Posts tagged with portfolio optimization

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.)

We showed in Chapter 6 that side information Y for the horse race X can be used to increase the growth rate by the mutual information I(X;Y). We now extend this result to the stock market.

Here, I(X;Y) is an upper bound on the increase in the growth rate, with equality if X is a horse race. We first consider the decrease in growth rate incurred by believing in the wrong distribution.

Thomas A. Cover & Joy A. Thomas, Elements of Information Theory

Gauging the frothiness of the webby/techy/san-fran VC market.

Source: Mark Suster. Propagated via one of tumblr’s owners, who added:

Based on the NVCA statistics on the venture capital industry, there are [approximately] 1,000 early stage financings every year….

And somewhere around 50 - 100 of them exit for more than \$100mm every year. So 5-10% of the companies financed by VCs end up exiting for more than \$100mm.

Mathematical PS: These are value-at-risk numbers, just upside-down.

hi-res

This book is something of a flourish — and I mean that in a pejorative way.  High symbology to wisdom ratio.

However, one really important idea I learned from it is that of one-sided risk measures.  This was apparently Markowitz’s idea, although I had believed he equated risk with statistical variance—illogical since gains are good.

The idea is intuitive, but wears an unusual mathematical structure:  the quasimetric.

hi-res