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Risk aggregation and stochastic claims reserving in disability insurance
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0002-6608-0715
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2014 (English)In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 59, 100-108 p.Article in journal (Refereed) Published
Abstract [en]

We consider a large, homogeneous portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using a conditional law of large numbers, we establish the connection between claims reserving and risk aggregation for large portfolios. Further, we derive a partial differential equation for moments of present values. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for solving the PDEs very efficiently. Finally, we give a numerical example where moments of present values of disability annuities are computed using finite-difference methods and Monte Carlo simulations.

Place, publisher, year, edition, pages
2014. Vol. 59, 100-108 p.
Keyword [en]
Disability insurance, stochastic intensities, condition al independence, risk aggregation, stochastic claims reserving
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-136257DOI: 10.1016/j.insmatheco.2014.09.001ISI: 000347501100010Scopus ID: 2-s2.0-84907835403OAI: oai:DiVA.org:kth-136257DiVA: diva2:675643
Note

QC 20150209. Updated from manuscript to article in journal.

Available from: 2013-12-04 Created: 2013-12-04 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Stochastic modelling in disability insurance
Open this publication in new window or tab >>Stochastic modelling in disability insurance
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers related to the stochastic modellingof disability insurance. In the first paper, we propose a stochastic semi-Markovian framework for disability modelling in a multi-period discrete-time setting. The logistic transforms of disability inception and recovery probabilities are modelled by means of stochastic risk factors and basis functions, using counting processes and generalized linear models. The model for disability inception also takes IBNR claims into consideration. We fit various versions of the models into Swedish disability claims data.

In the second paper, we consider a large, homogeneous portfolio oflife or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic environment. Using a conditional law of large numbers, we establish the connection between risk aggregation and claims reserving for large portfolios. Further, we derive a partial differential equation for moments of present values. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for solving the PDEs very efficiently. Finally, we givea numerical example where moments of present values of disabilityannuities are computed using finite difference methods.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. 9 p.
Series
Trita-MAT, ISSN 1401-2286 ; 2013:02
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-134233 (URN)978-91-7501-964-2 (ISBN)
Presentation
2013-12-19, rum 3721, Institutionen för Matematik, Lindstedtsvägen 25, KTH, Stockholm, 15:15 (English)
Opponent
Supervisors
Note

QC 20131204

Available from: 2013-12-04 Created: 2013-11-20 Last updated: 2013-12-04Bibliographically approved
2. Topics in life and disability insurance
Open this publication in new window or tab >>Topics in life and disability insurance
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers, presented in Chapters A-E, on topics in life and disability insurance. It is naturally divided into two parts, where papers A and B discuss disability rates estimation based on historical claims data, and papers C-E discuss claims reserving, risk management and insurer solvency.In Paper A, disability inception and recovery probabilities are modelled in a generalized linear models (GLM) framework. For prediction of future disability rates, it is customary to combine GLMs with time series forecasting techniques into a two-step method involving parameter estimation from historical data and subsequent calibration of a time series model. This approach may in fact lead to both conceptual and numerical problems since any time trend components of the model are incoherently treated as both model parameters and realizations of a stochastic process. In Paper B, we suggest that this general two-step approach can be improved in the following way: First, we assume a stochastic process form for the time trend component. The corresponding transition densities are then incorporated into the likelihood, and the model parameters are estimated using the Expectation-Maximization algorithm.In Papers C and D, we consider a large portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using the Conditional Law of Large Numbers (CLLN), we establish the connection between claims reserving and risk aggregation for large portfolios. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for computing reserves and capital requirements efficiently. Paper C focuses on claims reserving and ultimate risk, whereas the focus of Paper D is on the one-year risks associated with the Solvency II directive.In Paper E, we consider claims reserving for life insurance policies with reserve-dependent payments driven by multi-state Markov chains. The associated prospective reserve is formulated as a recursive utility function using the framework of backward stochastic differential equations (BSDE). We show that the prospective reserve satisfies a nonlinear Thiele equation for Markovian BSDEs when the driver is a deterministic function of the reserve and the underlying Markov chain. Aggregation of prospective reserves for large and homogeneous insurance portfolios is considered through mean-field approximations. We show that the corresponding prospective reserve satisfies a BSDE of mean-field type and derive the associated nonlinear Thiele equation.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. 21 p.
Series
TRITA-MAT-A, 2015:09
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-175334 (URN)978-91-7595-701-2 (ISBN)
Public defence
2015-11-06, F3, Lindstedtsvägen 26, Kungliga Tekniska högskolan, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20151012

Available from: 2015-10-12 Created: 2015-10-12 Last updated: 2015-10-13Bibliographically approved

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Djehiche, Boualem

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