This book is the definitive treatment of 'Fortune's Formula,' also described as 'The Kelly Criterion', used by gamblers and investors alike to determine the optimal size of a series of bets.

This volume provides the definitive treatment of fortune’s formula or the Kelly capital growth criterion as it is often called. The strategy is to maximize long run wealth of the investor by maximizing the period by period expected utility of wealth with a logarithmic utility function. Mathematical theorems show that only the log utility function maximizes asymptotic long run wealth and minimizes the expected time to arbitrary large goals. In general, the strategy is risky in the short term but as the number of bets increase, the Kelly bettor’s wealth tends to be much larger than those with essentially different strategies. So most of the time, the Kelly bettor will have much more wealth than these other bettors but the Kelly strategy can lead to considerable losses a small percent of the time. There are ways to reduce this risk at the cost of lower expected final wealth using fractional Kelly strategies that blend the Kelly suggested wager with cash. The various classic reprinted papers and the new ones written specifically for this volume cover various aspects of the theory and practice of dynamic investing. Good and bad properties are discussed, as are fixed-mix and volatility induced growth strategies. The relationships with utility theory and the use of these ideas by great investors are featured.

Kelly Capital Growth Investment Criterion

“This is a fantastic reference covering the theory and practice of the field beautifully organized and produced. I have already used it and I will refer it to my students and colleagues.”

—Professor David G Luenberger, Stanford University

“This volume provides a fascinating historical account and critical assessment of the Kelly criterion (expected logarithmic utility maximization) as a universal criterion for the tradeoff between risk and return in portfolio management and gambling.”

—George M Constantinides, Leo Melamed Professor of Finance, The University of Chicago, USA

Kelly Capital Growth Investment Criterion, The: Theory And Practice (World Scientific Handbook In Financial Economics Series 3) Reprint Edition, Kindle Edition by Leonard C MacLean (Author), Edward O Thorp (Author), William T Ziemba (Author). Introduction to the Kelly Capital Growth Alumni Professor Sauder School of Business University of British Columbia Investment Strategies 49 ell o y Criterion and Samuelson’s Objections to it The Kelly capital growth criterion, which maximizes the expected log of final.

“This book provides a comprehensive survey of research and applications on the Kelly growth optimal strategy that maximizes the expected utility of the log of final wealth…This book provides a fine coverage of these topics from original sources and recent research publications.”

—Quantitative Finance

“For those who have heard of the Kelly mythos and want to explore the science behind it, this book will be an instant classic. The editors have collected all the pivotal original papers, spanning centuries and the rarely bridged gulf between theory and practice. This book is indispensable for anyone interested in Kelly’s legacy.”

—William Poundstone, Author of Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street

“The present handbook assembles in an impressive way the classical papers and also provides the link to modern research. It also presents important papers with a critical view towards the Kelly criterion.”

—Professor Walter Schachermayer, Faculty of Mathematics, University of Vienna

Introduction

J.L.Kelly, in his seminal paper A New Interpretation of Information Rate (Bell System Technical Journal, 35, 917-926 see below) asked the interesting question: how much of my bankroll should I stake on a bet if the odds are in my favor? This is the same question that a business owner, investor, or speculator has to ask themself: what proportion of my capital should I stake on a risky venture?

Kelly did not, of course, use those precise words — the paper being written in terms of an imaginary scenario involving bookies, noisy telephone lines, and wiretaps so that it could be published by the prestigious Bell System Technical journal.

Assuming that your criterion is the same as Kelly's criterion — maximizing the long term growth rate of your fortune — the answer Kelly gives is to stake the fraction of your gambling or investment bankroll which exactly equals your advantage. The form below allows you to determine what that amount is.

Disclaimer

• The Kelly Strategy Bet Calculator is intended for interest only.
• We don't recommend that you gamble.
• We don't recommend that you place any bets based upon the results displayed here.
• We don't guarantee the results.
• Use entirely at your own risk.

