Conditional VaR (CVaR)

Conditional Value at Risk (CVaR), also known as Expected Shortfall (ES), measures the expected loss given that the loss exceeds the VaR threshold. It answers: "When things go really bad, how bad on average?"

CVaR vs. VaR:

Feature	VaR	CVaR
Question	"What's the maximum loss at X% confidence?"	"What's the average loss beyond VaR?"
Tail Risk	Ignores severity beyond threshold	Captures full tail severity
Example (95%)	"5% chance of losing more than $X"	"In that worst 5%, average loss is $Y"
Coherent Risk Measure	No	Yes

Example:

• Portfolio VaR (95%): -$100,000
• Meaning: 5% chance of losing more than $100K
• CVaR (95%): -$150,000
• Meaning: In the worst 5% of outcomes, average loss is $150K

Why CVaR Is Preferred:

• VaR tells you the threshold but not how bad losses get beyond it
• Two portfolios can have the same VaR but very different CVaR
• CVaR satisfies mathematical properties (subadditivity) that VaR violates
• Regulators increasingly require CVaR/ES reporting (Basel III)

Practical Application:

• Use CVaR to compare strategies with similar VaR but different tail behaviors
• A strategy with lower CVaR is preferable — it means smaller losses in extreme scenarios
• CVaR is particularly important for strategies involving options or leveraged positions