Optimal XL-insurance under Wasserstein-type ambiguity

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.

Section: Corollary 32 the Optimal Marginal Indemnity Function That mentioning

confidence: 98%

Optimal Insurance Contracts Under Distortion Risk Measures With Ambiguity Aversion

Jiang¹,

Escobar‐Anel

Ren

2020

“…The following works, among others, have investigated or touched this issue in various insurance contexts: Cairns (2000), Chen and Su (2009), Peters et al (2009), Zhao and Zhu (2011) , Landsman and Tsanakas (2012), Robert and Therond (2014), Liu and Wang (2017), Fujii et al (2017), and Jiang et al (2020). More applications to practical problems in insurance have been proposed during recent years but are still sporadic; see Wozabal (2014), Pflug et al (2017), Lam and Mottet (2017), Asimit et al (2017Asimit et al ( , 2019, Ghossoub (2019aGhossoub ( , 2019b, , Pflug (2019), andBirghila et al (2020), among others. We hope that our work can draw attention from actuaries and financial analysts to incorporate the consideration of partial ambiguity when addressing the ambiguity issue in practical problems.…”

Section: A Brief Literature Reviewmentioning

confidence: 99%

Worst-case moments under partial ambiguity

Tang

Yang²

2023

The model uncertainty issue is pervasive in virtually all applied fields but especially critical in insurance and finance. To hedge against the uncertainty of the underlying probability distribution, which we refer to as ambiguity, the worst case is often considered in quantifying the underlying risk. However, this worst-case treatment often yields results that are overly conservative. We argue that, in most practical situations, a generic risk is realized from multiple scenarios and the risk in some ordinary scenarios may be subject to negligible ambiguity so that it is safe to trust the reference distributions. Hence, we only need to consider the worst case in the other scenarios where ambiguity is significant. We implement this idea in the study of the worst-case moments of a risk in the hope to alleviate the over-conservativeness issue. Note that the ambiguity in our consideration exists in both the scenario indicator and the risk in the corresponding scenario, leading to a two-fold ambiguity issue. We employ the Wasserstein distance to construct an ambiguity ball. Then, we disentangle the ambiguity along the scenario indicator and the risk in the corresponding scenario, so that we convert the two-fold optimization problem into two one-fold problems. Our main result is a closed-form worst-case moment estimate. Our numerical studies illustrate that the consideration of partial ambiguity indeed greatly alleviates the over-conservativeness issue.

“…In reinsurance design, Hu et al (2015) studied optimal reinsurance with stop-loss contracts and incomplete information on the loss distribution in the sense that only the first two moments of the loss are known. See Pflug et al (2017), Birghila and Pflug (2019) and Gavagan et al (2022) for the design of an optimal insurance policy with the uncertainty set defined by the Wasserstein distance. Asimit et al (2017) considered model uncertainty in insurance contract design by maximizing over a finite set of probability measures.…”

Section: Introductionmentioning

confidence: 99%

Distributionally robust reinsurance with expectile

Xie

Liu²,

Mao³

et al. 2023

We study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.