Expected shortfall (CVaR)

What it means

A loss threshold tells you where an unusually bad outcome begins; expected shortfall asks how large losses average once you enter that tail. It is also called conditional value at risk, or CVaR. Always state the time horizon, confidence level, currency, and method used to estimate possible losses.

For a one-day, 95% expected shortfall, the relevant tail is the worst 5% of modeled one-day outcomes. The result is an average loss within that tail, not the largest possible loss. By contrast, 95% value at risk (VaR) is a loss threshold associated with that tail, not its average severity.

How it is calculated

For N equally weighted, simulated loss outcomes, sort losses from largest to smallest. If the worst-tail fraction produces a whole number of outcomes, use:

ES_c = (L_1 + L_2 + ... + L_k) / k; k = N × (1 − c)

• ES_c is expected shortfall at confidence level c, in the same units as the losses, such as US$.
• c is the confidence level as a decimal; 0.95 means 95%.
• N is the number of equally weighted loss outcomes.
• k is the number of outcomes in the worst tail. This simple formula assumes k is a whole number.
• L_1 through L_k are the k largest losses, ordered from largest downward. A gain can be recorded as a negative loss.

With unequal probabilities, the losses must instead be weighted by their probabilities in the tail. Estimation methods can also handle a tail fraction that does not contain a whole number of observations. Results depend on the assumed distribution or the outcomes used; historical data cannot capture every possible future event.

Worked example

Suppose a hypothetical US$10,000 stock basket has 20 equally weighted, modeled one-day outcomes. Record each outcome as a dollar loss, allowing negative numbers for gains. These numbers are illustrative, not a forecast.

1. Set the confidence level to 95%, leaving a 5% tail.
2. Calculate k = 20 × (1 − 0.95) = 1 outcome.
3. Suppose the largest modeled loss is US$600, and the second-largest is US$400. The worst 5% contains only the US$600 loss.
4. The estimated one-day 95% expected shortfall is therefore US$600 / 1 = US$600, or 6% of the US$10,000 basket value.

With just one observation in the tail, this estimate is especially sensitive to that observation. More outcomes do not remove model risk, but they can make a tail average less dependent on a single result.

Why it matters before you trade

VaR and expected shortfall answer different questions. A VaR threshold identifies a boundary for adverse outcomes under a model; expected shortfall examines the modeled losses beyond that boundary. Two baskets can have similar VaR estimates but different tail averages if one has much more severe losses in its worst scenarios.

In a pre-trade check, that distinction can help frame exposure to large losses before an order is placed. The estimate needs a relevant horizon and assumptions about the basket's holdings and risks. Sudden price gaps, thin trading, changing correlations, and data limitations can make realized losses worse than a modeled tail average. Expected shortfall is not a cap on losses or a probability of a successful trade.

How PreTrAIde uses it

PreTrAIde's stated quantitative measures include one-day 95% VaR, daily volatility, liquidity measures, market-impact estimates, and basket concentration—not expected shortfall. It would be incorrect to describe its VaR figure as CVaR or to claim it calculates a tail average.

Its pre-trade analysis evaluates a user-built stock basket before any order is placed and returns basket-level and line-level verdicts, risk assessments, and plain-language reasoning. Those outputs are decision support, not investment recommendations; they do not establish an expected-shortfall estimate.

Common mistakes

• Calling VaR the average tail loss: VaR is a threshold; expected shortfall averages losses in the specified tail.
• Omitting the horizon or confidence level: A one-day 95% figure is not interchangeable with a ten-day or 99% figure.
• Treating an estimate as a worst-case limit: Outcomes outside the modeled data or assumptions can be more severe.
• Comparing inconsistent methods: Estimates based on different scenarios, probability weights, or definitions of loss may not be directly comparable.