| A new method for estimating the threshold of a three-parameter lognormal density function is presented and compared to other methods currently in use. The new method is used to construct a statistical model of a pertinent business problem--measuring the financial risk of being an underwriting member at Lloyd's of London.; The central theme is measuring the risk of insuring underwriting members, who are personally responsible for losses. Relevant details about insurance and Lloyd's are explained. Actual data is presented, and problems of quantity, cyclicality, and dependence are illustrated.; Using a theoretical development and goodness-of-fit testing, the three-parameter lognormal is chosen to model the distribution of loss ratios. Basic properties of this density are presented and recent mathematical literature is summarized. Maximum likelihood, quantile, Cohen's, and Kane's order statistic methods of estimating the parameters from sample data sets are explored. A new method based upon minimizing the sum of errors between observed and theoretical values is developed. On an appropriate domain this sum attains a minimum value. Several methods are applied to lognormally distributed sample data sets generated using Kane's procedures. From these results and Kane's results it is observed that the new method possesses several advantages--greater applicability, greater accuracy as measured by mean squared error, and a narrower dispersion of estimates about the true threshold.; Reproductive properties of the lognormal are used to relate syndicate results to global results and lessen the problems of quantity and dependence. The new estimation method is used to construct a model of syndicate behavior. Numerical methods are used to determine the number of statistical trials necessary for various degrees of accuracy. A risk charge associated with uncertainty in parameter estimation is developed, and a profit loading is related to capital required to support the new coverage. These loadings are used to produce an appropriate gross premium charge, which is acceptable for successfully marketing the product. Questions which the model has helped to answer are discussed as are generalizations of the model to other areas. |
