Dependence Modelling in Insurance via Copulas with Skewed Generalised Hyperbolic Marginals

Vitali Alexeev; Katja Ignatieva; Thusitha Liyanage

doi:10.1515/snde-2018-0094

Article

Dependence Modelling in Insurance via Copulas with Skewed Generalised Hyperbolic Marginals

Vitali Alexeev , Katja Ignatieva and Thusitha Liyanage

Published/Copyright: December 20, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Studies in Nonlinear Dynamics & Econometrics Volume 25 Issue 2

Abstract

This paper investigates dependence among insurance claims arising from different lines of business (LoBs). Using bivariate and multivariate portfolios of losses from different LoBs, we analyse the ability of various copulas in conjunction with skewed generalised hyperbolic (GH) marginals to capture the dependence structure between individual insurance risks forming an aggregate risk of the loss portfolio. The general form skewed GH distribution is shown to provide the best fit to univariate loss data. When modelling dependency between LoBs using one-parameter and mixture copula models, we favour models that are capable of generating upper tail dependence, that is, when several LoBs have a strong tendency to exhibit extreme losses simultaneously. We compare the selected models in their ability to quantify risks of multivariate portfolios. By performing an extensive investigation of the in- and out-of-sample Value-at-Risk (VaR) forecasts by analysing VaR exceptions (i.e. observations of realised portfolio value that are greater than the estimated VaR), we demonstrate that the selected models allow to reliably quantify portfolio risk. Our results provide valuable insights with regards to the nature of dependence and fulfils one of the primary objectives of the general insurance providers aiming at assessing total risk of an aggregate portfolio of losses when LoBs are correlated.

Keywords: copula; dependence modelling; insurance losses; skewed generalised hyperbolic distribution

JEL Classification: G22; C46; C15

Appendix: copula estimation

Generally, the maximum likelihood technique is applied to estimate parametric copulas. From (1), the density of the random vector X = (X ₁, …, X_d )^⊤ is given by

(22) f ( x 1 , … , x d ; δ 1 , … , δ d , θ ) = c { F X 1 ( x 1 ; δ 1 ) , … , F X d ( x d ; δ d ) ; θ } ∏ j = 1 d f j ( x j ; δ j ) ,

where

(23) c ( u 1 , … , u d ) = ∂ d C ( u 1 , … , u d ) ∂ u 1 … ∂ u d

denotes a copula density. Denoting a vector of parameters α = (δ ₁, …, δ _d, θ)^⊤ ∈ ℝ^d+1, the likelihood function is given by

(24) L ( α ; x 1 , … , x T ) = ∏ t = 1 T f ( x 1 , t , … , x d , t ; δ 1 , … , δ d , θ ) .

Combining (22) and (24), the corresponding log-likelihood function is given by

(25) ℓ ( α ; x 1 , … , x T ) = ∑ t = 1 T ln ⁡ [ c { F X 1 ( x 1 , t ; δ 1 ) , … , F X d ( x d , t ; δ d ) ; θ } ] + ∑ t = 1 T ∑ j = 1 d ln ⁡ [ f j ( x j , t ; δ j ) ] .

To maximize this log-likelihood numerically, we perform the inference for marginals (IFM) method, which is a sequential two-step maximum likelihood method, see e.g. McLeish and Small (1988) and Joe (1997). Parameters from the marginals are estimated in the first step as

(26) δ ^ j = arg max δ ⁡ ℓ j ( δ j ) ,

where

(27) ℓ j ( δ j ) = ∑ t = 1 T ln ⁡ f j [ x j , t ; δ j ]

denotes the log-likelihood function for each of the marginal distribution j = 1, …, d. Their estimates are then substituted into the copula to obtain the pseudo log-likelihood function

(28) ℓ ( θ , δ ^ 1 , … , δ ^ d ) = ∑ t = 1 T ln ⁡ [ c { F X 1 ( x 1 , t ; δ ^ 1 ) , … , F X d ( x d , t ; δ ^ d ) ; θ } ] ,

which is then maximised with respect to θ to obtain the estimator θ ^ . The estimates α ^ I F M = ( δ 1 ^ , … , δ d ^ , θ ^ ) ⊤ solve the first order condition

(29) ( ∂ ℓ 1 / ∂ δ 1 , … , ∂ ℓ d / ∂ δ d , ∂ ℓ / ∂ θ ) = 0.

