Non-Life Insurance Mathematics and Credibility Theory
Non-life questions cover frequency-severity modelling, experience rating and IBNR estimation. In the Iranian market, compulsory motor third-party rates are still heavily influenced by tariff decisions, so credibility and trend adjustments to class rates are examined as practical, not purely theoretical, tools.
Core rules
- Aggregate loss S = Σ X_i for N claims has E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)E[X]².
- Bühlmann credibility: Z = n/(n+k) with k = EPV/VHM, where EPV is the expected process variance and VHM the variance of hypothetical means. The estimate is Z·(observed mean) + (1−Z)·(prior mean).
- Limited fluctuation credibility targets full credibility via n ≥ (z/κ)²·(σ/μ)².
- Chain ladder multiplies cumulative losses by development factors; Bornhuetter-Ferguson blends an expected loss ratio with reported development, stabilising immature years.
- GLMs with a Poisson log link for frequency and gamma or Tweedie for severity, with exposure as an offset, are standard.
Worked example
Three years of a motor class give observed mean severity 120. EPV = 400, VHM = 200, so k = 400/200 = 2 and Z = 3/5 = 0.6. With a prior (class) mean of 110, the credibility estimate is 110 + 0.6(120 − 110) = 116.
Traps
- Swapping EPV and VHM, which inverts the credibility weight.
- Using the raw variance of observed means instead of the variance of hypothetical means.
- Chain-ladder factors distorted by zero or negative cumulative values; exclude or smooth.
- Ignoring exposure and trend when on-levelling historical losses.
- Treating the credibility estimate as a posterior mean when the Bayesian model differs.