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

24 sample cards from the ASTAM — Advanced Short-Term Actuarial Mathematics deck

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

1. Compare stop-loss reinsurance with excess-of-loss reinsurance.

   Excess-of-loss applies a retention to EACH individual loss: reinsurer pays $\sum (X_i - d)_+$. Stop-loss applies a retention to AGGREGATE losses: reinsurer pays $(S - d)_{+}$ where $S$ is total losses. Stop-loss protects against accumulation of many claims; excess-of-loss against large single claims.

2. Give the formula for the expected cost of stop-loss reinsurance with aggregate retention $d$.

   $$E[(S-d)_{+}] = E[S] - E[S \wedge d] = \int_{d}^{\infty} S_{S}(x)\,dx$$ where $S_{S}$ is the survival function of aggregate losses $S$. For discrete $S$, $E[(S-d)_+]=\sum_{s>d}(s-d)f_S(s)$.

3. State the recursive (one-step) relationship for the stop-loss premium when $S$ is integer-valued, moving the retention from $d$ to $d+1$.

   $$E[(S-(d+1))_{+}] = E[(S-d)_{+}] - S_{S}(d) = E[(S-d)_{+}] - \big(1 - F_{S}(d)\big).$$ Each unit increase in retention reduces the stop-loss premium by the survival probability at $d$.

4. What is the fundamental quantity (credibility premium) in limited fluctuation versus greatest accuracy credibility?

   Both compute a credibility-weighted estimate $P_{c} = Z\bar{X} + (1-Z)M$, where $\bar X$ is the observed experience, $M$ the prior/manual mean, and $Z$ the credibility factor. Limited fluctuation sets $Z$ to control fluctuation; greatest accuracy (Bühlmann) derives $Z$ to minimize expected squared error.

5. Define the structural parameters $\mu$, $v$ (EPV), and $a$ (VHM) in Bühlmann credibility.

   Let $\Theta$ be the risk parameter. Hypothetical mean $\mu(\theta)=E[X\mid\theta]$, process variance $v(\theta)=\mathrm{Var}(X\mid\theta)$. Then: collective mean $\mu = E[\mu(\Theta)]$; Expected Process Variance $v = E[v(\Theta)]$; Variance of Hypothetical Means $a = \mathrm{Var}(\mu(\Theta))$.

6. Give the Bühlmann credibility factor $Z$ and the Bühlmann constant $k$ for $n$ observations.

   $$k = \frac{v}{a}, \qquad Z = \frac{n}{n+k} = \frac{n}{n + v/a}.$$ As $n\to\infty$ or $a$ grows relative to $v$, $Z\to 1$ (experience trusted).

7. Write the Bühlmann credibility premium for the next period given sample mean $\bar{X}$.

   $$P_{c} = Z\bar{X} + (1-Z)\mu, \qquad Z = \frac{n}{n+k},\ k=\frac{v}{a}.$$ This is the linear least-squares (best linear) approximation to the Bayesian premium.

8. How does the Bühlmann-Straub model differ from the basic Bühlmann model?

   Bühlmann-Straub allows different exposure/weights $m_{ij}$ per observation (varying volume by period). Process variance scales as $v/m_{ij}$. Credibility uses total exposure $m=\sum m_{j}$: $$Z = \frac{m}{m + k},\quad k=\frac{v}{a},$$ and the experience mean is the exposure-weighted average.

9. In Bühlmann-Straub, how is the credibility-weighted average $\bar{X}$ computed across periods with exposures $m_{j}$?

   $$\bar{X} = \frac{\sum_{j} m_{j} X_{j}}{\sum_{j} m_{j}}$$ the exposure-weighted mean of the per-unit observations $X_j$ (losses divided by exposure).

10. What does empirical Bayes (nonparametric) credibility estimate, and why is it called 'empirical'?

    It estimates the structural parameters $\mu$, $v$, and $a$ from the data itself rather than assuming a parametric prior. 'Empirical' because $\hat\mu$, $\hat v$, $\hat a$ come from observed sample means and variances across risks/policyholders.

11. Give the nonparametric empirical Bayes (Bühlmann) estimators of $\mu$ and $v$ for $r$ policyholders each with $n$ observations.

    $$\hat\mu = \bar{X} = \frac{1}{rn}\sum_{i=1}^{r}\sum_{j=1}^{n} X_{ij},\qquad \hat v = \frac{1}{r}\sum_{i=1}^{r}\frac{1}{n-1}\sum_{j=1}^{n}(X_{ij}-\bar X_{i})^{2}.$$ $\hat v$ averages each risk's sample variance (the within-risk variance).

12. Give the nonparametric empirical Bayes (Bühlmann) estimator of $a$ (VHM) for $r$ equal-sized risks.

    $$\hat a = \frac{1}{r-1}\sum_{i=1}^{r}(\bar X_{i} - \bar X)^{2} - \frac{\hat v}{n}.$$ It is the between-risk variance of the $\bar X_i$ minus the within-risk noise term. If negative, $\hat a$ is set to 0.

