Risk Measures Based on Benchmark Loss Distributions

Abstract

We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), that is, a function that associates to each potential loss a maximal acceptable probability of occurrence. The corresponding risk measure, called Loss VaR (LVaR), determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first-order stochastic dominance. We investigate the main theoretical properties of LVaR with a focus on their comparison with VaR and ES and discuss applications to capital adequacy, portfolio risk management, and catastrophic risk.

Is the Inf-convolution of Law-invariant Preferences Law-invariant?

“…In other words, the acceptance set of ρ 1 is A 1 = {Y ∈ X : Y st F B } and that of ρ 2 is A 2 = {Y ∈ X : Y st F U }. Such ρ 1 and ρ 2 belong to the class of risk measures based on benchmark loss distributions studied by Bignozzi et al (2019). One can easily check that ρ 1 and ρ 2 are both law-invariant monetary risk measures.…”

A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations

“…A further generalization of the VaR is a class of quantile-based risk measures called risk measures based on benchmark loss distributions (BLD), which was recently introduced by Bignozzi et al (2018).…”

“…For RM based on BLDs, the risk manager does not require additional capital reserves when the inequality P(L ≤ l) ≥ α(l) holds for every l ≥ 0. Bignozzi et al (2018) show that these measures result as a solution of the following maximization problem:…”

“…In contrast to the LVaR, RMBLDs are not elicitable. However, Bignozzi et al (2018) point out some interesting alternative backtesting methods and show the robustness in the sense of Cont et al (2010).…”

Adjusted Expected Shortfall

“…We refer to early work on ES in Acerbi and Tasche (2002) Acerbi (2002) , Frey and McNeil (2002) , and Rockafellar and Uryasev (2002) (where ES was called Conditional VaR). Some recent contributions to the broad investigation on whether and to what extent VaR and ES meet regulatory objectives are Koch-Medina and Munari (2016) , Embrechts et al (2018) , Weber (2018) , Bignozzi et al (2020) , Baes et al (2020) , and Wang and Zitikis (2021) . For robustness problems concerning VaR and ES, see, e.g., Cont et al (2010) and Krätschmer et al (2014) , and for their backtesting, see, e.g., Ziegel (2016) , Du and Escanciano (2017) , and Kratz et al (2018) .…”