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

What is the $(a,b,0)$ class of frequency distributions and its defining recursion?

Distributions satisfying $\frac{p_k}{p_{k-1}} = a + \frac{b}{k}$ for $k=1,2,\dots$. The three members are Poisson ($a=0$), binomial ($a<0$), and negative binomial ($a>0$).

How do you identify Poisson, binomial, and negative binomial from the variance-to-mean relationship?

Poisson: variance $=$ mean. Binomial: variance $<$ mean (underdispersed). Negative binomial: variance $>$ mean (overdispersed). This ratio is a quick diagnostic for frequency model selection.

What is the $(a,b,1)$ class and what are zero-truncated and zero-modified distributions?

The $(a,b,1)$ class satisfies the recursion $\frac{p_k}{p_{k-1}}=a+\frac{b}{k}$ only for $k \geq 2$, allowing $p_0$ to be set freely. Zero-truncated sets $p_0=0$; zero-modified sets $p_0^{M}$ to an arbitrary value, rescaling the rest by $\frac{1-p_0^{M}}{1-p_0}$.

Give the probability generating function (PGF) and its key use for frequency distributions.

$P_N(z) = E[z^{N}] = \sum_{k=0}^{\infty} p_k z^{k}$. Moments come from derivatives at $z=1$ (e.g. $P_N'(1)=E[N]$), and it identifies compound/aggregate distributions since $P_S(z)=P_N(P_X(z))$.

How does an ordinary deductible $d$ modify a loss $X$ under the per-payment vs per-loss perspective?

Per-loss variable: $Y^{L}=(X-d)_{+}=\max(X-d,0)$, equal to 0 when $X\le d$. Per-payment variable: $Y^{P}=X-d \mid X>d$, defined only when a payment occurs. $E[Y^{L}]=E[Y^{P}]\cdot P(X>d)$.

What is the formula for expected cost per loss with an ordinary deductible $d$?

$E[(X-d)_{+}] = E[X] - E[X \wedge d]$, where $E[X\wedge d]=\int_0^d S(x)\,dx$ is the limited expected value. This is the expected payment per loss event.

How does a policy limit $u$ and a maximum covered loss differ, and what is the limited expected value?

A policy limit $u$ caps the payment at $u$: payment is $X \wedge u = \min(X,u)$. The limited expected value is $E[X\wedge u]=\int_0^{u} S(x)\,dx$. With deductible $d$ and maximum covered loss $u$, payment per loss is $E[X\wedge u]-E[X\wedge d]$.

What is the effect of a coinsurance factor $\alpha$ combined with deductible $d$ and maximum covered loss $u$?

The per-loss payment is $\alpha\,[\,(X\wedge u)-(X\wedge d)\,]$, so $E[\text{payment}] = \alpha\,[E[X\wedge u]-E[X\wedge d]]$. With inflation factor $(1+r)$, replace $d,u$ by $\frac{d}{1+r},\frac{u}{1+r}$ and multiply by $(1+r)$.

Define the loss elimination ratio (LER) for a deductible $d$.

$\mathrm{LER}(d) = \frac{E[X\wedge d]}{E[X]}$, the proportion of expected losses eliminated by imposing an ordinary deductible $d$.

How does an ordinary deductible affect claim frequency in an aggregate model?

It thins the frequency: the probability that a loss exceeds the deductible is $v = P(X>d)$, so the number of payments is a thinned version of the number of losses. For Poisson losses with rate $\lambda$, payments are Poisson with rate $\lambda v$.

What are the two main models for aggregate losses $S$, and how do they differ?

Collective risk model: $S=\sum_{i=1}^{N} X_i$ with random claim count $N$ and i.i.d. severities $X_i$ independent of $N$. Individual risk model: $S=\sum_{j=1}^{n} X_j$ over a fixed number $n$ of policies, each $X_j$ possibly zero.

Give the mean and variance of aggregate losses $S=\sum_{i=1}^N X_i$ in the collective risk model.

$E[S]=E[N]\,E[X]$ and (compound variance formula) $\operatorname{Var}(S)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)\,E[X]^{2}$.

For a compound Poisson aggregate model with $N\sim\text{Poisson}(\lambda)$, simplify the mean and variance of $S$.

$E[S]=\lambda\,E[X]$ and $\operatorname{Var}(S)=\lambda\,E[X^{2}]$ (since for Poisson $E[N]=\operatorname{Var}(N)=\lambda$).

