Aggregate Risk and Capital
Combine frequency, severity, heterogeneity, dependence, and reinsurance into portfolio loss distributions.
Aggregate Risk and Capital
The previous modules modelled claim pieces. Risk theory combines them into a portfolio total and asks how its centre and tail change under dependence and reinsurance.
Two constructions answer different questions:
| Model | Random sum | Natural use |
|---|---|---|
| Collective model | portfolio count and severity are primary | |
| Individual model | identified risks have distinct claim probabilities/benefits |
Both can be extended with covariates, multiple claims, dependence, inflation, and reinsurance. Their simplifying assumptions should be selected for the decision, not because one formula is familiar.
The modelling chain
Rendering diagram…
Questions before calculation
- Is the horizon month, accident year, underwriting year, or ultimate runoff?
- Does count claims, policies, events, or payments?
- Are severities iid and independent of count? If not, conditional on what state might they be?
- Is loss gross or net of policy terms and reinsurance?
- Are inflation, expenses, premium, and investment income included?
- Is the output an expected cost, a pricing margin, economic capital evidence, or a regulatory input?
Learning sequence
- Derive compound-sum moments in Collective Risk.
- Retain risk-level heterogeneity in Individual Risk.
- Reconcile formulas and simulation in the Computation Lab.
- Add tail measures and common shocks in Tail Risk and Capital.
Before trusting a simulation, derive every moment that is available analytically. A mismatch usually reveals a parameterisation, unit, or coding error.