The Axiomatic Approach to Risk Measures for Capital Determination

Abstract

The quantification of downside risk in terms of capital requirements is a key issue for both regulators and the financial industry. This review presents the axiomatic approach, which is based on monetary risk measures. These provide a unifying mathematical framework for the determination of capital requirements, for economic indices of riskiness, and for the analysis of preferences in the face of risk and Knightian uncertainty. In the special case of distribution-based risk measures, we review recent advances in characterizing their statistical properties such as elicitability and robustness.

The joint impact of bankruptcy costs, fire sales and cross-holdings on systemic risk in financial networks

“…We investigate these in the section "Capital requirements" in more detail. For surveys of the literature on monetary risk measures, we refer to Schied (2011) andWeber (2015). A less sophisticated approach than monetary risk measures are capital adequacy ratios (CAR) based on risk-weighted assets.…”

Asymptotic stability of empirical processes and related functionals

“…This paper is motived by the study of the asymptotic stability of the random sequence τn τ (mn), n ≥ 1, when the variables ξn encode the historical data of a certain financial risk factor and τ : M1(E) → T is a certain statistic assessing the downside risk of the related exposure. Indeed, the study of the asymptotic behaviour of the estimators τn is crucial to gauge the risk properly, see Cont et al [3], Föllmer and Weber [8] and Krätschmer et al [18,19].…”

Utility‐based shortfall risk: Efficient computations via Monte Carlo

“…Bellini and Bignozzi (2015) showed that SR is the only convex risk measure that possesses elicitability, which is a property relevant to the context of backtesting. For a more detailed discussion about SR and elicitability, readers are referred to Weber (2006), Bellini and Bignozzi (2015), Föllmer and Weber (2015), Delbaen, Bellini, Bignozzi, and Ziegel (2016) and Bellini and Bernadino (2017).…”