Associate of the Society of Actuaries (ASA/FSA) Exam FAM — Fundamentals of Actuarial Mathematics Syllabus

Exam FAM — Fundamentals of Actuarial Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam FAM — Fundamentals of Actuarial Mathematics in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Severity, Frequency, and Aggregate Models

   3 topics

   • Loss Severity Distributions
     • Parametric severity models
     • Tail weight and moments
     • Creating new distributions (mixtures, splicing)
   • Frequency Distributions
     • (a, b, 0) and (a, b, 1) classes
     • Poisson, binomial, negative binomial
   • Aggregate Loss Models
     • Collective risk model
     • Compound distribution moments
     • Recursive and convolution methods
2. 2

   Coverage Modifications and Pricing

   3 topics

   • Policy Adjustments
     • Deductibles (ordinary and franchise)
     • Policy limits and coinsurance
     • Loss elimination ratio
   • Per-Loss vs. Per-Payment Variables
     • Expected costs with modifications
     • Effect of inflation on modified coverage
   • Risk Measures
     • Value at Risk (VaR)
     • Tail Value at Risk (TVaR)
3. 3

   Parametric Estimation and Credibility

   3 topics

   • Maximum Likelihood Estimation
     • Complete and censored/truncated data
     • Variance of MLE estimators
   • Model Selection
     • Goodness-of-fit tests
     • Information criteria
   • Credibility Theory
     • Limited fluctuation (classical) credibility
     • Buhlmann and Buhlmann-Straub credibility
     • Bayesian credibility
4. 4

   Survival Models and Life Contingencies

   3 topics

   • Survival Distributions
     • Survival function and force of mortality
     • Life tables and fractional-age assumptions
     • Select and ultimate mortality
   • Life Insurance Present Values
     • Whole life, term, and endowment insurances
     • Actuarial present value and variance
   • Life Annuities
     • Whole life and temporary annuities
     • Recursion relationships with insurances
5. 5

   Premiums and Reserves

   3 topics

   • Net Premium Calculation
     • Equivalence principle
     • Fully discrete and fully continuous premiums
   • Gross (Expense-Loaded) Premiums
     • Expense assumptions
     • Profit considerations
   • Policy Reserves
     • Net premium reserves (prospective and retrospective)
     • Recursive reserve calculations
     • Reserves at fractional durations

Exam FAM — Fundamentals of Actuarial Mathematics flashcards for Associate of the Society of Actuaries (ASA/FSA)

22 of 50 cards from the Exam FAM — Fundamentals of Actuarial Mathematics deck — real questions with worked answers.

1. What is the support and key parameter interpretation of the exponential distribution as a loss severity model?

   The exponential distribution has support $x \geq 0$ with density $f(x) = \frac{1}{\theta} e^{-x/\theta}$, mean $E[X] = \theta$, and variance $\operatorname{Var}(X) = \theta^{2}$. It is memoryless: $E[X - d \mid X > d] = \theta$ for any deductible $d$.

2. For a loss random variable $X$, define the $k$-th raw moment and the limited expected value $E[X \wedge u]$.

   The $k$-th raw moment is $E[X^{k}] = \int_{0}^{\infty} x^{k} f(x)\, dx$. The limited expected value at limit $u$ is $$E[X \wedge u] = \int_{0}^{u} x f(x)\, dx + u\,[1 - F(u)] = \int_{0}^{u} S(x)\, dx,$$ where $S(x) = 1 - F(x)$ is the survival function.

3. How do you classify a loss distribution as having a heavy (fat) tail relative to another?

   Distribution $X$ has a heavier tail than $Y$ if $\lim_{x \to \infty} \frac{S_X(x)}{S_Y(x)} = \infty$, i.e. its survival function decays more slowly. Equivalently, an increasing mean excess (mean residual life) function $e(d) = E[X - d \mid X > d]$ indicates a heavy tail; an increasing hazard rate indicates a light tail.

4. Define the hazard rate (force of mortality) $h(x)$ and its relationship to the survival function.

   The hazard rate is $$h(x) = \frac{f(x)}{S(x)} = -\frac{d}{dx}\ln S(x).$$ Conversely, $S(x) = \exp\!\left(-\int_{0}^{x} h(t)\, dt\right)$. An increasing $h(x)$ implies a lighter tail.

5. What characterizes the (a, b, 0) class of frequency distributions, and which distributions belong to it?

   The $(a,b,0)$ class satisfies the recursion $$\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \quad k \geq 1.$$ The only members are the Poisson ($a=0$), Binomial ($a<0$), and Negative Binomial ($0<a<1$, including Geometric).

6. For the Poisson distribution with mean $\lambda$, give the $(a,b,0)$ recursion parameters and the mean–variance relationship.

   For Poisson, $a = 0$ and $b = \lambda$, so $p_k = \frac{\lambda}{k} p_{k-1}$. The mean equals the variance: $E[N] = \operatorname{Var}(N) = \lambda$ (equidispersion).

7. Compare the mean–variance relationship for Poisson, Binomial, and Negative Binomial frequency models.

   Poisson: variance $=$ mean (equidispersion). Binomial: variance $<$ mean (underdispersion). Negative Binomial: variance $>$ mean (overdispersion). This dispersion ordering helps select a frequency model from data.

