Optimal premium policy of an insurance firm: Full and partial information

Herein, we study the optimization problem faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels by selecting an appropriate premium policy. Our problem is totally new and has three distinguishable features: (1) both full and partial information cases are investigated here; (2) the state is subject to terminal constraint; (3) a forward-backward stochastic differential equation formulation is given which is more systematic and mathematically advanced. This formulation also enables us to continue further research in a generalized stochastic recursive control framework (see Duffie and Epstein (1992), El Karoui et al. (2001), etc.). The optimal premium policy with the associated optimal objective functional are completely and explicitly derived. In addition, a backward separation technique adaptive to forward-backward stochastic systems with the state constraint is presented as an efficient and convenient alternative to the traditional Wonham’s (1968) separation principle in our partial information setup. Some concluding remarks are also given here.     

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.     

含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

Approximations for optimal Laplace transform inversion

A function (p) of the Laplace transform operatorp is approximated by a finite linear combination of functions (p+ _r ), where (p) is a specific function ofp having a known analytic inverse (t), and is chosen in accordance with various considerations. Then parameters _r ,r=1, 2,...,n, and then corresponding coefficientsA _r of the (p + _r ) are determined by a least-square procedure. Then, the corresponding approximation to the inversef(t) of (p) is given by analytic inversion of _r=1 ⁿ A _r (p+ _r ). The method represents a generalization of a method of best rational function approximation due to the author [which corresponds to the particular choice (t)1], but is capable of yielding considerably greater accuracy for givenn.The computations for this paper were carried out on the CDC-6600 computer at the Computation Center of Tel-Aviv University. The author is grateful to Dr. H. Jarosch of the Weizmann Institute of Science Computer Center for use of their Powell minimization subroutine (Ref. 1).     

Optimal investment-reinsurance policy for an insurance company with VaR constraint

This paper investigates an investment-reinsurance problem for an insurance company that has a possibility to choose among different business activities, including reinsurance/new business and security investment. Our main objective is to find the optimal policy to minimize its probability of ruin. The main novelty of this paper is the introduction of a dynamic Value-at-Risk (VaR) constraint. This provides a way to control risk and to fulfill the requirement of regulators on market risk. This problem is formulated as an infinite horizontal stochastic control problem with a constrained control space. The dynamic programming technique is applied to derive the Hamilton-Jacobi-Bellman (HJB) equation and the Lagrange multiplier method is used to tackle the dynamic VaR constraint. Closed-form expressions for the minimal ruin probability as well as the optimal investment-reinsurance/new business policy are derived. It turns out that the risk exposure of the insurance company subject to the dynamic VaR constraint is always lower than otherwise. Finally, a numerical example is given to illustrate our results.     

Effects of pollution restrictions on dynamic investment policy of a firm

The purpose of this paper is to determine the effects of different pollution standards on the firm's resource allocation decisions. To do so, a dynamic model of the firm is developed in which it is assumed that production causes pollution as an inevitable byproduct. Concerning its investment policy, we suppose that the firm can choose between investing in productive capital goods and investing in abatement efforts.It is shown that, in some cases, future abatement expenses have a negative impact on the present level of productive investment, even if the pollution standard is not binding at the moment. This implies a really dynamic optimal investment policy for the firm, which cannot be obtained within a comparative static analysis.This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. Comments by Frank van der Duyn Schouten and Piet Verheyen (Tilburg University) and by Raymond Gradus (Dutch Ministry of Finance, The Hague) are gratefully acknowledged.     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

Optimal volume-corrected laplace-metropolis method

The article provides a refinement for the volume-corrected Laplace-Metropolis estimator of the marginal likelihood of DiCiccioet al. The correction volume of probability α in DiCiccioet al. is fixed and suggested to take the value α=0.05. In this article α is selected based on an asymptotic analysis to minimize the mean square relative error (MSRE). This optimal choice of α is shown to be invariant under linear transformations. The invariance property leads to easy implementation for multivariate problems. An implementation procedure is provided for practical use. A simulation study and a real data example are presented.     

Optimal investment policy: An application of Stokes' theorem

An application of the Stokes' theorem is illustrated by solving the two-state problem, with inequality constraints, of Dobell and Ho concerning the optimal investment of resources. Whenever applicable, the Stokes' theorem approach seems to be elegant and parsimonious.     

