Abstarct: Gini index is a well-known tool in economics that is often used for measuring income inequality. In insurance, the index and its modifications have been used to compare the riskiness of portfolios, to order reinsurance contracts, and to summarize insurance scores (relativities). In this thesis, wee stablish several stochastic orders between the Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. The comparison of the Gini indexes (of empirically estimated risk measures) presented in this thesis provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985). In order to determine an appropriate amount of premium, statistical goodness-of-fit criteria must be supplemented with actuarial ones when assessing performance of a given candidate pure premium. In this thesis, concentration curves and Lorenz curves are shown to provide actuaries with effective tools to evaluate whether a premium is appropriate or to compare competing alternatives. The idea is to compare the premium income for sub-portfolios gathering low risks (identified as low by means of the premiums under consideration) to the true one, or equivalently, to the actual losses. Numerical illustrations performed on hypothetical data and real ones demonstrate the usefulness of the proposed approach. Finally, using Generalized linear models with the concentration and Lorenz curves, we examined the factors affecting the Tehran Stock Exchange (TSE) index, and selected the model that provided a better forecast of the TSE index.