State the Panjer recursion for computing the aggregate distribution when $N$ is in the $(a,b,0)$ class.

For a discrete severity on $0,1,2,\dots$, $f_S(s)=\frac{1}{1-a f_X(0)}\sum_{j=1}^{s}\left(a+\frac{b j}{s}\right) f_X(j) f_S(s-j)$, started with $f_S(0)=P_N(f_X(0))$.

What does the individual risk model assume and how is its distribution often approximated?

$S=\sum_{j=1}^{n} X_j$ with independent (not identical) risks; each $X_j=I_j B_j$ where $I_j$ indicates a claim. Because $n$ is fixed, $E[S]=\sum E[X_j]$ and $\operatorname{Var}(S)=\sum \operatorname{Var}(X_j)$; the total is often approximated by a normal distribution.

Compare a method-of-moments estimator with a maximum likelihood estimator.

Method of moments equates sample moments to theoretical moments and solves for parameters — simple but not generally efficient. MLE maximizes the likelihood $L(\theta)=\prod f(x_i;\theta)$; it is asymptotically unbiased, efficient (achieves the Cramér–Rao bound), and asymptotically normal.

What is the Kaplan–Meier (product-limit) estimator used for?

It is a nonparametric estimator of the survival function $S(t)$ from right-censored data: $\hat{S}(t)=\prod_{t_i \le t}\left(1-\frac{d_i}{r_i}\right)$, where $d_i$ is deaths and $r_i$ is the risk set at each event time $t_i$.

What does the Nelson–Aalen estimator estimate, and what is its formula?

It nonparametrically estimates the cumulative hazard function: $\hat{H}(t)=\sum_{t_i \le t}\frac{d_i}{r_i}$. A corresponding survival estimate is $\hat{S}(t)=e^{-\hat{H}(t)}$.

Define right-censoring and left-truncation in survival/severity data.

Right-censoring: the exact value is unknown but known to exceed some value (e.g. policy limit $u$). Left-truncation: observations below a threshold (e.g. deductible $d$) are not recorded at all; the data are conditional on exceeding $d$.

What is the asymptotic variance of the MLE and how is it estimated?

$\operatorname{Var}(\hat\theta)\approx \frac{1}{I(\theta)}$ where $I(\theta)=-E\!\left[\frac{\partial^{2}\ln L}{\partial\theta^{2}}\right]$ is the Fisher information. In multiple dimensions, the covariance matrix is the inverse of the information matrix, estimated by the negative inverse Hessian of the log-likelihood at $\hat\theta$.

Describe the Kolmogorov–Smirnov goodness-of-fit test statistic.

$D = \max_{x}\left| F_n(x) - F^{*}(x) \right|$, the largest absolute difference between the empirical CDF $F_n$ and the fitted CDF $F^{*}$. Large $D$ leads to rejection; it applies to continuous, individual (non-grouped) data.

What does the Anderson–Darling test emphasize compared with Kolmogorov–Smirnov?

Anderson–Darling weights differences by $\frac{1}{F^{*}(x)[1-F^{*}(x)]}$, placing more emphasis on fit in the tails of the distribution, whereas K–S treats all points equally and is most sensitive near the center.

Give the chi-square goodness-of-fit statistic and its degrees of freedom.

$\chi^{2}=\sum_{j=1}^{k}\frac{(O_j - E_j)^{2}}{E_j}$, comparing observed $O_j$ and expected $E_j$ counts in $k$ groups. Degrees of freedom $= k - 1 - (\text{number of estimated parameters})$.

Define the likelihood ratio test for comparing nested models.

Test statistic $\Lambda = 2\big(\ln L_{\text{full}} - \ln L_{\text{reduced}}\big)$, asymptotically $\chi^{2}$ with degrees of freedom equal to the difference in the number of parameters. Reject the simpler model if $\Lambda$ exceeds the critical value.

What are the AIC and (Schwarz) BIC criteria for model selection?

$\mathrm{AIC}=-2\ln L + 2k$ and $\mathrm{BIC}\;(\text{SBC})=-2\ln L + k\ln n$, where $k$ is the number of parameters and $n$ the sample size. Lower values are preferred; BIC penalizes complexity more heavily for large $n$.

What is the limited fluctuation (classical) credibility standard for full credibility of claim counts?

