Case Studies from Pricing Insurance Risk
Introduction to the Four PIR Case Studies
PIR presents four standard case studies to illustrate the theory. Using the new case page, users can also create their own custom case studies and produce the standard set of exhibits by specifying the stochastic model, capital standard and cost of capital, and reinsurance. See new case instructions for more details.
The four Case Studies are:
- Simple Discrete Example
- Tame: two tame lines
- Catastrophe and Non-Catastrophe Case Study
- Hurricane/Severe Convective Storm
The Cases aim to help practitioners develop an intuition for how each method prices business, informing their selection of an appropriate method for an intended purpose without resorting to trial and error.
The Cases share several common characteristics.
- Each includes two units, one lower risk and one higher.
- Reinsurance is applied to the riskier unit.
- Total unlimited losses are calibrated to ¤100. (The symbol ¤ denotes a generic currency.)
- Losses are in ¤millions, although the actual unit is irrelevant.
For each Case Study we produce a standard set of exhibits. The website supplements these with some etended exhibits that vary by case.
Simple Discrete Example
Results for the Discrete Example.
Ins Co. writes two units taking on loss values X1 = 0, 8, or 10, and X2 = 0, 1, or 90. The units are independent and sum to the portfolio loss X = X1 + X2. The outcome probabilities are 1/2, 1/4, and 1/4 respectively for each marginal. The 9 possible outcomes, with associated probabilities, are shown below. The output is typical of that produced by a catastrophe, capital, or pricing simulation model—albeit much simpler.
| \(X_1\) | \(X_2\) | \(X\) | \(X_1\) | \(\Pr(X_1)\) | \(\Pr(X_2)\) | \(\Pr(X)\) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1/2 | 1/2 | 1/4 | |
| 0 | 1 | 1 | 1/2 | 1/4 | 1/8 | |
| 0 | 90 | 90 | 1/2 | 1/4 | 1/8 | |
| 8 | 0 | 8 | 1/4 | 1/2 | 1/8 | |
| 8 | 1 | 9 | 1/4 | 1/4 | 1/16 | |
| 8 | 90 | 98 | 1/4 | 1/4 | 1/16 | |
| 10 | 0 | 10 | 1/4 | 1/2 | 1/8 | |
| 10 | 1 | 11 | 1/4 | 1/4 | 1/16 | |
| 10 | 90 | 100 | 1/4 | 1/4 | 1/16 |
Tame Case Study
Results for the Tame Case Study.
In the Tame Case Study, Ins Co. writes two predictable units with no catastrophe exposure. We include it to demonstrate an idealized risk-pool: it represents the best case—from Ins Co.’s perspective. It could proxy a portfolio of personal and commercial auto liability.
It uses a straightforward stochastic model with gamma distributions.
The Case includes a gross and net view. Net applies aggregate reinsurance to the more volatile unit B with an attachment probability 0.2 (¤56) and detachment probability 0.01 (¤69).
Catastrophe and Non-Catastrophe Case Study
Results for the Cat/NonCat Study.
In the Cat/Non-Cat Case Study, Ins Co. has catastrophe and non-catastrophe exposures. The non-catastrophe unit proxies a small commercial lines portfolio. Balancing the relative benefits of units considered to be more stable against more volatile ones is a very common strategic problem for insurers and reinsurers. It arises in many different guises:
- Should a US Midwestern company expand to the East coast (and pick up hurricane exposure)?
- Should an auto insurer start writing homeowners?
- What is the appropriate mix between property catastrophe and non-catastrophe exposed business for a reinsurer?
This Case uses a stochastic model similar to the Tame Case. The two units are independent and have gamma and lognormal distributions.
The Case includes a gross and net view. Net applies aggregate reinsurance to the Cat unit with an attachment probability 0.1 (¤41) and detachment probability 0.005 (¤121).
Hurricane and Severe Storm Case Study
Results for the Hu/SCS Case Study.
In the Hu/SCS Case Study, Ins Co. has catastrophe exposures from severe convective storms (SCS) and, independently, hurricanes (Hu). In practice, hurricane exposure is modeled using a catastrophe model. We proxy that using a very severe lognormal distribution in place of the gross catastrophe model event-level output. Both units are modeled by an aggregate distribution with a Poisson frequency and lognormal severity.
The Case includes a gross and net view. Net applies aggregate (see Errata) reinsurance to the HU unit with an occurrence attachment probability 0.05 (¤40) and detachment probability 0.005 (¤413).