# 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.