# Acceptance of an underwriting decision on risk in conditions...

## Adopting an underwriting decision about risk under uncertainty

Often, the underwriter does not (and can not) know all the characteristics of the proposed risk for insurance and is forced to make a decision in the face of uncertainty (uncertainty) of all circumstances related to risk. Decision making is one of the most important aspects of various areas of people's life and work. Obviously, to make an effective decision, a rigorous scientifically based approach, adequate forecast models and a serious mathematical apparatus are necessary.

One of the main problems of making a decision under these conditions is the disclosure of uncertainty. In the study of operations, special mathematical methods designed for quantitative justification of solutions under conditions of uncertainty have been developed. In many frequently occurring situations, these methods provide auxiliary material that allows you to better understand the complex situation and evaluate each possible solution from different, sometimes contradictory points of view, weigh its advantages and disadvantages and ultimately make a decision, if not the only correct one, at least, to the end thought-out.

One of the most commonly used methods for justifying a decision under uncertainty is game theory , based on conflict modeling.

In general, both the conditions for performing some operation and the actions of other participants in the operation, for example, competitors or unscrupulous insurers, may be uncertain. But often the uncertainty is connected not with conscious opposition to our intentions, but simply with our insufficient knowledge of the objective environmental conditions in which the managerial decision is to be made and implemented. In the study of operations, such an objective environment is usually called "nature", and the corresponding situation is "playing with nature". Applied to the problem being solved as a player the underwriter making the decision is considered, and under the "nature" means the surrounding insurance company external, including natural, environment, full of risks. The player acts consciously, trying to take the most advantageous decision for himself, while assuming that the "nature" is a disinterested instance, whose behavior, although unknown, does not at all contain elements of conscious opposition to the player's plans. " The section of game theory that deals with games with nature is directly connected with the theory of statistical solutions and mathematical statistics. Deciding games with nature is reduced to the formalization of the strategy of the participants in the game, the choice of the evaluation criterion, the compilation and analysis of the payment matrix.

The payment matrix is ​​a table in which the possible player strategies are Ai, the columns are possible strategies of the "nature" Pj, and the values ​​lying at the intersection of rows and columns are the results of the "games" a ij.

To select the optimal strategy (decision) for the player in conditions of uncertainty of the actions of the "nature" criteria that ensure a certain guaranteed win (or a minimum loss) are applied regardless of the actions of the "nature". The most common criterion of pessimism is Hurwitz's optimism:

, (3.2)

where q is the coefficient chosen between zero and one based on additional information about the state of nature and characterizing the degree of pessimism.

The Hurwitz criterion is applied if the player has unformalized information, which allows making a qualitative assumption about the state (strategies) of the "nature" between pessimism and optimism, but not sufficient to quantify the probability of success of a particular strategy.

If q = 1 (extreme pessimism), then the Hurwitz criterion turns into Wald's maximal criterion, which directs the player to the maximum possible of the minimum wins, i.e. to worse conditions.

Another variant of the extreme pessimism criterion is Savage's minimax criterion, which orient the player to the lowest possible of the maximum losses r ij.

(3.3)

If a winning matrix is ​​specified, the loss can be defined as the difference between the maximum possible win and the actual win for each player's strategy:

If q = 0 (extreme optimism), the Hurwitz criterion turns into a maximal criterion that orient the player to the maximum possible of the maximum wins, i.e. on the best conditions.

When choosing an optimization criterion, it is necessary to clearly represent the state and trends of the change in the external normative environment. The art of modeling using a seemingly simple game model consists in isolating from the mass of contradictory information the facts that allow one to formulate various possible states of the "nature", to correctly choose the evaluation criteria and, most importantly, to understand the nature of the "nature" relationship. to the object of research.

In accordance with the main theorem of the theory of games for finite games, there is always at least one optimal solution, perhaps in mixed strategies.

