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Expected Shortfall from frequency and severity distribution table

delta123

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FRM 2012  PART 1: MOCK EXAM D  22
www.bionicturtle.com Question 13:
13. For a certain operational process, the frequency distribution below implies a 20% probability of two (2) loss events during the month. For each loss event that occurs, the severity distribution is given by: 50% probability of $1,000 loss (i.e., the loss amount conditional on a loss occurrence), 20% probability of $10,000 loss, and 10% probability of $100,000 loss:

Assuming the frequency and severity are independent, what is the 99.0% monthly expected shortfall (ES) of the aggregate (tabulated) loss?
a) $110,000
b) $128,000
c) $140,000
d) $200,000
Answer & Explanation:
13. B. $128,000
The worst loss of $200,000 occurs with p.d.f. probability of 20%*10%*10% = 0.20%.
The next worse loss of $110,000 occurs with p.d.f. probability of 20%*10*20% + 20%*20*10% = 0.40% + 0.40% = 0.80%; this total loss counts twice because, order matters, and we can experience either a $10,000 loss followed by a $100,000 loss, or a $100,000 loss followed by a $10,000 loss.
These two loss amounts constitute the 1.0% tail (0.20% + 0.80%) and:
The average loss (conditional on the loss exceeding 99%) = ($200,000*0.20% + $110,000*0.80%)/1% = $128,000; i.e., the weighted average of this 1% loss tail.

Can someone help me understand the calculation or point to a video/link. Appreciate it.
 
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