Degree of risk - Actuarial calculations

Degree of risk

For an approximate initial assessment of the appropriateness of taking a risk, calculate a coefficient called in actuarial calculations the degree of risk:


where M ( Y) and σ ( Υ) is the mathematical expectation and the mean square deviation of the damage under the contract.

The degree of risk is, from the point of view of mathematical statistics, nothing more than coefficient of variation - the ratio of the standard deviation of the damage to its mathematical expectation, the dimensionless measure of the measure of the dispersion of damage values. As the volume of the portfolio of contracts grows, this coefficient decreases (the financial stability of the insurer rises) - the degree of risk decreases.

According to its values, the insurer performs a primary assessment of the appropriateness of accepting/not accepting the risk. In the actuarial literature, it is generally accepted that the risk is acceptable if Κ (Υ ) & lt; 0.3.

The degree of risk of a contract that has distributed damage in the event of an insured event:


If in the event of an insured event the damage is fixed:


In a portfolio of n homogeneous contracts, the number of insurance cases in which satisfies the binomial distribution law:

where K - the degree of risk in one insurance contract; K - the degree of risk in a portfolio of n homogeneous contracts.

With an increase in any portfolio of independent contracts in n times the risk level decreases by V/F times:



The insurance company was approached by a potential insurer and proposed a new risk for the insurer. Insurance sum S = C = = 20 000 cu The insurer estimated the probability p of the insured event to be 0.001. Under the terms of the contract, if the insured event occurs, the insurance company pays the insurance amount in full.

a) Determine if the insurer is interested in taking this risk. What is the risk of accepting such an agreement?

b) Find the conditions when the proposed new type of risk insurance company can take with a recommended risk of 0.3.


a) Since the damage is fixed, we calculate the risk premium for a new type of risk by the formula

For taking a risk, the insurer, in addition to the average payout, must also estimate the deviation from the average. Due to the fact that the damage is fixed, the number of insurance events that occurred during the year in such a "portfolio" (from 1 contract), obeys the binomial distribution law with the following parameters:

To determine the amount of deviation of the payment from its average, the insurance amount must be multiplied by the standard deviation:

The risk level of the insurer will be:

This value can be easily obtained from the formula (2.23) deduced by us for one contract with fixed damage:

We found out that it is reasonable for an insurer to refuse this type of risk, since the degree of risk is more than 100 times higher than the allowable value (0.3).

b) We know that the risk level is decreasing with the growth of the portfolio, so the task is to determine the volume when the insurer can take a new look at an acceptable level of risk equal to 0.3. Using formula (2.24), we can write:

In order for the insurance company to implement a new type of insurance at an acceptable level of risk, it is necessary to sell at least 11 100 similar policies.

Answers :

a) since the risk level is K (Y) = 31.6, then the insurance company must refuse to accept such a single risk to insurance;

b) in order to introduce a new type of insurance and ensure payments only to collected premiums, it is necessary to increase the portfolio volume to n = 11 100 similar risks.

There is a fundamental difference in the role of the standard deviation (SDE) in property insurance and life insurance. In property insurance, this role is very large, since the risk is estimated by the degree of risk K. In life insurance, the actuary relies on stable patterns expressed by the survival curve, therefore the role of the RMS is negligible. The amount of payments in life insurance is usually fixed, and in property insurance - distributed, therefore the degree of risk in property insurance is higher than in life insurance.

The maximum size of the accepted risk. Decision accept/not accept The new risk is based on the principle of reducing the degree of risk: not to take risks that worsen the situation in the portfolio. The maximum is determined by relying on the principle of equality of risk before and after taking a new risk. Therefore, the maximum size of the accepted risk depends on the degree of risk and the size of the company's insurance portfolio.

The effect of the risk level on the risk premium is such that the higher the risk level (coefficient K), the higher the risk premium should be.

Comparison of the previously obtained formulas for the relative risk premium calculated by the quantile method and the degree of risk leads to their linear dependence:


Thus, with an increase in the portfolio of contracts in n times both the degree of risk and the risk premium will decrease in ψη times. This, as we have already stressed, allows large insurance companies that collect portfolios of large contracts to offer their customers lower tariffs, increasing their competitiveness, while not losing in reliability (the probability of inconsistency).

At the same time, the amount of the risk premium can not exceed reasonable limits, since there is an insurance market and the average price of services on it. Therefore, the insurer is forced to

either refuse the client, or offer to take some of the risk to yourself. In the latter case, treaties with proportional compensation for damage, under the rule of the first risk, treat contracts with a franchise, i. various contracts of partial insurance.


