Associate of the Society of Actuaries (ASA/FSA) ASTAM — Advanced Short-Term Actuarial Mathematics Syllabus

ASTAM — Advanced Short-Term 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 ASTAM — Advanced Short-Term Actuarial Mathematics in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Advanced Loss and Aggregate Modeling

   3 topics

   • Severity Model Construction
     • Mixture and spliced models
     • Tail behavior and extreme value considerations
   • Aggregate Distribution Computation
     • Panjer recursion
     • Discretization methods
     • Normal and lognormal approximations
   • Risk Measures and Capital
     • Coherent risk measures
     • VaR and TVaR in aggregate context
2. 2

   Reinsurance and Coverage Modifications

   3 topics

   • Reinsurance Arrangements
     • Proportional (quota share, surplus)
     • Non-proportional (excess of loss, stop-loss)
   • Effect of Reinsurance on Distributions
     • Ceded and retained losses
     • Impact on moments and tail risk
   • Deductibles, Limits, and Inflation
     • Layered loss costs
     • Trending modified coverages
3. 3

   Credibility and Experience Rating

   3 topics

   • Greatest Accuracy (Buhlmann) Credibility
     • Structural parameters and credibility factor
     • Buhlmann-Straub for varying exposures
   • Empirical Bayes Estimation
     • Nonparametric estimation of structural parameters
     • Semiparametric methods
   • Bayesian Credibility
     • Conjugate prior families
     • Predictive distributions
4. 4

   Claims Reserving and Ratemaking

   3 topics

   • Loss Reserving Methods
     • Chain-ladder (development triangle) method
     • Bornhuetter-Ferguson method
     • Expected loss ratio method
   • Stochastic Reserving
     • Mack model
     • Bootstrap reserve variability
   • Ratemaking Fundamentals
     • Loss cost and expense provisions
     • Trend and on-level adjustments

ASTAM — Advanced Short-Term Actuarial Mathematics flashcards for Associate of the Society of Actuaries (ASA/FSA)

20 of 49 cards from the ASTAM — Advanced Short-Term Actuarial Mathematics deck — real questions with worked answers.

1. In loss modeling, what is the relationship between the survival function $S(x)$ and the cumulative distribution function $F(x)$?

   $S(x) = 1 - F(x) = \Pr(X > x)$. It gives the probability the loss exceeds $x$.

2. Define the hazard (failure) rate function $h(x)$ for a severity random variable $X$.

   $$h(x) = \frac{f(x)}{S(x)} = -\frac{d}{dx}\ln S(x)$$ It represents the instantaneous rate of decrement (death/failure) at $x$ given survival to $x$.

3. What is the raw $k$-th moment of a severity distribution $X$, and how do you compute the $k$-th limited expected value at limit $u$?

   Raw moment: $E[X^{k}] = \int_{0}^{\infty} x^{k} f(x)\,dx$. Limited expected value: $$E[(X \wedge u)^{k}] = \int_{0}^{u} x^{k} f(x)\,dx + u^{k} S(u)$$

4. How is a severity distribution classified as having a heavy tail versus a light tail using moments?

   A heavier tail has fewer finite moments. If $E[X^{k}]$ exists only for $k$ up to some finite value, the tail is heavy. A distribution whose moments all exist (e.g., gamma) is light-tailed; a Pareto (only lower moments finite) is heavy-tailed.

5. What is a coherent risk measure, and what four properties must it satisfy?

   A risk measure $\rho$ is coherent if it satisfies: (1) Subadditivity: $\rho(X+Y) \leq \rho(X)+\rho(Y)$; (2) Monotonicity: $X \leq Y \Rightarrow \rho(X) \leq \rho(Y)$; (3) Positive homogeneity: $\rho(cX) = c\,\rho(X)$ for $c>0$; (4) Translation invariance: $\rho(X+c) = \rho(X)+c$.

6. Define Value-at-Risk $\mathrm{VaR}_{p}(X)$ at security level $p$.

   $\mathrm{VaR}_{p}(X) = \pi_{p}$, the $100p$-th percentile of the loss distribution, i.e. the smallest value satisfying $$F(\pi_{p}) = \Pr(X \leq \pi_{p}) = p.$$ It is the $p$-quantile of losses.

7. Define Tail-Value-at-Risk $\mathrm{TVaR}_{p}(X)$ and state how it relates to VaR.

   $$\mathrm{TVaR}_{p}(X) = E[X \mid X > \mathrm{VaR}_{p}(X)] = \frac{1}{1-p}\int_{p}^{1} \mathrm{VaR}_{u}(X)\,du$$ Equivalently $\mathrm{TVaR}_{p} = \mathrm{VaR}_{p} + e(\mathrm{VaR}_{p})$, the VaR plus the mean excess over it. TVaR is coherent; VaR generally is not (fails subadditivity).

