What Is Conditional Value at Risk (CVaR)? VaR vs. CVaR
By Markets & Investing DeskUpdated Nov 10, 2026
Definition
Conditional value at risk (CVaR), also called expected shortfall, is the average loss a portfolio suffers in the worst cases, those beyond a chosen value at risk (VaR) threshold.
Key takeaways
- CVaR is always equal to or greater than VaR at the same confidence level because it averages the whole tail.
- Unlike VaR, CVaR is subadditive, so combining positions never shows more risk than the sum of the parts.
- In the hypothetical example, a 95% VaR of $45,000 sits beside a 95% CVaR of $63,000 on a $1,000,000 portfolio.
- Basel's Fundamental Review of the Trading Book moved bank capital from 99% VaR to expected shortfall at 97.5%.
Conditional value at risk (CVaR) is the expected loss of a portfolio in the worst cases, those beyond a chosen value at risk (VaR) threshold. If a portfolio’s one-day 95% VaR is $50,000, the one-day 95% CVaR answers a second question: on the 5% of days when losses exceed that level, how much is lost on average? CVaR is also called expected shortfall (ES). Some texts use “conditional VaR,” “average value at risk” or “tail VaR” for the same idea, though definitions can differ slightly for distributions with jumps.
VaR vs. CVaR
VaR is a quantile. A 95% one-day VaR of $50,000 means that, under the model, losses should exceed $50,000 on no more than 5% of days. It says nothing about what happens on those days. A loss of $51,000 and a loss of $500,000 look the same to VaR, because both are simply “beyond the threshold.”
CVaR looks inside that tail. It averages every outcome past the VaR level, so a portfolio exposed to rare but severe losses shows a CVaR far above its VaR, while a portfolio whose bad days cluster just past the cutoff shows a CVaR only slightly higher.
| VaR | CVaR (expected shortfall) | |
|---|---|---|
| Question answered | How bad can losses get at a given confidence level? | How bad are losses, on average, when they go past that level? |
| Treatment of the tail | Ignores everything beyond the cutoff | Averages the whole tail |
| Relationship | Lower bound of the tail | Always ≥ VaR at the same level |
| Coherent risk measure | No (can fail subadditivity) | Yes |
The last row matters for portfolios. A risk measure is subadditive if the risk of a combined portfolio is never greater than the sum of the risks of its parts, which is what diversification should deliver. VaR can violate this, especially with options or credit exposures, so merging two positions can appear to increase VaR. CVaR satisfies subadditivity, which is one reason it is classed as a coherent risk measure.
CVaR formula
Define loss L as a positive number (a 3% decline is a loss of 3%). At confidence level α (for example 95%):
CVaR_α = E[L | L ≥ VaR_α]
That is, the expected loss given that the loss is at least VaR. For a continuous distribution, an equivalent form averages VaR over every confidence level in the tail:
CVaR_α = (1 / (1 − α)) × ∫ from α to 1 of VaR_u du
In practice, CVaR is estimated in one of three ways:
- Historical simulation: sort past returns, take the worst (1 − α) share, and average them.
- Parametric: assume a distribution (often normal or Student’s t) and compute the tail mean analytically.
- Monte Carlo: simulate many scenarios from a model and average the simulated losses beyond VaR.
Example of conditional value at risk
Hypothetical example. A $1,000,000 portfolio has 100 days of return history. At 95% confidence, the tail is the worst 5% of days, here 5 observations. Those five daily losses are 9%, 7%, 6%, 5% and 4.5%.
- VaR (95%): using the convention that VaR is the smallest loss in the tail, VaR is 4.5%, or $45,000.
- CVaR (95%): (9 + 7 + 6 + 5 + 4.5) / 5 = 31.5 / 5 = 6.3%, or $63,000.
The gap between $45,000 and $63,000 is the information VaR leaves out. If the worst day had been a 25% loss instead of 9%, VaR would not move, but CVaR would jump to 9.5% ($95,000). The figures assume a one-day horizon, equal weighting of each day and one VaR convention; interpolating between the 5th and 6th worst days, for instance, gives a slightly different VaR. Five observations is also a very thin sample, a problem covered under limitations.
How CVaR is used in risk management
Portfolio optimization. CVaR can replace variance as the risk term in portfolio construction, giving “mean-CVaR” optimization: either minimize CVaR for a target expected return, or maximize expected return subject to a CVaR limit. Rockafellar and Uryasev showed in 2000 that minimizing CVaR over scenario data can be written as a linear program, which makes it tractable even with many assets and scenarios. Unlike mean-variance, it penalizes tail losses rather than all volatility, including upside moves.
Regulation. Under the Basel Committee’s Fundamental Review of the Trading Book, bank market-risk capital moved from 99% VaR to expected shortfall at 97.5% confidence, a direct response to VaR’s blindness to tail severity.
Comparing assets. Lower-volatility holdings such as high-grade bonds or large-cap stocks often show CVaR not far above VaR. Small-cap stocks, leveraged strategies and derivatives can show CVaR well above VaR. Assets with large upside potential often carry large CVaRs, so a small CVaR is not automatically better; weigh it against return, for instance with the Sharpe ratio.
Limitations of conditional value at risk
- Model dependence. CVaR is built on the same inputs as VaR. A normal-distribution assumption understates fat tails, and a calm historical window understates crisis losses.
- Few tail observations. By definition, CVaR depends on rare events. At 99% confidence with 250 days of data, the estimate rests on two or three observations, so it can swing sharply when one extreme day enters or leaves the window.
- Harder to backtest. VaR can be checked by counting breaches. Verifying an average of tail losses needs more data and more complex tests.
- Not a worst case. CVaR is an average. Individual losses in the tail can be, and in the example above are, much larger than CVaR itself.
- Horizon and liquidity. A one-day CVaR assumes positions can be adjusted within a day. Illiquid holdings may take longer to exit, which a short-horizon figure does not capture.
The Bottom Line
CVaR matters to portfolio managers, risk teams and bank regulators who need to know how severe losses get once VaR is breached. It captures tail severity that VaR ignores and supports mean-CVaR optimization, but it relies on the same model inputs and very few tail observations. Treat it as an average tail loss rather than a worst case, and weigh it against expected return.
