Chapter 13 · Risk Measures, Ambiguity, and Robustness
Part IV · Risk, Robustness, and Equilibrium
Chapter at a glance
Modern Finance Laboratory · Module 13 · Week 13
Module 13 has four panels. The axioms panel lets the user define a risk functional from a menu (quantiles, tail means, spectral weights, worst-of-scenarios) and fires the four coherence tests against it live, exhibiting counterexamples on failure—the VaR subadditivity example preloaded. The measure panel computes VaR and ES across confidence levels for Gaussian, Student-𝑡, and empirical inputs, with the Rockafellar– Uryasev objective (13.3) plotted so the reader watches its minimum land at the quantile. The robustness panel sweeps ES over user-defined (𝜇, 𝜎) boxes and entropy balls, reproducing Figure 13.3(a) with the October loss overlaid. T
Learning Outcome Statements
LOS 13.1 State the coherence axioms for risk measures and interpret each as a governance principle.
LOS 13.2 Define Value-at-Risk, compute it in Gaussian and Student-𝑡 models, and prove its failure of subadditivity by counterexample.
LOS 13.3 Define expected shortfall, prove its coherence via the Rockafellar–Uryasev representation, and compute it in closed form.
LOS 13.4 State and prove (in finite settings) the representation of coherent risk measures as worst-case expectations over scenario sets.
LOS 13.5 Backtest risk models by exception counting and state the limits of tail backtests.
LOS 13.6 Incorporate parameter and model ambiguity into risk limits: worst-case risk over uncertainty sets, with de-smoothed inputs.
Laboratory · Module 13 (book §13.11)
Module 13: The Risk Office
Course Website · Week 13
Guided experiment (supports LOS 13.1–13.6). (i) Feed the axioms panel the twobond example and reproduce the (−4, 98) violation; confirm ES passes. (ii) On the measure panel, verify the Meridian panel ($426M, $546M, $625M, $737M) and locate the RU minimum. (iii) Sweep the robustness box and reproduce $708M; find the smallest box for which October’s loss falls inside. (iv) On the audit panel, understate true volatility by 15% and report how many years the exception test needs to reject at 95% power—then write one sentence on what that implies for governance.
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Exercises
Exercises are grouped A–E throughout the book: A concept checks, B computations, C proofs and extensions, D modeling and application, E laboratory. Starred exercises (∗) are on the advanced track.
Part A — Concept checks
Exercise 13.1. For each axiom (A1)–(A4), give one sentence of governance meaning and one realistic situation in which a committee might deliberately want it violated (e.g. liquidity for (A4)), naming the section of this book that treats the violation.
Exercise 13.2. A desk head argues: “Our VaR has never been exceeded more than expected; the measure is fine.” Give the two-part reply: which defect of VaR a clean exception count cannot detect (blindness beyond the quantile), and the incentive problem it creates for exactly this desk head.
Exercise 13.3. Explain in three sentences why (13.4) implies that “statistical risk measure” and “stress-testing regime” are not alternatives but the same object, and what the ES scenario set’s density bound 1/(1 − 𝛼) means in stress-book language. 296 13 Risk Measures, Ambiguity, and Robustness
Part B — Computations
Exercise 13.4. Reproduce the Meridian panel: monthly mean and volatility from the Chapter 9 inputs; VaR95 = $426M; ES95 = $546M; ES97.5 = $625M; and the 𝑡 5 figure $737M using the Student-𝑡 ES formula. State October’s loss as a z-score.
Exercise 13.5. Work the two-bond example end to end: the state table, both VaRs, the portfolio VaR, all three ES values, and the verification of subadditivity for ES. Then find the default probability at which the VaR violation disappears and explain why.
Exercise 13.6. Verify (13.3) numerically for the standard Gaussian: plot or tabulate 𝑔(𝑐) on [1, 3], confirm the minimizer 1.96 and minimum 2.338 at 𝛼 = 97.5%, and repeat at 𝛼 = 95%.
Exercise 13.7. Compute the robustness sweep: ES97.5 at the four box corners of Section 13.6; the corner ranking; and the additional capital (in $M and in percent) that honest inputs demand over the flattered point.
Part C — Proofs and extensions
Exercise 13.8. Prove that VaR satisfies (A1), (A2), and (A4)—so its failure is only subadditivity—and prove that for jointly Gaussian positions VaR is subadditive (compute VaR of a sum via the variance triangle inequality), explaining why the defect lives in skewed and default-like exposures.
Exercise 13.9. Complete Theorem 13.1’s proof in the general (atomic) case: define ES by the integral form in (13.2), handle the atom at the quantile with the standard tail-splitting correction, and verify (13.3) on a three-point distribution where the naive conditional-expectation form fails.
Exercise 13.10. Verify the ES scenario set: on a finite space, show that ES 𝛼 (𝑋) = max{EQ [−𝑋] : 𝑑Q/𝑑P ≤ 1−1 𝛼 } by exhibiting the maximizing Q (reweight the worst (1−𝛼) mass to the bound) and checking both inequalities—then confirm Theorem 13.2’s formula reproduces (13.2). ′
Exercise 13.11. ∗ Prove the composition claim behind (13.5): if each 𝜌 P is coherent ′ with scenario set QP′ , then maxP′ ∈ M 𝜌 P is coherent with scenario set the closed convex Ð hull of QP′ ; and show that the entropic measure with penalty 𝛽(Q) = 1𝜃 KL(Q∥P) arises as the convex-dual of the exponential certainty equivalent of Chapter 9.
Part D — Modeling and application
Exercise 13.12. The trustee packet contains a competing proposal: cap the portfolio’s annualized volatility at 11% instead of any tail measure. Write the one-page assessment: 13.12 Exercises 297 which axioms a volatility cap satisfies (careful: monotonicity fails); what it cannot see (the two-density figure); when it and ES agree (elliptical worlds); and your recommendation for its role, if any, alongside the ES limit.
Exercise 13.13. Draft the replacement limit for the minutes, one page: the statement (measure, level, horizon, number); the computation standard (de-smoothed inputs per Chapter 12, the ambiguity box with its documented corners, the heavy-tail cross-check); the audit regime (exception counting with the acceptance region, clustering diagnostic, exceedance sizing) with escalation triggers at both the loss and the model level; and the three-sentence defense the risk officer reads aloud, each sentence resting on a theorem of this chapter.
Part E — Laboratory
Exercise 13.14. (Laboratory Module 13; supports LOS 13.1–13.6.) Perform the fourpart guided experiment of Section 13.11. Submit: (a) the axioms-panel verdicts on the two-bond book; (b) the Meridian panel with the RU minimum; (c) the box sweep with the smallest October-covering box; (d) the detection-power result and its governance sentence.
Exercise 13.15. (Course Website, Week 13, Notebook 13.) Extend the notebook: implement minimum-ES portfolio optimization via the linear-programming form of (13.3) on 1,000 simulated scenarios of the Chapter 9 three-asset market; compare the minimum-ES and minimum-variance frontiers; and report where and why they diverge (skew the equity scenarios with the Chapter 6 jump model to force a divergence).
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 13; they are not posted here. Problem-set files are on the Course Website, Week 13.