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), i.e. 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 provide a comprehensive study of the main finance theoretical and statistical properties of LVaR with a focus on their comparison with VaR and ES. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented.

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.…”

An axiomatization of $Λ$-quantiles

“…From a financial point of view, as discussed in Frittelli et al (2014), Corbetta and Peri (2018) and Hitaj et al (2018), it is sensible allowing a dependence of the "acceptability" probability level λ on the corresponding amount of the loss x. A related idea has been investigated in Bignozzi et al (2019), that constructed translation invariant risk measures based on benchmark loss distributions. At this level of generality, Λ-quantiles satisfy only a few properties that are collected in the following proposition.…”

Star-shaped acceptability indexes

“…A example of star-shaped risk measure that is neither positively homogeneous nor convex is the Benchmark loss VaR (LVaR) introduced in Bignozzi et al (2020). It is defined as LV aR θ (X) = sup t∈R + {V aR θ(t) (X)−t}, where θ : R + → [0, 1] is increasing and right-continuous.…”