Casualty Actuarial Society Credentials (ACAS/FCAS) MAS-I — Modern Actuarial Statistics I Syllabus

MAS-I — Modern Actuarial Statistics I syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for MAS-I — Modern Actuarial Statistics I in Casualty Actuarial Society Credentials (ACAS/FCAS), not a summary of it.

1. 1

   Probability Models and Stochastic Processes

   3 topics

   • Poisson Processes
     • Homogeneous and non-homogeneous Poisson processes
     • Interarrival and waiting time distributions
     • Compound Poisson processes and thinning/superposition
   • Markov Chains
     • Transition probability matrices and Chapman-Kolmogorov equations
     • Classification of states and chain irreducibility
     • Stationary and limiting distributions
     • Absorbing states and expected time to absorption
   • Reliability and Survival Models
     • Survival functions and hazard rates
     • System reliability: series and parallel structures
2. 2

   Severity, Frequency, and Aggregate Loss Models

   4 topics

   • Severity (Loss Size) Distributions
     • Parametric families: gamma, Pareto, lognormal, Weibull
     • Tail behavior and heavy-tailed distributions
     • Mixture and spliced distributions
   • Frequency (Count) Distributions
     • (a, b, 0) and (a, b, 1) class distributions
     • Zero-modified and zero-truncated distributions
   • Coverage Modifications
     • Deductibles, policy limits, and coinsurance
     • Loss elimination ratio and payment per loss vs. per payment
     • Effect of inflation on modified coverages
   • Aggregate Loss Models
     • Collective vs. individual risk models
     • Compound distributions and recursive (Panjer) computation
     • Stop-loss premiums and aggregate deductibles
3. 3

   Statistical Estimation and Credibility

   3 topics

   • Parametric and Nonparametric Estimation
     • Maximum likelihood for complete and censored data
     • Kaplan-Meier and Nelson-Aalen estimators
     • Variance estimation and the delta method
   • Hypothesis Testing and Model Selection
     • Goodness-of-fit tests (chi-square, Kolmogorov-Smirnov)
     • Likelihood ratio tests, AIC, and BIC
   • Credibility Theory
     • Limited fluctuation (classical) credibility
     • Buhlmann and Buhlmann-Straub credibility
     • Bayesian credibility and conjugate priors
     • Empirical Bayes parameter estimation
4. 4

   Simulation

   3 topics

   • Random Variate Generation
     • Inverse transform method
     • Generating from specific distributions
   • Monte Carlo Applications
     • Estimating expected values and quantiles via simulation
     • Bootstrap resampling and variance reduction
   • Simulation in Actuarial Modeling
     • Simulating aggregate loss distributions
     • Assessing estimator and model uncertainty via simulation

MAS-I — Modern Actuarial Statistics I flashcards for Casualty Actuarial Society Credentials (ACAS/FCAS)

19 of 57 cards from the MAS-I — Modern Actuarial Statistics I deck — real questions with worked answers.

1. What three conditions define a (homogeneous) Poisson process $\{N(t), t \geq 0\}$ with rate $\lambda$?

   (1) $N(0)=0$; (2) independent increments; (3) the number of events in any interval of length $t$ is Poisson distributed with mean $\lambda t$, i.e. $P[N(t+s)-N(s)=n] = \frac{(\lambda t)^{n} e^{-\lambda t}}{n!}$. It also has stationary increments.

2. In a Poisson process with rate $\lambda$, what is the distribution of interarrival times and of the waiting time until the $n$th event?

   Interarrival times are i.i.d. exponential with mean $\frac{1}{\lambda}$. The waiting time $S_n$ until the $n$th event is Gamma (Erlang) distributed: $S_n \sim \text{Gamma}(n, \lambda)$ with mean $\frac{n}{\lambda}$.

3. What is a nonhomogeneous Poisson process, and what is the distribution of $N(t)$?

   A Poisson process whose rate varies with time, $\lambda(t)$. The number of events in $(0,t]$ is Poisson with mean $m(t)=\int_{0}^{t}\lambda(s)\,ds$, so $P[N(t)=n]=\frac{m(t)^{n}e^{-m(t)}}{n!}$. Increments are independent but not stationary.

4. State the decomposition (thinning) property of a Poisson process.

   If each event of a Poisson process with rate $\lambda$ is independently classified as type $i$ with probability $p_i$ (where $\sum p_i = 1$), then the type-$i$ events form independent Poisson processes with rates $\lambda p_i$.

5. What is the superposition property of independent Poisson processes?

   The sum of independent Poisson processes with rates $\lambda_1, \lambda_2, \dots, \lambda_k$ is itself a Poisson process with rate $\lambda_1 + \lambda_2 + \cdots + \lambda_k$.

