Description: This is a concluding and encompassing part of our research on the project entitled \Weighted Premium Calculation Principles and Risk Capital Allocations, which has been supported by the Actuarial Education and Research Fund AERF and the Society of Actuaries Committee on Knowledge Extension Research CKER. We explore the concept of weighted distributions and their role in various phenomena occurring in insurance and finance. In particular, we relate weighted distributions to actuarial and economic premium calculation principles, and also to the capital asset pricing model CAPM. Imitating the latter, we propose a weighted insurance pricing model WIPM. Although general in formulation, we show that the WIPM can successfully be evaluated in a variety of situations, which we illustrate with a number of examples.

Description: The conditional tail expectation [CTE] is a popular actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the non-parametric CTE estimator has already been established in the literature. That result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, this presentation offers a solution to the problem by suggesting a new CTE estimator, which is applicable when claims losses have finite means but infinite variances.

Description: Many quantities arising in non-life insurance depend on claim severity distributions, which are usually modeled assuming a parametric form. Obtaining good estimates of the quantities, therefore, reduces to having good estimates of the model parameters. However, the notion of 'good estimate' depends on the problem at hand. For example, the maximum likelihood estimators [MLE] are efficient, but they generally lack robustness. Since outliers are common in insurance loss data, it is therefore important to have a method that allows one to balance between efficiency and robustness. This presentation suggests a general estimation method that we call the method of trimmed moments [MTM]. This method is appropriate for various model fitting situations including those for which a close fit in one or both tails of the distribution is not required. The MTM estimators can achieve various degrees of robustness, and they also allow the decision maker to easily see the actions of the estimators on the data, which makes them particularly appealing.

Description: This is the abstract for the research on robust and efficient fitting of claim severity distributions.

Description: This is the abstract for the reseach on the actuarial CTE risk measure for heavytailed losses: a new estimator and confidence intervals.c