The Quantitative Risk Management Exercise Book
Bibliographic record
Abstract
Exercise 2.6 (VaR and expected shortfall) a) Give mathematically precise definitions of value-at-risk VaR (L) and expected shortfall ES (L) for a random loss L at confidence level (0, 1).b) Explain the relative advantages of each risk measure over the other.Exercise 2.7 (Superadditivity scenarios for VaR) Describe some models for financial losses that can lead to situations where VaR is superadditive.Exercise 2.8 (Additivity for two linearly dependent random variables) Consider an arbitrary random variable X and let Y = aX + b for constants a > 0 and b BasicExercise 2.9 (Risk-neutral valuation for interest-rate derivatives) Consider a two-period model.Denote by r t , t {0, 1}, the simple interest rate from t to t + 1, so that 1 monetary unit invested at t is worth 1 + r t at t + 1. Assume that r 0 is 1.5% and that r 1 takes the values 1% and 2% with probability 1/2.Denote by p(t, T ) the price at t of a zero-coupon bond with maturity T and face value 1.a) Write down p(0, 1) and p(1, 2) for the cases r 1 = 0.01 and r 1 = 0.02.b) Suppose a long zero-coupon bond with maturity T = 2 and face value 1 is traded for 0.969729 at t = 0.In this setup an equivalent martingale measure Q is characterized by the probability q = Q(r 1 = 0.01).Compute q from p(0, 2).c) Apply risk-neutral valuation to price a stylized floor contract which pays an amount of 1 if r 1 < r 0 .Note.In general, a floor contract is an option which provides protection against low interest rates.Exercise 2.10 (Mapping of a stock portfolio affected by exchange rates) Consider a portfolio P consisting of two stocks S t,1 , S t,2 , where S t,1 denotes the value of stock 1 in EUR and S t,2 denotes the value of stock 2 in CHF.Let e CHF t denote the CHF/EUR exchange rate at time t.In other words, 1 CHF is worth e CHF t EUR at t. Furthermore, denote by 1 and 2 the number of shares in stocks 1 and 2 in P, respectively.a) Derive the value V t in EUR of P at time t in terms of the risk factors Z t,j = log S t,j , j {1, 2}, and. What is the corresponding mapping?b) Derive the value V t+1 of P at time t + 1 and the one-period loss L t+1 .c) Derive the linearized one-period loss L t+1 and express it in terms of portfolio weights w 1 , w 2 (the values of each stock investment relative to the value V t of the overall portfolio).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.204 | 0.079 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".