Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Abstract. In this paper we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy-tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

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“…Note that by the Karamata theorem, H(u) = 1 − H(u) is regularly varying with exponent −φ + 1; see [19]. Then by Proposition 4.5.1 in [8],…”

Section: Infinite Dimensional Phase-type Distributionsmentioning

confidence: 98%

“…Estimates of ruin probabilities for reserves u larger than those that could be handled by the proposed recursion scheme, might be carried out by the asymptotic expansions developed by [8]. We re- …”

Section: Infinite Dimensional Phase-type Distributionsmentioning

confidence: 99%

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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“…We write F = 1 − F for the survival function of F and I{·} for the indicator function, and denote by ⌈α⌉ the smallest integer l such that α ≤ l. In order to derive higher-order tail asymptotics of S n (c) we shall assume that F is a smoothly varying function, defined below as in Barbe and McCormick [7]. Definition 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

Tail asymptotic expansions for L-statistics

2014

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“…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%

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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Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Cited by 28 publications

References 46 publications

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Tail asymptotic expansions for L-statistics

Understanding operational risk capital approximations: First and second orders

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