13. What is semiparametric empirical Bayes estimation, and when is it used?

    Semiparametric estimation assumes a parametric form for the process (e.g., Poisson frequency, giving $v=\mu$, so $\hat v = \bar X$) but estimates the structure of $a$ empirically from the data. Used when the conditional distribution family is known but the prior is not.

14. In semiparametric empirical Bayes with Poisson claim counts, how is $\hat{v}$ obtained?

    For Poisson, the process variance equals the mean, so $v(\theta)=\mu(\theta)$ and $\hat v = \hat\mu = \bar X$. Then $\hat a = \hat{\mathrm{Var}}(\bar X_i) - \hat v / n$, using the overall sample mean for $\hat v$.

15. Define the Bayesian premium and contrast it with the Bühlmann premium.

    Bayesian premium $= E[X_{n+1}\mid \mathbf{X}]$, the predictive mean using the full posterior $\pi(\theta\mid\mathbf{x})$. The Bühlmann premium is the best LINEAR approximation to this. They coincide exactly when the model/prior pair is a linear-exponential conjugate (e.g., Poisson-gamma, Bernoulli-beta).

16. For the Poisson-gamma Bayesian model (Poisson$(\lambda)$ likelihood, gamma$(\alpha,\theta)$ prior on $\lambda$), what is the posterior distribution after observing $n$ counts summing to $\sum x_{i}$?

    The posterior of $\lambda$ is gamma with shape $\alpha^{*} = \alpha + \sum x_{i}$ and rate updated so the new scale is $\theta^{*} = \frac{\theta}{1 + n\theta}$. The predictive (Bayesian) mean is $\alpha^{*}\theta^{*}$.

17. Why does exact (Bayesian) credibility equal Bühlmann credibility for conjugate linear-exponential families?

    For these families the posterior mean is linear in the observations $\bar X$, so the optimal estimator is already linear. Since Bühlmann is the best linear estimator, it matches the exact Bayesian posterior mean: $E[X_{n+1}\mid\mathbf{X}] = Z\bar X + (1-Z)\mu$.

18. What is a conjugate prior, and name two common conjugate pairs used in Bayesian credibility.

    A conjugate prior yields a posterior in the same family as the prior. Common pairs: Poisson likelihood with gamma prior (gamma posterior); Bernoulli/binomial likelihood with beta prior (beta posterior); normal-normal; exponential/gamma likelihood with inverse-gamma prior.

19. Explain the chain-ladder method for loss reserving.

    Using a run-off triangle of cumulative paid/incurred losses by accident year and development period, compute age-to-age (development) factors $f_{j} = \frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}}$, then multiply the latest diagonal forward to project ultimate losses. The reserve is ultimate minus paid-to-date.

20. Define an age-to-age (loss development) factor and the cumulative development factor (CDF).

    Age-to-age factor $f_{j}=\frac{C_{\cdot,j+1}}{C_{\cdot,j}}$ measures growth from development period $j$ to $j+1$. The cumulative development factor to ultimate is $$F_{j} = \prod_{k=j}^{n-1} f_{k},$$ which scales a year's current cumulative loss up to its ultimate value.

21. Describe the Bornhuetter-Ferguson (BF) reserving method and its key advantage over chain-ladder.

    BF reserve $= \text{(expected ultimate)} \times (1 - 1/\text{CDF})$, i.e. an a-priori expected ultimate (often premium $\times$ expected loss ratio) times the unreported/undeveloped proportion. Advantage: more stable than chain-ladder for immature/volatile years because it does not rely heavily on the noisy latest diagonal.

22. Compare the chain-ladder and Bornhuetter-Ferguson reserve estimates as a development year matures.

    Chain-ladder relies entirely on reported losses to date (volatile when little is reported). BF blends an a-priori expectation with reported data via the percent-reported $1/F_j$. As a year matures ($1/F_j \to 1$), BF converges to the chain-ladder result; for green years BF leans on the a-priori.

23. What is the Mack model, and what does it add to the deterministic chain-ladder?

    The Mack model is a distribution-free STOCHASTIC chain-ladder that provides the standard error (prediction uncertainty) of reserve estimates. It assumes $E[C_{i,j+1}\mid C_{i,j}]=f_j C_{i,j}$ and $\mathrm{Var}(C_{i,j+1}\mid C_{i,j})=\sigma_j^2 C_{i,j}$, yielding mean-squared-error estimates without specifying a full distribution.

24. How does the over-dispersed Poisson (ODP) bootstrap produce a distribution of reserves in stochastic reserving?

    Fit an ODP GLM (whose mean reserves reproduce the chain-ladder estimate), compute residuals, then resample residuals to create many pseudo-triangles, refit each, and project reserves. The collection of projected reserves forms a predictive distribution, giving percentiles, standard error, and capital estimates.

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ASTAM — Advanced Short-Term Actuarial Mathematics flashcards FAQ