Risk Theory (I)

The two chapters Risk Theory I and Risk Theory II are obviously related. Together, they cover the individual and collective risk models, but in addition to the mathematical treatment there is also a lot of general insurance general knowledge. Hence, Risk Models 1 starts with an explanation of the desirable features of a risk portfolio, from a non-numerical perspective.

This beginning material about Risk theory emphasises stochastic processes which model counts – in order to model a count of claims on an insurer, known as a Poisson process – and how they can be combined with other statistical distributions – which can model the size of the claims.
Three compound distributions are introduced to help model these processes – the compound Poisson distribution, compound binomial distribution and compound negatvie binomial distribution. Each is good for in different conditions

Poisson is a good all purpose count distribution

Binomial is good where the portfolio size sets an upper limit on the possible number of claims

Negative binomial is an alternative to the Poisson in that it doesn’t set an upper limit, but allows for a variance greater than the mean.

In each case we are trying to model an integer number of claims, which can be extended by considering the severity of a claim to model the total amount payable across all claims. Hence, for example, arrive at the compound Poisson distribution, where the number of claims is modelled by the Poisson distribution, and the severity is modelled independently and separately for each claim by another distribution. For example, you could potentially model a team’s score in an AFL football match by using a Poisson distribution to model the number of scoring shots, and a Bernoulli trial on each scoring shot to determine if it was a goal worth six points or a behind worth one point. This would obviously simplify to a Binomial distribution, with the ‘n’ variable itself a random variable with a Poisson distribution.