On optimal control of capital injections by reinsurance and investments

Eisenberg, Julia

doi:10.1007/s11857-010-0124-0

Cited by 9 publications

(8 citation statements)

References 45 publications

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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%

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%

Mixed Band Control of Mutual Proportional Reinsurance

Taksar¹,

Liu²,

Yuan³

2013

JMF

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“…Using a different criterion to assess the performance of an insurance portfolio, Eisenberg (2010) thoroughly covers a variety of capital injection minimization problems under both the classical risk model and its diffusion approximation where the insurer has the possibility to dynamically reinsure its risk. The incorporation of dynamic reinsurance to the classical problem of maximizing the dividend payouts of an insurance company prior to ruin in a compound Poisson framework was treated by Azcue and Muler (2005) for general reinsurance schemes and by Mnif and Sulem (2005) for excess of loss reinsurance.…”

Section: Introductionmentioning

confidence: 99%

An optimal reinsurance problem in the Cramér–Lundberg model

Cani

Thonhauser

2016

Math Meth Oper Res

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In this article we consider the surplus process of an insurance company within the Cramér-Lundberg framework with the intention of controlling its performance by means of dynamic reinsurance. Our aim is to find a general dynamic reinsurance strategy that maximizes the expected discounted surplus level integrated over time. Using analytical methods we identify the value function as a particular solution to the associated Hamilton-Jacobi-Bellman equation. This approach leads to an implementable numerical method for approximating the value function and optimal reinsurance strategy. Furthermore we give some examples illustrating the applicability of this method for proportional and XL-reinsurance treaties.

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“…Inspired by these literatures, we study the optimization problem of portfolio selection and capital injection for an insurance company by minimizing the expected value of the total discounted capital injection costs. In the model set-up of previous works, we note that there exists only a risky asset but without a riskfree asset in the financial market, for example, see Browne (1995) and Eisenberg (2010). Specifically, it was assumed that part of a company's surplus can be invested in the risky asset, and that the remaining surplus is kept in cash.…”

Section: Introductionmentioning

confidence: 99%

Portfolio Selection by Minimizing the Present Value of Capital Injection Costs

Zhou

Yuen

2014

ASTIN Bull.

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This paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black-Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

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On optimal control of capital injections by reinsurance and investments

Cited by 9 publications

References 45 publications

Mixed Band Control of Mutual Proportional Reinsurance

Mixed Band Control of Mutual Proportional Reinsurance

An optimal reinsurance problem in the Cramér–Lundberg model

Portfolio Selection by Minimizing the Present Value of Capital Injection Costs

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