| Abstract | Rad opisuje model mješovitog životnog osiguranja pomoću Markovljevog procesa skokova \(Y(t), t\geq 0\) sa stanjima S := {AKTIVAN, DOŽIVLJENJE, SMRT, OTKUP, STORNO, KAPITALIZACIJA} koji ovisno o trajanju članstva može prijeći u neka od navedenih stanja iz skupa stanja S. Opisani proces \(Y(t), t\geq 0\) nije vremenski homogen, ali se može dekomponirati na dva takva lanca, ovisno o duljini trajanja članstva. U radu je opisana statistika za policu \(i\) unutar dobnog intervala \([x,x+1]\) i unutar vremena opažanja \([t_0, t_1]\) s pretpostavkom da su intenziteti prijelaza unutar dobnog intervala \([x,x+1]\) konstantni. Prikazani su svi mogući scenariji sa pripadajućim lexis dijagramima. Dan je primjer od 9 polica sa opisom na koji način su police obrađene za statistiku. Procijenjeni su intenziteti prijelaza pomoću metode maksimalne vjerodostojnosti. Pomoću Markovljevog svojstva nađena razdioba stanja za proces. Izračunate su procjene aproksimativnog 95% pouzdanog intervala za sve intenzitete prijelaza. Pomoću njih intenziteti prijelaza su izglađeni odgovarajućim funkcijama (linearni filter kroz 5 točaka, pravac, parabola). Grafički je prikazana procjena svakog pojedinog intenziteta prijelaza sa pripadnim intervalom pouzdanosti i izglađenim vrijednostima. Izračunato je očekivano vrijeme boravka u stanju AKTIVAN i KAPITALIZIRAN sa izglađenim stopama prijelaza za prvi odnosno drugi period trajanja članstva. Na kraju je izračunata sadašnja vrijednost police u trenutku \(x+k\) koja je sklopljena u trenutku \(x (_k V_x)\) kao očekivanje od slučajne sadašnje vrijednosti police u trenutku x + k koja je sklopljena u trenutku \(x (_k L_x)\). U poglavlju (3.9) na stranici 77 nalazi se primjer izračuna očekivane slučajne sadašnje vrijednosti police u trenutku \(x + k\) koja je sklopljena u trenutku x = 20 na n = 5 godina. |
| Abstract (english) | The thesis describes a model of life insurance model in case of death or endowment using a Markov Jump Process \(Y(t), t\geq 0\) with states S := {ACTIVE, ENDOWMENT, DEATH, WITHDRAWAL, CANCELLATION, CAPITALIZATION} which depending on the duration of membership may exceed one of the specified S states. The process \(Y(t), t\geq 0\) described is not time-homogeneous, but can be decomposed into two such chains, depending on the length of membership. The thesis describes statistics for contract \(i\) within the age interval \([x,x+1]\) and within the observation time \([t_0, t_1]\) with the assumption that the transition intensities are within the age interval \([x,x+1]\) constant. All possible scenarios with associated lexis diagrams are shown. An example of 9 contracts is given, describing how the contracts are processed for statistics. The transition intensities were estimated using the maximum likelihood method. The distribution of states for the process was found using the Markov property. Estimates of the approximate 95% confidence interval for all transition intensities were calculated. With these transitions, the intensities of the transitions are smoothed by appropriate functions (linear filter through 5 points, direction, parabola). An estimate of each individual transition intensity with the associated confidence interval and smoothed values ?? is shown graphically. The expected residence time in the state ACTIVE and CAPITALIZATION was calculated with smoothed transition rates for the first and second membership periods, respectively. Finally, the present value of the contract at time \(x + k\), which was concluded at time \(x (_k V_x)\), was calculated as the expectation of the random present value of the contract at time \(x+k\)which was concluded at time \(x (_k L_x)\). The section (3.9) on page 77 provides an example of calculating the expected random present value of a contract at time \(x+k\) that was entered at the age x = 20 for n = 10 years. |
