The Aggregate Risk and Completely Mixable Distributions

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The aggregate risk is of general interest in actuarial science and quantitative risk management. An important and challenging scenario is when the individual risks have known marginal distributions but an unknown dependence structure. This scenario arises from the fact that marginal distributions can be relatively easily modeled with statistical and financial tools in practice, but the dependence structure is much more difficult to capture accurately through available data. With the dependence structure unknown, it remains difficult for the banks and insurance companies to offset the aggregated risks according to industrial regulations. Due to such a concern, researchers are trying to quantify optimal bounds on risk measures of aggregate risk. Such problems were considered extensively with various assumptions in the past few decades and they were proven to be very challenging. Recently, a new tool, called the Completely Mixable Distributions (CMD), was introduced to deal with those problems. Copulas are multivariate functions that are used to model dependence structure, which also play an important role in quantifying aggregate risk. The measures considered in this proposal include important risk quantities such as the Value-at-Risk, Conditional Tail Expectation, the variance, the expected ultility and the stop-loss premium of the aggregate risk. This proposal includes research to 1) develop the theory of CMD; 2) quantify aggregate risk; 3) develop the theory of copulas. The main mathematical tools that will be used include probability distribution theory, copulas, combinatorics and mass transportation. Data analysis and numerical calculation will also play an important role. The proposed study will have significant value in probability, actuarial science and risk management. Graduate students will be intensively involved in the research process. They will gain skills in conducting research and numerical calculation, learn knowledge in mathematics, actuarial science and risk management, and fulfill the research requirements in their graduate programs through the proposed research.