Multivariate risk measures based on conditional expectation and systemic risk for exponential dispersion models. (English) Zbl 1446.91073
Summary: Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event”. We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.
MSC:
| 91G05 | Actuarial mathematics |
| 91G70 | Statistical methods; risk measures |
| 91G45 | Financial networks (including contagion, systemic risk, regulation) |
Keywords:
multivariate risk measures; conditional expectation; systemic risks; capital allocation; exponential dispersion modelsSoftware:
QRMReferences:
| [1] | Acharya, V. V.; Pedersen, L. H.; Philippon, T.; Richardson, M., Measuring systemic risk, Rev. Financ. Stud., 30, 2-47 (2017) |
| [2] | Alai, D. H.; Landsman, Z.; Sherris, M., A multivariate Tweedie lifetime model: Censoring and truncation, Insurance Math. Econom., 64, 203-213 (2015) · Zbl 1348.62230 |
| [3] | Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Math. Finance, 9, 203-228 (1999) · Zbl 0980.91042 |
| [4] | Bäuerle, N.; Shushi, T., Risk management with tail conditional certainty equivalents (2019), arXiv preprint arXiv:1902.06941 |
| [5] | Biagini, F.; Fouque, J. P.; Frittelli, M.; Meyer-Brandis, T., A unified approach to systemic risk measures via acceptance sets, Math. Finance, 29, 1, 329-367 (2019) · Zbl 1411.91633 |
| [6] | Blough, D. K.; Madden, C. W.; Hornbrook, M. C., Modeling risk using generalized linear models, J. Health Econ., 18, 153-171 (1999) |
| [7] | Brownlees, C. T.; Engle, R. F., SRISK: A conditional capital shortfall measure of systemic risk, Rev. Financ. Stud., 30, 1, 48-79 (2017) |
| [8] | Butler, R. W.; Wood, A. T., Saddlepoint approximation for moment generating functions of truncated random variables, Ann. Statist., 32, 2712-2730 (2004) · Zbl 1068.62015 |
| [9] | Cai, J.; Wang, Y.; Mao, T., Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures, Insurance Math. Econom., 75, 105-116 (2017) · Zbl 1394.91197 |
| [10] | Cossette, H.; Mailhot, M.; Marceau, E.; Mesfioui, M., Vector-valued tail value-at-risk and capital allocation, Methodol. Comput. Appl. Probab., 18, 3, 653-674 (2016) · Zbl 1349.91319 |
| [11] | Cousin, A.; Di Bernardino, E., On multivariate extensions of value-at-risk, J. Multivariate Anal., 119, 32-46 (2013) · Zbl 1287.91090 |
| [12] | Cousin, A.; Di Bernardino, E., On multivariate extensions of conditional-tail-expectation, Insurance Math. Econom., 55, 272-282 (2014) · Zbl 1296.91149 |
| [13] | De Jong, P.; Heller, G. Z., Generalized Linear Models for Insurance Data (2008), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1142.91046 |
| [14] | Denuit, M.; Dhaene, J.; Goovaerts, M. J.; Kaas, R., Actuarial Theory for Dependent Risks: Measures, Orders, and Models (2005), John Wiley & Sons |
| [15] | Detlefsen, K.; Scandolo, G., Conditional and dynamic convex risk measures, Finance Stoch., 9, 539-561 (2005) · Zbl 1092.91017 |
| [16] | Dhaene, J.; Laeven, R. J.A.; Zhang, Y., Systemic risk: conditional distortion risk measures (2019), arXiv:1901.04689 |
| [17] | Di Bernardino, E.; Fernández-Ponce, J. M.; Palacios-Rodríguez, F.; Rodríguez-Griñolo, M. R., On multivariate extensions of the conditional value-at-risk measure, Insurance Math. Econom., 61, 1-16 (2015) · Zbl 1314.91243 |
| [18] | Dickson, D. C., Insurance Risk and Ruin (2016), Cambridge University Press · Zbl 1060.91078 |
| [19] | Feinstein, Z.; Rudloff, B.; Weber, S., Measures of systemic risk, SIAM J. Financial Math., 8, 672-708 (2017) · Zbl 1407.91284 |
| [20] | Jørgensen, B., Exponential dispersion models, J. R. Stat. Soc. Ser. B Stat. Methodol., 127-162 (1987) · Zbl 0662.62078 |
| [21] | Jouini, E.; Meddeb, M.; Touzi, N., Vector-valued coherent risk measures, Finance Stoch., 8, 531-552 (2004) · Zbl 1063.91048 |
| [22] | Klebaner, F.; Landsman, Z.; Makov, U.; Yao, J., Optimal portfolios with downside risk, Quant. Finance, 1-11 (2017) |
| [23] | Kleinow, J.; Moreira, F.; Strobl, S.; Vahamaa, S., Measuring systemic risk: A comparison of alternative market-based approaches, Finance Res. Lett., 40-46 (2017) |
| [24] | Laeven, R. J.; Stadje, M., Entropy coherent and entropy convex measures of risk, Math. Oper. Res., 38, 2, 265-293 (2013) · Zbl 1297.91090 |
| [25] | Landsman, Z.; Makov, U.; Shushi, T., Multivariate tail conditional expectation for elliptical distributions, Insurance Math. Econom., 70, 216-223 (2016) · Zbl 1373.62523 |
| [26] | Landsman, Z.; Valdez, E. A., Tail conditional expectations for exponential dispersion models, ASTIN Bull.: J. IAA, 35, 189-209 (2005) · Zbl 1099.62122 |
| [27] | Mainik, G.; Schaanning, E., On dependence consistency of CoVaR and some other systemic risk measures, Stat. Risk Model., 31, 1, 49-77 (2014) · Zbl 1305.91248 |
| [28] | McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques and Tools, Vol. 3 (2005), Princeton university press: Princeton university press Princeton · Zbl 1089.91037 |
| [29] | Merakli, M.; Kucukyavuz, S., Vector-valued multivariate conditional value-at-risk, Oper. Res. Lett., 46, 3, 300-305 (2018) · Zbl 1525.91183 |
| [30] | Shi, P.; Feng, X.; Boucher, J. P., Multilevel modeling of insurance claims using copulas, Ann. Appl. Stat., 10, 834-863 (2016) · Zbl 1400.62238 |
| [31] | Shushi, T., The generalized exponential family of distributions and its characteristics, Comm. Statist. Theory Methods, 7, 1-7 (2017) |
| [32] | Smyth, G. K.; Jørgensen, B., Fitting Tweedie’s compound Poisson model to insurance claims data: dispersion modelling, ASTIN Bull.: J. IAA, 32, 143-157 (2002) · Zbl 1094.91514 |
| [33] | Torres, R.; Lillo, R. E.; Laniado, H., A directional multivariate value at risk, Insurance Math. Econom., 65, 111-123 (2015) · Zbl 1348.91295 |
| [34] | Valdez, E. A.; Dhaene, J.; Maj, M.; Vanduffel, S., Bounds and approximations for sums of dependent log-elliptical random variables, Insurance Math. Econom., 44, 385-397 (2009) · Zbl 1162.91440 |
| [35] | Zhou, M.; Dhaene, J.; Yao, J., An approximation method for risk aggregations and capital allocation rules based on additive risk factor models, Insurance Math. Econom., 79, 92-100 (2019) · Zbl 1401.91218 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
