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P2.T7.307. Operational risk distributions



307.1. Analyst Sally develops the following model for her bank's daily operational risk losses, which includes the assumption that the daily frequency is a Bernoulli with probability of only 6.0% (several minor losses are expected but model ignores losses below $3,000) and, further, an assumption that loss frequency and severity are independent:

Which is nearest to the estimate of daily unexpected loss (UL) with 99.0% confidence?

a. $2,017
b. $4,633
c. $9,754
d. $20,000

307.2. Peter is using the loss distribution approach (LDA) and he is trying to select a distribution for the severity of his bank's operation losses. He will be calibrating his model, consistent with Basel, at 99.9% confidence over a one year horizon. Each of the following is at least plausible, but which of the following is LEAST likely to be true about this selection?

a. The severity distribution is a single empirical distribution (body and tail) without extrapolation beyond internal data
b. The severity distribution is sub-exponential
c. The severity distribution is lognormal or Weibull or Generalized Pareto (GPD using POT)
d. The severity distribution is piecewise; i.e., the body uses a different distribution than the tail

307.3. Randy has collected internal data from his bank in order to calibrate operational loss severity and frequency distributions. He decides that he will not complement his dataset with external data, although he is concerned that the bank's IT systems may not have captured many of the high-frequency, low-severity (HFLS) losses. Therefore, he is likely to encounter each of the following biases or problems, EXCEPT which is he LEAST likely to encounter if he relies solely on this internal data?

a. Survivorship bias
b. Truncation bias
c. Scale bias
d. Data insufficiency problem

Dear Fran, Could you be so kind to narrate how to solve the first question 307.1. I got an answer c. $9,754 but I`m dubious that it`s the right one. Can you please depict the formula also. Thanks beforehand...

David Harper CFA FRM

David Harper CFA FRM
Staff member
@[email protected]

The link (Here In Forum) goes to the source question with the answer, but it's protected to paid members. We try to be helpful, so here is the answer I have:

Expected loss = 6% * (2%*80,000 + 5%*20,000 + 7%*10,000 + 12%*5,000 + 74%*3,000) = $367.20.
The worst 1.560% tail of this distribution includes the worst four tabulations:
$80,000 loss with probability = 6%*2% = 0.120%;
$20,000 loss with probability = 6%*5% = 0.300%;
$10,000 loss with probability = 6%*7% = 0.420%;
$5,000 loss with probability = 6%*12% = 0.720%;
$3,000 loss probability = 6% * 74% = 0.440%.

As the $5,000 loss falls at the 1 - (0.120% + 0.300% + 0.420%) = 99.160% quantile, and
The $3,000 loss falls at the 1 - (0.120% + 0.300% + 0.420% + 0.720%) = 98.440%, it follows that:
the 99.0% quantile occurs at the loss of $5,000; i.e., where the tail "to the right" is only 0.840%.

Therefore, UL at 99.0% = OpVaR of $5,000 - EL of $367.20 = $4,632.80

Then a customer asked if we can find it by counting up "from the left" instead of "from the right", which you can:
Hi AlokS, Sure, you can work it from the other direction (from zero), which is:
94% prob of loss = 0
6%*74% = 4.44% = 0.04440 prob of loss = $3,000
6%*12% = 0.72% = 0.00720 prob of loss = $5,000
which gets us to the 99% VaR as 94%+4.44%+ 0.72% = 99.160%, so 99% VaR is $5,000
... The rest must sum to 100%
Thank you very much for your help! I plan to register for the November 2013 FRM Part 1 exam and now I`m trying not only to repeat and revise but to read as much relevant literature as I can. Exam is very vast and the main problem is - the boundaries of the FRM program are very vague and obscure. Thank you very much David!