Estimation of the Marginal Expected Shortfall: the Mean When a Related Variable is Extreme

Cai, Juan-Juan; Einmahl, J.H.J.; Haan, Leo de; Zhou, Chen

doi:10.1111/rssb.12069

Cited by 83 publications

(142 citation statements)

References 22 publications

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“…Note that, in applications, extreme losses correspond to tail probabilities

τ_{n}

at an extremely high level that can be even larger than 1−1/ n ; see for instance Embrechts and Puccetti () who studied extreme operational bank losses, Cai et al . () for an application to extreme loss returns of banks in the US market and El Methni and Stupfler (2017a, b) who estimated excess‐of‐loss risk measures on automobile insurance data and the average value of a catastrophic loss in commercial fire risk. Therefore, estimating the corresponding quantile‐based risk measures is a typical extreme value problem.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Tail Risk Based on Extreme Expectiles

Daouia

Girard

Stupfler³

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

106

144

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We use tail expectiles to estimate alternative measures to the Value at Risk (VaR) and Marginal Expected Shortfall (MES), two instruments of risk protection of utmost importance in actuarial science and statistical finance. The concept of expectiles is a least squares analogue of quantiles. Both are M-quantiles as the minimizers of an asymmetric convex loss function, but expectiles are the only M-quantiles that are coherent risk measures. Moreover, expectiles define the only coherent risk measure that is also elicitable. The estimation of expectiles has not, however, received any attention yet from the perspective of extreme values. Two estimation methods are proposed here, either making use of quantiles or relying directly on least asymmetrically weighted squares. A main tool is to first estimate large values of expectile-based VaR and MES located within the range of the data, and then to extrapolate the obtained estimates to the very far tails. We establish the limit distributions of both of the resulting intermediate and extreme estimators. We show via a detailed simulation study the good performance of the procedures, and present concrete applications to medical insurance data and three large US investment banks.

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“…Note that, in applications, extreme losses correspond to tail probabilities

τ_{n}

Section: Introductionmentioning

confidence: 99%

Estimation of Tail Risk Based on Extreme Expectiles

Daouia

Girard

Stupfler³

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

106

144

View full text Add to dashboard Cite

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“…Secondly, using an extrapolation method based on proposition by Cai et al () applied to the bivariate vector ( X i , Z ) and Equation , we have that, for n 1 p = o ( k ) as n 1 → ∞ ,

θ_{p}^{i} \sim \frac{U_{i} false(1 false/ p false)}{U_{i} false(n_{1} false/ k false)} θ_{\frac{k}{n_{1}}}^{i} \sim {((), \frac{k}{n_{1} p})}^{γ^{i}} θ_{\frac{k}{n_{1}}}^{i} .

…”

Section: Resultsmentioning

confidence: 99%

“…Asymptotic dependence seems to be a necessary condition, as illustrated in Section 4.3, where we provide an example for which the assumption of asymptotic dependence is violated. The interested reader is also referred to Cai et al (). The assumption γ i ∈ (0,1/2) is necessary for Theorem . A careful reading of the proof of auxiliary Proposition shows that the result does not hold true anymore when γ i = 1/2.…”

Section: Resultsmentioning

confidence: 99%

“…Specific values for k ( n 1 ) and k i ( n 1 ) are chosen for each sample size n 1 , according to the selection procedure described by Cai et al () Indeed, a usual practice to choose the intermediate sequence k i is to select a range corresponding to the first stable region of the estimator

{\overset{γ}{true}}^{i}

. Then, to gain stability, we average the estimations

{\overset{γ}{true}}^{i}

corresponding to k i ( n 1 ) in the selected range.…”

Section: Simulation Studymentioning

confidence: 99%

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Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets

Bernardino

Prieur

2018

Environmetrics

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This paper deals with the problem of estimating the multivariate version of the conditional tail expectation introduced in the recent literature. We propose a new semiparametric estimator for this risk measure, essentially based on statistical extrapolation techniques, well designed for extreme risk lev els. We prove a central limit theorem for the obtained estimator. We illustrate the practical properties of our estimator on simulations. The performances of our new estimator are discussed and compared with the ones of the empirical Kendall's process‐based estimator, previously proposed by the authors. We conclude with two applications on real data sets: rainfall measurements recorded at three stations located in the south of Paris (France) and the analysis of strong wind gusts in the northwest of France.

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“…For instance, in order to estimate expectations of the form (3.1) in a bivariate set-up, [9] assume that large values of g(X) correspond to high values of h(X) (in their case h(X) = X i ). Under these constraints, the authors use results from Extreme Value Theory (EVT) to derive an estimator of (3.1) and study some of its properties.…”

Section: Simulation Methods To Calculate Conditional Expectations Of mentioning

confidence: 99%

Sequential Monte Carlo Samplers for Capital Allocation Under Copula-Dependent Risk Models

2014

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In this paper we assume a multivariate risk model has been developed for a portfolio and its capital derived as a homogeneous risk measure. The Euler (or gradient) principle, then, states that the capital to be allocated to each component of the portfolio has to be calculated as an expectation conditional to a rare event, which can be challenging to evaluate in practice. We exploit the copuladependence within the portfolio risks to design a Sequential Monte Carlo Samplers based estimate to the marginal conditional expectations involved in the problem, showing its efficiency through a series of computational examples.

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Estimation of the Marginal Expected Shortfall: the Mean When a Related Variable is Extreme

Cited by 83 publications

References 22 publications

Estimation of Tail Risk Based on Extreme Expectiles

Estimation of Tail Risk Based on Extreme Expectiles

Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets

Sequential Monte Carlo Samplers for Capital Allocation Under Copula-Dependent Risk Models

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