AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT

TLDR

The optimized certainty equivalent (OCE) is a decision‑theoretic criterion based on a utility function introduced in 1986. This paper re‑examines and extends the OCE, exploring its properties and positioning it relative to recent risk‑measure concepts. Using duality theory, the authors link the OCE to φ‑divergence, derive variational formulas, and establish conditions under which related certainty equivalents become coherent or convex risk measures. The negative OCE generates a broad class of convex risk measures, subsuming known examples such as conditional value‑at‑risk and bounded shortfall risk, and the paper provides illustrative examples of its flexibility.

Abstract

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re‐examines this fundamental concept, studies and extends its main properties, and puts it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the ϕ‐divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.