УДК 330.46:51-75

*Sergiy Illichevskyy*

** **

**THE IMPLEMENTATION OF BAYESIAN NETWORKS FOR**** ****MODELING OF INSURANCE MARKET**

*Annotation: this article is devoted to the research and development of the new
type of method for modeling of risks of insurance companies. This approach implements
Bayesian networks as a main tool for modeling.*

*Key words:** Bayesian networks, insurance company, ruin probability.*

* *

*Анотація:** дана стаття присвячена дослідженню і розробці нового
типу методів управління ризиками страхових компаній. Даний
підхід застосовує Байжсівські мережі як основній інтрумент моделювання. *

*Ключові слова:** Байєсівські мережі, страхова компанія, ймовірність
банкрутства.*

** **

** **

**I. Introduction.** Today it is
impossible to imagine a market economy without risks. They are involved almost
in every economic activity. There is a great need in measuring, predicting and
minimizing risks. Insurance services are one of the industries, which
permanently experience risks of bankruptcy. That is why calculating the ruin
probabilities for insurance companies are one of the problems that need
well-developed mathematical models. Nowadays Ukrainian insurance companies are searching
for new ways of profitability and competitiveness. Western European insurance
companies has an option of investing their fund for additional profit. That is
way there is a great necessity of creation and development of the actuarial
models for Ukrainian insurance to provide them the possibility of investing
their fund for additional profit.

**II. ****The analysis of the main researches and publications**. One of the first studies in this area was conducted in
the beginning of the twentieth century. Since then, the
mathematical methods of ruin probability calculation developed and accumulated
a great variety of models and approaches. While the permanent growing of
economic needs, insurance services increase steadily in the economies of all
developed countries. Insurance services are one of the youngest
industries any economy, which experience a stage of active development. In
global practice of developed countries, well organized insurance services are
involved in many economic sectors like investment activity of insurance
companies. This article studies how the actuarial mathematical tools can
positively affect the theoretical and practical development of insurance. The development of theoretical,
methodological, organizational and legal bases
of insurance market have been contributed by
many economists, such as: Bazylevych, V [1, 2], Chernyak,
O. [7-10], Pikus, R. [1], Starostina, A. [1], Filoniuk, O. [1], Kaminsky, A. [6],
Shpyrko, V. [1], Kalashnikov, V. [6] and others.

**III. ****Unsolved issues: **one
of the main problems at present for actuarial analysis of the Ukrainian
insurance market is the lack of large statistical base, which is necessary for
any econometric modeling. That is way there is a great necessity of actuarial
models that involve fewer statistical information. We analyze methods
of calculation of ruin probabilities for insurance company. We consider an
insurance company in the case when the premium rate is a bounded by some
nonnegative random function and the capital of the insurance company is
invested in a risky asset whose price follows a geometric Brownian.

**IV. Formulation of the problem.**** **The goal of the article is creation new types of
models of the analysis of ruin probabilities using Bayesian networks that can
be helpful for Ukrainian insurance companies. There are
different methods for approximating the distribution of aggregate claims and
their corresponding stop-loss premium by means of a discrete compound Poisson
distribution and its corresponding stop-loss premium. This discretization is an
important step in the numerical evaluation of the distribution of aggregate
claims, because recent results on recurrence relations for probabilities only apply to discrete distributions.
The discretization technique is efficient in a certain sense, because a
properly chosen discretization gives raise to numerical upper and lower bounds
on the stop-loss premium, giving the possibility of calculating the numerically
estimates for the error on the final numerical results.

**V.Results.** We consider an insurance company in the
case when the premium rate is a bounded nonnegative random function and the capital of the insurance
company is invested in a risky asset whose price follows a geometric Brownian
motion with mean return and volatility. If we
find exact the asymptotic upper and lower bounds for the ruin probability as
the initial endowment tends
to infinity, i.e. we show that for sufficiently
large Moreover if with we
find the exact asymptotics of the ruin probability, namely. If,
we show that for any .
We investigate the problem of consistency of risk measures with respect to
usual stochastic order and convex order. It is shown that under weak regularity
conditions risk measures preserve these stochastic orders. This result is used
to derive bounds for risk measures of portfolios. As a by-product, we extend
the characterization of coherent, law-invariant risk measures with the property
to unbounded random variables. A surprising result is that the trading strategy
yielding the optimal asymptotic decay of the ruin probability simply consists
in holding a fixed quantity (which can be explicitly calculated) in the risky
asset, independent of the current reserve. This result is in apparent
contradiction to the common believe that `rich' companies should invest more in
risky assets than `poor' ones. The reason for this seemingly paradoxical result
is that the minimization of the ruin probability is an extremely conservative
optimization criterion, especially for `rich' companies [3, p. 35].

