The contemporary risk management practice emphasizes the interplay of multilevel risks, of which the systematic and systemic risks are considered the main culprits of catastrophic losses. With this in mind, this thesis investigates three important topics in quantitative risk management, in which the systematic and systemic risks play a devastating role.
First of all, we center on the design of reinsurance policies that accommodate the joint interests of the insurer and reinsurer by drawing upon the celebrated notion of Pareto optimality in the context of a distortion-risk-measure-based model. Such a topic is of considerable practical interest in the current post financial crisis era when people have witnessed the significant systemic risk posed by the insurance industry and the vulnerability of insurance companies to systemic events. Specifically, we characterize the set of Pareto-optimal reinsurance policies analytically and introduce the Pareto frontier to visualize the insurer-reinsurer trade-off structure geometrically. Another enormous merit of developing the Pareto frontier is the considerable ease with which Pareto-optimal reinsurance policies can be constructed even in the presence of the insurer's and reinsurer's individual risk constraints. A strikingly simple graphical search of these constrained policies is performed in the special cases of value-at-risk and tail value-at-risk.
Secondly, we propose probabilistic and structural characterizations for insurance indemnities that are universally marketable in the sense that they appeal to both policyholders and insurers irrespective of their risk preferences and risk profiles. We begin with the univariate case where there is a single risk facing the policyholder, then extend our results to the case where multiple possibly dependent risks co-exist according to a mixture structure capturing policyholder's exposure to systematic and systemic risks.
Next, we study the asymptotic behavior of the loss from defaults of a large credit portfolio. We consider a static structural model in which latent variables governing individual defaults follow a mixture structure incorporating idiosyncratic, systematic, and systemic risks. The portfolio effect, namely the decrease in overall risk due to the portfolio size increase, is taken into account. We derive sharp asymptotics for the tail probability of the portfolio loss as the portfolio size becomes large and our main finding is that the occurrence of large losses can be attributed to either the common shock variable or systematic risk factor, whichever has a heavier tail.
Finally, we extend the asymptotic study of loss from defaults of a large credit portfolio under an amalgamated model. Aiming at investigating the dependence among the risk components of each obligor, we propose a static structural model in which each obligor's default indicator, loss given default, and exposure at default are respectively governed by three dependent latent variables with exposure to idiosyncratic, systematic, and systemic risks. The asymptotic distribution as well as the asymptotic value-at-risk and expected shortfall of the portfolio loss are obtained. The results are further refined when a specific mixture structure is employed for latent variables.