Weighted premium calculation principles

A prominent problem in actuarial science is to define, or describe, premium calculation principles (pcp’s) that satisfy certain properties. A frequently used resolution of the problem is achieved via distorting (e.g., lifting) the decumulative distribution function, and then calculating the expectation with respect to it. This leads to coherent pcp’s. Not every pcp can be arrived at in this way. Hence, in this paper we suggest and investigate a broad class of pcp’s, which we call weighted premiums, that are based on weighted loss distributions. Different weight functions lead to different pcp’s: any constant weight function leads to the net premium, an exponential weight function leads to the Esscher premium, and an indicator function leads to the conditional tail expectation. We investigate properties of weighted premiums such as ordering (and in particular loading), invariance. In addition, we derive explicit formulas for weighted premiums for several important classes of loss distributions, thus facilitating parametric statistical inference. We also provide hints and references on non-parametric statistical inferential tools in the area.     

On convex principles of premium calculation

The classical definition of a principle of premium calculation is generalized: risks with identical distributions do not necessarily lead to the same premium. In the first part (Sections 1–3) the theoretical properties of convexity are discussed; in particular, the gradient of a principle is introduced. It is noted that the more common principles are all convex. In the second part these notions are applied to solve two problems under rather general assumptions: (1) Optimal purchase of reinsurance: If the first insurer knows how the reinsurer determines his premium, what form and degree of reinsurance should he choose? (2) Optimal cooperation: How should n companies split up a given risk to minimize the total premium? The case where the optimal decompostion consists of constant quotas is described in detail. In general, there is a close connection with Pareto optimality on the one hand, and no trade equilibria on the other.     

On robust premium principles

Premium principles are investigated with respect to their behavior on extremely large claims. A measure for describing the reaction on such a large claim is defined and calculated for various principles. Modified principles are introduced, which are robust to large claims.     

On the robustness of premium principles

In the present note, the robustness of broad classes of premium calculation principles is investigated. It turns out that common principles of premium calculation are not robust.     

A family of premium principles based on mixtures of TVaRs

Risk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the risk loading. Members of this family are particular distortion premium principles that can be represented as mixtures of TVaRs, where the weights in the mixture reflect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles.     

A new premium calculation principle based on Orlicz norms

A multiplicative equivalent of the zero utility premium calculation principle is introduced. If the utility function happens to be a normalized Young function the new premium calculation principle is related to Orlicz norms.     

Properties and calculation of Paired Haar transform

Different properties of recently introduced Paired Haar transform have been shown. Nonpolynomial Haar Pxpansion of incompletely specified Boolean functions has been presented. Based on the above properties and expansion some applications of Paired Haar spectrum have been proposed. Algorithm for the calculation of Haar Pair spectrum from disjoint cubes for systems of incompletely specified Boolean functions has also been developed.     

Properties and calculation of Paired Haar transform

Different properties of recently introduced Paired Haar transform have been shown. Nonpolynomial Haar Pxpansion of incompletely specified Boolean functions has been presented. Based on the above properties and expansion some applications of Paired Haar spectrum have been proposed. Algorithm for the calculation of Haar Pair spectrum from disjoint cubes for systems of incompletely specified Boolean functions has also been developed.     

Motor premium rating

This paper outlines a very simple structure for the practical analysis of motor premiums from a UK point of view. It indicates that it is possible to model different types of claims cost. These results are combined together with some simple economic assumptions, to arrive at premiums. Then the paper develops a ‘points system’ which is similar to a number of premium systems operated by UK Insurance Companies. This ‘points system’ is used to compare different sets of assumptions. Finally, an analysis of surplus is described with an example.     

Properties and calculation of multivariate risk measures: MVaR and MCVaR

A recent paper by Prékopa (Ann. Oper. Res. 193(1):49–69, 2012) presented results in connection with Multivariate Value-at-Risk (MVaR) that has been known for some time under the name of p-quantile or p-Level Efficient Point (pLEP) and introduced a new multivariate risk measure, called Multivariate Conditional Value-at-Risk (MCVaR). The purpose of this paper is to further develop the theory and methodology of MVaR and MCVaR. This includes new methods to numerically calculate MCVaR, for both continuous and discrete distributions. Numerical examples with recent financial market data are presented.     

指数保费原理下的经验厘定

在经典的信度理论中,信度保费是在净保费原理下得到的. 但是, 保险商业中, 保险公司要求制定的保费必须适用于某合适的保费原理以适应具体的保险商业的需要. 本文建立了指数保费原理下的完全经验厘定模型, 得到了风险保费的信度估计和经验Bayes 信度估计, 并讨论了结构参数的估计及其性质. 最后证明了多合同模型的经验Bayes 信度估计的渐近最优性     

Calculation of Bayes premium for conditional elliptical risks

In this paper we discuss the calculation of the Bayes premium for conditionally elliptical multivariate risks. In our framework the prior distribution is allowed to be very general requiring only that its probability density function satisfies some smoothness conditions. Based on the previous results of Landsman and Nešlehová (2008) and Hamada and Valdez (2008) we show in this paper that for conditionally multivariate elliptical risks the calculation of the Bayes premium is closely related to the Brown identity and the celebrated Stein’s lemma.