# Law of large numbers - Insurance

## The law of large numbers

The totality of all insurance contracts concluded by the company is called an insurance portfolio. For each contract there is a risk of loss. That is, payment under the contract is a random amount. The insurance portfolio is a set of such random variables.

Within the framework of the theory of probability, general regularities are studied, to which sets of random events obey. In particular, it is proved that the cumulative effect of a large number of random variables under certain conditions leads to a result that is almost independent of the case. Theorems describing this pattern are collectively called law of large numbers & quot ;.

Similar non-random the behavior of the result of the impact of a large number of random variables can be explained by mutual compensation of their deviations from some expected "average" values. For example, the result of insurance for any particular contract is random. If the amount of payments on it is less than the premium paid, the result for the insurance company will be positive. Otherwise - negative. But within the framework of the collective of contracts, the summation of such positive and negative deviations reduces the spread of the overall portfolio result. The cumulative result of insurance operations loses a random character, becomes more predictable, natural. This manifests the effect of the so-called collective balance.

When establishing insurance fees, it is very important to correctly determine the price of risk that the participant transfers to the fund. According to the law of large numbers for a significant set of risks, the amount of losses will be highly likely to aspire to its expected value. This means that when calculating the insurance price for a quantitative risk assessment, you can use the expected value of payments.

Example. Combining Risks

Suppose that the probability of theft of a car of a certain brand worth 500 LLC rubles. is 0.03 per year (that is, on average three cars of 100 are stolen). The possibility of theft is always there. And although it occurs only with 3% of car owners, for each of them this event can become a disaster. In essence, this risk represents a "game", where a probability of 3% can lose 500,000 rubles, but you can not win. Therefore it is quite logical that people want to have protection from such events.

If the car owner wants to protect himself, he must create a reserve of funds to compensate for losses, i.e. to purchase a new car. How much money needs to be deferred to be sure of protection? For an individual owner, the answer is obvious - 500 000 rubles. Any other amount, less than this, for the purchase of a similar machine is simply not enough. But not everyone can create such a reserve.

Now assume that 1000 motorists decided to create a joint insurance fund for payments to those who were stolen by the car. In accordance with the law of large numbers, the average frequency of thefts in a given team of risks with a high probability will tend to its theoretical value of 0.03. That is, for 1000 cars will be expected 30 hijackings. If you divide the total cost of cars that are supposed to be stolen to all participants, then each will be enough to collect 500,000 o 30/1000 = 15,000 rubles! For this fee, the car owner has the right to expect a full refund of a loss of \$ 500,000.

With this method of protection, the participant of the fund donates a relatively small (in comparison with a possible loss) sum in exchange for the assurance that when stealing he will be reimbursed for the cost of the car. This became possible as a result of the pooling of a significant number of participants exposed to one risk in one insurance fund whose performance results are almost non-random due to the action of the law of large numbers.

In real life, it is almost impossible for such a large number of people to unite and correctly calculate the economic activity of the fund. Therefore, insurance funds are created and managed by special insurance organizations - commercial insurance companies and non-profit mutual insurance societies.

When using the law of large numbers to combine even a huge number of random variables, it should be understood that it does not guarantee the equality of the observed average results to the expected theoretical values. The law allows us only to say that for large sets, serious relative deviations of the actual results from the expected values ​​are less likely. For large risk associations, the possible amount of losses is more predictable than for small ones. Therefore, the growth in the number of contracts in the insurance company's portfolio is necessary primarily to ensure reliability.

To assess the risk, it is necessary to know the probabilities and expected amounts of losses. But the objectively existing theoretical values ​​of these parameters are unknown. There are only data on insurance cases in the past years, which represent the results of the study of random events. If there are many of them, then according to the law of large numbers, the observed average values ​​will almost certainly be close to expected. Thus, the law makes it possible to use statistical data to assess the likelihood of occurrence of risks and expected losses.

Thus, with regard to insurance, the law of large numbers theoretically justifies:

- the possibility of applying the expected values ​​to the risk assessment;

- the need to increase the number of contracts in the portfolio to reduce relative deviations in results and ensure the stability of the insurance fund;

- the possibility of using statistical data to estimate the probabilities and amounts of losses.

Underlining such a wide application of the provisions of the law of large numbers to insurance problems, it is often called the fundamental law of insurance.

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