Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Premium and Policy Valuation for Long-Term Coverages Practice Questions

A is correct. The equivalence principle (also called the net premium principle) requires that at the time of policy issue the EPV of the premium stream exactly equals the EPV of the benefit obligations: $\text{EPV(premiums)} = \text{EPV(benefits)}$. This makes the expected profit at issue equal to zero — no explicit loading for profit or expenses is built into the net premium.
A is correct: adding a safety loading above the EPV of benefits describes a gross premium or office premium, not the net premium under the equivalence principle.
C is incorrect: dividing the benefit by the expected future lifetime ignores the time value of money and the mortality distribution; it would only be correct in the trivial case of a level annuity with no discounting.
D is incorrect: earning the risk-free rate on reserves is a consequence of the reserve mechanics, not the definition of the net premium.
E is incorrect: the EPV of future benefits at issue equals the net single premium, not the initial reserve (which equals the EPV of benefits minus the EPV of future premiums, and starts at zero for a standard policy).

D is correct. Because the forces $\mu = 0.02$ and $\delta = 0.05$ are constant (age-independent), the prospective quantities $\bar{A}_{x+t}$ and $\bar{a}_{x+t}$ are the same at every age: $\bar{A}_{x+t} = 0.02/0.07$ and $\bar{a}_{x+t} = 1/0.07$. Therefore: $${}_{10}V = 100{,}000 \cdot \bar{A}_{50} - P \cdot \bar{a}_{50} = 100{,}000 \times \frac{0.02}{0.07} - 2{,}000 \times \frac{1}{0.07}$$ $$= \frac{2{,}000}{0.07} - \frac{2{,}000}{0.07} = 0$$ The reserve is identically zero at every policy duration under constant forces. B (14,286) is $100{,}000 \times 0.02/0.07 - 2{,}000/0.07 \times 0.5$ — an error in the annuity calculation. C (28,571) equals $100{,}000 \times 0.02/0.07$ alone, ignoring the future premium offset. D (5,000) results from incorrectly computing $P \times 10 \times e^{\delta \cdot 10}$ as an accumulated premium without adjusting for mortality. E (20,000) results from $P \times 10$, simply accumulating premiums without interest or mortality, which describes neither the prospective nor the retrospective reserve correctly.

C is correct. First compute the net single premium (NSP):
$$\text{NSP} = 1{,}000[vq_x + v^2 p_x q_{x+1} + v^3 {}_{2}p_x q_{x+2}]$$
$$= 1{,}000[0.01/1.05 + 0.99 \times 0.012/1.05^2 + 0.99 \times 0.988 \times 0.015/1.05^3]$$
$$= 1{,}000[0.009524 + 0.010773 + 0.013215] = 1{,}000 \times 0.033512 = 33.51$$
The premium annuity $\ddot{a} = 1 + v p_x + v^2 {}_{2}p_x = 1 + 0.99/1.05 + 0.99 \times 0.988/1.05^2 = 1 + 0.942857 + 0.886984 = 2.8299$. Net premium $P = 33.51/2.8299 = 11.843$. Prospective reserve at duration 1:
$${}_{1}V = 1{,}000[v q_{x+1} + v^2 p_{x+1} q_{x+2}] - P(1 + v p_{x+1})$$
$$= 1{,}000[0.012/1.05 + 0.988 \times 0.015/1.05^2] - 11.843(1 + 0.988/1.05)$$
$$= 1{,}000[0.011429 + 0.013469] - 11.843 \times 1.94095 = 24.898 - 22.993 = 1.905$$
Actual values are small for short-term policies. A correctly describes the prospective method; the exact numbers depend on full computation. B uses straight-line amortization. C claims reserves are always zero. D and E swap or misstate the reserve pattern.