VaR vs CVaR: What Is the Difference?

Value at Risk (VaR) vs Conditional VaR (Expected Shortfall)

Value at Risk is a loss threshold: a 95% VaR is the loss level a portfolio is expected to exceed only 5% of the time over a stated horizon. Conditional VaR, also called expected shortfall, is the average loss across those worst 5% of outcomes. VaR marks where the loss tail begins, CVaR measures how deep it runs. CVaR is always at least as large as VaR at the same confidence level, and unlike VaR it is a coherent risk measure that never penalizes diversification.

Value at Risk (VaR) vs Conditional VaR (Expected Shortfall)
	Value at Risk (VaR)Value at Risk (VaR)	Conditional VaR (CVaR)Conditional VaR (CVaR)
What the Number Reports	The loss threshold at the edge of the tail	The average loss inside the tail beyond that threshold
Question Answered	How much can I lose at this frequency?	How much do I lose on average once the threshold breaks?
Definition	The quantile loss exceeded only (1 − c) of the time	Average loss in the worst (1 − c) fraction of outcomes, weighting the boundary to fill that tail
Normal-Distribution Formula	V × (z × σ − μ)	V × [σ × φ(z) / (1 − c) − μ], with μ defined as mean return
Multiplier on Volatility at 95%	1.645	About 2.063
Same Portfolio, 95% Monthly, $1M at 5% Vol	$82,243	$103,136
Tail Sensitivity	Blind to tail depth; two books can share a VaR and differ wildly beyond it	Distinguishes tail shapes because it averages the deep losses
Coherent Risk Measure	No; can violate subadditivity and report a diversified book as riskier than its parts	Yes; combined CVaR never exceeds the sum of the parts
Regulatory Status for Trading-Book Capital	VaR and stressed VaR remain in the current U.S. market-risk capital rules	The Basel FRTB internal-model standard uses 97.5% expected shortfall
Backtesting	Simple; each period is a breach or not	Harder; severity is more difficult to validate against realized outcomes
Main Weakness	Says nothing about losses past the cutoff	Depends on thin tail data, so the estimate is noisy at high confidence

Value at Risk (VaR)Value at Risk (VaR)

Conditional VaR (CVaR)Conditional VaR (CVaR)

What the Number Reports

The loss threshold at the edge of the tail

The average loss inside the tail beyond that threshold

Question Answered

How much can I lose at this frequency?

How much do I lose on average once the threshold breaks?

Definition

The quantile loss exceeded only (1 − c) of the time

Average loss in the worst (1 − c) fraction of outcomes, weighting the boundary to fill that tail

Normal-Distribution Formula

V × (z × σ − μ)

V × [σ × φ(z) / (1 − c) − μ], with μ defined as mean return

Multiplier on Volatility at 95%

1.645

About 2.063

Same Portfolio, 95% Monthly, $1M at 5% Vol

$82,243

$103,136

Tail Sensitivity

Blind to tail depth; two books can share a VaR and differ wildly beyond it

Distinguishes tail shapes because it averages the deep losses

Coherent Risk Measure

No; can violate subadditivity and report a diversified book as riskier than its parts

Yes; combined CVaR never exceeds the sum of the parts

Regulatory Status for Trading-Book Capital

VaR and stressed VaR remain in the current U.S. market-risk capital rules

The Basel FRTB internal-model standard uses 97.5% expected shortfall

Backtesting

Simple; each period is a breach or not

Harder; severity is more difficult to validate against realized outcomes

Main Weakness

Says nothing about losses past the cutoff

Depends on thin tail data, so the estimate is noisy at high confidence

What is the difference between VaR and CVaR?

Value at Risk (VaR) and conditional Value at Risk (CVaR) read the same loss distribution at different places. VaR reports a threshold: at 95% confidence, the loss expected to be exceeded only 5% of the time over a stated horizon. CVaR, also called expected shortfall, reports the average loss in that worst 5% tail. VaR marks where the tail begins; CVaR measures how deep it runs.

CFA Institute defines VaR as the minimum loss that would be expected to be incurred a certain percentage of the time over a certain period, given assumed market conditions. Every VaR statement carries three parts: a horizon, a confidence level, and a loss amount. On the glossary’s $500,000 equity book at 1.5% daily volatility, a one-day 95% VaR of $12,337.50 means a loss larger than $12,337.50 is expected on roughly one trading day in twenty.

Conditional VaR averages the worst (1 - c) fraction of the loss distribution. At 95% confidence, it averages the worst 5%. For a continuous distribution this matches the mean loss beyond the VaR cutoff. For discrete or historical observations, include the fraction of the boundary observation needed to fill the selected tail.