6. In a compound Poisson process $X(t)=\sum_{i=1}^{N(t)} Y_i$ with $N(t)$ Poisson rate $\lambda$ and i.i.d. jumps $Y_i$, what are the mean and variance of $X(t)$?

   $E[X(t)] = \lambda t\, E[Y]$ and $\operatorname{Var}[X(t)] = \lambda t\, E[Y^{2}]$.

7. Define the one-step transition probability $P_{ij}$ of a discrete-time Markov chain and the Markov property.

   $P_{ij} = P[X_{n+1}=j \mid X_n = i]$. The Markov property states the future depends only on the present: $P[X_{n+1}=j \mid X_n=i, X_{n-1},\dots] = P[X_{n+1}=j \mid X_n=i]$.

8. State the Chapman–Kolmogorov equations for an $n$-step Markov chain transition matrix.

   $P_{ij}^{(n+m)} = \sum_{k} P_{ik}^{(n)} P_{kj}^{(m)}$, equivalently $\mathbf{P}^{(n)} = \mathbf{P}^{n}$ (the $n$-step matrix is the one-step matrix raised to the $n$th power).

9. How is the stationary (limiting) distribution $\boldsymbol{\pi}$ of an ergodic Markov chain found?

   Solve $\boldsymbol{\pi}\mathbf{P} = \boldsymbol{\pi}$ subject to $\sum_i \pi_i = 1$, i.e. $\pi_j = \sum_i \pi_i P_{ij}$. It gives the long-run proportion of time in each state.

10. Classify Markov chain states: what are recurrent, transient, and absorbing states?

    A state is recurrent if it is revisited with probability 1; transient if there is positive probability of never returning; absorbing if $P_{ii}=1$ (once entered, never left). Recurrent states with finite mean return time are positive recurrent.

11. What does it mean for a Markov chain state to be periodic, and what is aperiodicity?

    State $i$ has period $d = \gcd\{n : P_{ii}^{(n)} > 0\}$. If $d=1$ the state is aperiodic. A chain that is irreducible, aperiodic, and positive recurrent is ergodic and has a unique limiting distribution.

12. In reliability theory, define the survival function $S(t)$ and the hazard (failure) rate $h(t)$.

    $S(t) = P[T > t] = 1 - F(t)$. The hazard rate is $h(t) = \frac{f(t)}{S(t)} = -\frac{d}{dt}\ln S(t)$, the instantaneous failure rate given survival to $t$.

13. Express the survival function in terms of the cumulative hazard $H(t)$.

    $S(t) = \exp\left(-\int_{0}^{t} h(u)\,du\right) = e^{-H(t)}$, where $H(t)=\int_0^t h(u)\,du$ is the cumulative hazard function.

14. What distribution has a constant hazard rate, and what key property results?

    The exponential distribution has constant hazard rate $h(t)=\lambda$. This gives the memoryless property: $P[T > s+t \mid T > s] = P[T > t]$.

15. Define the mean residual life (mean excess) function $e(t)$ for a lifetime $T$.

    $e(t) = E[T - t \mid T > t] = \frac{\int_{t}^{\infty} S(u)\,du}{S(t)}$, the expected remaining lifetime given survival to time $t$. For the exponential it equals the constant mean $\frac{1}{\lambda}$.

16. Distinguish IFR and DFR distributions and give an example of each.

    IFR (Increasing Failure Rate): $h(t)$ increases (e.g. Weibull with shape $\tau > 1$). DFR (Decreasing Failure Rate): $h(t)$ decreases (e.g. Weibull with $\tau < 1$, or Pareto). Constant hazard (exponential) is the boundary, often called memoryless/no aging.

17. What is the hazard rate of a Weibull distribution with shape $\tau$ and scale $\theta$?

    With $S(t)=\exp[-(t/\theta)^{\tau}]$, the hazard rate is $h(t)=\frac{\tau}{\theta}\left(\frac{t}{\theta}\right)^{\tau-1}$. It is increasing for $\tau>1$, decreasing for $\tau<1$, constant for $\tau=1$.

18. How do you create a scale family of severity distributions, and which distributions are scale families?

    $X$ is a scale parameter $\theta$ if $cX$ has the same distributional form with parameter $c\theta$. Most continuous severity distributions (exponential, gamma, Pareto, lognormal, Weibull) have a scale parameter $\theta$, which is essential for inflation/currency adjustments.

19. What characterizes a heavy-tailed severity distribution, and how can tails be compared?

    Heavy tails decay slower than exponential, giving larger probability of extreme losses. Compare tails via the ratio $\lim_{x\to\infty}\frac{S_1(x)}{S_2(x)}$, by existence of moments (Pareto has only finite low-order moments), or by hazard rate (decreasing hazard implies heavy tail). Pareto/lognormal are heavy-tailed; gamma/Weibull($\tau>1$) are light.

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