Indy Telecom & Industrial Media

PD recently recommended two excellent books by Nassim Nicholas Taleb (homepage), the mathematician turned philosopher turned epistemologist. I am half way done with one of them, Fooled by Randomness and I have a brand new copy of The Black Swan sitting right here on my desk.

While I was reading the books, one thing that stuck me was this — within a given system, how many such quant funds (such as Taleb’s old firm, Empirica Capital LLC) does it take for Taleb’s risk-based hedge funds to succeed? Or inversely, how many is too many that would cause the system to fall down upon itself?

To find a way to answer these questions, I am interested in starting with finding out the possible success rates for these firms.

Let me explain myself. Let us assume that there are 100 quant firms that invest the way Taleb’s old firm did. This would mean that you have a hundred firms slowly bleeding to death, waiting for that one Black Swan, when things work out for them.

Now, for the sake of argument, let us assume that only half of these firms are going to succeed to some extent while the other half are going to fail.

The probability of all fifty of these firms having the exact same degree of success is quite low. In fact, it is more likely that within their given domain, these fifty firms have some sort of distribution of success-rates.

Let’s assume that two kinds of distributions are possible here — the first, obviously, is a Bell-curve, assuming a normal distribution. Going this route, we could try and apply the 68-95-99.7 rule and get the following distribution (courtesy Wikipedia):

So, let us assume that 68.2% of the quant firms are good enough for us to work with. This would leave us with:

This leaves us with about 34 firms which may perform well enough. Of course, the other interesting thing would be to see how much longer they can bleed themselves (assuming a decaying function) and seeing what percentage of these 34 companies actually “decay” themselves out. But that’s for later.

Now going back to our problem, the second distribution is more interesting. What if we had a log-normal distribution? Or better yet, let us assume a Log Levy distribution, which is more indicative of financial models (and of the stock market). This would be one where they have sharp peaks (for success) and long-tails (for the long periods of failure).

Since the stocks themselves are likely to show a Levy distribution, it would not be wrong to assume that our quants mirror this pattern.
If that is the case, our graph for a stable Levy distribution would look something like this (courtesy Wikipedia):

Now, we have a few cases to consider. If we consider then it becomes a normal distribution, which we just talked about. On the other hand, for all other values of , we could have heavy-tailed distributions.

So, here are my questions –

• What would be a logical assumption among the various values of alpha for our Levy distributions?
• What does this entail for our “functioning quant firm” numbers, i.e. where does that leave ?
• Assuming a standard decay rate for both normal and Levy distributions, which would be more effective (i.e., a better density function)?

Quite obviously, I am trying to back-track from a series of assumptions to evaluate the feasibility of having more than an optimal number of quant-based hedge funds. And I am also interested in evaluating the success and failure thresholds for such numbers.

Finally, I am looking for the Black Swan that will throw these numbers into quant hell.

(Yes, I am quite well aware that this post has a lot of inconsistencies and assumptions, and that I have completely ignored the injection of new blood into the system — quite obviously, this is an exercise in theory, so forgive me while I meander.)