How Superadditive Can a Risk Measure Be?

Wang, Ruodu; Bignozzi, Valeria; Tsanakas, Andreas

doi:10.1137/140981046

Cited by 30 publications

(15 citation statements)

References 42 publications

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“…The following simple lemma is a combination of theorem 4.1 and corollary 4.2 of Wang et al. (). Lemma The smallest law‐invariant coherent risk measure dominating

ρ^{v}

exists, and it is given by

ρ^{v *} false(X false) = \underset{μ \in {scriptP}_{v}}{trueprefixsup} \{0true \int_{0}^{1} {ES}_{p} false(X false) d μ false(p false)\}, X \in L^{1},

where

{scriptP}_{v} = false{μ \in scriptP : v (μ) < + \infty false}

.…”

Section: Asymptotic Equivalence For Convex Risk Measuresmentioning

confidence: 86%

“…It was given in Wang et al. () for right continuous

h \in H

; however, from there, it is a simple exercise to see that the lemma holds for all

h \in H

. In the latter paper, it is also shown that

ρ_{h^{*}}

is the smallest law‐invariant coherent risk measure dominating

ρ_{h}

.…”

Section: Asymptotic Equivalence For Distortion Risk Measuresmentioning

confidence: 88%

“…() for VaR and ES, but also those in Wang et al. () for general risk measures in the homogeneous setting. More importantly, our methods unify the two streams of research in this field.…”

Section: Introductionmentioning

confidence: 95%

“…The main results in Wang, Bignozzi, and Tsanakas () imply that holds in the homogeneous model (

F_{1} = F_{2} = \dots

) if ρ is a distortion risk measure or a convex risk measure. The assumption of homogeneity is nice for mathematical analysis; however, it is not a realistic assumption for practical applications.…”

Section: Introductionmentioning

confidence: 96%

“…A significant mathematical challenge arises as the method in Wang et al. () relies on the study of the quantity

{normalΓ}_{ρ} false(X false) = \underset{n \to \infty}{trueprefixlim} \frac{1}{n} trueprefixsup false{ρ (S) : S \in S_{n} (F, \dots, F) false}, X \sim F,

which cannot be naturally generalized to an inhomogeneous setting. In this paper, we use an alternative method by constructing a specific

S_{n} \in {scriptS}_{n}

such that

ρ (S_{n})

and

ρ^{*} false(S_{n} false)

are close.…”

Section: Introductionmentioning

confidence: 99%

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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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“…The following simple lemma is a combination of theorem 4.1 and corollary 4.2 of Wang et al. (). Lemma The smallest law‐invariant coherent risk measure dominating

ρ^{v}

exists, and it is given by

ρ^{v *} false(X false) = \underset{μ \in {scriptP}_{v}}{trueprefixsup} \{0true \int_{0}^{1} {ES}_{p} false(X false) d μ false(p false)\}, X \in L^{1},

where

{scriptP}_{v} = false{μ \in scriptP : v (μ) < + \infty false}

.…”

Section: Asymptotic Equivalence For Convex Risk Measuresmentioning

confidence: 86%

“…It was given in Wang et al. () for right continuous

h \in H

; however, from there, it is a simple exercise to see that the lemma holds for all

h \in H

. In the latter paper, it is also shown that

ρ_{h^{*}}

is the smallest law‐invariant coherent risk measure dominating

ρ_{h}

.…”

Section: Asymptotic Equivalence For Distortion Risk Measuresmentioning

confidence: 88%

“…() for VaR and ES, but also those in Wang et al. () for general risk measures in the homogeneous setting. More importantly, our methods unify the two streams of research in this field.…”

Section: Introductionmentioning

confidence: 95%

“…The main results in Wang, Bignozzi, and Tsanakas () imply that holds in the homogeneous model (

F_{1} = F_{2} = \dots

Section: Introductionmentioning

confidence: 96%

“…A significant mathematical challenge arises as the method in Wang et al. () relies on the study of the quantity

{normalΓ}_{ρ} false(X false) = \underset{n \to \infty}{trueprefixlim} \frac{1}{n} trueprefixsup false{ρ (S) : S \in S_{n} (F, \dots, F) false}, X \sim F,

which cannot be naturally generalized to an inhomogeneous setting. In this paper, we use an alternative method by constructing a specific

S_{n} \in {scriptS}_{n}

such that

ρ (S_{n})

and

ρ^{*} false(S_{n} false)

are close.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

Cai

Liu

Wang

2016

Mathematical Finance

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Application Quantile-Based Risk Measures in Sector Portfolio Analysis—Warsaw Stock Exchange Approach

Trzpiot

2019

Effective Investments on Capital Markets

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Dual utilities on risk aggregation under dependence uncertainty

Wang

2019

Finance Stoch

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Finding the worst-case value of a preference over a set of plausible models is a well-established approach to address the issue of model uncertainty or ambiguity.In this paper, we study the worst-case evaluation of Yaari's dual utility functionals of an aggregate risk under dependence uncertainty along with its decision-theoretic implications. To arrive at our main findings, we introduce a technical notion of conditional joint mixability. Lower and upper bounds on dual utilities with dependence uncertainty are established and, in the presence of conditional joint mixability, they are shown to be exact bounds. Moreover, conditional joint mixability is indeed necessary for attaining these exact bounds when the distortion functions are strictly inverse-S-shaped. A particular economic implication of our results is what we call the pessimism effect. We show that a (generally non-convex/non-concave) dual utility-based decision maker under dependence uncertainty behaves as if she had a more pessimistic risk-averse dual utility but free of dependence uncertainty.

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How Superadditive Can a Risk Measure Be?

Cited by 30 publications

References 42 publications

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

Application Quantile-Based Risk Measures in Sector Portfolio Analysis—Warsaw Stock Exchange Approach

Dual utilities on risk aggregation under dependence uncertainty

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