Tail Risk Calculator

Tip: click Load Example to copy these values into the form.

1) Historical simulation

• Let α = 1 − confidence. Downside VaR uses the α-quantile of returns.
• VaR = −Q_α (reported as a positive loss).
• ES is the average return in the tail beyond Q_α, then sign-adjusted.

2) Parametric (Normal)

• Horizon scaling: μ_h = μ·h, σ_h = σ·√h, where h is holding days.
• Quantile: Q_α = μ_h + σ_h·z_α, with z_α from the inverse normal.
• Downside ES: ES = −( μ_h − σ_h·φ(z_α)/α ).

3) Cornish–Fisher VaR adjustment

• Adjust z using skewness and excess kurtosis to better reflect asymmetry and fat tails.
• VaR uses z_CF. ES is shown as an approximation using the same tail shape.

1. Choose a tail direction and confidence level.
2. Select a method: historical for data-driven tails, or parametric for model-based estimates.
3. Paste a returns series, or enable manual moments to input mean and volatility.
4. Set the holding period to align with your risk horizon.
5. Click Calculate Tail Risk and review VaR, ES, and tail probability.
6. Use the download buttons to export a table for reporting.

Why Tail Risk Matters Beyond Volatility

Standard deviation assumes typical moves dominate outcomes, yet many portfolios fail in the worst 1–5% of days. Tail risk focuses on those extremes. A 95% confidence level targets the 5th percentile of returns, while 99% targets the 1st percentile. If daily volatility is 1.2%, a two‑sigma event is about 2.4%, but real markets can print 4–8% moves, making fat tails costly.

Inputs That Drive the Output

This calculator converts your returns series (or manual mean and volatility) into tail metrics. Portfolio value scales the percentage risk into currency impact, so a 3.0% VaR on 100,000 implies about 3,000 at risk. Holding period applies μ·h and σ·√h scaling, so moving from 1 day to 10 days multiplies volatility by √10 ≈ 3.16. Drift can be disabled when short horizons make mean negligible.

Reading VaR, ES, and Tail Probability

VaR is a threshold: with 95% downside VaR, losses worse than VaR are expected about 5% of the time. Expected Shortfall (ES) goes further by averaging losses beyond the VaR cutoff, so ES is typically larger than VaR for the same confidence. Tail probability at a chosen threshold answers a practical question such as “How often do we lose 5% or more?” based on your selected method.

Comparing Historical and Parametric Views

Historical simulation uses empirical quantiles, which can capture regime shifts if the sample includes them, but it is sensitive to small datasets. The normal parametric method smooths noise and supports longer horizons, yet it can understate risk when returns are skewed or have excess kurtosis. Cornish–Fisher adjusts the normal z‑score using sample skewness and kurtosis to reflect asymmetry and heavier tails.

Using Results for Limits and Hedging

Use VaR for reporting limits and ES for capital buffers, because ES better reflects what happens after the breach. Track the tail ratio (P95/|P05|) to see whether upside and downside tails are balanced; values above 1 suggest larger upside extremes relative to downside. Re‑run scenarios with different horizons and confidence levels to stress test drawdown tolerance and evaluate hedges. For governance, archive exports monthly and compare results against realized drawdowns and breaches.

1) What is the difference between VaR and ES?

VaR is the loss threshold at your confidence level. ES (Expected Shortfall) is the average loss when returns fall beyond that threshold, so ES usually exceeds VaR and better reflects extreme outcomes.

2) Should I enter returns as percent or decimal?

Use percent for values like -0.8 and 0.4. Use decimal for -0.008 and 0.004. Keep the same format for manual mean and volatility so the calculator scales results correctly.

3) How many observations should I provide?

For stable historical quantiles, more is better. Aim for at least 250 daily returns, or several years of weekly data. The calculator accepts smaller samples, but tail estimates can jump when one outlier dominates.

4) When is it reasonable to turn drift off?

For short horizons, drift is tiny versus volatility, so setting drift to zero can avoid overconfidence. Many risk teams disable drift for 1–10 day VaR and re-enable it for longer, strategic horizons.

5) Why can Cornish–Fisher give higher risk than Normal?

Normal VaR uses a symmetric bell curve. Cornish–Fisher shifts and stretches the z-score using skewness and excess kurtosis, which can increase downside VaR when returns are left-skewed or fat-tailed.

6) Is the tail probability at threshold the same as the confidence level?

No. Confidence sets where VaR is measured, like the 5% left tail at 95%. Tail probability answers how often returns cross your chosen threshold (for example, -5%), which may be larger or smaller than α.