Risk Measure Examination for Large Losses

The risk measures such as value at risk, and conditional values at risk do not always account for the sensitivity of large losses with certainty, as large losses often break the homogeneity especially seen in an illiquidity risk. In this study, we examine the characteristics of large-loss sensitivity more holistically, including small probability, within the framework of risk measures. The analysis incorporates the certainty equivalent, generation of the optimal certainty equivalent formulation, divergence utility, and general utility functions in their original form, and their relationship with expectiles and elicitability. The discussion provides a summary in the understanding of risk measure status and sensitivity involving small probably cases. Additionally, we evaluate large-loss sensitivity in risk-sharing scenarios using the convex conjugation of the divergence utility. By clarifying the conditions affecting large-loss sensitivity, the findings highlight the limitations of existing risk measures and suggest directions for future improvement. Furthermore, these insights contribute to enhancing the stability of risk-sharing business models.