## Risk assessment and the basis of actuarial calculations in insurance

As a result of mastering the material in Chapter 4, the student must:

* know *

* • * the fundamentals of actuarial calculations;

* be able to *

* • * calculate the insurance rate for typical risks;

* own *

* • * the main methods of insurance tariffing.

* Key terms : * risk map; probability, expectation, variance of loss; risk distribution function; the problem of insolvency of the insurer; homogeneity of risks; mass types of insurance; catastrophic risk; the scenario of the accident; mortality tables; tariff policy.

## Methods of assessing insurance risks

The insurance risk must be computable, because otherwise the insurer will not be able to determine the payment for it and will not take this risk on its own responsibility. Therefore, risk assessment is one of the most important issues for an insurer.

Strictly speaking, in the notion of "statistical risk" The method of its estimation is based on the statistics of manifestation.

Risk maps are often used to identify and qualify risk assessment (Figure 4.1), representing a graphical and textual description of a limited number of risks located in a rectangular table, one "axis" which indicates the force of the impact of the risk (loss), and on the other - the probability or frequency of its occurrence.

** Fig. 4.1. ** An approximate risk map

For preliminary quantitative risk assessments, the probability * p * of its occurrence is applied, the mathematical expectation M [u] of the random value of loss (damage) and its variance D [u].

* * In probability theory, ** random variable ** is a quantity that in practice can take a certain value in practice (risk manifestation), and it is not known in advance what it is. The random value of the loss can take values from 0 (the loss for a given risk occurrence has not occurred) to * u * max, which corresponds, for example, to the total loss of property and in value terms to its value.

The most complete risk is characterized by the law of distribution of the random value of loss, which establishes the relationship between the possible values of the random value of the loss and the corresponding probabilities. The ** Integral distribution function ** of a random variable is a function of the probability distribution of an event that a random value (for example, loss) does not exceed a certain value:

F (u) = P [u * & lt; * U],

where * U * is some current (variable) loss value * and. *

From the integral distribution function of a random loss by differentiating with the variable * U *, you can get its ** density function **, which makes it easy to calculate the probability of any loss.

The density function of the random distribution of property losses for most risks is in the form of a diminishing curve: the larger the amount of loss, the less its probability, i.e. small losses are much more common than large losses. The simplest form of the density function (the so-called Heinrich triangle) is shown in Fig. 4.2.

P ** Is. 4.2. ** Heinrich's Triangle

To quantify the parameters of the distribution law, statistics of losses by type of risk and known methods of statistical calculations are used. The actual distribution of random losses is obtained by ranking statistical material. If necessary and for the convenience of further studies of the loss, these distributions can be approximated by the known laws of distribution of random variables. The random distribution of property losses in insurance is usually approximated by the following laws:

• Normal distribution;

• log-normal distribution;

• Rayleigh distribution;

• Student's distribution;

• Pareto distribution, etc.

Logarithmically normal (lognormal) distribution is used more often than others because it approximates the distribution of losses in insurance of small property objects (cottages, cottages, cars), which is characterized by high probability of relatively small losses (Figure 4.3).

** Fig. 4.3. ** The density function * f (x) * and the integral function * F (x) * for the logarithmically normal random loss distribution

However, not all insurance risks can be considered statistical. There are rare risks, but lead to destructive consequences (earthquakes, tsunamis, nuclear accidents, etc.). Such risks are called catastrophic. The statistics of such risks, due to their rarity, are practically non-existent, therefore, for the quantitative estimation of them, analogy methods, systems stability theory, etc. are used.

The analogy method is used primarily to assess natural risks. Studying the historical references to various rare natural disasters that happened in the past, it is possible to estimate quite accurately the possible damage from such phenomena in our time. The question of assessing the likelihood of such a natural catastrophe is still open. For its approximate evaluation, the following methods are used.

1. Statistical, assuming that these or other catastrophic phenomena form a single statistical aggregate, for example, recurring approximately once in 100-120 years catastrophic floods on large rivers. Disadvantage of the method: significant variance and low accuracy of statistical estimates due to small volumes of statistical sampling and heterogeneity of the statistical material itself.

2. Mathematical modeling of catastrophic phenomena at the global level, for example earthquakes. The disadvantage of the method: the lack of a qualitative theory of the phenomenon, the lack of computational power, the difficulty of collecting the initial information for modeling.

To assess the probability of catastrophic accidents on large, complex technical systems, known methods of stability theory are used. In the system, chains of elements are allocated, failure (destruction) of which will lead to the destruction of the entire system and causing damage, as a result of such destruction, to others.

These methods are discussed in more detail in Section 4.4.

## thematic pictures

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