You're facing an FRM Part I question: A portfolio holds two short, out-of-the-money, uncorrelated digital options. Without calculating, which property of a "coherent" risk measure will Value at Risk (VaR) almost certainly violate for this portfolio? Many candidates know VaR isn't always coherent. But knowing it fails the subadditivity axiom in this specific case, because the combined risk of two low-probability, high-impact events can be much greater than their individual VaRs, is the judgment that separates a pass from a fail.
Value at Risk (VaR) is the maximum potential loss at a given confidence level, while Expected Shortfall (ES) is the average loss if that VaR threshold is breached. ES is considered a superior, coherent risk measure because it better captures the severity of extreme tail-risk events.
Key facts
- FRM Pass Rate
- Approximately 45-50% for Part I candidates (GARP).
- Relevant Domain
- Valuation and Risk Models (30% of Part I exam).
- Key Concepts
- Value at Risk (VaR), Expected Shortfall (ES), Coherent Risk Measures (Monotonicity, Subadditivity, Positive Homogeneity, Translation Invariance).
- Calculation Methods
- Parametric (e.g., Normal Distribution), Historical Simulation, Monte Carlo Simulation.
- Recommended Study Hours
- GARP suggests 200-250 hours of dedicated study for Part I.
- Official Body
- Global Association of Risk Professionals (GARP).
What is Value at Risk (VaR)?
Value at Risk (VaR) is a single number that answers the question: "What is the most I can expect to lose over a set time horizon, at a given level of confidence?" If a portfolio has a 1-day 99% VaR of $10 million, it means that on 99 days out of 100, you would not expect to lose more than $10 million. There is a 1% chance that the loss will be $10 million or more.
Think of it as a line in the sand. It tells you the threshold of a bad outcome, but it gives you no information about what happens if you cross that line. This is its single biggest weakness. A 1% chance of losing "$10 million or more" could mean losing $10.1 million or it could mean losing $100 million. VaR is blind to the severity of losses in the tail of the distribution.
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This limitation is why regulators and risk managers have increasingly looked for a better measure.
What is Expected Shortfall (ES)?
Expected Shortfall (ES) answers the question VaR leaves open: "If I do have a bad day (i.e., breach my VaR), what is my average expected loss?" ES is also known as Conditional VaR (CVaR) or Expected Tail Loss (ETL).
Using the previous example, if the 1-day 99% VaR is $10 million, the 1-day 99% ES might be $15 million. This means that on the 1% of days when your losses exceed $10 million, the average loss you can expect is $15 million.
ES is inherently more conservative than VaR because it considers not just the threshold of loss, but the magnitude of the losses beyond it. This makes it a more robust measure for capital adequacy and for understanding the true nature of extreme risks.
VaR vs. ES: A Side-by-Side Comparison
Choosing between VaR and ES is a core judgment tested on the FRM exam. This table breaks down their key differences.
| Feature | Value at Risk (VaR) | Expected Shortfall (ES) |
|---|---|---|
| Question Answered | "What's my maximum loss at a confidence level?" | "If I exceed my max loss, what's the average loss?" |
| Focus | Loss threshold (percentile). | Average loss in the tail. |
| Coherence | Not always coherent (can fail subadditivity). | Generally coherent. |
| Diversification | May penalize diversification for certain assets. | Properly rewards diversification. |
| Tail Risk Insight | Blind to the size of losses beyond the threshold. | Quantifies the expected size of extreme losses. |
| Regulatory Preference | Historically popular but being replaced. | Increasingly the standard (e.g., Basel III). |
What Makes a Risk Measure "Coherent"?
A risk measure is considered "coherent" if it behaves in a sensible and predictable way. The concept, developed by Artzner, Delbaen, Eber, and Heath, requires a measure to satisfy four specific axioms. ES satisfies them; VaR does not always.
- Monotonicity: If Portfolio A will always perform worse than Portfolio B, its risk must be greater. (If A always loses more, it's riskier).
- Translation Invariance: Adding a cash amount
kto a portfolio should reduce its risk by exactlyk. (Holding more cash makes you safer). - Positive Homogeneity: If you double the size of a portfolio, you double its risk. (Scaling up a position scales up the risk proportionally).
- Subadditivity: The risk of two portfolios combined cannot be greater than the sum of their individual risks.
Risk(A+B) ≤ Risk(A) + Risk(B).
This last axiom, subadditivity, is the most important one and the one VaR often fails. It is the mathematical representation of diversification. A failure of subadditivity means a risk measure might suggest that merging two businesses could create more risk than running them separately, which defies financial logic.
How Are VaR and ES Calculated?
The FRM exam requires you to be proficient in two main calculation methods, with a conceptual understanding of a third. Try VoraPrep's adaptive FRM questions to drill these calculation-heavy topics.