Optimal investment and consumption decision of a family with life insurance

We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.     

Optimal investment and life insurance strategies under minimum and maximum constraints

We derive optimal strategies for an individual life insurance policyholder who can control the asset allocation as well as the sum insured (the amount to be paid out upon death) throughout the policy term. We first consider the problem in a pure form without constraints (except nonnegativity on the sum insured) and then in a more general form with minimum and/or maximum constraints on the sum insured. In both cases we also provide the optimal life insurance strategies in the case where risky-asset investments are not allowed (or not taken into consideration), as in basic life insurance mathematics. The optimal constrained strategies are somewhat more complex than the unconstrained ones, but the latter can serve to ease the understanding and implementation of the former.     

Optimal investment of DC pension plan under short-selling constraints and portfolio insurance

In this paper we investigate an optimal investment problem under short-selling and portfolio insurance constraints faced by a defined contribution pension fund manager who is loss averse. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the S-shaped utility, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.     

Application of differential transform method to non-linear oscillatory systems

In this paper, the differential transform method is proposed for solving non-linear oscillatory systems. These solutions do not exhibit periodicity, which is the characteristic of oscillatory systems. A modification of the differential transform method, based on the use of Padé approximants, is proposed. We use alternative technique by which the solution obtained by the differential transform method is made periodic. The method is described and illustrated with examples. The results reveal that the method is very effective and convenient.     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.     

Optimal investment and reinsurance policies in insurance markets under the effect of inside information

In this paper, we study the problem of optimal investment and proportional reinsurance coverage in the presence of inside information. To be more precise, we consider two firms: an insurer and a reinsurer who are both allowed to invest their surplus in a Black–Scholes‐type financial market. The insurer faces a claims process that is modeled by a Brownian motion with drift and has the possibility to reduce the risk involved with this process by purchasing proportional reinsurance coverage. Moreover, the insurer has some extra information at her disposal concerning the future realizations of her claims process, available from the beginning of the trading interval and hidden from the reinsurer, thus introducing in this way inside information aspects to our model. The optimal investment and proportional reinsurance decision for both firms is determined by the solution of suitable expected utility maximization problems, taking into account explicitly their different information sets. The solution of these problems also determines the reinsurance premia via a partial equilibrium approach. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory

A novel optimal preventive maintenance policy for a cold standby system consisting of two components and a repairman is described herein. The repairman is to be responsible for repairing either failed component and maintaining the working components under certain guidelines. To model the operational process of the system, some reasonable assumptions are made and all times involved in the assumptions are considered to be arbitrary and independent. Under these assumptions, all system states and transition probabilities between them are analyzed based on a semi-Markov theory and a regenerative point technique. Markov renewal equations are constructed with the convolution of the cumulative distribution function of system time in each state and corresponding transition probability. By using the Laplace transform to solve these equations, the mean time from the initial state to system failure is derived. The optimal preventive maintenance policy that will provide the optimal preventive maintenance cycle is identified by maximizing the mean time from the initial state to system failure, and is determined in the form of a theorem. Finally, a numerical example and simulation experiments are shown which validated the effectiveness of the policy.     

Optimal decision rule in forming an insurance portfolio

The paper is devoted to finding an optimal decision rule for accepting/rejecting potential insureds when the demand for the insurance provision is a stochastic variable. A criterion to be maximized is the mean-variance utility function of the insurer. It is shown that the optimal decision rule is a stopping rule with some finite protection level.     

Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy

In this paper, we consider an optimal financing and dividend control problem of an insurance company. The management of the insurance company controls the dividends payout, equity issuance and the excess-of-loss reinsurance policy. In our model, the dividends are assumed to be paid out continuously, which is of interest from the perspective of financial modeling. The objective is to find the strategy which maximizes the expected present values of the dividends payout minus the equity issuance up to the time of ruin. We solve the optimal control problem and identify the optimal strategy by constructing two categories of suboptimal control problems.     

On a compound Poisson risk model with delayed claims and random incomes

The main focus of this paper is to analyze the Gerber-Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber-Shiu penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber-Shiu penalty function is derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber-Shiu penalty function.