reader to confront the world implied by the assumptions, not just the elegance of the math.

UNDERSTANDING UNCERTAINTY

Knight makes a sharper claim than most readers might expect: real businesses rarely operate in a purely “risky” world governed by a priori probabilities. In practice, firms operate within complex, adaptive systems where uncertainty is unavoidable. Business activity often depends on that uncertainty. It creates the possibility of profit and loss, and forces decisions in the face of incomplete knowledge. Profit is not a reward for bearing calculable risk; it is a reward for acting under uncertainty.

Risk management literature acknowledges the distinction between risk and uncertainty but often proceeds by absorbing uncertainty into probability-based models. It breaks risk into qualitative and quantitative categories while treating uncertainty as error absorbed into probability estimates. That move can make uncertainty less visible in the model, even though it drives both losses and the dispersion of outcomes.

Knight takes the opposite approach. He does not try to measure risk because, in his framework, risk exists only when probabilities are known in advance. Few, if any, real businesses operate under those conditions. In a fully competitive, risk-only world, outcomes are predictable, and margins disappear. There would be little left to manage in a traditional sense.

Uncertainty, by contrast, captures the gap between what is known and what unfolds. It reflects the limits of knowledge, shifting conditions and the evolution of the system itself. In my opinion, calling the discipline “risk management,” therefore, may not fully capture the broader challenge. The real task, from this perspective, is managing uncertainty.

KNIGHTIAN UNCERTAINTY MANAGEMENT

This article treats Knight’s framework as a lens rather than a settled prescription. My aim is not to dismiss risk management or probability-based modeling but to examine what may be lost when uncertainty is treated primarily as measurable risk.

One way to manage uncertainty is through specialization and the steady expansion of organizational knowledge. Knight makes this concrete; a manager’s role is not to do the work, but to recognize, hire and coordinate those who can. Advantage comes from a dense internal network of specialized knowledge and the ability to collaborate effectively. Optimally, some organizations localize decision-making and swarm problems, reducing uncertainty and preserving the conditions for pure profit.

As each layer specializes, knowledge can compound, uncertainty narrows and outcomes become more predictable. That is why I believe cost-cutting through layoffs may, in some cases, reduce long-term organizational knowledge. While it may lift short-term accounting profit, it can erode accumulated knowledge and potentially reduce the firm’s capacity to generate pure profit over time.

Knight also rejects two extremes: treating every situation as identical or as unique. Specialization trains people to identify what matters, abstract common features and make situations comparable. This process enables the use of statistical estimates, not as a substitute for uncertainty, but to bound it. The goal generally is to improve estimates of probability by providing tighter ranges and more accurate forecasts. Managers may also diversify into projects that are orthogonal to the core business to offset estimation errors and reduce exposure to any single outcome.

The key distinction is that uncertainty is context- and environment-specific. Therefore, managers may need to continuously underwrite uncertainty with the firm’s evolving knowledge. Once uncertainty is bound, risk can be measured and managed, but never in isolation. Conventional risk management might bypass this, assume stable probabilities, and project a level of certainty that does not exist. If outcomes are truly predictable, what management are you really doing?

ESTIMATING PROBABILITIES

Let’s take a look at two branches of probability estimation: Frequentist (objective) and Bayesian (subjective). They differ in perspective, but their ultimate weakness stems from both being based on Kolmogorov’s theoretical, additive-probability measure, i.e., a priori probability. Here is information from “The Geometry of Uncertainty” book:

“Probability theory’s frequentist interpretation is utterly incapable of modeling ‘pure’ data (without ‘designing’ the experiment which generates it). In a way, it cannot even properly model continuous data (because, under measure-theoretical probability, every point of a continuous domain has zero probability). It must resort to the ‘tail event’ contraption to assess its own hypotheses. Scarcely available data can only be modeled effectively asymptotically.”

Figure 1: Kolmogorov Probability

There are limitations to Bayesian reasoning as well: It just cannot model ignorance (absence of data); it cannot model pure data (without artificially introducing a prior, even when there is no justification for doing so and no clear-cut criterion for doing it); ‘uninformative’ priors can be dangerous, i.e., they may bias the reasoning process so badly that it can recover only asymptotically; it cannot model ‘uncertain’ data because it assumes the new evidence comes in the form of certainty, i.e., information not in the form of propositions of the kind ‘A is true’; and it can model scarce data only asymptotically.”⁷

What is the alternative? Belief functions. Developed in Dempster–Shafer theory, they extend classical Kolmogorov probability by representing uncertainty with lower and upper bounds (belief and plausibility) rather than single-point probability estimates. Instead of forcing a precise value, they assign belief to sets of outcomes, capturing both supporting evidence and unresolved uncertainty. Where probability states P(A) = 0.7, belief functions express a range, such as support between 0.6 and 0.9, making ignorance explicit rather than hidden.

Figure 2: Belief Functions

This framework distinguishes between what is known and what is not. It assigns belief mass to sets of possibilities rather than to individual outcomes. In place of a traditional event space, it uses a frame of discernment, the set of all possible answers to a problem, one of which must be correct. The belief measure Bel(A) represents the evidence supporting a set A and serves as a lower bound on the probability of A. The corresponding upper-bound probability, plausibility, is defined as Pl(A) = 1 − Bel(Aᶜ), capturing the extent to which the evidence does not contradict A.

Consider a familiar example: the Sharpe ratio, which provides a point estimate of excess return per unit of risk. A belief-function framing instead asks how much excess return the evidence supports—and within what bounds—shifting the focus from a single estimate to a range that explicitly reflects uncertainty.

While only an introduction, this section points toward a broader approach. I believe belief functions represent one possible approach to working with uncertainty directly rather than forcing it into precise probabilities. If you are interested, researchers have conducted studies on applying belief functions to portfolio evaluation and trading strategies.

IN CLOSING

Knight pushes equilibrium thinking to its logical limit by clarifying its underlying assumptions. By sharply distinguishing risk from uncertainty, he shows that a fully predictable, uncertainty-free world converges to perfect competition, where profits are competed away and economic activity becomes largely mechanical. Risk, in this framework, belongs to controlled, repeatable environments. Markets, by contrast, run on uncertainty.

This insight tends to challenge modern economics and risk management. When models assume stable probabilities, they describe a world that strips away the very force that drives real outcomes. The result may be precision that is, in practice, difficult to apply.

Knight’s deeper contribution is practical; in this view, firms don’t eliminate uncertainty—they organize to act under it. Specialization and coordination make judgment possible, but they do not remove uncertainty. Managing uncertainty, not risk, is the central economic task here.

Frameworks like Dempster–Shafer theory move in that direction by making uncertainty explicit rather than compressing it into precise probabilities. They replace single-point estimates with bounded belief and allow for incomplete knowledge.

If risk management begins after uncertainty is removed, it may have started too late. The key challenge is to face uncertainty directly, constrain it where possible, and build organizations capable of operating within it.

Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries or the respective authors’ employers.

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