Risk measures with the CxLS property

Delbaen, Freddy; Bellini, F.; Bignozzi, Valeria; Ziegel, Johanna F.

doi:10.1007/s00780-015-0279-6

Cited by 64 publications

(39 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Cont et al (2010) use the Lévy distance and show that there is a partial conflict between coherent risk measures (including CVaR) and Hampel's classical notion of robustness, Krätschmer, Schied, and Zähle (2014) and Delbaen, Bellini, Bignozzi, and Ziegel (2016) consider continuity with respect to the ϕ-weak topology. We refer the interested reader to Emmer et al (2015) for a brief summary on the topic.…”

Section: Robustness Of Risk Measuresmentioning

confidence: 99%

Robust and Pareto Optimality of Insurance Contracts

et al. 2017

Self Cite

View full text Add to dashboard Cite

a b s t r a c tThe optimal insurance problem represents a fast growing topic that explains the most efficient contract that an insurance player may get. The classical problem investigates the ideal contract under the assumption that the underlying risk distribution is known, i.e. by ignoring the parameter and model risks. Taking these sources of risk into account, the decision-maker aims to identify a robust optimal contract that is not sensitive to the chosen risk distribution. We focus on Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)-based decisions, but further extensions for other risk measures are easily possible. The Worst-case scenario and Worst-case regret robust models are discussed in this paper, which have been already used in robust optimisation literature related to the investment portfolio problem. Closed-form solutions are obtained for the VaR Worst-case scenario case, while Linear Programming (LP) formulations are provided for all other cases. A caveat of robust optimisation is that the optimal solution may not be unique, and therefore, it may not be economically acceptable, i.e. Pareto optimal. This issue is numerically addressed and simple numerical methods are found for constructing insurance contracts that are Pareto and robust optimal. Our numerical illustrations show weak evidence in favour of our robust solutions for VaR-decisions, while our robust methods are clearly preferred for CVaR-based decisions.

show abstract

Section: Robustness Of Risk Measuresmentioning

confidence: 99%

Robust and Pareto Optimality of Insurance Contracts

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…We show that when ⇢ is a convex risk measure, the corresponding extreme-aggregation measure ⇢ is the smallest coherent risk measure dominating ⇢, and a robust representation of ⇢ is thereby provided for ⇢ satisfying the Fatou property. In the specific case of shortfall risk measures, we show that ⇢ is identified with an expectile (Newey and Powell, 1987), which is the only coherent shortfall risk measure (Weber, 2006), as well as the only elicitable coherent risk measure (Ziegel, 2014;Delbaen et al, 2015).…”

Section: Introductionmentioning

confidence: 91%

“…For p > 1/2 the loss function`p is convex, such that e p is a shortfall risk measure and is also coherent . In fact, {e p , p > 1/2} is the only class of coherent shortfall risk measures (Weber, 2006) and the only class of elicitable coherent risk measures (Delbaen et al, 2015) 4 .…”

Section: Extreme-aggregation Measures Induced By Shortfall Risk Measuresmentioning

confidence: 99%

How Superadditive Can a Risk Measure Be?

Wang

Bignozzi

Tsanakas

2015

SIAM J. Finan. Math.

Self Cite

View full text Add to dashboard Cite

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extremeaggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex shortfall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management. Permanent repository link

show abstract

“…Developments on risk measures with convex level sets can be found in Weber (2006), Lambert (2012), Ziegel (2015), Bellini and Bignozzi (2014), Kou and Peng (2014) and Delbaen et al (2014). The work of Steinwart et al (2014) shows that convex level sets are also a sufficient criterion for elicitability under some weak regularity assumptions on ρ; see also Lambert (2012).…”

Section: Elicitable Distortion Risk Measuresmentioning

confidence: 98%