The original paper defining Conditional Value at Risk = CVaR = Expected Tail Loss.

### Pessimism & Probability Distributions

What’s “the best” statistic? If you couldn’t see an entire probability distribution, but you could ask one question of it and get one number out, what question would you ask?

In an interview given to EDGE magazine, Bart Kosko explains how great the `median` is. He used to think the `mean` was the statistic to look at (cf., Francis Galton’s story of the crowd average guessing correctly the weight of the prize hog to a tenth of a unit) but the `median` is more robust and so on.

My opinion is that the most important statistic for many practical purposes is something like the `25% CVaR` or `50% CVaR`. I think that’s the essence of “What do you stand to lose?” as people mean it in normal English.

In other words, I think the way people think about risk in everyday, non-finance terms, basically boils down to

• the observed `minimum` (theoretically possible `minimum` if you’re a lawyer) and
• the `CVaR` (expected loss) for some wide-ish (likely) swath of the bad outcomes.

The reason the `CVaR` is so intuitive is that it smoothly interweaves both

• egregiously bad, low probability outcomes (“You could die with a .01% probability!” is actually a good reason to avoid something)
• and likely bad outcomes (“After you graduate you might not find a job in your field”).
• Compare to: “Standard & Poor’s rated structured credit products based solely on the probability that they would pay off less than 100% of their principal plus interest, and not at all based on the expected loss if that happened.”

So obviously there isn’t “one best” question to ask. It depends what you want to know—if it’s the value of the gravitational constant, `median` may be a great statistic. On the other hand, if you’re looking at the salaries that might result from  your law degree or MBA—that is, if you’re looking for a sensible measure of risk and downside—then I’d suggest a `CVaR`.

(I actually emailed Dr Kosko and got a response—but he linked to a 20-page paper he had written and I never got around to reading it and felt bad responding without reading all of his response.)

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