The owner of the apartment insured her against damage in case of flooding for 2000 USD. for a period of 1 year. Consider a proportional protection agreement with an insurer's liability of 60% of the damage if the probability of an insured event is estimated at 5% and the amount of damage is distributed:

1) uniformly from 0 to the cost of the object 2 thousand cu;

2) according to the truncated exponential law with density

Find for each option:

a) a one-time risk premium;

b) the relative risk premium (with the help of the quantitative principle), which ensures the probability of the insurer performing its obligations at least 78%, as well as the net premium and gross premium, with the burden on handling business and profits 15% of the tariff, if the insurer has 210 such contracts in its portfolio;

c) what would be the relative risk premium, net premium and gross premium, if the portfolio volume increases by 7 times.


1. Since the insured has chosen a contract with proportional protection with the liability of the insurer 60% of the damage, according to the definition of this type of contract, the insurer reimburses an amount equal to 0.6X for all losses that have occurred. Therefore, the payments of the insurer are equal: Y = 0.6 • X.

Recall that a risk premium is a mathematical expectation of damage. Let's find the conditional mathematical expectation of payments of the insurer provided that the insured event occurred, according to the formula (2.5):

1а. The probability density function according to the uniform distribution law:


The variance of payments of the insurer, provided that the insured event occurred, according to (2.7) is:

To proceed to the unconditional distribution of damage, we calculate the total mathematical expectation (2.8) and variance (2.9) of payments:

One-time risk premium of the insurer under such agreement РП = 30 cu

1b. To find the risk premium, we use a different formula in comparison with Example 2.4-namely, the derived formula (2.25), which relates the relative risk premium to the degree of risk.

Calculate the coefficient of variation - an indicator of the degree of risk for the study agreement:

The coefficient of variation (& gt; 0.3 many times) indicates a very high degree of risk in one contract.

Now we define the relative risk premium Θ. It ensures that the insurer is not insolvent at a given level. The probability of fulfillment of obligations by the insurer by the condition of the task:

We obtained that the distribution function of the standard normal law Φ (t, _E) = 0.78, hence (Appendix 1), the quantile of the required level t] _ε = 0.77.

So, the relative risk premium for these conditions is (2.25):

And the degree of risk in the portfolio of 210 such contracts decreased to (2.24):

It has become almost acceptable.

The size of the net premium is easily determined from (2.10), using the previously obtained calculations:

Then the gross premium, taking into account the workload of 15%, is (2.1):

1c. Now consider the option of increasing the portfolio volume by 7 times. Then the relative risk premium will be:

The degree of risk will decrease:

The net premium will also decrease:

The gross premium with the increase in the portfolio volume will be:

Thus, with an increase in the portfolio volume by 7 times, the relative risk premium will decrease in time (almost 2.65 times), and, accordingly, decrease and net -, and gross-nremii, thereby increasing the competitiveness of the insurer.

2. Similarly, arguing, we will calculate the characteristics of the insurer's damage if the density function is subordinated to a truncated exponential distribution law.

The truncated exponential distribution (truncated exponential distributum) is determined by the distribution function:

This is a mixed type distribution. It has, as shown in the figure, a continuous part with a probability density function , and a discrete part - "bunch" Probability mass (probability mass in the original) at the point C.

The graph of this distribution function looks like:

In our case, the exponential distribution parameter is λ = 0.001, the probability mass at the point C:

This value corresponds to the probability of all damages exceeding the maximum value equal to the insurance sum of the contract of 2000 USD, at which payments are cut off at the level C.

The moments of such a mixed random variable are found as the sum of the moments for the discrete and continuous component:

Now let's proceed to finding the conditional numerical characteristics of the damage values ​​- the mathematical expectation (2.5), taking into account the partial payments of the insurance company - the proportional protection contract:

Similarly, by formula (2.7), by integration by parts there is a conditional variance:

Now we calculate the unconditional distribution of damage and the degree of risk using formulas (2.8) and (2.9):

In this version of the damage distribution function, the coefficient of variation is even higher than in the first version and also greater than 0.3 many times, which indicates a very high risk inherent in one such contract.

Reasoning similarly to the first part of the solution, we get the following values ​​of the risk premium, net premium and gross premium for a portfolio of 210 contracts:

If the volume of the insurer's portfolio is increased by 7 times, the corresponding values ​​will change as follows:

With a 7-fold increase in the portfolio, the relative risk premium will also decrease in {7 times (almost 2.65 times), and both net and gross premiums, thus increasing the competitiveness of the insurer.


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