Asymptotics Of Several Risk Measures For Portfolio Loss Under Dependent Structures

Under dependent structures,we give the asymptotics of several risk measures for portfolio loss,respectively.Accordingly,some relevant examples are also given in order to illustrate the main results.The explicit details are split into two parts as follows:(1)In the setting of bivariate Eyraud-Farlie-Gumbel-Morgenstem copula and heavy tails characterized by the power law of tail decay,the asymptotics of value-at-risk for portfolio loss is given as the confidence level tends to one.It can be seen from the obtained asymptotics that diversification decreases the value-at-risk of portfolio loss for the tail index greater than one and increases the value-at-risk of portfolio loss for the tail index less than one.In addition,the correspond-ing asymptotics of tail conditional expectation is also shown.To illustrate the obtained results,some relevant examples are presented.(2)Under the framework of multivariate regular variation,we obtain the first-order asymptotics of tail distortion risk measure for portfolio loss by a differen-t method from the one in Zhu and Li[1]when the confidence level tends to one.Further,based on the notion of second-order regular variation,the second-order asymptotics of tail distortion risk measure for portfolio loss is also derived.It is obvious that the derived second-order asymptotics is more accurate than the obtained first-order one.Moreover,for the particular multivariate regularly vary-ing case,we give also the corresponding first-and second-order asymptotics of tail distortion risk measure for portfolio loss.In order to exemplify the main obtained findings,some relevant examples are shown.