Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Practice Questions

B is correct. A universal life policy lapses when the account value is reduced to zero or below after charges are applied and the policyholder fails to make a sufficient premium payment within the grace period (typically 61 days) to restore a positive account value. This is the defining feature of UL's flexible premium structure: the policy persists only as long as the account value is sufficient to cover ongoing COI and expense charges.

A is incorrect: the guaranteed minimum credited rate is a contractual floor; if the insurer fails to credit at least that rate, it is a contract breach — not a lapse trigger. The credited rate falling below the guarantee is an insurer obligation problem, not a policyholder lapse event.
C is incorrect: switching between death benefit options is a policyholder right under the contract and does not cause lapse; no new underwriting is required.
C is incorrect: the NAR exceeding the face amount is definitionally impossible under standard Type A (NAR = DB − AV < DB for positive AV) and under Type B the NAR always equals the face amount.
E is incorrect: expense charges exceeding the monthly premium deplete account value but do not independently trigger lapse without the account value actually reaching zero.

C is correct. The key practical critique of the Markov assumption in long-term care (and disability) insurance is that it ignores duration-dependence. In reality, a person who has been disabled for 5 years has a very different recovery probability than someone newly disabled — the Markov property says only the current state (Disabled) matters, not the time spent there. This duration-dependence is well-documented empirically and is the primary motivation for semi-Markov or duration-dependent extensions.
B is incorrect: the Markov property does NOT require constant intensities — intensities can be arbitrary functions of current age $x + t$; the requirement is only that they not depend on past states or sojourn times.
A is incorrect: the Markov framework is entirely flexible in the number of states; adding states (e.g., partial disability, hospitalization) is a standard model extension that preserves the Markov structure.
D is incorrect: Kolmogorov's forward equations are specifically derived for continuous-time Markov chains — they depend on the Markov property and do not apply to non-Markov processes without modification.
E is incorrect: MLE is fully compatible with Markov models; the likelihood factorizes nicely by state because the Markov property implies that transitions from a given state are independent of the history prior to entering that state.

A is correct. The last-survivor status is alive whenever at least one of $(x)$ or $(y)$ is alive, so $T(\bar{x}\bar{y}) = \max(T(x), T(y)) \geq T(x)$ and $T(\bar{x}\bar{y}) \geq T(y)$ almost surely. Since the whole life insurance APV is an increasing function of the future lifetime, $A_{\bar{x}\bar{y}} \geq A_x$ and $A_{\bar{x}\bar{y}} \geq A_y$ must hold under any dependence structure. With the given inputs, $A_{\bar{x}\bar{y}} = 0.15 < 0.25 = A_x < 0.30 = A_y$, which is impossible. Furthermore, $A_{xy} = 0.40 > \min(A_x, A_y) = 0.25$ is itself a violation: since $T(xy) \leq T(x)$ always, we require $A_{xy} \geq A_x$ The inconsistency arises purely from $A_{\bar{x}\bar{y}} = 0.15 < A_y$. Choice D claims the result is valid under strong positive dependence — the ordering $A_{\bar{x}\bar{y}} \geq A_y$ is a mathematical identity, not an assumption that depends on the dependence structure. Choice B invents a product formula with no standard derivation. Choice C claims the ordering fails for term insurance — incorrect; the stochastic ordering implies APV ordering for any increasing benefit function. Choice E claims plausibility under perfect negative dependence, but the constraint $A_{\bar{x}\bar{y}} \geq A_y$ holds universally, not just under positive dependence.