Normal approximation of aggregate damage by portfolio, Modeling...

Normal approximation of aggregate damage by portfolio

If the number of contracts in the portfolio is large enough, we can use the central limit theorem of probability theory to find the numerical values ​​of the distribution of the sum of independent random variables using the following algorithm:

1) find the mean and variance of the random variables, for each random variable;

2) sum them to find the average and variance for the entire portfolio as a whole;

3) use the normal (Gaussian) approximation.

If the random variables are independent and identically distributed with the average μ and variance , then the central limit theorem:

(3.10)

where is the distribution function of the standard normal distribution law (Appendix 1).

The function of distribution of the value of the aggregate damage on the portfolio :

(3.11)

EXAMPLE 3.5

Using the data in Example 3.1, find out how many such contracts an insurer has to collect so that, given the reliability of 1-ε = 0.95 given in clause c), the relative risk premium does not exceed 20% (use normal approximation) and the given probability of non-disruption is ensured only by collected premiums.

Solution

According to (3.11)

According to (2.25), the relative risk premium is:

Convert this formula and express the number of contracts n :

(3.12)

By the condition, it is necessary to provide the probability of inconsistency F (t) = = 0.95. Hence, Φ (t) = 0.95 = & gt; according to the table of Appendix 1 of the distribution function of the standard normal law I 1,645.

Using the built-in function Excel , you can immediately find a level quantifier of 0.95 of the standard normal law: t = = NORMSINV (0,95) = 1,64485.

In Examples 3.1 and 3.3, we already found the mathematical expectation and variance for individual risks X,:

Now we can find the required number of contracts by (3.12):

So, the insurer must collect at least 440 such contracts so that for a given reliability of 1 - ε = 0,95 the relative risk premium does not exceed 20%.

Simulation of the collective loss of a group of risks within the framework of an individual model

The main actuarial task is to simulate the cumulative distribution function - knowing it, we know all about the total damage to the portfolio and can find any required probabilities, calculate tariffs, etc.

The bulk of the probabilities in the insurance portfolio is concentrated at the zero point of the risk loss (Υ -, = 0), because in almost all types of risk insurance the overwhelming majority of risks (in a particular year) does not generate losses. Hence, the distribution of the random variable V | far from the normal distribution.

With increasing number of independent and identically distributed risks, the standardized aggregate loss (Y - M (Y))/o (Y) according to the central limit law, really becomes more and more like a normally distributed quantity (in the sense of convergence of distribution). But, in view of the extreme asymmetry of the distribution of the quantity V, -, in most risk groups this law works only with a very large number of risks.

In order to obtain an acceptable approximation of the distribution of the cumulative loss Y, a small (as is typical for practice) risk group approximates the distribution of the total loss Y, the individual risk i by a continuous distribution that allows an explicit calculation of the convolutions. For a rough approximation of the distribution of the quantity Y, it is sufficient to know that the majority of the probabilities are at zero. In the subsequent convolution process, the approximation inaccuracy is quickly leveled out and very close to reality models of the cumulative loss are achieved (at least for the bulk of the probabilities).

The most well-known distribution on the interval (0; 00), which makes it possible to calculate the convolutions in explicit form, is the gamma distribution.

Gamma distribution

To model the loss distribution in actuarial calculations, it is usual to use not the usual gamma distribution, but with parametrization involving the expectation μ.

The probability density function of such a distribution is:

where a is the form parameter (a & gt; 0).

Since it is important that the approximating distribution of the magnitude of the cumulative loss has the largest probabilistic weight near the zero point, only the gamma distribution with the shape parameter a <1> 1 is taken into account in the actuarial calculations (Fig. 3.1).

Γ (Ω) is the Euler gamma function:

The gamma function has the following properties:

1) Γ (α + 1) = aT (a) for any a> 0;

2) if a is a natural number, then: T (a + 1) = a;

3)

The main characteristics of a gamma distribution with parameterization:

- the mathematical expectation is equal to M (x) = μ;

- variance ;

is the coefficient of variation

- the coefficient of asymmetry

Estimates of the parameters by the method of moments on the sample data are found as:

where - the average of the (selective average) damage;

- selective variance of damage.

Then it is desirable to refine the estimates using the maximum likelihood method.

