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COMMON SHOCK MODELS FOR CLAIM ARRAYS

Published online by Cambridge University Press:  08 June 2018

Benjamin Avanzi ,
Greg Taylor  and
Bernard Wong
Benjamin Avanzi
Affiliation:
School of Risk and Actuarial Studies, UNSW Sydney Business School, UNSW Sydney, NSW 2052, Australia Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Quebec H3T 1J4, Canada, E-Mail: [email protected]
Greg Taylor*
Affiliation:
School of Risk and Actuarial Studies, UNSW Sydney Business School, UNSW Sydney, NSW 2052, Australia
Bernard Wong
Affiliation:
School of Risk and Actuarial Studies, UNSW Sydney Business School, UNSW Sydney, NSW 2052, Australia, E-Mail: [email protected]
*
E-Mail: [email protected]
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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.

Keywords

Claim arrayclaim dependencycommon shockcorrelationloss reservingG22C52C55
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 1109 - 1136
Copyright
Copyright © Astin Bulletin 2018 

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References

Abdallah, A., Boucher, J.-P. and Cossette, H. (2015) Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bulletin, 45 (3), 577–599.Google Scholar
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), 184–200.Google Scholar
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, 63–78.Google Scholar
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), 225–263.Google Scholar
Braun, C. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34, 399–423.Google Scholar
De Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10 (2), 28–38.Google Scholar
De Jong, P. (2012) Modeling dependence between loss triangles. North American Actuarial Journal, 16 (1), 74–86.Google Scholar
Happ, S., Maier, R. and Merz, M. (2014) Multivariate Bühlmann–Straub credibility model applied to claims reserving for correlated run-off triangles. Variance, 8, 23–42.Google Scholar
Hess, K.T., Schmidt, K.D. and Zocher, M. (2006) Multivariate loss prediction in the multivariate additive model. Insurance: Mathematics and Economics, 39 (2), 185–191.Google Scholar
IAA (2009) Note on Enterprise Risk Management for Capital and Solvency Purposes in the Insurance Industry. Sydney, Australia: International Actuarial Association.Google Scholar
Jørgensen, B. (1997) The theory of dispersion models. In Monographs on Statistics and Applied Probability, vol. 76. London: Chapman & Hall.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J. (2008) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika, 95 (4), 987–991.Google Scholar
Luenberger, D. (2014) Investment Science. Oxford, UK: Oxford University Press.Google Scholar
Merz, M., Wüthrich, M.V. and Hashorva, E. (2013) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7, 3–25.Google Scholar
Meyers, G.G. (2007) The common shock model for correlated insurance losses. Variance, 1 (1), 40–52.Google Scholar
Meyers, G.G. (2016, Winter) Dependencies in stochastic loss reserve models. Casualty Actuarial Society Forum.Google Scholar
Renshaw, A. (1989) Chain ladder and interactive modelling (claims reserving and GLIM). Journal of the Institute of Actuaries, 116, 559–587.Google Scholar
Sharpe, W.F. (1963) A simplified model for portfolio analysis. Management Science, 9 (2), 277–293.Google Scholar
Shi, P. (2014) A copula regression for modeling multivariate loss triangles and quantifying reserving variability. ASTIN Bulletin, 44 (1), 85–102.Google Scholar
Shi, P., Basu, S. and Meyers, G.G. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16 (1), 29–51.Google Scholar
Shi, P. and Frees, E.W. (2011) Dependent loss reserving using copulas. ASTIN Bulletin, 41 (2), 449–486.Google Scholar
Taylor, G. (2000) Loss reserving: An actuarial perspective. In Huebner International Series on Risk, Insurance and Economic Security. Dordrecht, Netherlands: Kluwer Academic Publishers.Google Scholar
Taylor, G. (2017) Existence and uniqueness of chain ladder solutions. ASTIN Bulletin, 47 (1), 1–41.Google Scholar
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), 70–88.Google Scholar
Tweedie, M. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions, pp. 579–604.Google Scholar
Wüthrich, M. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester, UK: John Wiley & Sons.Google Scholar
Wüthrich, M.V. and Merz, M. (2015) Stochastic claims reserving manual: Advances in dynamic modeling. SSRN Manuscript no. 2649057.Google Scholar
Zhang, Y. (2010) A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46 (3), 588–599.Google Scholar
Zhang, Y. and Dukic, V. (2013) Predicting multivariate insurance loss payments under the Bayesian copula framework. Journal of Risk and Insurance, 80 (4), 891–919.Google Scholar