The Parametric Method (Normal Distribution)
This method assumes portfolio returns follow a specific statistical distribution, usually the normal distribution. It's fast but only as accurate as its assumption.
- Required Inputs: Portfolio value (P), mean return (μ), standard deviation (σ), and a Z-score corresponding to the confidence level.
- VaR Formula:
VaR = - (μ + Zα × σ) × P - ES Formula:
ES = - (μ + σ × [φ(Zα) / (1-α)]) × P, whereφ(Zα)is the value of the standard normal probability density function (PDF) at the Z-score, andαis the confidence level (e.g., 0.99).
The Historical Simulation Method
This method makes no distributional assumptions. It simply assumes the recent past is a good guide to the near future.
- Process:
- Collect a history of actual portfolio returns (e.g., the last 251 days).
- Sort the returns from worst to best.
- The VaR is the return at the percentile corresponding to your confidence level. For a 99% VaR with 500 observations, you'd find the (1 - 0.99) × 500 = 5th worst return.
- The ES is the average of all returns that are worse than or equal to the VaR.
The Monte Carlo Simulation Method
This method is a hybrid approach. You specify distributions for risk factors (like the parametric method) but then use those to simulate thousands of possible future outcomes to generate a return distribution (like the historical method). It's powerful and flexible but computationally intensive and less likely to require a full calculation on the exam.
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Parametric vs. Historical Simulation: Key Differences
Your choice of method can drastically change your risk estimate. Understanding the trade-offs is crucial.
| Aspect | Parametric Method | Historical Simulation Method |
|---|---|---|
| Assumption | Assumes returns follow a known distribution (e.g., normal). | Assumes the past is representative of the future. |
| Data Needs | Mean, standard deviation, correlation. | A long series of historical returns. |
| Pros | Fast, simple, and only needs a few parameters. | Captures fat tails and non-normality present in the data. |
| Cons | Fails if the distribution assumption is wrong (understates tail risk). | Can't capture events not present in the historical data. |
| Best For | Portfolios with well-behaved, linear returns. | Portfolios with options, non-linear payoffs, or complex assets. |
Worked Example: From Calculation to Analysis
Let's calculate the 1-day 99% VaR and ES for a $100 million portfolio using both methods.
Given Data:- Portfolio Value (P): $100,000,000
- Confidence Level (α): 99%
- Parametric Inputs:
- Mean return (μ): 0%
- Standard deviation (σ): 1.5%
- Z-score for 99% confidence (Zα): -2.33
- PDF at Z=-2.33, φ(Zα): 0.0264
- Historical Inputs (last 200 daily returns, sorted):
- 1st worst return: -5.1%
- 2nd worst return: -4.8%
- 3rd worst return: -4.2%
- ...and so on.
-
VaR = - (μ + Zα × σ) × P -
VaR = - (0.00 + (-2.33) × 0.015) × $100,000,000 -
VaR = - (-0.03495) × $100,000,000 -
VaR = $3,495,000
-
ES = - (μ + σ × [φ(Zα) / (1-α)]) × P -
ES = - (0.00 + 0.015 × [0.0264 / (1-0.99)]) × $100,000,000 -
ES = - (0.015 × [0.0264 / 0.01]) × $100,000,000 -
ES = - (0.015 × 2.64) × $100,000,000 -
ES = $3,960,000
- We need the observation at the 1% tail of 200 observations.
- Position = (1 - 0.99) × 200 = 2nd worst observation.
- The 2nd worst return from our data is -4.8%.
-
Historical VaR = - (-0.048) × $100,000,000 -
Historical VaR = $4,800,000
- ES is the average of all returns in the 1% tail. The 1% tail consists of the 1st and 2nd worst observations.
-
Average Tail Return = (-5.1% + -4.8%) / 2 = -4.95% -
Historical ES = - (-0.0495) × $100,000,000 -
Historical ES = $4,950,000
Common Exam Traps and Self-Check
Avoid losing easy points by watching for these common mistakes.
- Confidence vs. Tail: For a 99% VaR, you are interested in the 1% tail. Don't mistakenly look at the 99th percentile from the bottom, which would be a gain.
- Time Scaling: You can only scale a 1-day VaR to a T-day VaR by multiplying by √T if returns are independent and identically distributed (i.i.d.) with a zero mean. The exam might give you a scenario where this assumption is violated.
- VaR vs. ES Definition: Remember, VaR is a threshold, ES is an average. A question might ask you to interpret the numbers, not just calculate them.
- Subadditivity Conditions: Don't just say "VaR is not coherent." Be able to state that it fails subadditivity and give an example of when (e.g., a portfolio of short options).
- Monotonicity
- Positive Homogeneity
- Subadditivity
- Translation Invariance
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