Insurance Mathematics / Risk Theory
Scope
This domain covers aggregate loss models, frequency-severity decomposition, compound distributions, ruin theory, reinsurance, and credibility. Korean exam items often give a frequency distribution and a severity distribution and ask for the mean, variance, or an approximation to a tail probability.
Core rules
- Collective risk model: S = X_1+...+X_N, with N independent of severities.
- E[S] = E[N]E[X]; Var(S) = E[N]Var(X) + Var(N)(E[X])^2.
- Compound Poisson: Var(S) = λE[X^2].
- Panjer recursion applies to (a,b,0) frequency classes; Poisson is (a,b,0) with a=0, b=λ.
- Lundberg inequality: ψ(u) ≤ e^{-Ru}, where R solves λ + cR = λM_X(R) for a Poisson surplus with premium rate c.
- Proportional reinsurance scales severity; excess-of-loss changes the severity distribution and usually requires integration.
Worked example
Claims arrive as a Poisson process with λ = 4 per year. Severity has mean 3 and second moment 25. Find the mean and variance of annual aggregate claims, and approximate P(S > 30).
E[S] = 4 × 3 = 12. Var(S) = λE[X^2] = 4 × 25 = 100, so SD = 10. Using a normal approximation with continuity ignored, z = (30 - 12)/10 = 1.8, so P(S > 30) ≈ 1 - Φ(1.8) = 0.0359. The key step is using the second moment, not the variance of severity: Var(X)=25-9=16, but the compound Poisson variance uses 25. The normal approximation is illustrative and may be poor for skewed severities.
Traps
Using Var(X) instead of E[X^2] in the compound Poisson variance is the classic error. Doubling λ is not the same as doubling claim size. For ruin problems, check whether premium income is continuous or discrete and whether the safety loading is stated as a proportion of expected claims.