Bounds on the value-at-risk for the sum of possibly dependent risks

Abstract

In this paper, explicit lower and upper bounds on the value-at-risk (VaR) for the sum of possibly dependent risks are derived when only partial information is available about the dependence structure and the individual behaviors. When the marginal distributions are known, a reformulation of a result of Embrechts et al. [Finan. Stoch. 7 (2003) 145-167] makes it possible, under some regularity conditions, to compute explicit bounds for the VaR under various dependence scenarios. In the case where only the means and the variances of the risks are available, explicit bounds are obtained from an optimization over all possible values of the correlation matrix associated with the vector of risks. Analytical and numerical investigations are presented in order to investigate the quality of these bounds.

Tail asymptotics for the sum of two heavy-tailed dependent risks

“…On the other hand, without any knowledge of the underlying dependence structure, G(x 1 , x 2 ) has to be replaced by the counter-monotone copula C W (F (x 1 ), F (x 2 )) = max{F (x 1 ) + F (x 2 ) − 1, 0}, in which form the above result is due to Makarov [21]. For an extension to best-possible bounds on the distribution of general non-decreasing functions of n dependent risks, see for instance Cossette et al [8], Embrechts & Puccetti [12] and Mesfioui & Quessy [23]. However, the above approach is not well-suited for asymptotic considerations and does not make use of the heavy-tail assumption directly.…”

Portfolio Value at Risk Bounds Using Extreme Value Theory

“…In order to establish explicit bounds for VaR of a portfolio composed by possibly dependent assets, we combine the Dekkers et al (1989) quantile estimator of large order with VaR bounds obtained by Mesfioui and Quessy (2005) and discussed in section 2. Moreover, following the methodology of Vermaat et al (2005), we provide location change invariant bounds for quantile estimator of large order.…”

“…It is obvious, in this specific framework, of unknown dependence structure, that an explicit portfolio VaR is unreachable but one is interested in finding lower and upper bounds on VaR. Recent contributions include (Denuit, Genest , and Marceau, 1999;Denuit, Dhaene, Goovaerts, and Kaas, 2005;Embrechts, Höing, and Puccetti, 2005;Embrechts and Puccetti, 2006a ;Kaas, Laeven, and Nelsen, 2009;Luciano and Marena, 2002;Mesfioui and Quessy, 2005) and refer to Embrechts and Puccetti (2006b) for an extension to the general case where the marginal distributions of each of the assets are different. In a such case, a numerical procedure is suggested to compute VaR bounds.…”

“…In a such case, a numerical procedure is suggested to compute VaR bounds. In this paper, we study VaR bounds for sums of possibly dependent financial assets using the recent contribution by Mesfioui and Quessy (2005) when no information on the dependence between them is available. The underlying dependence involving the portfolio constituents is captured through the empirical copula.…”

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AbstractThe aim of this paper is to apply a semi-parametric methodology developed by Mesfioui and Quessy (2005) to derive the Value-at-Risk (VaR) bounds for portfolios of possibly dependent financial assets when the marginal return distribution is in the domain of attraction of the generalized extreme value distribution while the dependence structure between financial assets remains unknown. However, These bounds are very sensitive to location changes and depend heavily on the actual location.

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