Conditional Value at Risk
Conditional Value at Risk (CVaR), also known as Expected Shortfall (ES) or Tail VaR (TVaR), is a risk measure that quantifies the expected loss of a portfolio in the worst (1−α) fraction of scenarios — the average loss conditional on losses exceeding the Value at Risk threshold — providing a more complete picture of tail risk than VaR alone.
Key takeaways
- CVaR is the expected loss given that losses exceed the VaR threshold; it is always greater than or equal to VaR at the same confidence level.
- Unlike VaR, CVaR is a coherent risk measure — it satisfies subadditivity (CVaR of a portfolio ≤ sum of CVaRs of components), making it theoretically superior for portfolio optimization.
- Basel III's Fundamental Review of the Trading Book (FRTB) replaced VaR with Expected Shortfall (ES) at 97.5% confidence for internal models, reflecting CVaR's superior tail sensitivity.
- CVaR can be estimated through historical simulation (averaging tail losses), parametric methods (integrating the normal distribution's tail), or Monte Carlo simulation.
- For non-normal return distributions with fat tails (common in hedge fund strategies), CVaR can be substantially higher than what a normal distribution would predict.
Explanation
Value at Risk has been the dominant risk measure since its popularization by J.P. Morgan's RiskMetrics in 1994. However, it has a fundamental limitation: it says nothing about the magnitude of losses beyond the VaR threshold. A portfolio might have a 99% VaR of $10 million while experiencing losses averaging $50 million in the 1% of worst cases — VaR is silent on this distinction. CVaR corrects this deficiency by explicitly targeting the expected magnitude of tail losses.
Mathematically, for a loss random variable L with distribution F:
VaR_α = inf{l : P(L > l) ≤ 1 − α} = F⁻¹(α) CVaR_α = E[L | L ≥ VaR_α] = (1/(1−α)) × ∫[α,1] VaR_u(L) du
For a normally distributed loss with mean μ and standard deviation σ:
CVaR_α = μ + σ × φ(Φ⁻¹(α)) / (1−α)
where φ is the standard normal PDF and Φ⁻¹ is the inverse standard normal CDF. At 99% confidence, CVaR ≈ μ + 2.665σ (versus VaR ≈ μ + 2.326σ), approximately 14% higher.
The coherence property of CVaR — specifically, subadditivity — is critical for portfolio risk management. Subadditivity means CVaR(A + B) ≤ CVaR(A) + CVaR(B), implying that diversification never increases risk as measured by CVaR. VaR famously lacks this property: two positions with identical VaRs can be combined to create a portfolio with higher VaR than either component. This makes VaR-based portfolio optimization pathological in certain distributions.
For hedge funds with negatively skewed or fat-tailed return distributions — common in short-volatility strategies, merger arbitrage, and credit strategies — CVaR captures the true character of risk far better than VaR. A strategy might show a benign VaR but a catastrophic CVaR if it operates normally most of the time but experiences extreme losses in rare but plausible scenarios. Risk managers use CVaR as a primary risk metric for strategy-level limits and as an input to portfolio optimization under the CVaR minimization framework developed by Rockafellar and Uryasev (2000).
Formula
CVaR_α = E[L | L ≥ VaR_α] = μ + σ × φ(Φ⁻¹(α)) / (1−α) [Normal distribution case]
Example
A short-volatility hedge fund generates daily returns normally distributed with mean 0.02% and standard deviation 0.5%, but with an empirical fat tail: in approximately 2% of days, the fund suffers losses drawn from a Pareto distribution with a mean tail loss of 3.8%. Parametric 99% VaR (assuming normality) = 0.02% + 2.326 × 0.5% = 1.183%. Parametric CVaR = 0.02% + 2.665 × 0.5% = 1.353%. However, the fat-tail-adjusted historical CVaR = 0.80 × 1.353% + 0.20 × 3.8% = 1.082% + 0.760% = 1.842% — a CVaR 36% higher than the parametric estimate, reflecting the crash risk embedded in the strategy. This discrepancy is why post-GFC regulators mandated CVaR over VaR for bank trading books.
Related terms
Arbitrage Component Var Diversification Expected Shortfall Greeks Hedging Hedge Fund Merger Arbitrage Parametric Var Portfolio Optimization Standard Deviation Stop Loss Tail Risk