CVaR / Expected Shortfall
The average loss in the worst-case scenarios beyond VaR.
Conditional Value at Risk (CVaR), also called Expected Shortfall, measures the average loss when losses exceed the VaR threshold. It provides information about the tail risk that VaR misses. For risk management, CVaR is considered more informative than VaR alone.
Formula
Where
=Confidence level (e.g. 95%)
Variables
| \text{CVaR}_{\alpha} | Conditional VaR / Expected Shortfall at α |
| n_{tail} | Number of observations in tail (worst 5%) |
| L_i | Loss in scenario i |
Assumptions
- Simple average of losses beyond VaR threshold
- More conservative than VaR (always ≥ VaR)
- Coherent risk measure (subadditive)
- Sensitive to tail distribution shape
vs. Industry Tools
Basel III — Required for market risk capital; called Expected Shortfall
Solvency II — Insurance uses similar tail-based risk measures
Related Terms
Value at Risk (VaR)
The maximum expected loss at a given confidence level — but doesn't tell you how bad the tail is.
Monte Carlo Simulation
Generating thousands of possible future scenarios through random sampling.
Stress Test (Rate Shock)
Estimates impact of large yield moves using duration and convexity.