8. Why is VaR not a coherent risk measure, and which property does it violate?

   VaR violates subadditivity: for some random variables $\mathrm{VaR}_{p}(X+Y) > \mathrm{VaR}_{p}(X) + \mathrm{VaR}_{p}(Y)$, so it can penalize diversification. The other three coherence properties hold for VaR.

9. Define the mean excess loss (mean residual life) function $e_{X}(d)$.

   $$e_{X}(d) = E[X - d \mid X > d] = \frac{\int_{d}^{\infty} S(x)\,dx}{S(d)} = \frac{E[X] - E[X \wedge d]}{S(d)}$$ It is the expected loss in excess of $d$ given a loss exceeds $d$.

10. Under the collective risk model, what are the mean and variance of aggregate losses $S = X_{1}+\cdots+X_{N}$ (compound distribution)?

    $E[S] = E[N]\,E[X]$ and $$\mathrm{Var}(S) = E[N]\,\mathrm{Var}(X) + \mathrm{Var}(N)\,(E[X])^{2}.$$ (Compound variance formula, assuming $N$ and the $X_i$ independent.)

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

    $E[S] = \lambda\,E[X]$ and $\mathrm{Var}(S) = \lambda\,E[X^{2}]$. (Since for Poisson $E[N]=\mathrm{Var}(N)=\lambda$, the variance collapses to the second raw moment of severity.)

12. State the Panjer recursion and the $(a,b,0)$ relationship it requires.

    If frequency satisfies $p_{n} = \left(a + \frac{b}{n}\right)p_{n-1}$ and severity is on $0,1,2,\dots$, then for the aggregate: $$f_{S}(s) = \frac{1}{1 - a f_{X}(0)}\sum_{x=1}^{s}\left(a + \frac{b\,x}{s}\right) f_{X}(x)\, f_{S}(s-x)$$ starting from $f_{S}(0)$.

13. What are the $(a,b,0)$ parameter values for the Poisson, binomial, and negative binomial frequency distributions?

    Poisson$(\lambda)$: $a=0,\ b=\lambda$. Binomial$(m,q)$: $a=-\frac{q}{1-q},\ b=(m+1)\frac{q}{1-q}$. Negative binomial$(r,\beta)$: $a=\frac{\beta}{1+\beta},\ b=(r-1)\frac{\beta}{1+\beta}$.

14. How do you classify an $(a,b,0)$ distribution by the sign of $a$?

    $a < 0 \Rightarrow$ binomial; $a = 0 \Rightarrow$ Poisson; $0 < a < 1 \Rightarrow$ negative binomial (or geometric when $b=0$). The sign/magnitude of $a$ identifies the family.

15. Under the individual risk model, what are the mean and variance of aggregate losses $S = \sum_{i=1}^{n} X_{i}$?

    For independent (not necessarily identical) risks: $E[S] = \sum_{i=1}^{n} E[X_{i}]$ and $\mathrm{Var}(S) = \sum_{i=1}^{n}\mathrm{Var}(X_{i})$. Each policy is a separate (possibly two-point) random variable.

16. Compare the collective risk model and the individual risk model.

    Collective: $S=\sum_{i=1}^{N} X_i$ with random claim count $N$ and i.i.d. severities; uses a compound distribution. Individual: $S=\sum_{i=1}^{n} X_i$ with fixed number $n$ of policies, each having its own (often zero-inflated) loss. Collective is convenient when only portfolio-level frequency/severity are known; individual fits per-policy data.

17. With an ordinary deductible $d$, what is the per-loss (payment per loss) and per-payment expected cost?

    Per loss: $E[(X-d)_{+}] = E[X] - E[X \wedge d]$. Per payment (given a payment occurs): $$\frac{E[X] - E[X \wedge d]}{S(d)} = e_{X}(d).$$

18. How does a franchise deductible $d$ differ from an ordinary deductible, and what is its per-loss expected payment?

    Under a franchise deductible, if the loss exceeds $d$ the insurer pays the FULL loss (not just the excess). Per-loss expected payment: $$E[X] - E[X \wedge d] + d\,S(d).$$ It exceeds the ordinary deductible cost by $d\,S(d)$.

19. With deductible $d$, policy limit $u$, and maximum covered loss $u+d$, write the expected insurer payment per loss.

    $$E[(X \wedge (u+d))] - E[(X \wedge d)]$$ where $u$ is the maximum payment (policy limit) and $u+d$ is the maximum covered loss. Adding coinsurance $\alpha$ multiplies the whole expression by $\alpha$.

20. How does uniform inflation at rate $r$ affect expected costs with a deductible $d$? Give the per-loss formula.

    With losses inflated to $(1+r)X$, the per-loss expected cost is $$(1+r)\left[E[X] - E\!\left[X \wedge \tfrac{d}{1+r}\right]\right].$$ Inflation deflates the deductible by $(1+r)$, then scales the result up by $(1+r)$, raising the effective cost more than proportionally.

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