Μελέτη μοντέλων κινδύνου με εξάρτηση και διάχυση
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Keywords
Φερεγγυότητα ; Θεωρία χρεοκοπίας ; Ruin theory ; Cramer Lundberg ; Sparre Andersen ; Ανανεωτικό μοντέλο ; Εξάρτηση ; Διάχυση ; Δυϊκό ; Gerber ShiuAbstract
For the modeling of an insurance portfolio, one of the most basic parameters that should be
considered is that of solvency. To qualify a portfolio as solvent, the insurer must have sufficient
capital reserves to be able to compensate the customers in the event of a large claim. As this is
different for each insurance company, sometimes it is regulated by some regulatory authority that
regulates the probability of repayment of the insurer's obligations (eg 95%). A very basic concept in
the insurance portfolio is that of insurance surplus. The insurance surplus expresses the insurer's
profit over time. When the surplus becomes zero or negative, this event is called bankruptcy. Of
course, it does not refer to the fact that given this happens, the insurer is bankrupt, as there are
various strategies that the insurer can follow to compensate for the loss (e.g. capital injections).
The theory that studies the possibility of ruin is called Ruin Theory. Ruin theory, using
mathematical tools, studies the probability that the insurance company will not be able to repay its
insurance obligations according to certain assumptions. The study is not only limited to the
possibility of repayment of its obligations, but also to additional "ruin measures" which provide
specific information on the course and behavior of the insurance surplus before and after it is written
off.
The first attempts to properly model the insurance surplus were made by the actuary Filip
Lundberg in 1903. In his published doctoral thesis with title: "Approximations of the Probability
Function/Reinsurance of Collective Risks", he introduced the Classical Model of Ruin Theory, which
Harald Cramér later included in the contemplative processes. This model is considered extremely
pioneering for its time, as contemplative processes in today's sense had not yet been established.
Since then, various generalizations of this model have been published from time to time, with
the most important perhaps being the renewal ruin model published in 1957 by Erik Sparre
Andersen. Erik Sparre Andersen's model generalizes the assumption about the distribution of
interarrival claim occurrence times, which in practice applies more effectively than the assumption
of Lundberg's model.
In 1997, in the paper with title “The joint distribution of the time of ruin, the surplus
immediately before ruin, and the deficit at ruin” by Hans Gerber and Elias Shiu they introduced the
discounted penalty function. The latter has been the subject of extensive study by actuaries for its
properties and the information it provides about some ruin measures.
In this thesis, the first chapter covers the basic concepts for the classical and the
corresponding renewal ruin model, and some basic numerical results will be provided using the
discounted Gerber-Shiu penalty function. In the following chapters, some of these generalizations
that have been proposed for the renewal ruin model will be explored. In the last chapter, the binary
model of bankruptcy will be studied, where it mainly responds to businesses and organizations
characterized by continuous outflow of expenses and occasional income.