Some topics in risk theory and optimal capital allocation problems

Abstract

In recent years, the Markov Regime-Switching model and the class of Archimedean

copulas have been widely applied to a variety of finance-related fields. The

Markov Regime-Switching model can reflect the reality that the underlying economy

is changing over time. Archimedean copulas are one of the most popular

classes of copulas because they have closed form expressions and have great flexibility in modeling different kinds of dependencies.

In the thesis, we first consider a discrete-time risk process based on the compound

binomial model with regime-switching. Some general recursive formulas

of the expected penalty function have been obtained. The orderings of ruin probabilities are investigated. In particular, we show that if there exists a stochastic

dominance relationship between random claims at different regimes, then we can

order ruin probabilities under different initial regimes.

Regarding capital allocation problems, which are important areas in finance

and risk management, this thesis studies the problems of optimal allocation of

policy limits and deductibles when the dependence structure among risks is modeled

by an Archimedean copula. By employing the concept of arrangement

increasing and stochastic dominance, useful qualitative results of the optimal

allocations are obtained.

Then we turn our attention to a new family of risk measures satisfying a set

of proposed axioms, which includes the class of distortion risk measures with

concave distortion functions. By minimizing the new risk measures, we consider

the optimal allocation of policy limits and deductibles problems based on the

assumption that for each risk there exists an indicator random variable which

determines whether the risk occurs or not. Several sufficient conditions to order

the optimal allocations are obtained using tools in stochastic dominance theory.