Time-Consistent Investment-Reinsurance Strategies for the Insurer and the Reinsurer under the Generalized Mean-Variance Criteria

Abstract

Most of the existing literature on optimal investment-reinsurance only studies from the perspective of insurers and also treats the investment-reinsurance decision as a continuous process. However, in practice, the benefits of reinsurers cannot be ignored, nor can decision-makers engage in continuous trading. Under the discrete-time framework, we first propose a multi-period investment-reinsurance optimization problem considering the joint interests of the insurer and the reinsurer, among which their performance is measured by two generalized mean-variance criteria. We derive the time-consistent investment-reinsurance strategies for the proposed model by maximizing the weighted sum of the insurer’s and the reinsurer’s mean-variance objectives. We discuss the time-consistent investment-reinsurance strategies under two special premium principles. Finally, we provide some numerical simulations to show the impact of the intertemporal restrictions on the time-consistent investment-reinsurance strategies. These results indicate that the intertemporal restrictions will urge the insurer and the reinsurer to shrink the position invested in the risky asset; however, for the time-consistent reinsurance strategy, the impact of the intertemporal restrictions depends on who is the leader in the proposed model.

Nash Equilibrium Investment-Reinsurance Strategies for an Insurer and a Reinsurer with Intertemporal Restrictions and Common Interests

“…(ii) Under the framework of generalized multi-period mean-variance, we first deduce Nash equilibrium investment-reinsurance strategies (which satisfy time consistency) rather than the traditional pre-commitment strategy and obtain the equilibrium value function. (iii) We first theoretically analyze the effect of intertemporal restrictions on Nash equilibrium strategies, and find that it depends on the value of model parameters, which is different from the existing literature that directly believes that the intertemporal restrictions can make investors avoid risks (e.g., [25,28], etc.). In particular, we find that the effect of intertemporal restrictions on Nash equilibrium investment strategies depends on the reduction speed of risk aversion parameters and the value of risk-free benefits.…”

“…At present, most of the literature considering intertemporal restrictions directly believe that intertemporal restrictions will lead investors to avoid risks and adopt more conservative investment strategies (for example, Zhou et al [28], Xiao et al [25], etc. ), but they do not give corresponding theoretical results.…”

“…Given T = 100, w 1 0 = w 2 0 = 1. Referring to the parameter setting in Li and Ng [33] and Xiao et al [25], for t = 0, 1, • • • , T − 1, the model parameter values in this paper are given by Table 3. The risk aversion coefficient is set to be in exponential form, η t = η T * x (T−t) , t = 0, 1, • • • , T − 1, where x > 1 and η T = 1.…”

“…Therefore, compared with the optimal investment-reinsurance research in continuous time, it is more appropriate and practical to study the optimal investment-reinsurance strategy under the framework of discrete-time. Xiao et al [25] first studied a generalized multi-period investment-reinsurance optimization problem in discrete time, in which the insurer and reinsurer were studied as two independent individuals. In view of the fact that many large financial institutions include both insurance and reinsurance sectors, we investigate the optimal investment-reinsurance optimization problem under the discrete-time framework when the reinsurer and insurer have common interest objectives.…”

“…But the classical mean-variance model only considers the terminal performance and ignores the intermediate performance, which leads to a high probability of bankruptcy in the early stage of investment (refer to Zhu et al [27], Zhou et al [28]). For more literature of portfolio optimization considering intertemporal restrictions can be referred to Costa and Nabholz [29], Costa and Araujo [30], Costa and de Oliveira [31], Cui et al [32], Zhou et al [28] and Xiao et al [25], and so on. Most of the above literature that considers the intertemporal restrictions generally directly believes that the intertemporal restrictions will lead investors to avoid risks and adopt more conservative investment strategies, but does not give the corresponding results in theory.…”