The expected number of claims needed for full credibility is $\lambda_F = \left(\frac{z_{(1+p)/2}}{k}\right)^{2}$, where $p$ is the probability of being within $\pm k$ of the mean. The standard ensures aggregate losses are within $\pm k$ of the mean with probability $p$.

State the partial credibility formula in classical (limited fluctuation) credibility.

Credibility-weighted estimate $= Z\bar{X} + (1-Z)M$, where $M$ is the manual/prior mean and $Z = \min\!\left(\sqrt{\frac{n}{n_F}},\,1\right)$, with $n$ the observed claim count and $n_F$ the full-credibility standard (the square-root rule).

Give the Bühlmann credibility factor $Z$ and define EPV and VHM.

$Z=\frac{n}{n+k}$ with $k=\frac{\text{EPV}}{\text{VHM}}$. EPV $=E[\operatorname{Var}(X\mid\Theta)]$ is the expected process variance; VHM $=\operatorname{Var}(E[X\mid\Theta])$ is the variance of the hypothetical means. The estimate is $Z\bar X + (1-Z)\mu$.

How does Bühlmann–Straub credibility differ from basic Bühlmann?

Bühlmann–Straub allows different exposure/volume $m_i$ in each period. The credibility factor becomes $Z=\frac{m}{m+k}$ with $m=\sum m_i$ the total exposure and $k=\frac{\text{EPV}}{\text{VHM}}$, weighting observations by their exposures.

What is Bayesian credibility, and when does it coincide with Bühlmann credibility?

Bayesian credibility uses the posterior mean $E[\theta \mid \text{data}]$ as the estimate, combining prior and likelihood. For linear exponential family likelihoods with conjugate priors (e.g. Poisson–gamma, normal–normal), the exact Bayesian estimate equals the linear Bühlmann credibility estimate.

Describe the inverse transform method for generating a random variate.

Generate $U\sim\text{Uniform}(0,1)$ and set $X=F^{-1}(U)$, the inverse of the target CDF. Then $X$ has CDF $F$. For example, an exponential variate is $X=-\frac{1}{\lambda}\ln(1-U)$ (or $-\frac{1}{\lambda}\ln U$).

Explain the acceptance–rejection method for random variate generation.

To sample from $f$, choose envelope $g$ with $f(x)\le c\,g(x)$. Generate $Y\sim g$ and $U\sim\text{Uniform}(0,1)$; accept $Y$ if $U \le \frac{f(Y)}{c\,g(Y)}$, else repeat. The expected number of trials per accepted value is $c$.

How are Monte Carlo estimates of a quantity $\theta=E[h(X)]$ and their standard error computed?

Simulate $n$ i.i.d. values and average: $\hat\theta=\frac{1}{n}\sum_{i=1}^{n} h(X_i)$. The standard error is $\frac{s}{\sqrt{n}}$, so accuracy improves at rate $\frac{1}{\sqrt{n}}$; quadrupling $n$ halves the error.

What is the purpose of variance reduction techniques in simulation, and name two common ones.

They reduce the variance of Monte Carlo estimates for a given number of trials, improving efficiency. Common methods: antithetic variates (use $U$ and $1-U$ to induce negative correlation) and control variates (adjust using a correlated variable with known mean). Others include importance sampling and stratified sampling.

How is a single value of aggregate loss $S$ simulated in the collective risk model?

First simulate the claim count $N$ from the frequency distribution; then simulate $N$ independent severities $X_1,\dots,X_N$ from the severity distribution; finally sum them: $S=\sum_{i=1}^{N} X_i$. Repeating many times builds the empirical aggregate distribution.

When simulating, how do you determine the number of trials $n$ needed for a target accuracy?

To estimate $\theta$ within tolerance with confidence, set $n$ so the half-width $z_{(1+p)/2}\frac{\sigma}{\sqrt{n}}$ meets the requirement, giving $n \geq \left(\frac{z_{(1+p)/2}\,\sigma}{\text{tolerance}}\right)^{2}$. Pilot runs estimate $\sigma$.

How can the bootstrap be used to assess estimator variability in actuarial modeling?

Resample the observed data with replacement many times, recompute the statistic $\hat\theta^{*}$ on each resample, and use the spread of the bootstrap replicates to estimate the standard error or confidence interval of $\hat\theta$ — a nonparametric Monte Carlo approach requiring no distributional assumption.