References

Abramowitz, M., and I. A. Stegun. 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications.Search in Google Scholar

Akaike, H. 1974. “A New Look at the Statistical Model Identification.” IEEE Transactions on Automatic Control 19 (6): 716–723.10.1007/978-1-4612-1694-0_16Search in Google Scholar

Albrecher, H., J. Beirlant, and J. L. Teugels. 2017. Reinsurance: Actuarial and Statistical Aspects, Series: Statistics in Practice, 1st ed. Hoboken, NJ: John Wiley & Sons.10.1002/9781119412540Search in Google Scholar

Alemany, R., C. Bolancé, and M. Guillén. 2013. “A Nonparametric Approach to Calculating Value-at-Risk.” Insurance: Mathematics and Economics 52: 255–262.10.1016/j.insmatheco.2012.12.008Search in Google Scholar

Alm, J. 2016. “Signs of Dependence and Heavy Tails in Non-Life Insurance Data.” Scandinavian Actuarial Journal 10: 859–875.10.1080/03461238.2015.1017527Search in Google Scholar

Barndorff-Nielsen, O., and R. Stelzer. 1997. “Exponentially Decreasing Distributions for the Logarithm of Particle Size.” Proceedings of the Royal Society London A 353: 410–419.Search in Google Scholar

Barndorff-Nielsen, O., and R. Stelzer. 2005. “Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Levy-Processes.” Scandinavian Journal of Statistics 32 (4): 617–637.10.1111/j.1467-9469.2005.00466.xSearch in Google Scholar

Bernardi, M., A. Maruotti, and A. Petrella. 2012. “Skew Mixture Models for Loss Distributions: A Bayesian Approach.” Insurance: Mathematics and Economics 51: 617–623.10.1016/j.insmatheco.2012.08.002Search in Google Scholar

Blanco, C., and G. Ihle. 1999. “How Good is your VaR? Using Backtesting to Assess System Performance.” Financial Engineering News 11: 1–2.Search in Google Scholar

Bolancé, C., M. Guillén, E. Pelican, and R. Vernic. 2008. “Skewed Bivariate Models and Nonparametric Estimation for CTE Risk Measure.” Insurance: Mathematics and Economics 43: 386–393.10.1016/j.insmatheco.2008.07.005Search in Google Scholar

Bolancé, C., Z. Bahraoui, and M. Artís. 2014. “Quantifying the Risk using Copulae with Nonparametric Marginals.” Insurance: Mathematics and Economics 58: 46–56.10.1016/j.insmatheco.2014.06.008Search in Google Scholar

Bolancé, C., R. Alemany, and A. E. Padilla-Barreto. 2018. “Impact of D-Vine Structure on Risk Estimation.” Journal of Risk 20: 1–32.10.21314/JOR.2018.384Search in Google Scholar

Breymann, W., A. Dias, and P. Embrechts. 2003. “Dependence Structures for Multivariate High-Frequency Data in Finance.” Quantitative Finance 3: 1–14.10.1080/713666155Search in Google Scholar

Chavez-Demoulin, V., and P. Embrechts. 2010. Copulas in Insurance. New Jersey, USA: John Wiley & Sons.10.1002/9780470061602.eqf21007Search in Google Scholar

Cherubini, U., E. Luciano, and W. Vecchiato. 2004. Copula Methods in Finance. Southern Gate, Chichester, West Sussex, England: Wiley Finance Series.10.1002/9781118673331Search in Google Scholar

Christoffersen, P. 1998. “Evaluating Interval Forecasts.” International Economic Review 39: 841–862.10.2307/2527341Search in Google Scholar

Cossette, H., P. Gaillardetz, E. Marceau, and J. Rioux. 2002. “On Two Dependence Individual Risk Models.” Insurance: Mathematics and Economics 30: 153–166.10.1016/S0167-6687(02)00094-XSearch in Google Scholar

D’Agostino, R., and M. Stephens. 1986. Goodness-of-Fit Techniques, Statistics: A Series of Textbooks and Monographs, volume 68, 1st ed. Marcel Dekker.Search in Google Scholar

De Jong, P. 2012. “Modeling Dependence Between Loss Triangles.” North American Actuarial Journal 16: 74–86.10.1080/10920277.2012.10590633Search in Google Scholar

Denuit, M., C. Genest, and E. Marceau. 1999. “Stochastic Bounds on Sums of Dependent Risks.” Insurance: Mathematics and Economics 25: 85–104.10.1016/S0167-6687(99)00027-XSearch in Google Scholar

Denuit, M., J. Dhaene, and C. Ribas. 2001. “Does Positive Dependence Between Individual Risks Increase Stop-Loss Premiums?” Insurance: Mathematics and Economics 28: 305–308.10.1016/S0167-6687(00)00079-2Search in Google Scholar

