Worst-case Conditional Value-at-Risk And Conditional Expected Shortfall Based On Moment Information

Modern risk management usually needs to evaluate the risk under various dimensions.Because financial institutions face the risks of interdependence,the supervision of financial institutions alone is often not enough to prevent financial crises.Therefore,after the financial crisis,people have become interested in measuring systemic risk.Many methods have been proposed to measure system risk,such as CoVaR(conditional value-at-risk)and CoES(conditional expected shortfall).Measuring systemic risk requires the knowledge of its probability distribution.In most practices,the exact form of distribution is often lacking.In this paper,we adopt the modeling paradigm of distributed robust optimization(DRO).We study the worst-case CoVaR and CoES in the situation where only partial information on the underlying probability distribution is available.The research of this paper starts from the theoretical part.Firstly,we describe the form of optimization problem and give the definitions of two kinds of ambiguity sets.Based on the previous results,we further study the relationship between CoVaR and copula function.In case of the first two marginal moments are known,the closed-form solution and the value of the worst-case CoVaR and CoES are derived.The worst-case CoVaR and CoES under mean and covariance information are also investigated.This enables us to find that when the correlation coefficient of X and Y is greater than 0,the worst-case CoVaR and CoES are consistent with the worst-case values without the restriction of correlation coefficient,that is,when the organization X suffers a heavy loss,the Y with which it has a certain positive correlation will also suffer a heavy loss.Finally,we verify the theoretical results in the simulation part.We study the worst-case CoVaR and CoES in the following two cases:(1)The real distribution is binary Gaussian distribution or binary t distribution with different correlation coefficients.(2)The real marginal distribution is Pareto distribution with different copula functions.In both cases,we include the worst-case CoVaR and CoES.The simulation results show that our theoretical results are robust when some information of the real probability distribution is known.This will help us make more stable decisions in real life.