Modern Actuarial Statistics II: Credibility, Mixed Models, Bayesian Computation and Simulation
What this domain tests
MAS-II is the second modern-statistics paper in the CAS sequence. It moves past "the data speaks for itself": you blend a risk's own experience with collateral data (credibility), model hierarchical and correlated data (linear mixed models), and quantify uncertainty by simulation or Bayesian computation (MCMC). The exam rewards structural reasoning - knowing which variance component drives a formula, not just plugging numbers.
Credibility answers two different questions
- Limited fluctuation (classical): how much data is enough? Full credibility requires n ≥ n_F, with n_F = (z/k)²(σ/μ)² for pure premium, and n_F = (z/k)² expected claims for Poisson frequency (z from the confidence level, k the tolerated error). Partial credibility uses Z = √(n/n_F).
- Greatest accuracy (Bühlmann): how should data be weighted? Z = n/(n + k) with k = v/a, where v = E[Var(X|θ)] is the process variance (EPV) and a = Var(E[X|θ]) is the variance of hypothetical means (VHM). Estimate = Z·X̄ + (1 − Z)·μ.
- Bühlmann–Straub replaces the count n with total exposure: Z = Σm_i/(Σm_i + k), same k = v/a, using X̄_w = Σm_iX_i/Σm_i.
The other threads
Normal–normal Bayesian credibility reproduces Bühlmann exactly (Z = nτ²/(nτ² + σ²)), which is why conjugate posteriors are fair game. MCMC (Gibbs sampling, Metropolis–Hastings) is tested through diagnostics: burn-in, trace plots, autocorrelation, effective sample size, Gelman–Rubin. Linear mixed models split variation into fixed effects and random effects, and the BLUP has the same shrinkage form as a credibility estimate.
Worked example
Three years of loss ratios 0.60, 0.75, 0.66 give X̄ = 0.67. The study supplies v = 0.02 and a = 0.01, so k = 0.02/0.01 = 2 and Z = 3/(3 + 2) = 0.60. With collective mean μ = 0.70, the credibility estimate is 0.6(0.67) + 0.4(0.70) = 0.402 + 0.28 = 0.682.
Traps that cost marks
- Inverting k: it is v/a (process over hypothetical), so a large a raises Z.
- Using the number of years instead of Σm_i when exposures are unequal.
- Reporting Z above 1 in limited fluctuation instead of capping at full credibility.
- Estimating a from raw dispersion without removing the process component: Σm_i(X_i − X̄_w)² has expectation (n − 1)v + a(M − Σm_i²/M), with M = Σm_i.
- Quoting an MCMC posterior mean from an unburned or non-converged chain.