Ordering of insurance risk

Comparing and ordering of risks is a basic problem of actuarial theory and practice. Risks are generally modeled by random variables or distribution functions. There is a great variety of stochastic models in use which reflect the diversity of insurances like theft insurance, car insurance, liability insurance, etc. The diversity of insured populations is reproduced in the stochastic models by introducing individual or collective risk models which include internal, external, and group risk factors. There are risk models with rare extreme events, and on the other hand models with moderate or even bounded risks. A particular problem in risk theory is dependence in a portfolio of insurance policies which may lead to a drastic increase of the risk of the portfolio.