Collective Risk Model In Insurance

    Collective Risk model in insurance: is intended to model the total loss amount Z for claims, which incurred during a time period of length T (e.g. during 1 year) in a given insurance portfolio. Individual risk model: deals with risks corresponding to particular (individual) insurance policies:

     

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(total claim amount in individual risk model: Y1, ..., Yk is a sequence of claims corresponding to particular insurance policies for an insurance portfo­lio consisting of k policies; typically, Yi are independent random variables (see Sect. 26.6))

Collective risk model: assumes that in homogeneous insurance portfolios (or tar­iff groups, see Sect. 21.1) the claims incurred due to particular insurance events are identically distributed (and mostly also independent) random variables:

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(total claim amount in collective risk model: Xi, ..., XN is a sequence of claims (unlike the individual risk model, the ordering of this sequence is arbitrary regardless of the corresponding policies), and N is the number of claims during a given period; typically, Xi are independent and identically distributed random variables which are independent of the random variable N)

Models for number of claims: assume a probability distribution of the random variable N (the number of claims during a given period, see thereinbefore) with values 0, 1, 2,...; the best used probability distributions of N are:

- Poisson distribution (see Sect. 26.4) N ~ P(X): is the discrete probability dis­tribution with one parameter X >0; such an N may model the number of claims for a large number of independent homogeneous policies with a small probability of the claim (so called "distribution of rare events"):

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- negative binomial distribution (see Sect. 26.4) N ~ NB(r, p): is the discrete probability distribution with two parameters r >0 and 0 < p <1; such an N may model (for r e N) the number of failures before the rth success in independent trials with probability p of success:


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- mixed Poisson distribution: is the Poisson distribution with random parameter X (X is interpreted as a random intensity with distribution function F(X)); it is applied to insurance portfolios with heterogeneous risks, where insurance policies with a small risk (or a large risk) have X with small values (or large values), respectively; in particular, the case of a constant X with P(X = X0) = 1 implies the distribution P(X0) of N (see thereinbefore), and the case of the gamma distribution (see Sect. 26.5) with parameters p/(1 - p) and r implies the distribution NB(r, p) of N (see thereinbefore):
C:\Users\88CC~1\AppData\Local\Temp\FineReader10\media\image3.jpeg 

Models for number of claims in K tariff groups ( Collective Risk Model ): N denotes the number of claims during a given period in K mutually independent tariff groups (i.e. N = N\ + ... + NK, where Ni is the number of claims during a given period in the ith tariff group):

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Models for claim amount: assume a probability distribution of the random vari­able X (usually the claim amount per one claim during a given period, see thereinbefore) with non-negative values; the best used probability distributions of X are:

- logarithmic normal distribution (see Sect. 26.5) X — LN(p, a2): is the con­tinuous distribution with two parameters -to < p < to and a > 0; it holds ln X — N(p, a2); such an X may model the claim amount e.g. in accident, private motor, fire, windstorm and other insurances:

 

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- exponential distribution (see Sect. 26.5) X — Exp(A): is the continuous distri­bution with one parameter A > 0; it is a special case of the gamma distribution and the Weibull distribution for a = 1 (see thereinbefore); the exponential dis­tribution is also used to model the lengths of periods between insurance claims (see Sect. 22.2):

C:\Users\88CC~1\AppData\Local\Temp\FineReader10\media\image10.jpeg

- beta distribution (see Sect. 26.5): is the continuous distribution with two parameters p >0, q > 0; the U-shaped probability density of the beta distri­bution for p <1, q < 1 is used to model claims e.g. in the fire insurance (either very small claims, or on contrary very large ones are typical for this insurance product):

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- Pareto distribution (see Sect. 26.5): is the continuous distribution with two parameters a >0, b >0; due to heavy tails, it is applied in situations with outlying extreme loss amounts, e.g. in the sickness, fire and other insurances:

 

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