Context: insurance, statistics, actuarial science

Risk theory is an important part of insurance mathematics. It is basically concerned with modelling the arrival and the amount insurance claims and how that affects the solvency of an insurance company. This is particularly important as one does not want an insurance company to run out of money to pay for the claims, a situation experienced by HIH.

Risk theory is broadly divided into two parts:

• Individual risk theory. Here, one is concerned with a fixed number of insurance policies, each with a probability of generating a claim (for each time period). Thus the random variables involved here are the vector of indicator variables of whether a claim has occurred, and the vector of claim amounts.
• Collective risk theory. Here, one not concerned so much with the number of policies, but the number of claims. The random variables here are the number of claims, and the vector claim amounts (given the number of claims).

From these, the following issues are considered:

• The distribution of aggregate claims. For each time period, we want to find out (hopefully analytically) the distribution of the total claim amount. Particuarly troubling at the moment are when:
• The surplus process or how much money an insurance company has left after collecting premiums and paying out claims.
• Ruin. This is an extremely heavily studied subject. Of concern are the probability of ruin (within a finite time period, and ultimate probability of ruin), the time of ruin if ruin accurs, the surplus at ruin and the claim amount that caused the ruin. This will affect the premium rates that an insurance company is going to set, the amount and type of reinsurance it acquires, and the amount of initial capital it requires when it starts a new line of business.

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