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Gap between tail measures

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A desk has $40$ equally likely scenarios for tomorrow. The $5$ largest losses are $21$ , $17$ , $12$ , $10$ and $8$ , and every other scenario loses less than $8$ . Take the $90%$ VaR as the smallest loss $x$ with $P (L \leq x) \geq 90%$ , and the $90%$ ES as the average of the worst $10%$ of scenarios. What is ES minus VaR? Give a whole number.

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Hint

Work out how many scenarios sit in the worst $10%$ .

Solution

The tail holds $40 \times 10% = 4$ scenarios, so $36$ scenarios must sit at or below the VaR. The $36$ th smallest loss is the fifth largest, so $VaR = 8$ .
The tail average is $\frac{21 + 17 + 12 + 10}{4} = 15$ .
So the gap is $15 - 8 = 7$ .

Answer: $7$

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