Statistics for general insurance

Structure

In this part, I would like to introduce some basic statistics for general insurance, which includes the following topics:

• Chain ladder method for loss reserving
• Loss distributions
• Risk theory
  • Collective risk model
  • Individual risk model
• Ruin theory

Basic concepts

Before we move on to the next topic. Let’s introduce some definitions and concepts in general insurance.

For the insurance company, the main goal is to estimate the ultimate losses, the total amount of claims that will be paid out for a given accident year. The ultimate losses can be divided into two parts:

• Paid losses: the amount of claims that have been paid out by the insurance company.
• Total reserves: the estimate amount of claims should be paid out in the future.

So we can have the following equation:

$$\text{Ultimate losses} = \text{Paid losses} + \text{Total reserves}$$

The Total reserves can also be separated into two parts:

• Case reserves: the estimate amount of claims should be paid out for the claims that have been reported but not settled yet (outstanding claims).
• IBNR reserves (Incurred But Not Reported): the estimate amount of claims should be paid out for the claims that have not been reported yet (IBNR claims).

In summary, we can show the relationship between these concepts in the following diagram with claims and losses/reserves:

In all, we can use the following expression to represent the relationship between these concepts:

$$\text{Ultimate losses} = \underbrace{\text{Paid losses}}_{\text{Paid claims}} + \overbrace{\underbrace{\text{Case reserves}}_{\text{Outstanding claims}} + \underbrace{\text{IBNR reserves}}_{\text{IBNR claims}}}^{\text{Total reserves}}$$

References

Most of the content in this documentation is based on the following references:

Boland, Philip J. 2007. Statistical and Probabilistic Methods in Actuarial Science. Interdisciplinary Statistics. Boca Raton, FL: Chapman & Hall/CRC. http://catdir.loc.gov/catdir/toc/fy0709/2007060501.html.

Kaas, R. 2008. Modern Actuarial Risk Theory: Using R. 2nd ed. Berlin: Springer Verlag. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=016685508&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.