11.9 Process, Parameter, and Model Risk

Process & Parameter Risk

\[\underbrace{\text{Variance}}_{\mathrm{Var}(X)} = \underbrace{\mathrm{E}[\text{Process Variance}]}_{\mathrm{E}_{\theta}[\mathrm{Var}[X|\theta]]}+\underbrace{\mathrm{Var}[\text{Hypothetical Mean}]}_{\mathrm{Var}_{\theta}[\mathrm{E}[X|\theta]]}\]

• Process risk: average variance of the outcomes from the expected result

• Parameter risk: variance due to the many possible parameters in the posterior distribution

  • Similar to the idea of “range of reasonable estimates”

• Typically the parameter risk is much larger than process risk

  • e.g. Total Risk = \(\mathrm{Var}\left[ \sum \limits_{w=1}^{10} C_{w,10} \right]\)

    Parameter Risk = \(\mathrm{Var}_{\theta}\left[\mathrm{E}\left[\sum \limits_{w=1}^{10} C_{w,10}|\theta\right]\right] = \mathrm{Var}\left[ \sum \limits_{w=1}^{10} e^{\mu_{w,10} + \frac{\sigma^2_{10}}{2}} \right]\)

Model Risk: Risk of not selecting the right model

• If possible models are “know unknowns”, we can turn the model risk into parameter risk

  • Formulate a model as a weighted average of the candidate models with the weights as parameters

  • If posterior distribution of the weights assigned to each model has significant variability, this indicates of model risk

  • And in this case, just a special case of parameter risk

• If we have a very large dataset to run this model on, the parameter risk will shrink towards 0 and any remaining risk such as model risk will be interpreted as process risk

  • This is mostly a academia thought experiment as most aggregated loss triangles are small datasets

  • This serves to illustrate the theoretical difficulties that occur when we try to work with parameter/process/model classification of risk

  \(\therefore\) Should focus on the total risk