A consistent estimator to the orthant-based tail value-at-risk
ESAIM: Probability and Statistics, Tome 22 (2018), pp. 163-177.

In this paper, we address the estimation of multivariate value-at-risk (VaR) and tail value-at-risk (TVaR). We recall definitions for the bivariate lower and upper orthant VaR and bivariate lower and upper orthant TVaR, presented in Cossette et al. [Eur. Actuar. J. 3 (2013) 321–357 or Methodol. Comput. Appl. Probab. (2014) 1–22]. Here, we present estimators for both these measures extended to an arbitrary dimension d ≥ 2 and establish the consistency of our estimator for the lower and upper orthant TVaR in any dimension. We demonstrate these results by providing numerical examples that compare our estimator to theoretical results for both simulated and real data.

DOI : 10.1051/ps/2018015
Classification : 62G05, 62G20, 62G32, 62H12, 62P05, 91B30
Mots clés : Multivariate estimators, risk measures, copulas.
Beck, Nicholas 1 ; Mailhot, Mélina 1

1 
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Beck, Nicholas; Mailhot, Mélina. A consistent estimator to the orthant-based tail value-at-risk. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 163-177. doi : 10.1051/ps/2018015. http://www.numdam.org/articles/10.1051/ps/2018015/

[1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk. Math. Finance 9 (1999) 203–228. | DOI | MR | Zbl

[2] O. Bardou, N. Frikha and G. Pagès, Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling. Monte Carlo Method. Appl. 15 (2009) 173–210. | DOI | MR | Zbl

[3] B. Brahimi, B. Fatah and Y. Djabrane, Copula conditional tail expectation for multivariate financial risks. Arab J Math. Seci. 24 (2018) 82–100. | MR | Zbl

[4] H. Cossette, M. Mailhot, É. Marceau and M. Mesfioui, Bivariate lower and upper orthant value-at-risk. Eur. Actuar. J. 3 (2013) 321–357. | DOI | MR | Zbl

[5] H. Cossette, M. Mailhot, É. Marceau and M. Mesfioui, Vector-valued tail value-at-risk and capital allocation. Methodol. Comput. Appl. Probab. 18 (2016) 653–674. | DOI | MR | Zbl

[6] A. Cousin and E. Di Bernardino, On multivariate extensions of value-at-risk. J. Multivar. Anal. 119 (2013) 32–46. | DOI | MR | Zbl

[7] A. Cuevas, W. González-Manteiga and A. Rodríguez-Casal, Plug-in estimation of general level sets. Aust. N. Z. J. Stat. 48 (2006) 7–19. | DOI | MR | Zbl

[8] E. Di Bernadino and T. Laloë, Estimating level sets of a distribution function using a plug-in method: a multidimensional extension. Preprint (2012). | arXiv

[9] E. Di Bernardino, T. Laloë, V. Maume-Deschamps and C. Prieur, Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory. ESAIM: Probab. Stat. 17 (2013) 236–256. | DOI | Numdam | MR | Zbl

[10] P. Embrechts and G. Puccetti, Bounds for functions of multivariate risks. J. Multivar. Anal. 97 (2006) 526–547. | DOI | MR | Zbl

[11] European Commission, Solvency II (2014), http://ec.europa.eu/finance/insurance/solvency/index_en.htm.

[12] E.W. Frees and E.A. Valdez, Understanding relationships using copulas. North Am. Actuar. J. 2 (1998) 1–25. | DOI | MR | Zbl

[13] E. Jouini, M. Meddeb and N. Touzi, Vector-valued coherent risk measures. Finance Stoch. 8 (2004) 531–552. | DOI | MR | Zbl

[14] E. Lépinette and I.B. Tahar, Vector-valued coherent risk measure processes. Int. J. Theor. Appl. Finance 17 (2014) 1450011. | DOI | MR | Zbl

[15] M. Mailhot and M. Mesfioui, Multivariate TVaR-based risk decomposition for vector-valued portfolios. Risks 4 (2016) 33. | DOI

[16] G. Mainik and E. Schaanning, On dependence consistency of covar and some other systemic risk measures. Stat. Risk Model. 31 (2014) 49–77. | DOI | MR | Zbl

[17] J. Nešlehová, On rank correlation measures for non-continuous random variables. J. Multivar. Anal. 98 (2007) 544–567. | DOI | MR | Zbl

[18] OSFI, Minimum capital test for federally regulated property and casualty insurance companies. (2015), http://www.osfi-bsif.gc.ca/Eng/fi-if/rg-ro/gdn-ort/gl-ld/Pages/mct2015.aspx.

[19] R.T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk (2000). | MR | Zbl

[20] R.T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions. J. Bank. Finance 26 (2002) 1443–1471. | DOI

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