8. What is the negative binomial distribution's mean and variance in $(r, \beta)$ parameterization?

   With parameters $r > 0$ and $\beta > 0$: $E[N] = r\beta$ and $\operatorname{Var}(N) = r\beta(1 + \beta)$. Since $1 + \beta > 1$, variance exceeds the mean (overdispersion).

9. How is the $(a, b, 1)$ class of frequency distributions defined and when is it used?

   The $(a,b,1)$ class satisfies $\frac{p_k}{p_{k-1}} = a + \frac{b}{k}$ for $k \geq 2$ (recursion starts at $k=2$), allowing $p_0$ to be set arbitrarily. It is used for zero-modified or zero-truncated distributions where the probability of zero claims differs from the standard member.

10. Give the formula for a zero-modified probability $p_k^M$ in terms of the corresponding standard $(a,b,0)$ probabilities.

    $$p_k^M = \frac{1 - p_0^M}{1 - p_0}\, p_k, \quad k \geq 1,$$ where $p_0^M$ is the chosen modified probability at zero and $p_0, p_k$ are the original $(a,b,0)$ probabilities. Zero-truncated is the special case $p_0^M = 0$.

11. Define the collective risk model for aggregate losses $S$ and state its mean.

    In the collective risk model $S = X_1 + X_2 + \cdots + X_N$, where $N$ is the claim count (frequency) and $X_i$ are i.i.d. severities independent of $N$. The mean is $E[S] = E[N]\,E[X]$ (a compound expectation).

12. State the variance of aggregate losses $S$ in the compound (collective risk) model.

    $$\operatorname{Var}(S) = E[N]\,\operatorname{Var}(X) + \operatorname{Var}(N)\,(E[X])^{2}.$$ This follows from the law of total variance with $N$ and $X$ independent.

13. For a compound Poisson aggregate model with Poisson mean $\lambda$, simplify the mean and variance of $S$.

    With $N \sim \text{Poisson}(\lambda)$: $E[S] = \lambda\,E[X]$ and $\operatorname{Var}(S) = \lambda\,E[X^{2}]$ (the second raw moment of severity), since $\operatorname{Var}(N) = E[N] = \lambda$.

14. What is the individual risk model for aggregate losses and how does it differ from the collective risk model?

    The individual risk model is $S = \sum_{i=1}^{n} X_i$ over a fixed number $n$ of policies, where each $X_i$ (often $0$ with probability $q_i$) is the loss for policy $i$. Unlike the collective model, $n$ is fixed (not random) and the $X_i$ need not be identically distributed.

15. How does an ordinary deductible $d$ affect the per-loss payment variable, and give its expected value.

    The per-loss variable is $Y^L = (X - d)_+ = \max(X - d, 0)$. Its expected value is $$E[Y^L] = E[X] - E[X \wedge d] = \int_{d}^{\infty} S(x)\, dx.$$ This counts zero payments for losses below the deductible.

16. Distinguish the per-loss variable $Y^L$ from the per-payment variable $Y^P$ under a deductible $d$.

    Per-loss $Y^L = (X-d)_+$ includes losses below $d$ as zero payments. Per-payment $Y^P = (X-d) \mid X > d$ conditions on a payment occurring. They relate by $E[Y^P] = \frac{E[Y^L]}{S(d)} = \frac{E[X] - E[X \wedge d]}{1 - F(d)}$.

17. How is a policy limit $u$ incorporated, and what is the maximum covered loss with a deductible $d$ and limit $u$?

    A policy limit $u$ caps the insurer payment, giving payment $X \wedge u$. With an ordinary deductible $d$ and a limit of $u$ on payments, the maximum covered loss is $u + d$, and the expected payment per loss is $E[X \wedge (u+d)] - E[X \wedge d]$.

18. Define a coinsurance factor $\alpha$ and write the per-loss payment with deductible $d$, coinsurance $\alpha$, and maximum covered loss $u$.

    With coinsurance $\alpha$ (insurer pays fraction $\alpha$), deductible $d$, and maximum covered loss $u$: $$E[Y^L] = \alpha\big(E[X \wedge u] - E[X \wedge d]\big).$$ The order of operations is: apply deductible, then limit, then coinsurance.

19. What is the loss elimination ratio (LER) for a deductible $d$?

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

20. How does uniform inflation at rate $r$ affect limited expected values and deductibles?

    If losses inflate by factor $(1+r)$, the new loss is $(1+r)X$. The limited expected value scales as $E[(1+r)X \wedge u] = (1+r)\,E\!\left[X \wedge \tfrac{u}{1+r}\right]$. A fixed deductible $d$ is effectively reduced to $\frac{d}{1+r}$ in pre-inflation terms, so expected payments rise faster than $r$.

21. Define Value at Risk (VaR) at security level $p$ for a loss random variable $X$.

    $\text{VaR}_p(X) = \pi_p$ is the $100p$-th percentile of the loss distribution: the smallest value satisfying $F_X(\pi_p) = p$ (i.e. $\pi_p = F_X^{-1}(p)$). It is the loss exceeded with probability $1-p$.

22. Define Tail Value at Risk (TVaR / CTE) at level $p$ and state how it relates to VaR.

    $$\text{TVaR}_p(X) = E[X \mid X > \text{VaR}_p(X)] = \frac{1}{1-p}\int_{p}^{1} \text{VaR}_u(X)\, du.$$ It equals the average of losses exceeding the VaR threshold, so $\text{TVaR}_p \geq \text{VaR}_p$ always.

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