Reducing model risk via positive and negative dependence assumptions

Bignozzi, Valeria; Puccetti, Giovanni; Rüschendorf, Ludger

doi:10.1016/j.insmatheco.2014.11.004

Cited by 39 publications

(14 citation statements)

References 30 publications

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“…This relates to the results and applications considered in Bignozzi et al. (), Puccetti, Rüschendorf, Small, and Vanduffel (), and Rüschendorf and Witting (). Specifically, we consider a 16‐dimensional homogeneous risk vector

X = (X_{1}, \dots, X_{16})

having log‐normally distributed marginals

F = F_{1}, \dots, F_{16}

with mean

μ = 0

and standard deviation

σ = 1

.…”

Section: Illustrations and Numerical Examplesmentioning

confidence: 66%

“…Similar to Bignozzi et al. (), we suppose that the subgroups are comonotonic, so that their copula is equal to the upper Fréchet–Hoeffding bound, hence

P (\cap_{i \in I_{j}} false{X_{i} \leq x_{i} false}) = \underset{i \in I_{j}}{trueprefixmin} false{F (x_{i}) false} .

Due to the homogeneity of the marginals and the assumption of comonotonicity within the groups, we obtain an explicit distribution for the sum

\sum_{i \in I_{j}} X_{i} = : X_{I_{j}}^{'}

, that is,

P (\sum_{i \in I_{j}} X_{i} \leq x) = F (\frac{x}{false| I_{j} false|}) = : F_{j}^{'} false(x false) for all x \in R, j = 1, \dots, m .

Moreover, we assume that model ambiguity is prevalent in form of an unknown copula between the individual subgroups. Along with the above argument, this is equivalent to the copula

C_{m}

(X_{I_{1}}^{'}, \dots, X_{I_{m}}^{'})

being unknown.…”

Section: Illustrations and Numerical Examplesmentioning

confidence: 73%

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Value‐at‐Risk bounds with two‐sided dependence information

Lux¹,

Rüschendorf

2018

Mathematical Finance

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Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.

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X = (X_{1}, \dots, X_{16})

having log‐normally distributed marginals

F = F_{1}, \dots, F_{16}

with mean

μ = 0

and standard deviation

σ = 1

.…”

Section: Illustrations and Numerical Examplesmentioning

confidence: 66%

“…Similar to Bignozzi et al. (), we suppose that the subgroups are comonotonic, so that their copula is equal to the upper Fréchet–Hoeffding bound, hence

P (\cap_{i \in I_{j}} false{X_{i} \leq x_{i} false}) = \underset{i \in I_{j}}{trueprefixmin} false{F (x_{i}) false} .

Due to the homogeneity of the marginals and the assumption of comonotonicity within the groups, we obtain an explicit distribution for the sum

\sum_{i \in I_{j}} X_{i} = : X_{I_{j}}^{'}

, that is,

P (\sum_{i \in I_{j}} X_{i} \leq x) = F (\frac{x}{false| I_{j} false|}) = : F_{j}^{'} false(x false) for all x \in R, j = 1, \dots, m .

Moreover, we assume that model ambiguity is prevalent in form of an unknown copula between the individual subgroups. Along with the above argument, this is equivalent to the copula

C_{m}

(X_{I_{1}}^{'}, \dots, X_{I_{m}}^{'})

being unknown.…”

Section: Illustrations and Numerical Examplesmentioning

confidence: 73%

Value‐at‐Risk bounds with two‐sided dependence information

Lux¹,

Rüschendorf

2018

Mathematical Finance

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“…However, partial dependence information is often available (e.g., through knowledge of the variance of the sum). This case (and thus a possibly smaller dependence uncertainty spread VaR α (L + ) − VaR α (L + )), is studied by [5][6][7][8]21], where the RA also has been shown to be a useful tool.…”

Section: Introductionmentioning

confidence: 99%

Improved algorithms for computing worst Value-at-Risk

Hofert

Memartoluie

Saunders

et al. 2017

Statistics &Amp; Risk Modeling

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Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating worst Value-at-Risk for portfolios with arbitrary marginal loss distributions are given. In particular, a novel Adaptive Rearrangement Algorithm is introduced and investigated. These algorithms are implemented using the R package qrmtools and may be of interest in any context in which it is required to find columnwise permutations of a matrix such that the minimal (maximal) row sum is maximized (minimized).

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“…It is common to calculate

{\bar{VaR}}_{p} false(S_{n} false)

by numerical calculation and a popular algorithm is the Rearrangement Algorithm in Embrechts, Puccetti, and Rüschendorf (). If partial dependence information is available, one can study the values of risk measures in constrained subsets of

S_{n}

; see Bernard, Rüschendorf, and Vanduffel (), Bernard and Vanduffel (), and Bignozzi, Puccetti, and Rüschendorf () for research along this direction. In this paper, we focus on the full set

S_{n}

, that is, no dependence information.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

Cai

Liu

Wang

2016

Mathematical Finance

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In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

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Reducing model risk via positive and negative dependence assumptions

Cited by 39 publications

References 30 publications

Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk bounds with two‐sided dependence information

Improved algorithms for computing worst Value-at-Risk

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

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