Pricing Insurance Risk. Stephen J. Mildenhall

also used with game theory, nonadditive probability, and behavioral science to make predictions and understand uncertainty. Insurance largely avoids pricing based on subjective probabilities, as the problems with creating markets for terrorism and cyber risks attest.

      The differences between objective and subjective probabilities, and their relations to the development of probability and adjacent theories in judgment, psychology, physics, inference, and statistics, are discussed in Diaconis and Skyrms (2018).

      It is usual to distinguish process risk in the face of a well-defined (objective) probability model from uncertainty when there is no probability model or even no clearly defined set of possible outcomes. Process risk is also called a aleatoric uncertainty, derived from the Latin alea or dice. Uncertainty is sometimes called Knightian uncertainty after the economist Frank Knight. Epistemic uncertainty is caused by a knowledge gap, possibly one that could in principle be filled.

      Parameter risk is intermediate between process risk and uncertainty. It refers to a known model with unknown parameters. Actuaries often use Bayesian models to introduce (and sometimes remove) parameter risk. There is a blurred line between parameter risk and uncertainty. For example, a parameter can be used to select between competing models.

      A situation of unknown unknowns represents an extreme form of uncertainty.

      These concepts and relationships are summarized in Figure 3.1.

Figure 3.1 Taxonomy of insurance-related risk.

Example 6 When the US National Science Foundation decided to fund the LIGO search for gravitational waves there was epistemic uncertainty since no one knew for sure whether gravitational waves existed. If they did exist, experiments to detect them would still be subject to process risk: the chance none would be detected because no events producing gravitational waves occur during the observational period. There is even operational risk, that the detector malfunctions and misses an event. P.S. It appears they do exist!

      3.4 Representing Risk Outcomes

      A risky outcome can be labeled explicitly by describing the facts and circumstances causing it. Or we can identify the outcome with its value. Or we can identify it with the probability of observing no larger value.

      This section explores the mechanics, pros, and cons of these three representations. We call them the explicit, implicit, and dual implicit representations, respectively.

      3.4.1 Explicit Representation

      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.

3.4.2 Implicit Representation

      The implicit representation identifies an outcome with its value, creating an implicit