Kelly Strategy Bet Calculator

Results

• The odds are in your favor, but read the following carefully:
• According to the Kelly criterion your optimal bet is about 5.71% of your capital, or $57.00.
• On 40.0% of similar occasions, you would expect to gain $99.75 in addition to your stake of $57.00 being returned.
• But on those occasions when you lose, you will lose your stake of $57.00.
• Your fortune will grow, on average, by about 0.28% on each bet.
• Bets have been rounded down to the nearest multiple of $1.00.
• If you do not bet exactly $57.00, you should bet less than $57.00.
• The outcome of this bet is assumed to have no relationship to any other bet you make.
• The Kelly criterion is maximally aggressive — it seeks to increase capital at the maximum rate possible. Professional gamblers typically take a less aggressive approach, and generally will not bet more than about 2.5% of their bankroll on any wager. In this case that would be $25.00.
• A common strategy (see discussion below) is to wager half the Kelly amount, which in this case would be $28.00.
• If your estimated probability of 40.0% is too high, you will bet too much and lose over time. Make sure you are using a conservative (low) estimate.
• Please read the disclaimer below.

More Information

The BJ Math site used to contain a great collection of papers on Kelly betting, including the original Kelly Bell Technical System Journal paper. Unfortunately it is now defunct, and only contains adverts for an online casino. However, you can find much of the content through the Wayback Machine archive. The Internet Archive also contains a copy of Kelly's original paper which appeared as A New Interpretation of Information Rate, Bell System Technical Journal, Vol. 35, pp917-926, July 1956. (If this link breaks — as it has done several time since this page was written — try searching for the article title).

We based the above calculations on the description given in the book Taking Chances: Winning With Probability by John Haigh, which is an excellent introduction to the mathematics of probability. (Note that there is a misprint in the formula for approximating average growth rate on p359 (2nd edition) and the approximation also assumes that your advantage is small. There is a short list of corrections which can be found through John Haigh's web page).

Note that although the Kelly Criterion provides an upper bound on the amount that should be risked, there are sound arguments for risking less. In particular, the Kelly fraction assumes an infinitely long sequence of wagers — but in the long run we are all dead. It can be shown that a Kelly bettor has a 1/3 chance of halving a bankroll before doubling it, and that you have a 1/n chance or reducing your bankroll to 1/n at some point in the future. For comparison, a “half kelly” bettor only has a 1/9 chance of halving their bankroll before doubling it. There's an interesting discussion of this (not aimed at a mathematical reader) in Part 4 of the book Fortune's Forumla which gives some of the history of the Kelly criterion, along with some of its notable successes and failures.

Jeffrey Ma was one of the members of the MIT Blackjack Team, a team which developed a system based on the Kelly criterion, card counting, and team play to beat casinos at Blackjack. He has written an interesting book The House Advantage, which examines what he learned about managing risk from playing blackjack. (He also covers some of the measures put in place by casinos to prevent the team winning!)

• Blackjack: Play Like The Pros: A Complete Guide to BLACKJACK, Including Card Counting, , Lyle Stuart, 2006.
• The Game Plan: How Casual Players Become Threats in No Limit Hold ’Em Tournaments, , Independently published, 2019.
• Blackbelt in Blackjack : Playing 21 as a Martial Art, , Cardoza, 2005.
• Beat the Dealer: A Winning Strategy for the Game of Twenty-One, , Vintage, 1966.
• High-Leverage Casino Gambling Systems: How to play like a minnow and score like a whale on your next casino visit, , CreateSpace Independent Publishing Platform, 2012.
• Gambling and War: Risk, Reward, and Chance in International Conflict, , Naval Institute Press, 2017.
• The World's Greatest Blackjack Book, , Crown, 1987.
• Math of Poker Basic: Pile Up Money and Professional Player: Essential Poker Math, , Independently published, 2020.

Kelly Growth Criterion Model

See also