Risk contagion under regular variation and asymptotic tail independence

Abstract

Risk contagion concerns any entity dealing with large scale risks. Suppose Z = (Z1, Z2) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of Z2 on Z1. This can be measured in a variety of ways. In this paper, we study two such measures: the quantity E[(Z1 − t)+|Z2 > t] called Marginal Mean Excess (MME) as well as the related quantity E[Z1|Z2 > t] called Marginal Expected Shortfall (MES). Both quantities are indicators of risk contagion and useful in various applications ranging from finance, insurance and systemic risk to environmental and climate risk. We work under the assumptions of multivariate regular variation, hidden regular variation and asymptotic tail independence for the risk vector Z. Many broad and useful model classes satisfy these assumptions. We present several examples and derive the asymptotic behavior of both MME and MES as the threshold t → ∞. We observe that although we assume asymptotic tail independence in the models, MME and MES converge to ∞ under very general conditions; this reflects that the underlying weak dependence in the model still remains significant. Besides the consistency of the empirical estimators we introduce an extrapolation method based on extreme value theory to estimate both MME and MES for high thresholds t where little data are available. We show that these estimators are consistent and illustrate our methodology in both simulated and real data sets.

Estimation of the marginal expected shortfall under asymptotic independence

“…For this purpose, we impose conditions A(1)-A(3). In Das and Fasen-Hartmann (2018), Assumption B was assumed to guarantee the integrability, and the conditions in Lemma 2.3 in Kulik and Soulier (2015) played a similar role. Note that if = 1, the integrand is a conditional probability and it is thus easier to show the integrability (cf.…”

“…Various approaches have been used to model asymptotic independence and none of them is superior to the others. Das and Fasen‐Hartmann () have constructed an estimator of MES and have shown the consistency of the estimator, assuming that the distribution of the pair ( X , Y ) possesses hidden regular variation, which is a similar setting with ours. However, from the modeling point view, Das and Fasen‐Hartmann () also assume a heavy right tail for the distribution of the conditioning variable, that is, Y in our notation, which also results in a different estimator of MES than ours.…”

Asymptotics for Systemic Risk With Dependent Heavy-Tailed Losses

“…Under the TAI structure, we obtain an asymptotic characterization for the SRs based on the popular measure MES. Unlike the pioneering works by Cai et al (2015), Asimit and Li (2016), Das and Fasen (2018), we consider the marginal excess of a systemic portfolio risks provided one line of business or legal entity is in a financial distress, where the risk capital allocation by Euler's principle is also taken into consideration. Distinguished from Asimit et al (2011), a nonpositive amount of capital is also allowed to be allocated to any line of business.…”

“…Asymptotic evaluations of the MES were investigated in Asimit and Li (2016). A variate of variants on MES including Conditional Tail Expectation, Marginal Mean Excess, and their asymptotic approximations have appeared in the insurance and actuarial science literature, see Asimit et al (2011), Hua and Joe (2011), Das and Fasen (2018), Asimit and Li (2018). The determinants found in the literature provide us a guidance for dependent heavy-tailed losses in our model.…”

“…Recently, Hua and Joe (2014) studied the behavior of conditional tail expectation in an asymptotic tail independence case. Das and Fasen (2018) investigated the asymptotic behavior of certain risk contagion measures with multivariate regularly varying, hidden regularly varying, or tail asymptotically independent risk vectors. In this paper, we address asymptotically independent risks via the tail asymptotic independence when investigating SRs.…”