Remodified Certainty Equivalent And Insurance Premium Principles

In this paper,we remodified risk measure and premium principle pattern based on modified certainty equivalent risk measure.for each risks are dependent, wo don't consider they are independent if we measure a portfolio risk and add the dependence into the risk measure.In statistics ,relations of each component can be described most completely by their joint distribution. ,but in practical we are easy to get the data of each component and get the marginal distribution ,it's difficulty to catch the joint distribution .How do we deduce the joint distribution from marginal distribution of each component ?This's a question which have great applied value though it is a direct ones theoritically. Sklar proposed that a Copula is a function that links univariate to their multivariate distribution, Professor Zhang yaoting calls it associated function.Recently Copula have been used in finance and insurance .In the case that have the distributions of individual risk,we get the joint distribution of each risk from the definition of Copula and Sklar theorem.Consequently ,we can have much more information when we give the risk measure and premium principle . So we apply the connective function Copula to them, which made the application of risk measure and premium principle more broad than ever, thus we consider the dependence of portfolio investment risks in risk measure and premium principle. Modified certainty equivalent risk measure is a special case in remodified certainty equivalent risk measure. Particularly we can get more ideograph risk measure and premium principle when detailed mixed distribution be given in individual risks, which can be made better choice for either investors or insurance companies.We organized our paper with five sections.The first section we introduced the definition of Copula connective functions and their basic properties, and Sklar theorem which connected marginal distributions with joint distribution that is. Copula function is a multivariate distribution function whose marginal distribution is a uniform distribution function in interval [0, 1]. Copula can be used to describe the connection of individual risks and entire portfolio investment risk. It may be more comprehensive that we consider risk measure and premium principle based on Copula, thus we can deduce a more reasonable conclusion and make a more efficient decision.The second section we introduced kinds of price principle of premium principle, which includes net premium principle, expected value principle, variance principle, deviation from standard principle, zero utility principle, average value principle, percentiles principle and maximum loss principle. First we list each principle here, and then we will apply Copula function to them in the following sections with the consideration of probability measure exchange on net premium principle and dependence of kinds of risks, which implied that we considered more information with them, and we made more comprehensive decision.The third section we proposed remodified premium principle, which gave different premium principles with separating risk into positive and negative stochastic variables, but whatever the positive or negative stochastic variable is, the proposed premium principle will satisfy following properties: net premium principle, additionty, consistency, and order relation, except for theorem 3.2, we can reach different property for risk stochastic variables' positive or negative:, we have inequalitywe have inequalityThe premium principle we proposed in this section can satisfy the basic properties of premium, also we considered the dependence of risks on the ground of the proposed premium principle. For investors and insurance companies, they can make reasonable decision for themselves.The forth section we proposed remodified certainty equivalent risk measure, both properties averseness and combinatorial variety of risk will be satisfied with risk measure model, price decision model and premium model. We proved the newly developed modified certainty equ...