Stohastički okvir metode ulančanih ljestvica

Cilj ovog rada bio je, kao što i sam naslov kaže, stohastički uokviriti dobro poznati algoritam metode ulančanih ljestvica i to kroz prizmu dva modela: Model ulančanih ljestvica bez pretpostavljene distribucije; Raspršeni Poissonov model. Naime, sam algoritam je deterministički proces koji ne pruža nikakve informacije o pouzdanosti rezultata. Zbog toga ga je potrebno detaljnije opisati matematičkim rječnikom kako bi se kvantificirala pripadna neizvjesnost. Prvi model prirodni je nastavak algoritma, koji stohastički prati njegovu osnovnu logiku. Njime se postepeno gradi put ka vremenskim nizovima pomoću kojih ćemo izmjeriti varijabilnosti uz sve nužne pretpostavke kao što je nezavisnost godina nastanka. Drugi model počiva na pretpostavci da su podaci distribuirani raspršenom Poissonovom distribucijom što je potpuno drugačiji pogled nego u Modelu ulančanih ljestvica. U pozadini pristupa je zapravo teorija generaliziranih linearnih modela uz ograničenje na Poissonovu distribuciju iz familije eksponencijalnih. Ovi modeli izabrani su za stohastički okvir zbog toga što daju identične očekivane pričuve šteta pri čemu se razlikuju u iznosu varijabilnosti odnosno razlikuju se u drugim i višim momentima. To je posljedica pretpostavki na kojima su izgrađeni. Također, oba imaju prirodnu podlogu za bootstrap metodu koja nam pruža dodatnu vrlo vrijednu informaciju - distribuciju pričuva šteta. Kako bi rezultati svih metoda bili usporedivi, za mjeru varijabilnosti korištena je uvjetna srednjekvadratna pogreška. U drugom dijelu rada izabran je skup podataka na kojem su primijenjene sve 4 metode: Model ulančanih ljestvica bez pretpostavljene distribucije, Raspršeni Poissonov model, Bootstrap metoda modela ulančanih ljestvica bez pretpostavljene distribucije i Bootstrap metoda raspršenog Poissonovog modela. Modeli su obradeni kroz nekoliko koraka počevši sa grafičkom analizom reziduala preko testiranja pretpostavki do zadnjeg dijela a to je prezentiranje i usporedba rezultata.The aim of this paper was to make stochastic framework of well-known chain ladder algorithm through these two models: Chain ladder model without distributional assumption; Overdispersed Poisson model. Chain ladder algorithm is deterministic process that does not provide any information about the reliability of the results. Because of that, it is necessary to describe it more detailed in terms of mathematical theory to achieve quantification of uncertainty. First model is natural extension of algorithm because it follows its logic through stochastic modelling. It is gradually built up to time series model which will help us measure variability with all assumptions needed for that, such as independence of accident year. Second model is based on assumption that all incremental claims have overdispersed Poisson distribution which is completely different view than the Chain ladder model. In the background of this approach is the theory of generalized linear models with a restriction on the Poisson distribution from the exponential family. These models were chosen for stochastic framework because they produce identical expected claim reserve with different variability, more precisely with different second and higher moments. This is consequence of the assumptions they are built on. Also, both are suitable for extension with bootstrap method which provides us additional valuable information - the distribution of claim reserves. To make all the results comparable, we used mean square error of prediction as a measure of variability . In the second part of paper specific data set was chosen on which all 4 models were applied: Chain ladder model without distributional assumption, Overdispersed Poisson model, Bootstrap method based on chain ladder model, Bootstrap method based on overdispersed Poisson model. The models were broke down through few stages beginning with graphical analysis of the residuals through testing specific assumptions to the last part which is presentation and comparison of the results