Actuarium

CAS MAS-IModern Actuarial Statistics I

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.

Probability models (stochastic processes, survival)

30%
  • Model claim occurrence using Poisson processes, including compound and mixed Poisson processes.
  • Apply Markov chain transition matrices to compute state probabilities over time.
  • Compute survival and hazard functions and apply them to time-to-event actuarial models.
  • Simulate stochastic processes to approximate quantities without closed-form solutions.

Statistics (estimation, testing, MLE)

32%
  • Derive maximum likelihood estimators and their asymptotic properties for common distributions.
  • Construct confidence intervals and perform hypothesis tests for means, variances and proportions.
  • Apply the delta method and likelihood ratio tests to actuarial estimation problems.
  • Evaluate estimator bias, consistency and efficiency using theoretical criteria.

Extended linear models (GLMs)

28%
  • Specify a GLM by choosing a distribution from the exponential family and an appropriate link function.
  • Fit a GLM via maximum likelihood and interpret estimated coefficients as relativities.
  • Assess model fit using deviance, AIC/BIC and residual diagnostics.
  • Apply offsets and weights appropriately when modeling frequency, severity or loss ratio.

Time series

10%
  • Identify and test for stationarity, and apply differencing to non-stationary series.
  • Fit and forecast AR, MA and ARIMA models using ACF/PACF identification.
  • Evaluate residual diagnostics and forecast accuracy for a fitted time-series model.
  • Apply time series methods to project loss trend or economic indices.

Lecture videos for this exam

Open the full video library →

[MATH 5639 Actuarial Loss Models] Lecture 1: Probability Exercise 1

Bin Z · Loss Models

[MATH 5639 Actuarial Loss Models] Lecture 41: Ch12.1 Moment and quantile matching methods

Bin Z · Loss Models

4.4. Actuarial Math: Survival Models D

Dr. Amjad Rabi · Survival Models

036. Introduction to Continuous Time Survival Analysis

Dr. Dylan Spicker · Survival Models

MIT 18.650 Statistics for Applications, Fall 2016

MIT OpenCourseWare · Statistics & Regression · playlist

Overview

MAS-I covers Poisson processes and Markov chains, survival and reliability, parametric estimation and hypothesis testing, GLMs with an actuarial lens, and ARIMA time-series basics. It is the first exam with an explicit modeling mindset.

Duration
4 hours
Questions
45 multiple-choice
Style
Computer-based, 5 choices
Passing
Pass mark set per sitting

Syllabus map

Probability models (stochastic processes, survival)
30%
Probability and Distributions for InsuranceLife Contingencies Primer
Statistics (estimation, testing, MLE)
32%
Statistics and Estimation for Actuaries
Extended linear models (GLMs)
28%
GLMs and Predictive Modeling in Pricing
Time series
10%
Forecasting Premium, Loss, and Profitability

Key formulas

Poisson process with rate λ\lambda: N(t)∼Poisson(λt)N(t)\sim\text{Poisson}(\lambda t); inter-arrivals ∼Exp(1/λ)\sim\text{Exp}(1/\lambda); thinning gives independent Poisson processes with rates piλp_i\lambda.

MLE asymptotics   θ^∼˙N ⁣(θ,  I(θ)−1)\;\hat\theta\dot\sim N\!\left(\theta,\;I(\theta)^{-1}\right), I(θ)=−E ⁣[∂2ℓ/∂θ2]I(\theta)=-E\!\left[\partial^2\ell/\partial\theta^2\right].

Likelihood ratio test   2(ℓ1−ℓ0)∼˙χΔp2\;2(\ell_1-\ell_0)\dot\sim\chi^2_{\Delta p}.

GLM   g(μi)=xi⊤β\;g(\mu_i)=\mathbf{x}_i^\top\boldsymbol\beta, deviance D=2ϕ(ℓsat−ℓmodel)D=2\phi(\ell_{sat}-\ell_{model}); AIC =2p−2ℓ=2p-2\ell, BIC =pln⁡n−2ℓ=p\ln n-2\ell.

AR(1)   Xt=ϕXt−1+εt\;X_t=\phi X_{t-1}+\varepsilon_t: Var⁡(X)=σε2/(1−ϕ2)\operatorname{Var}(X)=\sigma^2_\varepsilon/(1-\phi^2), ρk=ϕk\rho_k=\phi^{k}.

Study strategy

  1. Build a one-page sheet of the Poisson-process results (thinning, superposition, compound, conditional uniform arrivals); they generate many questions.

  2. Derive MLEs by hand for exponential, Poisson, gamma-with-known-shape, and normal until the score equation is second nature.

  3. For GLMs, know the canonical link and variance function of each exponential-family member.

  4. Learn to read ACF/PACF patterns to identify AR vs MA order.

Common traps

  • Mixing rates per hour and per day in Poisson-process questions.

  • Reporting I(θ)I(\theta) instead of its inverse as the variance of the MLE.

  • Using degrees of freedom equal to the number of parameters in a nested test instead of the difference.

  • Stationarity conditions: AR(1) needs ∣ϕ∣<1|\phi|<1; MA(q) is always stationary.

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