An old article written for Wall Street Survivor reposted here. J.L. Kelly Jr. was a very interesting member of the famed Bell Labs. There is a new book published this year, 2012 about Bell Labs incidentally called: The Idea Factory, by John Gertner. Peguin 2012 | http://www.amazon.com/The-Idea-Factory-American-Innovation/dp/1594203288

Enjoy!

By Alwin Tong, for Wall Street Survivor University, published February 18, 2009

Looking to maximize growth in your WSS portfolio? The Kelly Criterion gives a mathematically secure method of establishing a betting scheme which maximizes the growth of your portfolio.

### The Kelly Criterion – Introduction

Looking to maximize growth in your WSS portfolio? The Kelly Criterion gives a mathematically secure method of establishing a betting scheme which maximizes the growth of your portfolio.

The world is full of serendipitous discoveries from the grand to the banal. An apple falls and inspires Newton, Columbus goes west and finds the people of America. Toxin is injected under the skin near the eyes and Botox treatments arrive.

The world of investing is not much different than this admittedly. It was in 1956 that J.L. Kelly, a member of the AT&T Bell Labs team, was assigned to find a way to figure out what to do with information that is passed through noisy telephone lines. Figure out phone lines, that was his job. What the world we got was a formula that described the mathematically perfect wagering algorithm of repeated games, which has been used in situations of betting known as the Kelly Criterion.

In about the equivalent realm of serendipity as the discover of using pantyhose as a fan belt, The Kelly formula which was original used to deal with long distance phone lines for AT&T and fate conspired to allow the clever Kelly to realize that this same formula could be used for many wagering games, including the financial markets.

The system of betting was first modelled as a game theory problem involving how to bet over time on horse races given that you were tipped off to the winner beforehand and looks like this:

*f** is the fraction of the current bankroll to wager;*b*is the odds received on the wager;*p*is the probability of winning;*q*is the probability of losing, (written as 1 −*p*.)

Don’t let this math scare you. What the formula states in a nutshell, is that you bet proportionally to your chances of winning. Before I go on, I should say that there are some caveats that I’ll mention later on in this article, but let’s take a look at some features and examples of pure betting under the Kelly Criterion. Let’s take a look at some everyday examples first.

### An Example: Basketball Game

If I go to a basketball game and know that the Knicks and Lakers are playing a best of 7 series. For hypothetical ease you only play only against each other and match bets. (equal payoffs of 1:1).

When the Knicks are playing at home, they have a 75% percent chance of winning, which means (without ties) that the Lakers are only at 25% chance of winning.

When the Knicks play away in L.A. they still have an advantage though let’s say the odds lower for the Knicks to 55% while the Lakers are at 45%.

Your Kelly Bet’s across the games should be like this.

Location |
Knick’s Odds |
Lakers’ Odds |
Formula |
Bet |
---|---|---|---|---|

NY |
75% | 25% | Bet% = (1).75 – .25 / (1) = .50 B is 1, because you’re playing in a 1:1 case. |
50% of bankroll |

LA |
55% | 45% | Bet% = (1).55 – .45 / (1) = .10 B is 1, because you’re playing in a 1:1 case. |
10% of bankroll |

If a third L.A. Lakers fan joins the wager, then the payoff will change, affecting the ** B **value of your formula. The payoff odds are now 2:1 Knicks to Lakers, under this scenario B will be equal 2, because for every amount you wager, you’ll receive twice that on a winning bet against the 2 others.

Location |
Knick’s Odds |
Lakers’ Odds |
Formula |
Bet |
---|---|---|---|---|

NY |
75% | 25% | Bet% = (2).75 – .25 / (2) = .50 B is 1, because you’re playing in a 1:1 case. |
62.5% of bankroll |

LA |
55% | 45% | Bet% = (2).55 – .45 / (2) = .10 B is 1, because you’re playing in a 1:1 case. |
32.5% of bankroll |

Those would be the growth optimal bets, taking into account both wins and losses. That’s Betting under the Kelly Criterion. It is also known by several other names such as: Geometric Mean Maximizing Portfolio Strategy and Fortune’s Formula. Pretty Simple right.

### Features of the Kelly Criterion

There are several nice features of Kelly betting that stand out.

- This Betting Scheme will not allow you to lose all of your money, because you would never bet all your money except in a 100% true win probability. In a true mathematical 100% win situation, a loss is not allowed in this case by definition. So the formula (in theory) does not allow you to go broke.
- Is optimized for growth allowing your growth profile to sit along a Logarithmic curve.
- States that you should not bet in any scenario that is less than 50%

### In Practice

The Kelly Criterion is a great theoretical tool for investors. However in Practice, following the strict criterion is often considered aggressive because it is maximized for making the largest fortune you can. It doesn’t take into account some findings of utility theory, where the same 100$ maybe more valuable to a person with very little money versus a person with already a vast sum. Also it doesn’t take into the increased benefit of portfolio diversification. Rather it is a tool in the mix of these others that will allow you make a decision in respect to your portfolio.

It is also very difficult to ascertain true odds of an event. And transaction costs.

Looking at some great investors, for example the legendary value investor Warren Buffet, we can see an asset allocation that is very close to following this criterion. (quote: www.**wilmott**.com/pdfs/050316_**ziemba**.pdf)

Also, to prevent the heavy swings associated with it, the amount invested can be scaled to a fraction. This is known as half-Kellying or third-Kellying. The rule to take away is simply to invest proportionally to your chances.

On WallStreetSurvivor.com because the portfolio you are dealing with is composed of fantasy dollars, the downside risk of losing real money is eliminated while the upside risk if you do well can be real cash. For this reason I would recommend betting with a Full Kelly implementation. Good Luck out there!