Optimal reinsurance under mean-variance premium principles

Kałuszka, Marek

doi:10.1016/s0167-6687(00)00066-4

Cited by 154 publications

(84 citation statements)

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“…19 Gajek and Zagrodny (2000). 20 Kaluszka (2001Kaluszka ( , 2005b Third, this is driven by tractability. While it is of interest to consider the optimal solutions among all general reinsurance treaties, the resulting VaR optimization problem is notoriously non-tractable due to the lack of convexity.…”

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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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confidence: 99%

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

Tan

Weng

2011

Geneva Risk Insur Rev

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“…The following Theorem 1 proves the existence of optimal reinsurance strategy for Problem (2). Moreover, the optimal reinsurance strategy comes from H 0 .…”

Section: Remark 2 Lemma 1 States That Problem (3) Is Equivalent Tomentioning

confidence: 77%

“…Problem (14) is another particular case of Problem (2). In fact, Problem (14) is reduced by Problem (2) by shrinking the probability measure set M 1 to the singleton {Φ}.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…The first and most classical optimization criterion is variance minimization. It is shown that the pure stop-loss reinsurance is the optimal reinsurance strategy because of the smallest variance of the insurer's retained loss among all the strategies with the same pure premium, see, Borch [1], Kaluszka [2], Kaas, et al [3], and so on. The second one is utility maximization, which is attributed to Arrow [4].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle

Chen

Wang

Ming

2016

Risks

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Abstract:In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer's risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., f * (x) = x ∧ c * + (x − d * ) + with c * and d * being constants such that 0 ≤ c * ≤ d * .

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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 reinsurance under mean-variance premium principles

Cited by 154 publications

References 10 publications

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

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

Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle

Optimal insurance in the presence of reinsurance

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