What is CVaR / Expected Shortfall?

Conditional VaR (CVaR), also called Expected Shortfall (ES), is the average loss in the tail beyond VaR. Where VaR tells you "on the worst 5% of days, losses are at least $X," CVaR tells you "on those worst 5% of days, the average loss is $Y." CVaR fixes VaR's key blind spot: VaR describes only the tail boundary, while CVaR describes the tail's severity. Regulators increasingly favor CVaR because it is a coherent risk measure — mathematically well-behaved in ways VaR is not.

Calculation type: Average of losses beyond VaR percentile Method version: 1.0 Date reviewed: 2026-07-04 Live scanner: /portfolio/risk/

Formula

CVaR_α = E[Loss | Loss > VaR_α]

"Expected loss, given the loss exceeds VaR at confidence level α." For historical simulation:

CVaR_α = −mean(worst (1−α) fraction of P&L observations)

For α = 95%: sort N days of portfolio P&L worst-to-best. CVaR₉₅ = negative of the average of the worst 5% of days (the first (1−0.95) × N = 5% × N observations).

Worked example

Same portfolio as VaR worked example: 100 shares SPY at $520 = $52,000 mark-to-market. Historical 500-day daily P&L series. The worst 25 days (5% of 500):

Position 1 (worst day): -$1,890
Position 2: -$1,720
Position 3: -$1,650
...
Position 25 (5th percentile): -$620

Average of the 25 worst days:

CVaR₉₅ 1-day = mean(worst 25 days) = -$1,100 → $1,100

Compare:

VaR₉₅ = $620 (boundary of the 5% tail)
CVaR₉₅ = $1,100 (average loss inside the 5% tail)

Interpretation: when the portfolio actually is in the worst 5% of outcomes, the average loss is $1,100 — much worse than the $620 VaR "boundary" implies. Similar relationship holds at 99%: CVaR₉₉ is always at least as large as VaR₉₉.

Why CVaR fixes VaR's blind spot

Consider two portfolios both with VaR₉₅ = $620:

Portfolio A: worst 5% of days cluster around $620-$700 loss. CVaR₉₅ = $660.
Portfolio B: worst 5% of days include a few extreme events: -$620, -$800, -$1,200, -$2,500, -$5,000. CVaR₉₅ = $2,024.

Both portfolios have identical VaR, but Portfolio B has 3× the CVaR. Under VaR alone, they look equivalent. Under CVaR, Portfolio B is clearly much riskier. This is the tail-severity blind spot that CVaR eliminates.

Coherent risk measure properties

CVaR is a coherent risk measure (Artzner et al., 1999) with four mathematical properties that VaR lacks:

Sub-additivity

CVaR(A + B) ≤ CVaR(A) + CVaR(B). Combining portfolios cannot increase total risk. VaR is not sub-additive: adding two portfolios can produce combined VaR exceeding the sum of individual VaRs. This is a well-known VaR pathology.

Monotonicity

If Portfolio A always loses at least as much as Portfolio B, then CVaR(A) ≥ CVaR(B). Both VaR and CVaR are monotonic.

Positive homogeneity

CVaR(λA) = λ × CVaR(A) for λ > 0. Doubling your position size exactly doubles your CVaR. Both VaR and CVaR satisfy this.

Translation invariance

CVaR(A + c) = CVaR(A) − c for constant cash c. Adding cash to a portfolio reduces its CVaR by exactly the cash amount. Both VaR and CVaR satisfy this.

How CVaR is used in practice

Regulatory adoption

Basel III (2019 Fundamental Review of the Trading Book) partially replaced VaR with CVaR (specifically ES_97.5) for regulatory capital calculations. Solvency II uses CVaR-related metrics for insurance capital. The trend is toward CVaR over VaR at institutional and regulatory levels.

Portfolio optimization

Because CVaR is sub-additive and convex, it can be used in convex optimization frameworks to find efficient portfolios that minimize tail risk for a given expected return. VaR cannot be used this way because it is not convex.

Retail risk management

For individual traders, the practical use is simple: CVaR is a better summary of "how bad could it get" than VaR. Compare your portfolio's CVaR₉₅ to comfort zone. If VaR₉₅ = 2% of capital but CVaR₉₅ = 8%, position sizing to VaR alone dramatically underestimates worst-case pain.

Common misconceptions

"CVaR is just a higher VaR"

No. CVaR is a different metric conceptually. VaR asks "what's the boundary of the tail?" CVaR asks "what's the average loss inside the tail?" These are different questions with different answers. CVaR is always at least as large as VaR (equality only when all tail losses are identical).

"Historical CVaR captures all future tail risk"

Same limitation as historical VaR: bounded by observed history. If a market regime shift produces losses outside the historical distribution, both VaR and CVaR will under-estimate future risk. This is why stress tests remain essential complements.

"CVaR replaces the need for stress tests"

No. CVaR is bounded by historical experience; stress tests are hypothetical extremes. Both are needed. Institutional risk management combines CVaR + stress tests + backtesting.

"CVaR is only for institutions"

The math is simple enough that retail traders can and should compute it. Any dataset of daily P&L can be sorted worst-to-best; taking the mean of the worst 5% or 1% is trivial. The portfolio VaR & stress test scanner does this automatically.

Related terms and tools

Live Portfolio VaR & Stress Test — compute VaR + CVaR for your positions.
Value at Risk glossary — the tail-boundary metric.
Portfolio Correlation Heatmap — concentration risk view.
Wheel Monte Carlo Simulator — forward-looking outcome distribution.
Beta-weighted delta — aggregate directional exposure.

Sources: Artzner, Delbaen, Eber, Heath "Coherent Measures of Risk" (Mathematical Finance, 1999). Basel III FRTB (2019). Hull "Risk Management and Financial Institutions" (Chapter 22). Educational only, not investment advice. Page last reviewed 2026-07-04.