Chapter 2 · Probability, Uncertainty, and Financial States
Part I · Uncertainty, Information, and Value
Chapter at a glance
Modern Finance Laboratory · Module 2 · Week 2
Module 2 puts this chapter’s objects under the reader’s hands in two panels. The distribution panel loads a daily equity-index return sample (the dataset behind Figure 2.2, downloadable from the Course Website, Week 2), fits normal and Student-𝑡 models by maximum likelihood, and displays densities, QQ-plots, and any requested tail probability or quantile under each fit side by side, with the empirical count alongside. The dependence panel is a copula sandbox: choose two marginals and a copula (Gaussian, 𝑡, and the asymmetric Clayton), set the correlation and tail parameters, and watch the simulated joint scatter, the joint-tail counts, and th
Learning Outcome Statements
LOS 2.1 Construct financial state spaces for portfolios, credit exposures, and macroeconomic scenarios, and specify events and probability measures on them.
LOS 2.2 Define random variables, distribution functions, and densities, and compute moments and quantiles for the standard distributions of finance.
LOS 2.3 Explain why expectation requires care beyond finite settings— heavy tails, nonexistent moments, events of probability zero—and interpret the Lebesgue view of expectation at an honest, intuitive level.
LOS 2.4 Analyze dependence between risks using covariance, correlation, and copulabased reasoning, and demonstrate the failure modes of correlation with concrete portfolios.
LOS 2.5 Quantify diversification and its limits through the algebra of variances and the phenomenon of tail dependence.
Laboratory · Module 2 (book §2.12)
Module 2: Probability and Distribution Lab
Course Website · Week 2
Guided experiment (supports LOS 2.2–2.5). (i) Reproduce both analysts’ numbers from Examples 2.5 and 2.11, then find the loss threshold at which the two models’ probabilities cross, and explain the crossing in one sentence. (ii) In the distribution panel, re-estimate the 𝑡’s degrees of freedom on rolling five-year windows and observe how unstable the tail parameter is—the estimation-risk theme of Chapter 12 in miniature. (iii) In the dependence panel, fix both marginals and the correlation at 0.6, switch the copula from Gaussian to 𝑡 4 , and record what happens to the portfolio’s 1% loss quantile;
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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 2.1. Design a sample space for each of the following, state its cardinality (or that it is a continuum), and give one financially relevant event that cannot be expressed in it, together with the refinement that repairs the omission: (a) one-year default analysis of Example 2.3’s twelve borrowers when the timing of default within the year matters for interest accrual; (b) a two-asset (equity, bond) annual allocation review; (c) a scenario analysis distinguishing recession/expansion but not inflation.
Exercise 2.2. A risk report states: “The strategy’s returns are uncorrelated with the market; it is therefore market-neutral.” Using the 𝑌 = 𝑋 2 construction of Section 2.5, explain precisely what is wrong with the inference, and name a familiar option position whose relation to the market has this character.
Exercise 2.3. In August 2007 a bank executive described witnessing moves of 25 standard deviations, “several days in a row.” Rewrite this statement so that it is a wellposed claim about a model rather than about the world, and state what the observation implied about that model. What does your answer suggest about the phrase “a one-inten-thousand-year event” in risk reporting?
Part B — Computations
Exercise 2.4. (The opening problem, completed.) Using the parameters of Examples 2.5 and 2.11: (a) verify all four probabilities reported by the first two analysts; (b) compute both models’ 1% quantiles of the annual return and interpret the difference for a risk 56 2 Probability, Uncertainty, and Financial States limit set at “no more than 1% chance of breaching”; (c) find (numerically) the loss threshold at which the two models assign equal probability, and explain its location using the shoulders-versus-tails redistribution of Example 2.11.
Exercise 2.5. Meridian’s private-credit sleeve experiences defaults at an average rate of 1.8 per year, modeled as Poisson. Compute the probability of (a) a default-free year; (b) at least four defaults in a year (answer: 0.109); (c) at least four defaults in a year, given at least one has occurred by mid-year with a half-year rate of 0.9. State the independence assumption used in (c) and one economic reason it may fail.
Exercise 2.6. Three assets have volatilities 18%, 22%, 30% and pairwise correlations 𝜌12 = 0.5, 𝜌13 = 0.2, 𝜌23 = 0.4. Compute the volatility of the portfolio with weights (0.5, 0.3, 0.2), and the marginal change in portfolio variance from shifting one percentage point of weight from asset 3 to asset 1.
Exercise 2.7. The equity book’s value follows 𝑆1 = 𝑆0 𝑒 𝑍 , where 𝑍 is normal with mean 𝜇 − 21 𝜎 2 and variance 𝜎 2 ; take 𝑆0 = 100, 𝜇 = 0.07, and 𝜎 = 0.175 (the lognormal model previewed for Chapter 7). Compute (a) E[𝑆1 ]; (b) P(𝑆1 > 120) (answer: 0.233); 2 (c) the variance of 𝑆1 , deriving the formula Var(𝑆1 ) = 𝑆02 𝑒 2𝜇 𝑒 𝜎 − 1 from the normal 2 2 moment-generating function E[𝑒 𝑡 𝑍 ] = 𝑒 𝑡 𝜇𝑍 +𝑡 𝜎 /2 .
