Expected shortfall as a threshold objective

For confidence c, ES equals the minimum of t+E[(L−t)+]/(1−c) over the threshold t.

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Separate threshold from outcome loss
Write the loss-side expression
Check a threshold in the original model
Allow a non-unique threshold
Keep implementation claims precise

Separate threshold from outcome loss

For any chosen threshold t, only the loss amount above t enters (L−t)+=max(L−t,0). This is an excess-loss amount, not an indicator probability and not the entire loss in an exceeding scenario.

Write the loss-side expression

\mathrm{ES}_c=\min_{t\in\mathbb{R}}\left[t+\frac{\mathbb{E}[(L-t)_+]}{1-c}\right]

For this finite integrable loss model; c is a confidence level.

Check a threshold in the original model

At 90% confidence, choose t=10. Only loss 50 has positive excess, equal to 40, with probability 5%. The objective is 10+(0.05×40)/0.10=30. Threshold t=0 gives 50; t=50 gives 50.

Allow a non-unique threshold

At exactly 95% confidence in this model, any threshold from 10 through 50 gives objective 50. A flat minimum interval is compatible with a unique ES value. Do not infer a unique optimized threshold from the ES number.

Keep implementation claims precise

The calculator allocates sorted worst-tail mass directly; it does not run an optimizer. This equivalent representation supplies an independent arithmetic check and clarifies what excess loss means.