Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Survival Models for Contingent Cash Flows Practice Questions

C is correct. The fundamental relationship between the force of mortality and the survival function follows from the definition of the force of mortality as:
$$\mu_{x+t} = -\frac{d}{dt}\ln {}_tp_x$$
Integrating from 0 to $t$ and using ${}_0p_x = 1$:
$${}_tp_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)$$
This is the general formula valid for any non-negative integrable force of mortality. B incorrectly subtracts the integral linearly, which would allow negative survival probabilities for large hazards. C is a discrete product formula with no limit passage and does not correspond to the continuous-time model. D is the form of a logistic hazard or odds-ratio model, not the standard actuarial survival model. E is the correct formula only under the special case of a constant force $\mu_{x+s} = \mu_x$ for all $s$.

C is correct. The key practical critique of the Markov assumption in long-term care (and disability) insurance is that it ignores duration-dependence. In reality, a person who has been disabled for 5 years has a very different recovery probability than someone newly disabled — the Markov property says only the current state (Disabled) matters, not the time spent there. This duration-dependence is well-documented empirically and is the primary motivation for semi-Markov or duration-dependent extensions.
B is incorrect: the Markov property does NOT require constant intensities — intensities can be arbitrary functions of current age $x + t$; the requirement is only that they not depend on past states or sojourn times.
A is incorrect: the Markov framework is entirely flexible in the number of states; adding states (e.g., partial disability, hospitalization) is a standard model extension that preserves the Markov structure.
D is incorrect: Kolmogorov's forward equations are specifically derived for continuous-time Markov chains — they depend on the Markov property and do not apply to non-Markov processes without modification.
E is incorrect: MLE is fully compatible with Markov models; the likelihood factorizes nicely by state because the Markov property implies that transitions from a given state are independent of the history prior to entering that state.

D is correct. The exact 10-year survival probability under Makeham's law is:
$${}_{10}p_{30} = \exp\!\left(-10A - \frac{Bc^{30}(c^{10}-1)}{\ln c}\right)$$
Computing each component:
- $10A = 10 \times 0.0005 = 0.005$
- $c^{30} = 1.1^{30}$. Note $1.1^{10} \approx 2.5937$, $1.1^{20} \approx 6.7275$, $1.1^{30} \approx 17.4494$
- $c^{10} - 1 = 2.5937 - 1 = 1.5937$
- $\ln c = \ln 1.1 \approx 0.09531$
- Gompertz term: $\frac{0.00005 \times 17.4494 \times 1.5937}{0.09531} = \frac{0.001390}{0.09531} \approx 0.01459$
- Total exponent: $-(0.005 + 0.01459) = -0.01959$
- ${}_{10}p_{30} = e^{-0.01959} \approx 0.9806$, closest to 0.9827 among options.
A uses only the constant term. C treats $\mu_{30}$ as constant over 10 years, overestimating cumulative hazard. D omits the constant term. E uses a midpoint approximation with incorrect exponentiation.