Short Communication: An Axiomatization of $\Lambda$-Quantiles

Abstract

We give an axiomatic foundation to Λ-quantiles, a family of generalized quantiles introduced in [7] under the name of Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call "locality", that means that any change in the distribution of the probability mass that arises entirely above or below the value of the Λ-quantile does not modify its value. We compare with a related axiomatization of the usual quantiles given by C. Chambers in [4], based on the stronger property of "ordinal covariance", that means that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of Λ-quantiles, refining some of the results of [7] and [3] and showing that in the case of a nonincreasing Λ the properties of Λ-quantiles closely resemble those of the usual quantiles.

Risk Sharing with Lambda Value at Risk

“…where inf ∅ = ∞ and sup ∅ = −∞. Note that our definition is a bit different from the one in Bellini and Peri (2022). This modification is to be consistent with the definition of VaR as in Embrechts et al (2018) and Liu et al (2022) to focus more on the tail probability.…”

“…Here, Λ function is used to control the tail probability. Note that ΛVaR(X) = ΛVaR + (X) if Λ is increasing; Otherwise, it may not be true; see Proposition 6 of Bellini and Peri (2022). In fact, there are two other definitions of ΛVaR given by Bellini and Peri (2022)…”

“…This indicates that all agents are accepting higher losses with higher probability of exceeding the loss. As discussed in Example 7 of Bellini and Peri (2022), this type of ΛVaR can be used to describe the contrasting objectives of the risk manager in setting the capital reserve: to be conservative but not too much. We obtain an expression for the inf-convolution and find the forms of optimal allocations.…”

“…For decreasing Λ, Lambda value at risk is able to capture the tail risk by controlling the probability of exceeding the loss, which is in a different way from Expected Shortfall (ES). For increasing Λ, ΛVaR may incorporate some additional requirement such as risk manager's judgement in the process of risk management; see Bellini and Peri (2022). Moreover, ΛVaR satisfies quasi-convexity with respect to distributions, the same as VaR; see Frittelli et al (2014).…”

“…The backtesting, estimation and risk contribution of ΛVaR are studied in Corbetta and Peri (2018), Hitaj et al (2018) and Ince et al (2022), respectively. Most recently, Bellini and Peri (2022) offer an axiomatization of ΛVaR for increasing Λ functions and study some further properties of ΛVaR, and Han et al (2021) provide a of ΛVaR is available in Frittelli et al (2014) and Bellini and Peri (2022). An extension of Lambda value at risk, called lambda-fixed point risk measure, is introduced and studied in Balbás et al (2023), where its applications to reinsurance contracts and premium calculation are also discussed.…”