Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

Albrecher, Hansjörg; Hipp, Christiane; Kortschak, Dominik

doi:10.1080/03461230902722726

Cited by 31 publications

(38 citation statements)

References 29 publications

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“…1−F (x) is of regular variation h ∈ RV −β for β ≥ 0, then as α → 1 one has for the inverse of the annual loss distribution the result (see [26] and [67]),…”

Section: The Journey Completes: Going From Compound Process Tail Asymmentioning

confidence: 99%

“…The second order single loss approximation as discussed in [65] and derived in [66] takes the form given by the Theorem 2.36, also see [67] [Proposition A3].…”

Section: Refinements and Second Order Single Risk Loss Process Asymptmentioning

confidence: 99%

“…• "Higher-Order" Single Loss Approximations: see discussions in [51] and recent summaries in [26] and references therein.…”

Section: Lemma 229 (Generalization To Insurance Mitigation)mentioning

confidence: 99%

“…We therefore then focus on key elements of heavy tailed loss process asymptotics with a view to understanding capital approximations. In doing so we draw together several disparate sources of information for practitioners from sources in both mathematics and risk literature, for example we consider results recently developed in actuarial literature for the heavy tailed case corresponding to the first order and second order asymptotic approximations, see comprehensive discussions in a general context in [26], [27] and the books, [28] and the forthcoming [29].…”

Section: A Brief Background On Loss Distributional Approach (Lda) Modelsmentioning

confidence: 99%

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Understanding operational risk capital approximations: First and second orders

Peters¹,

Targino²,

Shevchenko³

2013

JGR

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We set the context for capital approximation within the framework of the Basel II / III regulatory capital accords. This is particularly topical as the Basel III accord is shortly due to take effect. In this regard, we provide a summary of the role of capital adequacy in the new accord, highlighting along the way the significant loss events that have been attributed to the Operational Risk class that was introduced in the Basel II and III accords. Then we provide a semi-tutorial discussion on the modelling aspects of capital estimation under a Loss Distributional Approach (LDA). Our emphasis is to focuss on the important loss processes with regard to those that contribute most to capital, the so called "high consequence, low frequency" loss processes.This leads us to provide a tutorial overview of heavy tailed loss process modelling in OpRisk under Basel III, with discussion on the implications of such tail assumptions for the severity model in an LDA structure. This provides practitioners with a clear understanding of the features that they may wish to consider when developing OpRisk severity models in practice.From this discussion on heavy tailed severity models, we then develop an understanding of the impact such models have on the right tail asymptotics of the compound loss process and we provide detailed presentation of what are known as first and second order tail approximations for the resulting heavy tailed loss process. From this we develop a tutorial on three key families of risk measures and their equivalent second order asymptotic approximations: Value-at-Risk (Basel III industry standard); Expected Shortfall (ES) and the Spectral Risk Measure. These then form the capital approximations.We then provide a few example case studies to illustrate the accuracy of these asymptotic captial approximations, the rate of the convergence of the assymptotic result as a function of the LDA frequency and severity model parameters, the sensitivity of the capital approximation to the model parameters and the sensitivity to model miss-specification.Key words: Basel II/III; Capital Approximation; Loss Distributional Approach; Capital Approximation; Value-at-Risk; Expected Shortfall; Spectral Risk Measure; Subexponential; Regularly Varying. The Changing Landscape of Capital AccordsIn juristictions in which active regulation is applied to the banking sector throughout the world, the modelling of Operational Risk (OpRisk) has progressively taken a prominent place in financial quantitative measurement. This has occurred as a result of Basel II and now Basel III regulatory requirements. For example in the context of banking regulation in Australia, the basic framework of Basel II/III is summarized in Figure 1. In this juristiction one observes that a large amount of the developments in quantitative methodology for estimation of OpRisk capital, development of OpRisk frame- There have been both numerical and simulation based approaches adopted as well as analytical mathematical developments of closed form approximatio...

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“…1−F (x) is of regular variation h ∈ RV −β for β ≥ 0, then as α → 1 one has for the inverse of the annual loss distribution the result (see [26] and [67]),…”

Section: The Journey Completes: Going From Compound Process Tail Asymmentioning

confidence: 99%

“…The second order single loss approximation as discussed in [65] and derived in [66] takes the form given by the Theorem 2.36, also see [67] [Proposition A3].…”

Section: Refinements and Second Order Single Risk Loss Process Asymptmentioning

confidence: 99%

“…• "Higher-Order" Single Loss Approximations: see discussions in [51] and recent summaries in [26] and references therein.…”

Section: Lemma 229 (Generalization To Insurance Mitigation)mentioning

confidence: 99%

Section: A Brief Background On Loss Distributional Approach (Lda) Modelsmentioning

confidence: 99%

See 2 more Smart Citations

Understanding operational risk capital approximations: First and second orders

Peters¹,

Targino²,

Shevchenko³

2013

JGR

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“…In Section 2 a refined analysis of singularities in the complex domain is used to establish an integral representation of ψ(u) for Pareto claim sizes with distribution function (2) and arbitrary positive parameter a > 1. Two methods of proof are provided.…”

Section: Introductionmentioning

confidence: 99%

On ruin probability and aggregate claim representations for Pareto claim size distributions

Albrecher

Kortschak

2009

Insurance: Mathematics and Economics

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We generalize an integral representation for the ruin probability in a Crámer-Lundberg risk model with shifted (or also called US-)Pareto claim sizes, obtained by Ramsay [14], to classical Pareto(a) claim size distributions with arbitrary real values a > 1 and derive its asymptotic expansion. Furthermore an integral representation for the tail of compound sums of Pareto-distributed claims is obtained and numerical illustrations of its performance in comparison to other aggregate claim approximations are provided.

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