Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Severity, Frequency, and Aggregate Models Practice Questions

Subadditivity is one of the four axioms of coherent risk measures (monotonicity, positive homogeneity, translation invariance, subadditivity).

Choice A is incorrect because VaR is not subadditive in general; there are well-known counterexamples where diversification increases VaR, violating the coherence requirement.
Choice B is incorrect because translation invariance gives $\rho(X+c) = \rho(X)+c$, not $\rho(c) = 0$; the axiom describes how adding a deterministic amount shifts the risk measure.
Choice D is incorrect because coherent risk measures can handle heavy-tailed distributions; in fact, TVaR is specifically valued for its sensitivity to tail behavior.
Choice E is incorrect because TVaR (also called CTE or Expected Shortfall) is a coherent risk measure that satisfies all four axioms, including subadditivity.

$\text{TVaR}_0 = E[X]$, while $\text{VaR}_0$ is the minimum possible value (often 0 for loss distributions). Since $E[X]$ generally does not equal the minimum, (E) is FALSE.

(A) True: TVaR satisfies subadditivity. (B) True: TVaR $\ge$ VaR always. (C) True for continuous distributions. (D) True: TVaR satisfies all four coherence axioms.

Find $\text{VaR}_{0.95}$. Set $S(x) = 0.05$:

$(2{,}000/(2{,}000+x))^3 = 0.05$, so $2{,}000/(2{,}000+x) = 0.05^{1/3} = 0.3684$.

$x = 2{,}000/0.3684 - 2{,}000 = 5{,}429 - 2{,}000 = 3{,}429$.

$\text{VaR}_{0.95} = 3{,}429$.

For the Pareto distribution, the mean excess loss at $d$ is $e(d) = (\theta + d)/(\alpha - 1)$:

$e(3{,}429) = (2{,}000 + 3{,}429)/(3-1) = 5{,}429/2 = 2{,}715$

$\text{TVaR}_{0.95} = \text{VaR}_{0.95} + e(\text{VaR}_{0.95}) = 3{,}429 + 2{,}715 = 6{,}144$

Equivalently: $\text{TVaR}_p = \frac{\alpha}{\alpha-1}\text{VaR}_p + \frac{\theta}{\alpha-1} = \frac{3}{2}(3{,}429) + \frac{2{,}000}{2} = 5{,}144 + 1{,}000 = 6{,}144$.

The answer is $\text{VaR}_{0.95} = 3{,}429$ and $\text{TVaR}_{0.95} = 6{,}144$.

Why other choices are wrong:
- Choice E (2,715) is the mean excess loss alone, not VaR.
- Choice C (4,429) and Choice D (6,429) use incorrect formulas for the Pareto quantile.