The use of the theory of games to justify the underwriting solution is considered in the example of the decision of the choice of underwriting policy for the insurance of suburban housing units. From history it is known that on large flat rivers about once in 10-12 years there are strong (5-10 times greater in the area of ​​flooding) floods, and once in 100-120 years - catastrophic. In modern conditions, this periodicity is violated because of anthropogenic and anthropogenic impact on nature.

Terms of the task

The insurer insures at home for 12 years (the number of existing contracts n = 100 000), including in river valleys, with an average tariff t = 1% of the insured amount and the level of unprofitableness K yb = 0,7. Long-term weather forecast for the planned year: snowy winter and early spring, i.e. a large flood and flooding of houses located near rivers (no less than 20% of the total number of insured houses) is possible. You need to set a tariff for the next year. An increase in the tariff for every 0.5% leads to a loss of up to 20% of insurance contracts. Refusal of insurance of houses located near rivers will lead to impossibility of renewal of insurance contracts in the future.

Additional data: the average insured amount is 0.3 million rubles, the total expenses for running the case (RVD) of the insurer for the year is 6-8 billion rubles, depending on the number of losses.

Preparing a solution

1. Estimation of the consequences of a possible flood: the tariff in the flooding zone is 3%, accounting for possible contract losses as a result of the tariff increase, taking into account the increase in the loss ratio due to the loss of contracts (increasing the concentration of losses) and the consequences of flooding, accounting for the increase in the costs of running the case (WFD) insurance losses.

2. Development and evaluation of underwriting strategies.

As an indicator of the results of the game, we take the forecasted underwriting financial result (FR) according to the simplified calculation formula:

where S is the total insurance premium; U sum - total insurance loss.

The following natural strategies are possible:

1) П1 - flooding within usual limits;

2) P2 - severe flooding with flooding of all houses in river valleys;

3) P3 - catastrophic flooding with flooding of 30% of all insured houses.

The insurer has a lot of strategies. For clarity, consider the most characteristic of them.

1. Α1 - preservation of all previous insurance conditions for the sake of contract preservation.

2. A2 is a weighted average tariff increase for all, taking into account the proportion of houses falling into the zone of severe flooding.

3. A3 - the application of an increased tariff to houses falling into zones of severe flooding and the previous tariff to other houses.

Given the accepted indicator of the results of the game as a criterion for the analysis of strategies, it is logical to take the maximized Wald criterion.

## Compiling and Solving the Payment Matrix

The initial data for the payment matrix is ​​shown in Table. 3.2, the payment matrix - in Table. 3.3.

Table 3.2. Estimated Performance of Applied Strategies

 A t ,% n, th homes S, billion rubles. To уб WFD, billion rubles. UZS, bn RUR. P1 P2 P3 P1 P2 P3 P1 P2 P3 A1 1 100 30.0 0.7 0.72 0.79 6 7 8 21 21.6 23.7 A2 1.4 70 29.4 0.75 0.76 0.825 6 7 8 22.05 22.34 24.26 A3 1 and 3 80 + 10 * 24 + 9 0.7 and 0.75 0.7 and 0.8 0.7 and 1.0 6 7 8 23.3 16.8 + 7.2 16.8 + 9

* By house group: outside the zone and in the flood zone.

Table 3.3. Billing Matrix

 Strategies Insurer Nature Strategies P1 P2 P3 A1 3.0 1.4 -1.7 A2 1.35 0.06 -2.86 A3 3.7 2.0 -0.9

Analysis of Table. 3.3 shows that according to the Wald criterion, strategy 3 is optimal: a differentiated tariff for houses with an increased risk of flooding in severe flooding. To improve the financial result for this strategy, it is necessary to reinsure the risk of exceeding the loss ratio.

When using similar methods of justifying solutions, one should not forget that game theory, like any mathematical model of a complex phenomenon, has its limitations. The most important of them is the artificial reduction of the winnings to a single number that does not always adequately represent all the consequences of the chosen solution. Conscious of these limitations, it is necessary to consider game models as one of the ways to streamline our ideas about reality and subsequently subject the resulting solutions to additional checks.

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