Risk Management
Exam 9's Risk Management domain tests how an insurer names, measures and manages the risks it faces, and - more heavily - whether you can tell a usable risk measure from an unusable one. Questions typically pair definitions with a short numerical tail calculation.
Classifying and treating risk
- Sources of risk: hazard (property, liability, personnel), financial (price, credit, liquidity), operational (people, process, systems, external events) and strategic.
- The management cycle is identify → quantify → treat → monitor. Treatments are avoidance, loss control, retention and transfer; the frequency–severity grid puts high-frequency/low-severity exposures on retention and low-frequency/high-severity exposures on transfer.
- Transfer is never complete: expect retentions and deductibles, basis risk, moral hazard, adverse selection, and counterparty credit risk on the reinsurer or hedge provider.
Properties of risk measures
- VaR at confidence α is the α-quantile of the loss distribution: the smallest x with P(L ≤ x) ≥ α. It is silent about how bad losses are once the threshold is breached, and it is not subadditive, so VaR(A+B) can exceed VaR(A)+VaR(B).
- TVaR (expected shortfall) averages losses in the tail beyond VaR and is coherent - subadditive, monotone, positively homogeneous, translation invariant - which is why it aggregates cleanly across business units.
- Standard deviation and semi-deviation describe dispersion; ruin probability describes solvency, not earnings volatility.
Pooling and diversification
- Pooling lowers the coefficient of variation of the average loss (roughly by √n for independent identical risks) but never lowers the expected loss.
- Variance benefit depends on correlation: σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Correlation below 1 always helps; correlation of 1 removes the benefit entirely.
Worked example
Losses: 90% chance of 0, 6% of 100, 3% of 500, 1% of 2,000; mean loss = 41. P(L > 100) = 4% ≤ 5%, but P(L > 0) = 10% > 5%, so VaR95 = 100. The worst 5% of outcomes contain all of the 500 and 2,000 mass plus 1% of the 100 mass: (0.03×500 + 0.01×2,000 + 0.01×100)/0.05 = 36/0.05 = 720. TVaR is more than seven times VaR here - that gap is exactly the tail information VaR discards.
Traps
Calling VaR a maximum loss; assuming VaR is subadditive; computing TVaR as the loss at the threshold rather than the tail average; confusing risk retention with risk avoidance; ignoring moral hazard when a policy is fully insured.