Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

Cai, Jun; Tan, Ken Seng

doi:10.2143/ast.37.1.2020800

Cited by 173 publications

(128 citation statements)

References 13 publications

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“…6 See Basak andShapiro, 2001;Krokhmal et al, 2002;Rockafellar and Uryasev, 2002;Alexander and Baptista, 2004; and Gaivoronski and Pflug, 2005. 7 Gaivoronski and Pflug (2005 Cai and Tan (2007). 12 Cai et al (2008).…”

Section: Introductionmentioning

confidence: 99%

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Tan

Weng

2011

Geneva Risk Insur Rev

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The quest for optimal reinsurance design has remained an interesting problem among insurers, reinsurers, and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an insurer and thereby enhance its value. This paper complements the existing research on optimal reinsurance by proposing another model for the determination of the optimal reinsurance design. The problem is formulated as a constrained optimization problem with the objective of minimizing the value-at-risk of the net risk of the insurer while subjecting to a profitability constraint. The proposed optimal reinsurance model, therefore, has the advantage of exploiting the classical tradeoff between risk and reward. Under the additional assumptions that the reinsurance premium is determined by the expectation premium principle and the ceded loss function is confined to a class of increasing and convex functions, explicit solutions are derived. Depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer, we establish conditions for the existence of reinsurance. When it is optimal to cede the insurer's risk, the optimal reinsurance design could be in the form of pure stop-loss reinsurance, quota-share reinsurance, or a combination of stop-loss and quota-share reinsurance. The Geneva Risk and Insurance Review (2012) 37, 109-140. doi:10.1057/grir.2011; published online 23 August 2011Keywords: value-at-risk (VaR); optimal reinsurance; expectation premium principle; linear programming in infinite dimensional spaces IntroductionThe importance of sound risk management for financial institutions and insurance enterprises has been dramatically highlighted by the subprime crisis. Risk professionals are constantly seeking better risk measures to quantify risks associated with market, credit, operational, catastrophic, and many others. Risk measures such as value-at-risk (VaR) and conditional VaR or conditional tail expectation (CTE) have been proposed. Among these risk The Geneva Risk and Insurance Review, 2012, 37, (109-140) 9 and Zhou and Wu 10 demonstrated that employing VaR as a constraint could assist an insured in determining his or her optimal insurance policy. By minimizing VaR or CTE of the total risk exposure of an insurer, Cai and Tan 11 and Cai et al. 12 derived explicitly the optimal reinsurance treaties for the insurer. 13 While analytic solutions have been derived in the above reinsurance models, these results can be criticized on the ground that the optimality is based exclusively on minimizing an insurer's risk exposure. In practice, an insurer is concerned not only with its exposure to risk but also its profitability of insuring the underlying risks. To elaborate this point, let us first note that when an insurer uses reinsurance to cede (or transfer) part of its loss to a reinsurer, the insurer is liable to pay reinsurance 1 Artzner et al. (1999). 2 Basak and Shapiro (2001). 3 Yamai and Yoshiba (2005). 4 See Basle Commi...

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Section: Introductionmentioning

confidence: 99%

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Tan

Weng

2011

Geneva Risk Insur Rev

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“…Along with the paper of Cai and Tan (2007) above, other interesting examples are Cai et al (2008), Balbás et al (2009) or Bernard and Tian (2009). The differ ences among their approaches are caused by the insurer behavior.…”

Section: Introductionmentioning

confidence: 99%

“…Other well known financial risk measures such as the VaR or the tail value at risk (TVaR) are also being considered. For example, Kaluszka (2005) uses the TVaR as a premium principle and Cai and Tan (2007) calculate the optimal retention for a stop loss reinsurance by considering the VaR and the conditional tail expectation risk measures (CTE), under the expected value pre mium principle.…”

Section: Introductionmentioning

confidence: 99%

Stable solutions for optimal reinsurance problems involving risk measures

Balbás

Heras

2011

European Journal of Operational Research

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a b s t r a c tThe optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others.This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a ''stable optimal retention'' that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.

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“…These pioneering results are later extended to situation where there is a more sophisticated objective function and/or more realistic premium principles (see, e.g. Young 1999, Gajek & Zagrodny 2000, 2004, Kaluszka 2001, 2005, Cai & Tan 2007, Balbás et al 2009, 2015, Chi 2012, Asimit et al 2013, 2015, Cai et al 2013, Forthcoming, Chi & Tan 2013, Cui et al 2013, Cheung et al 2014, 2015, Bernard et al 2015, Cheung et al 2015, Boonen et al 2016. In the above-mentioned papers, the risk of the insurer is typically given and the objective boils down to determining an optimal strategy of transferring part of its risk to a reinsurer.…”

Section: Introductionmentioning

confidence: 99%

Optimal insurance in the presence of reinsurance

Zhuang

Boonen

Tan

et al. 2016

Scandinavian Actuarial Journal

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This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution. ARTICLE HISTORY

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Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

Cited by 173 publications

References 13 publications

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Stable solutions for optimal reinsurance problems involving risk measures

Optimal insurance in the presence of reinsurance

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