Θεωρία ασάφειας με εφαρμογές στη θεωρία αξιοπιστίας και αποθεματοποίηση ζημιών
Fuzzy theory with applications to credibility and loss reserving
KeywordsInsurance ; Claims reserving ; Fuzzy logic ; Fuzzy set theory ; Fuzzy numbers ; Fuzzy arithmetic ; Fuzzy regression ; Claim provisions ; Chain ladder ; Run-off triangle ; Geometric separation method ; Bornhuetter-Ferguson ; Credibility ; Bühlmann ; Hachemeister
The prediction of an adequate amount of claim reserves is crucial for the financial stability of insurance companies. Although many different deterministic and stochastic methods based on statistical analyses are used, the wide range of unquantifiable factors which increase the uncertainty should be taken under serious consideration when using any, based on statistical concepts, method to estimate claim provisions based on historical data. Therefore, in a state of uncertainty which is intrinsic in the nature of many actuarial and financial problems, when adequate and solvent data is not held, the use of fuzzy set theory becomes very attractive due to the tolerance of imprecision and uncertainty without loss of performance and effectiveness. In this thesis, initially we extend the classical chain ladder method using fuzzy methods. Therefore, we derive new estimators for claim development factors. Secondly, is presented the extension of London chain ladder method using hybrid fuzzy least-squares regression. Thirdly, is proposed the fuzzy extension of chain ladder with exponential model by applying hybrid fuzzy least-squares regression. In a fourth stage, hybrid fuzzy least-squares regression is used to predict future claim costs by utilizing the concept of a geometric separation method. Finally, is shown how Bornhuetter-Ferguson claims reserving method can be extended by applying fuzzy methods. The a priori information for the ultimate claims derives from market statistics and might contain vagueness . Likewise, the parameters of the claims development pattern can be vague or are adapted, retrospectively, due to subjective judgment. For every method we extend using fuzzy set theory we also provide an estimator for the weighted average depending on risk factor and an estimator for the uncertainty of the ultimate claims for single and aggregated accident years. Furthermore the fundamental credibility models of Bühlmann and Hachemeister are extended by utilizing the instruments that fuzzy set theory has to offer. Implementation of these fuzzy credibility models is conducted on property and casualty portfolio, in order to derive predictions for next period's claim amounts, through the extended models, for every section included in portfolio.