It is well-known that the analysis of activity of an insurance company in conditions of uncertainty is of great importance [9, p. 162]. Starting from the classical papers of Cramer and Lundberg which first considered the ruin problem in stochastic environment, this subject has attracted much attention. Recall that, in the classical Cramer-Lundberg model satisfying the Cramer condition and, the positive safety loading assumption, the ruin probability as a function of the initial endowment decreases exponentially. The problem was subsequently extended to the case when the insurance risk process is a general Levy process.

It is clear that, risky investment can be dangerous: disasters may arrive in the period when the market value of assets is low and the company will not be able to cover losses by selling these assets because of price fluctuations. Regulators are rather attentive to this issue and impose stringent constraints on company portfolios. Typically, junk bonds are prohibited and a prescribed (large) part of the portfolio should contain non-risky assets (e.g., Treasury bonds) while in the remaining part only risky assets with good ratings are allowed. The common notion that investments in an asset with stochastic interest rate may be too risky for an insurance company can be justified mathematically.

We deal with the ruin problem for an insurance company investing its capital in a risky asset specified by a geometric Brownian motion:

, (1)

where is a standard Brownian motion and .

It turns out that in this case of small volatility, i.e., the ruin probability is not exponential but a power function of the initial capital with the exponent. It will be noted that this result holds without the requirement of positive safety loading. Also, for large volatility, i.e., the ruin probability equals 1 for any initial endowment.

In all these papers the premium rate was assumed to be constant. In practice this means that the company should obtain a premium with the same rate continuously. We think that this condition is too restrictive and it significantly bounds the applicability of the above mentioned results in practical insurance settings.

The numerical calculation of finite time ruin probabilities for two particular insurance risk models are being analyzed. The first model allows for the investment at a fixed rate of interest of the surplus whenever this is above a given level. Our second model is the classical risk model but with the insurer's premium rate depending on the level of the surplus.

Our methodology for calculating finite time ruin probabilities is to bound the surplus process by discrete-time Markov chains; the average of the bounds gives an approximation to the ruin probability.

Our primary purpose in this paper is to discuss the numerical calculation of finite time ruin probabilities for two particular insurance risk models. Both models are extensions of the classical risk model. For each model, is a random variable which denotes the surplus at time , so that is a continuous time stochastic process; the aggregate claims in [0,t] are denoted , where has a compound Poisson distribution with Poisson parameter λ; individual claim amounts have , and mean. We assume that, so that all claims are positive. We assume without loss of generality that .

It would be possible to have more than two bands for the surplus
with a different rate of premium income at time *t *depending on the
band in which lies. However, all our
numerical examples assume just two bands and so we have presented the model in
this way.

An essential feature of the two models studied in this paper is that
they are time-homogeneous Markov processes; the level of the surplus at any
given time is sufficient to determine probabilistically its level at any time *h *later.
This is the feature that we will exploit in this paper to obtain bounds for the
finite time ruin probabilities for our two models. We do not need to assume any
form of 'net profit condition' for our two models, but we do need to assume
that and .

Our aim is to produce bounds for this probability; approximate values of the probability can be calculated by averaging the upper and lower bounds. However it is not always possible to produce absolute bounds.

The surplus process of an insurance portfolio is defined as the wealth obtained by the premium payments minus the reimbursements made at the times of claims. When this process becomes negative (if ever), we say that ruin has occurred. The general setting is the Gambler's Ruin Problem. We address the problem of estimating derivatives (sensitivities) of ruin probabilities with respect to the rate of accidents. Estimating probabilities of rare events is a challenging problem, since naive estimation is not applicable.

