Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

15.4.3 LOGNORMAL MODEL FOR SEVERITIES

In general, one can use the methodology summarized by Equation (15.68) to develop a model combining external data, internal data, and expert opinion for estimation of the severity. For illustration purposes, this section considers the LogNormal severity model; the Pareto severity model is developed in the next section.

Consider modeling severities X₁, …, X_K ,… using LogNormal (μ, σ ²), where X = (X₁,…, X_K) are the losses over past T years. Here, we take an approach considered in Section 15.2.4, where μ is unknown and σ is known. The unknown μ is treated under the Bayesian approach as a random variable Θ_μ. Then combining external data, internal data, and expert opinions can be accomplished using the following model.

Model Assumptions 15.6 (LogNormal–Normal–Normal)

Let us assume the following severity model for a risk cell in one bank:

1. Let Θ_μ ~ Normal be a normally distributed random variable with parameters μ₀ σ₀, which are estimated from (external) market data, that is, π(θ) in (15.68) is the density of Normal ;
2. Given Θ_μ = μ, the losses X₁, X₂,… are conditionally independent with a common LogNormal distribution:
   
   where σ is assumed to ...