Robust and Pareto optimality of insurance contracts

Asimit, Alexandru V.; Bignozzi, Valeria; Cheung, Ka Chun; Hu, Jiejun; Kim, Eun-Seok

doi:10.1016/j.ejor.2017.04.029

Cited by 66 publications

(38 citation statements)

References 38 publications

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“…If p = 0, then Problems (19) and (21) still allow us to find approximated solutions of (16), but the optimality conditions (20) and (22) will fail. The main reason is that the dual space of L 0 may be "too small."…”

Section: Robust V@r Computation and Optimizationmentioning

confidence: 99%

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V@R representation theorems in ambiguous frameworks

Balbás

Charron

2019

Appl Stoch Models Bus & Ind

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The value at risk (V@R) is a very important risk measure with significant applications in finance (risk management, pricing, hedging, portfolio theory, etc), insurance (premium principles, optimal reinsurance, etc), production, marketing (newsvendor problem), etc. It also plays a critical role in regulation about risk (Basel, Solvency, etc), it is very appreciated by practitioners due to its intuitive interpretation, and it is the unique popular risk measure remaining finite for heavy tailed risks with unbounded expectation. Besides, ambiguous frameworks are becoming more and more usual in applications of risk analysis. Lack of data or committed errors may provoke discrepancies between real probabilities and estimated ones. This paper combines both V@R and ambiguous settings, and a new representation theorem for V@R is given. Consequently, inspired by previous studies dealing with coherent risk measures and their representation, we will give new methods to compute and optimize V@R under ambiguity. This seems to be a relevant finding because the analytical properties of V@R are very weak if one compares with a coherent risk measure. Indeed, V@R is neither continuous nor convex, which makes it very complicated to deal with it in mathematical approaches. Nevertheless, the results of this paper will allow us to transform computation and optimization problems involving V@R into continuous and differentiable problems. KEYWORDSambiguity, heavy tail, representation theorem, value at risk INTRODUCTIONRisk measures beyond the standard deviation are becoming more and more important in many applications of risk analysis. Indeed, most of them can be interpreted in terms of potential capital losses, while the standard deviation does not satisfy this property in many situations. Besides, the standard deviation is not consistent with the second-order stochastic dominance and the standard utility functions in the presence of asymmetries, 1 and this drawback is overcome by other risk measures.Among many other examples, the coherent risk measures 2 and the expectation bounded risk measures 3 are very popular for practitioners and researchers since they have good analytical properties such as subadditivity, convexity, and continuity, making it feasible to deal with them in complex mathematical approaches. Important examples of such measures are the conditional value at risk (CV@R) and the weighted conditional value at risk. 3 With respect to the coherent and expectation bounded risk measures above, the value at risk (V@R) presents several shortcomings. Indeed, it is neither continuous nor convex, making it complex many analytical studies. Nevertheless, V@R 414

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Section: Robust V@r Computation and Optimizationmentioning

confidence: 99%

“…Let us address problem (25) above by means of Theorem 4 and its Remarks 2 and 3. With some minor manipulations, (21) shows that (25) may be replaced by…”

Section: Financial Examplementioning

confidence: 99%

V@R representation theorems in ambiguous frameworks

Balbás

Charron

2019

Appl Stoch Models Bus & Ind

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“…Risk measures are functionals mapping random variables to the real line (Artzner et al, 1999;Szegö, 2005). Risk measures are used in a variety of operations research and risk analysis applications, with Value-at-Risk (VaR) and Expected Shortfall (ES -also known as CVaR) particularly popular choices; indicatively see Rockafellar and Uryasev (2002); Tapiero (2005); Gotoh and Takano (2007); Ahmed et al (2007); Asimit et al (2017).…”

Section: Introductionmentioning

confidence: 99%

Reverse sensitivity testing: What does it take to break the model?

Pesenti

Millossovich

Tsanakas

2019

European Journal of Operational Research

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Sensitivity analysis is an important component of model building, interpretation and validation. A model comprises a vector of random input factors, an aggregation function mapping input factors to a random output, and a (baseline) probability measure. A risk measure, such as Value-at-Risk and Expected Shortfall, maps the distribution of the output to the real line. As is common in risk management, the value of the risk measure applied to the output is a decision variable. Therefore, it is of interest to associate a critical increase in the risk measure to specific input factors. We propose a global and model-independent framework, termed 'reverse sensitivity testing', comprising three steps: (a) an output stress is specified, corresponding to an increase in the risk measure(s); (b) a (stressed) probability measure is derived, minimising the Kullback-Leibler divergence with respect to the baseline probability, under constraints generated by the output stress; (c) changes in the distributions of input factors are evaluated. We argue that a substantial change in the distribution of an input factor corresponds to high sensitivity to that input and introduce a novel sensitivity measure to formalise this insight. Implementation of reverse sensitivity testing in a Monte-Carlo setting can be performed on a single set of input/output scenarios, simulated under the baseline model. Thus the approach circumvents the need for additional computationally expensive evaluations of the aggregation function. We illustrate the proposed approach through a numerical example of a simple insurance portfolio and a model of a London Insurance Market portfolio used in industry.

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“…This paper considers the scenario of an ambiguity-averse DM seeking the optimal insurance contract that minimizes its risk level according to some distortion risk measure. Asimit et al (2017) considered such a problem from the point of view of robust control. They focused on Value-at-Risk (VaR) and Conditional Value-at-Risk-based risk measures and applied the worst-case scenario and worst-case regret models to determine the "robust" optimal polices.…”

Section: Introductionmentioning

confidence: 99%

Optimal Insurance Contracts Under Distortion Risk Measures With Ambiguity Aversion

Jiang¹,

Escobar‐Anel

Ren

2020

ASTIN Bull.

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This paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.

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Robust and Pareto optimality of insurance contracts

Cited by 66 publications

References 38 publications

V@R representation theorems in ambiguous frameworks

V@R representation theorems in ambiguous frameworks

Reverse sensitivity testing: What does it take to break the model?

Optimal Insurance Contracts Under Distortion Risk Measures With Ambiguity Aversion

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