VaR vs CVaR: Measuring Tail Risk Two Different Ways

Key Takeaways

• VaR is a threshold: the loss a portfolio will not exceed with a given confidence over a horizon (e.g., "95% one-day VaR of $1m").
• CVaR (also called expected shortfall) is the average loss in the tail beyond the VaR point, it measures the severity of the bad outcomes VaR only bounds.
• VaR says nothing about how large losses become once the threshold is breached; CVaR is designed to capture exactly that, so it is the better gauge of fat-tailed risk.
• Regulators moved from VaR to expected shortfall (CVaR) for market-risk capital under Basel's FRTB because CVaR is coherent and tail-sensitive.

What It Is

Value at Risk (VaR) is the maximum loss expected over a set horizon at a set confidence level. A 95% one-day VaR of $1 million means: on 95% of days the loss should be no worse than $1 million; on the worst 5% of days it could be worse, VaR does not say by how much.

Conditional Value at Risk (CVaR), or expected shortfall, is the expected loss given that the loss has already exceeded the VaR threshold. It is the average of all the outcomes in that tail. By construction, CVaR is always at least as large as VaR at the same confidence level.

The Intuition

VaR draws a line at a percentile and reports where the line sits. Its blind spot is everything past the line: a portfolio whose worst 5% of days cluster just past VaR and one whose worst 5% include catastrophic blowups can share the same VaR. CVaR removes that blind spot by averaging what happens beyond the line. If VaR is "the height of the seawall," CVaR is "the average depth of the water when the wall is overtopped."

How It Works

Both are computed the same three ways, historical simulation, parametric (variance-covariance), or Monte Carlo, and both need a confidence level and a horizon.

• VaR = the loss at the chosen percentile of the loss distribution. For 95% VaR, sort outcomes and read the 5th-percentile loss.
• CVaR = the average of all losses worse than that percentile. For 95% CVaR, average the worst 5% of outcomes.

Because CVaR averages the whole tail, it is coherent in the technical sense, notably, it respects diversification (the CVaR of a combined portfolio is never more than the sum of parts). VaR is not coherent: in some cases it can perversely rise when you diversify, which is one reason regulators moved away from it.

Worked Example

A desk simulates 1,000 equally likely daily P&L outcomes. Sorted from worst to best, the 50 worst days (the worst 5%) are losses ranging from −$1.0m to −$6.0m, and the 50th-worst is −$1.0m.

• 95% VaR = the loss at the 5th percentile = $1.0m. Interpretation: on 95% of days losses stay within $1.0m.
• 95% CVaR = the average of those 50 worst days. Suppose they average −$2.3m. CVaR = $2.3m.

Two portfolios could both show a $1.0m VaR, yet one has a $2.3m CVaR and another (with a few extreme days) a $4.0m CVaR. VaR calls them equally risky; CVaR reveals the second is far more dangerous. That is the entire point of preferring CVaR for tail-heavy books.

Common Mistakes

1. Reading VaR as a worst case. It is a threshold, not a maximum. Losses beyond VaR are expected 5% of the time and can be far larger, CVaR is what quantifies them.
2. Comparing VaR and CVaR at different confidence levels. A 99% VaR and a 95% CVaR are not comparable; fix the confidence level and horizon before comparing.
3. Trusting parametric estimates on fat tails. Assuming normal returns understates both VaR and CVaR when returns are skewed or kurtotic; historical or Monte Carlo methods usually capture tails better.
4. Ignoring model risk and regime change. Both measures are backward-looking calibrations. Pair them with stress testing for scenarios the historical window never contained.

Frequently Asked Questions

Q: What is the difference between var vs cvar in simple terms? VaR is the loss you should not exceed on, say, 95% of days. CVaR is the average loss on the bad days when you do exceed it. VaR bounds the tail; CVaR measures its severity.

Q: Why is CVaR considered better than VaR? CVaR captures the size of losses beyond the VaR threshold and is "coherent," meaning it always rewards diversification. VaR ignores tail magnitude and can behave inconsistently, which is why regulators shifted to expected shortfall.

Q: Is CVaR always higher than VaR? Yes, at the same confidence level and horizon. CVaR averages outcomes in the tail beyond VaR, so it is always greater than or equal to VaR.

Q: How do I calculate var vs cvar from data? Build the loss distribution (historical, parametric, or Monte Carlo). VaR is the loss at your chosen percentile; CVaR is the average of all losses worse than that percentile.

Q: Which should a risk manager report, var vs cvar? Report both. VaR is the familiar, regulator-legacy threshold; CVaR communicates tail severity and is now the Basel standard for market-risk capital. Together they describe both the edge and the depth of the tail.

Sources

1. Investopedia. "Value at Risk (VaR)." https://www.investopedia.com/terms/v/var.asp
2. Investopedia. "Conditional Value at Risk (CVaR)." https://www.investopedia.com/terms/c/conditional_value_at_risk.asp
3. Bank for International Settlements. "Minimum Capital Requirements for Market Risk." https://www.bis.org/bcbs/publ/d457.htm

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.