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Stochastic Approximation Schemes for Economic Capital and Risk Margin Computations

David Barrera, Stéphane Crépey, Babacar Diallo, Gersende Fort, Emmanuel Gobet () and Uladzislau Stazhynski
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David Barrera: CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique
Stéphane Crépey: LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - INRA - Institut National de la Recherche Agronomique - UEVE - Université d'Évry-Val-d'Essonne - ENSIIE - CNRS - Centre National de la Recherche Scientifique
Babacar Diallo: LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - INRA - Institut National de la Recherche Agronomique - UEVE - Université d'Évry-Val-d'Essonne - ENSIIE - CNRS - Centre National de la Recherche Scientifique
Gersende Fort: IMT - Institut de Mathématiques de Toulouse UMR5219 - CNRS - Centre National de la Recherche Scientifique - PRES Université de Toulouse - UPS - Université Toulouse III - Paul Sabatier - Université Fédérale Toulouse Midi-Pyrénées - UT2J - Université Toulouse - Jean Jaurès - UT1 - Université Toulouse 1 Capitole
Emmanuel Gobet: CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique
Uladzislau Stazhynski: CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique

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Abstract: We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

New Economics Papers: this item is included in nep-cmp, nep-ias and nep-rmg
Date: 2019
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01710394
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Published in ESAIM: Proceedings and Surveys, EDP Sciences, In press

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