Expected Shortfall Calculator
Measure the tail risk VaR misses. Enter your portfolio value, expected annual return, annual volatility, and confidence level to get both Value at Risk and Expected Shortfall (conditional VaR): the average loss when losses exceed VaR, computed parametrically under the normal distribution.
Parametric estimates under a normal return distribution. Real return tails are fatter than normal, so treat these as a baseline, not a worst case. Not investment advice.
Worked example, checked against this exact tool
$1,000,000 portfolio, 8% expected annual return, 15% annual volatility, 95% confidence. z = 1.6449. VaR = 1,000,000 × (1.6449 × 0.15 - 0.08) = $166,728.04. ES = 1,000,000 × (0.15 × 0.1031 ÷ 0.05 - 0.08) = $229,406.92: in the worst 5% of outcomes the average loss is $229,406.92, which is $62,678.88 beyond VaR.
VaR vs expected shortfall
VaR answers "what is the most I lose 95% of the time": a threshold. Expected shortfall answers "when I breach that threshold, how bad is it on average": the tail's center of gravity. ES is always at least as large as VaR, and unlike VaR it is a coherent risk measure, which is why Basel's market-risk framework moved to 97.5% ES.
The parametric formulas
Under normally distributed returns with mean μ and volatility σ, at confidence α with z = Φ-1(α): VaR = P × (zσ − μ) and ES = P × (σφ(z)/(1−α) − μ), where φ is the standard normal density. The ES formula averages the whole tail beyond z rather than stopping at it.
The normal-distribution caveat
Real return tails are fatter than normal: crashes happen more often than the bell curve says. Parametric ES is a baseline for comparing portfolios and sizing buffers, not a worst-case promise. Historical-simulation ES on real return data is the usual complement.
What competing calculators miss
Reference pages define ES but rarely compute it; the previous version of this page simply echoed an input. This page derives both VaR and ES from portfolio statistics with the full substitution steps shown.
Frequently asked questions
What is expected shortfall?
The average loss in the outcomes worse than VaR: E[loss | loss > VaR]. Also called conditional VaR.
How is expected shortfall calculated?
Parametrically: ES = P × (σφ(z)/(1−α) − μ) with z = Φ-1(α) under normal returns.
What is the difference between VaR and ES?
VaR is the loss threshold at a confidence level; ES is the average loss beyond that threshold. ES ≥ VaR always.
Why do regulators prefer expected shortfall?
ES is a coherent risk measure that captures tail severity; Basel III uses 97.5% ES for market risk.
What does 97.5% expected shortfall mean?
The average loss in the worst 2.5% of outcomes.
Can expected shortfall be negative?
With a high expected return and low volatility, the formula can go negative, meaning no expected loss at that confidence: read it as zero tail risk in the model.
What are the limitations of parametric ES?
It assumes normal returns; real tails are fatter, so it understates crash risk.
How does volatility change ES?
Linearly in the parametric form: double the volatility and ES roughly doubles.
What is conditional VaR?
Another name for expected shortfall.
Is this investment advice?
No: a risk-measurement illustration for education, not advice.