First though a short diversion:
Repost: Dreamtime Finance (and the Kelly Criterion)
First posted June 13, 2011.And from Compounding My Interests, the title post:
Original post:
I've been meaning to write about Kelly for a couple years and keep forgetting. Today I forget no more.
In probability theory the Kelly Criterion is a bet sizing technique used when the player has a quantifiable edge.
(When there is no edge the optimal bet size is $0.00)
The criterion will deliver the fastest growth rate balanced by reduced risk of ruin.
You can grow your pile faster but you increase the risk of ending up broke should you, for example bet 100% of your net worth in a situation where you have anything less than a 100% chance of winning.
The criterion says bet roughly your advantage as a percentage of your current bankroll divided by the variance of the game/market/sports book etc..
Variance is the standard deviation of the game squared. In blackjack the s.d. is 1.15 so the square is 1.3225.
As blackjack is played in the U.S. the most a card counter can hope for is a 1/2% to 1% average advantage with much of that average accruing from the fact that you can get up from a negative table.
Divide by 1.3225 and you've got your bet size.
It's a tough way to grind out a living but hopefully this exercise will stop you from pulling a Leeson, betting all of Barings money and destroying the 233 year old bank.
I'll be back with more later this week.In the meantime here's a UWash paper with the formulas for equities investment....MORE
My earlier post laid out some important lessons on behavioral economics learned from Santa Fe Institute’s conference on Risk: the Human Factor. The specific lecture that first caught my eye when I saw the roster was Edward Thorp’s discussion on the Kelly Capital Growth Criterion for Risk Control. I had read the book Fortune’s Formula and was fascinated by one of the core concepts of the book: the Kelly Criterion for capital appreciation. Over time, I have incorporated Kelly into my position-sizing criteria, and was deeply interested in learning from the first man who deployed Kelly in investing. It's been mentioned that both Warren Buffett and Charlie Munger discussed Kelly with Thorp and used it in their own investment process. Thus, I felt it necessary to give this particular lecture more attention.*Previously:
In its simplest form, the Kelly Criterion is stated as follows:
The optimal Kelly wager = (p*(b+1)—1) / b where p is the probability (% chance of an event happening) and b is the odds received upon winning ($b per every $1 wagered).
It was Ed Thorp who first applied the Kelly Criterion in blackjack and then in the stock market. The following is what I learned from his presentation at SFI.
Thorp had figured out a strategy for counting cards, but was left wondering how to optimally manage his wager (in investing parlance, we’d call this position sizing). The goal was a betting approach which would allow for the strategy to be deployed over a long period of time, for a maximized payout. With the card counting strategy, Thorp in essence was creating a biased coin (a coin toss is your prototypical 50/50 wager, however in a biased coin, the odds are skewed to one side). This question was approached from a position of how does one deal with risk, rationally? Finding such a rational risk management strategy was very important, because even with a great strategy in the casino, it was all too easy to go broke before ever attaining successful results. In other words, if the bets were too big, you would go broke fast, and if the bets were too small you simply would not optimize the payout.
Thorp was introduced to the Kelly formula by his colleague Claude Shannon at MIT. Shannon was one of the sharpest minds at Bell Labs prior to his stint at MIT and is perhaps best known for his role in discovering/creating/inventing information theory. While Shannon was at Bell Labs, he worked with a man named John Kelly who wrote a paper called “New Interpretation of Information Rate.” This paper sought a solution to the problem of a horse racing gambler who receives tips over a noisy phone line. The gambler can’t quite figure out with complete precision what is said over the fuzzy line; however, he knows enough to make an informed guess, thus imperfectly rigging the odds in his favor.
What John Kelly did was figure out a way that such a gambler could bet to maximize the exponential rate of the growth of capital. Kelly observed that in a coin toss, the bet should be equal to one’s edge, and further, as you increase your amount of capital, the rate of growth inevitably declines.
Shannon showed this paper to Thorp presented with a similar problem in blackjack, and Thorp then identified several key features of Kelly (g=growth below):
This chart illustrates the points:
- If G>0 then the fortune tends towards infinity.
- If G<0 0.="0." fortune="fortune" li="li" tends="tends" the="the" then="then" towards="towards">
- If g=0 then Xn oscillates wildly.
- If another strategy is “essentially different’ then the ratio of Kelly to the different strategy tends towards infinity.
- Kelly is the single quickest path to an aggregate goal.
The peak in the middle is the Kelly point, where the optimized wager is situated. The area to the right of the peak, where the tail heads straight down is in the zone of over-betting, and interestingly, the area to the left of the Kelly peak corresponds directly to the efficient frontier.
Betting at the Kelly peak yields substantial drawdowns and wild upswings, and as a result is quite volatile on its path to capital appreciation. Therefore, in essence, the efficient frontier is a path towards making Kelly wagers, while trading some portion of return for lower variance. As Thorp observed, if you cut your Kelly wager in half, then you can get 3/4s the growth with far less volatility.
Thorp told the tale of his early endeavors in casinos, and how the casinos scoffed at the notion that he could beat them. One of the most interesting parts to me was how he felt emotionally despite having confidence in his mathematical edge. Specifically, Thorp felt that the impact of losses placed a heavy psychological burden on his morale, while gains did not have an equal and opposite boost to his psyche. Further, he said that he found himself stashing some chips in his pocket so as to avoid letting the casino see them (despite the casino having an idea of how many he had outstanding) and possibly as a way to prevent over-betting. This is somewhat irrational behavior amidst the quest for rational risk management...MORE
March 2008
Markets, Risk and Gambler's Ruin
From the Wall Street Journal:August 2011
Old Pros Size Up the Game
Thorp and Pimco's Gross Open Up on Dangers
Of Over-Betting, How to Play the Bond Market...MORE
Journal of Investment Consulting: Interview With Edward O. Thorp
One of the Quantfathers.Finally, a quick mention in:
Here's the interview (10 page PDF) with the HT to Infectious Greed who writes:
Thoughtful new interview with gambler, professor, investor and quant godfather Ed Thorp. There are many nuggets in this interview, so this is just an excerpt...
Cassandra's (Not so) Golden Rules About Investing (And Not Investing)
#21. NEVER double-down (except when you have material non-public information and deep pockets) or if you're Ed Thorp, or if you're playing at The Martingale Room.
