Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Joint Life Insurance and Annuities Practice Questions

B is correct. The joint-life status fails within one year if at least one of the two lives dies. Under independence:
$$q_{xy} = 1 - p_{xy} = 1 - p_x \cdot p_y = 1-(1-q_x)(1-q_y) = 1-(0.95)(0.96) = 1-0.912 = 0.088$$
Equivalently by inclusion-exclusion: $q_x + q_y - q_x q_y = 0.05+0.04-0.002 = 0.088$. A sums individual rates, double-counting the event both die. B uses only $q_x$. C uses an arithmetic average. D gives $q_x q_y$, the probability the last-survivor status fails in year 1 (both die), not the joint-life status.

C is correct. The prospective reserve equals the expected present value of future benefits minus the expected present value of future premiums, both conditioned on both lives surviving to time $t$ with attained ages $x+t$ and $y+t$:
$${}_tV_{xy} = A_{x+t:y+t} - P_{xy}\ddot{a}_{x+t:y+t}$$
At $t=0$: $A_{xy} - P_{xy}\ddot{a}_{xy} = 0$ by the equivalence principle. For $t>0$, the attained ages change and the reserve grows. A uses inception values rather than attained ages, always giving zero. B reverses the sign. C uses the last-survivor insurance identity incorrectly for the joint-life insurance. D sums individual reserves, an incorrect decomposition — the joint reserve is not the sum of marginal reserves.

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.