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

24 sample cards from the Exam FAM — Fundamentals of Actuarial Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. How do you construct the likelihood contribution for a left-truncated observation at deductible $d$?

   A left-truncated observation $x$ (only losses above deductible $d$ are observed) contributes the conditional density $\dfrac{f(x; \theta)}{S(d; \theta)}$ to the likelihood, normalizing by the probability the loss exceeded $d$.

2. What is the MLE of $\theta$ for an exponential distribution given a complete sample $x_1, \dots, x_n$?

   For the exponential density $f(x) = \frac{1}{\theta}e^{-x/\theta}$, the MLE is the sample mean: $$\hat\theta = \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.$$

3. State the asymptotic distribution of the MLE and the role of Fisher information.

   The MLE is asymptotically normal: $\hat\theta \approx N\!\left(\theta, \frac{1}{I(\theta)}\right)$, where $I(\theta) = -E\!\left[\frac{\partial^{2}}{\partial\theta^{2}}\ln f(X;\theta)\right]$ is the Fisher information. The estimated variance is the inverse of the observed/expected information.

4. Describe the likelihood ratio test for nested models.

   For nested models, the test statistic is $$\Lambda = 2\big(\ell_{\text{full}} - \ell_{\text{reduced}}\big),$$ which is asymptotically $\chi^{2}$ with degrees of freedom equal to the difference in the number of free parameters. Large $\Lambda$ rejects the simpler model.

5. Define the Schwarz Bayesian Criterion (SBC/BIC) used for model selection and the decision rule.

   $$\text{SBC} = \ell(\hat\theta) - \frac{r}{2}\ln n,$$ where $r$ is the number of estimated parameters and $n$ the sample size. Choose the model with the highest SBC; it penalizes added parameters more heavily than the AIC.

6. State the Akaike Information Criterion (AIC) and how it compares to SBC as a penalty.

   $$\text{AIC} = \ell(\hat\theta) - r,$$ with $r$ the number of parameters. Choose the largest AIC. SBC penalizes by $\frac{r}{2}\ln n$, which exceeds the AIC penalty of $r$ whenever $\ln n > 2$ (i.e. $n > 7$), so SBC favors simpler models for large samples.

7. What does the Kolmogorov–Smirnov (K–S) test measure for model selection?

   The K–S statistic is the maximum absolute difference between the empirical CDF $F_n(x)$ and the fitted CDF $F^{*}(x)$: $$D = \max_{x} \big| F_n(x) - F^{*}(x) \big|.$$ Smaller $D$ indicates a better fit; it is most sensitive to discrepancies near the center of the distribution.

8. What is the limited fluctuation (classical) credibility full-credibility standard for claim counts?

   The standard for full credibility on the number of claims (Poisson assumption) is $$\lambda_F = \left(\frac{z_{(1+p)/2}}{k}\right)^{2},$$ the expected number of claims needed so that observed counts are within $\pm k$ of the mean with probability $p$.

9. In limited fluctuation credibility, what is the partial credibility factor $Z$ and the credibility-weighted estimate?

   The square-root rule gives $Z = \sqrt{\frac{n}{n_F}}$ (capped at 1), where $n$ is the observed exposure/claims and $n_F$ the full-credibility standard. The estimate is $$\hat\mu = Z\,\bar{X} + (1 - Z)\,M,$$ with $M$ the prior/manual mean.

10. In Bühlmann (greatest accuracy) credibility, define the Expected Process Variance (EPV) and Variance of Hypothetical Means (VHM).

    $\text{EPV} = E_\Theta\!\left[\operatorname{Var}(X \mid \Theta)\right]$ (average within-risk variance) and $\text{VHM} = \operatorname{Var}_\Theta\!\left(E[X \mid \Theta]\right)$ (variance of the hypothetical means across risks). These drive the credibility factor.

11. Write the Bühlmann credibility factor $Z$ and the credibility premium.

    With $k = \frac{\text{EPV}}{\text{VHM}}$ and $n$ observations, $$Z = \frac{n}{n + k}.$$ The Bühlmann credibility estimate is $\hat\mu = Z\bar{X} + (1-Z)\mu$, where $\mu$ is the overall (collective) mean.

12. How does Bühlmann–Straub credibility differ from basic Bühlmann?

    Bühlmann–Straub allows unequal exposures $m_i$ per period. The credibility factor uses total exposure $m = \sum m_i$: $$Z = \frac{m}{m + k}, \quad k = \frac{\text{EPV}}{\text{VHM}},$$ and uses an exposure-weighted average of observed claim experience.

