How many scenarios are needed for reliable ES optimization?

At least 1/(1-alpha) scenarios should fall in the tail. For 99% ES, at least 100 scenarios are needed for stability. In practice, 1,000-10,000 scenarios are used for reliable estimates.

Can ES be estimated from historical data?

Historical ES = mean of the worst (1-alpha)*T returns. With T=500 daily returns and alpha=0.99, ES is the mean of the worst 5 returns. This is the historical simulation approach.

Expected Shortfall Linear Programming Optimizer

Minimize Expected Shortfall (ES) via linear programming over a discrete scenario set. ES is the average loss exceeding VaR, providing a risk measure sensitive to tail severity.

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Frequently Asked Questions

How many scenarios are needed for reliable ES optimization?

Can ES be estimated from historical data?