Control Theory and Insurance Systems

The primary target of the current thesis is the establishment of a link (bridge) between the abstract concepts of control theory and the practical applications connected with different insurance systems. The existence of such a link has been identified by several authors in the last two decades who have focused on the use of conventional control theory (and optimal control techniques) in actuarial problems. In our approach, after providing a short reference guide to the modern control theory in chapter 2 and a selective review of the literature in chapter 3, we propose three distinct models covering the most important areas of insurance applications (i.e. Life, General Insurance & Pensions). From the control point of view we actually use all the potential tools i.e. Multiple Input - Multiple Output, Time-Varying format, Nonlinear equations, Stability Analysis, Root Locus Method, Optimal design for the parameters involved and optimal control of a dynamic system. The thesis deals with the basic concept of an insurance system, "the premium" and aims to answer the critical question "How to calculate and control the premium rating process?". In the first model (chapter 4), we examine the general process of insurance pricing, using the standard equation which connects the three major variables involved i.e. premiums, claims & surplus. Starting from the roots of actuarial science and the static point of view and passing to Lundberg’s revolution with his dynamic view, we arrive at the modern alternative view of control theory with respect to pricing models. We concentrate to the concept of stability rather than to the traditional concept of ruin. In the second model (chapter 5), we construct a dynamic system which describes a special reinsurance arrangement (multinational pooling) which may only be handled using the modern control theory (as it refers to a multivariable system). Actually it is an extension of the previous model to a multi-system which consists of different subsystems, as described in chapter 4 considering also the interaction between them. Finally in the third model (chapter 6) we investigate the philosophy and mechanisms of the Social Security System and the PAYG funding method. It may also be seen as an extension of the first model to the area of pensions and the necessity of calculating and controlling the respective contribution rate and the age of normal retirement. At the end of the sixth chapter, a simulation is carried out for the projected population data of Greece up to the year 2020

An Application of Control Theory to the Individual Aggregate Cost Method

The paper investigates the individual aggregate cost method (also known as the individual spread-gain method), which is normally applicable in small pension funds or fully contributory schemes, using a control theoretical framework. We construct the difference equations describing the mechanisms of the respective funding method and then calculate the optimal control path of the contribution rate assuming (first) a stochastic and (second) a deterministic pattern for the future investment rates of return. For the first case, the optimal solution is achieved through a linear approximation and using stochastic optimization techniques. It is proved that the contribution rate is (optimally) controlled through the control of the valuation rate (which is determined incorporating a certain feedback mechanism of the past contribution rate). The optimal solution for the deterministic case is obtained using standard calculus and the method of Lagrange multipliers

Optimal Premium Pricing for a Heterogeneous Portfolio of Insurance Risks

The paper revisits the classical problem of premium rating within a heterogeneous portfolio of insurance risks using a continuous stochastic control framework. The portfolio is divided into several classes where each class interacts with the others. The risks are modelled dynamically by the means of a Brownian motion. This dynamic approach is also transferred to the design of the premium process. The premium is not constant but equals the drift of the Brownian motion plus a controlled percentage of the respective volatility. The optimal controller for the premium is obtained using advanced optimization techniques, and it is finally shown that the respective pricing strategy follows a more balanced development compared with the traditional premium approaches