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

Profit analysis on SOA Exam ALTAM tests profit measures (NPV of profits, profit margin, IRR), embedded value, asset share calculations, and the impact of experience deviations on product profitability.

111 Questions
45 Easy
55 Medium
11 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
In profit testing, the term px+t−1⋅tVp_{x+t-1} \cdot {_t}V appears in the profit recursion. What does this term represent?
Solution
A is correct.

The profit recursion tallies all cash outflows at year-end. For each policy that survives to year-end — a fraction px+t−1p_{x+t-1} of those in force at year-start — the insurer must hold the reserve tV{_t}V. The expected total reserve held for survivors is px+t−1⋅tVp_{x+t-1} \cdot {_t}V, which is a cash outflow (from the profit perspective) representing the liability being carried forward.
Question 2 Medium
Year-4 gains: q(b)=0.018q^{(b)}=0.018, q(a)=0.012q^{(a)}=0.012, NAR=35,000NAR=35{,}000; i(b)=8%i^{(b)}=8\%, i(a)=6%i^{(a)}=6\%, fund =20,000=20{,}000; e(b)=160e^{(b)}=160, e(a)=130e^{(a)}=130. Compute total gain.
Solution
B is correct.

Mortality gain (actual 0.0120.012 lighter than basis 0.0180.018 — favorable):
Gμ=(0.018−0.012)×35,000=0.006×35,000=210G_\mu = (0.018-0.012) \times 35{,}000 = 0.006 \times 35{,}000 = 210

Interest gain (actual 6%6\% below basis 8%8\% — unfavorable):
Gi=(0.06−0.08)×20,000=−0.02×20,000=−400G_i = (0.06-0.08) \times 20{,}000 = -0.02 \times 20{,}000 = -400

Expense gain (actual 130130 below basis 160160 — favorable):
Ge=(160−130)(1.08)=30×1.08=32.40G_e = (160-130)(1.08) = 30 \times 1.08 = 32.40

Gtotal=210+(−400)+32.40=−157.60G_{\text{total}} = 210 + (-400) + 32.40 = -157.60
Question 3 Hard
A 5-year profit signature: Π0∗=−2,000\Pi_0^*=-2{,}000, Πt∗=600−20(t−1)\Pi_t^*=600-20(t-1) for t=1,…,5t=1,\ldots,5. Compute NPV at r=16%r=16\% and estimate the IRR.
(Python:numpy.irr([−2000,600,580,560,540,520])≈0.178)(Python: numpy.irr([-2000,600,580,560,540,520]) \approx 0.178)
Solution
E is correct.

Compute NPV at r=16%r=16\%:
NPV=−2,000+517.24+430.98+358.76+298.25+247.58=−147.19<0NPV=-2{,}000+517.24+430.98+358.76+298.25+247.58=-147.19<0
Since NPV(16%)<0NPV(16\%)<0, IRR<16%<16\%, not 17.8%17.8\%. The Python result of 17.8%17.8\% would require a positive NPV at 16%16\%. Testing at r=5%r=5\%:
NPV=+433.43>0NPV=+433.43>0
Interpolating between 5%5\% and 16%16\%: IRR≈5+11×433.43/(433.43+147.19)=5+8.21=13.2%IRR\approx5+11\times433.43/(433.43+147.19)=5+8.21=13.2\%.
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