A Bayesian extreme value approach to the optimal reinsurance problem in a multivariate risk setting

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dc.contributor.advisorHarvey, Justinen_ZA
dc.contributor.authorSteenkamp, Shaun Francoisen_ZA
dc.contributor.otherStellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.en_ZA
dc.date.accessioned2023-12-01T13:22:54Z
dc.date.accessioned2024-01-08T20:48:18Z
dc.date.available2023-12-01T13:22:54Z
dc.date.available2024-01-08T20:48:18Z
dc.date.issued2023-12
dc.descriptionThesis (MCom)--Stellenbosch University, 2023.en_ZA
dc.description.abstractENGLISH SUMMARY: This thesis investigates a Bayesian extreme value theory approach to analyse the optimal reinsurance problem, more specifically the optimal layer selection of an excess of loss reinsurance contract. This thesis suggests a simulation approach to the optimization of the layer selection. This thesis proposes a multivariate excess of loss (XL) reinsurance structure, referred to as the simultaneous XL reinsurance structure and applies the developed optimization algorithm to this structure in several numerical examples. The approach takes a particular focus on extreme risks, thereby investigating the optimal reinsurance contract that best protects the insurance company from rare large claims. The methodology is explained for a univariate risk case, thereafter the model is extended to the bivariate and the multivariate risk cases. The optimal reinsurance agreement can be investigated using a variety of different models. This thesis develops a risk measure minimization model, with a focus on the conditional tail expectation (CTE) riskmeasure. The model allows for the insurance company’s reinsurance budget as a constraint in the optimization problem. Bayesian techniques are especially useful in problems where data is sparse, therefore this thesis suggests utilizing a Bayesian approach to the optimal reinsurance problem where rare large claims are considered. A Bayesian extreme value theory approach could improve the process of investigating the optimal reinsurance problem by utilising Markov Chain Monte Carlo (MCMC) methods to supplement the information from the data that the insurance company has available. The approach is extended into the bivariate and multivariate risk cases where a fictitious insurer, involved in various lines of business is considered. The dependence structure is modelled using a copula approach. Numerical examples are examined, and the results are interpreted. This thesis takes a focus on the tail of the data, thereby evaluating the optimal excess of loss reinsurance contract for very large claims with very small probabilities. The research suggests an algorithm for evaluating the optimal reinsurance strategy in a multivariate risk environment for insurance companies involved in different lines of business. The analysis will improve understanding and assist decision making on the reinsurance strategy from the insurer’s perspective.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Hierdie tesis ondersoek 'n Bayes-benadering tot ekstreem waarde teorie om die optimale herversekerings probleem te analiseer, meer spesifiek die optimale laag seleksie van 'n oorskot van verlies herversekeringskontrak. Die benadering fokus veral op ekstreem risiko's en ondersoek sodoende die optimale herversekeringskontrak wat die versekeringsmaatskappy die beste teen seldsame groot eise beskerm. Die metodologie word vir 'n eenveranderlike risiko geval verduidelik, daarna word die model uitgebrei na die tweeveranderlike en die meerveranderlike risiko gevalle. Die optimale herversekeringsooreenkoms kan met behulp van 'n verskeidenheid verskillende modelle ondersoek word. Hierdie tesis ontwikkel 'n risiko maatstaf minimaliserings model, met 'n fokus op die voorwaardelike stert verwagting risiko maatstaf. Die model maak voorsiening vir die versekeringsmaatskappy se herversekerings begroting as 'n beperking in die optimaliserings probleem. Bayes tegnieke is veral nuttig in probleme waar data skaars is, daarom stel hierdie tesis voor om 'n Bayes-benadering tot die optimale herversekerings probleem te gebruik waar skaars groot eise oorweeg word. 'n Bayes ekstreem waarde teorie benadering kan die proses waar ondersoek van die optimale herversekering gedoen word verbeter deur Markov-ketting Monte Carlo (MCMC) metodes te gebruik om die inligting van die data wat die versekeringsmaatskappy beskikbaar het aan te vul. Onlangse tegnologiese vooruitgang het dit moontlik gemaak om modelle te simuleer, te ontleed en te ondersoek op 'n skaal wat nooit voorheen moontlik was nie. Hierdie tesis stel 'n simulasiegebaseerde benadering voor om die optimale herversekeringskontrak te ontleed. Die benadering word uitgebrei na die tweeveranderlike en meerveranderlike risiko gevalle waar ondersoek gedoen word vir 'n fiktiewe versekeraar wat betrokke is by verskeie versekerings lyne. Die afhanklikheidstruktuur tussen hierdie risiko's word gemodelleer deur 'n copula-benadering. Voorbeelde word bespreek en die resultate daarvan word geinterpreteer. Hierdie tesis fokus op die stert van die data, en evalueer daardeur die optimale oorskot van verlies herversekeringskontrak vir groot eise met klein waarskynlikhede. Die navorsing stel 'n algoritme voor om die optimale herversekering strategie in 'n veel veranderlike risiko-omgewing te evalueer vir versekeringsmaatskappye wat by verskillende versekering lyne betrokke is. Die ontleding sal begrip verbeter en besluitneming oor die herversekering strategie vanuit die versekeraar se perspektief ondersteun.af_ZA
dc.description.versionMasters
dc.format.extentxiii, 191 pages : illustrations, includes annexures
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/129056
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch University
dc.rights.holderStellenbosch University
dc.subject.lcshExtreme value theory -- Mathematical modelsen_ZA
dc.subject.lcshBayesian statistical decision theoryen_ZA
dc.subject.lcshExtreme value theoryen_ZA
dc.subject.lcshMultivariate analysisen_ZA
dc.subject.nameUCTD
dc.titleA Bayesian extreme value approach to the optimal reinsurance problem in a multivariate risk settingen_ZA
dc.typeThesisen_ZA
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