Actuarium

CAS MAS-IIModern Actuarial Statistics II

Preliminary
4 hours·45 multiple-choice·≈350 study hours

Syllabus learning objectives

Official syllabus

Paraphrased from the CAS syllabus so the study-plan builder and practice sets track the topics you will actually be examined on. Weights are the official topic ranges.

Credibility (limited fluctuation, Bühlmann, Bühlmann–Straub)

22%
  • Determine the standard for full credibility and compute partial credibility factors under limited fluctuation credibility.
  • Compute Bühlmann credibility premiums from the expected process variance and variance of hypothetical means.
  • Apply Bühlmann-Straub credibility to risks with varying exposure across periods.
  • Estimate credibility parameters nonparametrically from sample data.

Bayesian analysis & MCMC

27%
  • Derive posterior distributions from a specified prior and likelihood for common conjugate families.
  • Compute Bayesian point and interval estimates (posterior mean, credible intervals).
  • Apply Markov Chain Monte Carlo methods (Metropolis-Hastings, Gibbs sampling) to sample from posterior distributions.
  • Compare Bayesian and frequentist estimates and interpret their practical differences.

Statistical learning (regularization, trees, boosting)

35%
  • Apply ridge, lasso and elastic net regularization to control model complexity and perform variable selection.
  • Fit decision trees and ensemble methods (random forests, gradient boosting) and tune hyperparameters via cross-validation.
  • Evaluate model performance using appropriate loss functions and out-of-sample validation.
  • Compare interpretability and predictive accuracy tradeoffs across statistical learning methods.

Linear mixed models

16%
  • Specify fixed and random effects in a linear mixed model for hierarchical/grouped actuarial data.
  • Estimate variance components using REML/ML and interpret shrinkage of random effects (empirical Bayes).
  • Apply mixed models to credibility-type problems with multiple levels of grouping.
  • Assess model fit and random-effect assumptions via residual diagnostics.

Lecture videos for this exam

Open the full video library →

Insurance Risk Pricing with GLM, GAM and XGBoost

Matthew Evans · Pricing & GLMs

Webinar: Machine Learning in Reserving on 15 July 2025

International Actuarial Association · Machine Learning in Actuarial Work

Overview

MAS-II is the credibility and modern-modeling exam: classical and Bayesian credibility, conjugate priors and MCMC diagnostics, penalized regression, tree ensembles, and linear mixed models as a generalization of Bühlmann–Straub.

Duration
4 hours
Questions
45 multiple-choice
Style
Computer-based
Passing
Pass mark set per sitting

Syllabus map

Credibility (limited fluctuation, Bühlmann, Bühlmann–Straub)
22%
Credibility Theory: Limited Fluctuation, Bühlmann, and Bayes
Bayesian analysis & MCMC
27%
Credibility Theory: Limited Fluctuation, Bühlmann, and Bayes
Statistical learning (regularization, trees, boosting)
35%
GLMs and Predictive Modeling in Pricing
Linear mixed models
16%
Credibility Theory: Limited Fluctuation, Bühlmann, and BayesStatistics and Estimation for Actuaries

Key formulas

Bühlmann credibility Z=nn+k,k=EPVVHM=E[Var⁡(X∣Θ)]Var⁡(E[X∣Θ])Z=\frac{n}{n+k},\qquad k=\frac{EPV}{VHM}=\frac{E[\operatorname{Var}(X\mid\Theta)]}{\operatorname{Var}(E[X\mid\Theta])}

Bühlmann–Straub with exposures mim_i: Zi=mimi+kZ_i=\dfrac{m_i}{m_i+k}.

Limited fluctuation full credibility (frequency, Poisson): nF=(z(1+p)/2r)2n_F=\left(\dfrac{z_{(1+p)/2}}{r}\right)^2; partial credibility Z=n/nFZ=\sqrt{n/n_F}.

Gamma–Poisson conjugacy: prior Γ(α,θ)\Gamma(\alpha,\theta), observe ∑x\sum x claims over nn periods ⇒\Rightarrow posterior Γ ⁣(α+∑x,  θ1+nθ)\Gamma\!\left(\alpha+\sum x,\;\dfrac{\theta}{1+n\theta}\right).

Metropolis–Hastings acceptance   α=min⁡ ⁣(1,π(y)q(x∣y)π(x)q(y∣x))\;\alpha=\min\!\left(1,\dfrac{\pi(y)q(x\mid y)}{\pi(x)q(y\mid x)}\right).

Ridge / lasso   min⁡β∥y−Xβ∥2+λ∥β∥22\;\min_\beta \|y-X\beta\|^2+\lambda\|\beta\|_2^2 (ridge), +λ∥β∥1+\lambda\|\beta\|_1 (lasso).

Study strategy

  1. Do Bühlmann problems in three fixed steps — hypothetical mean, process variance, then EPV/VHM — every time.

  2. Know the conjugate-prior table (Poisson–gamma, binomial–beta, normal–normal, exponential–inverse-gamma) and the fact that Bühlmann equals Bayes in these cases.

  3. For statistical learning, focus on bias–variance intuition and what each tuning parameter does (λ, depth, learning rate, mtry).

  4. Practice reading MCMC output: trace plots, R^\hat R, effective sample size, burn-in.

Common traps

  • Using total observations instead of per-risk exposure in Bühlmann–Straub.

  • Forgetting that limited-fluctuation standards for aggregate losses add the severity CV² term.

  • Confusing lasso (sparse) with ridge (shrinks but keeps all).

  • Treating a random intercept model's variance components as fixed-effect coefficients.

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