Dias, A. 2004. Copula Inference for Finance and Insurance. Doctoral Thesis. Unpublished doctoral dissertation, Swiss Federal Institute of Technology (ETH), Switzerland.Search in Google Scholar

Dias, A., and P. Embrechts. 2010. “Modelling Exchange Rate Dependence at Different Time Horizons.” Journal of International Money and Finance 8: 1687–1705.10.1016/j.jimonfin.2010.06.004Search in Google Scholar

Diers, D., M. Eling, and S. D. Marek. 2012. “Dependence Modelling in Non-Life Insurance Using the Bernstein Copula.” Insurance: Mathematics and Economics 50: 430–436.10.1016/j.insmatheco.2012.02.007Search in Google Scholar

Dowd, K., and D. Blake. 2006. “After VaR: The Theory, Estimation, and Insurance Applications of Quantile-Based Risk Measures.” Journal of Risk and Insurance 73: 193–229.10.1111/j.1539-6975.2006.00171.xSearch in Google Scholar

Eberlein, E., and U. Keller. 1995. “Hyperbolic Distributions in Finance.” Bernoulli 1: 131–140.10.2307/3318481Search in Google Scholar

EIOPA. 2014. “The Underlying Assumptions in the Standard Formula for the Solvency Capital Requirement Calculation.” The European Insurance and Occupational Pensions Authority. https://eiopa.europa.eu/Publications/Standards/EIOPA-14-322_Underlying_Assumptions.pdf.Search in Google Scholar

Eling, M. 2012. “Fitting Insurance Claims to Skewed Distributions: Are the Skew-Normal and Skew-Student Good Models?” Insurance: Mathematics and Economics 51: 239–248.10.1016/j.insmatheco.2012.04.001Search in Google Scholar

Embrechts, P. 2009. “Copulas: A Personal View.” Journal of Risk and Insurance 76 (3): 639–650.10.1111/j.1539-6975.2009.01310.xSearch in Google Scholar

Embrechts, P., A. McNeil, and D. Straumann. 2001. “Correlation and Dependency in Risk Management: Properties and Pitfalls.” In Risk Management: Value at Risk and Beyond, edited by M. Dempster and H. Moffatt. Cambridge: University Press.10.1017/CBO9780511615337.008Search in Google Scholar

Embrechts, P., A. McNeil, and D. Straumann. 2002. “Correlation and Dependency in Risk Management: Properties and Pitfalls.” In Risk Management: Value at Risk and Beyond, edited by U. Press. Cambridge: M. Dempster and H. Moffatt.10.1017/CBO9780511615337.008Search in Google Scholar

Fang, H., K. Fang, and S. Kotz. 2002. “The Meta-Elliptical Distributions with Given Marginals.” Journal of Multivariate Analysis 82 (1): 1–16.10.1006/jmva.2001.2017Search in Google Scholar

Frahm, G., M. Junker, and A. Szimayer. 2003. “Elliptical Copulas: Applicability and Limitations.” Statistics and Probability Letters 63: 275–286.10.1016/S0167-7152(03)00092-0Search in Google Scholar

Genest, C., M. Gendron, and M. Bourdeau-Brien. 2009. “The Advent of Copulas in Finance.” The European Journal of Finance 15 (7–8): 609–618.10.1080/13518470802604457Search in Google Scholar

Hu, L. 2006. “Dependence Patterns Across Financial Markets: A Mixed Copula Approach.” Applied Financial Economics 16: 717–729.10.1080/09603100500426515Search in Google Scholar

Hult, H., and F. Lindskog. 2002. “Multivariate Extremes, Aggregation and Dependence in Elliptical Distributions.” Advances in Applied Probability 34 (3): 587–608.10.1239/aap/1033662167Search in Google Scholar

Ignatieva, K., and Z. Landsman. 2015. “Estimating the Tails of Loss Severity via Conditional Risk Measures for the Family of Symmetric Generalised Hyperbolic Distributions.” Insurance: Mathematics and Economics 65: 172–186.10.1016/j.insmatheco.2015.09.007Search in Google Scholar

Ignatieva, K., and Z. Landsman. 2019. “Conditional tail risk measures for the skewed generalised hyperbolic family." Insurance: Mathematics and Economics 86: 98–114.10.1016/j.insmatheco.2019.02.008Search in Google Scholar

