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

E 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: $q_x + q_y - q_x q_y = 0.05+0.04-0.002 = 0.088$.

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.

D is correct.

A whole life insurance of 1 on a status pays when the status fails at time $T$, so its APV $E[v^{T}]$ falls as the future lifetime lengthens. The joint-life status fails at the first death and the last-survivor status at the second, so on every outcome $T(xy) = \min(T(x), T(y))$ and $T(\bar{x}\bar{y}) = \max(T(x), T(y))$. Discounting reverses the lifetime order:

$$A_{\bar{x}\bar{y}} \leq \min(A_x, A_y) \leq \max(A_x, A_y) \leq A_{xy}$$

The given joint value fits this ordering: $A_{xy} = 0.40 \geq 0.30 = \max(A_x, A_y)$.

On every outcome the pair $T(xy), T(\bar{x}\bar{y})$ is the pair $T(x), T(y)$ in sorted order, so $v^{T(xy)} + v^{T(\bar{x}\bar{y})} = v^{T(x)} + v^{T(y)}$. Taking expectations gives $A_{xy} + A_{\bar{x}\bar{y}} = A_x + A_y$ with no assumption about dependence between the lives, so

$$A_{\bar{x}\bar{y}} = 0.25 + 0.30 - 0.40 = 0.15$$

and $0.15 \leq \min(A_x, A_y) = 0.25$, as the ordering requires. The student's computation is valid.

Reading the ordering as $A_{\bar{x}\bar{y}} \geq \max(A_x, A_y)$ reverses it: the status that lasts longest pays latest and so has the smallest APV. The ratio $A_x A_y / A_{xy} = 0.1875$ is not a last-survivor formula; the additive identity is exact. The same pathwise argument applies to term benefits and whole life benefits alike, and to dependent lives as well as independent ones.