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Expected shortfall of a loan pair

StatisticsHard

Each of $2$ independent loans loses $100$ with probability $3%$ and otherwise loses nothing. The $95%$ ES is the average loss over the worst $5%$ of outcomes, counting only part of an outcome if the tail ends inside it. What is the $95%$ ES of the pair? Give an exact decimal.

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Hint

List the chances of losing $200$ , $100$ and $0$ , then fill the tail from the worst end.

Solution

Both loans lose $100$ , $200$ in all, with chance $0.0 3^{2} = 0.0009$ , and exactly one does with chance $2 \times 0.03 \times 0.97 = 0.0582$ .
The tail of $0.05$ takes all of $0.0009$ at $200$ , and the rest, $0.05 - 0.0009 = 0.0491$ , at $100$ .
So $\frac{0.0009 \times 200 + 0.0491 \times 100}{0.05} = 101.8$ .

Answer: $101.8$

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