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

A is correct. The profit recursion tallies all cash outflows at year-end. For each policy that survives to year-end — a fraction $p_{x+t-1}$ of those in force at year-start — the insurer must hold the reserve ${_t}V$. The expected total reserve held for survivors is $p_{x+t-1} \cdot {_t}V$, which is a cash outflow (from the profit perspective) representing the liability being carried forward. A describes discounting the reserve, which is not what this term does. C describes the reserve released by deaths, which is captured by the $q_{x+t-1}$ term on the death benefit. D confuses this term with a prospective premium EPV. E invents a risk-charge credit that does not appear in standard profit testing.

A is correct. With ${_0}V=0$, the profit recursion is:
$$(P-e_1)(1+i)-q_{50}(b-{_1}V)-p_{50}\cdot{_1}V=0$$
Net premium: $P-e_1=3{,}500-1{,}600=1{,}900$.
Accumulated: $1{,}900\times1.055=2{,}004.50$.
Expanding:
$$2{,}004.50=0.006(150{,}000-{_1}V)+0.994{_1}V=900-0.006{_1}V+0.994{_1}V=900+0.988{_1}V$$
$$0.988{_1}V=1{,}104.50$$
$${_1}V=1{,}117.91$$
A rounds without actuarial basis. B treats the fund as the reserve, ignoring death claims. C states $1{,}111.17$ from a subtraction error in the coefficient. E uses the premium as a proxy.

D is correct. Compute NPV at $r=18\%$:
$$NPV=-1{,}500+600/1.18+580/1.3924+560/1.6430=-1{,}500+508.47+416.63+340.84=-234.06<0$$
Since $NPV(18\%)<0$, IRR $<18\%$ — the IRR criterion fails.

DPP at $r=18\%$:
Year 1: $-1{,}500+508.47=-991.53<0$
Year 2: $-991.53+416.63=-574.90<0$
Year 3: $-574.90+340.84=-234.06<0$

The cumulative NPV never reaches zero within 3 years — DPP exceeds 3 years, also failing the target. A uses undiscounted sums, an incorrect basis for DPP. B asserts the IRR criterion passes despite $NPV<0$. D asserts the DPP criterion passes but the cumulative NPV is negative at year 2. E overstates the difficulty.