Foundations

VaR vs CVaR (Expected Shortfall)

VaR is a loss quantile; CVaR is the average loss beyond that quantile.

beginnerrisktail riskexpected shortfallloss distribution

VaR locates a tail threshold, while CVaR summarizes the losses beyond that threshold.

VaR at level α is the loss threshold that is only exceeded with probability 1 − α over a given horizon.

CVaR (Expected Shortfall) is the conditional average loss given that losses breach the VaR threshold.

INTERACTIVE INTUITION

Explore the mechanism

What information is gained by averaging losses after the VaR threshold has been crossed?

This diagram shows a loss distribution with the VaR at level α as a red quantile and the CVaR (Expected Shortfall) as an amber line at the average loss beyond that quantile.

α: 95.0% · tail prob: 5.0%

Confidence level α: 95.00%Loss volatility σ (scale): 2.00%

Numbers

VaR_95.0% ≈ 3.31% of notional

CVaR_95.0% ≈ 4.06% of notional

Tail severity CVaR − VaR ≈ 0.75%

CVaR / VaR ≈ 1.23 · tail probability ≈ 5.00%

Interpretation

The red dashed line is the VaR: only a fraction 1 − α of the loss distribution lies to its right. The shaded area under the blue density shows those tail outcomes.

The amber line is the CVaR, the average loss within the shaded region. As you increase α or σ, both VaR and CVaR move to the right, but CVaR typically grows faster because it reacts to how heavy the tail is. The mental model: VaR tells you “how far the tail starts”; CVaR tells you “how bad it is, on average, once you are in the tail”.

UNDERSTAND

What to notice

For continuous right-tailed loss distributions, CVaR is always at least as large as VaR and is more sensitive to extreme tail behaviour.

FORMALIZE

The quantitative structure

Notation

L: portfolio loss
alpha: confidence level
VaR_alpha: alpha loss quantile

Value at Risk

VaR_alpha(L) = inf{l : P(L <= l) >= alpha}

VaR locates the loss threshold reached at the selected confidence level.

Expected Shortfall

ES_alpha(L) = E[L | L >= VaR_alpha(L)]

For a continuous loss distribution, expected shortfall averages losses in the tail beyond VaR.

USE

A small example

Same VaR, different tails

Two portfolios both have 99% VaR of 10. Beyond that threshold, portfolio A loses 11 on average and portfolio B loses 30 on average.

Their VaR reports are identical at 10.
Their expected shortfalls are 11 and 30.

ResultExpected shortfall reveals the much heavier residual tail of portfolio B.

TRANSFER

Where it appears

tail-risk reporting
portfolio optimization with loss constraints
comparing portfolios with similar VaR but different tail severity

CHALLENGE

Where the intuition breaks

What this view leaves out

Both measures depend on the selected horizon, confidence level, loss model, and data; CVaR does not remove model risk.

CONNECT

Where this concept sits

Follow the strongest prerequisites, applications, extensions, and contrasts.

VaR vs CVaR (Expected Shortfall)YOU ARE HERE

Contrasts withValue at Risk (VaR)

VaR identifies a loss threshold, while Expected Shortfall summarises losses beyond that threshold.

SupportsDistributionally Robust Optimisation

Tail-risk objectives such as Expected Shortfall can be embedded in robust portfolio optimisation problems.

Contrasts with · Value at Risk (VaR)

VaR identifies a loss threshold, while Expected Shortfall summarises losses beyond that threshold.

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SOURCES

References

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