Use the original finite model
Losses −10/0/10/50 carry probabilities 30/40/25/5 percent. At 90% confidence, the selected worst-tail mass is 10%. The questions retain the common loss unit and horizon.
Check the mass before dividing
Include the largest losses first, keep each included probability within the available mass and ensure the selected total matches the complementary confidence. These are original teaching questions.
Try the questions
How much probability at loss 10 is included in the original 90% ES tail?
- A.
All 25%
- B.
5%
- C.
None
Answer and explanation
Answer: B
The selected total must be 1−0.90=0.10.
- A
This includes too much probability after the 5% at loss 50.
- B
It fills the remaining selected mass after loss 50.
- C
Then only 5% is selected rather than the required 10%.
The selected weighted loss sum is 3 and selected probability is 0.10. What is ES?
- A.
3
- B.
30
- C.
2
Answer and explanation
Answer: B
Divide by the selected probability, not by one.
- A
This leaves an unnormalised selected contribution.
- B
3/0.10 produces the selected-tail mean.
- C
This is the mean over the whole original model.
What is the original mean loss conditional on L≥10?
- A.
16.666667 approximately
- B.
30
- C.
50
Answer and explanation
Answer: A
The inclusive event contains 30% probability.
- A
(0.25×10+0.05×50)/0.30 uses the inclusive event.
- B
This is the exact worst-10% average, a different selection.
- C
This is the strict-above-10 mean in this model.
For losses 0/100/1,000 with probabilities 99/0.9/0.1 percent, what is 95% ES?
- A.
0
- B.
1.9
- C.
38
Answer and explanation
Answer: C
The remainder of the worst 5% is filled by zero loss.
- A
The zero VaR does not remove severe outcomes.
- B
This is the whole-model mean.
- C
The weighted positive-loss sum 1.9 is divided by the selected 5% mass.