Conditional VaR (CVaR/ES)

CVaR is superior to VaR for several interconnected reasons. First, CVaR is coherent: it satisfies subadditivity (diversification always helps), while VaR does not. Second, CVaR provides information about loss severity beyond the threshold, not just the probability of exceeding it. VaR says 'you'll lose no more than X with probability α' — but gives no information about how bad losses are when they do exceed X. Third, CVaR is convex and can be incorporated into portfolio optimization frameworks using linear programming. Fourth, CVaR is more sensitive to the shape of the tail — fat-tailed distributions produce materially higher CVaR, giving an honest picture of tail risk that VaR can obscure.

The Fundamental Review of the Trading Book (FRTB), Basel III's market risk framework, replaced 99% VaR with 97.5% Expected Shortfall as the primary capital metric. Key requirements include: ES must be calculated using a 10-business-day liquidity horizon (not the old 10-day scaled from 1-day); different asset classes have different liquidity horizons (from 10 days for large-cap equities to 120 days for some credit products); ES must be calibrated to a stressed historical period; and banks must also calculate a non-stressed ES for comparison. The framework took effect for large internationally active banks in 2023, with Basel III full implementation completed through 2025–2028 in various jurisdictions.

For normally distributed returns, the ES/VaR ratio is a fixed function of confidence level. At 95%, ES is approximately 25% higher than VaR. At 99%, ES is approximately 14% higher than VaR. At 99.9%, ES is about 8% higher. However, for fat-tailed (leptokurtic) distributions — which better describe actual financial returns — the ratio is substantially higher. A Student's t-distribution with 4 degrees of freedom produces ES/VaR ratios 50–100% higher than the normal case. This is why historical or Monte Carlo simulation methods typically produce higher CVaR than parametric normal models.

CVaR increases dramatically during stress periods because: (1) portfolio volatility rises, shifting the entire loss distribution rightward; (2) correlations between assets increase toward 1.0 during crises, reducing diversification benefits; (3) return distributions become more negatively skewed and fat-tailed, amplifying tail losses beyond what volatility alone would predict. During the 2008 crisis, typical daily VaR estimates were breached 10–20 times more frequently than the implied 1% probability — demonstrating that historical CVaR estimates calibrated to tranquil periods severely understate tail risk in crises. This is why FRTB requires stressed-period calibration.

Yes, and this is one of CVaR's major practical advantages over VaR. Rockafellar and Uryasev (2000) showed that CVaR minimization can be formulated as a linear program, making it computationally tractable even for large portfolios. The mean-CVaR portfolio optimization framework minimizes CVaR for a given expected return (analogous to mean-variance Markowitz optimization), but incorporates tail risk into the objective function. This produces portfolios that are more robust to extreme events compared to mean-variance optimal portfolios, at a modest cost in expected return. Many institutional risk managers now use mean-CVaR optimization as their primary portfolio construction framework.

Backtesting VaR is straightforward: count how many days actual losses exceeded the VaR estimate and compare to the expected frequency (e.g., 1% for 99% VaR). CVaR backtesting is more challenging because CVaR is an average conditional on tail events — and tail events are rare, making statistical inference difficult with limited data. Tests include: (1) conditional coverage tests — do losses beyond VaR average to the predicted CVaR? (2) Acerbi-Szekely spectral tests — comparing the entire tail distribution. Because tail events are rare, CVaR backtests have low statistical power and require many years of data to detect even moderate underestimation. This is an active area of research in quantitative risk management.

CVaR is a statistical estimate of the average tail loss given the current distributional model. Stress testing examines specific worst-case scenarios — often historical crises (2008 GFC, COVID-19 crash, 1987 Black Monday) or hypothetical events (geopolitical shock, central bank policy reversal, sector collapse). CVaR is model-dependent and captures average tail behavior under the modeled distribution; stress tests capture specific scenario losses regardless of their statistical probability. Best practice combines both: CVaR for ongoing daily risk monitoring and capital allocation, stress testing for scenario-specific preparedness and board-level risk communication.