Part C — Proofs and extensions
Exercise 2.8. From the three axioms of Definition 2.2 alone, prove the following: (a) the complement rule, P( 𝐴𝑐 ) = 1 − P( 𝐴); (b) monotonicity: if 𝐴 ⊆ 𝐵 then P( 𝐴) ≤ P(𝐵); (c) P( 𝐴 ∪ 𝐵) = P( 𝐴) + P(𝐵) − P( 𝐴 ∩ 𝐵). Identify where each proof uses additivity.
Exercise 2.9. (Chebyshev, and what model-freeness costs.) Prove Chebyshev’s inequality on a finite probability space: for any 𝜆 > 0, Var(𝑋) P |𝑋 − E𝑋 | ≥ 𝜆 ≤ . 𝜆2 (Hint: apply monotonicity, Proposition 2.7(ii), to 𝜆2 1 { | 𝑋−E𝑋 | ≥𝜆} ≤ (𝑋 − E𝑋) 2 .) Then apply it to the opening problem’s return (𝜇 = 0.07, 𝜎 = 0.175) to bound P(𝑅 < −0.35), compare the bound with both analysts’ answers, and draw the moral about distributionfree bounds versus distributional assumptions.
Exercise 2.10. Complete the equality analysis of Proposition 2.13: show that |𝜌 𝑋𝑌 | = 1 implies the affine relation 𝑌 = 𝑎𝑋 + 𝑏, where the slope is 𝑎 = Cov(𝑋, 𝑌 )/Var(𝑋); identify 𝑏. Then prove that for independent 𝑋, 𝑌 (finite case), E[𝑋𝑌 ] = E[𝑋]E[𝑌 ], hence 𝜌 𝑋𝑌 = 0, and exhibit (with proof) the converse’s failure using 𝑌 = 𝑋 2 .
Exercise 2.11. ∗ (Tail dependence, computed.) For continuous marginals, define the lower tail-dependence coefficient 𝜆 𝐿 := lim𝑢↓0 P 𝑌 ≤ 𝑞 𝑢 (𝑌 ) 𝑋 ≤ 𝑞 𝑢 (𝑋) . (a) Show that 𝜆 𝐿 depends only on the copula of (𝑋, 𝑌 ). (b) For the comonotone copula (𝑌 an increasing function of 𝑋), show 𝜆 𝐿 = 1; for the independence copula, 𝜆 𝐿 = 0. (c) Cite 2.13 Exercises 57 (or, harder, sketch) the fact that the Gaussian copula has 𝜆 𝐿 = 0 for every 𝜌 < 1 while the 𝑡 𝜈 copula has 𝜆 𝐿 > 0, and connect this asymptotic dichotomy to the finite-sample joint-tail counts of Figure 2.3.
Part D — Modeling and application
Exercise 2.12. Draft the two-page scenario matrix that Meridian’s risk team should attach to its answer to the CIO: four macro states (Figure 2.1(b)) with committeedefensible probabilities, conditional return assumptions for the equity book and the private-credit sleeve in each state, and the implied unconditional loss probabilities. State explicitly which entries of your matrix are frequencies, which are model outputs, and which are judgment.
Exercise 2.13. Write the one-page memo recommending whether Meridian’s risk reporting should move from normal to Student-𝑡 distributional assumptions. Your memo must include: the tail-versus-shoulder crossing of Example 2.11 explained for trustees; which reported numbers would change and in which direction; one governance risk of the change (hint: parameter instability, Laboratory experiment (ii)); and a recommendation with a review trigger.
Part E — Laboratory
Exercise 2.14. (Laboratory Module 2; supports LOS 2.2–2.4.) Perform the four-part guided experiment of Section 2.12. Submit: (a) a table of both models’ probabilities at the −15% and −35% thresholds with the crossing threshold from Exercise 2.4(c); (b) the rolling estimates of 𝜈 with one sentence on their stability; (c) the portfolio 1% loss quantile under the Gaussian and 𝑡4 copulas at 𝜌 = 0.6; (d) one paragraph: which single number in this exercise would most change a committee decision, and why.
Exercise 2.15. (Course Website, Week 2, Notebook 2.) Extend the notebook to simulate 50,000 draws from the Gaussian and 𝑡 4 copulas at 𝜌 ∈ {0.3, 0.6, 0.9} with standard normal marginals, tabulate the joint 1%-tail counts for all six cases, and verify numerically the claim of Laboratory experiment (iv) that raising the Gaussian correlation is a poor substitute for tail dependence. Report Monte Carlo standard errors for your counts.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 2; they are not posted here. Problem-set files are on the Course Website, Week 2.