Keywords : risk process, ruin probability, Lundberg’s inequality, survival probability, De Vylder’s method, maximum aggregate loss.
Pratiwi, Hasih*)
LPPM UNS, Penelitian, DP2M, Hibah Kompetitif Penelitian Sesuai Prioritas Nasional, 2009
İn terms of plate tectonic setting the Indonesian archipelago is situated in the triple junction of the three major plates; the Indo-Australian plate, the Eurasian plate, and the Pasific plate, so that earthquake happened in Indonesia every year. Loss of physical generated by earthquake is death and human victim as well as damage of building and area. Insurance companies are in the business of risks. They exist to pool together risks faced by individuals or companies who in the event of a loss are compensated by the insurer to reduce the financial burden. In its simplest form, when certain events occur, an insurance contract will provide the policyholder the right to claim all or a portion of the loss. In exchange for this entitlement, the policyholder pays a specified amount called the premium and the insurer is obligated to honor its promises when they come due.
In this paper, we simplify a real life insurance operation by assuming that the insurer starts with some non-negative amount of money, collects premiums, and pays claims as they occur. Our model of an insurance surplus process is then deemed to have three components: initial surplus (or surplus at time zero), premiums received and claims paid. If the insurer’s surplus falls at zero or below, we say that ruin occurs. The aim of this work is to derive a risk model by determining a ruin probability in risk process.
The risk process is a very important model for understanding how the capital or surplus of an insurance company evolve over time. The risk model generally can be expressed as a ruin probability that can be derived using survival probability (u). We find the solution for with assuming Lundberg’s inequality applies. All that is required to apply De Vylder’s approximation is that the first three moments of the individual claim amount distribution exist. In situation when the adjustment coefficient exists, the method provides good approximations when ruin probabilities are small. However the approximation is inaccurate for small values of u, especially u = 0. Generally, the method is not particularly accurate when the adjustment coefficient does not exist. We also derive an upper bound and a lower bound for ruin probability by considering the distribution of maximum aggregate loss.
Concerning with the risk model, we can estimate of earthquake risk if the Poisson parameter λ and the distribution of claim amount f(x) are known. These parameters can be obtained using the maximum likelihood estimation. We then can find the aggregate claim amount and finally we can determine the survival probability or the ruin probability.