CVaR — what it says about tail risk that VaR misses

VaR identifies a quantile in a specified loss distribution; it does not guarantee next month’s result. CVaR, called Expected Shortfall here, averages the worst specified tail probability mass, with boundary treatment for discrete data. This article shows a small hypothetical example and separates its arithmetic from bank capital rules.

How CVaR differs from VaR and why that difference is not cosmetic

Use L = −return, so positive values mean losses and negative values mean gains. VaR₉₅ is the smallest loss value whose cumulative model probability is at least 95%. In a continuous model, it leaves a 5% upper tail; in discrete data a boundary can have nonzero mass. A historical quantile is not a verified future probability, and a twelve-month sample cannot have exactly 95% of its equal-weight observations below a threshold.

For a continuous loss distribution without probability mass at the VaR boundary, CVaR can be written as the mean of the worst tail beyond VaR. For discrete scenarios or empirical samples, average the specified tail probability mass and include an appropriate fraction of any boundary mass. Simply averaging only observations strictly beyond VaR can give a different result.

Expected Shortfall in the Basel bank-capital framework

The Basel Committee’s January 2019 market-risk standard revised its 2016 framework and retains an internal-models approach using Expected Shortfall. Its MAR33 specifies a one-tailed 97.5% level, a base liquidity horizon of ten days with regulatory adjustments, and stress calibration. VaR still appears elsewhere, including default-risk and backtesting requirements. These are bank-capital rules; this historical document does not establish current implementation dates in every jurisdiction or a retail-account limit.

With a common loss distribution, horizon and confidence level, ES is subadditive: ES(L_A + L_B) ≤ ES(L_A) + ES(L_B). This is a property of that measure, not a cap on realised portfolio loss. The diagram below shows why separate tail averages need not equal the tail of the sum. VaR is not generally subadditive.

How to compute CVaR on a retail account — an illustrative example

Treat the following twelve hypothetical monthly returns as equally weighted scenarios. Set L = −return and retain monthly percentages as the data unit. The initial USD 50,000 balance is only an optional reference for a scenario amount, not the changing denominator behind every observed month.

At 95%, the desired tail mass is 5%, while each of twelve observations has mass 8⅓%. Correct boundary weighting takes 5% mass from the worst 14% loss, so this empirical ES₉₅ is 14%. It is mathematically defined but entirely dependent on one observed extreme. The displayed 11.5% averages the worst two observations and corresponds to α = 10/12, not 95%. Forty equally weighted observations give exactly two observations in a 5% tail; that count alone does not establish estimation reliability.

A tail estimate from twelve monthly returns is unstable because so few observations define it. Monthly returns include the net effect of gaps and intraday moves on month-end equity, but they can hide losses that were recovered before month-end. Review higher-frequency equity and drawdowns separately. Changes in market conditions can also make historical tail estimates unrepresentative.

Per-trade budgets and portfolio tails answer different questions

A chosen per-trade percentage expresses a modeled cash-loss budget. Portfolio ES describes a tail distribution over the specified horizon, which need not be monthly. Neither a 1% setting nor an ES estimate is a guaranteed loss cap. The two calculations require different inputs and can be reported together without prescribing a universal fraction.

A historical monthly tail average of 11.5% alone cannot identify a dollar-shock outcome or its cause. To assess overlapping positions, define common scenarios, sum the losses in each one and then select the portfolio tail. Inspect the exposure, costs and dependence assumptions rather than automatically reducing leverage from that single number.

The most common misreadings of CVaR

• Confusing confidence with loss frequency. ES95 averages the worst 5% model probability mass; it does not mean a specified loss occurs in 95% of future periods. Discrete boundary weighting matters.
• Rescaling a daily tail estimate mechanically. Square-root-of-time volatility scaling requires assumptions about stable variance and time dependence. Scaling VaR or expected shortfall adds distributional assumptions; normally distributed independent increments provide one common model. Do not apply the multiplier to empirical CVaR merely because you know the number of trading days.
• Treating CVaR as a guarantee. It summarises the selected empirical distribution or model. An individual future loss can exceed it; the source may omit scenarios that later occur.
• Skipping the Sharpe and Sortino context. CVaR on its own will not tell you whether a strategy is sound. Pair it with the Sharpe ratio and the Sortino ratio so you see the return-to-risk relationship rather than only the size of the tail.

What to do tomorrow to make CVaR work for you

1. Choose a horizon, sign convention, quantile method and window. For twelve equally weighted observations, a 5% empirical tail takes only part of the worst observation’s mass. If instead you average the worst two, label the 2/12 tail explicitly. Neither choice establishes future tail probabilities.
2. Keep ES separate from maximum drawdown and the chosen per-trade budget. ES averages horizon losses; drawdown depends on a path and running peaks. Do not treat either historical metric as an execution-loss limit.
3. Match position losses within common scenarios before calculating portfolio ES. Correlation alone does not specify tail dependence or loss size; a sum of separate ES values is not generally the ES of the combined portfolio.
4. Recompute the tail estimate over a consistently defined window. State whether you report signed returns or positive loss magnitudes: a deeper signed tail becomes more negative, while a positive-loss CVaR rises. A worsening estimate can reflect new observations, volatility, exposure or sampling noise; it does not identify only two possible causes.
5. A historical-return simulation can generate additional paths under a stated dependence model, but it does not create new independent evidence of unobserved shocks. Keep model and sampling assumptions explicit; see Monte Carlo strategy simulation.

For more on the regulatory context of the measure, see the forex-basics.com risk management section.