Joe, H. 1993. “Parametric Families of Multivariate Distributions with Given Margins.” Journal of Multivariate Analysis 46 (2): 262–282.10.1006/jmva.1993.1061Search in Google Scholar

Joe, H. 1997. Multivariate Models and Dependence Concepts. London: Chapman & Hall.10.1201/9780367803896Search in Google Scholar

Jorion, P. 2007. Value-at-Risk. The New Benchmark for Managing Financial Risk. New York, USA: The McGraw-Hill Companies.Search in Google Scholar

Kaas, R., J. Dhaene, and M. Goovaerts. 2000. “Upper and Lower Bounds for Sums of Random Variables.” Insurance: Mathematics and Economics 27: 151–168.10.1016/S0167-6687(00)00060-3Search in Google Scholar

Lane, M. 2000. “Pricing Risk Transfer Transactions.” ASTIN Bulletin, The Journal of the International Actuarial Association 30 (2): 259–293.10.2143/AST.30.2.504635Search in Google Scholar

Lee, S. C. K., and S. X. Lin. 2012. “Modeling Dependent Risks with Multivariate Erlang Mixtures.” ASTIN Bulletin, The Journal of the International Actuarial Association 42: 153–180.Search in Google Scholar

Lindskog, F., A. McNeil, and U. Schmock. 2003. “Kendall’s Tau for Elliptical Distributions.” in Credit Risk: Contributions to Economics, 149–156. Springer, Berlin, Germany: Springer-Verlag.10.1007/978-3-642-59365-9_8Search in Google Scholar

Lopez, J. 1998. “Regulatory Evaluation of Value-at-Risk Models.” Economic Policy Review 4 (3): 119–124.10.21314/JOR.1999.005Search in Google Scholar

Madan, D., and E. Seneta. 1990. “The Variance Gamma Model for Share Market Returns.” Journal of Business 63: 511–524.10.1086/296519Search in Google Scholar

McLeish, D. L., and C. G. Small. 1988. The Theory and Applications of Statistical Inference Functions. Lecture Notes in Statistics, volume 44. New York: Springer-Verlag.10.1007/978-1-4612-3872-0Search in Google Scholar

McNeil, A. 1997. “Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory.” ASTIN Bulletin, The Journal of the International Actuarial Association 27 (1): 117–137.10.2143/AST.27.1.563210Search in Google Scholar

McNeil, A., R. Frey, and P. Embrechts. 2004. Quantitative Risk Management: Concept, Techniques and Tools. Princeton, USA: Princeton University Press.Search in Google Scholar

McNeil, A., R. Frey, and P. Embrechts. 2015. Quantitative Risk Management: Concepts, Techniques and Tools, Rrevised Edition. Princeton University Press.Search in Google Scholar

Merz, M., M. Wüthrich, and E. Hashorva. 2013. “Dependence Modelling in Multivariate Claims Run-Off Triangles.” Annals of Actuarial Science 7: 3–25.10.1017/S1748499512000140Search in Google Scholar

Miljkovic, T. and B. Grün. 2016. “Modeling Loss Data using Mixtures of Distributions.” Insurance: Mathematics and Economics 70: 387–396.10.1016/j.insmatheco.2016.06.019Search in Google Scholar

Nelsen, R. 2006. An Introduction to Copulas, 2nd ed. Portland, OR, USA: Springer.Search in Google Scholar

Praetz, P. D. 1972. “The Distribution of Share Price Changes.” Journal of Business 45: 49–55.10.1086/295425Search in Google Scholar

Salzmann, R., and M. V. Wüthrich. 2012. “Modelling Accounting Year Dependence in Runoff Triangles.” European Actuarial Journal 2: 227–242.10.1007/s13385-012-0055-3Search in Google Scholar

Stephens, M. 1974. “EDF Statistics for Goodness of Fit and Some Comparisons.” Journal of the American Statistical Association 69: 730–737.10.1080/01621459.1974.10480196Search in Google Scholar

Vernic, R. 2006. “Multivariate Skew-Normal Distributions with Applications in Insurance.” Insurance: Mathematics and Economics 38: 413–426.10.1016/j.insmatheco.2005.11.001Search in Google Scholar

Wu, F., A. Valdez, and M. Sherris. 2007. “Simulating from Exchangeable Archimedean Copulas.” Communications in Statistics – Simulation and Computation 36: 1019–1034.10.1080/03610910701539781Search in Google Scholar

Published Online: 2019-12-20

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/snde-2018-0094

Keywords for this article

copula; dependence modelling; insurance losses; skewed generalised hyperbolic distribution