Assume the following operational process:

• A Bernoulli distribution characterizes the frequency of losses. The probability of no operational loss = 95%: P(0) = 95%, P(1) = 5%
• The loss severity is characterized by a PMF: P(-$50,000 loss) = 2%, P(-$30,000 loss) = 18%, P(-$10,000 loss) = 24%, and P(-$5,000 loss)  = 56%. (Note this is a discrete distribution where the outcomes are mutually exclusive and cumulatively exhaustive; however, a typical severity distribution is continuous)

What is the 99% and 95% operational value at risk (VaR)?

(Don't peek before trying!)

 

 

 

(Don't peek before trying!)

Answer:

99% Operational VaR = Unexpected loss = Loss @ 99% - Expected Loss (EL).

Start with the worst loss and works backwards:

• P($50,000 loss) = (5%)(2%) = 0.1%
• P($30,000 loss) = (5%)(18%) = 0.9%
• Cumulatively, that is 1% (0.1% + 0.9% = 1.0%). So, the loss @ 99% is $30,000

The expected loss = 5% * [(2%)(50,000)+(18%)(30,000)+(24%)(10,000)+(56%)(5,000)] = $580

So, the 99% Operational VaR = $30,000 (loss @ 99%) - $580 = $29,420

Similarly, the 95% Operational VaR = $4,420

The calculations are shown below in EditGrid.

A few points about this:

• VaR is not the worst loss
• This is the operational LDA approach: a loss frequency distribution is compounded with a loss severity distribution (but a severity distribution is typically continuous, unlike my discrete example here)
• I called the discrete severity a PMF versus a PDF per Gujarati definitions. Know the difference, please.
• Just like with a credit portfolio, there is an expected loss (EL). The VaR is the unexpected loss: VaR = Loss @ 9x% - EL.
• If you look at the EditGrid below, the two distributions compound into a single PMF. Next to it, the running total is the CDF. Each discrete outcome in the PDF is simply the product of [Frequency][Severity]. Notice the analogy to credits where EL = PD * LGD. In operational terms, EL = PE (probability of event, frequency) * LGE (loss given event, severity). Finally, back to Gujarati, that's a joint probability.

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