## Saturday, February 23, 2008

### Survey Course

Probability 105 – This introductory course uses readings, exercises, and perspectives from disparate fields such as film, literature, mathematics, and finance to place probability theory in a general intellectual context.

Required for all undergraduates. No auditors. Pass/fail only.
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Literature

Once initiated in the mysteries of Baal, every free man automatically participated in the sacred drawings, which took place in the labyrinths of the god every sixty nights and which determined his destiny until the next drawing. The consequences were incalculable. A fortunate play could bring about his promotion to the council of wise men or the imprisonment of an enemy (public or private) or finding, in the peaceful darkness of his room, the woman who begins to excite him and whom he never expected to see again. A bad play: mutilation, different kinds of infamy, death.

— Jorge Luis Borges, The Lottery in Babylon

Mathematics

Assume a coin toss game, using a fair coin, which results in an outcome of either "heads" (H) or "tails" (T), each of which has a one-in-two chance of occurring. The outcome of any one coin toss is independent of any and all other coin tosses. Answer the following questions.

Q: What is the probability, ex ante, of tossing the following sequence?:
HHHHHHHHHHHHHHHHHHHH
A: 1/220, or less than one in a million.

Q: Having tossed the sequence in the preceding question, what is the probability of tossing heads on the next throw? Tails?

A: 50%. 50%.

Q: Which of the two following independent sequences of results is more likely?:
HHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHT
A: Neither. They are equally likely.

Q: Now assume that Steve Cohen of SAC Capital is tossing the coin. Does this fact change any of your preceding answers?

A: Hey! That's not a fair question!

Correct.

Film

"You need to call it. I can't call it for you. It wouldn't be fair."

"I didn't put nothin' up."

"Yes, you did. You've been putting it up your whole life. You just didn't know it. ... You know what date is on this coin?"

"No."

"1958. It's been traveling 22 years to get here. And now it's here. And it's either heads or tails. And you have to say. Call it."

No Country for Old Men

Finance

Ultimi Barbarorum:

So for my third and final major point, we have to recognise that we don’t really know, in the absence of any measurable standard of beta, what is fake and what is true alpha. We don’t really know what it is that drives certain individual traders’ persistent returns over long and [ephemeral] returns over short periods. We come back to the older distinction of “luck” and “skill”, and defer to Napoleon’s Law, that, in the absence of empirical data on skill, persistent luck is the best quality to look for in a general, and in a trader. Which makes it all the more reasonable to pay up for it, and, on the part of the bank, to pay up if a single trader has a good year, for now that trader is more likely to be one of the few who may do well the next, who have that magic persistence. That fat bonus is the bank paying up for the option on that guy, rolling him over to next year by paying his opportunity cost, and in that sense is perfectly rational behaviour.

Even simple coin toss games are subject to rather longer and more frequent runs of sequential heads or tails than we naively expect. Ask someone to manufacture what they suppose to be a random series of coin tosses, and most people will create a sequence which alternates very frequently between heads and tails and which has a rather even distribution of results. Actual coin toss sequences usually look far less random than their invented cousins.

Investing is clearly a game that is far more complex and subject to dramatically more causal factors than tossing a coin. However, no sane person would deny that chance must play some sort of role in an investor's results. Baruch identifies a factor he calls "persistence" in a trader's superior returns. Does that mean that outcomes in sequential investing games are not independent? That winners tend to win? If so, why? Is this the result of skill, momentum, reputation, confidence?

Or are we looking at a dramatic case of survivorship bias, where the most successful (lucky) investors are the few among many that we focus on, send money to, and try to emulate simply because they have been successful? Are these wizards of finance only one or two coin tosses away from failure, ignominy, disgrace?

It's happened before.

The roll call of former investing greats is long and getting longer. Julian Robertson, George Soros, etc. once bestrode the markets like titans. Now they invest in golf courses and harangue mid-level NGO delegates in sleepy Swiss ski-towns. What happened to these former greats? Did their apparently formidable investing skills falter in the face of changing market conditions? Did their luck run out? Did they just get tired?

They have been replaced in the investing pantheon by Young (and not-so-young) Turks who seem to crop up reliably to titillate the masses and sell newspapers: Steve Cohen, Ken Griffin, John Paulson. Even now, as these few bask in the sun, thousands of thirty-something would-be übertraders are jostling on the Market Escalator for the right to become one of the new bold-faced names with billions at his command and the breathless admiration and envy of all the world. But there are only a few slots available. Who will make it to the top? Who has the skill? Who has the luck?

Who are you going to bet on?

Call it, Friend-o.