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In this paper, following the generalization of Delta Normal VaR to Delta Mixture Elliptic VaR in Sadefo-Kamdem [3], we give and explicit formula to estimate linear VaR and ES when the risk factors changes with the mixture of t-Student distributions. In particular, we give rise to Delta-Mixture-Student VaR and the Delta-Mixture-Elliptic ES.

Generally, in the financial literature, the notion of quadratic VaR is implicitly confused with the Delta-Gamma VaR, because more authors dealt with portfolios that contained derivatives instruments. In this paper, we postpone to estimate both the expected shortfall and Value-at-Risk of a quadratic portfolio of securities (i.e equities) without the Delta and Gamma Greeks, when the joint log-ret...

The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure for right-tail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase type distribution, and derive explicit formulas of several condi...

This paper compares three value-at-risk approximation methods suggested in the literature: Cornish-Fisher (1937), Sillitto (1969), and Liu (2010). Simulation results are obtained for three families of distributions: student-t, skewed-normal, and skewed-t. We recommend the Sillitto approximation as the best method to evaluate the value at risk when the financial return has an unknown, skewed, an...

rule ClassWithOperationsContributingClass merge c : Core!EClass with s : ObserverPattern!EClass into t : Target!EClass { for (sop in s.eOperations) { var op : new Target!EOperation; op.name := sop.name; t.eOperations.add(op); for (sopp in sop.eParameters) { var p : new Target!EParameter; p.name := sopp.name; p.eType ::= sopp.eType; op.eParameters.add(p); } } } auto rule ClassWithObserver merge ...

Two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivity for random variables, and the insurance risk measure, which assumes additivity for comonotonic random variables. We propose a new, data based, risk measure, called natural risk statistic, that is characterized by a new set of axioms. The new axioms only require subadditivity for comonotonic ran...

We estimate Value-at-Risk for sums of dependent random variables. We model multivariate dependent random variables using archimedean copulas. This structure allows one to calculate the asymptotic behaviour of extremal events. An important application of such results are Value-at-Risk estimates for sums of dependent random variables.