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Catastrophe, Ruin and Death - Some Perspectives on Insurance Mathematics
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0001-9746-0756
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis gives some perspectives on insurance mathematics related to life insurance and / or reinsurance. Catastrophes and large accidents resulting in many lost lives are unfortunatley known to happen over and over again. A new model for the occurence of catastrophes is presented; it models the number of catastrophes, how many lives that are lost, how many lost lives that are insured by a specific insurer and the cost of the resulting claims, this  makes it possible to calculate the price of reinsurance contracts linked to catastrophic events. 

Ruin is the result if claims exceed inital capital and the premiums collected by an insurance company. We analyze the Cramér-Lundberg approximation for the ruin probability and give an explicit rate of convergence in the case were claims are bounded by some upper limit.

Death is known to be the only thing that is certain in life. Individual life spans are however random, models for and statistics of mortality are imortant for, amongst others, life insurance companies whose payments ultimatley depend on people being alive or dead. We analyse the stochasticity of mortality and perform a variance decomposition were the variation in mortality data is either explained by the covariates age and time, unexplained systematic variation or random noise due to a finite population. We suggest a mixed regression model for mortality and fit it to data from the US and Sweden, including prediction intervals of future mortalities.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2014. , 36 p.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-103165ISBN: 978-91-7447-935-5 (print)OAI: oai:DiVA.org:su-103165DiVA: diva2:716005
Public defence
2014-06-05, room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: In press. Paper 4: Submitted.

Available from: 2014-05-14 Created: 2014-05-07 Last updated: 2014-05-09Bibliographically approved
List of papers
1. Pricing catastrophe risk in life (re)insurance
Open this publication in new window or tab >>Pricing catastrophe risk in life (re)insurance
2014 (English)In: Scandinavian Actuarial Journal, ISSN 0346-1238, E-ISSN 1651-2030, Vol. 2014, no 4, 352-367 p.Article in journal (Refereed) Published
Abstract [en]

What is the catastrophe risk a life insurance company faces? What is the correct price of a catastrophe cover? During a review of the current standard model, due to Strickler, we found that this model has some serious shortcomings. We therefore present a new model for the pricing of catastrophe excess of loss cover (Cat XL). The new model for annual claim cost C is based on a compound Poisson processof catastrophe costs. To evaluate the distribution of the cost of each catastrophe, we use the Peaks Over Threshold model for the total number of lost lives in each catastrophe and the beta binomial model for the proportion of these corresponding to customers of the insurance company. To be able to estimate the parameters of the model, international and Swedish data were collected and compiled,listing accidents claiming at least twenty and four lives, respectively. Fitting the new model to data, we find the fit to be good. Finally we give the price of a Cat XL contract and perform a sensitivity analysis of how some of the parameters affect the expected value and standard deviation of the cost and thus the price.

Place, publisher, year, edition, pages
London: Taylor & Francis, 2014
Keyword
catastrophe excess of loss; life reinsurance; catastrophe model; catastrophe data; Cat XL; POT-model; Solvency II; internal models
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-103151 (URN)10.1080/03461238.2012.695747 (DOI)000333884000004 ()
Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2017-12-05Bibliographically approved
2. Coupling and Explicit Rate of Convergence in Cramer-Lundberg Approximation for Reinsurance Risk Processes
Open this publication in new window or tab >>Coupling and Explicit Rate of Convergence in Cramer-Lundberg Approximation for Reinsurance Risk Processes
2011 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 40, no 19-20, 3524-3539 p.Article in journal (Refereed) Published
Abstract [en]

A classical result in risk theory is the Cramer-Lundberg approximation which says that under some general conditions the exponentially normalized ruin probability converges. In this article, we state an explicit rate of convergence for the Cramer-Lundberg approximation for ruin probabilities in the case where claims are bounded, which is realistic for, e. g., reinsurance models. The method, used to get the corresponding results, is based on renewal and coupling arguments.

Keyword
Coupling method, Cramer-Lundberg approximation, Rate of convergence, Reinsurance risk process, Reinsurance ruin probability
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-66850 (URN)10.1080/03610926.2011.581176 (DOI)000294892100011 ()
Note
authorCount :2Available from: 2011-12-27 Created: 2011-12-21 Last updated: 2017-12-08Bibliographically approved
3. Analysis of the Stochasticity of Mortality Using Variance Decomposition
Open this publication in new window or tab >>Analysis of the Stochasticity of Mortality Using Variance Decomposition
2014 (English)In: Modern Problems in Insurance Mathematics / [ed] Dmitri Silvestrov and Anders Martin-Löf, Springer Publishing Company, 2014, 199-222 p.Chapter in book (Refereed)
Abstract [en]

We analyse the stochasticity in mortality data from the USA, the UK and Sweden, and in particular to which extent mortality rates are explained by systematic variation, due to various risk factors, and random noise. We formalise this in terms of a mixed regression model with a logistic link function, and decomposethe variance of the observations into three parts: binomial risk, the variance due to random mortality variation in a finite population, systematic risk explained by the covariates and unexplained systematic risk, variance that comes from real changes in mortality rates, not captured by the covariates. The fraction of unexplained variance caused by binomial risk provides a limit in terms of the resolution that can be achieved by a model. This can be used as a model selection tool for selecting the number of covariates and regression parameters of the deterministic part of the regression function, and for testing whether unexplained systematic variation should be explicitly modelled or not. We use a two-factor model with ageand calendar year as covariates, and perform the variance decomposition for a simple model with a linear time trend on the logit scale. The population size turns out to be crucial, and for Swedish data, the simple model works surprisingly well, leaving only a small fraction of unexplained systematic risk, whereas for the UK and the USA, the amount of unexplained systematic risk is larger, so that more elaborate models might work better.

Place, publisher, year, edition, pages
Springer Publishing Company, 2014
Series
EAA, 7879
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-103153 (URN)10.1007/978-3-319-06653-0_13 (DOI)9783319066523 (ISBN)
Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2014-07-03Bibliographically approved
4. Multivariate Time Series Modeling, Estimation and Prediction of Mortalities
Open this publication in new window or tab >>Multivariate Time Series Modeling, Estimation and Prediction of Mortalities
(English)Article in journal (Refereed) Submitted
Abstract [en]

We introduce a mixed regression model for morality data whichcan be decomposed into a deterministic trend component explainedby the covariates age and calendar year, a multivariate Gaussian timeseries part not explained by the covariates, and binomial risk. Datacan be analyzed by means of a simple logistic regression model whenthe multivariate Gaussian time series component is absent and there isno overdispersion, as in Ekheden and Hössjer (2014). In this paper werather allow for overdispersion and the mixed regression model is ttedto mortality data from the United States and Sweden, with the aim toprovide prediction and condence intervals for future mortality, as wellas smoothing historical data, using the best linear unbiased predictor.We nd that the form of the Gaussian time series has a large impact onthe width of the prediction intervals, and it poses some new questionson proper model selection.

Keyword
Best linear unbiased predictor, generalized least squares, longevity, mortality prediction, multivariate time series, overdispersion
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-103163 (URN)
Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2014-05-07Bibliographically approved

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