Optimal dividend and risk control policies in the presence of a fixed transaction cost

Abstract

In this paper, we consider a large insurance company whose cumulative cash flow process is described by a drifted Brownian motion. The preference of the insurer is to maximize his/her firm’s value, which corresponds to the expected present value of the dividend payments up to the ruin time. In the business process, the insurer has the option to draw up a dividend payment policy and to purchase proportional reinsurance at a certain point in time. In view of some typical practical expenses (e.g., consultant commission), it is assumed that a fixed transaction cost occurs at the beginning of the reinsurance commitment which, once made, is irreversible. This leads to a mixed stochastic control problem of optimal stopping time and singular control. For this mixed problem, we derive closed-form solutions for the optimal time to purchase reinsurance, the optimal retained risk proportion, the optimal dividend barrier, and the value function. The optimal solution shows that reinsurance is valueless to the insurer when the fixed cost is larger than a threshold, and comes into play when the fixed cost is less than this threshold. We also perform some numerical calculations to assess the impacts of fixed costs on the value function and the optimal policies.

Introduction

In practice, an insurance company is typically faced with making decisions related to an optimization problem. Two examples of frequently-used optimization objectives are: (1) minimize the ruin probability to improve solvency by controlling the reinsurance proportion, and (2) maximize the dividend payments to enhance the insurance company’s value by choosing the most favorable dividend payout policy. Since dividend payouts and reinsurance are important ways to control risk and enhance firm value, the optimal dividend and reinsurance policies are very popular problems in mathematical insurance and actuarial science. For example, [1], [2], [3] studied optimal proportional reinsurance problems under the diffusion process by means of minimizing ruin probability, while [4], [5] considered maximizing firm value by choosing optimal reinsurance and dividend policies. These pioneer works suggest that the risk control policy of reinsurance does play a significant role in reducing the probability of ruin and increasing an insurance firm’s value. Afterwards, the optimal reinsurance and dividend problems are being widely studied in different models, such as the model with dividend transaction costs in [6] and the multiple reinsurers models in [7], [8]. Optimal reinsurance problems are also investigated with other control variables such as investment and capital injection. For example, [9], [10], [11], [12], [13], [14], [15] considered optimal investment–reinsurance problems with respect to different risk models, and [16], [17], [18] investigated optimal reinsurance problems in the presence of capital injection. In addition, optimal dividend is a classical problem of actuarial mathematics and attracts a lot of attention (see, [19], [20], [21]). Recently, optimal dividend problem is extended from univariate risk theory to two-dimensional setup. For example, [22] studied the optimal dividend and paying the deficit of the partner strategies of two collaborating companies, and [23] studied the optimal dividend and transferring money strategies of a company with two business lines with or without proportional transaction costs.

The early pioneering literature in this field investigated the issue of optimal reinsurance problems under the assumption that reinsurance takes place at an initial time without any fixed transaction cost. Transaction costs are inevitable in many decisions, but the previous studies focused mostly on the direct costs of the business. For example, [24] considered the fixed and proportional costs for each dividend payment in the diffusion process, while [25] studied the proportional and fixed costs for each injection in a dual model. These fixed and proportional costs of each payment typically lead to an impulse-regular control problem. However, indirect expenses that often exist may be nonnegligible (i.e., consultant commissions), but there are very few research results concerning this problem. High early transaction costs, which are fixed costs, are inevitable in the reinsurance business. High early transaction costs lead to a singular control/stopping time problem because they occur only at the beginning of reinsurance. The issue of fixed transaction costs in reinsurance was first studied by [26], where the authors chose to maximize the difference between the expected total utility of an insurer’s surplus and the discounted value of the insolvent fixed costs, since the existence of fixed transaction costs does affect insurers’ decision of both the time of purchasing reinsurance and the proportion of reinsurance. This has inspired us to consider the fixed costs of reinsurance within the broader optimization problems in insurance companies.

In this paper, we consider mixed optimal dividend and reinsurance problems in the presence of a fixed cost. Under this assumption, the decision maker has to choose when to start the reinsurance, the retained risk exposure of reinsurance, and the dividend policy to maximize the firm’s value. This leads to a mixed singular control/optimal stopping time problem. The main contribution of this paper is to consider the interaction between dividend and reinsurance decisions as a singular control/optimal stopping time problem. More specifically, fixed costs offer companies the opportunity to choose the start time of reinsurance that affects each firm’s dividend decisions. We solve this mixed singular control/optimal stopping time problem by establishing a connection with an optimal stopping problem. Our results show the interaction of the fixed costs on dividends and reinsurance, namely, when the fixed cost is higher than a certain threshold, the dividend is the priority, otherwise, reinsurance is the priority.

The rest of this paper is organized as follows: In Section 2 we formulate the optimization problems and we present two benchmarks useful to solve the mixed singular control/ optimal stopping problem. In Section 3, we solve the optimization problem in two cases. We first find a necessary and sufficient condition, under which the reinsurance is valueless and the problem is reduced to a pure optimal dividend problem. The other case (i.e., that does not satisfy this condition) is the principal mixed control/optimal stopping problem. We determine the optimal dividend and reinsurance polices as well as the value function by employing the optimal stopping theory and the dynamic programming principle. To further explain the sensitivities of the fixed costs, we provide some numerical examples in Section 4. Finally, in Section 5 we give some concluding remarks.