fortunes-formula

I came across so many references to this book that I finally decided to take a peek in to it. At the heart of it, the book is about Kelly’s criterion.A simple example made me motivate to read this book.

If you know that you gain 50% with prob = 0.6 and lose 50% with prob = 0.4…AND…if you have X dollars, how much do you bet ?A naive strategy of using expectation over one step is dangerous. Expected gain over 1 step = 0.6*0.5-0.4*0.5=0.1 Expected profit >0 . Should you bet your entire $X ? Common sense tells us that it cannot be the answer as there is a high probability that you would be out of the game before law of large numbers kicks in. Kelly was interested in knowing what fraction of the wealth should one bet, given the odds of gains. Well, the answer for the riddle is f=2p-1 where f is the fraction and p is probability of a gain. This is all the matters if you want to have a 10,000 ft view. However if you want to know the STORY behind the criterion, this book is a wonderful way to spend time and understand the story which William Poundstone has thoroughly documented. This book is pretty readable..It just tells the story behind the criterion and it tells it in a fantastic way.

PART 1 – ENTROPY :

The book starts like a crime thriller. It tells a story of John Payne who starts a wire service to communicate horse track results to the bookies. In a strange twist of events, AT&T rise and Payne’s original network converge to the establishment of Bell Labs, a place which is credited to have been the place of innumerable famous personalities. The first part of the book introduces some famous personalities .

 

Image (Shannon) Image(Thorpe) Image(John Kelly)

Firstly, about Claude Shannon, one of the few brilliant scientists who was single handedly responsible for bringing “Information theory” to the world. His contribution was immediately applied to a wide range of fields. Edward Thorpe , a brilliant empiricist who sets Shannon in to a direction where they team up to build a tool to estimate probabilities at a roulette wheel. In their adventure, they realize that it is very important to have a betting strategy in place. The simple martingale strategy of betting twice on loss until you win is a risky proposition as the gambler might be bankrupt before he manages to win.

Shannon’s key idea was the essence of a message is its unpredictability. John Kelly’s , another brilliant scientist’s key insight in to this idea was:
Greedy-though prudent better is faced a similar situation as a receiver of a noisy message in shannon’s case. Kelly extrapolated the same shannon’s ideas to the horse betting scheme. The key equation spelled out by Kelly is : Gmax = R where Gmax is the growth rate of gambler’s money and R is the information transmission rate (Shannon’s theory). It is also popularly known as edge/odds kelly criterion which is often quoted in the media.Ed Thorpe used Kellys ideas in BlackJack and made a killing in the vegas casinos.

Henry Latene idea : If in each period, the investor chooses the alternative with the largest geometric mean across possible states, the geometric mean strategy is going to dominate all the strategies. I am facing a paucity of time but I would love to read more on this. STREETWISE, a book having a collection of papers relating to portfolio management has one of the papers on Henry Latene. May be I will read it some day.

The first part of book titled entropy covers the people behind kelly criterion and gives a non-mathematical introduction to the same. After reading this part ,it is difficult not to go beyond this introduction and explore the ideas of risk management. Let’s say I have a statarb strategy to pick stocks. How do you manage the risk of the portfolio becomes extremely important ? I should somehow find time to read about these issues and work on its implementation.


PART 2 – BLACK JACK :

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This part is a journalistic account of Thorpe using Kelly Criterion in Las Vegas to make money. The story highlights one key idea that is recurrent in various forms through out the book.

Law of Large numbers is largely misunderstood by gamblers. The probabilities are realizations over the long run and you need to a criterion to bet , to avoid Gambler’s ruin. Kelly’s criterion is one such criterion, edge/odds tells you the way you should bet on successive outcomes. This part of the book compares 4 strategies , Bet it all, Martingale, fixed wager system, Kellys system. At a first glance , Martingale system looks good but there is a gambler’s ruin in the pursuit of the strategy, meaning, there is a chance of ruin before the law of large numbers strikes. This is where kelly criterion , which is a geometric criterion does exceedingly well. Even though the trajectory path is jittery, the kelly criterion beats all the strategies.

The above reliance on the way to make law of large numbers is exactly what Shannon used in Information theory. Sometimes reading all these beautiful ways to look at things, makes me feel that I should never try to lose this habit of reading regularly. However I have seen that once I start working in a company / startup, i find it difficult to devote time to read in silence. I hope to change that pattern this time around!

One thing that will make any math inclined reader to ponder is , “How to use Kelly criterion in stock trading strategies ?” I am certain that there is a ton of literature out there. I should find some time to go over it sometime!. Also a few of the books mentioned in this part of the book are
“Beat the dealer”, “Beat the market”. I guess the ideas of delta hedging was put to use by Thorpe in Newport Ventures, a successful company, much before Black scholes used it in the famous Black scholes formula

Part 3 – Arbitrage

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This is the part of the book I love the most because it is a topic close to my heart. It talks about Thorpe’s attention to convertible bond arbitrage and the way he devised a risk neutral strategy for warrants. The delta hedging technique which Black Scholes popularized years later was being practically implemented years earlier by Thorpe. This book talks about the rise of random walk in finance literature and the popularization of efficient market hypothesis .Samuelson, Fama, Sharpe, all touted EMH. Shannon on the other hand believed that the only way to make money was through arbitrage. However he himself did not use the technique in the earlier years to make money. His success came from picking few stocks.

