Expected Utility Theory And Optimal Reinsurance: Insights From Insurance Verbund

Since Expected Utility theory (EUT for short.) was axiomatized by Von Neumann and Oskar Morgenstern in 1940's, its applications in economics and practice are given much more concerns and deep investigation. From 1950's on, EUT has been developed extensively and thereof a set of concepts and theoretic analysis frames come down such as Arrow-Pratt's Risk aversion and MMEU, as well as Yaari's generalized expected utility theory in 1980's and so on. MMEU model is mainly used to analyze how decision makers act to maximize the effectiveness of risk and uncertainty in a risky environment.In the context of this thesis, both insurers and reinsurers are regarded as risk-averse, and they both make up the overall insurance Verbund. Based on this assumption, the optimization of the decisions of both insurers and reinsurers is done by means of EUT.As known, from the point of view of the insurance company, and regardless of its own insurable risks, its risk surplus process goes as follows: Here S(t)=(?) is the aggregate claims of the insurer, and p(t) is the total premium income until the time t.From the model above, we can find without much difficulty that traditional consideration on the insurance and reinsurance is expanded only from unilateral, that is, either only from the insurer's view or from the reinsurer's alone. More often than not, almost all discussions take only the insurer into account, and so the insured's are ignored;So is the situation of the insurer and the reinsurer, attention is more paid on one and the other is lost in sight. In fact, the insurance is an economic behavior done by both the insurer and the insured, or by the insurer and the reinsurer. And that is not the all. It is indisputable that the insurance Verbund, as a whole, is a Game-equilibrium or the balance of bargaining by all sides of the market. In the sight of this view, we take both sides of the insurer and the reinsurer into account. And so, whatever the insurer is of outward reinsurance or inward reinsurance, any insurance company ought to share all risks in the insurance Verbund. From this point of view, any insurance company reinsurance outward and inward meanwhile. Any party of the market would cede its risks to others, and retrocede the risks from other companies. As a result of this view, a new model of systemic co-insurance by n insurance companies is developed in Chapter 3 based on classical L-C risk surplus process. At the end of this thesis, the concepts of risk transferring coefficients and risk transferring matrix are put out. Details will be shown below in the corresponding chapters and sections respectively.In Chapter 2, an available glimpse of the fundamental frame of EUT is introduced to put forward some premium principles as well as their properties.In Chapter 3, the general theories are brought out about insurance and reinsurance in the risky or uncertainty environment. Basic types, forms and functions of reinsurance are advanced here, especially about quate share reinsurance, no proportional reinsurances such as XLR, SLR, Co-insurance, LCR and ECOMOR, etc.In Chapter 4, based on the work of Arrow's,Borch's and Gerber's, all risk-averse parts of the insurance Verbund have the same objective of MMEU. A systemic co-insurance model of many insurers and reinsurers is introduced along with the analysis approach out of Borch's insurance market equilibrium theory built on MMEU. In this context, no matter what is the company in the insurance Verbund it is taken as an insurance company and a reinsurance one again.In Chapter 5, from the perspective of MMEU, we derive the lower and upper bounds of risk premiums of both the insurer and the re-insurer if they are both of exponential utility functions with exponential individual claims in compound Poisson process according to proportional reinsurance and XLR and SLR respectively.In Chapter 6, we give some preliminaries about large claims reinsurance and ECOMOR.In the last Chapter, concentration is put on a simplified co-insurance model in which we focus on proportional co-reinsurance with exponential individual claims in compound Poisson process and then the ruin probability of the ith company is obtained without much trouble, and thereby Risk Transferring Coefficients and Risk Transferring Matrix, and Fractional Effect are proposed as very useful cocepts.