An Optimal Choice of Insurance and Reinsurance Policies in the Risk Process
The chapter studies a dynamic risk model defined on infinite time interval, where both insurance and per-claim reinsurance policies are chosen by the insurer in order to minimize a functional of the form of variation coefficient under constraints imposed with probability one on insured's and reinsurer's risks. We show that the optimum is achieved at constant policies, the optimal reinsurance is a partial stop loss reinsurance and the optimal insurance is a combination of stop loss and deductible policies. The results are illustrated by a numerical example involving uniformly distributed claim sizes.
The paper studies a problem of optimal insurer’s choice of a risk-sharing policy in a dynamic risk model, so-called Cramer-Lundberg process, over infinite time interval. Additional constraints are imposed on residual risks of insureds: on mean value or with probability one. An optimal control problem of minimizing a functional of the form of variation coefficient is solved. We show that: in the first case the optimum is achieved at stop loss insurance policies, in the second case the optimal insurance is a combination of stop loss and deductible policies. It is proved that the obtained results can be easily applied to problems with other optimization criteria: maximization of long-run utility and minimization of probability of a deviation from mean trajectory.
An optimal control problem is formulated for a class of nonlinear systems for which there exists a coordinate representation transforming the original system into a system with a linear main part and a nonlinear feedback. In this case the coordinate transformation significantly changes the form of original quadratic functional. The penalty matrices become dependent on the system state. The linearity of the transformed system structure and the quadratic functional make it possible to pass over from the Hamilton–Jacoby–Bellman equation (HJB) to the state dependent Riccati equation (SDRE) upon the control synthesis. Note that it is rather difficult to solve the obtained form of SDRE analytically in the general case. In this study, we construct the guaranteed control method from the point of view of the system quality based on feedback linearization of the nonlinear system; the transformation of the cost function upon linearization is examined, as well as the system behavior in the presence of disturbance and the control synthesis for this case. The presented example illustrates the application of the proposed control method for the feedback linearizable nonlinear system.
Asset liability management has received much attention lately among other financial mathematics problems. Optimal investment with constraints is a distinctive feature of this class of problems. The paper presents solution of the constrained optimal control problem for a specific market model and optimal criterion. The proposed model has correlated dynamics of assets in a general form and allows for a closed form solution of the problem.
Book include abstracts of reports presented at the IX International Conference on Optimization Methods and Applications "Optimization and applications" (OPTIMA-2018) held in Petrovac, Montenegro, October 1 - October 5, 2018.
Proceedings include extended abstracts of reports presented at the III International Conference on Optimization Methods and Applications “Optimization and application” (OPTIMA-2012) held in Costa da Caparica, Portugal, September 23—30, 2012.
We study the risk sharing implications that arise from introducing a disaster relief fund to the cat insurance market. Such a form of intervention can increase efficiency in the private market, and our design of disaster relief suggests a prominent role of catastrophe reinsurance. The model predicts buyers to increase their demand in the private market, and the seller to lower prices to such an extent that her revenues decrease upon introduction of disaster relief. We test two predictions in the context of the Terrorism Risk Insurance Act (TRIA). It is already known the introduction of TRIA led to negative abnormal returns in the insurance industry. In addition, we show this negative effect is stronger for larger and for low risk-averse firms -- two results that are consistent with our model. The seller's risk aversion plays an important role in quantifying such feedback effects, and we point towards possible distortions in which a firm may even be overhedged upon introduction of disaster relief.