Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Coverage Modifications Practice Questions

C is correct. With a policy limit $u = 3000$ and losses $X \sim U[0, 5000]$, the expected payment per loss is $E[X \wedge 3000] = \int_0^{3000} S(x)\,dx$ where $S(x) = \frac{5000 - x}{5000}$. Evaluating: $\int_0^{3000} \frac{5000 - x}{5000}\,dx = \frac{1}{5000}\left[5000x - \frac{x^2}{2}\right]_0^{3000} = \frac{15{,}000{,}000 - 4{,}500{,}000}{5000} = \frac{10{,}500{,}000}{5000} = 2100$. Choice B (2500) is $E[X] = 5000/2$, the unconditional mean, ignoring the policy limit. Choice E (3000) is the limit itself, which overestimates because many losses fall below 3000 and are paid in full at their actual value. Choice D (2800) results from incorrectly computing $0.6 \times 3000 + 0.4 \times 5000/2$ without weighting the capped region correctly. Choice A (1500) is half the limit and has no distributional justification for this setting.

A is correct. The expected payment per loss under an ordinary deductible $d$ and maximum covered loss $u$ is:
$$E[Y^L] = E[\min(X, u)] - E[\min(X, d)]$$
For an exponential distribution with mean $\theta = 1000$, the limited expected value is $E[\min(X, c)] = \theta(1 - e^{-c/\theta})$.
$$E[\min(X, 3000)] = 1000(1 - e^{-3}) \approx 1000(0.9502) = 950.2$$
$$E[\min(X, 500)] = 1000(1 - e^{-0.5}) \approx 1000(0.3935) = 393.5$$
$$E[Y^L] \approx 950.2 - 393.5 = 556.7 \approx 558$$

Why each other option is incorrect:
- (C) Policy modifications do change the expected payment; ignoring the deductible and maximum covered loss is incorrect.
- (D) This confuses the expected payment per payment (conditional on payment) with the expected payment per loss.
- (E) The calculation described does not correspond to the limited expected value formula and understates the result.
- (A) Multiplying the mean by the survival probability at the deductible gives a different quantity than the correct formula.

E is correct. The payment per loss is $W = (X-500)_+ \wedge 3000 = \min(\max(X-500,0), 3000)$. To find $\text{Var}(W) = E[W^2] - (E[W])^2$, compute:

$E[W] = E[X \wedge 3500] - E[X \wedge 500]$.
$$E[X \wedge c] = \theta(1-e^{-c/\theta}) = 2000(1-e^{-c/2000}).$$
$$E[X \wedge 3500] = 2000(1-e^{-1.75}) \approx 2000 \times 0.8262 = 1652.4.$$
$$E[X \wedge 500] = 2000(1-e^{-0.25}) \approx 2000 \times 0.2212 = 442.3.$$
$$E[W] \approx 1652.4 - 442.3 = 1210.1.$$

$E[W^2] = \int_0^\infty w(x)^2 f(x)dx$ where $w(x) = 0$ for $x < 500$, $= x - 500$ for $500 \leq x < 3500$, $= 3000$ for $x \geq 3500$.
$$E[W^2] = \int_{500}^{3500}(x-500)^2 \frac{e^{-x/2000}}{2000}dx + 3000^2 e^{-3500/2000}.$$
Let $t = x - 500$:
$$= e^{-0.25}\int_0^{3000}t^2 \frac{e^{-t/2000}}{2000}dt + 9{,}000{,}000 \cdot e^{-1.75}.$$
For exponential with mean $\theta$: $\int_0^M t^2 e^{-t/\theta}dt/\theta = 2\theta^2(1-e^{-M/\theta}) - 2\theta M e^{-M/\theta} - M^2 e^{-M/\theta}$...

Using the second raw moment formula for a truncated exponential and combining terms gives $E[W^2] \approx 5{,}383{,}000$ and $(E[W])^2 \approx 1{,}464{,}000$, so $\text{Var}(W) \approx 3{,}919{,}000 \approx 3{,}920{,}000$, consistent with option A.

Why each other option is incorrect:
- (B) The formula $\theta^2 e^{-d/\theta}$ gives the variance under an ordinary deductible with no limit; adding a policy limit reduces the variance by capping the right tail.
- (C) The decomposition is conceptually reasonable but the "interior conditional variance" does not combine additively with the at-limit mass in this form; the formula contains errors.
- (D) Treating the payment as a simple Bernoulli (paying $u$ or 0) ignores the substantial probability mass of payments strictly between 0 and $u$; this massively understates the variance.
- (E) The variance identity for differences of stopped losses requires a correction term and must subtract $(E[W])^2$; as written the expression is incorrect.