Minimising expected discounted capital injections by reinsurance in a classical risk model

Eisenberg, Julia; Schmidli, Hanspeter

doi:10.1080/03461231003690747

Cited by 42 publications

(39 citation statements)

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“…Another approach should be developed, as can be seen indirectly in the work of Eisenberg [18] a isenberg and Schmidli [5], where a nd E ed for 6. doi:10.1287/opre.26.4.620 similar (although not identical) problem is consider the case of a surplus process modeled via the classical Cramer-Lundberg model.…”

Section: )mentioning

confidence: 99%

“…The mutual proportional reinsurance model developed in this paper is a generalization of the proportional reinsurance models (e.g., Cadenillas et al [3], Hojgaard and Taksar [4], Eisenberg and Schmidli [5], Løkka and Zervos [9]) and is modified with the two differing characteristics noted above. More specifically, the proportional reinsurance rate can be adjusted in continuous time, and the underlying mutual reserve process is regulated by a two-sided impulse control in terms of a contingent dividend payment (i.e., a downside impulse control to decrease the mutual reserve level) and contingent call for contributions (i.e., an upside impulse control to increase the mutual reserve level).…”

Section: Introductionmentioning

confidence: 99%

“…Reinsurance has been long investigated as an intrinsic part of commercial insurance, of which the mainstream modeling framework is profit maximization with the one-sided impulse control of an underlying reserve process. There are two types of one-sided impulse control: downside-only impulse control (such as a dividend payment) with a fixed cost K  (e.g., Cadenillas et al [3], Hojgaard and Taksar [4]) and upside-only impulse control (such as inventory ordering) with a fixed cost K  (e.g., Bensoussan et al [2], Eisenberg and Schmidli [5], Sulem [6]). In this paper, we examine mutual proportional reinsurance-a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as the P&I Clubs in marine mutual insurance (e.g., Yuan [7]) and the reserve banks in the US Federal Reserve (e.g., Dawande et al [8]).…”

Section: Introductionmentioning

confidence: 99%

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Mixed Band Control of Mutual Proportional Reinsurance

Taksar¹,

Liu²,

Yuan³

2013

JMF

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In this paper, we investigate the optimization of mutual proportional reinsurance-a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual insurance and reserve banks in the US Federal Reserve, where a mutual member is both an insurer and an insured. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control   , , , a A B b a with and a A coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of , the reserve should immediately be raised to level A; when the reserve reaches an upper boundary of b, it should immediately be reduced to a level B. An interesting finding produced by the study reported in this paper is that there exists a situation such that if the upside fixed cost is relatively large in comparison to a finite threshold, then the optimal band control is reduced to a downside only (i.e., dividend payment only) control in the form of   , B b 0, 0; with 0 a A   . In this case, it is optimal for the mutual insurance firm to go bankrupt as soon as its reserve level reaches zero, rather than to jump restart by calling for additional contingent funds. This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mixed Band Control of Mutual Proportional Reinsurance

Taksar¹,

Liu²,

Yuan³

2013

JMF

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“…The problem of minimising the expected discounted capital injections in the Cramér-Lundberg model and in its diffusion approximation with dynamic reinsurance has already been solved in [3] and [4]. Optimal reinsurance strategies with respect to ruin probabilities have been considered in [10] (see also [12]), and in [2] with respect to dividends.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, the ruin probability does not take into account the time of ruin nor the severity of ruin. 734 J. EISENBERG AND H. SCHMIDLI Eisenberg and Schmidli [3], [4] introduced an alternative measure of risk. They proposed to valuate the capital injections.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

Eisenberg

Schmidli

2011

Journal of Applied Probability

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We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0,b]. Here b =b means 'no reinsurance' and b = 0 means 'full reinsurance'. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t } t≥0 , i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a 'weak' solution and calculate the value function numerically.

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2017

Reinsurance

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Minimising expected discounted capital injections by reinsurance in a classical risk model

Cited by 42 publications

References 7 publications

Mixed Band Control of Mutual Proportional Reinsurance

Mixed Band Control of Mutual Proportional Reinsurance

Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

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