Why the gap between VaR and CVaR matters

VaR is silent about depth. Two portfolios can share an identical 95% VaR while one loses slightly more past the cutoff and the other loses catastrophically. A book that sells options is the classic case: it shows modest standard deviation and a comfortable VaR while carrying a heavy left tail that only appears when the threshold breaks.

CVaR separates those two books. Using the glossary’s worked example, a $1,000,000 portfolio with returns near zero mean and 5% monthly volatility has a 95% monthly VaR of 1.6449 × 5% × $1,000,000, or about $82,243. The normal-distribution CVaR multiplier is the standard normal density at the 95% quantile divided by 0.05, which is 2.0627, giving a 95% CVaR of about $103,136. The $20,893 difference is the tail severity VaR never reports. On a fat-tailed real return series the gap is wider than the normal shortcut suggests.

How VaR and CVaR differ in practice

Both measures are computed the same way, from the same distribution, using the parametric, historical simulation, or Monte Carlo method. VaR reads the percentile; CVaR averages past it. Most risk systems produce both from one run.

Expected shortfall satisfies monotonicity, positive homogeneity, translation invariance, and subadditivity. For two positions, combined expected shortfall does not exceed the sum of their separate expected shortfalls. VaR does not share that general subadditivity property. The Basel FRTB internal-model standard therefore uses expected shortfall to capture tail severity and support coherent aggregation.

The trade runs the other way on estimation quality. VaR is easy to backtest, because each period either breaches the threshold or does not, and Federal Reserve analysis of bank backtesting exceptions during the March 2020 crash tracks exactly that binary count. CVaR is harder to validate, and at 99% confidence a one-year daily sample offers only a handful of tail observations, so a single new extreme can move the estimate materially.

When to use VaR vs CVaR

Start with the payoff shape. For a roughly linear equity book, VaR is a fast and comparable number that supports position sizing and account-level loss limits. Adding a position that pushes portfolio VaR above a preset ceiling is a signal to trim, hedge, or pass.

For nonlinear exposure, CVaR is the measure that governs. Short-premium and leveraged strategies are precisely the ones a variance-based budget underprices, and stating the risk budget in expected shortfall terms restrains them. CVaR also ranks strategies with similar Sharpe ratios by how badly they lose in stress, which is what determines whether a trader can survive the drawdown and keep the system running.

Use the two measures together. VaR sets the loss threshold. Expected shortfall measures the average severity inside that tail. For trading decisions, compare both with the position’s modeled payoff and stress scenarios.

Bottom line

VaR locates the loss threshold. Expected shortfall measures the severity of the selected tail. Report both from the same loss scenarios, horizon, and confidence level, then use the position’s payoff shape to decide how each informs sizing.

Value at Risk (VaR) vs Conditional VaR (Expected Shortfall): FAQ

Is CVaR always higher than VaR?

Not always higher, but never lower: at the same confidence level and horizon, expected shortfall is at least as large as VaR because it averages the worst selected fraction of losses. Use the correct tail weight for discrete outcomes. If loss is $0 with 96% probability and $100 with 4% probability, 95% VaR is $0 and expected shortfall is $80: the worst 5% combines all 4% of the $100 loss with 1% of the $0 outcome.

Why did regulators move from VaR to expected shortfall?

The Basel FRTB internal-model standard uses expected shortfall to capture the severity of tail losses. VaR identifies a quantile; expected shortfall averages the selected loss tail and satisfies subadditivity. National rules determine when that standard applies. The United States retains VaR-based requirements in its current rule, and the UK schedules its FRTB internal-model approach for January 2028.

Do VaR and CVaR use different data or models?

No. Both use a loss distribution built from parametric, historical, or simulated scenarios. VaR reads the selected quantile. CVaR averages the worst selected fraction of outcomes, including the required weight at a discrete boundary. Produce both from the same scenarios to compare the threshold and tail severity.

Does CVaR tell me my maximum possible loss?

It tells you the average loss past the VaR cutoff, and some losses in that tail run larger than the average. CVaR gives a better sense of expected tail severity than VaR does; for the worst case, run stress tests and scenario analysis on outcomes outside the modeled distribution.

Which should an options trader use?

Options books usually need CVaR because their payoffs are nonlinear. A short-premium strategy can show modest volatility and a benign VaR while carrying a heavy left tail, and VaR will not flag it. CVaR pulls that tail into the number that drives sizing. For either measure, use Monte Carlo simulation rather than the parametric shortcut, because a normal approximation misses the curvature that gamma and vega introduce.

Which confidence level should each measure use?

Both measures should use the same stated confidence level when they are compared. Common internal choices include 95% and 99%, and the Basel FRTB internal-model standard specifies 97.5% expected shortfall. At 95%, VaR marks the threshold of the worst 5% and expected shortfall averages that same 5% tail, so the two numbers describe one tail: where it begins and how deep it runs.