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COMMON SHOCK MODELS FOR CLAIM ARRAYS

  • Benjamin Avanzi (a1) (a2), Greg Taylor (a1) and Bernard Wong (a3)
Abstract

The paper is concerned with multiple claim arrays. In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, we develop a methodology for the construction of such correlation structures (to any dimension). Indeed, the literature does not document any methodology by which practitioners, who often parameterise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner.

We construct a broad and flexible family of models, where dependency is induced by common shock components. Models incorporate dependencies between observations both within arrays and between arrays. Arrays are of general shape (possibly with holes), but include the usual cases of claim triangles and trapezia that appear in the literature. General forms of dependency are considered with cell-, row-, column-, diagonal-wise, and other forms of dependency as special cases. Substantial effort is applied to practical interpretation of such matrices generated by the models constructed here.

Reasonably realistic examples are examined, in which an expression is obtained for the general entry in the correlation matrix in terms of a limited set of parameters, each of which has a straightforward intuitive meaning to the practitioner. This will maximise chance of obtaining a reliable matrix. This construction is illustrated by a numerical example.

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Abdallah, A., Boucher, J.-P. and Cossette, H. (2015) Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bulletin, 45 (3), 577599.
Abdallah, A., Boucher, J.-P., Cossette, H. and Trufin, J. (2016) Sarmanov family of bivariate distributions for multivariate loss reserving analysis. North American Actuarial Journal, 20 (2), 184200.
Avanzi, B., Taylor, G.C., Vu, P.A. and Wong, B. (2016a) Stochastic loss reserving with dependence: A flexible multivariate Tweedie approach. Insurance: Mathematics and Economics, 71, 6378.
Avanzi, B., Taylor, G.C. and Wong, B. (2016b) Correlations between insurance lines of business: An illusion or a real phenomenon? Some methodological considerations. ASTIN Bulletin, 46 (2), 225263.
Braun, C. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34, 399423.
De Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10 (2), 2838.
De Jong, P. (2012) Modeling dependence between loss triangles. North American Actuarial Journal, 16 (1), 7486.
Happ, S., Maier, R. and Merz, M. (2014) Multivariate Bühlmann–Straub credibility model applied to claims reserving for correlated run-off triangles. Variance, 8, 2342.
Hess, K.T., Schmidt, K.D. and Zocher, M. (2006) Multivariate loss prediction in the multivariate additive model. Insurance: Mathematics and Economics, 39 (2), 185191.
IAA (2009) Note on Enterprise Risk Management for Capital and Solvency Purposes in the Insurance Industry. Sydney, Australia: International Actuarial Association.
Jørgensen, B. (1997) The theory of dispersion models. In Monographs on Statistics and Applied Probability, vol. 76. London: Chapman & Hall.
Kuang, D., Nielsen, B. and Nielsen, J. (2008) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika, 95 (4), 987991.
Luenberger, D. (2014) Investment Science. Oxford, UK: Oxford University Press.
Merz, M., Wüthrich, M.V. and Hashorva, E. (2013) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7, 325.
Meyers, G.G. (2007) The common shock model for correlated insurance losses. Variance, 1 (1), 4052.
Meyers, G.G. (2016, Winter) Dependencies in stochastic loss reserve models. Casualty Actuarial Society Forum.
Renshaw, A. (1989) Chain ladder and interactive modelling (claims reserving and GLIM). Journal of the Institute of Actuaries, 116, 559587.
Sharpe, W.F. (1963) A simplified model for portfolio analysis. Management Science, 9 (2), 277293.
Shi, P. (2014) A copula regression for modeling multivariate loss triangles and quantifying reserving variability. ASTIN Bulletin, 44 (1), 85102.
Shi, P., Basu, S. and Meyers, G.G. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16 (1), 2951.
Shi, P. and Frees, E.W. (2011) Dependent loss reserving using copulas. ASTIN Bulletin, 41 (2), 449486.
Taylor, G. (2000) Loss reserving: An actuarial perspective. In Huebner International Series on Risk, Insurance and Economic Security. Dordrecht, Netherlands: Kluwer Academic Publishers.
Taylor, G. (2017) Existence and uniqueness of chain ladder solutions. ASTIN Bulletin, 47 (1), 141.
Taylor, G. and McGuire, G. (2007) A synchronous bootstrap to account for dependencies between lines of business in the estimation of loss reserve prediction error. North American Actuarial Journal, 11 (3), 7088.
Tweedie, M. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions, pp. 579–604.
Wüthrich, M. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester, UK: John Wiley & Sons.
Wüthrich, M.V. and Merz, M. (2015) Stochastic claims reserving manual: Advances in dynamic modeling. SSRN Manuscript no. 2649057.
Zhang, Y. (2010) A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46 (3), 588599.
Zhang, Y. and Dukic, V. (2013) Predicting multivariate insurance loss payments under the Bayesian copula framework. Journal of Risk and Insurance, 80 (4), 891919.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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