Calculate expected shortfall, confidence level, or value at risk under a zero-mean normal loss model. The supported confidence range is 0.500001–0.999999; the normal-tail assumption may understate extreme losses.

Estimate the average loss in the upper tail.

Model: zero-mean normal losses, same loss units and horizon. Normal tails may understate extreme risk.

For example, 0.95 means 95%. Supported: 0.500001–0.999999.

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Expected Shortfall Formula

Expected shortfall (ES), also called conditional value at risk (CVaR), is the average loss over the worst (1 − CL) fraction of the loss distribution. For continuous distributions this equals the conditional mean above VaR; discrete distributions require allocation of probability mass at the cutoff. If you assume a normal distribution with mean 0, ES and VaR can be related using the standard normal PDF/CDF, which is the approximation used by the calculator above.

ESCL = (1) / (1 - CL)∫CL¹VaRᵤ du \ (Normal approx., mean 0) ES = VaR × (varphi(zCL)) / ((1 - CL) zCL), zCL = Phi⁻¹(CL)

Variables:

  • ES is the expected shortfall
  • VaR is the value at risk
  • CL is the confidence level

In general, ES is computed from the loss distribution (for example using historical data, Monte Carlo simulation, or a parametric model) by averaging the tail losses beyond the VaR threshold. Under the normal-approximation shown above, ES can be estimated from VaR and CL using the standard normal z-score and density.

What is Expected Shortfall?

Expected shortfall, also known as conditional value at risk (CVaR), is a risk measure used in finance to assess the risk of extreme losses in a portfolio. It estimates the average loss in the worst (1 − CL) fraction of outcomes beyond a chosen confidence level. Unlike value at risk (VaR), which gives a loss threshold (a quantile) at a specific confidence level, expected shortfall accounts for the severity of losses in the tail of the loss distribution, offering a more comprehensive view of tail risk. This makes it a valuable tool for risk management and regulatory compliance in financial institutions.

How to Calculate Expected Shortfall?

The following steps outline how to calculate the Expected Shortfall.

  1. First, determine the value at risk (VaR) for the portfolio at your chosen horizon and confidence level.
  2. Next, choose the confidence level (CL) for the risk assessment (commonly 0.90, 0.95, or 0.99).
  3. If using the normal-approximation used by the calculator, compute the z-score zCL = Φ-1(CL) and the standard normal density φ(zCL).
  4. Finally, estimate expected shortfall using ES = VaR · φ(zCL) / ((1 − CL) · zCL), then check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Value at risk (VaR) = 100,000

Confidence level (CL) = 0.95. Under the zero-mean normal model, z ≈ 1.644853627 and φ(z) ≈ 0.1031356404. Using unrounded values gives ES ≈ 100,000 × φ(z) / (0.05 × z) ≈ 125,404.03 loss units.