Defining Risk

The financial literature frequently mentions risk, but it lacks a widely accepted definition of risk. This omission is no coincidence. Risk is an intuitive notion that resists formal definition. Adopting a largely historical perspective, this article draws on ideas that emerged during the 20th century and uses them to formalize specific limits to our ability to ever define the notion of risk. In doing so, it offers insights into the nature of risk. The financial literature frequently mentions risk, but it lacks a widely accepted definition of risk. Applications—such as risk limits, trader performance-based compensation, portfolio optimization, and capital calculations—increasingly depend on metrics of risk, but do these metrics reflect true risk? With financial decisions and compensation hanging in the balance, debates flare.Adopting a largely historical perspective, I draw on ideas that emerged during the 20th century to analyze the nature of risk. These include subjective probability and the philosophy of operationalism, both of which have origins in the empiricism of David Hume.Frank Knight's famous definition of risk is problematic for being unrelated to common usage. It depends upon an objectivist interpretation of probability and is somewhat parochial, being more useful in certain fields than in others. Harry Markowitz was careful to not define risk in his groundbreaking work on portfolio theory. This may have been due to his subjectivist perception of probabilities.Risk, as that notion is commonly understood, comprises two components: exposure and uncertainty. According to Percy Bridgman's philosophy of operationalism, only that which can be perceived can be defined. At best, we may define perceived exposure or perceived uncertainty. Accordingly, we can only hope to define perceived risk. There is no such thing as true risk.As practitioners of finance, we use subjective probabilities to operationally define perceived uncertainty. We use utility or state preferences to operationally define perceived exposure. Because perceived risk takes so many forms, it is not easy to operationally define. To simplify this task, we may operationally define some aspects of perceived risk. We adopt risk metrics—such as variance of return or maximum likely credit exposure—to identify specific aspects of perceived risk.What is risk? How can we quantify risks that cannot be perceived? If a trader or business manager has knowledge not reflected in a risk metric, does the risk metric misrepresent risk? In the absence of true risk, these questions are empty. A more practical question is whether a risk metric is useful. Used in a given application, will it promote behavior that management considers desirable?