Risk Theory and Credibility
Risk Theory and Credibility
This domain tests aggregate claim models, ruin concepts, and credibility weighting. Items often ask for the mean and variance of aggregate claims or the credibility factor from variance components.
Core rules and concepts
- Collective risk model: S=Σ_{i=1}^N X_i with N independent of severities X_i.
- E[S]=E[N]E[X]; Var(S)=E[N]Var(X)+Var(N)(E[X])².
- Compound Poisson: N~Poisson(λ), so E[S]=λE[X] and Var(S)=λE[X²].
- Cramér-Lundberg surplus U(t)=u+ct−S(t); ruin probability decreases with initial surplus and premium loading.
- Limited fluctuation credibility chooses Z large enough for full-credibility standards.
- Bühlmann credibility: Z=n/(n+k), k=EPV/VHM, where EPV is expected process variance and VHM is variance of hypothetical means.
- Credibility premium=Z×sample mean+(1−Z)×prior mean.
Worked example
Claim counts have E[N]=4 and severities have E[X]=500 and E[X²]=400,000. Under compound Poisson, Var(S)=λE[X²]=4×400,000=1,600,000, so SD(S)=1,264.91. Using Var(S)=E[N]Var(X) alone would understate risk because it omits claim-count variability.
Traps that cost marks
- Omitting the Var(N)(E[X])² term.
- Using the number of observations as Z instead of n/(n+k).
- Confusing EPV and VHM in k.
- Treating the credibility prior mean as the overall mean without justification.
- Applying ruin formulas outside their assumptions, such as ignoring the premium loading.