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:



(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:


(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"):


- 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:


- 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):

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):


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:



- 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):


- 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):


- 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:



Add comment

Security code


o Microsoft 43.10 ▲0.05 (0.11%)
o Google 574.01 ▲0.64 (0.11%)
o Yahoo 43.76 ▼0.23 (-0.52%)


Company ID [NASDAQ:MSFT] Last trade:43.10 Trade time:12:50PM EST Value change:▲0.05 (0.11%)


Company ID [NASDAQ:GOOG] Last trade:574.01 Trade time:12:50PM EST Value change:▲0.64 (0.11%)


Company ID [NASDAQ:YHOO] Last trade:43.76 Trade time:12:50PM EST Value change:▼0.23 (-0.52%)

Capital Market Expectations:


The survey method of expectations setting involves asking a group of experts for their expectations and using the responses in capital market formulation. If the group queried and providing responses is fairly stable, the analyst in effect has a panel of experts and the approach can be called a panel me...

Tuesday, 17 May 2011

Fixed Income Manager:

Emerging Market Debt

Emerging markets comprise those nations whose economies are considered to be developing and are usually taken to include Latin America, Eastern Europe, Africa, Russia, the Middle East, and Asia excluding Japan. Emerging market debt (EMD) includes sovereign bonds (bonds issued by a national government) as well as debt securities issued ...

Friday, 20 May 2011

Alternative Investment:

Real Estate Market. Types of Real Estate

As one of the earliest of the traditional alternative investments, real estate plays an important role in institutional and individual investor portfolios internationally. The focus of our discussion is equity investments in real estate (covered in the definition given earlier)....

Wednesday, 25 May 2011

Equity Manager:

Long-Short Portfolio

Whereas style investing is concerned with portfolio characteristics (low P/E, high earnings growth, etc.), long—short investing focuses on a constraint. Essentially, many investors face an investment policy and/or regulatory constraint against selling short stocks. Indeed, the constraint is so common and pervasive that many investors do not even recognize it ...

Tuesday, 24 May 2011