Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 43

Finance theory is based on the notion of a security which pays one monetary unit in just one particular state of the world, known as an Arrow-Debreu security (see Section 8.6 for more). Of course, the state of the real world at any instant in time would be unimaginably complex to try to describe, so in practice, abstractions are used.

Example 7 We can imagine a narrower context where actuaries are trying to understand the loss experience of their personal auto physical damage portfolio over the past year. A spreadsheet describes each claim in terms of

 Policy number

 Date and time of loss

 Dollar value of damage

 Policy terms: deductible, limit

 GPS location of accident

 Vehicle make, model, year, and VIN

 Driver name, gender, age

 Other vehicle(s) involved

 Description of accident

 Link to photos of damage

 Link to adjuster report

 Link to police report

This is already getting unwieldy, but it is still not enough to fully explain what happened. Fortunately, the impossible is not necessary. To adequately identify each event, it is enough to note

 Policy number

 Vehicle VIN

 Date and time of loss

 GPS location of accident

because this is enough to put a unique identifier on each claim. More information may be required to assess the adequacy of the existing rate plan, but that’s a different question.

With sufficient detail of identifying variables, we have an explicit representation of risk outcomes. Mathematically, we represent the set of all possible variable value combinations as a set Ω called the sample space, and we characterize one particular combination of values as an element or sample point ω∈Ω. Other attributes not necessary for event identification, such as the amount of damage and driver name, are functions of that unique event identifier ω. If such functionally dependent information exists somewhere in a claim database or other data source, then it can be retrieved and associated with the event.

Example 8 As another example, consider the actuaries responsible for the commercial multiperil line of business. They are looking at their exposure to hurricanes and earthquakes. A simulation study results in a spreadsheet with the following columns:

 Nine-digit simulated catastrophe identifier

 Hurricane/earthquake flag

 Hurricane landfall lat-lon, velocity vector, wind speed, and radius to maximum winds

 Earthquake epicenter lat-lon, peak ground acceleration, and Modified Mercali Index value

 Multiperil portfolio gross loss

 Multiperil portfolio net loss after reinsurance

This information enables the actuaries to start to understand where their peak exposures are.

On another floor of the same building, the actuaries responsible for commerical auto physical damage have a similar file, except losses refer to the commercial auto portfolio. But the same nine digit catastrophe identifier is used. This means that the Enterprise Risk Management folks (in yet another building) can take those two spreadsheets and merge them together to see results across the entire commercial property book of business. Without this linkage, the dependence of results across the two lines of business would be a mystery.

The strength of explicit event representation is that it enables outcomes to be linked across a book of business, thus dependence risk can be modeled without making assumptions. It is useful when events are not too numerous and affect significant portions of the portfolio. When events are very numerous and affect only small portions of the portfolio, explicit event representation does not provide enough benefit to justify its greater complexity.

To summarize: the explicit representation uses a random variable X(ω) defined on a sample space Ω of interpretable sample points, such as typhoon landfall and windspeed, or earthquake epicenter and magnitude. It is the most detailed representation and allows for easy aggregation, critical in reinsurance and risk management. It can distinguish between different events even if they cause the same loss outcome. It suffers from being arbitrary, especially regarding the detail communicated by the sample points, and the complexity of defining events, especially for high volume lines where it is unrealistic to tie an event to each individual policyholder.