Solution approaches are very recent, mostly through the use of Importance Sampling techniques. Sensitivity estimation is an even harder problem for these situations. We study different methods for estimating ruin probabilities: one via importance sampling (IS), and two others via indirect simulation: the storage process (SP), which restates the problems in terms of a queuing system, and the convolution formula (CF). To estimate the sensitivities, we apply the RPA method to importance sampling, the IPA method to storage process and the Score Function method to convolution formula. Simulation methods are compared in terms of their efficiency, a criterion that appropriately weighs precision and CPU time. As well, we indicate how other criteria such as set-up time and prior formulas development may actually be problem-dependent. The canonical model in Risk Theory assumes that claims due to accidents arrive according to a Poisson process of rate . The successive claim amounts, denoted , are i.i.d. random variables with general distribution and premiums are received at a constant rate .

The event epochs of the process are denoted by ,and are the interarrival times. The cumulative claims process:

(2)

is a compound Poisson process. We shall often write to denote the embedded discrete event process and , with an obvious abuse of notation. If we set then the ruin probability is [9, p. 217]. Call . If ,then for all initial endowment . As a consequence of this result, it is common to assume that premiums satisfy .

**VI. Conclusions. **We analyzed methods of calculation of ruin probabilities for
insurance company calculated by Bayesian networks. We considered an insurance
company in the case when the premium rate is a bounded by some nonnegative
random function and the capital of the insurance company is invested in a risky
asset whose price follows a geometric Brownian.

The weak development of insurance market in Ukraine is explained by the low incomes of Ukrainians and their disinterest in spending money on insurance, although some cases.

The analyzed economic and mathematical models are recommended to be used in for Ukrainian insurance companies for increasing profitability and diversification of ruin risks.

**VII. The prospects for further development of the problem. **Although there are several methods for calculation the ruin
probabilities for insurance companies, this study may enrich existing methods,
for cases of investment activities for Ukrainian insurance companies.

**References**

** **

1. Базилевич К.С. Страхування: Підручник // Базилевич В.Д., Базилевич К.С., Пікус Р.В., Філонюк О.Ф., Черняк О.І., Старостіна А.О., Шпирко В.В. // В.Д.Базилевич (ред.). – К. : Знання, 2008. – С. 1019.

2. Базилевич В.Д. Страховий ринок України. Монографія. – К.: Товариство “Знання”, КОО, 1998. – 374 с.

3. Камінський А.Б. Моделювання фінансових ринків: [монографія] // А.Б.Камінський. - К.: Видавничо-поліграфічний центр „Київський університет”, 2006. – 304 c.

4. Анісімов В.В., Черняк О.І. Математична статистика: Навч. посібник для студентів вузів.− К.:МП „Олеся”, 1995. −104 с.

5. Гихман И.И., Скороход А.В., Ядренко М.И. Теория вероятностей и математическая статистика. − Київ: Вища школа, 1998.

6.
Kalashnikov, V., Norberg, R. (2002) Power tailed
ruin probabilities in the presence of risky investments. *Stochastic Process.
Appl. *98. P.** **211-228.

7. Chernyak O.I. Limit theorems for dependent summation schemes/| O.I. Chernyak, J. Sztrik // Random Operators and Stochastic Equations.-1993. - v.1, N 1. - Р.29-36. VSP, Utrecht, The Netherlands.

8. Chernyak O.I. Asymptotic behaviour of a complex renewable standby system with fast repair // O.I. Chernyak, J. Sztrik // Problems of Control and Information Theory.- 1991.-v.20(1).-Р.37-44.Budapest,Hungary.

9. Pavlenko T. Credit risk modeling using bayesian networks / T.Pavlenko, O.Chernyak // International Journal of Intelligent Systems.- 2010. -Volume 25, Issue 4.-Wiley Periodicals, Inc., A Wiley Company.- P.326 – 344 (USA).

10. Chernyak O. Classification of Financial Conditions of the Enterprises in Different Industries of Ukrainian Economy Using Bayesian Networks / O.Chernyak , Y.Chernyak // Proceedinds of 5 International Conference On Information Technologies in Agriculture, Food and Environment (HAICTA-2011), September 8-11, 2011, Skiathos Island, Greece.-Volume II. –P.519-530.

*
Стаття надійшла до редакції
22.01.2012
р.*