However Thorpe who had already used Kelly criterion, used warrants to make money, wanted to expand and get more money to manage. One way he did was to popularize his concepts in “Beat the market”. The book’s popularity brought him investors who were eager to put in money. Thorpe’s Princeton-Newport ventures did a tremendous business which is still a sort of benchmark for all the money managers. A dollar invented in 1968 would have grown to $14.78 in 1988. Over 19 years this was a return of 15.1 %(S&P averaged 8.8%). The most interesting thing was about the standard deviation which was 4%, meaning a Sharpe ratio of 3.7, a bloody good Sharpe ratio by any standards. If you ever think of managing money, this ratio should always be in your mind to remind you that there was guy who beat the market hands down with a Sharpe ratio of 3.7. WOW!, till this day, money managers talk about this performance. If you read this part of the boo,k, you cannot help but start thinking about kelly criterion, delta hedging, information ratios. The meat of the book lies in this and the next part

On a side note, there is a new term that I came across in this part of the book:

Paul Samuelson coined the term , Performance Quotient. Like IQ, this measures a portfolio manager’s ability to generate alpha.A PQ of 100 is average.Sameulson theorized that if such people existed, they would all be invisible. :You would not find them working for Ibanks. They have too high an IQ for that.They would operate by stealth, investing their own money or their friends money.They would keep the system to themselves.”

 

Part 4 – St. Petersburg Wager

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This is the famous wager problem where expected value of the wager is infinite. So, the question is that no one would pay infinite amount to play the game. whats the fair value of the game. This paradox created a lot of interest and lead to popularization of utility function ,Bernoulli’s masterpiece where he mentions that risky ventures should be evaluated based on the geometric mean of the outcomes. This ultimately lead to the involvement of Henry Latane, Markowitz, Kelly and finally it was being accepted that mean-variance analysis does not talk about compounding of investments. But Kelly’s world is about compounding of returns, the winnings are being reinvested continuously.Even though Samuelson and Merton were strict opponents of geometric criterion, other people like Henry Latane, Thorpe believed in Kelly criterion.

The world to this day is divided I guess between Kelly and Mean-Variance approach. However Thorpe, the man with one of the highest PQs believed in kelly(half-kelly to be conservative). Kelly has no log normal assumption or distribution assumption. It has no utility function behind it. It is a plain simple formula to avoid ruin. Will it work in specific markets ? Can some on use Kelly in India for stock selection? I do not know whether people are using it. But it will be fun to play with the money and see whether it works.

Part 5 – RICO

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This part gives a historical recount of RICO, a tax evasion law that became attached to Princeton-Newport ventures. Most of the partners got indicted. Thorpe was one person who was found innocent. The whole issue with princeton-Newport ventures was that they were parking stocks at firms and buying back at a price so as to offset short term gains. Tax evasion!! was good in the short run until RICO hit them. However this did not happen during the stellar performance of the fund where it had a Sharpe ratio of 3.7. So , for all the aspiring fund managers, that metric can still be considered as a benchmark for your performance !

Part 5 – Blowing up

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This details the blow up that took place at LTCM. Ironically , the fund failed to be long term!. The point in illustrating this case by the author is to bring the relevance of Kelly criterion and how the criterion would have saved LTCM from a blow up. Thorpe brilliant point about convergence trades made me think about the relevance of some trading strategies that I know about.

Part 6 – Signal Vs Noise

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Well , the last part of the book is a short collection of diverse views on Kelly . Shannon, it turns out was more than a stock picker than arbitrage exploiter. Thorpe believed in relative value and made a killing in various markets. The book ends with the author not taking any side to the debate on Kelly. Left to the imagination of the reader and interpretation, I feel that somebody soon , the heavy weights(PhDs) from ivy/other schools with get behind the Kelly bandwagon and make it popular. Ideas about mean variance will die soon and may be then , people will start giving importance to Kelly criterion.

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This book has made me inquisitive about the actual implementation of Kelly formula, how to use it in a long-short portfolio.

Lets say I manage to identify some long-short trades, is my current implementation of risk management sound ? May be I should simulate P&L with Kelly criterion and see how it behaves!. The following are some of the wonderful references which I hope to read some day:

  • Beat the market – Thorpe
  • Streetwise – Peter Bernstein
  • Portfolio Choice and Kelly Criterion
  • Henry Latane’s paper on geometric mean
  • Shannon’s basic funda relating to Information theory
  • Implementation of Kelly Function
  • How to implement Gmax = R

Wow!! its a long way to go for me, before before understanding all the stuff clearly..

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