Pricing Insurance Risk. Stephen J. Mildenhall
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Название: Pricing Insurance Risk
Автор: Stephen J. Mildenhall
Издательство: John Wiley & Sons Limited
Жанр: Банковское дело
isbn: 9781119756521
isbn:
3.3.3 Types of Uncertainty
Probabilities can be objective or subjective. Objective probabilities are amenable to precise determination. They are usually based on physical symmetry (coin toss, dice roll) or repeated observations. Objective probability applies the law of large numbers, the central limit theorem, Bayesian statistics, and credibility theory to make precise predictions about samples. Insurance is largely based on objective probabilities from repeated observations (loss data).
Subjective probabilities provide a way of representing a degree of belief. They follow the same rules as objective probabilities and have proven a powerful way to impose order and consistency in economics and finance. Subjective probabilities are applied to nonrepeatable events: an election, a horse race, or an economic outcome. Subjective probability is 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.
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