Beyond Static Models: Applying Randomized Distortions to Insurance Risk Management

1. Introduction: The Challenge of Fixed Risk Measures

In actuarial science and insurance, risk measures are fundamental tools for quantifying potential financial losses. Practitioners and regulatory bodies frequently rely on well‑known measures such as Value at Risk (VaR) and Average Value at Risk (AVaR) to set capital requirements and manage solvency. These measures are typically forms of distortion risk measures, meaning they operate by applying a fixed mathematical function‑known as a distortion function‑to the probabilities of outcomes to arrive at a single value for risk.

This article explores a sophisticated approach detailed in a recent research paper by Ohood Aldalbahi and Miryana Grigorova that leverages "randomized" distortions of these measures. While the concept of random distortion functions has been explored in prior literature, this paper's key contribution is extending the framework to handle situations of model ambiguity‑where the true probability of losses is uncertain‑through the use of capacities. It provides crucial representation theorems for these more generalized risk measures, offering a formal structure for creating more dynamic and context‑aware risk models.

2. The Core Concept: What is a "Randomized" Risk Measure?

The core of this approach is the "G‑random distortion function." In simple terms, this is a mathematical tool that allows the parameters of a risk measure to change depending on a specific state of the world or a particular piece of information available to a decision‑maker.

Think of a common scenario in a large insurance firm: multiple experts are tasked with assessing the same risk, but they disagree on the correct parameter to use for a calculation. For example, they might agree on using VaR but disagree on the acceptable level of risk (the α parameter).

The "randomization" detailed in the paper allows a decision‑maker to formally model their confidence in different expert opinions under different circumstances. The structure of the model is determined by G, a sub‑σ‑algebra that formally represents the specific information the decision‑maker possesses about the experts' credibility in different states of the world.

3. Application 1: A "Randomized" Value at Risk (VaR)

The research paper provides a practical example of how this framework can be applied to Value at Risk.

Scenario Setup

The scenario involves two experts and a single decision‑maker, modeled as follows:

• The Disagreement: Two experts agree on using VaR as the risk measure but disagree on the acceptable risk level. The first proposes a level of α, while the second proposes a more conservative level of β, where α < β.
• The Model: A "random distortion function" is created that applies the VaR with level α in some situations (when an event ω occurs in a set A) and with level β in others (when ω occurs in the complementary set Ac).
• The Interpretation: The set A represents the scenarios where the decision‑maker considers the first expert to be "right," while Ac represents the scenarios where they consider the second expert to be "right." This is specifically relevant in cases where the two chosen risk levels would lead to different outcomes.

Outcome Analysis

For a potential loss X that occurs with probability p, the randomized VaR produces a clear set of outcomes that depend on where the probability p falls relative to the experts' chosen risk levels.

4. Application 2: Extending the Concept to Average Value at Risk (AVaR)

The same "randomization" principle can be applied to the Average Value at Risk (AVaR), another cornerstone of modern risk management. The paper outlines a similar scenario where two experts agree on using AVaR but disagree on the parameter (α vs. β).

However, the nature of their disagreement is more complex than in the VaR example. In the AVaR case, even when the probability of loss p is below both thresholds (p ≤ 1 - β), the experts still assign different non‑zero risk values: the first expert calculates the risk as p/(1‑α), while the second calculates it as p/(1‑β). This contrasts sharply with the VaR scenario, where they agreed on a risk value of zero in the equivalent situation.

This makes the framework's utility even more compelling. The randomly distorted Choquet integral‑a tool that generalizes standard integration to account for ambiguity captured by capacities‑provides a formal method to "randomise" the risk measure across all states where the experts' chosen parameters lead to disagreement, allowing the model to reflect this uncertainty rather than forcing a single, artificial consensus.

5. Advanced Applications: Handling Multiple Experts and Model Ambiguity

The framework presented by Aldalbahi and Grigorova is not limited to two experts. It can be extended to handle more complex and realistic scenarios.

1. Multiple Experts: The model can be generalized to accommodate n different experts, each suggesting a different risk parameter (α1, α2, · · · , αn). The model then constructs a risk measure that depends on which of the n experts a decision‑maker trusts in any given situation, providing a sophisticated tool for aggregating diverse expert opinions.
2. Model Ambiguity (The Capacity Framework): This approach is particularly powerful when there is ambiguity about the true probability distribution of potential losses. In such cases, a mathematical object called a "capacity" is used instead of a single probability measure to assess random events. Using randomized risk measures within this capacity framework makes them even more robust, as they can formally account for uncertainty in the underlying model itself, not just in its parameters.

6. Conclusion: A More Flexible Future for Risk Modeling

The randomly distorted risk measure framework offers a significant benefit for the insurance industry. It provides a formal, rigorous mathematical structure for handling common real‑world challenges like expert disagreement and model ambiguity. By moving beyond static, one‑size‑fits‑all models, this approach allows for the creation of more dynamic, nuanced, and realistic risk management tools that better reflect the complex nature of financial and actuarial risk.