Questions: Expected Shortfall and Tail Risk Measurement

Question 1

Portfolio A has a 99% VaR of $10M with a maximum possible loss of $12M. Portfolio B also has a 99% VaR of $10M, but its maximum possible loss is $200M. How do their Expected Shortfall values compare?

• A. Both have the same ES since they have identical VaR at the same confidence level
• B. Portfolio A has higher ES because its tail is more concentrated near the VaR threshold
• C. Portfolio B has higher ES because its tail extends far beyond the VaR threshold
• D. ES cannot be compared without knowing the exact shape of each distribution

Answer: (C)

VaR only marks where the tail begins — it says nothing about what is inside the tail. ES averages all losses beyond the VaR threshold. Portfolio A's tail spans $2M above VaR; Portfolio B's tail spans $190M above VaR. ES integrates over the entire tail, so Portfolio B's ES is dramatically higher despite identical VaR. This is precisely the failure VaR has that ES corrects: two radically different risk profiles appear identical under VaR.

Question 2

Two portfolios each have a 99% VaR of $3M. A risk manager combines them into a single portfolio. What does VaR's violation of subadditivity imply about the combined VaR?

• A. The combined VaR must be exactly $6M by the additivity of risk measures
• B. The combined VaR must be at most $6M because diversification always reduces risk
• C. The combined VaR could theoretically exceed $6M, violating the intuition that diversification helps
• D. The combined VaR must be less than $6M because correlations are never perfectly positive

Answer: (C)

VaR violates subadditivity — the property that the risk of a combined portfolio should be no greater than the sum of its parts. It is mathematically possible to construct two portfolios where their individual VaRs are low but their combined VaR is high. This is not just theoretical: it could incentivize splitting portfolios to game capital requirements. Expected Shortfall is subadditive by construction, making it a more defensible basis for capital allocation.

Question 3

Expected Shortfall is preferred over VaR for capital allocation in part because ES captures how severe losses are in extreme scenarios, not just how likely they are to exceed a threshold.

• T. True
• F. False

Answer: True

This is the core distinction. VaR answers 'how likely is a loss bigger than X?' — ES answers 'when losses are extreme, how extreme are they on average?' For capital adequacy, you need to hold enough capital to absorb the actual severity of bad outcomes. ES computes E[Loss | Loss > VaR], integrating over the entire tail. VaR only marks the threshold. Regulators (Basel III/IV) shifted to ES precisely because tail severity, not just tail probability, determines how much capital is needed.

Question 4

Two portfolios with identical VaR at the same confidence level should have identical risk profiles.

• T. True
• F. False

Answer: False

VaR gives no information about the shape or severity of losses beyond the threshold. Two portfolios can have identical VaR but radically different tails — one with losses clustering just above the threshold, another with a small probability of catastrophic losses far into the tail. ES distinguishes these; VaR does not. This is why identical VaR is not sufficient evidence of equivalent risk, and why ES is increasingly required for regulatory capital calculations.

Question 5

Explain why VaR fails to distinguish between a 'mild tail' and a 'catastrophic tail,' and how Expected Shortfall corrects this.

Model answer: VaR marks the loss level exceeded a given percentage of the time — it tells you where the tail begins, not what is inside it. A portfolio with maximum losses of $6M and one with maximum losses of $600M can have identical VaR if both exceed the threshold with equal frequency. Expected Shortfall corrects this by computing E[Loss | Loss > VaR]: the average loss across all scenarios in the tail. A heavier, more severe tail produces a higher ES even when VaR is identical, because ES integrates over tail severity rather than just marking the threshold.

Practically, this matters most for capital adequacy and stress testing. The question is not just 'how often will we lose more than $X?' but 'when we do, how much do we lose on average?' ES forces institutions to model and hold capital against tail severity. This is why the Basel framework shifted from VaR to ES — a bank that holds capital based on VaR is prepared for the frequency of large losses, but not necessarily their magnitude when they occur.