Іллічевський С.О.

Київський національний університет імені Тараса Шевченка, Україна

Contemporary Risk Modeling for Insurance Company

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 [1, p. 351].

It is well-known that the analysis of activity of an insurance company in conditions of uncertainty is of great importance [2, 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 [5, p. 1092]. 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:

,

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 .

Conclusions. We analyzed methods of calculation of ruin probabilities for insurance company in presents of its investing activity. 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.

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