Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Reserving and Pricing for Short-Term Insurance Coverages Practice Questions

B is correct. Allocated Loss Adjustment Expenses (ALAE) are claim-specific costs that can be directly assigned to an individual claim file, including defense counsel fees, expert witness costs, independent medical exams, and outside adjuster fees retained for that particular claim. Because they are identifiable at the claim level, ALAE is included in the total loss cost used in pure premium calculations.

Why each other option is incorrect:
- (A) describes Unallocated Loss Adjustment Expenses (ULAE) — overhead costs such as staff salaries and general claims department expenses that cannot be tied to individual claims.
- (B) confuses reinsurance premium allocation with loss adjustment expenses; ceded reinsurance premiums are not LAE.
- (D) describes general company overhead or administrative expenses, which are not classified as ALAE regardless of which department incurs them.
- (E) describes the netting of salvage/subrogation and reinsurance recoveries, which is a different concept unrelated to the definition of ALAE.

Mack's model (1993) makes three key assumptions:
1. $E[C_{i,k+1} \mid C_{i,1}, \ldots, C_{i,k}] = f_k \cdot C_{i,k}$ — the expected cumulative losses at the next age depend only on the current cumulative losses through a proportional relationship.
2. $\text{Var}(C_{i,k+1} \mid C_{i,1}, \ldots, C_{i,k}) = \sigma_k^2 \cdot C_{i,k}$ — variance is proportional to current cumulative losses.
3. $\{C_{i,k}\}_{k \geq 1}$ are independent across accident years $i$.
A is correct.

Why each other option is incorrect:
- (A) Mack's variance assumption is $\text{Var} \propto C_{i,k}$ (not the square of the mean); describing it as proportional to the square is the ODP model's implicit structure.
- (B) This describes the cross-classified Poisson (England-Verrall ODP) model, not Mack's model.
- (D) Mack's model explicitly produces a variance formula, so development factors are treated as random estimators with associated uncertainty.
- (E) Mack's model makes no distributional assumption beyond the first two moments; lognormality is not assumed.

A is correct. In Mack (1993), the total MSEP for the aggregate reserve includes cross-terms:
$$\widehat{\text{MSEP}}(\hat{R}) = \sum_i \widehat{\text{MSEP}}(\hat{R}_i) + 2 \sum_{i < i'} \hat{R}_i \hat{R}_{i'} \sum_{k = n-i'+1}^{n-1} \frac{\hat{\sigma}_k^2}{\hat{f}_k^2 S_k}$$

The sum over $k$ starts at $n - i' + 1$ — the first future development interval for the more mature accident year $i'$ — and runs to the last interval. This captures the parameter error correlation: both accident years $i$ and $i'$ will be projected using the same estimated factors $\hat{f}_k$ for those future intervals, so their reserve errors are positively correlated.

Why each other option is incorrect:
- (A) includes development intervals already observed for both accident years; parameter error correlation arises only from future intervals where the same estimated factors are applied, not from intervals with actual data for both years.
- (B) uses a product rather than a sum over future intervals; Mack's cross-term is a sum of per-interval parameter variance contributions divided by column sums, not a product.
- (C) equates the cross-term to the sum of squared reserves, which corresponds to the case of perfect positive correlation — a different (and overstated) formula that ignores the actual covariance structure derived by Mack.
- (D) squares the sum of per-interval variances; the cross-term is proportional to the covariance between future reserve estimates, not the square of the individual variance sums.