13. Define the survival function $S_x(t)$ and the actuarial notation ${}_tp_x$ and ${}_tq_x$.

    For a life aged $x$, ${}_tp_x = S_x(t) = \Pr[T_x > t]$ is the probability of surviving $t$ more years, and ${}_tq_x = 1 - {}_tp_x = \Pr[T_x \leq t]$ is the probability of dying within $t$ years. Note ${}_tp_x + {}_tq_x = 1$.

14. Express ${}_tp_x$ in terms of the force of mortality $\mu_{x+t}$.

    $${}_tp_x = \exp\!\left(-\int_{0}^{t} \mu_{x+s}\, ds\right),$$ where $\mu_{x+s} = \frac{-S_x'(s)}{S_x(s)}$ is the force of mortality (hazard rate) at age $x+s$.

15. State the complete and curtate expectation of life formulas.

    Complete expectation: $\mathring{e}_x = \int_{0}^{\infty} {}_tp_x\, dt$. Curtate expectation: $e_x = \sum_{k=1}^{\infty} {}_kp_x$. Approximately $\mathring{e}_x \approx e_x + \frac{1}{2}$ under uniform distribution of deaths.

16. Give the actuarial present value (EPV) of a fully discrete whole life insurance $A_x$.

    $$A_x = \sum_{k=0}^{\infty} v^{k+1} \, {}_kp_x \, q_{x+k},$$ where $v = \frac{1}{1+i}$ is the annual discount factor. It pays $1$ at the end of the year of death.

17. Write the actuarial present value of a fully continuous whole life insurance $\bar{A}_x$.

    $$\bar{A}_x = \int_{0}^{\infty} v^{t}\, {}_tp_x\, \mu_{x+t}\, dt = \int_{0}^{\infty} e^{-\delta t}\, {}_tp_x\, \mu_{x+t}\, dt,$$ where $\delta = \ln(1+i)$ is the force of interest. It pays $1$ at the moment of death.

18. State the relationship between the EPV of whole life insurance $A_x$ and the whole life annuity-due $\ddot{a}_x$.

    $$A_x = 1 - d\,\ddot{a}_x, \quad\text{equivalently}\quad \ddot{a}_x = \frac{1 - A_x}{d},$$ where $d = \frac{i}{1+i}$ is the annual discount rate. This is the fundamental annuity–insurance identity (discrete case).

19. Give the variance of the present-value random variable for fully continuous whole life insurance.

    $$\operatorname{Var}(Z) = {}^{2}\bar{A}_x - \left(\bar{A}_x\right)^{2},$$ where ${}^{2}\bar{A}_x$ is the insurance EPV computed at force of interest $2\delta$ (double the force of interest).

20. Write the EPV of a whole life annuity-due $\ddot{a}_x$ and a continuous life annuity $\bar{a}_x$.

    Discrete: $\ddot{a}_x = \sum_{k=0}^{\infty} v^{k}\, {}_kp_x$ (payments at the start of each year). Continuous: $\bar{a}_x = \int_{0}^{\infty} v^{t}\, {}_tp_x\, dt = \int_{0}^{\infty} e^{-\delta t}\, {}_tp_x\, dt$.

21. State the equivalence principle for calculating a net premium.

    The net premium $P$ is set so that the EPV of future premiums equals the EPV of future benefits at issue: $$\text{EPV(premiums)} = \text{EPV(benefits)}.$$ Equivalently, the EPV of the loss-at-issue random variable $L_0$ is zero: $E[L_0] = 0$.

22. Give the net annual premium for a fully discrete whole life insurance of 1, $P_x$.

    $$P_x = \frac{A_x}{\ddot{a}_x} = \frac{d A_x}{1 - A_x} = \frac{1}{\ddot{a}_x} - d,$$ funding a benefit of $1$ paid at end of year of death with level premiums payable at the start of each year.

23. How does a gross (expense-loaded) premium differ from a net premium, and state its equivalence equation?

    The gross premium $G$ funds benefits plus expenses (and possibly profit). By the equivalence principle: $$\text{EPV(gross premiums)} = \text{EPV(benefits)} + \text{EPV(expenses)}.$$ Expenses include per-policy, per-premium (percentage), and acquisition (first-year) costs, so $G \geq P$ (the net premium).

24. Define the prospective policy reserve and give the formula for a fully discrete whole life policy at duration $t$.

    The prospective reserve is EPV(future benefits) $-$ EPV(future net premiums). For fully discrete whole life: $$\,{}_tV_x = A_{x+t} - P_x\,\ddot{a}_{x+t} = 1 - \frac{\ddot{a}_{x+t}}{\ddot{a}_x}.$$ It measures the insurer's accumulated liability per surviving policyholder.

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