Posts tagged with portfolio selection

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 is the best quant finance book I’ve yet read.  The symbols on the cover may look daunting, but the text actually keeps notation simple.  Many topics are covered quickly and accessibly; this is a maths book you can actually skim, or skip around in.  I think that’s due to good writing.

Also:  I stand firmly in the Robust camp.  After my class with Karen Kafadar, I’m confident that Robust models are easier to explain and more reliable.  Her typical example was to mis-type just one of the data by repeating a digit or moving the decimal place — and how likely is that! — and see how much the output changed.  Ideally your real-world recommendation shouldn’t change too much based on just one data point.  (If that’s unavoidable, you should withdraw any recommendation.)

So many mathematical questions or ideas yield up a flowering of possible tweaks and adjustments that can be made to a model, with no recommendation of which parameter value to use.  A good answer is:  whatever is most stable across different potential scenarios.

There is a wide variance among the Frank J. Fabozzi series (Advanced Stochastic Optimization, for example, is way worse than this).  If you only have time to read one, read this one.

hi-res