Gamma distribution can be considered a completely realistic model for aggregate and normalized losses of a group of identically distributed independent risks.

In addition, the gamma distribution has a favorable property: the sum of independent gamma-distributed risks has a gamma distribution even if the parameters μ, • and q; not the same

The probability density of the gamma distribution as a function of the shape parameter a

Fig. 3.1. The probability density of a gamma distribution as a function of the form parameter a

for all risks, by constantly their ratio μ (• /a - v This makes it possible to model with the help of gamma distributions the aggregate loss of a group of risks with different insurance sums.

The inverse Gaussian distribution

The inverse normal (Gaussian) distribution is used to model non-negative random variables for which the right side of the distribution curve is more gentle than the left one. While a normal random variable can also take negative values.

In actuarial calculations, the reverse Gaussian distribution with the parameterization involving the expectation μ, whose probability density has the form (fig 3.2):

A random variable distributed according to the inverse Gaussian law can take only positive values.

The parameter μ is a position parameter that coincides with the mathematical expectation of a random variable in the same way as for a normal distribution law.

The parameter a is a parameter of the form, with increasing (a- + °°) the curve of the density of the inverse normal distribution becomes more similar to the normal distribution. Therefore, in actuarial calculations, as in the case of a gamma distribution, a <1>

is used.

Fig. 3.2. The probability density of the inverse Gaussian distribution

The main characteristics of the inverse Gaussian distribution with parametrization:

- the mathematical expectation is ;

- variance ;

is the coefficient of variation

- the coefficient of asymmetry

Estimates of parameters by the method of moments are determined by the formulas

The inverse Gaussian distribution can also be considered as a completely realistic model for the aggregate and normalized losses of a group of identically distributed independent risks and in many respects similar to the gamma distribution.

Like the gamma distribution, the inverse Gaussian is preserved as a result of convolution even when the parameters and , are not the same for all risks, but their ratio is always , so it is also suitable for modeling the distribution of the cumulative loss with different insured sums.

One of the advantages over a gamma distribution is the ability to express the distribution function through a standard normal distribution and its tabulated distribution function (Appendix 1):

Lognormal distribution

The continuous random variable X has logarithmically normal ( lognormal ) distribution with the parameters μ and σ if its logarithm is subordinate to the normal law, and the probability density function looks like (Figure 3.3 ):

The logarithmically normal value assumes only positive values.

Since inequalities and are equivalent, then the distribution function of the lognormal distribution coincides with the normal distribution function of the random variable In.k

where Φ (/) is the distribution function of the standard normal quantity of the form

The parameter μ is the scale parameter. If in the normal distribution law the parameter μ acts as the average value of the random variable X, then in the lognormal - as the median of the random variable X.

As in the case of a normal distribution, the probability density of a lognormal distribution can not be integrated to obtain the probability distribution function in explicit form. However, the values ​​of the integral function of the lognormal distribution can be found using the values ​​of the integral function of the standard normal distribution (Appendix 1).

The lognormal distribution has a steep left and flat right descent, i.e. positive asymmetry. As the parameter μ increases, the probability density distribution curve will shift to the right, approaching a normal curve.

The parameter σ is the standard deviation of In. d is the shape parameter. The smaller the σ, the more asymmetric the curve, po -

Probability density of lognormal distribution

Fig. 3.3. Probability density of lognormal distribution

In the actuarial calculations, the lognormal distribution is applicable only if σ takes small values, less than the parameter μ.

The main quantitative characteristics of the lognormal value:

- the mathematical expectation is ;

- variance ;

is the coefficient of variation ;

is the asymmetry coefficient ;

Statistical estimates of the parameters t and σ lognormal distribution on the basis of the sample data can be determined by the method of moments:

The lognormal distribution is formed as a result of multiplying a large number of independent or weakly dependent nonnegative random variables, each of which is small in variance with the variance of the result. The logarithmically normal distribution is based on the multiplicative process of forming random variables; such that the effect of each additional factor on a random quantity is proportional to its achieved level.

The lognormal distribution is not invariant with respect to convolution, unlike the gamma distribution and the inverse Gaussian distribution, and is well suited for modeling the size of losses in a separate insured event.

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