Jinzhu Li

Asymptotics for a Discrete-time Risk Model with the Emphasis on Financial Risk

This paper focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account. We study the asymptotic behaviour of the ruin probability and the tail probability of the aggregate risk amount. Precise asymptotic formulas are derived under weak moment conditions on involved risks. The main novelty of our results lies in the quantification of the impact of the financial risk. Keywords: asymptotics; financial risk; insurance risk; regular variation; ruin probability Mathematics Subject Classification: Primary 62P05; Secondary 62E10, 91B30 We are concerned with the asymptotic behaviour of the tail probabilities P(Sn > x) and P(Mn > x) as x ! 1, in which P(Mn > x) coincides with the insurers finite-time ruin probability within period n given that the initial wealth is x. In the literature {Xi;i � 1} and {Yi;i � 1} are usually called the insurance risk and the financial risk, respectively. Under certain independence or identical distribution assumptions imposed on Xis and Yis, the asymptotic tail behaviour of Sn and Mn has been extensively studied by many researchers. See, e.g., Tang and Tsitsiashvili (2003, 2004), Konstantinides and Mikosch (2005), Tang (2006), Zhang et al. (2009), Chen (2011), and Yang and Wang (2013) for some recent findings. Since the products of Yis appearing in (1.1) essentially cause technical problems in the derivation of explicit asymptotic formulas, most of existing works assumed that the financial risk is dominated by the insurance risk, i.e., the tails of Yis are lighter than the tails of Xis, usually through imposing strong moment conditions on Yis. Then the problem becomes relatively tractable and the final results are mainly determined by the tails of Xis. However, as shown by empirical data and the most recent financial crisis, the financial risk may impair the insurers solvency as seriously as does the insurance risk and, hence, it should not be underestimated

Systemic Risk: An Asymptotic Evaluation

The systemic risk (SR) has been shown to play an important role in explaining the financial turmoils in the last several decades and understanding this source of risk has been a particular interest amongst academics, practitioners, and regulators. The precise mathematical formulation of the SR is still scrutinised, but the main purpose is to evaluate the financial distress of a system as a result of the failure of one component of the financial system in question. Many of the mathematical definitions of the SR are based on evaluating expectations in extreme regions and therefore, Extreme Value Theory (EVT) represents the key ingredient in producing valuable estimates of the SR and even its decomposition per individual components of the entire system. Without doubt, the prescribed dependence model amongst the system components has a major impact over our asymptotic approximations. Thus, this paper considers various well-known dependence models in the EVT literature that allow us to generate SR estimates. Interestingly, our findings reveal sensible results. That is, the SR has a significant impact under asymptotic dependence, while weak tail dependence, known as asymptotic independence, produces an insigniOcant loss over the regulatory capital.