Partial boundary mass in expected shortfall

At 90% confidence, the original model’s ES uses 5% probability at loss 50 and 5% at loss 10.

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Record the supplied loss distribution
Allocate the worst 10%
Compute the selected weighted average
Divide by selected probability
Reconcile the selected mass

Record the supplied loss distribution

Original same-horizon loss model
Loss units	Probability	Cumulative probability
−10	30%	30%
0	40%	70%
10	25%	95%
50	5%	100%

Allocate the worst 10%

Start with the largest loss: 50 has 5% probability, so include all of it. Another 5% mass is needed. The next loss 10 has 25% probability, so include only 5% of that 25%, or one fifth of its probability mass.

Compute the selected weighted average

Tail-mass bridge
Loss	Included probability	Probability × loss
50	0.05	2.5
10	0.05	0.5
Other outcomes	0	0
Sum	0.10	3

Divide by selected probability

The selected weighted average is 3/0.10=30 loss units. The lower quantile is 10, and the model maximum is 50. Each result answers a different question. The boundary allocation changes its inclusion weight, not its economic loss amount.

Reconcile the selected mass

Sum the included probabilities before interpreting the average. Using a total of 30% or 5% would describe a different conditional subset rather than the selected 10% tail.