Module XV·Article II·~3 min read
Expected Shortfall
Portfolio Risk Management
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Expected Shortfall: measuring risk in the tail
Expected Shortfall (ES), also known as Conditional VaR (CVaR), answers the question: “If something bad happens, how bad is it on average?”
ES eliminates the main drawback of VaR — ignoring the severity of losses in the tail of the distribution.
VaR vs Expected Shortfall
| Metric | Question | Answer |
|---|---|---|
| VaR | Where does the tail begin? | Threshold (boundary) |
| Expected Shortfall | If we end up in the tail — how much do we lose on average? | Average loss in the tail |
Visual interpretation
For a 95% confidence level:
VaR (95%) = the point to the left of which lies 5% of the distribution
ES (95%) = the mean value of those worst 5%
Formula for Expected Shortfall
$ ES = E[\text{Loss} \mid \text{Loss} > VaR] $
For the normal distribution:
$
ES = VaR \times \frac{\varphi(z)}{1-\alpha}
$
where $\varphi$ is the density of the normal distribution, $\alpha$ is the confidence level.
Example: two portfolios
| Metric | Portfolio A | Portfolio B |
|---|---|---|
| VaR (95%) | $1,000,000 | $1,000,000 |
| ES (95%) | $1,200,000 | $3,500,000 |
| Tail Risk | Thin tail | Fat tail |
| Asset examples | Diversified equity | Short options, leveraged |
VaR is the same, but the real risk of Portfolio B is much higher!
Advantages of ES over VaR
| Property | VaR | ES |
|---|---|---|
| Tail information | No | Yes |
| Subadditivity | No (can be violated) | Yes (always) |
| Coherent risk measure | No | Yes |
| Sensitivity to outliers | Low | High |
| Regulatory preference | Basel II | Basel III/IV |
Subadditivity: why is it important
For rational risk management:
Portfolio Risk ≤ Sum of Individual Risks
VaR can violate this property:
| Scenario | Position A VaR | Position B VaR | Combined VaR |
|---|---|---|---|
| Possible with VaR | $1M | $1M | $2.5M (!) |
| With ES always | $1.2M | $1.2M | ≤ $2.4M |
Basel III/IV requirements
The regulator switched from VaR to ES:
| Parameter | Basel II (VaR) | Basel III/IV (ES) |
|---|---|---|
| Metric | VaR 99% | ES 97.5% |
| Horizon | 10 days | Variable (10-250 days) |
| Stress period | None | Stressed ES |
ES for different assets
| Asset | VaR (95%, 1-day) | ES (95%, 1-day) | ES/VaR ratio |
|---|---|---|---|
| S&P 500 | 1.5% | 2.2% | 1.47 |
| EM Equities | 2.5% | 4.0% | 1.60 |
| High Yield | 1.2% | 2.5% | 2.08 |
| Bitcoin | 8% | 15% | 1.88 |
| Short Put Options | 5% | 25% | 5.00 |
Conclusion: High ES/VaR ratio = fat tail, tail risk.
Practical applications
- Capital allocation — ES basis for risk capital
- Limit setting — ES limits for trading desks
- Performance attribution — risk-adjusted returns
- Portfolio optimization — minimize ES instead of VaR
CIO recommendations
- Use ES alongside VaR — full risk picture
- Monitor the ES/VaR ratio — tail risk indicator
- Stress test ES — ES under stressed volatility assumptions
- Report to board — ES is more understandable to non-technical stakeholders
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