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

Joint life insurance and annuities on SOA Exam ALTAM cover joint-life and last-survivor statuses, common shock mortality models, reversionary annuities, and multiple life contingent cash flows.

120 Questions
45 Easy
56 Medium
19 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
For two independent lives with qx=0.05q_x = 0.05 and qy=0.04q_y = 0.04, calculate qxyq_{xy}, the probability the joint-life status fails within one year.
Solution
E is correct.

The joint-life status fails within one year if at least one of the two lives dies. Under independence: qxy=1−pxy=1−px⋅py=1−(1−qx)(1−qy)=1−(0.95)(0.96)=1−0.912=0.088q_{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: qx+qy−qxqy=0.05+0.04−0.002=0.088q_x + q_y - q_x q_y = 0.05+0.04-0.002 = 0.088.
Question 2 Medium
The prospective reserve at time tt for a joint-life whole life insurance with net annual premium PxyP_{xy} is tVxy{}_tV_{xy}. Which of the following is the correct prospective formula?
Solution
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 tt with attained ages x+tx+t and y+ty+t: tVxy=Ax+t:y+t−Pxya¨x+t:y+t{}_tV_{xy} = A_{x+t:y+t} - P_{xy}\ddot{a}_{x+t:y+t} At t=0t=0: Axy−Pxya¨xy=0A_{xy} - P_{xy}\ddot{a}_{xy} = 0 by the equivalence principle. For t>0t>0, the attained ages change and the reserve grows.
Question 3 Hard
Given Ax=0.25A_x = 0.25, Ay=0.30A_y = 0.30, and Axy=0.40A_{xy} = 0.40, a student computes the last-survivor APV as Axˉyˉ=Ax+Ay−Axy=0.15A_{\bar{x}\bar{y}} = A_x + A_y - A_{xy} = 0.15. Which statement correctly evaluates this result?
Solution
A is correct.

The last-survivor status is alive whenever at least one of (x)(x) or (y)(y) is alive, so T(xˉyˉ)=max⁡(T(x),T(y))≥T(x)T(\bar{x}\bar{y}) = \max(T(x), T(y)) \geq T(x) and T(xˉyˉ)≥T(y)T(\bar{x}\bar{y}) \geq T(y) almost surely. Since the whole life insurance APV is an increasing function of the future lifetime, Axˉyˉ≥AxA_{\bar{x}\bar{y}} \geq A_x and Axˉyˉ≥AyA_{\bar{x}\bar{y}} \geq A_y must hold under any dependence structure. With the given inputs, Axˉyˉ=0.15<0.25=Ax<0.30=AyA_{\bar{x}\bar{y}} = 0.15 < 0.25 = A_x < 0.30 = A_y, which is impossible. Furthermore, Axy=0.40>min⁡(Ax,Ay)=0.25A_{xy} = 0.40 > \min(A_x, A_y) = 0.25 is itself a violation: since T(xy)≤T(x)T(xy) \leq T(x) always, we require Axy≥AxA_{xy} \geq A_x. The inconsistency arises purely from Axˉyˉ=0.15<AyA_{\bar{x}\bar{y}} = 0.15 < A_y.
Create a Free Account to Access All 120 Questions →

About FreeFellow

FreeFellow is an AI-native exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, GARP FRM, IRS Enrolled Agent, IMA CMA, and FINRA / NASAA securities licensing candidates — built around modern AI as a core capability rather than as a bolt-on. Every lesson ships with AI-narrated audio. Every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback. Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass — question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking — is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.